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Renormalization group analysis of the standard model, the minimal supersymmetric extension of the standard model, and the effective action

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Renormalization group analysis of the standard model, the minimal supersymmetric extension of the standard model, and the effective action
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Arason, Haukur, 1962-
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vi, 102 leaves : ill. ; 29 cm.

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Mass ( jstor )
Neutrino masses ( jstor )
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Thesis (Ph. D.)--University of Florida, 1993.
Bibliography:
Includes bibliographical references (leaves 96-101).
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Haukur Arason.

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RENORMALIZATION GROUP ANALYSIS OF THE STANDARD MODEL, THE MINIMAL SUPERSYMMETRIC EXTENSION OF THE
STANDARD MODEL, AND THE EFFECTIVE ACTION






By

HAUKUR ARASON


















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA

1993














ACKNOWLEDGEMENTS

I would like to offer my sincere thanks to my advisor, Pierre Ramond, for his generous support and kind encouragement during my tenure as his student. I have benefited immensely from his teaching because of his extensive knowledge and friendly character.

I would also like to acknowledge those who were my collaborators at one point or another. In particular I am indebted to Diego Castafio, Eric Piard, and Brian Wright for many educating and entertaining discussions.

I thank Profs. C. Thorn, P. Sikivie, R. Field, and C. Stark for serving on my committee.

Last but not least I want to thank my wife for her continued support throughout this work.






















11















TABLE OF CONTENTS



Page

ACKNOWLEDGEMENTS .................... ii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . v

CHAPTERS

1 INTRODUCTION ................ ..... . 1

2 RG STUDY OF THE SM AND THE MSSM . . . . . . . . . . 5 2.1 Signatures of Structures Beyond the SM . . . . . . . . .. 5
2.2 Initial Value Extraction from Data . . . . . . . ..... . . . 8
2.3 Quantitative Analysis of the SM ... .. .. ... . . . . 28
2.4 Quantitative Analysis of the MSSM ... . . . . . . . . . 38

3 A MULTISCALAR SM EXTENSION . ............ 52
3.1 The M odel . . . . . . . . . . . . . . . . . . . . . ... 52
3.2 Unitary Gauge . . .. . ................ 60
3.3 Majorana Neutrino Masses . .... .. . . ....... . . . . 64
3.4 Experimental constraints . . . . . . . . . . . . . . . . . 67
3.5 Solar Neutrino Phenomenology . . ... . . . . . . . 69

4 THE RG IMPROVED EFFECTIVE ACTION ... . . . . . . 73 4.1 Approximating the EP . . . . . . .. . . . . . . . . . 73
4.2 One Massless Scalar . . . . ............... 77
4.3 A Massive Scalar . ....... . .. . . . . . . . . . . . .78
4.4 Two Massive Scalars .. ...... . . . .... . . . . .80

5 CONCLUSIONS ...... ........... .. . . 84

APPENDICES

A THE SM f FUNCTIONS ................... 87

B THE MSSM 1f FUNCTIONS . . . . . . . . . . . . . . . . 91
iii








C EXPLICIT FORM OF 6(p) ................. 94

D THE SHIFTED FIELD METHOD . ............. 95

REFERENCES . .............. . . ...... . 96

BIOGRAPHICAL SKETCH ........ .... . ...... 102












































iv











Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

RENORMALIZATION GROUP ANALYSIS OF
THE STANDARD MODEL, THE MINIMAL SUPERSYMMETRIC EXTENSION OF THE
STANDARD MODEL, AND THE EFFECTIVE ACTION



By

Haukur Arason

December 1993
Chairman: Pierre Ramond
Major Department: Physics

We present some examples of the uses of the Renormalization Group (RG) in particle physics. The Standard Model (SM) is analyzed in great detail, and in particular Grand Unification is ruled out in the Standard Model. Similar analysis is performed in the Minimal Supersymmetric extension of the Standard Model (MSSM). In that case we show that Grand Unification is possible. In the context of Grand Unified Theories (GUT), there arise certain relations among the masses and mixing angles. These relations are explored both in the Standard Model and in the MSSM. In particular we show that they can all be satisfied in the MSSM at a plausible GUT scale if the masses of the Higgs boson and the top quark meet certain constraints.

The scalar sector is experimentally the least constrained aspect of the Standard Model. We explore this freedom by introducing a multiscalar extension of the Standard Model, that incorporates the Peccei-Quinn solution to the strong CP problem and at the same time induces neutrino masses that solve the solar neutrino problem.
V








In the Standard Model the Effective Potential (EP) plays a crucial role as it can provide the mechanism for electroweak symmetry breaking. The Renormalization Group is an indispensable tool for analyzing the quantum corrections to the EP. We discuss in detail how the Renormalization Group is used to find the leading logarithmic correction to the EP. Motivated partly by the experimental uncertainty of the scalar sector of the Standard Model and partly by the Supersymmetric extensions of the Standard Model, we evaluate the quantum corrections to the Effective Potential in a model containing two scalars and attempt to solve the very difficult problem of finding the leading log contribution to the Effective Potential.
































vi















CHAPTER 1
INTRODUCTION

The Standard Model has survived all experimental tests so far and incorporates all known particle physics down to scales of fractions of millifermis. The Standard Model is also completely consistent down to the Planck scale, where quantum gravity would take over. In spite of this success, the Standard Model has some unsatisfactory features such as a large number of parameters, three chiral families, and three distinct gauge structures. Consequently, physicists have searched for a simpler underlying structure that would break down to the Standard Model in the low energy limit. The observed pattern of the quantum numbers of the elementary particles has given rise to the idea of a Grand Unified Theory (GUT) [1,2,3,4]. which simplifies the gauge structure of the Standard Model. Experiments can not access the high energies where the GUT predictions deviate from the prediction of the Standard Model; it is however possible to use the Renorminalization Group to extrapolate the Standard Model parameters to smaller scales [5]. The purpose is to find if those parameters satisfy GUT relations at shorter distances. This is what we will do both for the Standard Model and for the Minimal Supersymmetric extension of the Standard Model (MSSM). This has been done before, but we will improve upon earlier projects by using newv and improved data, by treating the full Yukawa sector, by using improved treatment of thresholds, and by using 2-loop/3-loop #-functions in the Standard Model. Our plan is as follows. We use experimental data to fix the parameters of the Standard Model at lower energies. We

1






2

then numerically integrated the MS Renormalization Group Equations [6,7] for the Standard Model and the MSSM to evolve the parameters of the models to Planck scale. Although it is not possible to analytically express CKM angles and the quark masses in terms of the Yukawa couplings, our approach is numerical so we run the Yukawa matrices and evaluate the relevant quantities by diagonalizing the matrices at every step of the Runge-Kutta method. In the Standard Model, we use 2-loop Renormalization Group Equations in evolving the couplings, except in pure QCD where 3-loop contributions are also significant and therefore included in the running of the strong coupling and of the quark masses in the low energy region. In the supersymmetric extension, we work to 1-loop. In each case, we include a proper treatment of thresholds [8]. Our incomplete knowledge of the Standard Model parameters forces us to repeat the analysis for a range of allowed values of the top quark and Higgs masses. Using these techniques, we examine interesting GUT inspired relations among the gauge couplings, and the masses and mixing angles of quarks and leptons in the Standard Model. We extend the analysis to the minimal supersymmetric extension to determine its effect on these relations.

In the Standard Model we do not see any clear evidence of Gauge Unification. The gauge couplings only semi-converge and unification between the bottom- and r-yukawa predicted by the simplest GUTs takes place at a scale many orders of magnitude away from the possible Grand Unification scale.

In the MSSM we find a remarkable agreement with GUTs. The gauge couplings unify at a scale which does not violate proton decay and we can satisfy all the GUT inspired mass and mixing angle relations at the GUT scale. For this to happen we have to place bounds on the masses of the top






3

quark and the Higgs boson. The details of this analysis are found in chapter

2.

Although neutrinos are necessarily massless in the Standard Model [9] and direct experimental searches for the consequences of massive neutrinos, e.g. oscillations, have so far failed to turn up anything, the deficit in the expected number of neutrinos from the sun [10,11] indicates that neutrinos might be massive [12]. Global symmetries, such as baryon number and lepton number, are not viewed in the same way as local symmetries; they are expected to be approximate symmetries, their apparent conservation being explained by the appearance of tiny ratios, as happens in Grand Unified Theories, axion theories, or theories that involve gravity. Since the mass of the W-boson is seventeen orders of magnitude smaller than the Planck mass, there is ample room for such ratios. In order to incorporate neutrino masses in the Standard Model, the lepton number has to be violated, either spontaneously or explicitly. In chapter 3 we introduce a model with explicit lepton number breaking where neutrino masses are generated radiatively at one loop. The model is a minimal extension of the Standard Model with invisible axion and neutrino masses, and for a reasonable range of parameters the model provides a solution to the strong CP and solar neutrino problems.

The Effective Potential plays a crucial role in analyzing models in particle physics as it determines the vacuum structure of the models. The EP may also be important for the development of the early universe, since it may lead to inflation. In the Standard Model electroweak braking take place at the classical level; however the study of the Effective Potential, as shown in a classic paper [13] by S. Coleman and E. Weinberg, may provide new possibilities for symmetry breaking at the quantum level. Thus quantum corrections might






4

be quite important in analyzing the EP. Since it is not possible to calculate the full EP, it is essential to have reliable methods for approximating the important contributions to the EP. Being an expansion in h, the loop-expansion provides an approximation to the Effective Potential. Unfortunately this approximation is only valid in a limited range of field space. It has been known for some time that in the case of one massless scalar [13] the Renormalization Group can be used to extend the validity of the loop-expansion and give the Renormalization Group Improved (RGI) Effective Potential. Intuitively it is easy to understand how the Renormalization Group (RG) can tell us how the Effective Potential, Veff , behaves as a function of the field, 0. The dimensionless quantity Veff/d4 can only depend on the Renormalization constant yp and 0 through the combination (0/ji), and since the Renormalization Group tells us how that quantity depends on p, it also tells us how it depends on 6. This argument also indicates that it might be harder to find the Renormalization Group Improved approximation to the Effective Potential in models with a massive scalar, where there are three dimensionful quantities Y, 0, and m, and in models containing more than one scalar. Recently this has been achieved in a model containing a massive scalar [14,15] and in chapter 4 we attempt to do the same in a model containing two scalars. The multiscalar case is complicated not only because it contains many dimensionful quantities, but also because field space is multidimensional.














CHAPTER 2
RG STUDY OF THE SM AND THE MSSM

We present a comprehensive analysis of the running of all the couplings of the Standard Model to two loops, including thresholds effects and taking full account of the Yukawa sector. Our purpose is to determine what the running of these parameters up to Planck mass may indicate for the physics of the Standard Model and its extensions. We examine several GUT inspired relations among the parameters of the Standard Model. We extend the analysis to the Minimal Supersymmetric extension to determine its effect on these mass relations. Remarkably Supersymmetry allows for all the GUT relations to be satisfied at a single grand unified scale. For this to happen we have to place bounds on the top quark mass, which remarkably satisfy the p-parameter bound; furthermore, using the minimal SUSY boundary condition on the scalar quartic coupling, we also obtain bounds on the Higgs mass.


2.1 Signatures of Structures Beyond the SM

The gauge structure of the Standard Model contains three distinct gauge groups SU(3)C x SU(2)" x U(1)Y; however the quantum numbers of the particles of the Standard Model allow for the possibility that this is a low energy manifestation of a Grand Unified Theory. A GUT includes only a single gauge structure such as SU(5), SO(10), or E6 [1,2,3,4], and thus requires the Standard Model gauge couplings to unify


91 = 92 = g3 (2.1.1)

5






6

at some large grand unification scale. This is the single most important signature of a GUT structure beyond the Standard Model.

In the context of the SU(5) GUT [2], several mass relations arise, based on simple assumptions for the possible Higgs structure. The mass term for the down quarks and leptons comes from the Yukawa interaction of the 5 and 10 of fermions. This leads to relations between the charge -1/3 quarks' and the charged leptons' Yukawa couplings. With only a 5 of Higgs, one obtains equality between the 7 lepton and bottom quark masses at the GUT scale:


7mb = m1 . (2.1.2) To the level of approximation used at the time, this relation was found to be consistent at experimental scales, after taking into account the running of the quark masses [16]. Similar relations apply to the lighter two families, but are clearly incompatible with experiment. To alleviate this, a new scheme was proposed [17] with a slightly more complicated Higgs structure (using a 45 representation in conjunction with the 5). It replaces the above with the more complicated relations for the two lighter families nfd = 3me ,
(2.1.3)
3ms = mp.

The situation concerning the mixing angles is equally intriguing. There happens to be a near numerical equality between the square of the tangent of the Cabibbo angle and the ratio of the down to the strange quark masses (determined from current algebra). This Gatto- Sartori-Tonin- Oakes (GSTO) relation [18] reads

tan Oc (. (2.1.4) n n'/s2





7

It has provided the central inspiration in the search for Yukawa matrices. Very general classes of matrices with judiciously chosen textures [19] (i.e., zeroes in the right places) could reproduce this relation, at least approximately.

In the context of SO(10) [3], these three different relations could all be obtained in one model [20], with the required texture enforced naturally by discrete symmetries at the GUT scale. In this model the mixing of the third family with the two lighter ones is dictated exclusively by the the charge 2/3 quarks' Yukawa matrix. There ensues an GSTO-like relation for the mixing of the second and third families [20]


VCb = t, (2.1.5) which provides a relation between the top quark mass and the lifetime of the B meson.

These four relations can all be obtained if one takes the Yukawa mixing matrices to be of the form [20] (shown here in a specific basis)


0 P0





0 0 0 0 T
Although some of these relation are derived with specific and sometimes complicated Higgs structures in mind (as in the SO(10) model), they may well prove sturdier than the theories which generated them. In the following, we will examine the relations in the context of the Standard Model at varying scales all the way to Planck scale. We will then extend the analysis to the minimal supersymmetric extension of the Standard Model, and compare the effect of this extension on their compatibility at some unified scale.






8

2.2 Initial Value Extraction from Data


al(M7) and a9(M7)

The determination of the SU(2)L x U(1)y couplings proceeds from the Standard Model relations: ) 1(-) C2 O(p)
4x cos20,y(p)
47r cs2 W )(2.2.1) a2(P) -
47r sin2 Ow()

where a(p) = e2(p)/47r and C2 is a normalization constant which equals 1 for the Standard Model and equals , when the Standard Model is incorporated in grand unified theories of the SU(N) and SO(N) type [5). What is required to specify these couplings are the values of a(p) and sin2 OwV(p) in the renormalization scheme we employ (i.e., MS). The electromagnetic fine structure constant (ac, I 137.030) is extrapolated from zero momentum scale to a scale p equal to MZ in our case. In pure QED with one species of fermion with mass m, the MS renormalized vacuum polarization function is given by


1 2 a~p /1 2 1 q2
I(q2) a(n 2 - 6 dx x(1 - x)ln[1 - x(1 -x)2]) (2.2.2)
3f m m
0

The renormalized coupling a(p) is related to the fine structure constant aem as follows

aem = (2.2.3)
1 + n(0)
In the Standard Model where there are many species of charged fermions and charged gauge bosons, Eq. (2.2.3) generalizes to [21]


-1(p) = a- _ 2 9 Q2( - mf)1n - (2.2.4) f mf 67






9

The effects of the strong interaction which enter as a hadronic contribution to the vacuum polarization function must be included also. The nonperturbative nature of the strong interaction at low momentum is handled by rewriting the hadronic contribution to the vacuum polarization at zero momentum as ih(0O) = (nh(0) - IIh(q2)) + Ih(q2) . (2.2.5)


If q2 is chosen large enough, IIh(q2) can be calculated perturbatively. The terms (IIh(O)-IIh(q2)) can then be related to the total cross section for e+e- -hadrons [211. Using the optical theorem, we can write


Im{iih(s)} = a(e+e- -- hadrons), (2.2.6) 4xaem

where s is the square of the center of mass energy. For the process ee- -+ P+p-, the cross section is calculated to be (taking nip = 0) a(e+e- -+ p+p -) 4 re (2.2.7) 3s

In terms of the ratio of these two cross sections, a(e +e- --* hadrons)
R(s) = , (2.2.8) a(e+e- -+ p-)

we can write Eq. (2.2.6)


Im{lhl((s) = aeR(s) . (2.2.9)
3

Using an unsubtracted dispersion relation for Ih(q2), the combination (IIh(O)IIh(q2)) can be expressed as


IIh()- II2) = q 2ae ds R(s) (2.2.10) 37r s(q2 - s)
4m2?





10

This can be evaluated using experimentally known data. This procedure yields a value

a- (Mz) = 127.9 � 0.3. (2.2.11) The process independent, renormalized weak mixing angle sin2 Ow of the on-shell scheme is defined to be sin2 Ow = 1 - , (2.2.12) M2

where My, and MZ are the physical masses of the W and Z gauge bosons. Knowing the precise values of the tV and Z boson masses and using the equation above provides one way of extracting the value of sin2 Owi. Alternatively, the bare relation involving the low energy Fermi constant measured in muon decay and the W boson mass G e
Gpo2 (2.2.13)
8 sin20 VoA12

may be corrected to order a and rewritten [22,23] ran,,, 1 1 (2.2.14) M z = IZ cos 8I- = ( ) i , (2.2.14) G sin0 :(1 - Ar)5

with (rraem/V9/G2) = 37.281 GeV and Ar is a parameter containing order a radiative corrections which depends on the mass of the top and Higgs. We can view the radiative corrections represented by Ar as accounting for the mismatch in the scales associated with the parameters of the relation. GP and aem are low energy parameters whereas .M11 and sin2 0w are associated with the electroweak scale. We can absorb the radiative effects using the renormalization group by replacing G1 and aem with corresponding running parameters at MZ:
7r )1 . (2.2.15) V'-Gu(MZ )Mn,2 sin2 Ow,






11

Combining Eqs. (2.2.14) and (2.2.15) gives Ar 1 - G(Mz) (2.2.16) a(Mz) Gu

Using Eq. (2.2.4) and the fact that Gp(Mz) a G1, [23,24] gives an estimate of the size of the radiative corrections Ar M 0.07. (2.2.17) For large values of Mt and MH (Mi, MH > MZ) [22,25]

aem 3aem, M2 11em M
Ar , 1 - S - - l+ In .N (2.2.18)
ca(Mg) 167 sin4 011 A12 487r sin2 0w M2

A third way of extracting sin2 0r, is from neutral current experiments, among which deep inelastic neutrino scattering appears to provide the best determination. A running sin2 09(p) may be defined in MS and differs from the above sin2 Ow by order a corrections. The MS running W boson mass mw(Lp) and the corresponding physical mass l1', identified as the simple pole at q2 = M of the W propagator, are related as follows r = m2.(p) + AWT ,(Mp, p) , (2.2.19)


where ATW is the transverse part of the W self-energy. A similar relation holds for the Z boson. In MhS renormalization, the following relation defines the running sin2 4 w( p)

sin2 WP() = 1 - )(2.2.20) Equation (2.2.19) and its Z analog may be combined with Eq. (2.2.20) to give

sin2 C) cos2 , A z(M, p) A (M ,) (2.2.21)
sin2 sin2 - M2 ) (2.2.21) An explicit expression relating sin2 1,V and sin2 Ow(Mw) is given in Ref. 26.






12

Another relation for sin2 0w(p) may be arrived at directly linking it to MZ [27] or MW [28]. In particular, if one chooses My as the input mass, then one introduces a radiative correction parameter A W such that


sin2 0W(M Z)(1 - A1W) = sin2 9W(1 - Ar) , (2.2.22) from which it follows that

(37.271)2
sin2((12) = (1 .2.23) Similarly one can introduce a, radiative correction AiZ if one chooses Ma as the input mass


sin2 w(M) cos2 (Z) = sin2 cos2)(1 - ) = sin2 A' cos2 (1 - Ar) . (2.2.24) A fit to all neutral current data gives sin2 0n,(MZ) = 0.2324 � 0.0011 , (2.2.25) for arbitrary Mrt [29). Using these values of a(Mz) and sin2 Ow(M) yields al(MZ) = 0.01698 � 0.00009, (2.2.26)
a2(Mz) = 0.03364 � 0.0002 .




The value of the strong coupling is known with less precision than most of the parameters of the Standard Model and it is by far the most uncertain of the three gauge couplings. This is due to large theoretical uncertainties arising from the nonperturbative nature of low energy QCD and the slow convergence of perturbation series in high energy QCD. Moreover, this uncertainty is hard to quantify.






13

In the extraction of as = g2/47r from a physical process, many obstacles arise. Since the convergence of the QCD perturbation theory series is not very fast, one must check higher order effects. Even if one chooses processes which do not involve hadronization, a most delicate problem in the extraction of as comes from working to finite order in perturbation theory. Physical quantities should of course be renormalization scheme independent, but the necessity of approximation introduces dependence on the renormalization scheme. Typically, the same physical quantity calculated in two different schemes to the nth order of as will differ by terms of order an+. As as is large, this difference may be large and thus may lead to renormalization scheme dependence problems. This problem manifests itself in the difficulty of choosing the renormalization scale p to use for the particular experiment from which one is extracting the strong coupling. Ideally one would like to choose p to minimize the unknown higher order terms, but that is of course not possible. Sometimes p is approximated by the scale at which the highest order known term vanishes or the scale at which that term gives a stationary prediction. However, the most frequent choice is p = E, where E is some characteristic energy scale of the experiment. This choice is plausible since it minimizes the typical terms that arise which involve ln(Q/p), with Q some momentum in the process, typically - E. All the processes from which the strong coupling is extracted suffer from this problem and thus each individual extraction of as has large uncertainties. To obtain the best estimate of the strong coupling, we shall take together the results from different processes. These include e+e- scattering into hadrons, heavy quarkonium decay, scaling violations in deep inelastic lepton-hadron scattering, and jet production in e+e- scattering.





14

We first consider the extraction of as from e+e- scattering into hadrons. The cross section is, ignoring finite quark mass effects [30],

a(e+e- - hadrons) = 4 3(1 + z)( e)
q
1+ a" + r A2 )2 + A( )3 + O(a) , (2.2.27)

where the effects from Z exchange have been put into the factor z. For nfl = 5 the numerical values of the coefficients are [31] A2 = 1.409 and A3 = -12.805. This determination of as has the advantage that it is inclusive, since there is no dependence on hadronization models. Its main drawback is that the effect is not very sensitive to as as the effect starts at zeroth order in as. The experimental error is relatively large and in fact dominates the theoretical error. The value of as has been extracted from the total cross section of e+einto hadrons by Gorishny et al. [31] who find as(34GeV) - 0.170 � 0.025. As an estimate of the error coming from cutting off the perturbation series we use the size of the highest order correction and estimate the relative cutoff error to be - 13(as/7r)2. Thus we find as(34GeV) 2- 0.170 � 0.025 � 0.006(cutoff) " 0.170 � 0.026, which using three-loop as and two-loop a running is equivalent to as(Mz) - 0.140 � 0.01S (recent LEP data [32] give essentially the same result).

The decay of heavy quarkonium is another process from which as can be extracted. The decay rates are sensitive to the strong coupling, the dominant modes going as a2 or a3, depending on the state of the q4 system. The decay rates can be calculated in the nonrelativistic approximation. The rates themselves depend on the wave function amplitude at the origin, which is unknown







but cancels out of branching ratios. The most useful of these branching ratios are [33]

F(T -. -GG) 4 aem M ( sb),
F(T -- GGG) 5 5as(Mb) 1 '
F(T -* GGG) 10 MT 2 (7.2 - 9)[as(Mb)]3 + 0.43asMb) F(T p+p-) 9 [2Mb a 7
(J/ - GGG) 5M 2 (72 - 9)[()3 as(Mc)
= 7r a s(Alc )]3 1 + 1.6 .ss m c)) r(J/ -+ p+p-) 8 2_jc 7ra2n + 1.6
(2.2.28)

The main uncertainty of this extraction of as is theoretical. The known higher order corrections are large so one expects the unknown corrections also to be large. In addition there are relativistic errors. Kwong et al. [331 have made a detailed analysis of as extraction from quarkonium decays. They find as(Mb) = 0.179 � 0.009 from F(T - -yGG)/F(T -- GGG). They have also estimated the relativistic corrections and find as(Mb) = 0.189 � 0.008 and as(Mc) = 0.29 � 0.02 by looking at F(T -+ GGG)/F(T -- + pp-) and F(J/o -+ GGG)/F(J/, --+ p+p-). The errors given do not include the cutoff and relativistic errors. In their analysis Kwong et al. parametrized the relativistic corrections by a factor (1 + Cv2/c2) in the branching ratios. They found C1 5 C S C2, with C1 - -3.5 and C2 - -2.9. Here (v2/c2)j/, = 0.24 and (v2/c2)T = 0.073. We estimate the relativistic error to be of the order as(Mb) (v2/c2)T (C2 - C1). Similarly, we estimate the cutoff error by the highest order corrections in Eqs. (2.2.28) . Using those estimates we find that as is most accurately determined from F(T -* GGG)/F(T -4 p+-). We estimate the cutoff error to be 2 as(Mlb)(0.43 as(Mb)/r) ~ 0.005 and the relativistic error to be as(Mb) (,2/c2)T (C2 - C1) 2 0.008 and conclude that as(Mb) = 0.189 � 0.008 � 0.005(cutoff) � 0.008(relativistic) = 0.189 � 0.012. This value is equivalent to as(MZ) = 0.111 � 0.005.





16

Analysis of the structure functions in deep inelastic scattering gives a similar value for as. The strong coupling affects the way the structure functions vary with energy. These effects show up as logarithmic corrections to the exact Bjorken scaling predicted by the simple parton model. Like the other methods mentioned so far, the measurements of the structure functions do not depend on fragmentation and hadronization. The scaling violations in the structure functions have been measured with beams of electrons, muons, neutrinos, and antineutrinos on targets of hydrogen, deuterium, carbon, and iron among others. Martin et al. [34] have analyzed the most recent data and found as(MZ) = 0.109 � 0.00S, including estimates of the truncation error.

Finally, we consider the extraction of as from e+e- scattering into jets. The production of multijets in E+e- scattering depends strongly on as. There, comparison of the QCD prediction and data introduces hadronization model dependence into the extraction of a(. The evaluation of as is further complicated by dependence on cutoffs between different jets and the usual problems of the unknown higher order terms. To reduce the jet resolution problem, event shape variables, such as the energy-energy correlations, the asymmetry of the energy-energy correlations, the oblateness, the thrust, etc., are used to extract as. As an example of an event shape variable we look at the energy-energy correlation (EEC) defined by [35]

1 dEEEC 1 do
- (x) = - -1 yivj dyidyj. (2.2.29)
a cos X a dyidyjd cos\

The EEC can be experimentally constructed as

1
Neveni~, : yiyj6(cos Oij - cos x), (2.2.30) events i,j






17

where x is the angle between calorimeter cells and yi = 2Eif/i are the center of mass energy fractions of the detected particles. The asymmetry of the energy-energy correlation (AEEC) is defined as

1 dEAEEC 1 dEEC 1 dEEC
( = - (Xr - X) - -(X)- (2.2.31) a dcos () ad cos adcos x A perturbative calculation of the asymmetry gives [35]

1 dEAEEC A(cos ) 1 + ) + O( (2.2.32)
a d cos 7 I ]

where functions A and R are calculated in perturbative QCD. The best data on the jet rates come from LEP. Recently those results have been extensively discussed in the literature [32.36,37]. Combining all the LEP data on jet distributions, including the full theoretical error, gives [32] as(MZ) = 0.115 � 0.008.

To summarize, the values of as(Mz) and its error are given in Table I.


Table I: Values of as at AMZ and its error Process as Aas e+(- ---+ hadrons 0.140 0.018 T decay 0.111 0.005 Deep inelastic scattering 0.109 0.008 Jet distribution in e+e-scattering 0.115 0.008




To pick the value of as(MZ) for our numerical studies we take the Gaussian weighted average of these values ([E(as/A 2)/ Z(1/Aa )] �[2(1/Aa)] ), and we find as(Mz) = 0.113 � 0.004.





18

Yukawa Couplings

To take full account of the Yukawa sector in running all the couplings, initial values for the Yukawa couplings are necessary. They must be extracted from physical data such as quark masses and CKM mixing angles. Furthermore, the interesting parameters must be determined step by step in the process of running to Planck mass. These two procedures are not unrelated and require the diagonalization of the up-type, down-type, and leptonic Yukawa matrices.

We use Machacek and Vaughn's [6] convention where the interaction Lagrangian for the Yukawa sector is


C = QLYU UR + QL 4Ydt dR + L Yet eR + h.c. (2.2.33) The Yukawa couplings are given in terms of 3x3 complex matrices. After electroweak symmetry breaking, these translate into the quark and lepton masses v2me 0 0
Ye =- m0 0 , v 0 0 m,
mnd 0 0
Yd - 0 ms 0 , (2.2.34) vt 0 0 mb
mu 0 0
Yu = 0 mc 0 V U, 0 0 mit)

where V is the CKM matrix which appears in the charged current


j ~ ULTVdL . (2.2.35)


It is a unitary 3 x 3 matrix often parametrized as follows:
( c1 S1c3 slS3
V = -Sl2 C12c3 - 2s3i C1C2S3 + s2c3ei6 , (2.2.36)
--Sl2 cls2c3 + c2s3 i6 cl 2s3 - c2c3ei~ where si = sin 0i and ci = cos Oi, i = 1,2, 3.




