Renormalization group study of the minimal supersymmetric extension of the standard model with softly broken supersymmetry


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Renormalization group study of the minimal supersymmetric extension of the standard model with softly broken supersymmetry
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vi, 131 leaves : ill. ; 29 cm.
Castaño, Diego J., 1961-
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Physics thesis Ph. D   ( lcsh )
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Thesis (Ph. D.)--University of Florida, 1993.
Includes bibliographical references (leaves 125-130).
Statement of Responsibility:
by Diego J. Castaño.
General Note:
General Note:

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A Lela

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in 2011 with funding from
University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation


I would like to acknowledge all of those who have influenced and helped

me in my physics career. I thank my fellow students, particularly Haukur

Arason, Sam Mikaelian, and Eric Piard, with whom I have shared illuminating

conversations, both in physics and outside. I have learned a lot of physics

from my talks with them. I thank all of my teachers and professors for their

excellent instruction through the years. I especially thank Pierre Ramond. I

have gained much from his tutelage. In being a great particle physicist, he has

given me a goal for which to strive. And last but not least, I thank my wife

and my family for their continual support, understanding, and encouragement.



ACKNOWLEDGEMENTS ..................... iii

ABSTRACT . . . vi


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


3 THE STANDARD MODEL ................. 8
3.1 aI(Mz), a2(Mz), and a3(Mz) . 8
3.2 Yukawas . . . 13
3.3 Quark and Lepton Masses . .. 15
3.4 Top and Higgs Masses ... .............. 17
3.5 Vacuum Expectation Value . .... 19
3.6 Thresholds . . 21
3.7 Analysis and Results . . 22

4 THE MSSM ...... ........... ...... .37
4.1 The Supersymmetric Standard Model . .. 37
4.2 Procedure .. . .. .. ....38
4.3 One Light Higgs Limit ... .. ....... .... 39
4.4 Initial Data . .. .. 40
4.5 Analysis and Results . . .. 41
4.6 Comments ....................... 43

5.1 Minimal Low Energy Supergravity Model . 49
5.2 Radiative Electro-Weak Breaking. . ... 51
5.3 Sfermion Masses .......... .. .. ...... .54
5.4 Higgs Masses .. ... ........ .... .... .54
5.5 Chargino Masses .. . 55

5.6 Neutralino Masses ...............
5.7 Boundary Conditions at M . .
5.8 Numerical Procedure . .
5.9 Thresholds . . .
5.9 Analysis and Results . .

6 CONCLUSIONS .................



B.1 Gauge Couplings ...............
B.2 Anomalous Dimension of the Scalar Field .
B.3 Yukawa Couplings .... ............
B.4 Thresholds . . .

C THE MSSM 0 FUNCTIONS ............




G 'METAPLECTONS' ...............
G.1 Introduction . . .
G.2 Anyons . . .
G.3 The Metaplectic Representations of OSp(r/2m, R)
G.4 New Representations of Anyons . .
G.5 Conclusion . . .


BIOGRAPHICAL SKETCH ........... ....

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. 85
. 88

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. .. 104

. .. 107

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.. 123

. 125

. 131

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



Diego J. Castafio

May 1993
Chairman: Pierre Ramond
Major Department: Physics

Using the renormalization group all the couplings of the Standard Model

and its minimal supersymmetric extension are run to two loops, taking full ac-

count of the Yukawa sector. It is found that in the standard model the gauge

couplings fail to unify, whereas extending the model through supersymmetry

achieves unification. Bounds are placed on the top quark mass by requiring

equality of the bottom and 7 Yukawa couplings at the scale of unification. In its

simplest form, the supersymmetric model has a degenerate superparticle spec-

trum, and for ISUSY = 1 TeV, and Mb = 4.6 GeV, one finds 139 < Mt < 194

GeV, which remarkably satisfy the p-parameter bound. The corresponding

bounds on the Higgs mass are found to be 44 < MH < 120 GeV. The model is

then coupled to supergravity yielding a richer superparticle spectrum and mak-

ing it more accountable to experiment. This model has the attractive feature

that the electro-weak symmetry is radiatively broken. For the special case in

which global supersymmetry breaking arises solely from soft gaugino masses,

Mt is expected to be less than ~ 130 GeV. A higher upper bound is predicted,

if A/h is at the lower end of its experimental uncertainty.


In the last few years, it has become apparent, using the ever increasing ac-

curacy in the measurement of the strong coupling, that supersymmetry (SUSY)

affords an elegant means to achieve gauge coupling unification [1] at scales con-

sistent with grand unified theories [2] (GUTs). Whereas in the standard model

(SM) the three gauge couplings unify "two by two" forming the "GUT trian-

gle," in the simplest minimal supersymmetric extension of the standard model

(MSSM), these gauge couplings spectacularly unify at a point (within the ex-

perimental errors in their values). Given that the scale of unification in these

models is generally above the lower bound set by proton decay, the so-called

SUSY-GUTs have gained increasing interest. Constraints coming from Yukawa

coupling unification in supersymmetric SU(5) and SO(10) models can be used

to yield interesting predictions for various low energy parameters including the

top quark mass [3,4].

This analysis employs the renormalization group (RG) [5] to extrapo-

late the parameters of the standard model and of its minimal supersymmetric

extension to unexplored scales [6]. With the ever increasing precision of ex-

periment, the inclusion of two loop effects is crucial. The complete Yukawa

sector contribution is also included. In the standard model case, thresholds

effects are implemented in both a simple and a more involved method. The

results of these two methods are discussed. Numerical methods are used to

evolve the parameters to different scales using the 3 functions found in the


literature [7], and the results are plotted for representative values of the Higgs

boson and top quark masses. The running of the quark and lepton masses and

of the Cabibbo-Kobayashi-Maskawa (CKM) angles is generally given through

the running of the Yukawa matrices (even to one loop). In this work, the quark

masses and CKM angles are evolved by diagonalizing the Yukawa matrices at

every step in the Runge-Kutta method used in solving the 03 functions. Often

it is assumed that the contribution of the Yukawa couplings matrix is given

essentially by the top quark Yukawa since it is much larger than the others.

Sometimes a better approximation is made by keeping only the diagonal en-

tries. The present numerical technique represents a small improvement over

these methods.

Chapter 2 consists of a basic introduction to renormalization and renor-

malization scheme dependence.

Chapter 3 addresses the standard model. A review of initial data extrac-

tion from experiments is presented. Many excellent reviews may be found in

the literature, e.g., Marciano's [8] or Peccei's [9]. The determination of the

standard model gauge couplings is discussed as is the initial data extraction of

the Yukawas and the CKM angles. The extraction of the quark masses from

data is also discussed. This is a complex issue well known to be marred by

the nonperturbative nature of QCD. Hence, in the low energy regime, the pure

QCD three-loop contribution is included in the analysis of the running of the

quark masses. Initial data for lepton masses follow the quark discussion. Then

the extraction of and constraints on the physical top and Higgs masses are

considered. The scale dependence of the renormalized scalar vacuum expecta-

tion value is addressed. The method used to obtain the values of all running


parameters at the same initial scale is described in Appendix E. In the fol-

lowing section, threshold effects are discussed. Finally, a quantitative analysis

of the results is presented. The effects of using one loop versus two loop 0

functions are contrasted and of including a proper versus a naive treatment of

thresholds. Plots of all the running parameters over the entire range of mass

scales are included and also used to display the effects discussed. Furthermore

some tables are presented with actual numerical differences associated with

these effects.

Chapter 4 addresses the minimal supersymmetric extension of the stan-

dard model. A brief discussion of supersymmetric models is presented. The

procedure by which top quark mass bounds are determined using the equality

of the bottom and tau masses at the scale of unification is discussed as well

as the one light Higgs limit employed. The initial data for gauge couplings

and quark masses are presented. This is followed by results on the top quark

and Higgs boson masses. Finally, a discussion on how to improve the results

is presented.

Chapter 5 implements some of the improvements discussed at the end of the

last chapter. Namely, the supersymmetric two loop 3 functions are included.

Also, soft symmetry breaking terms are added. These lead to a nondegenerate

superparticle spectrum as well as to the radiative breaking of the electro-weak

symmetry. Similar analyses have appeared in the literature but use one loop

3 functions and the tree level Higgs potential [10]. In Chapter 5, a brief

discussion of the effective one loop potential is presented. The mass formulas

for the sfermions, Higgses, charginos, and neutralinos are given. The boundary

conditions at the unification scale in these minimal low energy supergravity

models are discussed. Next the numerical procedure employed is described.


The treatment of thresholds and the "special" form of the 0 functions needed

is discussed next. Finally, some preliminary results are presented.

Chapter 6 contains the conclusions of this work.

Appendix A contains all the needed two loop 3 functions for renormaliza-

tion group studies of the standard model. Appendix C contains those of the

minimal supersymmetric extension of the standard model. Many of these 3

functions have yet to appear in the literature in as general a form. Appendix

B presents some examples in calculation of the3 functions of Appendix C.

Appendix D deals with issue of the vacuum expectation value's 3 function. A

toy model is used to gather insight into the problem. The numerical solution

routines used extensively throughout this work are discussed in Appendix E.

Appendix F contains a cumbersome formula needed in the extraction of the

Higgs boson physical mass. Finally Appendix G presents a novel use of the

spinor representations of the orthosymplectic Lie supergroup OSp(r/2m, R)



Renormalization is a reparametrization of a theory which renders Green

functions and physical quantities finite order by order in perturbation theory.

A specific choice of renormalized parameters defines a renormalization scheme.

The physics is, of course, independent of how the theory is renormalized. A

common way of relating bare and renormalized parameters is

go = g g (2.1)

where go is the bare parameter, g is the renormalized parameter, and 6g is

the counterterm. Fixing the counterterms by requiring them to consist only

of the infinite terms needed to render the theory finite defines the minimal

subtraction (MS) prescription [12]. A feature of the MS scheme is a mass

scale [ which enters in the process of regularizing divergent integrals using

dimensional regularization. Furthermore the unit of mass [i is used to keep

couplings dimensionless when continuing to d dimensions in the dimensional

regularization procedure. For example, if Eq. (2.1) represents any of the three

gauge couplings of the Standard Model, then p is introduced as follows to keep

them dimensionless

go(e )-~ = g 6g (2.2)

where g is a constant parametrizing the arbitrariness in the finite parts of diver-

gent integrals in dimensional regularization and e = (4 d)/2. Equation (2.2)

defines a family of MS schemes. Choosing e = 1 is the simplest MS scheme



which was described above. Choosing e2 = eYET/4r, where yE = 0.5722...

is the Euler-Mascheroni constant, defines the so-called modified minimal sub-

traction (MS) prescription [13]. This scheme is the most commonly employed

in QCD calculations, and it is the one adopted here. The free parameters of

the Standard Model in the MS schemes are 1 dependent. Their p evolution

is governed by the / functions of the renormalization group. Moreover, these

running parameters are not in general equal to their corresponding physical

values (consequently, for the masses, a convention is adopted wherein upper

case M refers to physical values and lower case m denotes MS values). This

is to be contrasted with the on-shell renormalization scheme in which, for

example, the renormalized masses equal their physical values and the renor-

malized electromagnetic coupling equals the fine structure constant. However,

the MS schemes have the attractive characteristic that the / functions are p

independent and therefore particularly simple to integrate. Physical quanti-

ties P({gi(pi)}, p) expressed in terms of p and the running parameters of the

theory, {gi(7)}, must be 1 independent

d 0 8
Y P({gi(t)}), ) = (i + 1it3 )P = 0, (2.3)

where the 3i are the 3 functions. The two-loop / functions of the Standard

Model have been collected in Appendix A.

As mentioned above physical quantities are renormalization scheme inde-

pendent. However, this assumes that calculations can be done without ap-

proximation. In reality, calculations are only perturbative approximations and

these do depend on the renormalization scheme.