19

The entries of the parametrized CKM matrix can be related simply to the experimentally known CKM entries. The particle data book [29] gives the following ranges of values (assuming unitarity) for the magnitudes of the elements of the CKM matrix:

0.9747-0.9759 0.218-0.224 0.001-0.007
VI = 0.218-0.224 0.9734-0.9752 0.030-0.058 . (2.2.37)
0.003-0.019 0.029-0.058 0.9983-0.9996

These ranges of values can be converted to bounds for si, i = 1, 2, 3, and sin 6. We arrive at these bounds by finding values for the four angles such that the entries of the CIKM matrix obtained from these satisfy the conditions imposed by Eq. (2.2.37). We find

0.2188 < sin 01 < 0.2235,

0.0216 < sin 02 < 0.0543 , (2.2.38)

0.0045 < sin 03 < 0.0290 .

However the accuracy with which IV I is known does not constrain sin 6. A set of angles {01, 02, 03, 6} was chosen that falls within the ranges quoted above. The initial data needed to run the Yukawa elements are extracted from the CKM matrix and the quark masses. A problem arises though for the mixing angles, which was solved for the quark masses, in that it is not clear at what scale the chosen initial values for these angles should be considered known. However, even when the top quark is taken as high as the p-parameter bound allow the variations in 02 and 03 between Ml11 and the Planck scale are at most 20%, and these variations are always negligible in the experimental region. Therefore, the exact knowledge of the initial scales for the angles is not as critical as might be feared a priori.






20

Known Quark Masses

As QCD is assumed to imply quark confinement, extraction of quark masses from experiment follows the same circuitous route as other QCD quantities such as as. In the past decade a variety of techniques have been developed and utilized to extract quark masses from the observed particle spectrum. Below, we shall briefly recount some such techniques. Furthermore, we shall present some values for the heavy quark masses based on the application of our numerical technique to three loops.

The light quark masses are the ones least accurately known. They are determined by a combination of chiral perturbation techniques and QCD spectral sum rules (QSSR). In the former case the light quark masses are directly expressible in terms of the parameters of the explicit SU(2) and SU(3) chiral symmetry breakings. One then considers an expansion of the form [38]


lbaryon = a + bmlight +-" (2.2.39) for the mass of a baryon from the 1+ octet, and one of the form e2n = Bmlight + (2.2.40)


for a typical member of the pseudoscalar octet. A parameter measuring the strength of the breaking of the more exact SU(2) chiral symmetry in comparison with the SU(3) one is the ratio ms - m
R - (2.2.41) md - mu

where
1
m 2(mu + md) . (2.2.42)
2






21

To lowest order in isospin splittings, this translates in the meson sector into R -= M- M (2.2.43) M 0 - M2
KO K+

and in the baryon sector into three different determinations of R, (M - M) - (M - M A) Ml, - Mp
R = (M - MN) - '(ME - MA) (2.2.44) M7- - M=0
M- - MN
R=
M- - My+

To make R compatible with all the above mass splittings one has to consider higher order corrections in Eqs. (2.2.39) and (2.2.40). Here infrared divergences emerge as one is expanding about a ground state containing Nambu-Goldstone bosons. Once such singularities are removed within the context of an effective chiral Lagrangian, one finds the following as the optimum value of R R = 43.5 � 2.2 . (2.2.45)


Together with the ratio [39] "- = -5.7 � 2.6 , (2.2.46) also determined by applying Eq. (2.2.40) to the physical masses of r, 17, and K, they imply the following renormalization group invariant mass ratio n7d - mu = 0.28 � 0.03 . (2.2.47) 2m'

Applied to the light quarks the QCD spectral sum rules imply [39]

mru + rid = 24.0 + 2.5 MeV . (2.2.48)






22

Together with Eq. (2.2.47) they reduce to dh, = 8.7 � 0.8 MeV ,
(2.2.49)
r'd = 15.4 � 0.8 MeV .
The parameter ri7 is a renormalization group invariant which to three loops is related to the MS running mass parameter m(I) via [40]

m(pu)= [-1s(tL) -L(/ 71 2 72 - )as(y)

1 02 71 2 2 71 72 3 71 73 s(P)
2 2 # )1 02 1 1 3 7r
(2.2.50)

where the f3i and the -i are the coefficients of the 0 functions for as and m given in Appendix A. From Eqs. (2.2.50) (to two loops) and (2.2.49) one may infer the following values
mnu(1 GeV) = 5.2 � 0.5 MeV , (2.2.51)
nod(1 GeV) = 9.2 � 0.5 MeV . In applying expression (2.2.50) it should be kept in mind that the continuity of m(p) across a quark mass threshold requires hA to depend on the effective number of flavors at the relevant scale, analogously to the QCD scale A. The strange quark mass is determined, averaging the value derived from Eqs. (2.2.46) and (2.2.48) with those obtained using Eq. (2.2.49) and the various QSSR values for ih, + rhis, to be [41]

ihs = 266 � 29 MeV , (2.2.52) corresponding to the running value ms(1 GeV) = 194 � 4 MeV . (2.2.53) For the heavier quarks, charm and bottom, one can make a more precise prediction. Here, the nonrelativistic bound state approximation may be applied. The physical mass M(q2 = M2) appearing in the Balmer series may






23

be identified with the gauge and renormalization scheme invariant pole of the quark propagator

S(q) = z(q)['y . q- A(q2)]-1 Corresponding to the above pole mass is its Euclidean version, T(-q2), which is not gauge invariant, although it is renormalization group invariant, and is therefore not physical. The Euclidean mass parameter is the one often employed in the J/24 and T sum rules, as it minimizes the radiative corrections in such sum rules. In the Landau gauge the two are related to two loops according to [42]


m(Af ) = M(Ai2) [1 - s()n4 . (2.2.54) Once the pole mass is determined from the Euclidean one, the running mass at the pole mass is obtained to three loops via

2 )(2 = M2) 711 - = AP(q M A2M) , (2.2.55)
1 + 7 +K( )

where K = 13.3 for the charm and K = 12.4 for the bottom quarks [43].

From the J/ti and T sum rules the following values have been extracted [39]
Mc(-q2 = Al2) = 1.26 � 0.02 GeV, (2.2.56)
b( --2 = Al2) = 4.23 � 0.05 GeV To obtain an accurate value for the corresponding pole masses, we numerically solve Eq. (2.2.54) with the above values inserted and the three-loop P function for as, to obtain the following pole masses AMc(q2 = A 2) = 1.46 1 0.05 GeV, (2.2.57)
Mb(q2 = A2) = 4.58 � 0.10 GeV . Recently [44], new values for the charm and bottom pole masses have been extracted from CUSB and CLEO II by analysis of the heavy-light B and B*




24

and D and D* meson masses and the semileptonic B and D decays with the results
Mcl(q2 = A2) = 1.60 � 0.05 GeV, (2.2.58)
(q = l) = 4.95 + 0.05 GeV . A weighted average of the values in Eqs. (2.2.57) and (2.2.58) yields AMc(q2 = M'2) = 1.53 � 0.04 GeV, (2.2.59)
Mb(2 2= .~2) = 4.89 � 0.04 GeV . The running masses at the corresponding pole masses follow from Eq. (2.2.55) mc(Mc) = 1.22 � 0.06 GeV ,
(2.2.60)
m7b(Mb) = 4.32 � 0.06 GeV .

With these taken as initial data along with the value of the strong coupling at MZ quoted earlier, we run (to three loops) the masses and as to obtain the following values at the conventionally preferred scale of 1 GeV mc(1 GeV) = 1.41 � 0.06 GeV , (2.2.61)
7mb(1 GeV) = 6.33 � 0.06 GeV

Our numerical approach does not make any more approximations than the ones assumed in the 3 functions and the mass equations used, apart from the approximation inherent in the numerical method itself, and is more in line with our program than using the "perturbatively integrated" form of the 0 functions. Thus we shall adopt the above values. It should be stressed that at the low scales under consideration the three-loop as corrections we have included in our mass and strong coupling 3 functions are often comparable to the two-loop ones and hence affect the accuracy of our final values noticeably. Nevertheless, it should be noted that the above expressions relating the various mass parameters are not fully loop consistent as to our knowledge Eq. (2.2.54) has only been computed to two loops.






25

In conclusion, it should be pointed out that although we opted for the QSSR extraction of masses, there are rival models, such as the nonperturbative potential models, which predict appreciably higher values of the heavy quark masses than the ones quoted here. These models, however, are not as fundamental as the approach considered here, and their connection to field theory is rather problematic.


Lepton Masses

The physical (pole) masses of the leptons are very well known [29] Ale = 0.51099906 � 0.00000015 MeV , l = 105.658387 � 0.000034 MeV , (2.2.62) Mr, = 1.7841+0.027 GeV
-0.0036

We use these values to determine initial data for the running masses. Some authors neglect QED corrections and use the physical values for the running values at -- MZ, which introduces only a small error. By calculating the oneloop self-energy corrections, one arrives at a QED relation between the running MS masses and the corresponding physical masses 3a(p) p' 4
171(0) = Al[1 - )(l1n + 4)]. (2.2.63) 4ir ml 3

Choosing p = 1 GeV as in the quark mass case and using Eqs. (2.2.63)and (2.2.4) yield the running lepton masses (taking mi = M in the log term above is an appropriate approximation to order a) nme(1 GeV) = 0.4960 MeV ,

rnp(1 GeV) = 104.57 MeV , (2.2.64) mr(1 GeV) = 1.7835 GeV .






26

Higgs Boson and Top Quark Masses

The Higgs boson and top quark masses have not been measured directly at present; however their values affect radiative corrections such as Ar. Consistency with experimental data on sin2 OW requires Mt < 197 GeV for MH = 1 TeV at 99% CL assuming no physics beyond the Standard Model [45]. Precision measurements of the Z mass and its decay properties combined with low energy neutral current data have been used to set stringent bounds on the top quark mass within the minimal Standard Model. A global analysis of this data yields t = 1221+4 GeV, for all allowed values of MhH [46]. Recent direct search results set the experimental lower bound Mt 110 GeV. As for the Higgs, the analysis of Ref. 46 gives the restrictive bound, M H 5 600 GeV, if Mt < 120 GeV, and MH < 6 TeV, for all allowed Mr. Since perturbation theory breaks down for JMH 1 TeV, the latter bound on the Higgs boson mass is not necessarily meaningful. LEP data set a lower bound on the Higgs boson mass of 48 GeV [47].

In our Standard Model analysis, initial values of the MS running top quark mass mt and of the scalar quartic coupling A at MZ are chosen arbitrarily (consistent with the bounds quoted above). As noted earlier, these running parameters are not equal to their physical counterparts. However, any reasonable prediction for the masses of the top quark and of the Higgs boson that may come from our analysis should be that of experimentally relevant, physical masses. Therefore, formulas similar to Eq. (2.2.63) relating MS running parameters to physical masses are needed. To calculate the physical or pole mass of the top quark, we use Eq. (2.2.55) in its general form

Mt 4 as(Mt) 5 AMi as(Mt) 2
=t 1 +- + [16.11- .04 (1 - )]( )2 , (2.265)
mt(Mt) 3 r. i= M r\






27

where Mi, i = 1,...,5, represent the masses of the five lighter quarks. Likewise the physical mass of the Higgs boson can be extracted from the following relation [48]:

A(P) = 2 (1 + 6(11)) , (2.2.66) where 6(p) contains the radiative corrections. Its form is rather elaborate and we relegate it to Appendix C. Equations (2.2.65) and (2.2.66) are highly nonlinear functions of M and MH, respectively, and we solve them numerically to find the masses of the top quark and the Higgs boson.

In the Minimal Supersymmetric extension of the Standard Model we treat the masses of the Higgs boson and the top quark somewhat differently. In that case we will be able to constrain the values of those masses, by imposing some of the relations that we discussed in Section 2.1.


Vacuum Expectation Value of the Scalar Field

The vacuum expectation value (vev) of the scalar field may be extracted from the well known lowest order relation v = (fG,)-2 = 246.22 GeV . (2.2.67) From the very well measured value of the muon lifetime, r = 2.197035 + .000040 x 10-6 s [29], the Fermi constant can be extracted using the following formula [49]

G2n + 3 m1 + a(mp_) 25 (2.2.68)
192 f )( 5 2 + 2x 4 where

f(x) = 1 - 8x + 8x3 - 4 - 12x2lnx , (2.2.69) giving

G, = 1.16637 � 0.00002 x 10-5 GeV-2 . (2.2.70)





28

This parameter may be viewed as the coefficient of the effective four-fermion operator for muon decay in an effective low energy theory e [ve_(1 - 75)e][pTy(1 - 75)Vp] . (2.2.71)


A direct calculation (e.g., in the Landau gauge) of the electromagnetic corrections yields that the operator is finitely renormalized (i.e., Gp does not run) [23,24]. Another way to see this is by using a Fierz transformation to rewrite the above expression

"[e73(1 - 7Y5)vZ,][-YO7(l - Y5)e] . (2.2.72) The neutrino current does not couple to the photon field, and the e - P current is conserved and is hence not multiplicatively renormalized.

We need an initial value for the running vacuum expectation value at some scale p. Wheater and Llewellyn Smith [50] consider muon decay to order a in the context of the full electroweak theory and derive an equation relating an MS running Gp to the experimentally measured value. From this formula we can extract a value for v(MgZ).

However, the formula is derived in the 't Hooft-Feynman gauge, and the evolution equation Eq. (A.18) of Appendix A for the vev is valid only in the Landau gauge. Nevertheless, motivated by the discussion of the previous paragraph, we choose the initial condition for the vev to be v(Mw) = 246.22 GeV. Using our numerical methods we arrive at v( M). We find that this procedure leads to no significant correction, and we therefore take v(Mz) = 246.22 GeV.


2.3 Quantitative Analysis of the SM

Let us summarize the most important features of the renormalization group running in the Standard Model. At the one loop level, the gauge couplings are





29

unaffected by the other couplings in the theory. On the other hand, the Yukawa couplings are affected at one loop by both the gauge and Yukawa couplings. Since the top Yukawa coupling is at least as big as the gauge couplings at low energy, that means the running of the Yukawas is sensitive to mostly the top Yukawa and the QCD gauge couplings. Thus we can expect the mass and mixing relations we described in section 2.1 to be sensitive to the value of the top quark mass. The Higgs quartic self-coupling enters in the running of the other couplings only at the two loop level, so that its effect on the other parameters is small. However, its own running is very sensitive to the top quark mass; it can become negative as easily as it can blow up, corresponding to vacuum instability or to strong self-interaction of the Higgs (triviality bound), respectively. The discovery of the Higgs with mass outside these bounds would be a signal for physics beyond the Standard Model. The graphs in Fig. 1 summarize these bounds for representative values of the top quark mass. For example, if Mt = 150 GeV, we see from the corresponding plot that a Higgs mass between 95 and 150 GeV need not imply any new physics up to Planck scale. However, if the Higgs were observed outside of this range, then some new physics must appear at the scale indicated by the curve, either because of vacuum instability if MyH < 95 GeV or because the Higgs interaction becomes too strong if MH > 150 GeV. It is amusing to note that it is for comparable values of the top and Higgs masses that these bounds are least restrictive, but it is important to emphasize that a high value of the top with a relatively low value of the Higgs necessarily indicates the presence of new physics within reach of the SSC. Subsequently, when examining the possible relations in the context of the Standard Model, we will make the choices for Mt and MH in our renormalization group runs consistent with these bounds. For a chosen value







30

of Mt, varying MH within the vacuum stability and triviality bounds does not affect any of our results, and we will therefore choose a corresponding,

representative value of MH.



M20= 100 GeV M,=125 GeV
Triviality - Vacuum -Triviality Stability
- 15 15 -10 Allowed 10 Allowed

5 5


0 11 I11 , i ii 0 FI i , E I I
50 100 150 200 50 100 150 200
(a) (b)


Mt=150 GeV Mt=200 GeV
20 20
Vacuum 11 -Trivialit
tability- 15-Stability- 15
15 Triviality

10 Allowed 10 Vacuum Stabilityo 5- 5


0 LI I 0 1 11111 111 11 ItIII
50 100 150 200 50 100 150 200 250
(c) MH (GeV) (d) MH (Gev)



Figure 1. Vacuum stability and triviality bounds on the Higgs
mass for (a) AM=100 GeV, (b) M,=125 GeV, (c) M=150 GeV, and (d) M,=200 GeV, giving scales of expected new
physics beyond the Standard Model.




We have studied the effects of changing the values of Mt and MH in our

analyses of the running of the other parameters. As expected we observed that, for any Mt between 100 GeV and 200 GeV, varying MH, while maintaining perturbativity and vacuum stability, did not affect appreciably the evolution of




31

any of the other parameters. However, changing Mt itself showed a significant difference in the running of the heavier quarks. This is illustrate in Fig. 3. We note that the point where mb = mr moves down to a lower scale for a higher top quark mass. This is expected since from Eq. (A.9) one can see that the bottom type Yukawas are driven down by an increased top Yukawa. This behavior is to be contrasted with the SUSY GUT case in which the bottom Yukawa 0 function (see Eq. B.2.3) is such that this crossing point is shifted toward a higher scale with an increased top mass. As we shall see in the SU(5) SUSY GUT model, the equality of the bottom and 7 Yukawas at the scale of unification can be used to get bounds on the top and Higgs masses.


STANDARD MODEL


-1
600



40
I o



-20

-1
CX3

0
0 5 10 15 20 logo(u/1GeV)



Figure 2. Running of the inverse gauge couplings showing their
propagated experimental errors.






32

In Fig. 2. we display the evolution of the inverse of each of the three gauge couplings. In this figure, we see that the three gauge couplings only semiconverge in forming the GUT triangle around 1013-16 GeV. Notice that that the uncertainties do not come close to filling the "GUT triangle", thus ruling out grand unification, assuming the Standard Model as an effective theory in the desert up to the Planck scale.

The relation mb = mr (Relation I) is the most natural one in the SU(5) theory, and it could be expected to be valid at scales where the Standard Model gauge couplings are the closest to one another. We examine its validity for three different physical values of the top and Higgs masses in the Standard Model. The results are summarized in Fig. 3.





----- Mt=100 GeV, MH=100 GeV 1.5 --Mt=150 GeV, MH=150 GeV
1.5
- Mt=190 GeV, MH=180 GeV




-





.5
0 5 10 15 20 logo(p/1GeV)



Figure 3. Plot of mb/mi as a function of scale in the Standard
Model for various top and Higgs masses.





33

The noteworthy feature of the figure is that this simplest of the SU(5) relations is valid at an energy scale many orders of magnitude removed from that at which the gauge couplings tend to converge. Our result is vastly different from that of the original investigations in Ref. 16. We have improved on their work by including two loop effects in the running of the quark Yukawas, by taking into account the full Yukawa sector, and most importantly by incorporating QCD corrections in the extraction of the bottom quark mass.

We now turn to the more complicated relations among masses of the two lighter families, rmd = 3me., 3ms = n m (Relations II). There are large theoretical uncertainties in the extraction of the masses of the three lightest quarks from experiment, although the mass ratios are known more accurately. Following Refs. [51], we take their values to be mnd/mu = 1.8 and ms/md = 21, so that specifying ms fixes 7md and mu. We note that ms/md and mp/me effectively do not run. Therefore, given this value for ms/md, we do not expect relations (II) to be both satisfied exactly, since mp/9me - 23. The uncertainties in the light quark masses are accounted for by examining the ratios md/3me and 3ms/m77 for a range of 75s(1 GeV) values from 140 to 250 MeV. We have run these same ratios for representative values of the top and Higgs masses but find the results to be fairly insensitive to the value of the top. Therefore, in Fig. 4., we only present results for top and Higgs masses of 190 GeV and 180 GeV, respectively. Unlike relation (I) which holds only at ~ 107 GeV, we see that relations (II) can hold within - 5% at 1016 GeV for acceptable values of the light quark masses.





34


----- md(1GeV)=12 MeV i- - md(1GeV)=6.6 MeV
3



1
~-- ----------0 5 10 15 20
(a) logo0(/1GeV)



----- m,(1GeV)=250 MeV 3-- m,(1GeV)=140 MeV


co 2
------------0 5 10 15 20
(b) logto(p/1GeV)


Figure 4. Plots of (a) md/3m, and (b) 3m,,/m, as a function of scale
in the Standard Model for M ,=190 GeV and MH=180 GeV.



We find the GSTO relation, tan Oc = Vmd/ms (Relation III), to be quite independent of scale. The reason is that the Cabibbo angle effectively does not run, and the ratio of light quarks is essentially unaffected by QCD, since both are far away from the Pendleton-Ross [52] infrared fixed point. Further, we observe that their numerical values are fairly independent of the value of the top quark mass and of the Higgs mass. The agreement is spectacular, hovering






35

around the 4% level. For example, for Mt = 100 GeV and Mn = 100 GeV, we find that tan 8c/ ld/ms = 1.038 from MZ to Planck scale.

Relation IV, Vcb = /-i,/mt, involves the top quark mass directly, which may thus be predicted from this relation. On the other side of the equation, the experimental value of the "23" element of the CKM matrix, Vcb, is known only to within - 10%, Vb = 0.043 � 0.006. The value of Vb at all scales is obtained by running the CKM angles. These numerical results do not depend on the value of the CP-violating phase. We note that because of the Pendleton-Ross fixed point, the ratio of the two quark masses runs appreciably in the infrared region. We find that for a top quark in its lower allowed range, 91 - 150 GeV, this relation fails over all scales. Accordingly, we present our results for values of Vcb and Mt for which the relation can be satisfied below Planck scale. We use for Vcb values ranging from its central value of .043 to .050. (As discussed earlier we take the value of m. at 1 GeV to be 1.41 GeV.)

The results of our runs can be summarized in Fig. 5. in which we plot both Vcb and V/-c/mt as a function of scale. From the first plot, we see that the top quark has to be at least 178 GeV for ie/mt to meet Veb at the Planck scale. The second plot shows that Alt = 180 GeV allows for this relation to be easily satisfied at 1016 GeV, if ;cb(MZ) = .05. This means that a few GeV difference in Alt changes the meeting of the curves (both of which are affected by Mt) by three orders of magnitude! Finally, in the third plot, we see that for this relation to be valid at the unification scale, using the central value of Vb, a 197 GeV top quark is needed. We conclude that, given the uncertainties in the value of 1'cb, this relation may well be valid as long as Mt > 175 GeV.






36
Mt=178 GeV; MH=160 GeV
.07 ]11 i2 1 ,-- Veb(Mz)=.050
------------------V~b(Mz)=.043
.06 - "V ,.43





.04 I
0 5 10 15 20
(a) log10(p/1GeV)

.07 M=180 GeV; MH=170 GeV .07 M=197 GeV; MH=190 GeV
.07 T 1-- 0 1 1 1 1 , , , I f I f I
- (rn/m)' - (m /m) I/ SVb(Mz)=.050 ---- Veb(Mz)=.050 S.06 ~ Vb(Mz)=.043 > .06 - --- Vb (M)= 043-- .05 .05


.04 I I .04 i I 1 1 -1
0 5 10 15 20 0 5 10 15 20
(a) logo(0/1 GeV) (b) logo( /1GeV)



Figure 5. Plot of Vd and n/mi as a function of scale in the
Standard Model for (a) M,=178 GeV, (b) M,=180 GeV, and (b) M,=197 GeV, for both the central value (.043)
and the maximal value (.050) of Vb(z).



There are other other possible mass relations. An interesting mass relation

involves the ratio of the determinants of the charge -1/3 to charge -1 mass matrices and should equal one if relations (I) and (II) are valid. We note that, independent of the top mass (91-200 GeV), this weaker (less predictive) relation (mdmsmb = mempmr) can be satisfied at 1016 GeV for quark masses

within the range stated above.

The relations considered so far have been motivated by specific theoretical models. There are other relations which are not similarly motivated but which






37

may nevertheless hint at an underlying unified structure. In our search for simple relations among the quark masses, we considered an appealing "geometric mean" relation in the up sector, namely mumt = m . (2.3.1)


This relation favors higher top quark masses and can be satisfied well at 1016 GeV in the Mt = 190 GeV, AMH = 180 GeV scenario with an up quark mass compatible with that needed to satisfy relations (II) at 1016 GeV.

A similar relation involving the down-type quarks was tested


n1d"7b = m . (2.3.2) This relation favors higher top quark masses as well. In fact, in order to satisfy Eqs. (2.3.1) and (2.3.2) at 1016 GeV, as well as relations (II), given a fixed value for ms(1 GeV) within the range cited, the top quark mass would have to be larger than 200 GeV. Fortuitously, such a value would also favor relation (IV). These geometric mean relations have been discussed in the literature recently [53].

To conclude our analysis of the Standard Model case, we see that it is hard to arrive at a unified picture. The gauge couplings only semi-converge and the scale at which relation (I) tends to be satisfied does not coincide with that at which the other relations are valid. Still, the disagreement is never too large, which leaves the possibility that small course corrections in the running of the parameters allow most if not all of these relations to hold simultaneously at a unified scale. It is remarkable that for a top quark at the upper reaches of its allowed range, the long life of the bottom quark lends plausibility to the SO(10)-inspired relation (IV).






38

2.4 Quantitative Analysis of the MSSM

As is well known, the Standard Model shows no apparent inconsistencies until perhaps the Planck scale, where quantum gravity enters the picture. The nature of the physics to be found between our scale and the Planck scale is a matter of theoretical taste. At one extreme, because of the values of the gauge couplings, new phenomena may be inferred every two orders of magnitude. At the other, there is the possible desert suggested by GUTs; however, the absence of new phenomena over many orders of magnitude cannot be understood (perturbatively) unless one generalizes the Standard Model in some way to solve the hierarchy problem. Supersymmetrizing the Standard Model at an experimentally accessible scale can accomplish this. This particular scenario is bolstered by the fact that with such "low energy" Supersymmetry, the three gauge couplings of the Standard Model meet at one scale (- 1016 GeV) at the perturbative value of - 1/26 (see Fig. 6) [54,55,56]. However this should not be viewed as proof of supersymmetry since given the values of al, a2, a3 at some scale, and three unknowns (the value of a at the unification scale, the unification scale, and an extra scale such as the SUSY scale) there is always a solution. The exciting aspect of this analysis is the numerical output, namely a low SUSY scale, Msusy, and a perturbative solution below the Planck scale which does not violate proton decay bounds




39
Msus = 1 TeV
60


-1




1- 1

S20


03


0 5 10 15 20 logo(u/1GeV)



Figure 6. Plot of the running of the inverse couplings. The dotted lines above and below the solid lines represent the experimental error for each coupling. Note the small region where all three couplings intersect. We found that this region reduced to a point when MsUsy = 8.9 TeV
and was non-existent above that scale.


The collapse of the GUT triangle in the supersymmetric extension fixes two scales, the one at which the gauge couplings unify, the other at the threshold of Supersymmetry. Minimal Supersymmetry implies two Higgs doublets and eliminates the quartic self-coupling of the Standard Model. But there appears an extra parameter, the ratio of the vacuum values of these two doublets, parametrized by an angle 3, tan = vu/lvd, where vu (vd) is the vacuum expectation value of the Higgs field that gives mass to the charge 2/3 (-1/3,

-1) fermions. In the following we examine the relations discussed in section 2.1 among masses and mixing angles in the context of the minimal supersymmetric




40

extension of the Standard Model. We limit our study to the case of only one light Higgs.

We will first restrict ourselves to an SU(5) SUSY-GUT [57] where yb and yr, the bottom and 7 Yukawa couplings, are equal at unification. The crossing of these renormalization group flow lines is sensitive to the physical top quark mass, Mt. This can be seen in the down-type Yukawa renormalization group equation (above MSUsY), from which we extract the evolution of Yb, since the top contribution is large and appears already at one loop through the up-type Yukawa dependence:

dYd Yd[ 3YdtYd + YutYu + Tr3YdtYd + YetYe}
dt 16x2 (2.4.1)
7 2 2 16 2 91 + 32 + 3

where Yu,d,e are matrices of Yukawa couplings. Demanding that their crossing point be within the unification region determined by the gauge couplings allows one to constrain Mr. This yields an upper and lower bound for Mt which is fairly restrictive.

We consider the simplest implementation of supersymmetry and run the couplings above MSUSY to one loop. The superpotential for the supersymmetric theory is


1W = uQYYufil + ldQ Yde + dYec + Wi" du , (2.4.2) where the hat denotes a chiral supermultiplet. We assume the MSSM above Msus, and a model with a single light Higgs scalar below it. This is done by integrating out one linear combination of the two doublets at Msesy, thereby leaving the orthogonal combination in the Standard Model regime as the "Higgs doublet":

(,,SA) = 4dcosg + fusino , (2.4.3)





41

where 1 = ir24*, and where tano is also the ratio of the two vacuum expectation values (vu/vd) in the limit under consideration. This sets boundary conditions on the Yukawa couplings at Msusy. Furthermore, in this approximation the quartic self coupling of the surviving Higgs at the SUSY scale is given by:

A(MSUSy) = (g+ g2)cos2(20) . (2.4.4) This correlates the mixing angle with the quartic coupling and thereby gives a value for the physical Higgs mass, MHiggs. Using the experimental limits on the MHiggs further constrains some of the results. By using the renormalization group we take into account radiative corrections to the light Higgs mass [58] and hence relax the tree level upper bound, MHiggs - MZ [59].

We determine the bounds on AM1 and MHiggs by probing their dependence on 0. In SUSY-SU(5), tan 3 is constrained to be larger than one in the one light Higgs limit. It seems natural to us to require that yl > yb up to the unification scale [60], thereby yielding an upper bound on tan 0.

To probe the dependence of our results on Mb we use both Mb = 4.6 GeV and Mb = 5 GeV in our study. We also investigate the effect of varying MsUSy. Given the values of the gauge couplings, we find unification up to a SUSY scale of 8.9 TeV, and as low as Mv. For empirical reasons we did not investigate solutions below that scale.