Consider massless QCD with the one dimensionless coupling, as. Suppose

a physical quantity, P. is calculated to nth order in perturbation theory in two


renormalization schemes, then the nth order approximation is given by

2 n-1
Pn = as(Po + Plas + P2a2 + + Pn-las) (2.4)

in one scheme and by

p t I + 2 in-(2.5)
Pn = s(Po + Plas + P'2as +. 2 + Pn-las ) (2.5)

in the other scheme. The two couplings, defined in their respective renormal-

ization schemes, can be related to each other

= as(1 + blas + b2a2 + + bn-an-1) (2.6)

Substituting Eq. (2.6) into Eq. (2.5) yields

Pn= s(p" + p'as + p2a + + Pn-la )
+Tas(pas +... +p2n-la"1 )(
As this is an approximation to the same quantity, P, the first n terms can be

identified with Pn. Therefore the two approximations differ by terms of higher

pn 2n- 1). (2.8)
P n- P= as(pns +' ... + P2n- (2.8)

In QCD where as is large, this difference may be large and thereby lead to

renormalization scheme dependence problems. In the Standard Model as in

QED where the couplings are small, this is not so great a problem. In QCD

where the strong coupling as is large, there will be renormalization scheme

dependence problems. In the electroweak model as in QED where the couplings

are small, this is not so great a problem.


3.1 ai(MZ), ag(Mz), and ca(MZ)

The determination of the SU(2)L x U(1)y couplings proceeds from the

Standard Model relations

a ) g ) _C2 Wa(P)
4 cs2 W() (3.1.1)
( = g2(j) g a(y)
a2 47 sin2 W() '

where a(y/) = e2(p)/47 and C2 is a normalization constant which equals 1

for the Standard Model and equals i when the Standard Model is incorpo-

rated 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(/y) and sin2 Ow(i)

in the renormalization scheme employed (i.e., MS). The electromagnetic fine

structure constant (ar-1 137.036) is extrapolated from zero momentum scale

to a scale p equal to Mg in the present case. In pure QED with one species

of fermion with mass m the MS renormalized vacuum polarization function is

given by

1(m) A2 q2
In(q2) = 3 (In 6 dx x(1 x)ln[l x( x) ]) (3.1.2)
37 m m-
The renormalized coupling a(y) is related to the fine structure constant aem

as follows

aem () (3.1.3)


In the Standard Model where there are many species of charged fermions and
charged gauge bosons, Eq. (3.1.3) generalizes to [14]

a-(p) = ae 2 Q2 0( mf)ln + (3.1.4)
37f mf 6Xr

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

nI(0) = (IIh(0) II(q2)) + h(q2) (3.1.5)

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 [14]. Using the optical theorem, one may write

Im{Ih(s)} = -s-- (e+e- hadrons) (3.1.6)

where s is the square of the center of mass energy. For the process e+e -+

+fi~-, the cross section is calculated to be (taking m, = 0)

(ee- -+ ) = (3.1.7)

In terms of the ratio of these two cross sections,

S(e+e- --+ hadrons)
R(s) = (e+ -+- -), (3.1.8)
o(e+e- -+ M+y-)

one may write Eq. (3.1.6)

Im h)} (3.1.9)_
Im{h (s)} = R(s). (3.1.9)


Using an unsubtracted dispersion relation for IIh (q2) the combination (IIh(0)-

IIh(q2)) can be expressed as
q2 aem0 R(s s)
nho) _h(q2) = ()(3.1.10)
37r s(q2 s)

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

a value

a-(Mz) = 127.9 0.3 (3.1.11)

The process independent, renormalized weak mixing angle sin2 OW of the

on-shell scheme is defined to be

sin2 O = 1 (3.1.12)
where MW and MZ are the physical masses of the W and Z gauge bosons.

Knowing the precise values of the W and Z boson masses and using the equa-

tion above provide one way of extracting the value of sin2 OW. Alternatively,

the bare relation involving the low energy Fermi constant measured in muon

decay and the W boson mass

Go e2
o e8sin(3.1.13)
v/2 8 sin2 Wo2 y! 0

may be corrected to order a and rewritten [15,16]

7raem 1 1
Mw = MzcosOw = (-- ) (3.1.14)
VGu sinyw(1 Ar)

with (raem/V'GM) = 37.281 GeV and Ar is a parameter containing order

a radiative corrections and which depends on the mass of the top and Higgs.

The radiative corrections represented by Ar can be viewed as accounting for

the mismatch in the scales associated with the parameters of the relation. G"

and aem are low energy parameters whereas .M1 and sin2 9y are associated


with the electroweak scale. One can absorb the radiative effects using the

renormalization group by replacing G, and aem with corresponding running

parameters at MZ
(MZ1 (3.1.15)
G'G(MZ)M2 v sin2 OW
Combining Eqs. (3.1.14) and (3.1.15) gives

em G,(MZ)
Ar. 1 e G(M) (3.1.16)
a(Mz) Ga

Using Eq. (3.1.4) and the fact that G,(Mz) ; Gp (see Section 2.8) gives an

estimate of the size of the radiative corrections

Ar ; 0.07 (3.1.17)

For large values of Mt and MH (Mt, MH > MZ) [15,17]

aem 3em Mt2 11aem M2
Ar 1 ae 3aem M + 1In (3.1.18)
a(Mz) 167rsin4 w M 487rsin2 w M(

A third way of extracting sin2 OW is from neutral current experiments, among

which deep inelastic neutrino scattering appears to provide the best determi-

nation. A running sin2 Ow(P) may be defined in MS and differs from the above

sin2 OW by order a corrections. The MS running W boson mass mw(Ji) and

the corresponding physical mass MW, identified as the simple pole at q2 = M

of the W propagator, are related as follows

M2 = m () + Aw(Mw, p) (3.1.19)

where A 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 sin2 OW(t)

2sin w (
sin2 0W(z) -- 1P (3.1.20)


Equation (physmw) and its Z analog may be combined with Eq.(3.1.20) to

sin2 OW( -) 1 cos2 Aw (MA, j(t) AT (M
s=w() = 1 s (W ) (3.1.21)
sin2 0W sin2 W M| M 2

An explicit expression relating sin2 BW and sin2 0y(My) is given in Ref. 18.

Another relation for sin2 OW(t) may be arrived at directly linking it to MZ
[19] or MW [20]. In particular, if one chooses MW as the input mass, then
one introduces a radiative correction parameter AW such that

sin2 Ow(Mz)(1 Ary) = sin2 Ow(1 Ar) (3.1.22)

from which it follows that

sin2 Ow(Mz) = .21 (3.1.23)
My(1 AiW)
Similarly one can introduce a radiative correction Aig if one chooses MZ as
the input mass

sin2 8w(MZ)cos2 O(MZ)(1 Aiz) = sin2W Bcos2 OW(1 Ar) (3.1.24)

A fit to all neutral current data gives

sin2 OW(MZ) = 0.2324 0.0011 (3.1.25)

for arbitrary Mt [21]. Using these values of a(Mz) and sin2 G8(Mz) yields

ai(Mz) = 0.01698 0.00009,
a2(MZ) = 0.03364 0.0002.
The value of the strong coupling has not been determined experimentally as
well as a1 and a2. The nonperturbative nature of low energy QCD leads to rel-

atively large uncertainties in its determination. Different processes for the ex-

traction of a3 yield significantly different results. Unless otherwise stated, the


value of a3 used in the runs will be that obtained from a Gaussian weighted av-

erage of the results of several processes, including e+e- scattering into hadrons

[22], heavy quarkonium decay [23], deep inelastic scattering '24], and e+e-
scattering into jets [25]. The value found is [6]

a3(Mz) = 0.113 0.004 (3.1.27)

3.2 Yukawas

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 to be plotted 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


Machacek and Vaughn's [7] convention are used where the interaction

Lagrangian for the Yukawa sector is

C = i~YutQ + dYdtQ + eYeDtL + h.c. (3.2.1)

The Yukawa couplings are given in terms of 3 x 3 complex matrices. After elec-

troweak symmetry breaking, these translate into the quark and lepton masses

V2 me 0 0)
Ye = 0 my 0 ,
v 0 mr
V2 mdd 0 0
Yd 0 m 0 (3.2.2)
v 0 0 mb
5 m 0 0 V,
Yu 0 me 0 V ,
v 0 0 mt


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

j ~ ,U L-y VdL (3.2.3)

It is a unitary 3 x 3 matrix often parametrized as follows

( cl 31c3 1i33
V = -lC2 clC2C3 s2s3ei cC2S3 + s2c3e6 (3.2.4)
-S1S2 clS2c3 + c2s3e6 clS2S3 c2c3ei

where si = sin0i and ci = cos i, i = 1,2,3.

The entries of the parametrized CKM matrix can be related simply to

the experimentally known CKM entries. The particle data book [21] 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
IVI= 0.218-0.224 0.9734-0.9752 0.030-0.058 (3.2.5)
\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.

These bounds are arrived at by finding values for the four angles such that the

entries of the CKM matrix obtained from these satisfy the conditions imposed

by Eq.(3.2.5). One finds

0.2188 < sin 0 < 0.2235 ,

0.0216 < sin 2 < 0.0543 (3.2.6)

0.0045 < sin 03 < 0.0290 .

However, the accuracy with which IVI is known does not constrain sin 6. A set

of angles {01,02, 3,6} 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 (see Section 2.5), in that it

is not clear at what scale the chosen initial values for these angles should


be considered known. However, it is observed that for the whole range of

initial values, the running of the mixing angles is quite flat, with a perceptible

increase in 02 between MW and the Planck scale for higher top masses. This

is in accordance with the angles being related to ratios of quark masses, and

therefore, the exact knowledge of that scale (or lack thereof) is not as critical

as might be feared a prior.

3.3 Quark and Lepton Masses

There are large theoretical uncertainties in the extraction of the masses

of the three lightest quarks from experiment. They are determined by chiral

perturbation techniques and QCD spectral sum rules [26,27]. In the following,

the MS running masses at 1 GeV of the three lightest quark will be taken to

be [6]
mu(1 GeV) = 5.2 0.5 MeV ,

md(l GeV) = 9.2 0.5 MeV (3.3.1)

ms(1 GeV) = 194 4 MeV ,
For the charm and bottom, the nonrelativistic bound state approximation may

be applied. One speaks of physical masses and associates these with the pole of

the quark propagator. A weighted average, based on results from J/1 and T

sum rules [28] and from heavy-light, B and B*, D and D* meson masses and

semileptonic B and D decays [29], yields the following physical and running

masses at 1 GeV for the charm and bottom [6]

Me = 1.60 0.05 GeV, mc(1 GeV) = 1.41 0.06 GeV ,
Mb = 4.89 0.04 GeV, mb(l GeV) = 6.33 0.06 GeV ,

Although the associated errors on these averages are restrictive, there is larger

uncertainty in the actual central values. For example, in the bottom quark,

values as small as mb(l GeV) = 5.7 GeV and as large as mb(l GeV) = 6.5


GeV are acceptable. In the case of the lighter quarks, their mass ratios are

known more accurately than the actual value of their respective masses. The

ratios md/mu = 1.8 and ms/md = 21 are widely accepted.

The physical (pole) masses of the leptons are very well known [21]

Me = 0.51099906 0.00000015 MeV,

My = 105.658387 0.000034 MeV (3.3.3)
M = 1.7841+00027 GeV .
M -0.0036

These masses represent an example of a physical quantity as discussed in Chap-

ter 2. Indeed expressed in terms of the renormalized parameters (i.e., masses

and couplings) of the theory, the physical mass is just the simple pole of the rel-

evant field's propagator. Suppose S(f; g, m, p) is the renormalized propagator

of some fermion with a simple pole at #j = M(g, m, pi) such that

S(O; g, m, P) = ,(g,m, ) (3.3.4)
# M(g, m, P)

with finite residue, z(f/ = M; g, m, p). The relevant renormalization group

equation is

G( + 0 + mym- + 27)S(Y; g, m, p) = 0 (3.3.5)

where 7 is the anomalous dimension and 7m is the mass anomalous dimension.

Inserting Eq. (3.3.4) in Eq.(3.3.5), then multiplying the resulting expression by

(/ M)2, and lastly setting M = M gives the renormalization group equation
for M

d 9 a a
-M(g, m, I) = ( + + mym )M(g, m, ) = 0 (3.3.6)
dy apy og am

These values are used to determine initial data for the running masses. Some

authors neglect QED corrections and use the physical values for the running


values at ~ MiZ, which introduces only a small error. By calculating the one-

loop self-energy corrections, one arrives at a QED relation between the running

MS masses and the corresponding physical masses

mt(p) = M[1 3(In)1n2 + 4 (3.3.7)
47r ml 3

Choosing p = 1 GeV as in the quark mass case and using Eqs. (3.3.7) and

(3.1.4) yields the running lepton masses (taking mi = MI in the log term

above is an appropriate approximation to order a)
me(1 GeV) = 0.4960 MeV ,

mp(l GeV) = 104.57 MeV (3.3.8)

mr(1 GeV) = 1.7835 GeV .

3.4 Top and Higgs Masses

The Higgs boson and top quark masses have not been measured directly at

present; however, their values affect radiative corrections such as Ar. Consis-

tency with experimental data on sin2 OW requires Mt < 197 GeV for MH = 1

TeV at 99% CL assuming no physics beyond the Standard Model [30]. Pre-

cision 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+41 GeV, for all allowed values of MH [31]. Recent

direct search results set the experimental lower bound Mt Z 91 GeV. As for

the Higgs, the analysis of Ref. 31 gives the restrictive bound, MH $i 600 GeV,

if Mt < 120 GeV, and MH < 6 TeV, for all allowed Mh. Since perturbation

theory breaks down for. MH ; 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 [32].


In the present analysis, initial values of the MS running top quark mass

mt and of the scalar quartic self-coupling A at MZ are chosen arbitrarily (con-

sistent with the bounds quoted above). As noted earlier in Chapter 2, 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 this analysis should be that of experimentally relevant,

physical masses. Therefore, formulas similar to Eq. (3.3.7) relating MS running

parameters to physical masses are needed. To calculate the physical or pole

mass of the top quark, the following equation is used [33]

1 4 as(Mt) MA as(Mt)2 (3.4.1)
Mt = 1+4 (M- [16.11 -1.04 (1 ()2 (3.4.1)
mt(Mt) 3 r i=1

where Mi, i = 1,..., 5, represent the masses of the five lighter quarks. Like-

wise the physical mass of the Higgs boson can be extracted from the following

relation [34]

A(Mp) = G MH(1 + 6)) (3.4.2)

where S(/p) contains the radiative corrections. Its form is rather elaborate and

it is relegated to Appendix B. Equations (3.4.1) and (3.4.2) are highly nonlin-

ear functions of Mt and MH, respectively. Their solution requires numerical

routines that are described in Appendix C.

At the one loop level, the gauge couplings are 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 one expects the mass and mixing relations just

described 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 quark and lepton 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 [35]. The discovery

of the Higgs with mass outside these bounds would be a signal for physics

beyond the Standard Model. The graphs in Figs. 1-4 summarize these bounds

for representative values of the top quark mass. For example, if Mt = 150 GeV,

one can 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 MH < 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.

3.5 Vacuum Expectation Value

The vacuum expectation value (vev) of the scalar field may be extracted

from the well known lowest order relation

v = ( G)-5 = 246.22 GeV (3.5.1)

From the very well measured value of the muon lifetime, -r = 2.197035

.000040 x 10-6 s [21], the Fermi constant can be extracted using the following

formula [36]

1 G 3 m2 a(m) 25 2
123 )( + 3mW12 2 + a~(m) 7r), (3.5.2)
1927r3 M 5 m 2x 4
J" W


f(x) = 1 8x + 8x3 4- 12x21nx, (3.5.3)


Gp = 1.16637 0.00002 x 10-5 GeV-2 (3.5.4)

This parameter may be viewed as the coefficient of the effective four-fermion

operator for muon decay in an effective low energy theory

2[e (1 75)e] [W (1 75)v] (3.5.5)

A direct calculation (e.g., in the Landau gauge) of the electromagnetic correc-

tions yields that the operator is finitely renormalized (i.e., G1 does not run)

[16,37]. Another way to see this is by using a Fierz transformation to rewrite

the above expression

S[ve77 (1 t5)v][YY(1 75)e] (3.5.6)

The neutrino current does not couple to the photon field, and the e p current

is conserved and is hence not multiplicatively renormalized.

An initial value is needed for the running vacuum expectation value at some

scale p. Wheater and Llewellyn Smith [38] 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

one can extract a value for v(MZ). However, the formula is derived in the 't

Hooft-Feynman gauge, and the evolution equation, Eq. (1.18) of Appendix A

for the vev, is valid only in the Landau gauge (see Appendix D). Nevertheless,

motivated by the discussion of the previous paragraph, the initial condition for

the vev is chosen to be v(MW) = 246.22 GeV. Using the initialization algorithm

(see Appendix C), one arrives at v(Mz). It is found that this procedure leads


to no significant correction, and one therefore takes, ab initio, v(Mz) = 246.22


3.6 Thresholds

In mass independent renormalization schemes, the running couplings are

unphysical. From the decoupling theorem [39] one expects the physics at en-

ergies below a given mass scale to be independent of the particles with masses

higher than this threshold. Therefore, for a correct interpretation of these run-

ning couplings, one must take into account the thresholds [40,41,42]. For the

electroweak threshold, one loop matching functions [42] are used with the two

loop beta functions valid in the Standard Model regime below the SUSY scale.

These matching functions are obtained in MS renormalization by integrating

out the heavy gauge fields in such a way that the remaining effective action is

invariant under the residual gauge group [41]. At the electroweak threshold,

near My, the heavy gauge fields and the top quark are integrated out. Below

this threshold there is an effective SU(3)C x U(1)EM theory. Thresholds in

this region are obtained by integrating out each quark to one loop at a scale

equal to its physical mass. At these scales the one loop matching functions

in the gauge couplings vanish and the threshold dependence appears through

steps in the number of quark flavors [43] as the renormalization group scale

passes each physical quark mass.

In the SM runs (see next section), in which an analysis of the relative

importance of including a proper treatment of thresholds effects instead of

using a simple step function technique, it is found that the former method

does not improve significantly over the latter [6].


3.7 Analysis and Results

The results of numerically integrating the 0 functions of the Standard

Model parameters from 1 GeV to Planck mass are depicted in the following

figures. For most of these plots, the arbitrary choice, Mt = MH = 100 GeV, is

made. Figure 5 displays the evolution of the inverse of each of the three gauge

couplings including the associated uncertainties in their values. In it, one sees

the "GUT triangle" signifying the absence of grand unification, assuming the

Standard Model as an effective theory in the desert up to the Planck scale.

Here, the differences between one- and two-loop evolution appear in the high

energy regime. Differences are also present for the strong coupling at low en-

ergies where it becomes large. Note that the uncertainties do not fill in the

"GUT triangle." Figures 6, 7, and 8 display the evolution of the light mass

fermions (me, mu, and md), the intermediate mass fermions (mp and ms), and

the heavy mass fermions (mr, me, and mb), respectively. The largest differ-

ences between one-loop and two-loop evolution occur in the bottom, charm,

and strange quark masses in these cases. In Fig. 9, the quartic self-coupling A

and the top Yukawa coupling yt for (Mt = 100 GeV, MH = 100 GeV) and for

(Mr = 200 GeV, MH = 195 GeV) are plotted. These two couplings are the

only unknown parameters of the Standard Model. The effects of changing the

values of Mt and MH in the analyses of the running of the other parameters

have been studied. It is 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 any of the other parameters. However,

changing Mt itself showed a significant difference in the running of the heav-

ier quarks. In particular, Fig. 10 shows that the intersection point between

the bottom quark and the r lepton moves down to a lower scale for higher


top quark masses. This is expected since from Eq. (1.9) one can see that the

bottom type Yukawas are driven down by an increased top Yukawa. This is

to be contrasted with the SUSY GUT case in which the bottom Yukawa /

function is such that this crossing point is shifted toward a higher scale with

an increased top mass. The relation mb = mr (I) is the most natural one in

the SU(5) theory [44], and it could be expected to be valid at scales where the

Standard Model gauge couplings are the closest to one another. Its validity

is examined for three different physical values of the top and Higgs masses in

the Standard Model. The noteworthy feature of Fig. 10 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. This re-

sult is vastly different from that of the original investigations [45]. This work

improves upon that work by including two loop effects in the running of the

quark Yukawas, by taking into account the full Yukawa sector, and most im-

portantly by incorporating QCD corrections in the extraction of the bottom

quark mass. Other mass relations were studied in the SM context [46]: (II)

md = 3me, 3ms = mp [47], (III) tan Cabbibo = (md/ms)1/2 [48], and (IV)

Vcb = (mc/mt)1/2 [49]. There is no scale at which all of these can be satisfied.
The scale at which relation (I) tends to be satisfied does not coincide with that

at which the others are valid. Still the disagreement is never too large. In an

SU(5) SUSY GUT model, the equality of the bottom and r Yukawas at the

scale of unification will be used to get bounds on the top and Higgs masses [3].

Lastly, the running of the CKM angles is displayed in Fig. 11. The initial data

used are sin01 = 0.2206, sin 2 = 0.0298, and sin 3 = 0.0106. Also 6 has been

taken to be 900, which corresponds to the case of maximal CP violation. As


mentioned in Section 2.4, the evolution curves for these angles are effectively


In the present case of the Standard Model, it is found that two-loop running

of the parameters does at times improve on the one-loop running. Indeed, the

differences of several parameters in their one- versus two-loop values at various

scales have been tabulated, for the cases (Mt = 100 GeV, MH = 100 GeV) and

(Mt = 200 GeV, MH = 195 GeV). Table 1 illustrates the difference between

one-loop and two-loop running in the ratio mb/mr, for the three scales 102

GeV, 104 GeV, and 1016 GeV. Clearly, the difference between one- and two-

loop results is more pronounced at higher scales, as expected. Over all these

scales the difference is never less than 5.5%. Note that the ratio becomes equal

to one well below the scale of grand unification as noted above in the discussion

of Fig. 10. Table 2 presents a similar comparison for the top Yukawa. Here,

two loops represent a smaller correction with the difference at all scales always

being less than 5%. Finally, Table 3 displays the same analysis for as for

the case Mt = MH = 100 GeV. No appreciable deviation from the tabulated

values is observed for any Mt ~ 200 GeV (except in the low energy regime

where the difference is at most ~ 4%). At scales MZ, the inclusion of two

loops is important in the evolution of the strong coupling (and of the quark

masses). Indeed, it is found that the pure QCD three-loop contribution is also

significant and therefore include it in the running of the strong coupling and of

the quark masses in the low energy region. As seen in this table, the combined

two and three loops in the low energy regime account for a 17% difference at

1 GeV in as.

Although in the cases considered in these last two tables there does not

appear to be a significant difference in two-loop over one-loop evolution at

scales above MZ, the first table does show a 10% difference at the scale 1016

GeV. The effects of using a naive step approximation vs. a proper treatment of

thresholds are numerically unimportant for the cases discussed above. Indeed

they are less important than the two-loop effects.

Table 1: mb/mr

Mt = 100 GeV Mt = 200 GeV
102 GeV 104 GeV 1016 GeV 102 GeV 104 GeV 1016 GeV
One loop 1.879 1.455 0.8081 1.868 1.392 0.6647
Two loop 1.782 1.348 0.7336 1.769 1.285 0.6047

Table 2: yt

Mt = 100 GeV Mt = 200 GeV
102 GeV 104 GeV 1016 GeV 102 GeV 104 GeV 1016 GeV
One loop 0.5405 0.4160 0.1928 1.133 0.9780 0.7145
Two loop 0.5405 0.4071 0.1842 1.143 0.9700 0.6816

Table 3: as

1 GeV 102 GeV 104 GeV 1016 GeV
One loop 0.3128 0.1118 0.07103 0.02229
Two and three loop 0.3788 0.1117 0.07039 0.02208








50 100 150 200
MH (GeV)

Figure 1. Vacuum stability and triviality bounds on the Higgs
mass for Mt = 100 GeV giving scales of expected new
physics beyond the Standard Model.