We determine that the lower end scale, AILT, of the unification region corresponds to an a3 value of 0.104 at MIZ, while the higher end scale, MHGUT, corresponds to a value of 0.108 at AMl for a3. We find that the unification region is insensitive to the range of top, bottom, and Higgs masses considered. In our analysis of the bounds for AMt, the values for a 1 and a2 are chosen to be the central values since their associated experimental uncertainties are less





42

significant than for a3. Demanding that Yb and y, cross at MLUT and taking a3 = 0.104 then sets a lower bound on Mt. Correspondingly, demanding that Yb and yr cross at MGHT and taking a3 = 0.108 yields an upper bound on Mt. These bounds are found for each possible value of f.


MsusY=1 TeV, Mb=4.6 GeV 200




Q 150




-100


MH -"

50

40 50 60 70 80 90 P (deg)



Figure 7. Plot of the top quark mass, Mt, and of the Higgs mass,
MHiggs, as a function of the mixing angle / for the highest value of a3 (high curves) and the lowest value
of a3 (low curves) consistent with unification.


Fig. 7. shows the upper and lower bound curves for both Mt and MHiggs as a function of # and for Msusy = 1 TeV and Mb = 4.6 GeV. When applicable we use the current experimental limit of 38 GeV on the light supersymmetric neutral Higgs mass [61], to determine the lowest possible Mt value consistent






43

with the model. We find 139 < Mt < 194 GeV and 44 < MHiggs < 120 GeV. We investigated the sensitivity of these results on MSUSy in the range, 1.0 � 0.5 TeV. We find that the bounds on Mt are not modified, but the upper bound on the Higgs is changed to 125 GeV, and the lower bound drops below the experimental lower bound.

For Mb = 5.0 GeV, we see an overall decrease in the top and Higgs mass bounds: 116 < Mt < 181 GeV, MHiggs < 111 GeV. We display the results of our analysis for the extreme case, Msusy = 8.9 TeV, in Fig. 8., with Mb = 4.6 GeV. This only significantly changes the upper bound on MHiggs to 144 GeV compared to the Msusy = 1 TeV case.


Msusy=8.9 TeV, Mb=4.6 GeV 200




150




o100 - H




50 i 1 l l
40 50 60 70 80 90
p (deg)



Figure 8. Same as Fig. 7 for MSUSy = 8.9 TeV and Mb = 4.6
GeV.




44
MsusY= 1 TeV
100


S30


Mb -4.6 GeV




1 Mb=5.0 GeV ".

0 10 20 30 40 50 60 P (deg)



Figure 9. Plot of the ratio of the top to bottom Yukawas, yt/yb, at
the GUT scale, for two different bottom masses (solid and dashed curves) as a function of tan / for the highest value of a3 (high curves) and the lowest value of a3 (low
curves) consistent with unification.


We have also run yt up to the unification region and compared it with yb and Yr to see what the angle / must be for these three couplings to meet [62], as in an SO(10) or E6 model [3.4] with a minimal Higgs structure. It is clear that this angle is precisely our upper bound on / as described earlier. In Fig. 9. we display Y /yb at the GUT scale as a function of tan / for MsUSy = 1 TeV and for the two bottom masses we have considered. If we demand that the ratio be one we can determine the mixing angles for the low and high ends of the unification region. Then going back to Fig. 9. we find as expected a much tighter bound on the masses of the top and of the Higgs. Indeed, for Mb = 4.6 GeV, we have 49.40 < tan /3 < 54.98, which yields 162 < Mt < 176 GeV and





45

106 < MHiggs < 111 GeV. When Mb = 5.0 GeV, we obtain 31.23 < tan I< 41.18, which gives 116 < Mt < 147 GeV and 93 < MHiggs < 101 GeV.

The strategy of the remaining part of the MSSM analysis is to exploit relations (II-IV) to constrain Mn and therefore 0 and MH. For MSUSY = 1 TeV, we treat two cases, the first where unification takes place at its lowest value (Low MGUT) and the second where it is at its highest value (High MGUT)- In the following, we will not discuss our results for upper limit of the Supersymmetry breaking scale, 8.9 TeV, since it adds nothing to our conclusions.

In the case of the mass relations among the light quark and lepton masses (relations (II)), we find that our plots do not depend on Mt, therefore we only display them for a representative value. Here we do not follow the strategy used in the Standard Model case (i.e., we do not vary ms), although we still keep the ratios md/mu and ms/mnd fixed. Instead, we look for that value of ms(l GeV) which gives us the best agreement for relations (II) both for the low and high AIGUT (we cannot expect exact agreement at one scale, since ms/md does not run). For instance, we can get the same value of ms(1 GeV) in two different ways, either by demanding that at low MGUT md/3me = 1.1 and 3ms/mp = 1 or at high MGUT that md/3me = 1 and 3ms/mp = .9. In both cases, the masses of the lightest quarks at 1 GeV are: m, = 3.80 MeV, md = 6.67 MeV, and ms = 141 MeV. These results are summarized in Fig. 10.




46
MsusY = 1 TeV
2.5 _ , l , I I l I i
--------- LOW MGUT
- HIGH MGUT 2
' md(1GeV)=6.7 MeV




S---- - --- --- --0 5 10 15 20
(a) logo(u/1GeV) Msus =1 TeV
2 .5 1 1 1 1 1 1 i i
--------- LOW MGUT
S -- HIGH MGUT 2
m,(1GeV)=141 MeV

S1.5


1 - ..------------------- .---------- .. ...- --- -0 5 10 15 20
(b) logl0(#/1GeV)



Figure 10. Plots of (a) md/3n, and (b) 3m,/m as a function of scale
in the SUSY case with Msusv'=1 TeV, for the low and high unification scales (no appreciable dependence on
M, was found).


In the above, our philosophy has been to take the known low energy data, and using the renormalization group, derive its implications at high energy. As we did for relation (I), we could impose both of relations (II) at one unification scale (low or high MGUT). This would fix ms/md at this scale, and since ms/md and map/me do not run, that would yield ms/md = 23.6, a value that





47

is only 12% larger than the value discussed in section 2.2. Furthermore, the results of our runs yield the masses at 1 GeV of the down and strange quarks to be md = 5.86 MeV, ms = 138 MeV, in the low MGUT case, and md = 6.49 MeV, ms = 153 MeV, in the high MGUT case. We note that this approach has also been taken by the authors of Ref. 63.

Before discussing relation (IV), let us note that, with Supersymmetry, relation (III) is again well satisfied at all scales.


Msusy=1 TeV; Mt=198 GeV .0 7I II l i l l 1 1 1 1/2 HIGH MGUT



' .05


.04 - Veb


.03
I I I I I I I I 1 1
0 5 10 15 20 log 0(pu/1GeV)


Figure 11. Plot of ~b and ,/inm, as a function of scale in the
SUSY case with Msusy=1 TeV for Mt=198 GeV and for
Veb( Mz)=.043.


We now turn to relation (IV). As we did in the Standard Model case we display our results both for the central value of Vb (.043) and for its upper value (.050). Then we look for values of M which give us agreement at the





48

unification scale (low or high MGUT). Using the central value for Vcb, we find no agreement at the unification scale. However, this relation is satisfied at Planck scale, if we use both a high MGUT and Mt = 198 GeV (the highest possible value consistent with relation (I)), as displayed in Fig. 11.

We have also made several runs with a higher value Vcb. There, the relation can actually be satisfied provided that we use the high MGUT scale and Mt = 198 GeV as shown in Fig. 12.


Msusy=1 TeV; Mt=198 GeV .07 1
P .o6 L(r/ rn l/2 HIGH MGUT








o .04


.03


0 5 10 15 20
log o0(/ 1GeV)


Figure 12. Same as Fig. 11 with Vcb(Mz)=.050.



In the low MGUT case, the two curves meet closer to the Planck scale. In fact, theory does not dictate to us the exact scale at which the SO(10)inspired relation is valid; it could be much higher than the scale of unification of the Standard Model's gauge couplings. To account for this, we now plot, in




49

Fig. 13., Vb as a function of M, assuming that relation (IV) is valid at MGUT, 10MGUT, and 100MGUT, and using the higher value of g3(Mz).


Msusy - 1 TeV
.06

16
MGUT MGUT=-1.26x10 GeV

.055 - 1OM/


100MGUT

.05




.045
150 160 170 180 190 200 Mt (GeV)



Figure 13. Plot of Vb(AMz) as a function of M, assuming relation
(IV) holds at various scales ZMGUT.


Given an initial value of Vb at MZ, Fig. 13 can be used to determine the needed Mt (and hence f) to satisfy relation (IV) at MGUT, 10WGUT, or 100MGUTWe can see from this figure that as long as Vcb is larger than its central value, then relation (IV) can be satisfied above the SU(5) GUT scale and still allow for a lower value of M .
The relation, mdunsmb = mempmr, involving the determinants of the charge -1/3 and charge -1 fermion mass matrices holds in the minimal supersymmetric model at 1014 GeV in the high MGUT case and at 1018 GeV in




50

the low MGUT case. In both cases, this relation holds within , 10% at 1016 GeV for 160 < Mt < 198 GeV.

A priori one might naively assume that mumt = mn could be easily satisfied because of the uncertainty in mt. However, two facts make the relation viable in the supersymmetric case. First, the value predicted for the top mass is within the range allowed by experiment and the p-parameter bound. Second, and most remarkable, is the fact that this top mass value is compatible with relations (I)-(III). In Fig. 14, we display the running of the ratio mumt/m2 for the low and high MGUT cases.


MsusY-1= TeV; Mt-160 GeV
2I I l I l I I HIGH MGUT

--------- LOW MGUT

V r mu(1GeV)=3.8 MeV




S 1




.5 IIiI I
0 5 10 15 20
log( (/1GeV)



Figure 14. Plot of mndmi,/,n. as a function of scale for the highest
value of aa(MZ) (high curve) and the lowest value of
a3(Mz) (low curve) and for M,=16o GeV.





51

We show the curves representing the lower Mt value of 160 GeV for which the relation is best satisfied at MGUT. This relation is incompatible with relation (IV) however, since the latter favors a higher top mass. We note that mc affects these two relations in an "inverse" manner. A lower experimental value for the charm quark mass favors relation (IV) whereas a higher experimental value favors the geometric mean relation.

One may also consider the geometric mean relation in the down sector. We find however that this relation fails to hold in the supersymmetric case. Other relations among the Yukawa couplings have been considered in the literature. Theoretical bias or numerology can lead to still other relations valid at some unifying scale. In all cases, a thorough renormalization group analysis will be required in investigations of a possible deeper structure.














CHAPTER 3
A MULTISCALAR SM EXTENSION

We present a minimal model with explicit lepton number breaking and an invisible axion. Neutrino masses are generated at one loop. For a reasonable range of parameters the model provides a solution to the strong CP and solar neutrino problems.

This chapter is organized as follows. In section 3.1 we will present this model exhibiting SU(2)"1 x U(1)" x U(1)PQ symmetry and incorporating explicit lepton number violation in the scalar sector. We provide some details of the minimization of the scalar potential and give conditions on parameters that insure electric charge conservation. We then discuss the requirements that the gauge hierarchy problem places on the couplings of the model. In section 3.2 we display the model in unitary gauge and present the scalar mass matrices and their mass eigenstates. In section 3.3 we compute the neutrino masses and diagonalize the Majorana mass matrix. In section 3.4 we discuss experimental constraints on parameters of our model. Finally in section 3.5 we show that the model can provide a solution to the solar neutrino problem via the Mikheyev-Smirnov-Wolfenstein (MSW) effect [12].


3.1 The Model

The model we present below is essentially that of DFSZ, augmented by a charged scalar singlet. Thus it extends only the scalar sector of the Standard Model, by adding another isodoublet and two isosinglets. By introducing two


52






53

standard Higgs doublets we can generate interesting new couplings. This extension of the Standard Model appears naturally in two popular types of theories. In the first, the global symmetry of the Standard Model includes a chiral phase symmetry, the Peccei-Quinn (PQ) symmetry [64], used to explain the lack of strong CP violation. The second is the N = 1 supersymmetric extension of the Standard Model. In the first case, the theory contains a pseudo-NambuGoldstone boson, the axion [65]. For the model to be viable, the axion must be very light, and it can only be "seen" by Sikivie-type detectors [66]. The model is implemented through the presence of an electroweak singlet neutral Higgs field which carries the PQ charge. This invisible axion [67,68] extension of the Standard Model has two Higgs doublets, Hu and Hd, with hypercharges 1 and

-1 respectively, the same PQ charge, as well as a neutral singlet 4. Their quantum numbers are linked through the quartic coupling Hu T2Hd , (3.1.1)


which shows that 4 carries unit PQ charge. This describes the standard DFSZ [67,68] invisible axion model. (The Zhitnitski! model differs only in using a cubic interaction instead of this quartic one.)

The new observation we make is that by adding only a single scalar field, the S+ field, we can form another quartic term which explicitly breaks lepton number by two units, since , carries no lepton number. It is given by


Ht HdS++ . (3.1.2) We note that S+ carries one unit of PQ charge, fixed through the leptons since the charged lepton masses are taken to arise from the Yukawa coupling to Hd. Thus we come to the conclusion that in the context of the Standard Model with





54

invisible axion, we need to add only one charged spinless particle to generate lepton number violation, and hence neutrino masses and mixing. We stress that much of what we have to say has already appeared in the literature, the S+ field was first advocated by Zee [69], invisible axion models originally by Kim [70] and Shifman, Vainshtein and Zakharov [71], and later by Zhitnitski [67] and Dine, Fischler, and Srednicki [68].

Before going into the details of our model, let us summarize the basic features of the invisible axion model. The invisible axion model "solves" the strong CP problem, at the cost of introducing a new PQ-breaking scale of the order of 1010-12GeV, the scale being bracketed by astrophysical and cosmological constraints. Thus the model contains the tiny ratio
_1
GF2
V 10- , (3.1.3) where VO = v*/ (b) . Unlike GUTs, this does not present any technical hierarchy problem [72], since 4 is not required to have any large couplings to the fields of the Standard Model. We add that there is still a physical hierarchy problem yet to be answered: how are such disparate scales generated? In fairness, we should also remark that the invisible axion model, originally devised to explain away the limit on the 0 parameter, 9 < 10-9, itself introduces a similar ratio!

In our model the Yukawa part of the DFSZ Lagrangian


iZ[yj LT a2jL + Y QiL a2jL 72Hd + i YjQ iLu2UjLT2Hu + h.c. , (3.1.4)


is augmented by

iz[ij]LLT a2r2LjLS+ + h.c. , (3.1.5) which is antisymmetric in family space, as is required by the Pauli principle and S+ is an isoscalar spinless field with Y = 2, and L = -2. With three






55

families, these Yukawa couplings break all but the total lepton number. The PQ symmetry insures that only one doublet couples to each charge sector of the quarks, insuring the absence of tree level flavor-changing neutral currents [73]. Here we have the weak isodoublets LiL = l , QiL = di L and diL and iiiL the left-handed isosinglet antiquarks of charge ( and -, respectively. The Higgs potential carries both the DFSZ term HuTr2HdI2 and our new term HutHdS++, in addition to all the normal quartic terms. It has no cubic terms. Note that the DFS quartic term is needed. Its absence from the potential would yield a theory with both L and PQ invariance. Upon 4 getting its vacuum value, the linear combination L + 2PQ would be preserved. Electroweak breaking would then yield a Majoron theory of a type already excluded by measurements of the Zo width. Thus both terms are needed to conform with experience. The most general potential of these fields consistent with SU(2)'1'x U(1)x U(1)PQ symmetry with the single explicit lepton number violating term is given by

V(Hu,Hd,, S) = VDFS(Hu,Hd, D) + V'(Hu, Hd,',S), (3.1.7) where [68]

VDFS(Hu,Hd,4) = AX(HuTH f )2 + A(HtHd - )2 (+ - 2 + AudHu t HuHd Hd + A dHutHdHd Hu + 6uHut HuAJ*4

+ EdHd'Hd~* + u Td(iH Tr2Hd 2 + h.c.),
(3.1.8)

and
V'(Hu, Hd, 4, S) = mfSS-S+ + As(S-S+)2 + AusHutHuS-S+

+ AdsHdtHdS-S+ + 6s,~S- S+ + A(HutHdS*4 + h.c.).
(3.1.9)






56

The A term is the only lepton number violating interaction. For a large range of parameters it is possible to minimize the potential in (3.1.7) so that the vacuum expectation values (VEVs) of the scalars align without breaking charge and have magnitudes consistent with the values of the vector boson masses. As in the DFSZ model

M gv , M = g2 + g'2 v, (3.1.10) where g and g' are the SU(2)"' and U(1)Y couplings respectively, and v = /2 + ,v2 where vu and - d are the magnitudes of the VEVs of the neutral components of Hu and Hd. We therefore require v - 250 GeV. For the model to be consistent with limits on axion couplings, the VEV of the P field, Vo, must be Z 109.5 GeV [74,75].

The analysis of the minimization of the potential works very much as in the DFSZ case. Assuming exact PQ symmetry (nonperturbative QCD effects turned off) and requiring that electric charge not be broken, we can without loss of generality parametrize the VEVs of the Higgs fields as

(H.) = i0e" (Hd) 0 ,d (1 , (3.1.11) by using SU(2)" x U(1)Yx U(1)PQ global transformations. All the parameters in (3.1.11) are real. (Note: The phase of 4 can only be rotated away by a U(1)PQ transformation. However, as the true effective potential, modified by QCD instanton effects, explicitly breaks PQ, this phase will be in principle calculable from the minimization of the full effective potential. Its presence is an indication of strong CP violation.)

Sufficient conditions for minimization are the requirements that the potential be bounded from below:

Au, Ad, As, A > 0,






57

,Aud > -2vrA1Ad, u > -2 /X--A, bd > -2VAdA,

Aus > -2 AXA,, Ads > -2v X-f,, b > -2 V A. (3.1.12) In order to maintain charge invariance of the vacuum it is sufficient that the following parameters be positive: 6d, Ad ' udm Aus, Ads, bs > 0,


and that the conditions

A< -, (3.1.14) v, VO
where
2 2 ds2 > 0,
A 2m2s+ bs V +AusvA + Ads'>
(3.1.15)
2 0 ,,2 u d 2
B = - 4Ad + 6d + (Aud + Ad)' > 0, be satisfied. Note that bud and A are chosen to be real and positive. This can always be done by phase redefinitions of the Higgs doublets and the S field. The minimization conditions for the potential in terms of these parameters are

4Af - 2A,- = 6,u2 + d- 26udvuvd, (3.1.16)

4u( - I), + ul2 - vudd V2 +Audvu = 0, (3.1.17)

2
4Ad( - f d2)d + 6dt'd+V2 - 5udvu V + A dVdV2 = 0. (3.1.18) Positive definiteness of the Hessian of the potential consistent with nonzero vu, vd and V put rather complicated upper bounds on Aud, bu and 6d which we omit. With bud > 0 the minimization gives 0 = 0. All these conditions are sufficient for minimization and charge invariance in the original DFSZ model.

We have noted that astrophysical bounds on axion couplings require VO Z 109.5 GeV, introducing a scale much larger than the electroweak scale into the theory. In general, without extreme fine-tuning of parameters, this separation






58

of scales will not be maintained when radiative corrections are included [72]. The philosophy of this problem is that all extremely small parameters of a theory, e.g. small ratios of mass scales, must obtain radiative corrections of the same order or smaller to avoid such fine-tuning. This is the concept of naturalness and it ensures that the properties of the theory are stable against small variations in the fundamental parameters [76]. This kind of problem arises if there are coupling constants which are much larger than these ratios, as happens in GUTs [5), where gauge couplings of order 10-2 create radiative corrections to ratios of the order of 10-13. It is not obvious how to assess the seriousness of the gauge hierarchy problem, since the origin of the parameters which enter in our effective low energy theories is unknown. One can envisage a scenario in which this effective low energy theory stems from a highly nonlinear theory, undoubtedly including gravity, which leaves us below Planck mass with a bunch of parameters which happen to be at the fixed points of the mother theory. This would mean that the input parameters obey peculiar relations among themselves, relations we would deem highly unnatural, although we would eventually find out that they have been fine-tuned by the mother theory!

In the DFSZ model the relevant small quantity is


~5 10-4. (3.1.19) This can be related to ratios of couplings via the minimization conditions of equations (3.1.17) - (3.1.18) :

t12 budt - bu
- 2 < 1, (3.1.20)
1 2Auc2 + Auds2 - 4Au(fu/v2)

where


c = cos Si , s = si d , t = tan d (3.1.21) ' V VU





59

Equation (3.1.20) is augmented by a similar expression with u and d indices interchanged. By assuming vu , fu v d " fd, we can satisfy these conditions with 6u, d, bud < 1, that is
- ~ - ,(3.1.22)
vA
where we have given the three couplings between the 4 field and the Higgs doublets the generic label 6 and the couplings Au, Ad and Aud are given the generic label A. It is reasonable to take A Z 10-2 since the A couplings get renormalized by gauge couplings. In fact all couplings with prefix A have this property. From the above we infer that 6 10-16. Naturalness requires that the corrections to 6 be multiplicative, 6 -+ (1 + O(10-2A)), 10-2 being a typical loop integration factor. From the diagrams of Fig. 15, we see that it amounts to demanding that 6, - 6 and A2 ~ 6A, ie. A E 10-9. Taking m, > 1 and using the above typical values of couplings, we find that the bound (3.1.14) is weaker than the hierarchy bound.





60

, , d , d



S 1
S :

u s

" IS S '













Figure 15. a) O(bs A,us) correction to 6, or O(bs AXds) correction to
6d and b) O(A2) correction to bu, bd


The coupling A does not get renormalized by gauge couplings. However, there is no reason to assume it is small as A couplings give rise to multiplicative renormalization of the 6's and can be O(A). The intuitively correct result is that if all cross-couplings to the (D field are extremely small for both our model and for the DFSZ model, there is no technical hierarchy problem.


3.2 Unitary Gauge

In this model we expect to have, in the neutral sector, three massive scalars, one massive pseudoscalar and a nearly massless axion, and in the charged sector, two massive scalars. It is convenient to work out their masses and other





61

properties in the unitary gauge where the Nambu-Goldstone bosons have been explicitly eaten. To find the unitary gauge we define new doublet fields: H1 = sHu + cHd,
(3.2.1)
H2 = cH, - sHd,

where H = ir2H*. Taking into account that after minimization 9 = 0 in (3.1.11), we find that

(H1) = 0, (H2) = (3.2.2) The physical unitary gauge is obtained by performing an SU(2) gauge transformation that factors out the SU(2)" x U(1)Y Nambu-Goldstone phases in H2, so that they may be eaten by the vector bosons. We define H1 ) (3.2.3) 10
H2' = UH2 = ;1 P2 , V where U is the SU(2) group element. If we write S(Pd + Xd + t'd) Hu +Hd = ) (3.2.4)
Hu (p, + i',N+v) h d then
h = chu - sh: and X2= cu- Xd (3.2.5) are the Nambu-Goldstone bosons of electroweak breaking. They will become the longitudinal gauge bosons while the orthogonal combinations

h' = sh( + ch:d and X1 = sX, + cXd, (3.2.6) will appear in the Lagrangian. Dropping primes in (3.2.3) and writing h0 1
--- - +(pl ix ),
(3.2.7)
4 =-(p + V )e





62

we can obtain the scalar masses in unitary gauge. The neutral mass term for P1, P2 and po is p M2p, where pT = (Pl, P2, PO). We can use the typical relations obtained in the analysis of the hierarchy problem, i.e. 6 - AX2 and A v"- AE where E = < 1, to write this matrix in the simplified form ae2 b2 CE3
M 2 Av(be2 a'E2 de3 . (3.2.8) c0 3 dE3 1

The constants a, a', b, c and d are of order 1. Mixings between po and P1,2 are extremely suppressed relative to pi and P2 mixing by a factor of e 5 10-7. The mass eigenstates are given by ; = O(P)p, where O(P) is an orthogonal matrix such that M2 = O(P)TM2 O(P), and is parametrized as
P p(diay)
clc2 -S3C1S2 - Slc3 --2C3Cl + s3sl O(p) = sic2 c3cl - s3s21 -c3s2 - s3c2 , (3.2.9) \2 3c2 c3c2 where ci = cosOi and si = sin i. The angle 01 ~ 0(1) while 02,3 " O(e). The physical fields are
l P 1cos 81 - P2 sin 81, P2 " Pl sin 01 + P2 cos 01, (3.2.10) PO "~ + Pl92 + P203, with masses

S2, 2 ~AV. (3.2.11) Thus two of the neutrals have masses similar to the mass of the Standard Model Higgs, while the other is superheavy for A - O(A). The xi and ao = 04VO mass matrix may be diagonalized to yield a massive pseudoscalar, X, with a Standard Model mass (due to (3.1.22) ) and the massless axion, a. These fields are
1
a = (2v,vdX1 - vVWa ), v2V + 4vvd
1 (3.2.12) X = (vVX1 + 2vuvda), v2V,2 + 4vvd
Vu<





63

with
v2 v2
71 = Ud(- + 2csv2) - bud (3.2.13) S 2cs 2cs We require that mx > 30 GeV [77] and impose the bounds 109.s GeV < VO < 1012 GeV, (3.2.14)


where the upper bound corresponds to a energy density of axions sufficient to close the universe [78]. This and the hierarchy bound gives the unrestrictive limits 10-16 Z 6ud Z 10-21 for cs - 0(1). As expected the axion is mostly the a� field.

Finally we work out the charged scalar mass matrix for applications to the calculation of the neutrino masses. The h+, S+ mixing from the A term in the potential gives rise to the mass matrix

'ud + EudV AvV, A Z' ( 1
(h, S+) 2 + +2 (i2 S (3.2.15) where As2 = Ausc2 + Ad.';2. The physical mass eigenstates are + = i +cos a - S+ sin a, (3.2.16)
+ = h+ sin a + S+ cos a, with the mixing angle given by (henceforth we take off our hats)


tan 2a = 12 2 (3.2.17)
_- (3.2.1,1
m 2 - 1 ud) + V s ud assuming mn22 > 77211, where



122 1 A v2 (3.2.18) 22 1





64

The masses are

m = (m1 + m22 V " -n - )2 + 47n )2)
h2 2 1 2212(3.2.19) M = (mm +22+ (m - 22) +4(m22). Note also the useful identity sin 2a 2m22 = (3.2.20)
-M -m A,82 - 172

In the limit m2 > m21 Av 2, the S+ mass is determined by the free parameter ms. In this limit the masses are M ~ 2, m-, 1 Av2, (3.2.21)


and a �5 1.


3.3 Majorana Neutrino Masses

The AL = 2 Majorana masses for the left-handed neutrinos are generated at one loop through the h+ - S+ mixing. To see this we write the charged scalar terms in the Yukawa couplings of (3.1.4) and (3.1.5) in terms of mass eigenstates of the charged fermions and scalars
lepton I -2
L = - i- cot 31iLM(e)eR(h+ cosa + S+ sin a)
v, (3.3.1)
- 2i(v )CZ'e(L(-h+ sina + S+ cosa) + h.c.,

where a was defined in (3.2.17) and we have suppressed family indices. Here Z' is a new antisymmetric matrix Z' = A*ZAt where y(e) = _VAtM(e)B and M(e) is the diagonal charged lepton mass matrix. Since 2' has three independent elements, z[ij], their phases can be absorbed by a redefinition of the phases of the lepton doublets. Hence we take 2' to be real. Note that there will be no CP violation coming from the neutrino mass matrix.




65

The one loop diagram contributing to the neutrino masses is shown in Fig. 16.








/
/ 1


I I


SLpi Li V LJ






Figure 16. One loop contribution to Majorana mass in the unbroken theory.


The induced neutrino mass matrix is [79,69]

0 zep(ze - X) zer(xe - 1) Cmo(v, , v) ze(x - xp) 0 Zp,(xp - 1) v, +h.c., Zer(.rc - 1) zr(x, - 1) 0 Vr L (3.3.2)

where, in the limit AL,. m1, > n, n?772 N92 no _ r cot3 sin 2aln( - ), (3.3.3) (47,)2 v n1 and e = , x = 7. This matrix can be conveniently diagonalized using perturbation in xc and xl~. In the limit xe, xp < 1 the matrix is ( 0 -zer
0 -zpr ), (3.3.4)
--er -ZPr 0





66

and can now be diagonalized by a real orthogonal matrix O such that M(V) = OTMdiag. Diagonalization yields one massless neutrino i1L and two degenerate Majorana neutrinos i/2,3 of mass


7m2,3= m0 4e + z2. (3.3.5) (The negative mass is made positive by a chiral redefinition of Majorana eigenvectors.) The corresponding Majorana eigenvectors are

i1 = vL cos Ov - vjL sin 8, + c.c.,
1
f2 = -(veL sin p + vL cos8O + VrL - c.c.), (3.3.6)

v3 = 1(-"eL sin 8 - VpL COS v + vrL + c.c.), where VL = OvL and


cos = . sin v = Zer (3.3.7) c r+ z Z2r + z These last two eigenstates can be combined to form a Dirac neutrino.