50 100 150 200

MH (GeV)

Figure 2. Vacuum stability and triviality bounds on the Higgs
mass for Mt = 125 GeV giving scales of expected new
physics beyond the Standard Model.








50 100 150 200

MH (GeV)

Figure 3. Vacuum stability and triviality bounds on the Higgs
mass for Mt = 150 GeV giving scales of expected new
physics beyond the Standard Model.






50 100 150 200


MH (GeV)

Figure 4. Vacuum stability and triviality bounds on the Higgs
mass for Mt = 200 GeV giving scales of expected new
physics beyond the Standard Model.




50 a -


I( 30 -1

20 20

0 -1

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

Figure 5. Running of the inverse gauge couplings using their prop-
agated experimental errors for the two-loop case only.

0.020 i i i

0.015 Mt=100 GeV
MH= 100 GeV


0.000 -

-0.005 I I I I
0 5 10 15 20

Figure 6. Light quark and lepton masses for Mt = 100 GeV and
MH = 100 GeV.

0.25 i i i' i-

0.20 Mt= 100 GeV
MH=100 GeV
Q 0.15 -


0.00 I I I I
0 5 10 15 20
logo10 (/1GeV)

Figure 7. Intermediate quark and lepton masses for Mt = 100
GeV and MH = 100 GeV.

M= 100 GeV
MH=100 GeV

mb -

0 m

0 5 10 15 20

Figure 8. Heavy quark and lepton masses for Mt = 100 GeV and
MH = 100 GeV.

.2 i l l i l l I I I I I I I I I I
: yI Mt=100 GeV, MZ=100 GeV
1.0 Mt=200 GeV, Mg=195 GeV


0.4 -

0.2 -

0.0 I I
0 5 10 15 20
log0 o(b//1GeV)

Figure 9. Top Yukawa and scalar quartic couplings.








Figure 10. Plot of mb/mr as a function of scale in the Standard
Model for various top and Higgs masses.

H \ / I I / 'T--
-- Mt=100 GeV, M= 0) GeV
Mt=150 GeV, MH=150 GeV -
-- --- Mt=190 GeV, MH=180 GeV

- .\-

- ........

0 5 10 15 20

log10 (,/1 GeV)


S- sin01
"' 0.2
SMt=100 GeV
. o.1 MH=100 GeV
- SlnUp
o sin__
S 0.0 sin -

0 5 10 15 20
loglo(/,/1 GeV)

Figure 11. CKM mixing angles for Mt = 100 GeV and MH = 100


4.1 The Supersymmetric Standard Model

In the minimal supersymmetric extension of the standard model (MSSM),

every particle has a supersymmetric partner, their spins differing by a half [50].

Also required is a second Higgs field with opposite hypercharge to the first as

the superpotential cannot contain both a field and its complex conjugate. The

second Higgs is also needed for anomaly cancellation and to give this sector a

mass. For renormalizable theories, the superpotential can have at most degree

three interactions. The superpotential for the MSSM is (suppressing the SU(2)

and Weyl metrics)

W = UYuDuQ + dYd dQ + -Ye dL + APud + h.c., (4.1.1)

where the hat indicates a chiral superfield and the overline denotes a left-

handed CP conjugate of a right-handed field, 7 = ia2,*. The usual Yukawa

interactions are accompanied by new Yukawa interactions among the scalar

quarks and leptons and the Higgsinos in the supersymmetric Lagrangian. There

are also new gauge Yukawa interactions involving the gauginos. The new purely

scalar interactions form the scalar potential which is positive definite in super-

symmetric theories. The scalar potential will be discussed in a subsequent


A remarkable aspect of supersymmetry is that all these new interactions

require no new couplings. The p term is only present to avoid a Peccei-Quinn



(PQ) symmetry. Omitting it would lead to exact PQ symmetry and to a
visible axion which is experimentally ruled out. An alternative way to break

the PQ symmetry is to omit the f term and add an explicit soft symmetry

breaking term, m3 uDd. The f can be interpreted dynamically as essentially

the vacuum expectation value of a singlet chiral superfield, N, through the

following additional interactions

ANu d + KNN (4.1.2)

This approach also provides a natural explanation for p ~ O(Mw). The

cubic term now explicitly breaks the PQ symmetry. Table 4 displays the

SU(3) x SU(2) x U(1) quantum numbers of the chiral (all left-handed) and

vector superfields of the MSSM.

4.2 Procedure

Bounds are presented for the mass of the top quark in a minimal supersym-

metric extension of the Standard Model (MSSM) with minimal Higgs structure

in the context of a grand unified theory (GUT) by numerically evolving the

couplings using their renormalization group equations. This analysis improves

on previous endeavors by taking full account of the Yukawa sector.

In the expectation that the Standard Model is only the low energy mani-

festation of some yet unknown GUT or of a possible supersymmetric (SUSY)

extension thereof, the three couplings g3, g2, and gl corresponding to the Stan-

dard Model gauge groups, SU(3)C x SU(2)L x U(1)y, should meet at some

large grand unification scale. Using the accepted values and associated errors

of these couplings unification is observed in the SUSY-GUT case but not in

the pure GUT case, as noted by several groups [1,51] (see Fig. 12). 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 the analysis of Ref. 1 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 [52].

Furthermore, in the context of a minimal GUT [44] there are constraints on

the Yukawa couplings at the scale of unification. One first restrict oneself to an

SU(5) SUSY-GUT [53] where yb and yr, 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, Mt. This can be seen in the down-

type Yukawa renormalization group equation (above MSUSY, for example),

from which the evolution of yb is extracted, since the top contribution is large

and appears already at one loop through the up-type Yukawa dependence
dYd 1
dt 162Yd[ 3YdtYd + YuYu + Tr{3YdtYd + YtYe)
7 2 16 (4.2.1)
(1gl + 392 +93) *

where Yu,d,e are the matrices of Yukawa couplings. Demanding that their

crossing point be within the unification region determined by the gauge cou-

plings allows one to constrain Mt. This yields an upper and lower bound for

fI( which nevertheless is fairly restrictive.

There is a threshold at MSUSyy. Here the matching condition is the naive

one of simple continuity due to the lack of knowledge about the superparticle

spectrum. The scale is taken to be variable to account for this ignorance.

4.3 One Light Higgs Limit

The simplest implementation of supersymmetry is considered and the cou-

plings are run above AMUSY to one loop. The MSSM is assumed 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


#(SM) = 'dcos3 + iusin/ (4.3.1)

where $ = irT2*, and where tan3 is also the ratio of the two vacuum ex-

pectation values (vu/vd) in the limit under consideration. This sets boundary

conditions on the Yukawa couplings at MSUSY. Furthermore, in this approx-

imation the quartic self-coupling of the surviving Higgs at the SUSY scale is

given by

A(MSy) = g 2cos2(2) (4.3.2)

This correlates the mixing angle with the quartic coupling and thereby gives

a value for the physical Higgs mass, MH. Using the experimental limits on

the MH further constrains some of the results. By using the renormalization

group one takes into account radiative corrections to the light Higgs mass [54]

and hence relax the tree level upper bound, MH MZ [55].

4.4 Initial Data

The bounds on Mt and MH are determined by probing these masses de-

pendence on 0. In SUSY-SU(5), tan f is constrained to be larger than one in

the one light Higgs limit. It seems natural to require that yj > yb up to the

unification scale [56], thereby yielding an upper bound on tan #. The initial

values at MZ for the gauge couplings are taken to be [1,57]

al = 0.016887 0.000040,

a2 = 0.03322 0.00025 (4.4.1)

3 = 0.109+04 ,
a3 -0.005 '


where GUT normalization for al is used. The following set of four quark

running masses defined at 1 GeV by the Particle Data book [21] are used

mu = 5.6 MeV, md = 9.9 MeV, ms = 199 MeV, and me = 1.35 GeV. For

the bottom mass, the Gasser and Leutwyler bottom mass value of 5.3 GeV at

1 GeV is used which translates into a physical mass of Mb = 4.6 GeV [26].

To probe the dependence of the results on Mb, the case Mb = 5 GeV is also

studied, this is the typical value obtained from potential model fits for bottom

quark bound states [58]. The effect of varying MSUSY is also investigated.

Given the values of the gauge couplings, unification holds for SUSY scales up

to 8.9 TeV and as low as MW. For empirical reasons solutions below MW were

not investigated.

4.5 Analysis and Results

The inclusion of supersymmetry collapses the GUT triangle. This is il-

lustrated in Fig. 12 taking MASUSY = 1 TeV. As mentioned above, a range

MyW MSUSY 5 9 TeV will achieve unification within one sigma error. From

Fig. 13 (the magnified unification region of Fig. 12), one determines that the

lower end scale, MGIjUT, of the unification region corresponds to an a3 value

of 0.104 at MZ, while the higher end scale, MHGT, corresponds to a value of

0.108 at MZ for a3. It is found that the unification region is insensitive to

the range of top, bottom, and Higgs masses considered. In the analysis of the

bounds for Mt, the values for a1 and a2 are chosen to be the central values

since their associated experimental uncertainties are less significant than for

a3. Demanding that Yb and yr cross at M UT and taking 03 = 0.104 then

sets a lower bound on Mt. Correspondingly, demanding that Yb and yr cross


at MgUT and taking a3 = 0.108 yields an upper bound on Mt. These bounds

are found for each possible value of /.

Figure 14 shows the upper and lower bound curves for both Mt and MH as

a function of f and for MgSSY = 1 TeV and Mb = 4.6 GeV. When applicable

the current experimental limit of 38 GeV on the light supersymmetric neutral

Higgs mass [59] is used to determine the lowest possible Mt value consistent

with the model. It is found that 139 < Mt < 194 GeV and 44 < MH < 120

GeV. The sensitivity of these results on MSUSY is investigated in the range,

1.0 0.5 TeV. It is found 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, an overall decrease in the top and Higgs mass bounds

is observed 116 < Mt < 181 GeV, MH < 111 GeV. Varying MSUSY as above

modifies the respective bounds. The top mass lower and upper bounds become

113 and 119 GeV, respectively. The upper bound on MH changes to 115 GeV.

The results of the analysis are displayed for the extreme case, MSUSy = 8.9

TeV, in Fig. 15, with Mb = 4.6 GeV. This only significantly changes the upper

bound on MH to 144 GeV compared to the MSUSY = 1 TeV case.

yt has also been run up to the unification region and compared with yb and

yr to see what the angle must be for these three couplings to meet [60], as in

an SO(10) or E6 model [61,62] with a minimal Higgs structure. It is clear that

this angle is precisely the upper bound on / as described earlier. In Fig. 16

yt/yb is displayed at the GUT scale as a function of tan 3 for MsuSY = 1
TeV and for the two bottom masses considered. If one demands that the ratio

be one, one can determine the mixing angles for the low and high ends of the

unification region. Then going back to Fig. 14. one finds as expected a much


tighter bound on the masses of the top and of the Higgs. Indeed, for Mb = 4.6

GeV, one has 49.40 < tan3 < 54.98, which yields 162 < Mt < 176 GeV and

106 < MH < 111 GeV. When Mb = 5.0 GeV, one obtains 31.23 < tan/3 <

41.18, which gives 116 < Mt < 147 GeV and 93 < MH < 101 GeV.

The four mass relations of Section 3.7 were also studied in the context of

the MSSM [46]. In this case, these relations can all be satisfied at the scale of

gauge coupling unification. However, for this to be true, several things must

happen: first Vcb must be larger than its presently measured central value of

0.043; second the top quark mass must be around 190 GeV (if it is lighter,

then agreement dictates that V4b should be larger still); third the Higgs boson

mass should be around 120 GeV. An analysis which recently appeared in the

literature has reached similar conclusions [63].


To improve on this analysis, one should implement the supersymmetric two

loop beta functions and the corresponding thresholds. The effects of soft SUSY

breaking terms should be investigated. Also, all the supersymmetric particles

have been integrated out at the same scale. It would be interesting to study the

effect of lifting this restriction. It should be noted that the bounds on the top

mass are very similar to those of Ref. 56, although the physics is very different.

These issues will be addressed in a subsequent chapter. However, given the

relative crudeness of the approximations made here, it is remarkable that the

experimental bounds on the p-parameter were satisfied. A sign which gives

credence to the program.








0 5 10 15 20

logo 10(//1GeV)

Figure 12. Plot of the running of the inverse couplings. The dot-
ted lines above and below the solid lines represent the
experimental error for each coupling.

Table 4: Particle Quantum Numbers in the MSSM

Q u d L T u 'i dA ra fB

U(1) + --1 +_ -+ +1 + o o o
SU(2) 2 1 1 2 1 2 2 1 3 1

SU(3) 3 3 3 1 1 1 1 8 1 1

I 00
I -l

2 8 I l|i I 1 | 1 1 I I Ii i I

27 .... .....
d. .. ............** .
....................... .



15.6 15.8 16 16.2 16.4


Figure 13. The plot depicts a blow-up of the area around the unifi-
cation point. (Note the small region where all three cou-
plings intersect. This region reduced to a point when
MSUSY = 8.9 TeV and was non-existent above that

MsuY = 1 TeV, Mb = 4.6 GeV






50 60 70 80


13 (deg.)

Figure 14. Plot of the top quark mass, Mt, and of the Higgs mass,
MH, 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 as per Fig. 12.



TeV, Mb

= 4.6 GeV

50 60 70 80

P (deg.)


Figure 15. Same as Fig. 14 for

MSUSY = 8.9 TeV and Mb = 4.6

MsusY = 8.9








1 TeV

0 10 20 30 40 50 60


Plot of the ratio of the top to bottom Yukawas, yt/Yb, 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) consis-
tent with unification as per Fig. 12.




Figure 16.



5.1 Minimal Low Energy Supergravity Model

Since no super particles have been observed experimentally, supersymme-

try, if truly present in nature, must be broken. One way to accomplish this

breaking is to couple the standard model to N = 1 supergravity (SUGRA).

In the minimal low energy supergravity model considered, supersymmetry is

explicitly broken by the addition of supergravity induced soft terms (including

gaugino mass terms)
2 (bt ( 2 + m l 1t + B A( ,u~ d h c .
Vsoft = "M.uu + md d + d + h.c.)
+ 2 t t 2 t -t 2 t +e 2m t-
+ mQi iQi mL ii +mui uii +md idi + meii )

+ Z( Aiy iiuQj + AYJYd'di;4dQj + AYeiei'tdLj + h.c.) ,
1 3
Vgaugino = M1AAl + h.c. ,

where Vgaugino is the Majorana mass terms for the gaugino fields, Al (suppress-

ing the group index), corresponding to U(1), SU(2), and SU(3), respectively.

From the supersymmetry algebra, one deduces that spontaneous symmetry

breaking occurs if and only if the vacuum energy is not zero. In global su-

persymmetric theories, the scalar potential is a sum of F and D terms. Su-

persymmetry is spontaneously broken if either the vacuum values of the F

term ( [64]) or D term ( [65]) are non-zero. A consequence of the spontaneous



symmetry breaking is a massless fermion in analogy with the breaking of an

ordinary global symmetry. We will assume that the spontaneous breaking of

the local N=1 supersymmetry is communicated to the "visible" sector by weak

gravitational interactions from some "hidden" sector. Spontaneous symmetry

breaking in supergravity occurs via the super-Higgs mechanism. The goldstone

fermion, or goldstino, associated with the breaking of global supersymmetry

is eaten by the gravitino thereby providing it with a mass. This spontaneous

symmetry breaking of supergravity manifests itself at low energy as explicit

soft breaking terms of global supersymmetry. This leads to a common (grav-

itino) mass, m0, for the scalars of the model and masses, MI, for the gauginos

at the GUT scale, MX. By assuming gauge coupling unification, we can take

the three gaugino masses equal. Furthermore, the trilinear soft couplings A?,

A and A" are all equal to a common value A0. The bilinear soft coupling BO

may be related to A0 (BO = A0 mo), if the SUGRA model has only canonical

kinetic terms for the chiral superfields. This scenario is to be contrasted with

one in which general soft breaking terms are added ad hoc to the Lagrangian.

In the most general case, there are sixty-three soft symmetry breaking param-

eters. Their number can be reduced by invoking certain symmetries such as

flavor and family blindness. In either case, all of these couplings will evolve to

different values under the renormalization group. The complete scalar poten-

tial appears as

V = VF + VD + Vsoft (5.1.2)

where VF contains the potential contributions from the F-terms

VF = ISYUQ + I'd + IdYdQ + ,eL + ,,2

+ IYQiul2 + IYd Qd + YeLd l2 (5.1.3)

+ iYuu + dYdd|2 + I Ye d2,


and VD contains the potential contributions from the D-terms

9 1 tr 1
VD = ( d Q d+ 4
26 3 3 2 2 U 2
+ ( t + Lt ++ + F+ d

+ ( QtQ *tu ~xd )2

where ? = (ri,r2,r3) are the SU(2) Pauli matrices and A = (A ,..., A) are

the Gell-Mann matrices. In general, one must impose constraints on the pa-

rameters to avoid charge and color breaking minima in the scalar potential.

Some necessary constraints have been formulated, such as

A2 < 3(m +m +m2 ) ,

A < 3(m + m + m ), (5.1.5)

A2 < 3(m + m + md).

However, these are in general neither sufficient nor indeed always necessary

[66]. They involve very specific assumptions about the spontaneous symmetry


5.2 Radiative Electro-Weak Breaking

An appealing feature of the models being considering is that they can

lead to the breaking of the electro-weak symmetry radiatively. The one loop

potential responsible for the breaking is

Vl-toop(Y) = Vtree(p) + Vi(p) (5.2.1)


1 m 3
S64r2 (_)2S(2sp + 1)m4(ln 2


where M2 is the field dependent squared mass matrix of the model and mp is

the eigenvalue mass of the pth particle of spin sp. The tree level potential is

Vtree(L) = m2 d d + m2 u + m3( u0d + h.c. )
'2 g2 (5.2.3)
+ -( tL ~d )+ 2 + ( + d )2

m1 = md + It

m2 = m2 + 2 (5.2.4)

m2 = Bf .
Minimization gives
1 2 m2 _- 2 tan2 p
-m = m (5.2.5)
2Z tan2 # 1
where m2 = (g2 + g2)v2/2, v2 = u + v2 and

Bf = 2 M) sin 23 (5.2.6)

where tan/3 = vu/vd, also

-2 2 AV1
m = m + (5.2.7)

where v = vd, v2 = vu and

9AV 1 Z( )a2 m2 am2
A = (-1)22s (2sp + 1)m2(ln-_ 1) (5.2.8)
av2- 327r2 P 2 1v,528
The parameters of the potential are taken as running ones, that is, they vary

with scale as dictated by the renormalization group. Because the one loop

correction to the potential is not negligible at all scales, an appropriate scale

must be chosen if the tree level formulas are to be valid [67]. Indeed, the tree

level analysis may lead to incorrect conclusions about the regions of parameter

space that yield electroweak breaking and consistent scenarios. When AV1 is

included, the value of I is not critical as long as p is in the neighborhood of

-ArI .


Although, as stated, the tree level results cannot always be trusted, one

can get some idea under what conditions electro-weak breaking occurs. The

renormalization group evolution of mn (see Appendix C) can be such that it

turns negative at low energies if the top quark mass is large enough, whereas

2d runs positive. From Eq. (5.2.3), the scale at which this occurs is set by

the condition

m (b) m2(pb) m (pb) = 0. (5.2.9)

If the free parameters are adjusted properly, then the correct value of the

ZO mass (MZ = 91.17 GeV) can be achieved. At tree level there is another

critical scale that must be considered. Again from Eq. (5.2.3), it is evident that

the potential becomes unbounded from below along the equal field (neutral

components) direction, if

m 2(s) + mi(ps) < 2m (I) (5.2.10)

Since mm2 m4 > 0 implies m2 + m2 > 2m, condition ((5.2.10)) can only

occur at scales lower than condition ((5.2.9)), so p, < Pb. As mentioned above,

when working with the tree level potential an appropriate scale to minimize it

must be used at which one loop corrections may be safely neglected. Gamberini

et al. [67] give a prescription for the choice of this scale.

In the present work, the one loop corrections are incorporated. Contribu-

tions from the third generation are included, that is, the top-stop, bottom-

sbottom, and tau-stau. The one loop effective potential should be constant

against the renormalization group to this order. The Zo mass is chosen as the

scale at which to evaluate the minimization conditions Eqs. (5.2.9)-(5.2.10).

Equation (5.2.5) can be written

m2 m tan2 9 1
i2 (MZ) d .- 2 (5.2.11)
tan2 3 1 (

where m-,d = mudA+a A1/v2ud and used to solve for p(Mz) (see Numerical

Procedure Section) given the value of all the relevant parameters at MZ. A

choice for the sign of p must be made (A is multiplicatively renormalized; see

Appendix C). It is evident that no adequate minimum exists if the parameters

are such that p2 < 0.

5.3 Sfermion Masses
The mass matrices for scalar matter are constructed from Eq. (5.1.2). For

example, in the up squark sector the relevant mass matrix appears as

/ ML2 M2l
m1 M-Li Lj La Rj\
M2 L M2 (5.3.1)
\ RLj LiRj /
where i,j = 1, 2,3 are flavor indices and
M2 2 +v2(v2 g)g2
LL,L = mQij + u2(YuYu)ij J )(Y(uL)g'2 T3(uL) ij,

M R, = i + v (YuY)ij J(vd vu)(Y(uR)g'2)j ,

M Lj = AvdY + vuAYu' ,
M2Ri = M2.
Li Rj Rj Li
Note in Table 4 that Y = Q T3 in the adopted notation. Similar matrices

follow for the other sfermions. These mass formulas as well as the ones to

follow are given in terms of running parameters. The domain of validity of

these formulas is at low energies (- MZ) with the parameters taking on their

renormalization group evolved values at this scale.

5.4 Higgs Masses
If the following notation is employed

'1= \2 2=

then the physical masses of the Higgs at tree level are calculated from the
following three matrices

1 2 Vtree 12 tan
-M sn( tanM 1
2 =9_9)~(si 2 = 1 cot 3
1 2 tree 1 2 (tan/ -1
2 ( )( ) M sin 2 2 -1 cot 0

+m1 M2 cot3 -15.4.1)
2 z + -1 tan
2Vtree 1 2 tan f 1
2a(7a(+si) 2 n2 1 cot )

where M 2 = m + m, M2 = M2 + m2 and m2 g2v2/2. The eigenval-

ues for the first matrix are 0 corresponding to the Goldstone boson and M2
corresponding to the CP odd scalar. The second matrix gives the masses of

the light and heavy Higgs bosons

MHi,h + 2mA) Z / + )2 4MA2m cos2 20] (5.4.2)

This tree level result predicts Mh < MZ. One loop calculations show that
this need not be the case [68]. The third matrix has eigenvalues 0 and M2

corresponding to a massless, charged Goldstone boson and a charged scalar. In-

cluding Eq. (5.2.2) in the calculations leads to corresponding one loop versions

of these masses [69,70].

5.5 Chargino Masses
The following four terms contribute to the chargino masses

-i29g2'u tr i- i 2t-i I 4d 2+ M2WiTWi + h.c.. (5.5.1)
2 d 2

The first two terms are the supersymmetric Yukawa-gauge terms. Letting

A = (Wt2 ilW)/V\2, the mass matrix follows

0 0 M2 -92vd A+
A(+ A- 0 0 2vu (5.5.2)
{A d ) ^ 2 M2 gu 0 0 A-
-92vd -A 0 0 7 d
Diagonalization yields two charged Dirac fermions with masses

M2Hh 1 2 2+2
2= [(M22 +2 + 2m ) (5.5.3)
y/(M2 + A2 + 2m2 )2 4(M2A m2 sin 20)2] .