The matrix O appears in interactions mediating processes that violate individual lepton number conservation and is given by cos , - sin 0v O 0= (-- V CO O L . (3.3.8) S sin0, - cos0 /


The neutrino masses and mixing to first order in xp, but still neglecting Xe, give rise to a mass for z1 given by m1, - mv.,ax sin 2v,, (3.3.9) where

a = ze cos Ov. (3.3.10) P 7r






67

Also v2,3 obtain a mass splitting equal to mu,, with v2 being the lighter neutrino of the two. The corrections to mixing angles in this approximation are worked out in Ref. 80. Finally let us approximate the natural scale of neutrino masses in this model. Using (3.2.20) for the mixing angle we have n2 2
M = )2AV4 cot ' ln( _s ). (3.3.11) Let us assume that zir O0(10-2) or 0(10-3), cot P -~ 0(1) and that AVO ~ Av - 2.5 GeV. Now if we take Ms, mh O0(100 GeV) with their mass squared difference of order (100 GeV)2 and take the logarithm in (3.3.11) to be 0(1), then the scale of the massive neutrinos is m2,3 ~ 10 - 100 eV. For Ms - 1 TeV, my23 ~ 0.1 - 1 eV. Note of course that we only know that A < 10-9 and it could well be as small as 6. In this case, for squared mass differences of order (10 GeV)2, we get m2,3 O0(10-3) or 0(10-4) eV.


3.4 Experimental constraints

In this section we give constraints on the parameters of our model that arise from phenomenological bounds.

As with any multiple Higgs doublet model, charged scalars will mediate both flavor-changing neutral currents and new contributions to CP violating processes at one loop level. In our model these take place through the quark Yukawa coupling

uark =_ - (h+ cos o + S+ sin a)(tan /38i M(u)VdL
v (3.4.1) + cot ILVM(d)dR) + h.c.,

where V is the standard CIKM matrix. There are various ways to constrain the ratio d = tan 3. The best constraints on tang come from an analysis of the B - B system [81] combined with the analysis of CP violation constraints






68

(Note that these bounds do not hold in the Supersymmetric extension of the Standard Model). The BdL - BdS mass difference is completely dominated by top quark contributions. A plot of mh vs. tan / appears in Fig. 7 of Ref. 82 and they obtain the approximate bound, 69.1 1.88
tan < 0.7 + 1.16 nh, (3.4.2) I i
for mt and "th in GeV and ml > 65 GeV. After including ( constraints they find tan fl 5 3.5 for 25 GeV 5 mhr < 300 GeV. We can carry over this analysis to our model in the limit a < 1 and Ms > mh. Without this limit the complications of mixing spoil the bounds.

We can also limit tan /3 from below. Here it is useful to look at semileptonic decays of B mesons which are mediated by the leptonic Yukawa couplings of (3.3.1) . The best bound arises by considering contributions of charged scalars to the rate for B -- 7vX [83]. Although there are no direct experimental bounds on this decay, one may still find a constraint on cot 0 given the experimental values for the total rate B -+ X and the rate B -+ evX. By requiring that B -- rvX does not saturate the total width, in contradiction to the observed B -4 evX branching ratio. The authors of Ref. 83 obtain cot G 0.9 e' (3.4.3)
1 GeV'

which is again applicable in our model in the limit of small h+-S+ mixing. If one further requires that the Yukawa couplings proportional to cot # do not exceed the perturbativity limit then cot / 5 130 [82]. We now have upper and lower bounds on this parameter.

The most stringent constraints on the lepton family number violation parameters, zir, come from electron and muon number violating processes such as -p a e7 and and nonstandard contributions to p --,+ 3eptons.






69

By considering the tree-level contribution to y decay and assuming that the fractional deviation of Cabibbo universality is no more than 10-3, we have the bound [9]

e: < 10 G-3GFAI, (3.4.4) where
- cos2 a sin2 a 1
s= M + sin a (3.4.5) For , = 100 GeV we get zep < 9 x 10-.

A similar bound on the parameter comes from modifications of the standard amplitude for vp-e scattering due to charged scalars [9] 2, 5 V '10-GFI, (3.4.6)


The gauge contribution to the process p -- ej, is suppressed by a GIM mechanism, the amplitude being proportional to the square of neutrino masses. The resulting rate relative to ('(p --+ evlvp) = 1G2Fm is

BR(p -- e) = _ 2 -2 < 5 x 10-11, (3.4.7) 487-,G2
F s

where we have given the present experimental upper bound on this branching ratio [84]. We therefore have an upper bound on a product of Yukawa couplings, 3-2
ZCerz r < 10-3GF2s. (3.4.8) given the value of Ms5 ( MsAl for Ms > mih). For Ms = 100 GeV we get zerZr < 10-4


3.5 Solar Neutrino Phenomenology

The solar neutrino problem is a long-standing problem to proponents of the standard electroweak and solar models and attempts to resolve it have fostered




70

many extensions of these models. For two decades Davis and collaborators [10] have recorded a deficit (by a factor of 3.8) of primarily higher energy 8B electron neutrinos from the sun, in their 37Cl experiment. A solar neutrino deficit has also recently been measured by the Kamiokande II experiment [11].

The most popular solution to this problem assumes the correctness of the standard solar model and modifies the Standard Model by giving the neutrinos masses and mixings. One may add massive right-handed partners for the usual left-handed neutrinos as in some grand unified models, some singlet Majoron models, and in gauged SU(2)L x SU(2)R models. The GUT models typically have a natural mechanism [85] for tiny neutrino masses. The left-handed neutrinos can also form Majorana masses as in other Majoron models and in our model. Just as in the model of Zee [69] we predict an extremely light electron neutrino and approximately degenerate pt and r neutrinos. The resulting neutrino oscillation, for certain ranges of parameters, is consistent with solar neutrino data.

Perhaps the most promising explanation of the solar neutrino problem in these scenarios is the .MSW effect [12]. There vacuum neutrino oscillations are enhanced by the index of refraction produced by neutrino-electron scattering in the sun. Recent analysis of Kamiokande II data together with 37CI data by several authors [86,87] indicates that the nonadiabatic Rosen-Gelb solution [88] is favored over the original adiabatic solution [89]. The more recent work of Ref. 87 indicates the following constraints on the neutrino masses and mixing angles for the relevant solutions

A = m- 2 10-eV2, (3.5.1)






71

with less than one order of magnitude uncertainty and sin2 298 " 10-1.5�05. (3.5.2)


Taking the nonadiabatic solution, we fit the parameters of our model to these values. Recalling the analysis of section 3.3 we consider the approximation of a massless electron neutrino. (Recall that my,, , m2ax., sin 208 where a < 1 so that m,, < 2.5 x 10-41m2 for nonadiabatic solutions.) From equations (3.3.5) and (3.3.7) we have


(Am2) = n0(:r + ,) 2 10 -3eV, (3.5.3)


and
sin 29 = 2zCTr ZP -0.75�0.25. (3.5.4) zEr ~�zT

Recall that to obtain such a small neutrino mass without having Ms > 1 TeV, we had to assume A Z 6, A larger only by an order of magnitude or so. It is simpler to consider the expression


(Am 2)7(sin 20,)~ 4)2 - cot 3AV V 2 n( ), (3.5.5) (47r)2 m- 2h as it can be related to the p -+ ce constraint. For the nonadiabatic solution this obeys the inequalities


3 x 10-4 e; < (Am2)(sin 2) < 6 x 10-4 eV. (3.5.6)


It is possible to find a "natural" range of parameters which satisfy this bound assuming that A - 6. where 6 5 10-16 came from our hierarchy considerations. One such solution takes mh = 100 GeV and takes the values of cot 0 and Ze7zlJr to be their upper bounds from (3.4.3) and (3.4.8) respectively. We






72

take VO = 109.5 GeV and work in the approximation a < 1. For this approximation to be valid we expect AM l 0(1 TeV). For these values of parameters we get
1 1 3 myMs Ms (sm2ln2-2 flI ). (3.5.7) (Am 2)(sin 29v)5 4 x 10- A12 _2 In( m (3.5.7) S - h mh

For Ms = 1 TeV, we get (Am2) ~ 4x10-3 /ze + zr and (Am2) (sin20

6 x 10-4 eV, just at the upper bound. The upper bound (3.4.8) on zeryZ is , 10-2 for this value of M. We can satisfy (3.5.4) and the other bounds by choosing say zer " 10-2 and Z,, ~ 10-1 so that the bound on their product is no longer saturated. Then (A 72)- 4 x 10-4 and (Am 2)(sin20,) a 10-4. By fitting parameters more specifically one can clearly obtain ranges of parameters that satisfy the nonadiabatic conditions. However due to the relatively large number of parameters to work with and weak or partial bounds on some of them, it is difficult to obtain ranges of allowed charged scalar masses.














CHAPTER 4
THE RG IMPROVED EFFECTIVE ACTION

We present a detailed discussion on the quantum corrections to the Effective Potential. In section 4.1 we discuss the loop-expansion and how the Renormalization Group can be used extend its validity. We discuss in detail the little known role that the unphysical constant term of the Effective Potential plays in the Renormalization Group Improvement, but this subtlety has been largely ignored in the literature. We review the calculation of the Renormalization Group Improved Potential for a massless and a massive scalar in sections 4.2 and 4.3 respectively. Finally in section 4.4 we attempt to extend those calculations to the case of two scalars.


4.1 Approximating the EP

Up to an unphysical additive constant, the full EP is the sum of all the One Particle Irreducible (1PI) diagrams, with each loop typically contributing a factor of hAln-. The loop expansion thus is an expansion in h and hence provides a natural way of approximating quantum corrections to the Effective Potential.

There are three different methods that can be used to evaluate the loop corrections to the EP. The most straightforward method [13] is to sum all the 1-loop diagrams, 2-loop diagrams, etc. The drawback is that rarely is it possible to add up the infinite series of diagrams that enter at each loop level. Another method is the Shifted Field Method [90] (SFM) (see Appendix D).


73




74

In this method one shifts the scalar fields by a constant O(x) ---+ (x) + �, leading to a theory with 0 dependent parameters. Then 0V/0 is given as the one point function for 4. Since this involves only a finite number of diagrams the method is frequently applicable. The SFM has the added virtue that it uncovers the logarithmic factors that enter the EP, even when it can not be used to evaluate the EP. The third method [91] to find the 1-loop contribution to the EP, involves using the (-function technique to evaluate functional determinants appearing in the saddle point approximation to the generating functional. The (-function method is sometimes the most straight forward method of evaluating the quantum corrections, as will be illustrated in the two scalar model to be discussed later in this chapter.

Whatever method is used to calculate the loop expansion, it only has limited applicability. Being essentially an expansion in hAlno it is only valid for a small value of the logarithm and thus only holds in a limited range of field space. Coleman and Weinberg [13] found a way of using the Renormalization Group to give an approximation without this limitation, the Renormalization Group Improved (RGI) EP. The renormalization process forces us to introduce an arbritrary scale p into the calculation of the quantum corrections to the EP. If the EP is a physical quantity it should not depend on the renormalization scale p and should obey a RGE

d a a = ( 1
PVeff = ( + / - 749 - "m2m m-2)Veff = 0 where

A 0c O2m2
T his -ea - tomnmh2 = P 0st (4.1.2) This equation has the solution






75

with t defined by

y(t) = le , (4.1.3) and

A(t)= +Jpdt',
0


(t) = C exp - Jcdt' , Ti2(t) = m2 exp - Jm2dt' . (4.1.4)
0 0
where V is the functional form of the EP at some particular scale, e.g.


'( p.A , O) = Veff It=o(P, A, 0)

So if we know the functional form of Veff at one scale t (or equivalently pi), we know it for all scales. provided -7, 7m-2, and / are known and can be solved to give 0(t) and A(t). Then one is free to choose whatever t (or y) one pleases, in particular one can choose a � dependent t that kills the higher order logarithmic terms, and thus extend the validity of the approximation to large regions in field space.

There is however a subtle problem associated with this procedure [14,15,92]. This has to do with the choice of boundary condition in solving the RGE for the EP. As a boundary condition for Ve ff one would like to use V1 = V(0) + V(1), but V(1) may contain a term constant in � that is usually ignored since it does not affect physics (except gravity). This constant term does depend on i and thus on t. When a 0 dependent t is chosen to kill the logarithm, this term will induce terms with 0 dependence. Thus one has to treat this term carefully when improving the potential. Being unphysical the constant term does not necessarily obey the RGE. It is however possible to explore the freedom to add





76

a constant to the EP to make it obey the RGE. Let "V be the sum of all the 1PI graphs. Then "V is the EP up to a constant Veff = (, A, m, ) + Q(1, A, m) where Q is a arbitrary constant in 0. Expressing V in terms of the 1PI Greens functions, F, we have

(I, A, M, ) = Q(, A, m) - nr() n=1
The Greens functions of course contain the physics of the potential, while Q(/p, A, m) = V(j1, A, m, 0 = 0) is the unphysical constant. It is important to note that 9 can get contributions from every loop order. The EP is Ve = - r(n) + + n=1
where 0 is calculated in the loop expansion, while Q is arbitrary. The Greens functions are of course independent of p and thus obey the RGE, so d d
LVeff = V( + 1) = Di + DQ, where
0 2
D = (itp + /3 - ,,m2 ) In order to make Veff satisfy the RGE, one has to choose Q such that DQ + Dt = 0. (4.1.5) By a clever choice of Q, one can find an approximation to the EP that can serve as a boundary condition for solving the RGE for the EP. One logical choice would be to eliminate the vacuum energy by choosing [92] Q = - l =v,






77

where v is the vacuum expectation value of the potential.

Another way of solving the problem is to treat the constant term just like another parameter of the model, include it in the Renormalization procedure making the zero-point Greens function, I(0) = -( + Q), obey the RGE. That is, we treat the unphysical part of the EP in the same way as the physical part. In this case the RGE for the EP becomes

d a a a 2 a
p+Veff = (G + - 4 -nm'm 2 2 / 0 )Veff = 0 (4.1.6) where

0i

The renormalization of the zero-point function guarantees that the loop expansion is the appropriate boundary function for the EP. In this case Eq.(4.1.5) is replaced by


(D + Oj )(fQ + Q) = Do + S3i = -(D + iS )0(0) = 0.



4.2 One Massless Scalar

Let us quickly review the procedure in the simplest case, that is in the model of a massless self interacting scalar [13], where the interaction is described by the classical potential V(0) 4
4!

The result of the one loop calculation is V2 4 2 3
- 2562(ln2 ) (4.2.1) 256r2 2 2

The approximation Veff " V(0) + V() is only valid for A and ln2 small.
7z





78

Notice however that the constant term is absent from V(1), so to the 1-loop order is

O=0

We can hence use V(O) + Vi() as a boundary condition for the RGE for the EP.

In the single scalar model we have at one loop

3
/3= A and y=0 and thus
A
A(t) and (t)= =

Putting this result back in to the one-loop approximation to Veff we have Vef = - +At 0 + 2 In2e2t - +0 A3, A21n 2 (4.2.2)
1 - T-7At 4! 25 et 22,2t

We now can choose t, and we have complete freedom in doing so; in particular we can choose t as a function of 0. Our guiding light in choosing t is of course to make Eq. (4.2.2) as good an approximation to the EP as possible. The terms that limit the applicability of the one loop approximation contain the factor Aln(02/p2e2t), and those terms we can eliminating by choosing t = In(o/p). We then get, (notice the redefinition p -p -3/4), 'eff -1
_1 - -A ln(62/p2) 4!

which is valid for large ln(02/p2). This is the leading log or the RGI i-loop approximation to the Effective Potential.


4.3 A Massive Scalar

For a massive scalar [14,15] the tree-level potential is V(O) A +4 +1m72 +2 , 4! 2





79

and the one-loop contribution to the potential is V1 + 2)2 14 + m2 3 64 A2 2 In 2� The one-loop approximation Veff ~ V(0) + V(1) suffers from the same limitations as in the massless case, e.g. it is only valid for small values of Aln 9 W � In principle one might worry about other logarithmic factors showing up at higher loops, since there are now three dimensionfull quantities involved, e.g. p, 0, and m. However the SFM shows us that this is the only possible logarithmic factor. (See Appendix D)

To extend the region of validity of our approximation to large values of the logarithm we repeat the steps taken in the massless case and put in the running parameters and choose p (or t) to kill the logarithmic factor. But in this case V(1) contains a constant term and we were careful to include a constant term in the tree level potential.

The 1-loop running of the parameters is given by 3 1
A 16X2 , ' = 16x , 1Gw r2 1672 10 = 0, $ 1 4 (4.3.1) 16r2 2

Notice that Q = 0 is not invariant under the RG. This is the key point. It is not consistent to ignore the unphysical part of the potential when renormalizing the theory.

Solving Eq. (4.3.1) gives the running parameters

(2t) = (t) = + 1/



( - 3-1/3 2 A 1 72)
(4.3.2)





80

Putting the running parameters back into the one-loop potential we find the RGI approximation to the EP
4 1 - 1 1 ~- 1 2+ 2 3
Veff + iTn + Q + 64 2( 2 +�T2)2(In Y2e2t ) where t is completely arbitrary. Now we can choose

1 -02 + i92
t = -In
2 p2

and kill all the logarithmic factors. Then by expanding t in terms of the parameters, we get an expression that is valid for small A and all values of 4.


4.4 Two Massive Scalars

The simplest model involving two interacting scalars is described by the tree level potential

V(o) = l + 12 + 2 + 2 2+ A 0&2 + H (4.4.1)
4! 11 4! 2 2 2 12

Using the SFM we see that in this theory two logarithmic factors will contribute to the EP. These are In(B2/,p) and In(B.2/p), where

B= M N M + (M - M2)2 + 16A2
/(4.4.2)
B 2 (k12 + N V(M - M)2 + 16A20 ) with
1 + A2,

m2 2 1 2(4.4.3) S= m2 + A22
The actual calculation of the one loop contribution to the EP can be performed using the C-function method [91]. In the saddle point approximation the path integral expression for the generating functional becomes

WE= Nexp(-So)J DODP2exp [-1 d4x(l,2)A( 0)]






81

with
(02 + --2 -a2 + M2 2A 1�2 84'21011 -0 22 A=V -2 � =) 201 2 02 + M


The Gaussian integral over the fields 01 and �2 can be done formally to yield


IVE = N exp(-So)(det A)-The evaluation of the functional determinant, det(A), is complicated by the fact that A is a 2 x 2 matrix differential operator; however this operator diagonalizes to
(02 + B2 0
0 -2 + B2 and thus

det(A) = det(-02 + B ) det(-02 + B2)

and we can use the result of the (-function calculation (for constant C)

1 C 3 det(-2 + C)= exp C2(In )We find
V(I)- 641 [B (n1 2)+B 4(n- 2 )] (4.4.4)

The first step in RGI the EP is to solve analytically the RGEs for the running of the parameters of the model. These are to 1-loop

16r = 0, 12002 = 0,
at Ot
167-2 Al = (3A2 + 12A2), 16 2 (3A + 12A2),
2t 2
16r2 = A(A1 + A2 + SA). (4.4.5)

167r t (m - 2Am), 167r2 (2m12 - 2Am1),
28H 1 (in+ ).
16xr -- = -(m + ).




82

Unfortunately the general solutions have so far eluded us.

In order to solve the problem, at least in some special case, one is tempted to explore the possible symmetries of the model. There are two symmetries of potential interest. If A1 = A2 = 6A and mi = m2, the model exhibit an 0(2) symmetry. In that case the problem essentially reduced to the case of one dimensional field space and thus looses much of its appeal. If A1 = A2 and mi = Mn2, the model is symmetric under the interchange 01 - 02 and the RGEs for the couplings reduce to


16r2 1 = (3A + 12A2) 167r = A(2A1 + SA) . (4.4.6)


This is an interesting case but unfortunately we have not been able to solve this simpler case either.

The third case of interest is case when the two scalars are weakly coupled, e.g. when

A << A1,A2 < 1. (4.4.7)




83

In this case we can solve the RGEs by expanding in A0 = A(t = 0). We find

t (t) , 1() = , 2(t) =2 ,

AI(t) = , + O( (t)




-2 ()= 10 2 0 1+O(),
(1- t 2A
2 m )-ot m2(t) = o- t + A10 (1 - +
1lo 3 16UrH(t) = Ho + ~ [~ (1 -(1 - )3+A (1 - (1 - 20
2 + 2AO 3 A

+2 (t)Aon10 o 1 ;(1 - -t 1 d+1 O(Ai) +

- 1 T6 0 20 (1 -0 ( t)
2 A2 3 +
2m20 1 1 -- t - 1 2)







(4.4.8)
where A10 = 1(0), A20 = A2(0). m0 = 5i (0), m0 = E2(0), and H0 = (0). We can now plug the running parameters back into (4.4.3), (4.4.2), (4.4.1), and


(4.4.4) and choose t = In(B1/PI) to kill the B1 terms, that is the terms including ln(B12//2). We are then left with an approximation to the EP that is good if the A 's and ln(B/B7 ) are small. The expression obtained is not pretty but


that is not to be expected for a Renormalization Group Improvement of the Effective Potential in a model with more than one scalar. In the weak coupling limit we have been partly successful as we have increased the region of validity of our approximation of the Effective Potential.














CHAPTER 5
CONCLUSIONS

The aim of this study has been to explore physics beyond the Standard Model. First in chapter 2 by studying patterns suggested by Grand Unified Theories, within the Standard Model and in its Minimal Supersymmetric Extension. We worked in the MIS renormalization scheme using the Standard Model two-loop renormalization group /3 functions. We evolved the parameters of the Standard Model, i.e., the gauge couplings, the quark and lepton masses, the Yukawa sector mixing angles and phase, and the scalar quartic coupling, from a mass scale of 1 GeV to Planck mass. We reviewed the extraction from experiment of the initial values for these parameters with specific emphasis on the extraction of the strong coupling constant and of the quark masses (especially those of the charm and bottom flavor). We treated the threshold effects appropriately. i.e., rather than using naive step function implementation of thresholds, we implemented one-loop matching conditions for both gauge boson and fermion mass thresholds. In the Standard Model case, there are many unsatisfactory features, not the least of which is the failure of the gauge couplings to unify within experimental error, forming a GUT triangle. The simplest of the SU(5) relations, 7nb = mr, can only be satisfied some ten orders of magnitude from the scale of the GUT triangle. The other relations md = 3me, 3ms = mp, tan Oc = vrnd/ms, and Vcb = Xmc/mt, could be satisfied at 1016 GeV. The geometric mean relations we considered can also be simultaneously satisfied, but this requires a top quark mass greater than 200 84





85

GeV. In the SUSY case the GUT triangle collapses and we achieved a striking agreement for the four GUT inspired relations considered. But, for this to be true, several things must occur: first Vb must be larger than its presently central value; second the top quark mass must be around 190 GeV (if it is a bit lighter, then agreement dictates that Veb should be larger still); third the Higgs mass should hover around 120 GeV. These conclusions are qualitatively correct if one demands maximum agreement. However, it is difficult to arrive at more definite numbers without an exhaustive analysis of the parameter space.

In chapter 2 we introduced a new extension of the Standard Model which has the minimal field content to include an invisible axion and massive neutrinos. The cost of such an extension is the appearance of a large number of additional parameters. The potential was minimized so that the hierarchy of VEVs, v = 250 GeV and V,, ' 109.5 GeV, is maintained. We have seen that most of the couplings in the potential fall into two classes with typical values A - O(10-2) and 6 , 0(10-16). These naturally lead to the following masses. There are five neutral particles: two scalars with Standard Model

(SM) type masses, one very massive scalar, one pseudoscalar with SM type mass and a massless axion (The axion will become massive due to instanton effects). The two charged scalars typically have masses of 0(102 - 103 GeV) and we generally assume M > mh/,. A notable exception to these two classes of couplings is A 5 10-9 which appears in the only explicit lepton number breaking term in the potential and enters in the charged scalar mixing angle. The ratio 1 = tan i is not fixed by the minimization but is constrained by the B0 - Bo systems from above and by semileptonic Bo decays from below so that 1 G tan a 5 O("'I ). The three new Yukawa couplings Z[ij] which induce violations of lepton family numbers are bounded from above, these






86

bounds depending critically on the value of Ms. By unitarity considerations, Ms should be less than 1 TeV although it may obtain a larger mass from a more fundamental theory for which our model is a low energy version. For Ms - 1 TeV we found zerZpr 5 10-2 from P -- e-y data, while zep 5 10-1 from violations of Cabibbo universality and modifications of the SM V - A structure. Arbitrarily fixing mih = 100 GeV, Ms = 1 TeV we found that the nonadiabatic Rosen-Gelb solution to the solar neutrino problem could be obtained for zer ~ 0(10-2), z, - 0(10-1) and also A O0(10-15 - 10-16) , 6 for small charged scalar mixing. There is no reason why A cannot be as small as the other hierarchy-suppressed couplings. Unfortunately, it will be some time before our model is more tightly constrained by experimental bounds on two Higgs doublet phenomenology as well as on a whole host of lepton number and lepton family number violating processes. We have yet to find a single Higgs boson. The verification of MSW oscillations in the sun would highly favor tiny, radiatively produced, neutrino masses. We add that grand unified models with a hierarchy of neutrino masses produced by a seesaw mechanism are also highly favored in explaining such small neutrino masses.

Finally in chapter 4 we discussed in the Renormalization Group Improvement of the loop-expansion of the Effective Potential, with special emphasis on largely ignored subtleties about the role of the vacuum term of the EP. We also made a partially successful attempt at improving the Effective Potential in a model containing two scalars. This effort was marred by technical and fundamental problems, that need to be (and will be) overcome.
















APPENDIX A
THE SM P FUNCTIONS In this appendix we compile the renormalization group P functions of the Standard Model. These have appeared in one form or another in various sources. We have confirmed their validity through a comparative analysis of the literature. Our main source is Ref. 6. Following their conventions,

1 t2
�= L~YU uR +-QL4iYd td + L4Yet en + h.c. - ,A( (A.1) where flavor indices have been suppressed, and where QL and e are the quark and lepton SU(2) doublets, respectively: QL =dL ( LL (A.2)

4 and i are the Higgs scalar doublet and its SU(2) conjugate: 4) = O , ' = i2. (A.3)


UR, dR, and eR are the quark and lepton SU(2) singlets, and Yu,d,e are the matrices of the up-type, down-type, and lepton-type Yukawa couplings.

The 0 functions for the gauge couplings are

dgl= - 1 b r (A.4) dt 1687




87






88

where t = In and I = 1, 2, 3, corresponding to the gauge group SU(3)C x SU(2)L x U(1)y of the Standard Model. The various coefficients are defined to be
4 1
bl = -ng - l 1
32 4 1
b2 = - ng - , (A.5)
4
b3 = 11- 3ng,

with ng = nffl.

In the Yukawa sector the # functions are

dYude = Yu,d,( 1 ud,e) , (A.6) dt TG 2 udd where the one-loop contributions are given by

2 (yuYu, - YdYd) -2(S)- ( 2 (+ g + 8g3)
= (YdYd - YuYu) + Y2(S) - ( g + 2 + 893), (A.7)
922 4 4
3() 3 tye Y+ Y2(S)- (g + g2), with

Y2(S) = Tr{3YutYu + 3YdtYd + YYee } . (A.8)

In the Higgs sector we present 13 functions for the quartic coupling and the vacuum expectation of the scalar field. Here we correct a discrepancy in the one-loop contribution to the quartic coupling of Ref. 6 dA 1 31) (A.9) dt - 162 A

where the one-loop contribution is given by

12A2 - ( g + 9g2)A + ( 3g + g2 9+ A )
A 5 2 4 25 15 12 2(A.10)
+ 4Y2(S)A - 4H(S),
with

H(S) = Tr{3(YutY,)2 + 3(YdtYd)2 + (YetYe)21 . (A.11)




89

The # function for the vacuum expectation value of the scalar field is

dlnvdt 1 (912 2


This expression was arrived at using the general formulas provided in Ref. 6 for the anomalous dimension of the scalar field, choosing the Landau gauge.

In the low energy regime the effective theory is SU(3)C x U(1)EM. We employ the general formula of Ref. 93 to arrive at the f functions for the respective gauge couplings:

d93 _ 3_ 38 9 3'
d= [(71u + Id) - 11] ) + [ (7u + 71d) - 102] )4
dt 3 (47-, 3 (47)

+ [ 2nu-l + d (A.13)
9 9 (4x)4
5033 325 2857 g7
+--[ (1 (u -- 7d) - (nau + 7id)2 _ (4)
18 54 (47r)6 and

de 16 4 4 C 64 4 e5
S= [-nu + -9fd + 311 4 + [ -7 u + -2-7d + 4n1] 4
32 (A.14) + [-4lu + -nd
9 9 (4r)4

where n,, nd, and nj are the number of up-type quarks, down-type quarks, and leptons, respectively. In Eq. (A.21) we have also included the three-loop pure QCD contribution to the ,3 function of 93 [94].

For the evolution of the fermion masses we used Ref. 95. It is known that there is an error in their printed formula [96]. Using the corrected expression, we compute the following mass anomalous dimension. The fermion masses in the low energy theory then evolve as follows: dm
di = '9)n>' l . (A.15)






90

where the 1 and q refer to a particular lepton or quark, and where
2 2 1 e 3 93 Y(1,q) = Y(1,q)(4T7)2 +(1,q) (47)2
[,,11l 4 33 4 9 13 2 2 1 + (1,q)e + (q))3 + ( 341eg (A.16)
6
333 93
( ) (47)6r

The superscript 1 and 3 refer to the U(1)ENt and SU(3)c contributions, respectively. Explicitly, the above coefficients are given by



3 0
3 2
7(1,q) = _621,q)


Y(q) = -8

13 33 = 0
Y(l) (1)
1Y -3Q4 0 220 0 2 (A.17)
7(1,q) = 3Q1,q) + ~" + 9 'd + 3 n ,q)
13 -4Q2
^7(q) = -4Qq)
33 404 40 (q) 3 9
333 2 140 (Nu + 7d)2 + (160((3)+ 2216 )(nu + nd)- 3747 ,
Y(q) - 2 + 9

where Q(I,q) is the electric charge of a given lepton or quark, and ((3) = 1.2020... is the Riemann zeta function evaluated at three. In the mass anomalous dimension for the quarks above, we have also included the three-loop pure QCD contribution -y333 [94].

















APPENDIX B THE MSSM P FUNCTIONS Using some of the notation of Falck [7], the superpotential and soft symmetry breaking potential are as follows:

w = +Yu, uQ + dYd(1d + TY dL + h.c. + p4ud d

2 t 2 t 2 t + 2 itL
Vsoft = mU 4,, + n d (o)i dd + mQ + n L

7+ ti itJ +m <+ m + Bp( 4d + h.c.)

+ 7(Ai 4,, j + .4i +idJdid j + AJYeiJ3 djLj+ + h.c.),

3
Vgaugino = A11 + .c..
1=1
(B.1)

Various a2's have been omitted and a sum over the number of generations is implied. Also, hats imply superfields, tildes the superpartners of the given fields, and overbars the charge conjugate.


B.1 Gauge Couplings

First the gauge couplings are

dgl 1 3

3
dt 162

- (16r2 2[ bkg -Tr{CYu Yu +CldYdtyd + CleYetYe]
k
(B.1.1)

91





92

where t = Inp and I = 1, 2, 3, corresponding to gauge group SU(3)C x SU(2)L x U(1)y of the Standard Model. The various coefficients are defined to be
3
bl = - 2ng ,
5
b2 = 5 - 2,ng , (B.1.2) b3 = 9 - 2ng ,


S38 6 88 9 9
(blk) = 14 8 , + -17 0 (B.1.3)
113 6 0 0 -54 and
/26 14 18

(Clf) = 6 2 , withf = u , d , e , (B.1.4)


with ng = n f.


B.2 Superpotential Parameters: p and the Yukawas

In the following we list the 3 functions for the parameters of the superpotential.

dln 1[ Tr{3YtYu + 3YdtYd + Yetye} - 3( g2 + g) ] (B.2.1)
dt 16r2 5

In the Yukawa sector the 3 functions are dYud,d u,d,e 1 1.) , (B.2.2) dt Yu6d e(1 ude)

where the one-loop contributions are given by

-) 3Y tYu + YdtYd + 3Tr{YutYu} - (1 + 3g2 +-) , flu u (-91 16 2(1) = 3YdtYd + YutY + Tr3YdtY d + YeYe) - ( + 392 + 163 1) = 3YetYe + Tr{3YdtYd + ye Ye) (12 + 392) (B.2.3)





93

B.3 Higgses' Vacuum Expectation Values

The evolution of the vacuum expectation values of the Higgs's is given by dlnVt,,Od 1 (1) dt - 1672" ' 'd

where the one-loop contribution is given by

Y (91 + g.) - 3Tr{YutY} (B.3. 1)

(1) 3 1 2 + g2) - 3Tr{YdtYd} - Tr{YetYe}
- 4 5(9
















APPENDIX C
EXPLICIT FORM OF 6(p)

In Ref. 48 the radiative corrections term b(p), from Eq. (2.2.66), is derived. In this appendix, we present its explicit form as it appears in this reference except for some minor notational changes. In the following, s and c refer to sinOyw and cos01v, respectively. Also, ( is defined to be the ratio M2I/M2.


) V2 { ) + fo( )( )} , (C.1)

where the various functions are defined as follows:

p2 3 1 1 c2 9 25 7r
fi(j(, p) =61n - n--Z(- - Z(-)- Inc2+ /
M2 2 9 39
2 M2 p2
fo((,p) = -61n7 [1 + 2c - 2 - + -2 + 2Z( ) M M - c 2 c2 3c21nc 2 2 15 + 4c2( )+ s2 + 12c21nc (1+2c2

2c2
2 _M 1 2 2

f-1(~') = 61n [1 + 2c - 4 ] - 6Z( )-2cZ( )- 12cnc M4 M2 M2t z z

+ s8(1 + 2c4) + 24 (In -t -2 + Z( t .1,14 M2 - M2& with
Z(z) = 2Atan-(1/A) , (z > )
Aln[(1 + A)/(1 -A)] , ( <) , (C.3) A - 4


94




Full Text

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RENORMALIZATION GROUP ANALYSIS OF THE STANDARD MODEL, THE MINIMAL SUPERSYMMETRIC EXTENSION OF THE STANDARD MODEL, AND THE EFFECTIVE ACTION By HAUKUR ARASON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1993

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ACKNOWLEDGEMENTS I would like to offer my sincere thanks to my advisor, Pierre Ramond, for his generous support and kind encouragement during my tenure as his student. I have benefited immensely from his teaching because of his extensive knowledge and friendly character. I would also hke to acknowledge those who were my collaborators at one point or another. In particular I am indebted to Diego Castano, Eric Piard, and Brian Wright for many educating and entertaining discussions. I thank Profs. C. Thorn, P. Sikivie, R. Field, and C. Stark for serving on my committee. Last but not least I want to thank my wife for her continued support throughout this work. ii

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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS " ABSTRACT ^ CHAPTERS 1 INTRODUCTION ^ 2 RG STUDY OF THE SM AND THE MSSM 5 2.1 Signatures of Structures Beyond the SM 5 2.2 Initial Value Extraction from Data 8 2.3 Quantitative Analysis of the SM 28 2.4 Quantitative Analysis of the MSSM 38 3 A MULTISCALAR SM EXTENSION 52 3.1 The Model 3.2 Unitary Gauge 60 3.3 Majorana Neutrino Masses ^'^ 3.4 Experimental constraints ^"^ 3.5 Solar Neutrino Phenomenology 69 4 THE RG IMPROVED EFFECTIVE ACTION 73 4.1 Approximating the EP ^3 4.2 One Massless Scalar "^"^ 4.3 A Massive Scalar 4.4 Two Massive Scalars SO 5 CONCLUSIONS 84 APPENDICES A THE SM 13 FUNCTIONS 87 B THE MSSM l3 FUNCTIONS 91 iii

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C EXPLICIT FORM OF <5(/i) 94 D THE SHIFTED FIELD METHOD 95 REFERENCES 96 BIOGRAPHICAL SKETCH 102 iv

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy RENORMALIZATION GROUP ANALYSIS OF THE STANDARD MODEL, THE MINIMAL SUPERSYMMETRIC EXTENSION OF THE STANDARD MODEL, AND THE EFFECTIVE ACTION By Haukur Arason December 1993 Chairman: Pierre Ramond Major Department: Physics We present some examples of the uses of the Renormalization Group (RG) in particle physics. The Standard Model (SM) is analyzed in great detail, and in particular Grand Unification is ruled out in the Standard Model. Similar analysis is performed in the Minimal Supersymmetric extension of the Standard Model (MSSM). In that case we show that Grand Unification is possible. In the context of Grand Unified Theories (GUT), there arise certain relations among the masses and mixing angles. These relations are explored both in the Standard Model and in the MSSM. In particular we show that they can all be satisfied in the MSSM at a plausible GUT scale if the masses of the Higgs boson and the top quark meet certain constraints. The scalar sector is experimentally the least constrained aspect of the Standard Model. We explore this freedom by introducing a multiscalar extension of the Standard Model, that incorporates the Peccei-Quinn solution to the strong CP problem and at the same time induces neutrino masses that solve the solar neutrino problem.

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In the Standard Model the Effective Potential (EP) plays a crucial role as it can provide the mechanism for electroweak symmetry breaking. The RenormaHzation Group is an indispensable tool for analyzing the quantum corrections to the EP. We discuss in detail how the Renormalization Group is used to find the leading logarithmic correction to the EP. Motivated partly by the experimental uncertainty of the scalar sector of the Standard Model and partly by the Supersymmetric extensions of the Standard Model, we evaluate the quantum corrections to the Effective Potential in a model containing two scalars and attempt to solve the very difficult problem of finding the leading log contribution to the Effective Potential. vi

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CHAPTER 1 INTRODUCTION The Standard Model has survived all experimental tests so far and incorporates all known particle physics down to scales of fractions of milliferniis. The Standard Model is also completely consistent down to the Planck scale, where quantum gravity would take over. In spite of this success, the Standard Model has some unsatisfactory featmes such as a large number of parameters, three chiral families, and three distinct gauge structures. Consequently, physicists have searched for a simpler underlying structure that would break down to the Standard Model in the low energy limit. The observed pattern of the quantum numbers of the elementary i)articles has given rise to the idea of a Grand Unified Theory (GUT) [1,2,3,4]. which simplifies the gauge structure of the Standard Model. Experiments can not access the high energies where the GUT predictions deviate from the prediction of the Standard Model; it is however possible to use the Renormalization Group to extrapolate the Standard Model parameters to smaller scales [5]. The purpose is to find if those parameters satisfy GUT relations at shorter distances. This is what we will do both for the Standard Model and for the Minimal Supersymmetric extension of the Standard Model (MSSM ). This has been done before, but we will improve upon earlier projects by using new and improved data, by treating the full Yukawa sector, by using improved treatment of thresholds, and by using 2-loop/3-loop ^-functions in the Standard Model. Our plan is as follows. We use experimental data to fix the parameters of the Standard Model at lower energies. We

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then numerically integrated the MS Renormahzation Group Equations [6,7] for the Standard Model and the MSSM to evolve the parameters of the models to Planck scale. Although it is not possible to analytically express CKM angles and the quark masses in terms of the Yukawa couplings, our approach is numerical so we run the Yukawa matrices and evaluate the relevant quantities by diagonahzing the matrices at every step of the Runge-Kutta method. In the Standard Model, we use 2-loop Renormahzation Group Equations in evolving the couplings, except in pure QCD where 3-loop contributions are also significant and therefore included in the running of the strong coupling and of the quark masses in the low energy region. In the supersymmetric extension, we work to 1-loop. In each case, we include a proper treatment of thresholds [8]. Our incomplete knowledge of the Standard Model parameters forces us to repeat the analysis for a range of allowed values of the top quark and Higgs masses. Using these techniques, we examine interesting GUT inspired relations among the gauge couplings, and the masses and mixing angles of quarks and leptons in the Standard Model. We extend the analysis to the minimal supersymmetric extension to determine its effect on these relations. In the Standard Model we do not see any clear evidence of Gauge Unification. The gauge couplings only semi-converge and unification between the bottomand r-yukawa predicted by the simplest GUTs takes place at a scale many orders of magnitude away from the possible Grand Unification scale. In the MSSM we find a remarkable agreement with GUTs. The gauge couplings unify at a scale which does not violate proton decay and we can satisfy all the GUT inspired mass and mixing angle relations at the GUT scale. For this to happen we have to place bounds on the masses of the top

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3 quaxk and the Higgs boson. The details of this analysis are found in chapter 2. Although neutrinos are necessarily massless in the Standard Model [9] and direct experimental searches for the consequences of massive neutrinos, e.g. oscillations, have so far failed to turn up anything, the deficit in the expected number of neutrinos from the sun [10,11] indicates that neutrinos might be massive [12]. Global symmetries, such as baryon number and lepton number, are not viewed in the same way as local symmetries; they are expected to be approximate symmetries, their apparent conservation being explained by the appearance of tiny ratios, as happens in Grand Unified Theories, axion theories, or theories that involve gravity. Since the mass of the W-boson is seventeen orders of magnitude smaller than the Planck mass, there is ample room for such ratios. In order to incorporate neutrino masses in the Standard Model, the lepton number has to be violated, either spontaneously or explicitly. In chapter 3 we introduce a model with explicit lepton number breaking where neutrino masses are generated radiatively at one loop. The model is a minimal extension of the Standard Model with invisible axion and neutrino masses, and for a reasonable range of parameters the model provides a solution to the strong CP and solar neutrino problems. The Effective Potential plays a crucial role in analyzing models in particle physics as it determines the vacuum structure of the models. The EP may also be important for the development of the early universe, since it may lead to inflation. In the Standard Model electroweak braking take place at the classical level; however the study of the Effective Potential, as shown in a classic paper [13] by S. Coleman and E. Weinberg, may provide new possibiUties for symmetry breaking at the quantum level. Thus quantum corrections might

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be quite important in analyzing the EP. Since it is not possible to calculate the full EP, it is essential to have reliable methods for approximating the important contributions to the EP. Being an expansion in the loop-expansion provides an approximation to the Effective Potential. Unfortunately this approximation is only valid in a limited range of field space. It has been known for some time that in the case of one massless scalar [13] the Renormalization Group can be used to extend the vahdity of the loop-expansion and give the Renormahzation Group Improved (RGI) Effective Potential. Intuitively it is easy to understand how the Renormalization Group (RG) can tell us how the Effective Potential, Vf//. behaves as a function of the field, (f>. The dimensionless quantity ^cffl^^ ^^^^ ^"^^ depend on the Renormalization constant /i and (j) through the combination ((?)///), and since the Renormalization Group tells us how that quantity depends on it also tells us how it depends on . This argument also indicates that it might be harder to find the Renormalization Group Improved approximation to the Effective Potential in models with a massive scalar, where there are three dimensionful quantities ^, (t>, and m, and in models containing more than one scalar. Recently this has been achieved in a model containing a massive scalar [14,15] and in chapter 4 we attempt to do the same in a model containing two scalars. The multiscalar case is complicated not only because it contains many dimensionful quantities, but also because field space is multidimensional.

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CHAPTER 2 RG STUDY OF THE SM AND THE MSSM We present a comprehensive analysis of the running of all the couplings of the Standard Model to two loops, including thresholds effects and taking full account of the Yukawa sector. Our purpose is to determine what the running of these parameters up to Planck mass may indicate for the physics of the Standard Model and its extensions. We examine several GUT inspired relations among the parameters of the Standard Model. We extend the analysis to the Minimal Supersymmetric extension to determine its effect on these mass relations. Remarkably Supersymmetry allows for all the GUT relations to be satisfied at a single grand unified scale. For this to happen we have to place bounds on the top quark mass, which remarkably satisfy the p-parameter bound; furthermore, using the minimal SUSY boundary condition on the scalar quartic coupling, we also obtain bounds on the Higgs mass. 2.1 Signatures of Structures Beyond the SM The gauge structure of the Standard Model contains three distinct gauge groups SU(3)^ X SU(2)^^ x however the quantum numbers of the particles of the Standard Model allow for the possibihty that this is a low energy manifestation of a Grand Unified Theory. A GUT includes only a single gauge structure such as SU(5), SO(IO). or Eq [1,2,3,4], and thus requires the Standard Model gauge couplings to unify g\= 92= 93 (2-1-1)

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at some large grand unification scale. This is the single most important signature of a GUT structure beyond the Standard Model. In the context of the SU(5) GUT [2], several mass relations arise, based on simple assumptions for the possible Higgs structure. The mass term for the down quarks and leptons comes from the Yukawa interaction of the 5 and 10 of fermions. This leads to relations between the charge -1/3 quarks' and the charged leptons' Yukawa couplings. With only a 5 of Higgs, one obtains equality between the r lepton and bottom quark masses at the GUT scale: mi, = iut (2.1.2) To the level of approximation used at the time, this relation was found to be consistent at experimental scales, after taking into account the running of the quark masses [16]. Similar relations apply to the lighter two families, but are clearly incompatible with experiment. To alleviate this, a new scheme was proposed [17] with a slightly more complicated Higgs structure (using a 45 representation in conjunction with the 5). It replaces the above with the more complicated relations for the two Hghter famihes md = 3me , (2.1.3) 3771s = . The situation concerning the mixing angles is equally intriguing. There happens to be a near numerical equality between the square of the tangent of the Cabibbo angle and the ratio of the down to the strange quark masses (determined from current algebra). This Gatto-Sartori-Tonin-Oakes (GSTO) relation [18] reads tan^c^A/^(2.1.4) V "^s

PAGE 13

7 It has provided the central inspiration in the search for Yukawa matrices. Very general classes of matrices with judiciously chosen textures [19] (i.e., zeroes in the right places) could reproduce this relation, at least approximately. In the context of 50(10) [3], these three different relations could all be obtained in one model [20], with the required texture enforced naturally by discrete symmetries at the GUT scale. In this model the mixing of the third family with the two hghter ones is dictated exclusively by the the charge 2/3 quarks' Yukawa matrix. There ensues an GSTO-like relation for the mixing of the second and third families [20] (2.1.5) which provides a relation between the top quark mass and the lifetime of the B meson. These four relations can all be obtained if one takes the Yukawa mixing matrices to be of the form [20] (shown here in a specific basis) / 0 P 0 \ Yu={P 0 Q , VO Q Vj R R s I) -35 f) Vo 0 0 (2.1.6) Although some of these relation are derived with specific and sometimes comphcated Higgs structures in mind (as in the 50(10) model), they may well prove sturdier than the theories which generated them. In the following, we will examine the relations in the context of the Standard Model at varying scales all the way to Planck scale. We will then extend the analysis to the minimal supersymmetric extension of the Standard Model, and compare the effect of this extension on their compatibility at some unified scale.

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8 2.2 Initial Value Extraction from Data a](M^) and aoiM?) The determination of the SU(2)l x U(1)y coupHngs proceeds from the Standard Model relations: 47r cos^%(//) (2 2 1) where a(/z) = e^(^)/47r and is a normalization constant which equals 1 for the Standard Model and equals | when the Standard Model is incorporated in grand unified theories of the SU(N) and SO(N) type [5]. What is required to specify these couphngs are the values of a{fi) and sin^^vvC/^) in the renormahzation scheme we employ (i.e., MS). The electromagnetic fine structure constant (a~^ ^ 137. 03G) is extrapolated from zero momentum scale to a scale fi equal to Mz in our case, hi pure QED with one species of fermion with mass m, the MS renormalized vacuum polarization function is given by 2 ^ 2 U(q'^) = 4^(ln^ G / dx x(l x)ln[l x(l x)\]) . (2.2.2) ' m^ J 0 The renormahzed coupling a{fi) is related to the fine structure constant Oem as follows = iTn(0) • ^^-^-^^ In the Standard Model where there are many species of charged fermions and charged gauge bosons, Eq. (2.2.3) generahzes to [21]

PAGE 15

9 The effects of the strong interaction which enter as a hadronic contribution to the vacuum polarization function must be included also. The nonperturbative nature of the strong interaction at low momentum is handled by rewriting the hadronic contribution to the vacuum polarization at zero momentum as n^o) = (n^o) n^Q^)) + n^g^) . (2.2.5) If hadrons) , (2.2.6) where .s is the square of the center of mass energy. For the process e+e" the cross section is calculated to be (taking m^x = 0) a(e + e--^.V) = ^^, (2-2-7) OS In terms of the ratio of these two cross sections, a(e+e hadrons) 9 R{s) = , , _ , , (2.2.8) we can write Eq. (2.2.6) Im{n^.)} = ^i?(^) . (2.2.9) Using an unsubtracted dispersion relation for U^iq^), the combination (n''(0)'n.^{q^)) can be expressed as -.2 n^o) n'v/) = f ds . (2.2.10) Ami

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10 This can be evaluated using experimentally known data. This procedure yields a value a-l(M^) = 127.9 ±0.3 . (2.2.11) The process independent, renormalized weak mixing angle sin^ dw of the on-shell scheme is defined to be sin2^,,.l-^, (2.2.12) where Mw and Mz are the physical masses of the W and Z gauge bosons. Knowing the precise values of the TV and Z boson masses and using the equation above provides one way of extracting the value of sin^ Alternatively, the bare relation involving the low energy Fermi constant measured in muon decay and the W boson mass rio ^ ^ (2.2.13) Ssin^^lVoA/iVo ' may be corrected to order a and rewritten [22,23] M„ = M,.o.e,y = (^^ ' (2.2.14) V2G^ sm^lv(l-Ar)2 with (7raem/v/2G^)5 = 37.2S1 GeV and Ar is a parameter containing order a radiative corrections which depends on the mass of the top and Higgs. We can view the radiative corrections represented by Ar as accounting for the mismatch in the scales associated with the parameters of the relation. and Oem are low energy parameters whereas il% and sin^ ^vv are associated with the electroweak scale. We can absorb the radiative effects using the renormalization group by replacing and Qem with corresponding running parameters at Mz'« 1 . (2.2.15) ^Gp(M2)M^ sin2^vr

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Combining Eqs. (2.2.14) and (2.2.15) gives A,«l__^54^. (2.2.16) a{Mz) Using Eq. (2.2.4) and the fact that G^{Mz) « [23,24] gives an estimate of the size of the radiative corrections Ar w 0.07 . (2.2.17) For large values of M< and Mh {Mt, Mh > Mz) [22,25] . ^ aem 3acw Mf ^ Uagm i^^^H (2 2 18) a{Mz) IGnsmU^yMl 4S7r sin^ % M| ' A third way of extracting sin"'^ 6\y is from neutral current experiments, among which deep inelastic neutrino scattering appears to provide the best determination. A running sin^ e^yi^i ) may be defined in MS and differs from the above sin^ 6w by order a corrections. The MS running W boson mass mw{fi) and the corresponding physical mass M\y, identified as the simple pole at = My^ of the W propagator, are related as follows Ml= m^iy{^l) + Aly^y{Mlr,li) , (2.2.19) where Ajy^y is the transverse part of the W self-energy. A similar relation holds for the Z boson. In MS renormalization, the following relation defines the running sin^ &w{^^) sin2^„.(,) = i-4rTEquation (2.2.19) and its Z analog may be combined with Eq. (2.2.20) to give sin2%.(/0 _ ^ cos^% 4^(A/|,^) >l^ty(M^,/x) ^ ^ ^2 2.21) sin^Ow sin^^iy M| An expHcit expression relating sin^ and sin^ d[y{Mi\:) is given in Ref. 26.

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Another relation for sin^ ^h'(A') J^ay be arrived at directly linking it to Mz [27] or Mw [28]. In particular, if one chooses M\y as the input mass, then one introduces a radiative correction parameter Arw such that sin^ e^:{Mz){l Afw) = sin^ %(1 Ar) , (2.2.22) from which it follows that sin^%^(M^)= .Jm'^a' ^ • (2.2.23) Similarly one can introduce a radiative correction Ar^ if one chooses Mz as the input mass sin2 ew{Mz) cos"^ Ow{Mz )(1 Af^) = sin^ 9^ cos^ dwO^ Ar) . (2.2.24) A fit to all neutral current data gives sm^0\Y{Mz) = 0.2324 ±0.0011 , (2.2.25) for arbitrary M< [29]. Using these values of q(M^) and sm^ew{Mz) yields ai(i\/2) = 0.01698 ±0.00009 , (2.2.26) a2{Mz) = 0.03364 ± 0.0002 . The value of the strong coupling is known with less precision than most of the parameters of the Standard Model and it is by far the most uncertain of the three gauge couplings. This is due to large theoretical uncertainties arising from the nonperturbative nature of low energy QCD and the slow convergence of perturbation series in high energy QCD. Moreover, this uncertainty is hard to quantify.

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13 In the extraction of as = gj/'^T^ from a physical process, many obstacles arise. Since the convergence of the QCD perturbation theory series is not very fast, one must check higher order effects. Even if one chooses processes which do not involve hadronization, a most delicate problem in the extraction of Os comes from working to finite order in perturbation theory. Physical quantities should of course be renormalization scheme independent, but the necessity of approximation introduces dependence on the renormalization scheme. Typically, the same physical quantity calculated in two different schemes to the n"' order of as will differ by terms of order q ^ +^ . As as is large, this difference may be large and thus may lead to renormalization scheme dependence problems. This problem manifests itself in the difficulty of choosing the renormalization scale fi to use for the particular experiment from which one is extracting the strong coupling. Ideally one would like to choose ^ to minimize the unknown higher order terms, but that is of course not possible. Sometimes ii is approximated by the scale at which the highest order known term vanishes or the scale at which that term gives a stationary prediction. However, the most frequent choice is ^ = where E is some characteristic energy scale of the experiment. This choice is plausible since it minimizes the typical terms that arise which involve ln(Q//i), with Q some momentum in the process, typically ~ E. All the processes from which the strong coupling is extracted suffer from this problem and thus each individual extraction of as has large uncertainties. To obtain the best estimate of the strong coupling, we shall take together the results from different processes. These include e+e~ scattering into hadrons, heavy quarkonium decay, scaling violations in deep inelastic lepton-hadron scattering, and jet production in e'^e~ scattering.

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14 We first consider the extraction of as from e+e~ scattering into hadrons. The cross section is, ignoring finite quark mass effects [30], a(e+e^ hadrons) = ^^3(1 + .)(^ e^) where the effects from Z exchange have been put into the factor z. For Ufi = 5 the numerical values of the coefficients are [31] A2 = 1.409 and ^3 = -12.805. This determination of Og has the advantage that it is inclusive, since there is no dependence on hadronization models. Its main drawback is that the effect is not very sensitive to as the effect starts at zeroth order in a^. The experimental error is relatively large and in fact dominates the theoretical error. The value of has been extracted from the total cross section of e+e~ into hadrons by Gorishny et al. [31] who find a5(34GeV) ~ 0.170 ± 0.025. As an estimate of the error coming from cutting off the perturbation series we use the size of the highest order correction and estimate the relative cutoff error to be ~ 13(Qs/7r)2. Thus we find Qs(34GeV) ~ 0.170 ± 0.025 ± 0.006(cutofr) ~ 0.170 ± 0.026, which using three-loop as and two-loop a running is equivalent to as{Mz) ^ 0.140 ± O.OIS (recent LEP data [32] give essentially the same result). The decay of heavy quarkonium is another process from which Qg can be extracted. The decay rates are sensitive to the strong coupling, the dominant modes going as q| or q^, depending on the state of the qq system. The decay rates can be calculated in the nonrelativistic approximation. The rates themselves depend on the wave function amplitude at the origin, which is unknown

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15 but cancels out of branching ratios. The most useful of these branching ratios are [33] r(T iGG) ^ 4 aem A _ 2 q^^AM^T) Ur GGG)~ bas{Mb)\ ' n J r(T -> GGG) _ 10 r(J/t/' GGG) ^ 5 (2.2.28) The main uncertainty of this extraction of Og is theoretical. The known higher order corrections are large so one expects the unknown corrections also to be large. In addition there are relativistic errors. Kwong et al. [33] have made a detailed analysis of as extraction from quarkonium decays. They find as{Mb) = 0.179 ± 0.009 from r(T -> ^'GG)/T{r GGG). They have also estimated the relativistic corrections and find Q;s(Mfc) = 0.189 ± 0.008 and as{Mc) = 0.29 ± 0.02 by looking at r(T GGG)/r(T ^ /x+zi") and T{J/xj^ -> GGG}/T{J/il' -» l^i'^l-')The errors given do not include the cutoff and relativistic errors. In their analysis Kwong et al. parametrized the relativistic corrections by a factor (1 + Cv^jc^) in the branching ratios. They found Ci ^ C ^ C2, with Ci ~ -3.5 and C2 2. -2.9. Here [v^lc^)ji^, = 0.24 and [v^lc^)^ = 0.073. We estimate the relativistic error to be of the order as{Mh) {v^/c^)^ {C2-C{). Similarly, we estimate the cutoff error by the highest order corrections in Eqs. (2.2.28) . Using those estimates we find that as is most accurately determined from r(T ^ GGG)/r(T ^ //+//"). We estimate the cutoff error to be ~ as(i\/fc )(0.43 asiMb)/n) ~ 0.005 and the relativistic error to be as{Mi,) {v^ /c'^)y {C2 Ci) ~ 0.008 and conclude that asiMh) = 0.189 ± 0.008 ± 0.005(cutoff) ± 0.008(relativistic) = 0.189 ± 0.012. This value is equivalent to as{Mz) = 0.111 ± 0.005. i

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16 Analysis of the structure functions in deep inelastic scattering gives a similar value for Qg. The strong coupling affects the way the structure functions vary with energy. These effects show up as logarithmic corrections to the exact Bjorken scaling predicted by the simple parton model. Like the other methods mentioned so far, the measurements of the structure functions do not depend on fragmentation and hadronization. The scaling violations in the structure functions have been measured with beams of electrons, muons, neutrinos, and antineutrinos on targets of hydrogen, deuterium, carbon, and iron among others. Martin et al. [34] have analyzed the most recent data and found as{Mz) = 0.109 ± O.OOS, including estimates of the truncation error. Finally, we consider the extraction of as from e+e~ scattering into jets. The production of multijets in e + e" scattering depends strongly on q^. There, comparison of the QCD prediction and data introduces hadronization model dependence into the extraction of q^. The evaluation of as is further complicated by dependence on cutoffs between different jets and the usual problems of the unknown higher order terms. To reduce the jet resolution problem, event shape variables, such as the energy-energy correlations, the asymmetry of the energy-energy correlations, the oblateness, the thrust, etc., are used to extract as. As an example of an event shape variable we look at the energy-energy correlation (EEC) defined by [35] 1 dS^^^, 1 V/" d(7 -ix) a d cos X The EEC can be experimentally constructed as = E / T-J^, ViVjdy^dyj. (2.2.29) (T r-i J dyjdyjd cos \ _i — ^ y,yj^( cos cos x), (2.2.30)

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17 where x is the angle between calorimeter cells and yi = 2Ei/y/s are the center of mass energy fractions of the detected particles. The asymmetry of the energy-energy correlation (AEEC) is defined as 1 di:^^^^ X = TT-x ---^ (x) o f/cosx cr dzosx a dcosx A perturbative calculation of the asymmetry gives [35] 1 dS^^^"^ (2.2.31) , -(x) = — --ilcosx) cr a cos x 1 + — i2(cosx) + 0(a2) (2.2.32) where functions A and i? are calculated in perturbative QCD. The best data on the jet rates come from LEP. Recently those results have been extensively discussed in the Hterature [32.30,37]. Combining all the LEP data on jet distributions, including the full theoretical error, gives [32] Qs(M^) = 0.115 ±0.008. To summarize, the values of ols{Mz) and its error are given in Table I. Table I: Values of at Mz and its error Process Ags t~ hadrons 0.140 0.018 T decay 0.111 0.005 Deep inelastic scattering 0.109 0.008 Jet distribution in e'''e~ scattering 0.115 0.008 To pick the value of a^k^^z) ^^r our numerical studies we take the Gaussian weighted average of these values ([^(ajAa^)/ E( l/^a^)] ± [E(l/Acts)]"^ ), and we find as{Mz) = 0.113 ± 0.004.