5.6 Neutralino Masses

Contributing to the neutralino masses are the terms in Eq. (5.5.1) and

-ig'v/2u(+1 )~u ig'd/20i(-2 df + MI BB (5.6.1)
2 2 2

The neutralino mass matrix follows

S-MI 0 0 _aV.
0 -M2 -3 2V
0.- 1 0 iW3
d2 2 0 0d


5.7 Boundary Conditions at My

In this work, the modified minimal subtraction scheme (MS) of renormal-

ization is employed. The parameters of the Lagrangian are not in general equal

to any corresponding physical constant. For example, in the case of masses,

except for those of the bottom and top quark (see Eq. (3.4.1)), all other physi-

cal masses will be determined from their corresponding running masses by the

simpler relation

S= m() =(5.7.1)


The renormalization group P functions of the gauge and Yukawa couplings

have been calculated to two loops without making any approximations in the

Yukawa sector of the model. These have been included in Appendix C.

Because SUGRA models make certain simplifying predictions about the

soft parameters at the unification (Planck) scale, the evolution of the renor-

malization group equations is initiated at this scale. It has been demonstrated

that the introduction of supersymmetry leads to gauge coupling unification at

approximately 1016 GeV. Therefore one takes AMX = 1016 GeV.

At the unification scale, MX, all scalars have a common mass

mQi(MAX) = mui(MY) = md (My) = mL (My)
= me,(MX) = r (M ) = mM (MX) = m0 ,

as do the gauginos

M1(Mx) = M2(Mx) = M3(Mx) = m (5.7.3)

The trilinear soft scalar couplings will be taken diagonal and equal at MX

Az'(Mx) = A" (Mx) = A?(Mx) = AO (5.7.4)

Also the bilinear soft scalar coupling and the mixing mass at MX are given by

B(Mx) = BO, p(Mx) = po (5.7.5)

Furthermore, to constrain the parameter space, the bottom and tau masses

will be assumed equal at My

mb(MX) = mr(MX) (5.7.6)

Also for the sake of CPU time, we will take the masses of the two lightest

families to be zero in all the runs.


5.8 Numerical Procedure

There are seven free parameters in the model considered. These are A0,

B0, mO, ml/2, O0, tan/3, and mt. The two minimization constraints (5.2.5)

and (5.2.6) reduce this set to five, which are taken to be A0, mi, m1/2, tan 3,

and mt. In the present framework, B0 and 40 will be determined using the

numerical solutions routines described in Appendix E in conjunction with the

minimization of the one loop effective potential at MZ. Minimization at MZ

will give B(MZ) and p(Mz). To arrive at B0 and to (their corresponding

values at MX), the solution routine are employed as follows. A guess for B0

and [0 is made at VMX and then the parameters of the model are run to MZ

at which scale the evolved value of B is compared to the minimization out-

put value for B at IZ. The same is done for Ap. If the compared values

agree to some set accuracy, then B0 and Po are the required values. Other

analyses that also extract B(MZ) and p(Mz) simply evolve these two param-

eters via their renormalization group equations back to MX to find B0 and

Po relying on their near decoupling from the full set of renormalization group

equations. Notice from Eq. (5.2.11) that a choice must be made for the sign

of p. To constrain the parameter space further, the bottom quark and tau

lepton masses will be taken equal at Mx. This equality is a characteristic of

many SUSY-GUTs. This constrains the model to four free parameters, A0,

mo, m1/2, and tan #. Demanding that mb(MX) = mr(MX) and achieving the
correct physical masses for the bottom quark and tau lepton fixes the mass of

the top quark which affects the evolution of the bottom Yukawa significantly.

Gauge coupling unification shall be assumed, an assumption which appears

reasonable when one considers SUSY models with SUSY breaking scales 6 10


TeV. The solution routines could be used to find the precise (and similar) val-

ues of al, a2, and a3 at Mx that will evolve to the experimentally known

values at MZ, however this increases the CPU time considerably. Therefore

some precision shall be sacrificed in their MZ values by taking them exactly

equal at MX. This is already a theoretical oversimplification since one does

not expect the gauge couplings to be exactly equal due to threshold effects

at the GUT scale. It is found that for all cases studied, the common value

a 1(Mx) = a2(Mx) = a3(My) = 25.31 leads to errors no bigger than

1%, 5%, and 10% in al(M.z), a2(MZ), and a3(Mz), respectively. This is not

so bad considering that the (combined experimental and theoretical) errors on

a3(MZ) from some processes can be as large as this.
It is well known that there is a fine tuning problem inherent in the ra-

diatively induced electro-weak models. For certain values of the parameters,

the top quark mass must be tuned to an "unnaturally" high degree of accu-

racy to achieve the correct value of MZ. This problem is generally handled

by rejecting models that require "too much" tuning. The amount of tuning is

usually defined quite arbitrarily. The usual procedure is to define fine tuning

x2 8M2
ci = M2 a (5.8.1)
Z i
where xi are parameters of the theory such as mg, ml/2, p, or mt. One then

demands that the ci be less than some chosen value that is typically taken to be

10. The differences in using the tree level vs. one loop effective potential were

analyzed to some extent. The results agree generally with those of Ref. 67.

Moreover it is found that the fine tuning problem is exacerbated in the tree

level analysis. The basis for the "theoretical" fine tuning problem can be seen,


if one makes some simplifying assumptions, in the dependence of MW on the

top quark Yukawa coupling yt [71]

MW ~ MXe-1Y (5.8.2)

In the tree level analysis, one encounters another fine tuning problem. The

vacuum expectation value coming from the minimization conditions changes

rapidly from 0 to infinity over the interval (ps, pb). Using the prescription of

Ref. 67 for the scale A at which to adequately minimize the tree level potential,

find v(A), and thereby arrive at a value for MZ, one finds that although a small

variation in yt(MX) may lead to a small variation in 11b, the steepness in the

tree level vacuum expectation value will lead to a large variation in the value

of v(A) and therefore in MZ. Hence, in the tree level analysis, solutions which

may be within the bounds of the "theoretical" fine tuning may nevertheless

display a fine tuning aspect because of this "tree level" fine tuning problem.

The use of the one loop effective potential levels off the vev around the MZ

scale. The vevs depend on scale through wave function renormalization effects

which are never large as can be seen from the form of the renormalization

group equations for the vevs in Appendix C.

In the present framework, solutions shall also be rejected based on fine

tuning considerations, however the present method differs somewhat from the

usual one in that it is incorporated in the solution routine described above.

The routine is an iterative one which determines the convergence properties

of the solution which reflect an inherent fine tuning. If the convergence is too

slow, the solution will be rejected. Effectively any solution which the computer

cannot pinpoint within an allotted number of iterations is rejected. Given

values for AO, mn, ml/2, tan and sign(p), the solution routines search for


the values of v(Mx), mb,r(MX), mt(Mx), B0, and A0. The process by which

B0 and 0 are found was described above. The remaining three parameters

are determined similarly. The routine makes a guess for v(Mx), mb,r(MX),

and mt(MX), then the full renormalization group equations are evolved to 1

GeV calculating superparticle threshold masses in the process and minimizing

the one loop effective potential at MZ. The merits of the guess for v(MX),

mb,r(MX), and mt(Mx) is assessed by comparing the resulting values of MZ,
mr(1 GeV), and mb(l GeV) with the expected ones. The process is iterated

until the correct values are achieved to within some tolerance.

5.9 Thresholds

In the minimal low energy supergravity model being considered, the super

particle spectrum is no longer degenerate as in the simple global supersymme-

try model in which all the super particles are given a common mass, MSUSY

In the simple case, one makes one course correction in the renormalization

group evolution at MSUSY. In the model with soft symmetry breaking, the

nondegenerate spectrum should lead to various course corrections at the super

particle mass thresholds. To this end, the renormalization group / functions

must be cast in a new form which makes the implementation of the thresholds

effects (albeit naive) evident (see Appendix B). Since the MS renormalization

group equations are mass independent, particle thresholds must be handled us-

ing the decoupling theorem [39], and each super particle mass has associated

with it a boundary between two effective theories. Above a particular mass

threshold the associated particle is present in the effective theory, below the

threshold the particle is absent.


The simplest way to incorporate this is to (naively) treat the thresholds as

steps in the particle content of the renormalization group fl functions. This

method is not always entirely adequate. For example, in the case of the SU(2)

gauge coupling there will be scales in the integration process at which there are

effectively a half integer number of doublets using this method. Nevertheless,

this method should yield the correct, general behavior of the evolution. It is

a simple means of implementing the smearing effects of the non-degenerate

super particle spectrum. The determination of the spectrum of masses is done

without iteration as is common in other analyses. The method employed in

this work deduces the physical masses by solving the equation m(p) = P for

each superparticle in the process of evolving from MX to 1 GeV. The usual

iterative method requires several runs to find a consistent solution.

5.9 Analysis and Results

The tremendous computing task involved in analyzing the full parameter

space of the soft symmetry breaking models, using the methods described

as designed, would be far too time consuming given the available computing

facilities. Therefore, in the following analysis, some simplifications will be made

in the procedural method. First, only the heaviest family of quarks and leptons

will have non-zero mass. Second, as stated previously, the value of the strong

coupling at MZ will be allowed to vary from its central value of .113 by at most

10%. This translates into a similar error in the bottom quark mass. Third,

the allotted number of Runge-Kutta steps, involved in numerically integrating

the renormalization group equations, will be cut down to ~ 100.

The present analysis will restrict itself to a subclass of soft symmetry break-

ing models that have two or three soft parameters equal to zero at MX. One


such class follows from the no-scale model [72] and has AO = m0 = 0. The

strict no-scale model has A0 = m0 = B0 = 0. Another class coming from

string derived models has A0 = B0 = 0. Since Bo is not a free parameter in

the procedure adopted in this work (see Section 5.8), B0 = 0 results must be


Because the GUT inspired constraint, mb(AX) = mr(MX), is enforced

in this analysis, the results will depend on the mass of the bottom quark.

Most results will be reported for the case mb(l GeV) = 6.00 GeV, but lower

mass (5.70 GeV) and higher mass (6.33 GeV) cases were also studied. The

running value of 6.00 GeV for mb(l GeV) corresponds to a physical bottom

mass Mb = 4.85 .15 GeV, with the uncertainty coming from the error in the

strong coupling, as discussed above. The results of Chapter 4, indicate that

the top quark mass predictions increase with decreasing bottom mass, and the

present analysis corroborates this.

Given the no-scale or string-inspired cases, the phase space is explored by

setting AO = m0 = 0 or A0 = 0, respectively, and coarse graining the remaining

hyperslice. tan j was most commonly coarse grained as 2, 5, and 10. The mass

of the lowest supersymmetric particle (LSP), when it is a neutralino, is observed

to be correlated with the value of ml/2. Cosmological considerations indicate

that the LSP must be neutral and colorless and have a mass less than ~ 200

GeV. Points in parameter space that lead to LSPs other than neutralinos with

masses less than 200 GeV (or sneutrinos) are rejected.

Figure 17 shows a slice of the available phase space, in the no-scale case,

plotted against the top quark mass. The present unofficial bound on the top

quark mass is 120 GeV. The figure indicates that the top quark cannot have

a mass greater than 132 GeV in this model, if mb(l GeV) = 6.00 GeV

(This upper bound is raised to ~ 160 GeV, if mb(l GeV) = 5.70 GeV, and the

model is ruled out, if mb(l GeV) = 6.33 GeV). The figure also indicates that

190 GeV r ml/2 < 265 GeV; however, nothing significant can be said about

the value of tan 3. The points to the right of the allowed region were found

to lead to charged LSPs (iR) and are not displayed. Points to the left of the

region do not lead to electro-weak breaking. Furthermore, as Figs. 14 and 15

indicated, the top quark mass does not grow monotonically with tan /, rather

it reaches a maximum for some high tan # value then decreases. For the choice

sign(I) = -, the top quark mass upper bound falls below the experimental

limit, so this case is ruled out.