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18 Yukawa Couplings To take full account of the Yukawa sector in running all the couplings, initial values for the Yukawa couplings are necessary. They must be extracted from physical data such as quark masses and CKM mixing angles. Furthermore, the interesting parameters must be determined step by step in the process of running to Planck mass. These two procedures are not unrelated and require the diagonalization of the up-type, down-type, and leptonic Yukawa matrices. We use Machacek and Vaughn's [6] convention where the interaction Lagrangian for the Yukawa sector is The Yukawa couphngs are given in terms of 3x3 complex matrices. After electroweak symmetry breaking, these translate into the quark and lepton masses where V is the CKM matrix which appears in the charged current C = Qj^^yJuh + QL^Yd^dR + ii^YjeR + h.c. (2.2.33) (2.2.34) (2.2.35) It is a unitary 3x3 matrix often parametrized as follows: (2.2.36) where Sj = sin^i and c,= cos^,-, i = 1,2,3.

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19 The entries of the parametrized CKM matrix can be related simply to the experimentally known CKM entries. The particle data book [29] gives the following ranges of values (assuming unitarity) for the magnitudes of the elements of the CKM matrix: 0.9747-0.9759 0.218-0.224 0.001-0.007 \ 0 218-0.224 0.9734-0.9752 0.030-0.058 . (2.2.37) 0 003-0.019 0.029-0.058 0.9983-0.9996/ These ranges of values can be converted to bounds for 5,-, i = 1,2,3, and sm^. We arrive at these bounds by finding values for the four angles such that the entries of the CKM matrix obtained from these satisfy the conditions imposed by Eq. (2.2.37). We find 0.218S < sin ^1 < 0.2235 , 0.021G < sin ^2 < 0.0543 , (2.2.38) 0.0045 < .sin ^3 < 0.0290 . However the accuracy with which is known does not constrain sin 6. A set of angles {^i, ^3. ^) ^as chosen that falls within the ranges quoted above. The initial data needed to run the Yukawa elements are extracted from the CKM matrix and the quark masses. A problem arises though for the mixing angles, which was solved for the quark masses, in that it is not clear at what scale the chosen initial values for these angles should be considered known. However, even when the top quark is taken as high as the p-parameter bound allow the variations in 62 and ^3 between M\y and the Planck scale are at most 20%, and these variations are always negligible in the experimental region. Therefore, the exact knowledge of the initial scales for the angles is not as critical as might be feared a priori.

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20 Known Quark Masses As QCD is assumed to imply quark confinement, extraction of quark masses from experiment follows the same circuitous route as other QCD quantities such as asIn the past decade a variety of techniques have been developed and utilized to extract quark masses from the observed particle spectrum. Below, we shall briefly recount some such techniques. Furthermore, we shall present some values for the heavy quark masses based on the application of our numerical technique to three loops. The hght quark masses are the ones least accurately known. They are determined by a combination of chiral perturbation techniques and QCD spectral sum rules (QSSR). In the former case the light quark masses are directly expressible in terms of the parameters of the explicit SU(2) and SU(3) chiral symmetry breakings. One then considers an expansion of the form [38] Mbaryon = « + bmnght + • (2.2.39) for the mass of a baryon from the ^'^ octet, and one of the form for a typical member of the pseudoscalar octet. A parameter measuring the strength of the breaking of the more exact SU(2) chiral symmetry in comparison with the SU(3) one is the ratio R = il^l^ , (2.2.41) md ruu where m' = ^(mu + mrf) . (2.2.42)

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21 To lowest order in isospin splittings, this translates in the meson sector into R^J^ri^, (2.2.43) and in the baryon sector into three different determinations of R, ^_ ^^{M^-Mn)-1(Mt:-Mx) ~ Mn Mp ^(MH-MAr)-l(Mv-AfA) (2.2.44) ~ A/hA/ho A/j;Aiv+ To make R compatible with all the above mass splittings one has to consider higher order corrections in Eqs. (2.2.39) and (2.2.40). Here infrared divergences emerge as one is expanding about a ground state containing Nambu-Goldstone bosons. Once such singularities are removed within the context of an effective chiral Lagrangian, one finds the following as the optimum value of R /? = 43.5 ±2.2. (2.2.45) Together with the ratio [39] ^ = 25.7 ±2.6, (2.2.46) m' also determined by applying Eq. (2.2.40) to the physical masses of tt, t/, and A', they imply the following renormalization group invariant mass ratio 'JlAZiJI^ = 0.28 ± 0.03 . (2.2.47) 2m' Applied to the light quarks the QCD spectral sum rules imply [39] mu + = 24.0 + 2.5 MeV . (2.2.48)

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Together with Eq. (2.2.47) they reduce to ihu = 8.7 ± 0.8 MeV , (2.2.49) rhd = 15.4 ± 0.8 MeV . The parameter m is a renormahzation group invariant which to three loops is related to the MS running mass parameter m{iJ.) via [40] 2[/?2V/3i 132) h^^l^^l /53^JL tt Ji' (2.2.50) where the fii and the 7, are the coefficients of the ^ functions for as and m given in Appendix A. From Eqs. (2.2.50) (to two loops) and (2.2.49) one may infer the following values mt,(l GeV) = 5.2 ±0.5 MeV , (2.2.51) 777 j(l GeV) = 9.2 ±0.5 MeV . In applying expression (2.2.50) it should be kept in mind that the continuity of m(/x) across a quark mass threshold requires m to depend on the effective number of flavors at the relevant scale, analogously to the QCD scale A. The strange quark mass is determined, averaging the value derived from Eqs. (2.2.46) and (2.2.48) with those obtained using Eq. (2.2.49) and the various QSSR values for rhu + rhs, to be [41] ms = 2G6 ± 29 MeV , (2.2.52) corresponding to the running value ms(l GeV) = 194 ± 4 MeV . (2.2.53) For the heavier quarks, charm and bottom, one can make a more precise prediction. Here, the nonrelativistic bound state approximation may be applied. The physical mass M{q^ = M2) appearing in the Balmer series may

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23 be identified with the gauge and renormalization scheme invariant pole of the quark propagator S{q)^z{q)['j-q-Miq'^T^ . Corresponding to the above pole mass is its Euclidean version, m(-g^), which is not gauge invariant, although it is renormalization group invariant, and is therefore not physical. The Euclidean mass parameter is the one often employed in the J/V" and T sum rules, as it minimizes the radiative corrections in such sum rules. In the Landau gauge the two are related to two loops according to [42] 1 5^1„4 (2.2.54) Once the pole mass is determined from the Euclidean one, the running mass at the pole mass is obtained to three loops via ,„,„2 _ ,/2) ""^l' = ^^') , (2.2.55) where A' = 13.3 for the charm and K = 12.4 for the bottom quarks [43]. From the J/t/' and T sum rules the following values have been extracted [39] md-q'^ = Mh = 1.26 ± 0.02 GeV , (2.2.56) m6(-ry2 = M^) = 4.23 ± 0.05 GeV . To obtain an accurate value for the corresponding pole masses, we numerically solve Eq. (2.2.54) with the above values inserted and the three-loop /3 function for as, to obtain the following pole masses Mciq^ = Mh = 1.46 ± 0.05 GeV , (2.2.57) Mfc(g2 = Ml) = 4.58 ± 0.10 GeV . Recently [44], new values for the charm and bottom pole masses have been extracted from CUSB and CLEO II by analysis of the heavy-Ught B and B*

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and D and D* meson masses and the semileptonic B and D decays with the results Mciq^ = Ml) = 1.60 ± 0.05 GeV , (2.2.58) Mtl^^ = Ml) = 4.95 ± 0.05 GeV . A weighted average of the values in Eqs. (2.2.57) and (2.2.58) yields MAq^ = Mh = 1.53 ± 0.04 GeV , (2.2.59) Mb(q'^ = Ml) = 4.89 ± 0.04 GeV . The running masses at the corresponding pole masses follow from Eq. (2.2.55) ,77c( Mc ) = 1.22 ±0.06 GeV , (2.2.60) miiMb) = 4.32 ±0.06 GeV . With these taken as initial data along with the value of the strong coupHng at Mz quoted earlier, we run (to three loops) the masses and Og to obtain the following values at the conventionally preferred scale of 1 GeV 7nc(l GeV) = 1.41 ±0.06 GeV , (2.2.61) mi(l GeV) = 6.33 ± 0.06 GeV . Our numerical approach does not make any more approximations than the ones assumed in the d functions and the mass equations used, apart from the approximation inherent in the numerical method itself, and is more in line with our program than using the "perturbatively integrated" form of the /3 functions. Thus we shall adopt the above values. It should be stressed that at the low scales under consideration the three-loop ag corrections we have included in our mass and strong coupling functions are often comparable to the two-loop ones and hence affect the accuracy of our final values noticeably. Nevertheless, it should be noted that the above expressions relating the various mass parameters are not fully loop consistent as to our knowledge Eq. (2.2.54) has only been computed to two loops.

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In conclusion, it should be pointed out that although we opted for the QSSR extraction of masses, there are rival models, such as the nonperturbative potential models, which predict appreciably higher values of the heavyquark masses than the ones quoted here. These models, however, are not as fundamental as the approach considered here, and their connection to field theory is rather problematic. Lepton Masses The physical (pole) masses of the leptons are very well known [29] Me = 0.5109990G ± 0.00000015 MeV , M^, = 105.658387 ± 0.000034 MeV , (2.2.62) Mr = 1.7S4llS:S2'3^ GeV . We use these values to determine initial data for the running masses. Some authors neglect QED corrections and use the physical values for the running values at ~ il/^, which introduces only a small error. By calculating the oneloop self-energy corrections, one arrives at a QED relation between the running MS masses and the corresponding physical masses m/(//) = M/[l-^(ln^ + |)]. (2.2.63) 47r mj «J Choosing /i = 1 GeV as in the quark mass case and using Eqs. (2.2.63)and (2.2.4) yield the running lepton masses (taking m/ = Mi in the log term above is an appropriate approximation to order a ) n7e(l GeV) = 0.4960 MeV , m^,{\ GeV) = 104.57 MeV , (2.2.64) 7?ir(l GeV) = 1.7835 GeV .

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26 Higgs Boson and Top Quark Masses The Higgs boson and top quark masses have not been measured directly at present; however their values affect radiative corrections such as Ar. Consistency with experimental data on sin^ 6^ requires M< < 197 GeV for M// = 1 TeV at 99% CL assuming no physics beyond the Standard Model [45]. Precision measurements of the Z mass and its decay properties combined with low energy neutral current data have been used to set stringent bounds on the top quark mass within the minimal Standard Model. A global analysis of this data yields Mt = 122+J^ GeV, for all allowed values of Mh [46]. Recent direct search results set the experimental lower bound Mt ^110 GeV. As for the Higgs, the analysis of Ref. 46 gives the restrictive bound, M^ ^ 600 GeV, if Mt < 120 GeV, and M^j < 6 TeV, for all allowed A/<. Since perturbation theory breaks down for Z 1 TeV, the latter bound on the Higgs boson mass is not necessarily meaningful. LEP data set a lower bound on the Higgs boson mass of 4S GeV [47]. In our Standard Model analysis, initial values of the MS running top quark mass mt and of the scalar quartic coupling A at Mz are chosen arbitrarily (consistent with the bounds quoted above). As noted earlier, these running parameters are not equal to their physical counterparts. However, any reasonable prediction for the masses of the top quark and of the Higgs boson that may come from our analysis should be that of experimentally relevant, physical masses. Therefore, formulas similar to Eq. (2.2.63) relating MS running parameters to physical masses are needed. To calculate the physical or pole mass of the top quark, we use Eq. (2.2.55) in its general form ^ = 1 + 1^ + 110.11 1.04^(1 ^)l(^)^ , (2.2.65) mt{Mt) 3 TT ^ M< TT

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27 where Mi, z = 1, . . . , 5, represent the masses of the five Hghter quarks. Likewise the physical mass of the Higgs boson can be extracted from the following relation [48]: A(^) = ^mJ(1 + %)) , (2.2.66) where 6{n) contains the radiative corrections. Its form is rather elaborate and we relegate it to Appendix C. Equations (2.2.65) and (2.2.66) are highly nonlinear functions of Mt and M // , respectively, and we solve them numerically to find the masses of the top quark and the Higgs boson. In the Minimal Supersymmetric extension of the Standard Model we treat the masses of the Higgs boson and the top quark somewhat differently. In that case we will be able to constrain the values of those masses, by imposing some of the relations that we discussed in Section 2.1. Vacuum Expectation Value of the Scalar Field The vacuum expectation value (vev) of the scalar field may be extracted from the well known lowest order relation V = (^/2G,,^^ = 246.22 GeV . (2.2.67) From the very well measured value of the muon lifetime, = 2.197035 ± .000040 X 10"*^ s [29], the Fermi constant can be extracted using the following formula [49] r^' = S4/(4)U + l4i^ + ^
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This parameter may be viewed as the coefficient of the effective four-fermion operator for muon decay in an effective low energy theory ^[l^eAl 75)e][7l7/?(l " (2.2.71) A direct calculation {e.g., in the Landau gauge) of the electromagnetic corrections yields that the operator is finitely renormahzed (i.e., does not run) [23,24]. Another way to see this is by using a Fierz transformation to rewrite the above expression ^{uel^l 75)e] . (2.2.72) v2 The neutrino current does not couple to the photon field, and the e-fi current is conserved and is hence not multiplicatively renormalized. We need an initial value for the running vacuum expectation value at some scale ^. Wheater and Llewellyn Smith [50] consider muon decay to order a in the context of the full electroweak theory and derive an equation relating an MS running Gn to the experimentally measured value. From this formula we Ccin extract a value for v{Mz ) However, the formula is derived in the 't Hooft-Feynman gauge, and the evolution equation Eq. (A. IS) of Appendix A for the vev is vahd only in the Landau gauge. Nevertheless, motivated by the discussion of the previous paragraph, we choose the initial condition for the vev to be v{Mw) = 246.22 GeV. Using our numerical methods we arrive at v{Mz). We find that this procedure leads to no significant correction, and we therefore take v{Mz) = 246.22 GeV. 2.3 Quantitative Analvsis of the SM Let us summarize the most important features of the renormalization group running in the Standard Model. At the one loop level, the gauge couplings are

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29 unaffected by the other coupHngs in the theory. On the other hand, the Yukawa coupHngs are affected at one loop by both the gauge and Yukawa couplings. Since the top Yukawa coupling is at least as big as the gauge couplings at low energy, that means the running of the Yukawas is sensitive to mostly the top Yukawa and the QCD gauge couplings. Thus we can expect the mass and mixing relations we described in section 2.1 to be sensitive to the value of the top quark mass. The Higgs quartic self-coupling enters in the running of the other couplings only at the two loop level, so that its effect on the other parameters is small. However, its own running is very sensitive to the top quark mass; it can become negative as easily as it can blow up, corresponding to vacuum instability or to strong self-interaction of the Higgs (triviality bound), respectively. The discovery of the Higgs with mass outside these bounds would be a signal for physics beyond the Standard Model. The graphs in Fig. 1 summarize these bounds for representative values of the top quark mass. For example, if Mt = 150 GeV, we see from the corresponding plot that a Higgs mass between 95 and 150 GeV need not imply any new physics up to Planck scale. However, if the Higgs were observed outside of this range, then some new physics must appear at the scale indicated by the curve, either because of vacuum instability if A/// < 95 GeV or because the Higgs interaction becomes too strong if M// > 150 GeV. It is amusing to note that it is for comparable values of the top and Higgs masses that these bounds are least restrictive, but it is important to emphasize that a high value of the top with a relatively low value of the Higgs necessarily indicates the presence of new physics within reach of the SSC. Subsequently, when examining the possible relations in the context of the Standard Model, we will make the choices for Mt and Mh in our renormalization group runs consistent with these bounds. For a chosen value

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30 of Mt, varying M// within the vacuum stability and triviaUty bounds does not affect any of our results, and we will therefore choose a corresponding, representative value of Mjj. 20 ^15 « 10 an o Mi=100 GeV I I I I I I I I Allowed I I I I I I I I I I I I I 20 15 10 5 M,=125 GeV n I I I I I I I / — Vacuum Stability Allowed I I I I I I I I I I I I (a) 20 15 10 ^ 5 50 100 150 200 M,=150 GeV III' -Vacuum M 1 1 1 1 \ ' ' ' 1 -Triviality^ -Stability Allowed = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~ (b) 20 15 10 5 0 50 100 150 200 Mt=200 GeV Jill -I 1 1 1 1 1 1 1 1 1 IA| 1 1 l"T^ Triviality-/— \ Vacuum / Stability W 1 1 1 1 1 L 1 1 1 1 1 1 1 1 1 r 50 (c) 100 150 M„ (GeV) 200 (d) 50 100 150 200 250 M„ (GeV) Figure 1. Vacuum stability and triviality bounds on the Higgs mass for (a) M,=100 GeV, (b) Af, = 125 GeV, (c) M, = 150 GeV, and (d) M,=200 GeV, giving scales of expected new physics beyond the Standard Model. We have studied the effects of changing the values of Mt and M// in our analyses of the running of the other parameters. As expected we observed that, for any M< between 100 GeV and 200 GeV, varying Mfj, while maintaining perturbativity and vacuum stability, did not affect appreciably the evolution of

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31 any of the other parameters. However, changing Mt itself showed a significant difference in the running of the heavier quarks. This is illustrate in Fig. 3. We note that the point where = mr moves down to a lower scale for a higher top quark mass. This is expected since from Eq. (A. 9) one can see that the bottom type Yukawas are driven down by an increased top Yukawa. This behavior is to be contrasted with the SUSY GUT case in which the bottom Yukawa ^ function (see Eq. B.2.3) is such that this crossing point is shifted toward a higher scale with an increased top mass. As we shall see in the SU(5) SUSY GUT model, the equality of the bottom and r Yukawas at the scale of unification can be used to get bounds on the top and Higgs masses. STANDARD MODEL 0 5 10 15 20 logio(/^/lGeV) Figure 2. Running of the inverse gauge couplings showing their propagated experimental errors.

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In Fig. 2. we display the evolution of the inverse of each of the three gauge coupHngs. In this figure, we see that the three gauge couphngs only semiconverge in forming the GUT triangle around lO^^"^^ GeV. Notice that that the uncertainties do not come close to filling the "GUT triangle", thus ruling out grand unification, assuming the Standard Model as an effective theory in the desert up to the Planck scale. The relation = (Relation I) is the most natural one in the SU{b) theory, and it could be expected to be vahd at scales where the Standard Model gauge couplings are the closest to one another. We examine its validity for three different physical values of the top and Higgs masses in the Standard Model. The results are summarized in Fig. 3. 1.5 B S 1 .5 1 L 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 \ -— Mt-100 GeV, Mh100 GeV Z \ -Mt=150 GeV, Mh= 150 GeV \ Mt=190 \^ GeV. Mh= 180 GcV N. s. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 5 10 15 20 logio(M/lGeV) Figure 3. Plot of mt/nir as a function of scale in the Standard Model for various top and Higgs masses.

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33 The noteworthy feature of the figure is that this simplest of the SU(5) relations is valid at an energy scale many orders of magnitude removed from that at which the gauge couplings tend to converge. Our result is vastly different from that of the original investigations in Ref. 16. We have improved on their work by including two loop effects in the running of the quark Yukawas, by taking into account the full Yukawa sector, and most importantly by incorporating QCD corrections in the extraction of the bottom quark mass. We now turn to the more complicated relations among masses of the two Hghter families, n?^/ = 37??e. 3j77s = (Relations II). There are large theoretical uncertainties in the extraction of the masses of the three Hghtest quarks from experiment, although the mass ratios are known more accurately. Following Refs. [51], we take their values to be m^lrnu 1.8 and mslm^ 21, so that specifying ms fixes m^i and mi,. We note that ms/rrid and m^/me effectively do not run. Therefore, given this value for mslm^. we do not expect relations (II) to be both satisfied exactly, since mf^/Qme « 23. The uncertainties in the light quark masses are accounted for by examining the ratios mj/3me and 3ms/m^ for a range of ??7s(l GeV) values from 140 to 250 MeV. We have run these same ratios for representative values of the top and Higgs masses but find the results to be fairly insensitive to the value of the top. Therefore, in Fig. 4., we only present results for top and Higgs masses of 190 GeV and 180 GeV, respectively. Unlike relation (I) which holds only at ~ 10^ GeV, we see that relations (II) can hold within ~ 5% at 10^^ GeV for acceptable values of the light quark masses.

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34 H CO \ 6 2 (a) I I I I I I I I I ^ ind(lGeV) = l'2 MeV md(lGeV)=6.6 MeV 5 10 15 logio(/^/lGeV) S 2 (b) —1111 — TT" I 1 1 1 1 1 1 1 1 1 1 1— ----m,(lGeV)-250 MeV— m,(lGeV)-140 MeV^ 1 1 1 1 1 1 1 1 1 1 1 ii~r~hT 1 5 10 15 log,o(M/lGeV) 20 Figure 4. Plots of (a) md/377i, and (b) 3m, /m^ as a function of scale in the Standard Model for il/, = 190 GeV and Mh = 180 GeV. We find the GSTO relation, tan^c = \/rnd/ms (Relation III), to be quite independent of scale. The reason is that the Cabibbo angle effectively does not run, and the ratio of light quarks is essentially unaffected by QCD, since both are far away from the Pendleton-Ross [52] infrared fixed point. Further, we observe that their numerical values are fairly independent of the value of the top quark mass and of the Higgs mass. The agreement is spectacular, hovering

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35 around the 4% level. For example, for Mt = 100 GeV and Mh = 100 GeV, we find that i&nOdyJm Jms = 1.03S from Mz to Planck scale. Relation IV, V^b = y/mc/mt, involves the top quark mass directly, which may thus be predicted from this relation. On the other side of the equation, the experimental value of the "23" element of the CKM matrix, Vd,, is known only to within ~ 10%, Vcb = 0.043 ± 0.006. The value of V'c6 at all scales is obtained by running the CKM angles. These numerical results do not depend on the value of the CP-violating phase. We note that because of the Pendleton-Ross fixed point, the ratio of the two quark masses runs appreciably in the infrared region. We find that for a top quark in its lower allowed range, 91-150 GeV, this relation fails over all scales. Accordingly, we present our results for values of Vcb and Mt for which the relation can be satisfied below Planck scale. We use for Vcb values ranging from its central value of .043 to .050. (As discussed earlier we take the value of lUc at 1 GeV to be 1.41 GeV.) The results of our runs can be summarized in Fig. 5. in which we plot both Vcb and y/nic/mt as a function of scale. From the first plot, we see that the top quark has to be at least ITS GeV for yjmdmt to meet Vcb at the Planck scale. The second plot shows that Mt = 180 GeV allows for this relation to be easily satisfied at 10^*^ GeV, if Vcb{Mz) = -05. This means that a few GeV difference in Mt changes the meeting of the curves (both of which are affected by Mt) by three orders of magnitude! Finally, in the third plot, we see that for this relation to be valid at the unification scale, using the central value of Vcb-, a 197 GeV top quark is needed. We conclude that, given the uncertainties in the value of Vcb-, this relation may well be valid as long as Mt > 175 GeV.

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36 .07 . .06 ^.05 M,=178 GeV; Mh=160 GeV 1 1 I I ' I I I I I I I I I I (iTle/mi) -V,,(Mz)=.050 V,,(Mz)=.043 (a) 0 5 10 15 log,o(M/lGeV) 20 .07 .06 .05 .04 M,=^180 GeV; Mh=17Q GeV I I I I I I I I I I I I I I I I I I (me/mj V,^(Mz) = .050 V,,(Mz) = .043 I I I I I I I I I I I I i I ! I I I I (a) 5 10 15 logio(M/lGeV) 20 .07 .06 ^.05 M,=^197 GeV; Mh^190 GeV I I I I I (me/mi) V,k(Mz)=.050 V,,(Mz)=.043 I I I I (b) 04 I I I I I I I I I I I I I I I I I I I 0 5 10 15 20 log,o(M/lGeV) Figure 5. Plot of Vet and \/mc/m, as a function of scale in the Standard Model for (a) A/,=178 GeV, (b) Mr=180 GeV, and (b) M, = 197 GeV, for both the central value (.043) and the maximal value (.050) of Vcb(Mz)There are other other possible mass relations. An interesting mass relation involves the ratio of the determinants of the charge -1/3 to charge -1 mass matrices and should equal one if relations (I) and (II) are vahd. We note that, independent of the top mass (91-200 GeV), this weaker (less predictive) relation (m^msmj = rriem^mT) can be satisfied at 10^^ GeV for quark masses within the range stated above. The relations considered so far have been motivated by specific theoretical models. There are other relations which are not similarly motivated but which

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37 may nevertheless hint at an underlying unified structure. In our search for simple relations among the quark masses, we considered an appealing "geometric mean" relation in the up sector, namely mumt = . (2.3.1) 1 6 This relation favors higher top quark masses and can be satisfied well at 10 GeV in the Mt = 190 GeV, Mh = ISO GeV scenario with an up quark mass compatible with that needed to satisfy relations (II) at 10^^ GeV. A similar relation involving the down-type quarks was tested nijinii = ml . (2.3.2) This relation favors higher top quark masses as well. In fact, in order to satisfy Eqs. (2.3.1) and (2.3.2) at 10^^ GeV, as well as relations (II), given a fixed value for ms{l GeV) within the range cited, the top quark mass would have to be larger than 200 GeV. Fortuitously, such a value would also favor relation (IV). These geometric mean relations have been discussed in the literature recently [53]. To conclude our analysis of the Standard Model case, we see that it is hard to arrive at a unified picture. The gauge couplings only semi-converge and the scale at which relation (I) tends to be satisfied does not coincide with that at which the other relations are valid. Still, the disagreement is never too large, which leaves the possibility that small course corrections in the running of the parameters allow most if not all of these relations to hold simultaneously at a unified scale. It is remarkable that for a top quark at the upper reaches of its allowed range, the long life of the bottom quark lends plausibihty to the SO(10)-inspired relation (IV).