Analysis of the available data indicates that the strict no-scale case lowers

the top quark mass upper bound slightly to ~ 128. However, there is an allowed

range of 3 9 for tan /. Figure 18 is a plot of the evolution of the left-handed

and right-handed stop, bottom, stau, and gaugino soft mass parameters for

the particular strict no-scale case ml/2 = 240 GeV, tan # = 8.3, sign(i) = +.

The resulting spectrum of super particle masses is presented in Table 5. The

LSP is a neutralino in this case and has a mass of 92 GeV. The top quark mass

is just above the experimental limit at 126 GeV, and the Higgs boson mass is

77 GeV.

The strict no-scale case is a special case of the string inspired one, therefore

the 128 GeV top quark mass upper bound is not expected to decrease but rather

to increase in the "stringy" case. The data indicate that top quark masses as

high as 150 GeV are possible. Finally, a representative stringy scenario is

presented. Figure 19 displays the variation of Mt and BO with tan 3 for the

slice of parameter space with A0 = 0, mo = 100 GeV, and ml/2 = 250 GeV.

The scenario inferred from the figure has Mt = 130 and tan / = 7.8.


Table 5

A0 0
m0 0

mr/l2 240
pto 171
tan 3 8.3

sign(p) +

Mt 126
d 473 506
ii 390 532

e 94 176

iL 180

7 92
Z, TV 153, 147

Ho 213, 271
fH* 270

h, H 77, 158

H, A 235, 222


Ao=O, mo=0, s(u,)=+







- I I I I I

* .

, I





ml/2 (GeV)

Figure 17. Slice of parameter space in no-scale case displaying Mt
against the common gaugino mass ml/2. The points
represent radiative electro-weak breaking solutions with
mb = mr at MX and a neutralino LSP with mass ; 200




I I,


600 i i '


400 -



0o I
0 5 10 15 20
logoo/ 1 GeV)

Figure 18. Evolution of squark (dashes), slepton (dots), and gaug-
ino (solid) soft mass terms for AO = 0, mo = 0, ml/2
240 GeV, and tan 3 = 8.3.



Ao=O, mo=100, ml/2=250

,I I

- \




I I I -


Figure 19. Mt (solid) and Bo (dashes) plotted against tan 3 for
A0 = 0, mo = 100 GeV, and ml/ = 250 GeV to infer
the BO = 0 case.












This work has studied, using one and two loop renormalization group 3

functions, both the standard model and its minimal supersymmetric extension.

The parameters of these models, i.e., the gauge couplings, the quark, squark,

lepton, and slepton masses, the Yukawa sector mixing angles and phase, the

scalar quartic coupling, and soft symmetry breaking parameters, were run over

scales ranging from 1 GeV to Planck mass. The aspects of the standard model

and its minimal supersymmetric extension were reviewed.

In the standard model case, thresholds effects were served well by naive step

functions. The more sophisticated implementation using one loop matching

functions did not significantly improve on the step function method. The

difference between two loops and one loop represented a more significant effect,

albeit sometimes small. Plots exhibiting these different features for all the

parameters of the Standard Model were included. Gauge coupling unification

was shown to fail, and some interesting mass and mixing angle relations were

considered, but failed to hold simultaneously at a common scale.

In the supersymmetric case, in which gauge coupling unification is achieved,

bounds on the top quark and Higgs boson masses were determined using the

SU(5) inspired constraint that the bottom and tau masses be equal at the

scale of unification. Remarkably, the top quark bounds were consistent with

the p parameter bounds. In supersymmetric context, the mass relations could


all be accommodated at the scale of unification provided the top quark mass

was high ~ 190.

Minimal low energy supergravity models were considered. They have the

appealing feature that the electro-weak symmetry is radiatively broken for cer-

tain ranges of the soft breaking parameters. The study of specific models, with

some soft parameter fixed, resulted in upper bounds for the top quark mass.

No-scale models in which only gaugino masses provide global supersymmetry

breaking yield top quarks with masses less than ~ 130 GeV. The results are

sensitive to the value of the bottom quark mass. Lower bottom quark masses,

within the experimental uncertainty, lead to higher top quark upper bounds.

In these models, the ratio of vacuum expectation values of the two Higgs fields

is expected to be larger than ~ 720.


In this appendix, the renormalization group P functions of the Standard
Model are compiled [6]. These have appeared in one form or another in various
sources. Effort has gone into confirming their validity through a comparative
analysis of the literature. The main source is Ref. 7. Following their conven-


S= +Yu6Q + Yd+tQ + YetL + h.c. IA(Itl)2, (A.1)

where flavor indices have been suppressed, and where Q and L are the left-
handed quark and lepton SU(2) doublets, respectively,

Q( UL) L (vL (A.2)

1 and i are the Higgs scalar doublet and its SU(2) conjugate

S= ( ) i2s. (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 3 functions for the gauge couplings are

dgl 3 2 3
1-b bkl 2
dt 167r2 (12)2
k (A.4)
S(l Tr{CluYutY + CldYdtYd + CieYeYe} ,


where t = Iny 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


(Cf) = (

4 1
b= -23ng 1,
22 4 1
b2- 3 3- ng- 6 ,
b3 = 11- f ng,
/19 1 11
-n 3 49 3
14 T 7 (

17 1 3\
TU 1 2
3 3 1 with =u d e ,

with ng = .

In the Yukawa sector the f functions are




dYu_,d,e 1 + (1) 1 (2)
dt Yu,d,e( 1Gi2 ude (162)2 u,d,e

where the one-loop contributions are given by

1) = 3(YuYu YdYd) + Y2(S) (1g + g2 + 8g2)

= (Ydd YYu) + Y2(S) -( 1 g2 + 9 + 8)
3Y1 + + 8g+)

S+YeYe + Y2(S)- (g +




Y2(S) = Tr{3YutYu + 3YdtYd + YeYe} ,


(bkl) 0 196
00 T 1002)

and the two-loop contributions are given by

2232 135 43 9
(2) = 3(YtY)2 YtYYr+r tydtyd + dty(YYd21

+ Y2(S)( Ydt Y Yutyu) X4(S) + 3 A2 2A(3YdtY, + YdtYd)
223 2 135 g7_ 9 2 9 2 2)ydtYd
+ (- 2 + + 16gY ( g + 163)Y
5 29 4 9 2 2 1922 35 4 2
+Y4(S) + (200+ 4ng)g1 Ng2 + 3 ( ng)g2 + 9g2gg
404 80 4
9( 3 -9 ng)3 'g
(d2) =- |(Ydtyd)2 YdtYdyutyu 1YutYuYdtYd + 11(YtY)2

SY2(S)( 5YutYu YdtY X4(S) + A2 2A(3YdtYd + YutYu)
4 4 92
1872 135 5 7 1 92 + 279
+ ( 9g + 9 22 + 16g3)YetY4 (_1 2gg + 16g3Y Y
5 29 1 4 2722 3122 35 _
S2 0 + n) 2 + 4 )
22 404 80 4

(2) (YetYe)2 |y2(S)YetYe X4(S)+ 3X2 6AYetYe
3872 135 5 51 11 4 2722
+(- 1 + g2)YetYe -y4(S) + ( + ng)g4 + 2 2
(- ng)4 ,
Y4(S) (1g2 + 9 + 8g1)Tr{YutYu}
10 4 (A.12)
+( g2+ g + 8g )Tr{YttYd} + (g + g2)Tr{Ye Ye
X4(S)= -Tr{3(Y ~Yu)2 + 3(YdtYd)2 + (YetYe)2
94 (A.13)
In the Higgs sector, the 0 functions for the quartic coupling and the vacuum
expectation of the scalar field are presented. Here a discrepancy in the one-loop
contribution to the quartic coupling of Ref. 7 is corrected
dA 1 (1) 1 (2) (A.14)
dt 16_2 + (16r2)2 A

where the one-loop contribution is given by
12A2 (99 21 + 92+ 9 3 4 2g2g4
1 2) -g1 1 2 2g4)
= 22 (5 + (A.15)

+ 4Y2(S)A 4H(S) ,
H(S) = Tr{3(YutYu)2 + 3(YdtYd)2 + (YYet)2} (A.16)

and the two-loop contribution is given by (discrepancies found by Ford et al.
[73] in Machacek and Vaughn [7] are corrected),
(2) 182 22 (313 11722
( -78A3 + 18(2 + 3g )A2 [( 10ng)g 92
9 229 50 4 497 6 3 97 8 4
+ + 4 ng)g1 ] (y- 8ng)g2 -54 + ng)glg2
9 239 40 4 2 27 59 40 g
( -ng)g lg2 "( -ng)g1
25 24 +9 ng)g 125 24 9 g
64gTTr{(YtYu)2 + (YdtYd)2}

SgTr{2(YutY)2 (YdtYd)2 + 3(YetYe)2}
4Y4(S) + 10A[ (11 9 2 + g2 +8g2)Tr{YutYu} (A.17)
+( + + 8g +8 )Tr{YdtY} + 3( + )Tr{Ye+Ye} i

+ g ?[(- g + 2192)Tr{YUtY,} +( 2 + 9)Tr{YtYd}
5 10 2
+ (- 9g2 + llg2)Tr{YYeYe} 24A2y2(S) AH(S)

+ 6ATr{YutYuYdtYd} + 20Tr{3(YutYu)3 + 3(YdtYd)3

+ (YetYe)3} 12Tr{YutYu(YutYu + YdtYd)YdtYd}
The / function for the vacuum expectation value of the scalar field is

nv 1 (1) 1 (2) (A.18)
dt -16ir2 (16r2)2

where the one-loop contribution is given by

(1) = ( + g2 ) Y2(S) (A.19)
7 9 9

and the two-loop contribution is given by

(2) _3A2 4(S)+ X4(S)
93 1 4 511 5 4 27 2 2(A.20
( + ng) + ( n 2 ng)g2 80192 "
These expressions were arrived at using the general formulas provided in Ref. 7

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. The

general formula of Ref. 74 is used to arrive at the / functions for the respective

gauge couplings
dg3 2 38
S[(nu + nd) 11] + [3(nu + d) -102]
dt 3 (4X)2 3 (47r)4
8 2 3e2
+ [-nu + -nd] (A.21)
9 9 (4r)4
5033 325 2 2857 g
+ [- (nu + nd)- (nu + nd)2 7] ,
18 54 2 (47r)6
de 16 4 4 e3 64 4 e5
-t [-nu + nd + -+ [nu + nd + 4nl
64 16 e3g2
9 9 (4r)4
where nu, nd, and nj are the number of up-type quarks, down-type quarks, and

leptons, respectively. In Eq. (A.21), the three-loop pure QCD contribution to

the / function of g3 have also been included [75].

For the evolution of the fermion masses, Ref. 76 is used. It is known that

there is an error in their printed formula [77]. This typographical error is

found and corrected. The calculation of the two-loop contribution to the mass

anomalous dimension in QCD using the corrected formula agrees with the

result obtained in Ref. 77. Equation (2.26) of Ref. 76 should read
i17 148
7m = 6ij(1 (C' + C C12 (C4C + C12C +
3 3 F FR
(C' C + C'RC ) + T(Cm 2Cj 2C )bi .


Using the corrected expression, the following mass anomalous dimension is

computed. The fermion masses in the low energy theory then evolve as follows

t = 7(l,q)m (A.24)

where the I and q refer to a particular lepton or quark, and where
2 2
1 e2 3 93
(1,q) = (,q) T2 + (l,q) ()2
+ [11 4 33 4 13 2 1
+ [7(1 q)e4 (1,3q)g3 + 2^g()e2 4 (A.25)
333 93
+^(1) (4r)6

The superscripts 1 and 3 refer to the U(1)EM and SU(3)C contributions,

respectively. Explicitly, the above coefficients are given by

7(l,q) 1-6Q,q)
7(3) = 0
3) -8

13 33
7() = 7(1) =
1(1 ) 3Q4 80 20 20 n 2 (A.26)
7(l,q) (-,q) +9 9 3 + 1(,q)
13 _4/Q2
7q)= -4q)
33 404 40
7( 3 + 9 (nu + nd)
333 2 140 2 2216
7) 3[7 (nu + id) + (160((3) + )(nu + nd) 3747]

where Q(l,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 anoma-

lous dimension for the quarks above, the three-loop pure QCD contribution

333 [75] have also been included.