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38 2.4 Quantitative Analysis of the MSSM As is well known, the Standard Model shows no apparent inconsistencies until perhaps the Planck scale, where quantum gravity enters the picture. The nature of the physics to be found between our scale and the Planck scale is a matter of theoretical taste. At one extreme, because of the values of the gauge couplings, new phenomena may be inferred every two orders of magnitude. At the other, there is the possible desert suggested by GUTs; however, the absence of new phenomena over many orders of magnitude cannot be understood (perturbatively) unless one generalizes the Standard Model in some way to solve the hierarchy problem. Supersymmetrizing the Standard Model at an experimentally accessible scale can accomplish this. This particular scenario is bolstered by the fact that with such "low energy" Supersymmetry, the three gauge couplings of the Standard Model meet at one scale (~ 10^^ GeV) at the perturbative value of ~ 1/26 (see Fig. 6) [54,55,56]. However this should not be viewed as proof of supersymmetry since given the values of ai, 02, 03 at some scale, and three unknowns (the value of a at the unification scale, the unification scale, and an extra scale such as the SUSY scale) there is always a solution. The exciting aspect of this analysis is the numerical output, namely a low SUSY scale, MsvsY-< and a perturbative solution below the Planck scale which does not violate proton decay bounds

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39 0 5 10 15 20 logio(M/lGeV) Figure 6. Plot of the running of the inverse couphngs. The dotted Hnes above and below the solid lines represent the experimental error for each coupling. Note the small region where all three couplings intersect. We found that this region reduced to a point when MguSY = ^-^ "^^^ and was non-existent above that scale. The collapse of the GUT triangle in the supersymmetric extension fixes two scales, the one at which the gauge couplings unify, the other at the threshold of Supersymmetry. Minimal Supersymmetry implies two Higgs doublets and eliminates the quartic self-coupling of the Standard Model. But there appears an extra parameter, the ratio of the vacuum values of these two doublets, parametrized by an angle 0, tan/3 = Vu/vj, where Vu (uj) is the vacuum expectation value of the Higgs field that gives mass to the cheirge 2/3 ( — 1/3, -1) fermions. In the foUoAving we examine the relations discussed in section 2.1 among masses and mixing angles in the context of the minimal supersymmetric

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40 extension of the Standard Model. We limit our study to the case of only one Hght Higgs. We will first restrict ourselves to an SU(5) SUSY-GUT [57] where and yT, the bottom and r Yukawa couplings, are equal at unification. The crossing of these renormalization group flow lines is sensitive to the physical top quark mass, MtThis can be seen in the down-type Yukawa renormalization group equation (above Msusy)^ ^o"^ ^^^^^ extract the evolution of yj, since the top contribution is large and appears already at one loop through the up-type Yukawa dependence: ^ ~Y,[ 3Y,tY, + Y„tY. + rr{3Y,tY, + Ye^Ye} dt (2.4.1) (^^1 + + -3-93) ] • where d e are matrices of Yukawa couplings. Demanding that their crossing point be within the unification region determined by the gauge couplings allows one to constrain i\/<. This yields an upper and lower bound for Mi which is fairly restrictive. We consider the simplest implementation of supersymmetry and run the couplings above MsuSY to one loop. The superpotential for the supersymmetric theory is W = ^uQYuu' + ^dQ^dd' + ^d'^ee' + fi^d^u , (2.4.2) where the hat denotes a chiral supermultiplet. We assume the MSSM above MsuSY^ and a model with a single light Higgs scalar below it. This is done by integrating out one linear combination of the two doublets at MsuSY^ thereby leaving the orthogonal combination in the Standard Model regime as the "Higgs doublet": ^(s.M) = ^dcos/? + ^usinlS , (2.4.3)

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41 where $ = 1x2$*, and where tan/5 is also the ratio of the two vacuum expectation values {vulvfj) in the Hmit under consideration. This sets boundary conditions on the Yukawa couplings at MsuSY • Furthermore, in this approximation the quartic self coupling of the surviving Higgs at the SUSY scale is given by: HMsusY ) = ^(^1 + 92^(213) . (2.4.4) This correlates the mixing angle with the quartic coupHng and thereby gives a value for the physical Higgs mass, M^iggs. Using the experimental limits on the M Higgs further constrains some of the results. By using the renormalization group we take into account radiative corrections to the light Higgs mass [58] and hence relax the tree level upper bound, Mijiggg ~ Mz [59]. We determine the bounds on M< and Mjnggg by probing their dependence on /3. In SUSY-SU(5), tan /i is constrained to be larger than one in the one light Higgs Hmit. It seems natural to us to require that y< > up to the unification scale [60], thereby yielding an upper bound on tan ,9. To probe the dependence of our results on we use both = 4.6 GeV and A/j = 5 GeV in our study. We also investigate the effect of varying MgusYGiven the values of the gauge couplings, we find unification up to a SUSY scale of 8.9 TeV, and as low as M^y. For empirical reasons we did not investigate solutions below that scale. We determine that the lower end scale, Mq^j, of the unification region corresponds to an 03 value of 0.104 at Mz-, while the higher end scale, Mq^j^ corresponds to a value of 0.108 at Mz for 03. We find that the unification region is insensitive to the range of top, bottom, and Higgs masses considered. In our analysis of the bounds for M<, the values for a\ and 02 are chosen to be the central values since their associated experimental uncertainties are less

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significant than for 03. Demanding that and yr cross at Mq^j^ and taking 03 = 0.104 then sets a lower bound on M(. Correspondingly, demanding that yb and yr cross at M^jjj^ and taking 03 = 0.108 yields an upper bound on M<. These bounds are found for each possible value of 13. MsusY=l TeV, Mb=4.6 GeV 200 1 1 1 1 M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ^ — V >^ .^150 M t ^-^^^^^^^^^^"'^^^ X "100 50 1 1 'T''l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 40 50 60 70 80 90 (3 (deg) Figure 7. Plot of the top quark mass, M<, and of the Higgs mass, Mjjiggs, as a function of the mixing angle ^ for the highest value of 03 (high curves) and the lowest value of 03 (low curves) consistent with unification. Fig. 7. shows the upper and lower bound curves for both Mt and Mfjiggg as a function of /9 and for MsusY = 1 TeV and = 4.6 GeV. When applicable we use the current experimental limit of 38 GeV on the hght supersymmetric neutral Higgs mass [61], to determine the lowest possible Mt value consistent

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43 with the model. We find 139 < Mt < 194 GeV and 44 < Mniggs < 120 GeV. We investigated the sensitivity of these results on Msus^' the range, 1.0 ± 0.5 TeV. We find that the bounds on Mt are not modified, but the upper bound on the Higgs is changed to 125 GeV, and the lower bound drops below the experimental lower bound. For Mb = 5.0 GeV, we see an overall decrease in the top and Higgs mass bounds: 116 < Mt < 181 GeV, Mniggs < m GeV. We display the results of our analysis for the extreme case, MsuSY = 8-9 TeV, in Fig. 8., with = 4.6 GeV. This only significantly changes the upper bound on M Higgs to 144 GeV compared to the MguSY — ^ TeV case. 200 — §150 X 400 50 M SUSY:8.9 TeV, Mh=4.6 GeV I I I I I I I I I I Mt Mh I ! ! I TT I I I I I I I I 40 50 60 70 iS (deg) 80 90 Figure 8. Same as Fig. 7 for MsuSY = 8-9 TeV and = 4.6 GeV.

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44 0 10 20 30 40 50 60 /S (deg) Figure 9. Plot of the ratio of the top to bottom Yukawas, ytlvh, at the GUT scale, for two different bottom masses (solid and dashed curves) as a function of tan for the highest value of a'3 (high curves) and the lowest value of 03 (low curves) consistent with unification. We have also run yt up to the unification region and compared it with and j/r to see what the angle 0 must be for these three couplings to meet [62], as in an SO(IO) or model [3,4] with a minimal Higgs structure. It is clear that this angle is precisely our upper bound on /? as described earher. In Fig. 9. we display j/
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45 106 < Mniggs < 111 GeV. When Mb = 5.0 GeV, we obtain 31.23 < tan/3 < 41.18, which gives 116 < M< < 147 GeV and 93 < Mniggs < 101 GeV. The strategy of the remaining part of the MSSM analysis is to exploit relations (IIIV) to constrain Mt and therefore (3 and M//. For MsuSY = 1 TeV, we treat two cases, the first where unification takes place at its lowest value (Low Mgut) and the second where it is at its highest value (High Mqut)In the following, we will not discuss our results for upper limit of the Supersymmetry breaking scale, 8.9 TeV, since it adds nothing to our conclusions. In the case of the mass relations among the light quark and lepton masses (relations (II)), we find that our plots do not depend on M<, therefore we only display them for a representative value. Here we do not follow the strategy used in the Standard Model case (i.e., we do not vary rris), although we still keep the ratios ni^/m^ and ms/?77^ fixed. Instead, we look for that value of ms(l GeV) which gives us the best agreement for relations (II) both for the low and high Mqut (we cannot expect exact agreement at one scale, since ms/m^ does not run). For instance, we can get the same value of ms(l GeV) in two different ways, either by demanding that at low MqUT mjZme = 1.1 and 3ms/mfj = 1 or at high Mqut that md/^me = 1 and 3ms /m^ = .9. In both cases, the masses of the lightest quarks at 1 GeV are: ruu = 3.80 MeV, = 6.67 MeV, and nig = 141 MeV. These results are summarized in Fig. 10.

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2.5 CO \ J 1.5 ! I I I I I I I 46 MsusY = 1 TeV I I I I I I I I I — -LOW Mgut HIGH Mgut . md(lGeV)=6.7 MeV : I I I I I I I I I I I I I I I I I I I (a) 2.5 5 10 15 20 logio(/i/lGeV) MsusY = 1 TeV I I I I I I I I I I I I I I I I LOW Mr GUT HIGH Mgut . ms(lGeV)=141 MeV (b) 5 10 15 logio(/i/lGeV) Figure 10. Plots of (a) md/^m, and (b) Zmjm^ as a function of scale in the SUSY case with Msusy = ^ TeV, for the low and high unification scales (no appreciable dependence on Mt was found). In the above, our philosophy has been to take the known low energy data, and using the renormalization group, derive its implications at high energy. As we did for relation (I), we could impose both of relations (II) at one unification scale (low or high Mgut)This would fix mslm^ at this scale, and since mslm^ and m^i/mc do not run, that would yield ms/md = 23.6, a value that

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47 is only 12% larger than the value discussed in section 2.2. Furthermore, the results of our runs yield the masses at 1 GeV of the down and strange quarks to be = 5.86 MeV, rris = 138 MeV, in the low Mqut case, and rud = 6.49 MeV, TTis = 153 MeV, in the high Mqut case. We note that this approach has also been taken by the authors of Ref. 63. Before discussing relation (IV), let us note that, with Supersymmetry, relation (III) is agciin well satisfied at all scales. .0? TeV; Mt=198 GeV 1/2 HIGH M GUT _ .03 I I I I I I I I I I I I I I I H 0 5 10 logio(M/lGeV) 15 20 Figure 11. Plot of V^t and \JmJmt as a function of scale in the SUSY case with Msvsy = \ TeV for M, = 198 GeV and for Vci(A/z)=.043. We now turn to relation (IV). As we did in the Standard Model case we display our results both for the central value of Vch (.043) and for its upper value (.050). Then we look for values of Mt which give us agreement at the

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48 unification scale (low or high Mcut)Using the central mlue for Vc6' ^e find no agreement at the unification scale. However, this relation is satisfied at Planck scale, if we use both a high McuT and Mt = 198 GeV (the highest possible value consistent with relation (I)), as displayed in Fig. 11. We have also made several runs with a higher value VcbThere, the relation can actually be satisfied provided that we use the high MquT scale and Mt = 198 GeV as shown in Fig. 12. 0 5 10 15 20 logio(M/lGeV) Figure 12. Same as Fig. 11 with V,i{Mz)=.ObO. In the low Mqut case, the two curves meet closer to the Planck scale. In fact, theory does not dictate to us the exact scale at which the 50(10)inspired relation is valid; it could be much higher than the scale of unification of the Standard Model's gauge couplings. To account for this, we now plot, in

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49 Fig. 13., as a function of Mt, assuming that relation (IV) is valid at Mqut^ \QMguTi '^^^Mgut^ and using the higher value of g^iMz).06 M SUSY = 1 TeV .055 3 > .05 .045 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 _Mgut\ MGUT=l-26xlO^^GeV _ 10Mgut\^\ — N. Si V IOOMgut \ ' i / . • 1 1 __ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 150 160 170 180 190 Mt (GeV) 200 Figure 13. Plot of Vcb(Mz) as a function of M, assuming relation (IV) holds at various scales 'ZMgvtGiven an initial value of V'c6 a* ^^Z^ Fig. 13 can be used to determine the needed Mi (and hence j3) to satisfy relation (IV) at Mqut^ '^OMqut^ or lOOMcuTWe can see from this figure that as long as F^i is larger than its central value, then relation (IV) can be satisfied above the SU{5) GUT scale and still allow for a lower value of Mt . The relation, rUfimsmi = mem^imT, involving the determinants of the charge —1/3 and charge —1 fermion mass matrices holds in the minimal supersymmetric model at 10^"^ GeV in the high Mquj case and at 10^^ GeV m

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50 the low Mgut case. In both cases, this relation holds within ~ 10% at 10^^ GeV for 160 < M< < 198 GeV. A priori one might naively assume that mumt = ml could be easily satisfied because of the uncertainty in m<. However, two facts make the relation viable in the supersymmetric case. First, the value predicted for the top mass is within the range allowed by experiment and the p-parameter bound. Second, and most remarkable, is the fact that this top mass value is compatible with relations (I)-(III). In Fig. 14, we display the running of the ratio mumt/m^ for the low and high Mquj cases. 0 5 10 15 20 logio(M/lGeV) Figure 14. Plot of m^mtlml as a function of scale for the highest value of aziMz) (high curve) and the lowest value of asiMz) (low curve) and for M, = 160 GeV.

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51 We show the curves representing the lower Mt value of 160 GeV for which the relation is best satisfied at McUTThis relation is incompatible with relation (IV) however, since the latter favors a higher top mass. We note that mc affects these two relations in an "inverse" manner. A lower experimental value for the charm quark mass favors relation (IV) whereas a higher experimental value favors the geometric mean relation. One may also consider the geometric mean relation in the down sector. We find however that this relation fails to hold in the supersymmetric case. Other relations among the Yukawa couplings have been considered in the hterature. Theoretical bias or numerology can lead to still other relations valid at some unifying scale. In all cases, a thorough renormalization group analysis will be required in investigations of a possible deeper structure.

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CHAPTER 3 A MULTISCALAR SM EXTENSION We present a minimal model with explicit lepton number breaking and an invisible axion. Neutrino masses are generated at one loop. For a reasonable range of parameters the model provides a solution to the strong C P and solar neutrino problems. This chapter is organized as follows. In section 3.1 we will present this model exhibiting SU(2)" x U(l)^ x 11(1)^*^ symmetry and incorporating explicit lepton number violation in the scalar sector. We provide some deteiils of the minimization of the scalar potential and give conditions on parameters that insure electric charge conservation. We then discuss the requirements that the gauge hierarchy problem places on the couplings of the model. In section 3.2 we display the model in unitary gauge and present the scalar mass matrices and their mass eigenstates. In section 3.3 we compute the neutrino masses and diagonalize the Majorana mass matrix. In section 3.4 we discuss experimental constraints on parameters of our model. Finally in section 3.5 we show that the model can provide a solution to the solar neutrino problem via the Mikheyev-Smirnov-Wolfenstein (MSW) effect [12]. 3.1 The Model The model we present below is essentially that of DFSZ, augmented by a charged scalar singlet. Thus it extends only the scalar sector of the Standard Model, by adding another isodoublet and two isosinglets. By introducing two 52

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53 standard Higgs doublets we can generate interesting new couplings. This extension of the Standard Model appears naturally in two popular types of theories. In the first, the global symmetry of the Standard Model includes a chiral phase symmetry, the Peccei-Quinn (PQ) symmetry [64], used to explain the lack of strong CP violation. The second is the A'^ = 1 supersymmetric extension of the Standard Model. In the first case, the theory contains a pseudo-NambuGoldstone boson, the axion [65]. For the model to be viable, the axion must be very light, and it can only be "seen" by Sikivie-type detectors [66]. The model is implemented through the presence of an electroweak singlet neutral Higgs field which carries the PQ charge. This invisible axion [67,68] extension of the Standard Model has two Higgs doublets, Hu and H^, with hypercharges 1 and -1 respectively, the same PQ charge, as well as a neutral singlet Their quantum numbers are Hnked through the quartic coupling H jr.jHd^'^ , (3.1.1) which shows that $ carries unit PQ charge. This describes the standard DFSZ [67,68] invisible axion model. (The Zhitnitskii model differs only in using a cubic interaction instead of this quartic one. ) The new observation we make is that by adding only a single scalar field, the S"*" field, we can form another quartic term which explicitly breaks lepton number by two units, since $ carries no lepton number. It is given by HjHdS+^ . (3.1.2) We note that S"*" carries one unit of PQ charge, fixed through the leptons since the charged lepton masses are taken to arise from the Yukawa coupling to Hj. Thus we come to the conclusion that in the context of the Standard Model with

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54 invisible axion, we need to add only one charged spinless particle to generate lepton number violation, and hence neutrino masses and mixing. We stress that much of what we have to say has already appeared in the hterature, the 5+ field was first advocated by Zee [69], invisible axion models originally by Kim [70] and Shifman, Vainshtein and Zakharov [71], and later by Zhitnitskii [67] and Dine, Fischler, and Srednicki [68]. Before going into the details of our model, let us summarize the basic features of the invisible axion model. The invisible axion model "solves" the strong CP problem, at the cost of introducing a new PQ-breaking scale of the order of 10^°"^^GeV, the scale being bracketed by astrophysical and cosmological constraints. Thus the model contains the tiny ratio _ 1 ^Fl ^ 10-9 ^ (3.1.3) where = V2 ($) . Unlike GUTs, this does not present any technical hierarchy problem [72], since $ is not required to have any large coupHngs to the fields of the Standard Model. We add that there is still a physical hierarchy problem yet to be answered: how are such disparate scales generated? In fairness, we should also remark that the invisible axion model, originally devised to explain away the limit on the 6 parameter, 9 < 10~^, itself introduces a similar ratio! In our model the Yukawa part of the DFSZ Lagrangian i{y^Ly2ejL + y\^^QjL^2djL]r2Hd + ry\fQlL<^2i^jLr2Hu + h.c. , (3.1.4) is augmented by iz[ij^L'fL(r2r2LjLS+ + h.c. , (3.1.5) which is antisymmetric in family space, as is required by the Pauli principle and 5+ is an isoscalar spinless field with Y = 2, and L = -2. With three

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55 families, these Yukawa couplings break all but the total lepton number. The PQ symmetry insures that only one doublet couples to each charge sector of the quarks, insuring the absence of tree level flavor-changing neutral currents [73]. Here we have the weak isodoublets and diL and un the left-handed isosinglet antiquarks of charge ^ and -3, respectively. The Higgs potential carries both the DFSZ term Hu"^ T2Hd^^ and our new term HJHjS'^^. in addition to all the normal quartic terms. It has no cubic terms. Note that the DFS quartic term is needed. Its absence from the potential would yield a theory with both L and PQ invariance. Upon $ getting its vacuum value, the linear combination L -f 2PQ would be preserved. Electroweak breaking would then yield a Majoron theory of a type already excluded by measurements of the Z° width. Thus both terms are needed to conform with experience. The most general potential of these fields consistent with SU(2)^^'x U(l )^ X U( 1 )^'^ symmetry with the single expHcit lepton number violating term is given by V{Hu,Hd,^,S) = VDFsiHu.Hd,^) + V'{Hu,Hd,^,S), (3.1.7) where [68] VoFsiHu^Hd,^) = Xu{hJHu -fif + UHj^Hd flf + A($*$ fl? + X^jHjHuHjHd + X'^jHjHdHjHu + SuHJHu^*^ + SdH^H^r^ + 8^d{iHlr2Hd^'^ + h.c), (3.1.8) and V'{Hu,Hd,^,S) = mlS-S^ + \s{S-S^f + \usHu'^HuS-S^ + XdsHj^HdS-S+ + (5s$*$5-5+ + AiHu^HdS-^^ + h.c). (3.1.9)

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56 The A term is the only lepton number violating interaction. For a large range of parameters it is possible to minimize the potential in (3.1.7) so that the vacuum expectation values (VEVs) of the scalars align without breaking charge and have magnitudes consistent with the values of the vector boson masses. As in the DFSZ model M^y=^\gv, Mz = \yJ7^^v, (3-1-10) where g and g' are the SU(2)"' and U(l)^ couphngs respectively, and v = ^vl + uj, where -^Vu and -^^vj are the magnitudes of the VEVs of the neutral components of Hu and Hj. We therefore require v ~ 250 GeV. For the model to be consistent with hmits on axion couplings, the VEV of the $ field, V^, must be Z 10^-^ GeV [74,75]. The analysis of the minimization of the potential works very much as in the DFSZ case. Assuming exact PQ symmetry (nonperturbative QCD effects turned off) and requiring that electric charge not be broken, we can without loss of generality parametrize the VEVs of the Higgs fields as by using SU(2)'^x U( 1 )^ x U( 1 global transformations. All the parameters in (3.1.11) are real. (Note: The phase of $ can only be rotated away by a U(l)'^'^ transformation. However, as the true effective potential, modified by QCD instanton effects, explicitly breaks PQ, this phase will be in principle calculable from the minimization of the full effective potenticil. Its presence is an indication of strong CP violation.) Sufficient conditions for minimization are the requirements that the potential be bounded from below: A„, A^, As, A > 0,

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57 \us>-2^/X^s, Arf, >-2v/AA, 6s>-2,/\sA. (3.1.12) In order to maintain charge invariance of the vacuum it is sufficient that the following parameters be positive: <5d, Kd^ >^'ud^rnl \us, 6, > 0, (3.1.13) and that the conditions A<^/45, (3.1.14) where . „ 0,9^ A = -Iml + SsV^ + Xusvi + Xds^'d > 0' (3.1.15) B = -4Xdfj + 6dV^ + {Xud + ^'ud)^'l>^^ be satisfied. Note that 6^d A are chosen to be real and positive. This can always be done by phase redefinitions of the Higgs doublets and the S field. The minimization conditions for the potential in terms of these parameters are AAfl 2hVl = Suvl + 8dvl 28^dVuVd, (3.1.16) 4A«(^ fl)vu + tuVuVl S^V^ + Kdvuvl = 0, (3.1.17) 4Arf(| fjV^d + ^dVdV} KdVuVl -f KdVdvl = 0. (3.1.18) Positive definiteness of the Hessian of the potential consistent with nonzero Vu, Vd and put rather complicated upper bounds on A^,^, 6u and Sd which we omit. With S^d > 0 ^^^^ minimization gives ^ = 0. All these conditions are sufficient for minimization and charge invariance in the original DFSZ model. We have noted that astrophysical bounds on axion coupUngs require Z 10^-^ GeV, introducing a scale much larger than the electroweak scale into the theory. In general, without extreme fine-tuning of parameters, this separation

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58 of scales will not be maintained when radiative corrections axe included [72]. The philosophy of this problem is that all extremely small parameters of a theory, e.g. small ratios of mass scales, must obtain radiative corrections of the same order or smaller to avoid such fine-tuning. This is the concept of naturalness and it ensures that the properties of the theory are stable against small variations in the fundamental parameters [76]. This kind of problem arises if there are coupling constants which are much larger than these ratios, as happens in GUTs [5], where gauge couplings of order 10"^ create radiative corrections to ratios of the order of 10"^^. It is not obvious how to assess the seriousness of the gauge hierarchy problem, since the origin of the parameters which enter in our effective low energy theories is unknown. One can envisage a scenario in which this effective low energy theory stems from a highly nonlinear theory, undoubtedly including gravity, which leaves us below Planck mass with a bunch of parameters which happen to be at the fixed points of the mother theory. This would mean that the input parameters obey peculiar relations among themselves, relations we would deem highly unnatural, although we would eventually find out that they have been fine-tuned by the mother theory! In the DFSZ model the relevant small quantity is This can be related to ratios of couplings via the minimization conditions of equations (3.1.17) (3.1.18) : Vl 2A,c2 + A,,.2 4A,(/2/t'2) ' ^ • ' where c = cos/3 = ^ , . = sin/^=^ ,< = tan^ = ^ . (3.1.21) V V Vu

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59 Equation (3.1.20) is augmented by a similar expression with u and d indices interchanged. By assuming Uu ~ /u ~ ~ fd^ we can satisfy these conditions with Su,Sci,8^id < 1, that is (3.1.22) where we have given the three couphngs between the $ field and the Higgs doublets the generic label 8 and the couplings Au, and X^d are given the generic label A. It is reasonable to take A ^ 10"^ since the A couplings get renormalized by gauge couplings. In fact all couplings with prefix A have this property. From the above we infer that 6 < 10~^^. Naturalness requires that the corrections to 6 be multiplicative, 6 ^ 6(1 + ©(IQ-^A)), 10"^ being a typical loop integration factor. From the diagrams of Fig. 15, we see that it amounts to demanding that Sg ~ 8 and ^ 8X, ie. A <> 10"^. Taking ^ > 1 and using the above typical values of couplings, we find that the bound (3.1.14) is wealcer than the hierarchy bound.

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60 (a) (b) ; Hi.u Figure 15. a) 0(Ss\us) correction to Su or 0{6s\ds) correction to 6d and b) 0(A2) correction to 6u, <5(f. The coupling A does not get renormalized by gauge couplings. However, there is no reason to assume it is small as A couplings give rise to multiplicative renormalization of the ^'s and can be 0(A). The intuitively correct result is that if all cross-couplings to the $ field are extremely small for both our model and for the DFSZ model, there is no technical hierarchy problem. 3.2 Unitary Gauge In this model we expect to have, in the neutral sector, three massive scalars, one massive pseudoscalar and a nearly massless axion, and in the charged sector, two massive scalars. It is convenient to work out their masses and other

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61 properties in the unitary gauge where the Nambu-Goldstone bosons have been expHcitly eaten. To find the unitary gauge we define new doublet fields: Hi = sHu + cHd, (3.2.1) H2 = cHu sHd, where H = iT2H* . Taking into account that after minimization ^ = 0 in (3.1.11), we find that (i^l) = 0, (^^2) = ^(°). (3.2.2) The physical unitary gauge is obtained by performing an SU(2)'^ gauge transformation that factors out the SU(2)"' x U(l)^ Nambu-Goldstone phases in H2, so that they may be eaten by the vector bosons. We define \'h ) (3.2.3) V2 V ^2 + ^' / where V is the SU(2)'^^ group element. If we write then h\ = c/i J s]r^ and X2 = cxu &Xd (3.2.5) are the Nambu-Goldstone bosons of electroweak breaking. They will become the longitudinal gauge bosons while the orthogonal combinations h\ = shf^ + c/jj and xi = + cxd, (3.2.6) will appear in the Lagrangian. Dropping primes in (3.2.3) and writing

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62 we can obtain the scalar masses in unitary gauge. The neutral mass term for PI, p2 and is ^p^M^p, where p^ = (pi, p2, P<^)We can use the typical relations obtained in the analysis of the hierarchy problem, i.e. 6 ~ Ae^ and A ^ \/6\ ~ Ae where e = < 1, to write this matrix in the simphfied form „ , ae^ be^ ce' . M2 = IKYl ( 6e2 a'e2 ) . (3.2.8) The constants a, a', 6, c and d are of order 1. Mixings between p^ and pi 2 are extremely suppressed relative to p\ and p2 mixing by a factor of e ^ 10~ . The mass eigenstates are given by p = O^P^ p, where O^^^ is an orthogonal matrix such that = O^^^^M^^^/.^^^jO'^), and is parametrized as /ciC2 -S3Ci.<*2 >siC3 -.^2C3Cl +^^35l\ O^''^ = I s\C2 C3C1 .'S352'S1 -C3'S2'5l -•^3^2 I , (3.2.9) \ .^2 -^3^2 C3C2 / where c,= cos^,and .s,' = sin^;. The angle ^1 ~ 0(1) while ^2,3 ~ 0{t). The physical fields are pi ~ pi cos ^1 — P2 sin^l , P2 ~ PI sin^i + P2 cos^l, (3.2.10) with masses m m2:.2AV;2. (3.2.11) Thus two of the neutrals have masses similar to the mass of the Standard Model Higgs, while the other is superheavy for A ~ 0(A). The xi and = 9^V^ mass matrix may be diagonalized to yield a massive pseudoscalar, x, with a Standard Model mass (due to (3.1.22) ) and the massless axion, a. These fields axe ^ ^ (3.2.12)

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63 with y2 y2 We require that > 30 GeV [77] and impose the bounds 10^-^ GeV < < 10^2 GeV, (3.2.14) where the upper bound corresponds to a energy density of axions sufficient to close the universe [78]. This and the hierarchy bound gives the unrestrictive Hmits 10"^^ Z i>ud ^ 10~^^ fo^^ ~ ^l^)expected the axion is mostly the field. Finally we work out the charged scalar mass matrix for applications to the calculation of the neutrino masses. The 5+ mixing from the A term in the potential gives rise to the mass matrix ^ ' ^AvV, + i6.V^ + iX,2v') [S-)' where A52 = ^usC^ + ^ds^"^The physical mass eigenstates are /)+ = cos o — 5"^ sin a, 5"*" = 1" sin Q + 5"*" cos a , with the mixing angle given by (henceforth we take off our hats) (3.2.16) tan2a = -^5 — = — r— 57f—^' (3.2.10 assuming > "i^^, where 2 ^ \' 2 I 1 c t2 '"11 = 0^'^^' + ^Ad^O^ ICS l,-.o 1 "4 =ml+'-8sVl+'-\s2r^. (3.2.18) "^"12 = ^A^'^'V

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64 The masses are m = ^(m?i + m^2 + \/Ki-">22)^+4(m22)2). Note also the useful identity (3.2.19) (3.2.20) In the limit ml » m]^ ~ Xv'^, the 5+ mass is determined by the free parameter 777.5. In this limit the masses are M]-ml ml^\\v\ (3.2.21) and Q w 4^ < 1. 3.3 Maiorana Neutrino Masses The AL = 2 Majorana masses for the left-handed neutrinos are generated at one loop through the /!| 5+ mixing. To see this we write the charged scalar terms in the Yukawa couphngs of (3.1.4) and (3.1.5) in terms of mass eigenstates of the charged fermions and scalars ^lepton ^ cot;ill7lM^'hR{h+cosa + S-^sma) ' V (0.6A) -It {uTf 2' f li-h'^ sin Q + 5+ cosq) + h.c, where a was defined in (3.2.17) and we have suppressed family indices. Here Z' is a new antisymmetric matrix Z' = A*Za'^ where y^^^ = -^AtM^^^B and M^^) is the diagonal charged lepton mass matrix. Since Z' has three independent elements, their phases can be absorbed by a redefinition of the phases of the lepton doublets. Hence we take Z' to be real. Note that there will be no CP violation coming from the neutrino mass matrix.

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65 The one loop diagram contributing to the neutrino masses is shown in Fig. 16. 'Li Figure 16. One loop contribution to Majorana mass in the unbroken theory. The induced neutrino mass matrix is [79,69] CeM-Te x,,) 0 z,r{x^ 1) 1 +h.c., --er(xe-l) ~>r(x;, 1) 0 ) K^r J L (3.3.2) where, in the hmit Ms. mi, > n?e, v/2 m? Mi 7770 = — cot /? sin 2aln( ) (3.3.3) and Xf = Xn = This matrix can be conveniently diagonahzed using perturbation in Xe and .r^^. In the Hmit Xe. x^ < 1 the matrix is 0 0 0 0 0 (3.3.4)

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66 and can now be diagonalized by a real orthogonal matrix O such that M^^^ = 0'^m{'^^ O. Diaeonalization yields one massless neutrino un and two degendiag ° erate Majorana neutrinos i>2,3 of mass ^2,3 = moyjz^r + ^It(3.3.5) (The negative mass is made positive by a chiral redefinition of Majorana eigenvectors.) The corresponding Majorana eigenvectors are ui = Ugi cos 9i/ — v^i sin 9i, -\c.c, h = ^"^^^ '^^ ^^^^^ ^ ^'^ " (3.3.6) v2 where (^1 = 0^1 and cos^.= ^. sin^. = -^=^^. (3.3.7) These last two eigenstates can l)e combined to form a Dirac neutrino. The matrix O appears in interactions mediating processes that violate individual lepton number conservation and is given by / cosdt/ —smSu ^ \ ^ 0= Tl''''^" Ti''''^' 72 . (3.3.8) l^--^sin^. -^cos^. The neutrino masses and mixing to first order in a;^, but still neglecting Xe, give rise to a mass for v\ given iDy mj,j ~ 777,/2<^rr^ sin2^;/, (3.3.9) where = zni cosdu. (3-3.10)

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67 Also 1/2,3 obtain a mass splitting equal to , with U2 being the lighter neutrino of the two. The corrections to mixing angles in this approximation are worked out in Ref. 80. Finally let us approximate the natural scale of neutrino masses in this model. Using (3.2.20) for the mixing angle we have mo = ^AVVcot/9-fLln(4)(3-3.11) {Any ^ Mi-mj^ mf^ Let us assume that ~ 0(10-2) or 0(10-^), cot /? ~ 0(1) and that AF^ ~ \v ~ 2.5 GeV. Now if we take A/s, vxh ~ 0(100 GeV) with their mass squared difference of order (100 GeV)^ and take the logarithm in (3.3.11) to be 0(1), then the scale of the massive neutrinos is mi,, ^ ~ 10 100 eV. For Ms ~ 1 TeV, mi/2 3 ~ 0.1-1 eV. Note of course that we only know that A < 10"^ and it could well be as small as 8. In this case, for squared mass differences of order (10 GeV)2, ^^.^ ^ 0(10"^) or 0(10"^) eV. 3.4 Experimental constraints In this section we give constraints on the parameters of our model that arise from phenomenological bounds. As with any multiple Higgs doublet model, charged scalars will mediate both flavorchanging neutral currents and new contributions to CP violating processes at one loop level. In our model these take place through the quark Yukawa coupling ^quark ^ _ ,-:^( /,+ cos a + 5+ sin Q )( tan iTrM^") Vc/i ^^^^^ + cot^T7lVM('''c//?) + h.c., where V is the standard CKM matrix. There are various ways to constrain the ratio ^ = tan/3. The best constraints on tan/3 come from an analysis of the J5 B system [81] combined with the analysis of CP violation constraints

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68 (Note that these bounds do not hold in the Supersymmetric extension of the Standard Model). The Bdi B^s iiiass difference is completely dominated by top quark contributions. A plot of m/, vs. tan/? appears in Fig. 7 of Ref. 82 and they obtain the approximate bound, o 69.1 1.88 ,„ , tan /3 < + -^"16 m/,, (3.4.2) for m< and m/, in GeV and m< > 65 GeV. After including ^ constraints they find tan/9 ^ 3.5 for 25 GeV ^ m/, < 300 GeV. We can carry over this analysis to our model in the limit q < 1 and Ms > m/,. Without this limit the complications of mixing spoil the bounds. We can also hmit tan 3 from below. Here it is useful to look at semileptonic decays of B mesons which are mediated by the leptonic Yukawa couplings of (3.3.1) . The best bound arises by considering contributions of charged scalars to the rate for B —> tuX [S3]. Although there are no direct experimental bounds on this decay, one may still find a constraint on cot /3 given the experimental values for the total rate B — ^ A' and the rate B — > euX . By requiring that B tuX does not saturate the total width, in contradiction to the observed B — > euX branching ratio. The authors of Ref. 83 obtain which is again apphcable in our model in the limit of small h'^-S'^ mixing. If one further requires that the Yukawa couplings proportional to cot 0 do not exceed the perturbativity limit then cot (3 ^ 130 [82]. We now have upper and lower bounds on this parameter. The most stringent constraints on the lepton family number violation parameters, ZiT, come from electron and muon number violating processes such as ^ — » 67 and and nonstandard contributions to 31eptons.

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69 By considering the tree-level contribution to fi decay and assuming that the fractional deviation of Cabibbo universality is no more than 10"^, we have the bound [9] ,l mi,). For Mg = 100 GeV we get Zer^fiT 10 ^. 3.5 Solar Neutrino Phenomenologv The solar neutrino problem is a long-standing problem to proponents of the standard electroweak and solar models and attempts to resolve it have fostered

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70 many extensions of these models. For two decades Davis and collaborators [10] have recorded a deficit (by a factor of 3.8) of primarily higher energy electron neutrinos from the sun, in their ^'Cl experiment. A solar neutrino deficit has also recently been measured by the Kamiokande II experiment [11]. The most popular solution to this problem assumes the correctness of the standard solar model and modifies the Standard Model by giving the neutrinos masses and mixings. One may add massive right-handed partners for the usual left-handed neutrinos as in some grand unified models, some singlet Majoron models, and in gauged SV{2)i xSU(2)/j models. The GUT models typically have a natural mechanism [85] for tiny neutrino masses. The left-handed neutrinos can also form Majorana masses as in other Majoron models and in our model. Just as in the model of Zee [69] we predict an extremely light electron neutrino and approximately degenerate // and r neutrinos. The resulting neutrino oscillation, for certain ranges of parameters, is consistent with solar neutrino data. Perhaps the most promising explanation of the solar neutrino problem in these scenarios is the ^IS^^" effect [12]. There vacuum neutrino oscillations are enhanced by the index of refraction produced by neutrino-electron scattering in the sun. Recent analysis of Kamiokande II data together with ^^Cl data by several authors [86,87] indicates that the nonadiabatic Rosen-Gelb solution [88] is favored over the original adiabatic solution [89]. The more recent work of Ref. 87 indicates the following constraints on the neutrino masses and mixing angles for the relevant solutions A7772 = m| m? ~ 10-^eV2, (3.5.1)

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71 with less than one order of magnitude uncertainty and sin2 2^^~10-^-^±°^ (3.5.2) Taking the nonadiabatic sohition, we fit the parameters of our model to these values. Recalling the analysis of section 3.3 we consider the approximation of a massless electron neutrino. (Recall that m,,^ « m^^ax^sm26i, where cr < 1 so that mj^i < 2.5 x 10~^mu2 for nonadiabatic solutions.) From equations (3.3.5) and (3.3.7) we have {Am^p = mo(4 + c2,)^ ~ lO-^eV, (3.5.3) and sin 2^, = ^ 10-0 -±0-2l (3.5.4) Recall that to obtain such a small neutrino mass without having Ms > 1 TeV, we had to assume A ^ <5, A larger only by an order of magnitude or so. It is simpler to consider the expression 1 1 M"^ (Am2)2(sin2^^)^ = cot liAV^-—^ ^^7V^n{-^), (3.5.5) as it can be related to the /./ — e-j constraint. For the nonadiabatic solution this obeys the inequalities ' ' • . ' 3 X 10"'' eV < (A77j^)^(sin2^i.)^ < 6 x 10"^ eV. (3.5.6) It is possible to find a '"natural" range of parameters which satisfy this bound assuming that A ~ 6. where 6 ^ 10~^^ came from our hierarchy considerations. One such solution takes 77?/, = 100 GeV and takes the values of cot /? and ZerZytT to be their upper bounds from (3.4.3) and (3.4.8) respectively. We

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72 take = 10^-^ GeV and work in the approximation a < 1. For this approximation to be valid we expect Ms ~ 0(1 TeV). For these values of parameters we get (Am2p(sin2^.)U4x 10-3-^^ln(4)(3-5.7) Mi ml mj; For Ms = lTeV,weget(A77T2)^ ~ 4x10-^ ^zj^ + 4r and ( Am2)i(sin 2^^.)^ ~ 6 X 10""* eV, just at the upper bound. The upper bound (3.4.8) on ZctZ^t is « 10"^ for this value of Ms. We can satisfy (3.5.4) and the other bounds by choosing say Zer ~ 10"^ and Zf,T ~ 10"^ so that the bound on their product is no longer saturated. Then (Ani^)^ k4x 10"^ and ( Am2)i(sin2^^)^ « lO"! By fitting parameters more specifically one can clearly obtain ranges of parameters that satisfy the nonadiabatic conditions. However due to the relatively large number of parameters to work with and weak or partial bounds on some of them, it is difficult to ol)tain ranges of allowed charged scalar masses.

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CHAPTER 4 THE RG IMPROVED EFFECTIVE ACTION We present a detailed discussion on the quantum corrections to the Effective Potential. In section 4.1 we discuss the loop-expansion and how the Renormalization Group can be used extend its validity. We discuss in detail the little known role that the unphysical constant term of the Effective Potential plays in the Renormalization Group Improvement, but this subtlety has been largely ignored in the literature. We review the calculation of the Renormalization Group Improved Potential for a massless and a massive scalar in sections 4.2 and 4.3 respectively. Finally in section 4.4 we attempt to extend those calculations to the case of two scalars. 4.1 Approximating the EP Up to an unphysical additive constant, the full EP is the sum of all the One Particle Irreducible (IPI) diagrams, with each loop typically contributing a factor of /lAln^. The loop expansion thus is an expansion in h and hence provides a natural way of approximating quantum corrections to the Effective Potential. There are three different methods that can be used to evaluate the loop corrections to the EP. The most straightforward method [13] is to sum all the 1-loop diagrams, 2-loop diagrams, etc. The drawback is that rarely is it possible to add up the infinite series of diagrams that enter at each loop level. Another method is the Shifted Field Method [90] (SFM) (see Appendix D). 73

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74 In this method one shifts the scalar fields by a constant {x) -* ^{x) + (f), leading to a theory with 4> dependent parameters. Then dY/d(f> is given as the one point function for ^. Since this involves only a finite number of diagrams the method is frequently applicable. The SFM has the added virtue that it uncovers the logarithmic factors that enter the EP, even when it can not be used to evaluate the EP. The third method [91] to find the 1-loop contribution to the EP, involves using the (-function technique to evaluate functional determinants appearing in the saddle point approximation to the generating functional. The C-function method is sometimes the most straight forward method of evaluating the quantum corrections, as will be illustrated in the two scalar model to be discussed later in this chapter. Whatever method is used to calculate the loop expansion, it only has limited applicabihty. Being essentially an expansion in hXlnp it is only valid for a small value of the logarithm and thus only holds in a limited range of field space. Coleman and Weinberg [13] found a way of using the Renormahzation Group to give an approximation without this limitation, the Renormahzation Group Improved (RGI) EP. The renormahzation process forces us to introduce an arbritrary scale fj into the calculation of the quantum corrections to the EP. If the EP is a physical quantity it should not depend on the renormahzation scale n and should obey a RGE where P = ,^. -7,0 = A^^. -7n.3m =,-^. (4.1.2) This equation has the solution V^^^ = f(77(/),A(0,^(t),m2(t))

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75 with t defined by and i 7^.dt'j . (4.1.4) where V is the functional form of the EP at some particular scale, e.g. V(//,A, dependence. Thus one has to treat this term carefully when improving the potential. Being unphysical the constant term does not necessarily obey the RGE. It is however possible to explore the freedom to add

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76 a constant to the EP to make it obey the RGE. Let V be the sum of all the IPI graphs. Then V is the EP up to a constant where Q is a arbitrary constant in (p. Expressing V in terms of the IPI Greens functions, F, we have V(/i, A, m, = A, m) i^-^''^^"^" n=l The Greens functions of course contain the physics of the potential, while d{fi,X,m) = V(//,A,m,(^ = 0) is the unphysical constant. It is important to note that Q, can get contributions from every loop order. The EP is n where Q is calculated in the loop expansion, while Q. is arbitrary. The Greens functions cire of course independent of // and thus obey the RGE, so dfi ' ' anwhere In order to make ^eff satisfy the RGE. one has to choose II such that DQ+D^l = 0. (4.1.5) By a clever choice of fi, one can find an approximation to the EP that can serve as a boundary condition for solving the RGE for the EP. One logical choice would be to eliminate the vacuum energy by choosing [92]

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77 where v is the vacuum expectation value of the potential. Another way of solving the problem is to treat the constant term just like another parameter of the model, include it in the Renormalization procedure making the zero-point Greens function, T^^^ = -(1^ + fi), obey the RGE. That is, we treat the unphysical part of the EP in the same way as the physical part. In this case the RGE for the EP becomes where The renormalization of the zero-point function guarantees that the loop expansion is the appropriate boundary function for the EP. In this case Eq.(4.1.5) is replaced by (D + + n) = D(i + ^^ = -[D + /^fi^)r^°^ = 0. 4.2 One Massless Scalar Let us quickly review the procedure in the simplest case, that is in the model of a massless self interacting scalar [13], where the interaction is described by the classical potential The result of the one loop calculation is (4.2.1) The approximation Vgyy ~ V^^^ + V^^^ is only vahd for A and In^ small.

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78 Notice however that the constant term is absent from V^^^ so to the 1-loop order is Cl = 0 We can hence use V^^) + V^^^ as a boundary condition for the RGE for the EP. In the single scalar model we have at one loop and thus \(t) = ^ and 2//i^e^^), and those terms we can ehminating by choosing t = ln(^//i). We then get, (notice the redefinition j^t ^ y. e""^/'*), V A ^'^^^l-3^Aln(02/A^2)4! which is valid for large j (i^ ) . This is the leading log or the RGI 1-loop approximation to the Effective Potential. 4.3 k Massive Scalar For a massive scalar [14,15] the tree-level potential is 4! 2

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79 and the one-loop contribution to the potential is The one-loop approximation \\jf ~ V(°) + V^^) suffers from the same limitations as in the massless case, e.g. it is only valid for small values of Aln-^-^^^^. In principle one might worry about other logarithmic factors showing up at higher loops, since there are now three dimensionfuU quantities involved, e.g. and m. However the SFM shows us that this is the only possible logarithmic factor. (See Appendix D) To extend the region of validity of our approximation to large values of the logarithm we repeat the steps taken in the massless case and put in the running parameters and choose i.i (or /) to kill the logarithmic factor. But in this case V(^) contains a constant term and we were careful to include a constant term in the tree level potential. The 1-loop running of the parameters is given by 3 • 1 Notice that Q = 0 is not invariant under the RG. This is the key point. It is not consistent to ignore the unphysical part of the potential when renormalizing the theory. Solving Eq. (4.3.1) gives the running parameters A (4.3.2)

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80 Putting the running parameters back into the one-loop potential we find the RGI approximation to the EP J . loo — 1 1-0 09 AA(j!>^ + m^ 3, V.// + + " + 3I^(2« ' 2' where t is completely arbitrary. Now we can choose 1. ^A^M^ t = -In ^ and kill all the logarithmic factors. Then by expanding t in terms of the parameters, we get an expression that is valid for small A and all values of (j). 4.4 Two Massive Scalars The simplest model involving two interacting scalars is described by the tree level potential = ^4 + Im?^? + ^t + + \^ll + H (4.4.1) Using the SFM we see that in this theory two logarithmic factors will contribute to the EP. These are In (BJ/zO and ln(B^///), where Bi \ = \ [\\\ + \i\ y(M;-M2)2 + 16A2<^2^2 (4.4.2) with \ (4.4.3) The actual calculation of the one loop contribution to the EP can be performed using the (-function method [91]. In the saddle point approximation the path integral expression for the generating functional becomes iy£ = iVexp(-5o) y Z)oiD(?)2exp ^ ^x{4>xA2)a(^^^

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81 with ^2 , d'-v d-v The Gaussian integral over the fields (f)\ and (i)2 can be done formally to yield U'^ = ^'exp(-5o)(det.4) The evaluation of the functional determinant, det(.4), is complicated by the fact that is a 2 X 2 matrix differential operator; however this operator diagonahzes to + D\ 0 0 + 5.2 and thus det(.4) = detl-c*"^ + B]) det(-c>2 + and we can use the result of the (-function calculation (for constant C) det(-c)2 + C) = exp C v(^) = -^[5j(h4-') + B^,(ln2|-|)] (4.4.4) We find B? 3 _ . . b:^ The first step in RGl the EP is to solve analytically the RGBs for the running of the parameters of the model. These are to 1-loop 167r' 167r IGtt IGtt 167r = 0, 16.2^ = 0. dt .5A2 di 2^X1 dt 2?^ = dt = (Ajn.2 2Am2). 16.2^ = (A.2m2 2Amf), at ut 1 1,4 4 , = (3Af + 12A^), IStt^-^ = (3A^ + 12A^), A(Ai + A-2 + SA). (4.4.5)

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82 Unfortunately the general solutions have so far eluded us. In order to solve the problem, at least in some special case, one is tempted to explore the possible symmetries of the model. There are two symmetries of potential interest. If Ai = A2 = 6A and mi = 7^2, the model exhibit an 0(2) symmetry. In that case the problem essentially reduced to the case of one dimensional field space and thus looses much of its appeal. If Ai = A2 and mi = m2, the model is symmetric under the interchange 01
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83 In this case we can solve the RGBs by expanding in Aq = \{t = 0). We find p{t) = fie^ , '^i(t) = (A2), = ^^0 + 2 m 10 + AoAio 2 10'"20 3 Ain 1 167:2 + m. 20 ^20 3 Aon i' 1-(1-T^^)^ 2m?nm2 Aio7 (i-i^OS ^10 i A20 J di (1 3 -^20 _ 1 (1 3 Ak -2 (4.4.8) where Aio = Ai(0), A20 = A2(0). m^g = m\[Q), m^Q = m^(0), and Hq = H{0). We can now plug the running parameters back into (4.4.3), (4.4.2), (4.4.1), and (4.4.4) and choose t = ln(B7/// ) to kill the Bi terms, that is the terms including ln(Bi^/7z2). We are then left with an approximation to the EP that is good if the A 's and ln(B^'/B7") are small. The expression obtained is not pretty but that is not to be expected for a Renormahzation Group Improvement of the Effective Potential in a model with more than one scalar. In the weak coupling Hmit we have been partly successful as we have increased the region of validity of our approximation of the Effective Potential.

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CHAPTER 5 CONCLUSIONS The aim of this study has been to explore physics beyond the Standard Model. First in chapter 2 by studying patterns suggested by Grand Unified Theories, within the Standard Model and in its Minimal Supersymmetric Extension. We worked in the MS renormalization scheme using the Standard Model two-loop renormalization group /3 functions. We evolved the parameters of the Standard Model, i.e., the gauge couplings, the quark and lepton masses, the Yukawa sector mixing angles and phase, and the scalar quartic coupling, from a mass scale of 1 GeV to Planck mass. We reviewed the extraction from experiment of the initial values for these parameters with specific emphasis on the extraction of the strong couphng constant and of the quark masses (especially those of the charm and bottom flavor). We treated the threshold effects appropriately, i.e.. rather than using naive step function implementation of thresholds, we implemented one-loop matching conditions for both gauge boson and fermion mass thresholds. In the Standard Model case, there are many unsatisfactory features, not the least of which is the failure of the gauge couplings to unify within experimental error, forming a GUT triangle. The simplest of the SU(5) relations, = 7777-, can only be satisfied some ten orders of magnitude from the scale of the GUT triangle. The other relations rrigi = 3me, 3ms = m^, tan^c = >/m.£//ms, and V^i = \/Tnc/mt, could be satisfied at 10^*^ GeV. The geometric mean relations we considered can also be simultaneously satisfied, but this requires a top quark mass greater than 200 84

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85 GeV. In the SUSY case the GUT triangle collapses and we achieved a striking agreement for the four GUT inspired relations considered. But, for this to be true, several things must occur: first Vcb must be larger than its presently central value; second the top quark mass must be around 190 GeV (if it is a bit hghter, then agreement dictates that Vd, should be larger still); third the Higgs mass should hover around 120 GeV. These conclusions are qualitatively correct if one demands maximum agreement. However, it is difficult to arrive at more definite numbers without an exhaustive analysis of the parameter space. In chapter 2 we introduced a new extension of the Standard Model which has the minimal field content to include an invisible axion and massive neutrinos. The cost of such an extension is the appearance of a large number of additional parameters. The potential was minimized so that the hierarchy of VEVs, V = 250 GeV and \ ^ Z lO'^-^ GeV, is maintained. We have seen that most of the couplings in the potential fall into two classes with typical values A ~ 0(10"^) and 6 ~ 0(10"^^). These naturally lead to the following masses. There are five neutral particles: two scalars with Standard Model (SM) type masses, one very massive scalar, one pseudoscalar with SM type mass and a massless axion (The axion will become massive due to instanton effects). The two charged scalars typically have masses of 0(10^ 10^ GeV) and we generally assume Ms > "?/,• A notable exception to these two classes of couplings is A ^ 10~^ which appears in the only explicit lepton number breaking term in the potential and enters in the charged scalar mixing angle. The ratio ^ = tan i3 is not fixed by the minimization but is constrained by the systems from above and by semileptonic B^ decays from below so that Q Q^^y ^ tana ^ O(^). The three new Yukawa couplings 2^,-^] which induce violations of lepton family numbers are bounded from above, these

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86 bounds depending critically on the value of MsBy unitarity considerations, Ms should be less than 1 TeV although it may obtain a larger mass from a more fundamental theory for which our model is a low energy version. For Ms ~ 1 TeV we found Zer^fiT ^ 10"^ from /i 67 data, while Zg/i ^ 10"^ from violations of Cabibbo universality and modifications of the SM V A structure. Arbitrarily fixing mi, = 100 GeV, Ms = 1 TeV we found that the nonadiabatic Rosen-Gelb solution to the solar neutrino problem could be obtained for ^er ~ 0(10-2), 2^^^ ^0(iO-l)andalsoA~ 0(10-15 _ iq-IG) ^ S for small charged scalar mixing. There is no reason why A cannot be as small as the other hierarchy-suppressed couplings. Unfortunately, it will be some time before our model is more tightly constrained by experimental bounds on two Higgs doublet phenomenology as well as on a whole host of lepton number and lepton family numl^er violating processes. We have yet to find a single Higgs boson. The verification of MSW oscillations in the sun would highly favor tiny, radiatively produced, neutrino masses. We add that grand unified models with a hierarchy of neutrino masses produced by a seesaw mechanism are also highly favored in explaining such small neutrino masses. Finally in chapter 4 we discussed in the Renormalization Group Improvement of the loop-expansion of the Effective Potential, with special emphasis on largely ignored subtleties about the role of the \'acuum term of the EP. We also made a partially successful attempt at improving the Effective Potential in a model containing two scalars. This effort was marred by technical and fundamental problems, that need to be (and will be) overcome.

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APPENDIX A THE SM /? FUNCTIONS In this appendix we compile the renormahzation group /? functions of the Standard Model. These have appeared in one form or another in various sources. We have confirmed their validity through a comparative analysis of the literature. Our main source is Ref. 6. Following their conventions, a = Q.^yJui, + Q.^Yd^dn + ^i$Ye^e« + h.c. ^K^^^f , (A.l) where flavor indices have been suppressed, and where Qi and are the quark and lepton SU(2) doublets, respectively: $ and $ are the Higgs scalar doublet and its SU(2) conjugate: ^={jO^ , $ = 772$. (A.3) Ufl, dn, and Cr are the quark and lepton SU(2) singlets, and Yy g are the matrices of the up-type, down-type, and lepton-type Yukawa couplings. The ^ functions for the gauge couplings are 87

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88 where < = In/i and / = 1, 2, 3, corresponding to the gauge group SU(3)c x SU(2)l X U(1)y of the Standard Model. The various coefficients are defined to be . = -3"^ ~ To ' 4 ^'3 = 11 3% ' with Ug = ^11 f IIn the Yukawa sector the 13 functions are where the one-loop contributions are given by ^1'^ = ^(Y.^Y. Y,/tY^) -f Y.2[S) + \gl + Sgl) , /?(^) = ^(Y,tY, Y.tY,) + l2(S) {\g\ + + 8,|) , (A.7) /3('U|YetYe + l2(S)-%'f+ff|), 2 with F2(5) = Tr{3Y,;Y, + 3Y^^Yd + Ye^Y^} . (A.8) In the Higgs sector we present 0 functions for the quartic coupling and the vacuum expectation of the scalar field. Here we correct a discrepancy in the one-loop contribution to the quartic coupling of Ref. 6 f = i'^i" where the one-loop contribution is given by /»l" = m^-(?.^o,|,A + 2,|,? + H.fp| + .,^) ^^^^^ + AY2{S)\-AH{S) , with H{S) = Tr{3(Y„tY„)2 + 3(Y^^Yd)2 (YetYe)^} . (A.ll)

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89 The /? function for the vacuum expectation value of the scalar field is This expression was arrived at using the general formulas provided in Ref. 6 for the anomalous dimension of the scalar field, choosing the Landau gauge. In the low energy regime the effective theory is SU(3)c x U(1)emWe employ the general formula of Ref. 93 to arrive at the 0 functions for the respective gauge couplings: ,5033^ , ^ 325^ ^ -2 2857 ffj and ,16 4 4 . ,64 4 It = t^"" ^ 9"^ + 3"' W ^ ^27"" 27"^' + w .64 16 , t^g'l (A.14) where n^, and ni are the number of up-type quarks, down-type quarks, and leptons, respectively. In Eq. (A. 21) we have also included the three-loop pure QCD contribution to the /i function of f/3 [94]. For the evolution of the fermion masses we used Ref. 95. It is known that there is an error in their printed formula [96]. Using the corrected expression, we compute the following mass anomalous dimension. The fermion masses in the low energy theory then e^•olve as follows:

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90 where the / and q refer to a particular lepton or quark, and where 9z 1 ^333 9S The superscript 1 and 3 refer to the U(1)em ^^'^ SU(3)c contributions, respectively. Explicitly, the above coefficients are given by 7f,) = 0 ^(,) = -8 7(0 = 7(0 = ° 7(",) = -30(1,,) + 1-9-"" + -^"<' + T'lK.,) >") = 33 404 40, ifgf = ^[^("« + "c/)'^ + (160C(3) + ^)(»" + "d) 3747] , where 5) is the electric charge of a given lepton or quark, and ((3) = 1.2020 ... is the Riemann zeta function evaluated at three. In the mass anomalous dimension for the quarks above, we have also included the three-loop pure QCD contribution 7^^^^ [94].

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APPENDIX B THE MSSM 0 FUNCTIONS ' Using some of the notation of Falck [7], the superpotential and soft symmetry breaking potential are as follows: + milieu + ml(fd + mi(U + Bui^y^d + h.c.) y gaugino = +^ MiXiXi + h.c. (B.l) Various 02 s have been omitted and a sum over the number of generations is implied. Also, hats imply superfields, tildes the superpartners of the given fields, and overbars the charge conjugate. B.l Gauge Couplings First the gauge couplings are 3 (B.1.1) 91

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92 where t = In/v and / = 1, 2, 3, corresponding to gauge group SU(3)c x SU(2)l x U(1)y of the Standard Model. The various coefficients are defined to be and with He 62 = 5 2ng , 63 = 9 2ng , / 38 6 88 \ / 9 9 0 \ (%) = 2 14 8 3 3 -17 0 11 VT5 3 68 1° 0 -54 j (Q/) = /•26 14 1S\ ' T T T ^ 6 6 2 4 4 0 , with f = u , d , e Infi. (B.1.2) (B.1.3) (B.1.4) B.2 Superpotential Parameters: // and the Yukawas In the following we list the functions for the parameters of the superpotential. ^ = Tr{3Y,tY, + 3Y,;Yrf + Ye^Ye} 3(^pf + g^) ] . (B.2.1) In the Yukawa sector the 3 functions are (B.2.2) where the one-loop contributions are given by ^1^) = SY^ty^ + Y^Y^ + 3Tr{Y,;Yu) (y^^l + + y^I) , ^i^^ = SYrftY^ + YjYy + Tr{3Y/Yrf + Ye^Ye) {-gj + 3gl + -^gj) , 16 = SYe^Ye + Tr{3Y^tY^; + y^ty^ j ihi + ^di) (B.2.3)

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93 B.3 Higgses' Vacuum Expectation Values The evolution of the vacuum expectation values of the Higgs's is given by where the one-loop contribution is given by 7i^^ = 7(75l+i?2)-3Tr{Y,;Y„} , '<^u 4^5^^ (B.3.1) = l^h' + 3Tr{Y,tY,) Tr{YetYe} •

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APPENDIX C EXPLICIT FORM OF <5(/i) In Ref. 48 the radiative corrections term 6{n), from Eq. (2.2.66), is derived. In this appendix, we present its explicit form as it appears in this reference except for some minor notational changes. In the following, 5 and c refer to sm6iv and cos6^y, respectively. Also, is defined to be the ratio Mfj/M^. %o = ^^{c.fi(^-^') + /o(^,/o + r\f-iU,/0} , (c.i) where the various functions are defined as follows: Mi'") = 61"^ + >i iZ(i ) Z( Inc' + ^(f ^ H /0(«..) = -6>ni|ll+2c^-2|H3.n^+2Z |) Uln[(l + A)/(1-A)] , {z<\) , (C.3) A=\l4~|^ . with 94

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APPENDIX D THE SHIFTED FIELD METHOD The Shifted Field Method [90] was used in this work. In this appendix we give an example of its uses. In the massive scalar model the tree-level potential is 4! 2 When the theory is shifted o 0 + O V(0) _ + A^^3 + 1( 1 ;^^2 ^ ,„2)^2 ^ ( ^ ^ ^^^^^^^ The linear term is -f and the cubic coupling is thus to one loop the one-point function is d(f> 3! 3! y (27r)^ 3! J (27r)4g2+ lA<^2 + „,2 This gives We also see that the only source of a logarithmic factor in the calculation at any loop order is the propagator p^ + M?, where M^ = ^\4>^ + m^, and thus the only logarithmic factor that can come into the Effective Potential is 95

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BIOGRAPHICAL SKETCH The author was born in Iceland on January 28, 1962. He received a B.S. degree in physics at the University of Iceland in 1985. He then continued his education in the United States, first at Columbia University in New York City for two years, where he earned a Master of Arts in physics in 1987, and then later at the University of Florida, where he developed a keen interest in theoretical particle physics. After completing his studies at the University of Florida he will return to his native country to teach physics. 102

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pierre MTKamond, Chairman Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. c Charles B. Thorn Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, m scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pierre bikivie Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Richarti D. i-'ield Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Christopher Stark _ Associate Professor of Mathematics ]

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This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1993 Dean, Graduate School


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