In this appendix, some useful results in calculating the / function of the

MISSM are included. Also some sample calculations are presented. Although,

to one loop the MSSM P functions have appeared in the literature, one is not

aware of the two loop ones' appearance except in some approximate form, such

as keeping only the contribution of the heaviest family.

There are at least two ways to proceed. References [78,79] give general

formulas valid to two loops to compute the 3 functions of gauge and Yukawa

couplings in a supersymmetric gauge theory. These were used to calculate

the two loop gauge and Yukawa coupling / functions appearing in the next

appendix. Reference [7] gives formulas valid to two loops to compute the 3

functions of gauge, Yukawa, and scalar quartic couplings in a general gauge

theory. For the purpose of including thresholds, this approach is more useful,

and it is this manner that the following examples are done. The results of this

second approach were checked against the first one, and they agreed (as they


Not all calculations are included, as this would be useless and not enlight-

ening. However, it is hoped that the examples will give some flavor of the

endeavor and that the results will prove useful to anyone wishing to pursue

such calculations further.

B.1 Gauge Couplings

B.1.1 Group Invariants
Tables 6 and 7 will be useful in the calculations of the/3 functions of the
MSSM. Also useful are the following group invariants

C2(R) = TA(R)TA(R) = C2(R) I(R) (B.1.1.1)

Using this definition and

Tr(TA(R)TB(R)} = T2(R)6AB (B.1.1.2)

the following identity is derived
Tr{C2(R)} = d(G)T2(R)

C2(R)Tr{I(R)} = rT2(R) (B.1.1.3)

C2(R)d(R) = T2(R)r ,

where r = d(G) is the rank of the group, and SAA = d(G). From this, it follows

C2(G) = T2(G). (B.1.1.4)

The Yukawa interactions arising in the SUSY Lagrangian can be written

S wa = -iNg[f* T- jAA Aj Ti ,i (B.1.1.5)

where TA are the group generators, and overlined fermi (Grassmann) fields
are right-handed. Now make the correspondence
Ya Y = -ivgT ,

jo(i) = +i gT
Written this way, the i (i.e., 0(i)) subscript represents the ith group component
of the scalar, and the j represents the group component of the fermion. The i

encodes the scalar information in SUSY whereas the a does so in the notation

of Ref. 7. When taking a Yukawa term from Ref. 7 and "supersymmetrizing" it,
the trace need not represent a trace unless all scalar indices (i.e., the superscript

on the Y's) are contracted. For example, in the one-loop contribution to the

scalar anomalous dimension

Tr{YaYbt} =[-i/gTA[iVgT

= 2g2(TATA)ij
= 2g2C2ij

= 2g2C2(S)6ij

Now to compute Y4(F).

Tr{C2(F)YaYat} = 2g2C2(F)Tr{ I TBTB}

= 2g2C2(F)Tr{TBTC}6CB
= 2g2C2(F)T2(R)6BC CB

= 2g2d(G)C2(F)T2(R)

Y4(F) Tr( C2(F) 3aat}
d(G) (B.1.1.9)
= 2g2C2(F)T2(R)
These results may be used to derive the SUSY gauge beta function. In the
general (but single simple group G) case, at one loop
(47r22(1)/3= 2 1 11
(47r)2p()/ = T 2(F) + T2(S) C2(G) (B.1.1.10)

where F, S, and G stand for the fermion, scalar, and adjoint representations,
respectively. To go to the SUSY from, the following correspondences are used

F--R +G .
SUSY (B.1.l.11)
S >R.


(47)21SY/3 = [T(R) + C2(G) + 1 2() C2(G)
( 3 3 32 (B.1.1.12)
= T2(R) 3C2(G) .
At two loops, the only extra complication is a Yukawa contribution in the
general case that must be supersymmetrized as discussed above

(4x)43(2)/g5 = [10 C2(G) + 2C2(F)IT2(F) + [ C2(G) + 4C2(S)]T(S)

-C2(G)2 Y4(F).
Applying the stated correspondences yields
(47r2(2) 10
(4w)2 S g5 = C2([2() + T2(G)] + 2C2(G)T2(R)
+ 2[C2(R)T2(R) + C2(2(G)T(G)] + 4C2(R)T2(R)

C2(G)2 [Y4(R) + Y4(G)]
= 4C2(G)T2(R) + 6C2(R)T2(R) 6C2(G)2

[2C2(R)T2(R) + 2C2(G)T2(R)]

= 2C2(G)T2(R) + 4C2(R)T2(R) 6C2(G)2

B.1.2 Fierzing
A matrix M can be expressed in terms of the group G's generators

M fATA + f0 I. (B.1.2.1)

Taking the trace of both sides yields

Tr{M} = f0d(R) (B.1.2.2)

Multiplying by TB and taking the trace gives

Tr{MTB} = fAT2(R)bAB


1 1
M = TrMTA TA + Tr{M} I. (B.1.2.4)
T2(R) d(R)
If M = 4(t, it follows that

t = 1 (4tTA 4)TA + 1 ( ) (B.1.2.5)
T2(R) d(R)

where use has been made of Tr{ TTA} = Tr{ftTAq} -= tTA[. Now
multiplying on the left by ,t and on the right by gives

($t$)2 = T (tTA)(tTA) (t2 (B.1.2.6)
T2(R) d(R)
Combining like terms yields

(tTAX)(tTA ) = T2(R)[1 ](t)2 (B.1.2.7)

In the fundamental representation of SU(n), T2(R) = 1/2 and d(R) = n.
Taking SU(2) as an example, with 4 a doublet, it follows that

(=tTA Xt)()TA) ( 2 (B.1.2.8)

It is also useful to have the analogous expression for the case when there are
two different SU(2) fields involved
_1 Dt T t,(a> I)(-,t Ta 2) + 1 l 2) ,
(~1T2)( T2(R) d(R)= ()

which follows immediately from Eq. (B.1.2.5).

B.2 Anomalous Dimension of the Scalar Field
Using the Machacek-Vaughn convention, the Yukawa sector of the SUSY
Standard Model is written
C = ULU~uQL + ULUulQL + UU~,uQL+

dLDddQL + dLDddQL + dLDdQL+

TLEDdLL + LEidLL + eLEidLL + c.c.

= ULiaUij uA(iU2)ABQLj3B oa3 + *uA(i'2)ABOQLjB Uj ULi +''
The ir2's involved in the Weyl spinor product have been suppressed and so
have the ia2's involved in the SU(2) product, although the ia2's are displayed
in the second step. The SU(3) product follows since the barred fields are anti-
quarks. All fermions are represented by Grassmann fields. The c.c. is explicitly
calculated in the example below
[XL(ir2)7L] = *(i72) *

= XL(i2)7L
= [XR(ir2)](ir2)[(--ir2)R]

= YR(i-2)7R = j(ir2)R,
where the identity, 4R = ir2~, has been used. The following identifications
can be made from Eq.(B.2.1)
Y+u((A) )ABi-u(A) a -6(ia 2)BAUi ,
[(uia).](Qj} +g a*(2)Asjij (O (Qjm3B)[,(uif -B]
y+u(uia) -u( ia)
[.( A)](QjI3B +ba 3(i2)ABUij Y(Qj/3B)[(4A) --3a(i'2)BA ji
y+u(Qji3B) ,-((Qji3B) (iT2)BAi
[(uia)(OA)]. =6ai2)ABUij Y[(A)(uia)] =- (2BA
y+d(fA) .r-d(fA) _(io2)BAD i
Y[(di)(Qjp3) +6aO(ia)ABDij Y (QB)[*(dia] -)BAD
y+d(dia) V-d(dia) -a(i2)BADi
[.*(A)](Qj- +#a)?(i2)ABDij (Qj3B)[(4A)i -3aQ~2BA ,
y+d(Qj#B) -d(Qjl3B) (io)
[(dia)(OA)]. = + 3(iT2)ABDij Y((A)(dia)] = 0a(2)BAD

+d(A) --d(OA) -(i2)BAEi
[(ei).(LB) = +(i2)ABEi (LB)(ei)]
+d(ei) ]L -d(kei) (io2)BAEi ,
](LjB)= +(iO2)ABEij Y(L[(A) -(
y+d(LjB) + 2)A j -d(LjB) -(i.)BEi
[(ei)(A)]. = +(i2)ABEi [( A)(ei)] BA

The + and on the Y's denote whether or not the 4 is complex conju-
gated, respectively. The upper index always denotes the scalar, and a bullet is

placed in its unoccupied lower position. Since two of the three fields are either
transposed or not (in the spinor, SU(2) doublet, SU(3) triplet, or generation
sense), they appear grouped in square brackets. Dots over Y's indicate that
the Yukawa coupling matrix it represents is daggerred.
Consider the following example from Ref. 7

X = Tr{YbytaycYtc + Tr{YbYtcyaytc} (B.2.6)

a quantity that enters the scalar 2-loop anomalous dimension. In the above
notation (suppressing some indices at first), the first term reads (in the fu
X(+A)(-B) = Tr{y+u(OA)y-u(B)YY} (B.2.7)

but to this must be added another term

(+A)(-B) = Tr{Y+u(A)y-u(B)YY} (B.2.8)

since there must be alternating daggerred Y's in these expressions, and both
equations (B.2.7) and (B.2.8) above satisfy this (the underlines on two Y's
indicate that their scalar indices are to be contracted). Furthermore, the result
must be symmetric, in this case, in A and B, so the next two terms must also
be included
X(-A)(+B) = Try-u(OA)y+u(OB)yy}
X(-A)(+B) = Tr {Y-u(4A)y+u(B)yy} .

Putting it all together, the first term in Eq. (B.2.6) is given by

X = ((X X+-) + (X-+ + X2-)].
2 1


Similarly the second term in Eq. (B.2.6) is


Now to compute these contributions (the flavor and color indices are suppressed
since these contract in a straightforward way)

X+- Tr{+u(A) -u(B) [+u(E),-u(E) y+u(QE)V-u(QE) I
1 =" u(QC) (QC).u u.(QD) (QD)*u u(4D). *(4D)u
= 3Tr{(UtU)2 (iT2)AC(-io2)CB[ (ia2)ED(-io2)DE + (i"2)DE(-iO2)ED

= 12Tr{(UtU)26AB ,

X+ =0,
xi = 0 ,
X-+ Tr --u(A) v+u(B) -u(-E) y+u(E) + -d(4E) +d(OE)
2 (QC).u u.(QD) (QD)*u u.(QC) (QD).d d.(QC)
+ -U(U) v+U(U) + -d(d) v+d(d)
(QD)(OE)* *( E)(QC) (QD)(OE)* (OE)(QC)
= 2.3Tr{(UtU)2UtUDtD}(-ia2)CA(ia2)BD(--i2)DE(i2)EC

= 6Tr{(UtU)2 + UtUDtD} AB
Some of the 3's come from the trace over the suppressed color delta functions.

X = 3Tr{3(UtU)2 + UtUDtD}AB .

y+- = Tr{Y+u(OA)-_y-u(OB)Y} = 0,

Y2+ = Tr{Y +u(A)Y-u(4B)Y} = 0,

Y-+ = Tr{Y-U(OA)YY+U(B)Y} = 0,

y-+ = Tr{fY-u(OA)YY+u(B)Y} = 0.




Y = [(Y1 + Y2)+- + (Y + Y2)-+]

The full answer is then

Xu = 3X + Y = Tr{3(UtU)2 + UtUDtD} (B.2.15)
2 2

B.3 Yukawa Couplings
The Yukawa interactions arising from the D-terms in the SUSY Lagrangian

can be written

YU = -ivzg[ Ti jR -i T j i R], (B.3.1)

where the overlined fermi (Grassmann) fields are right-handed. Now make the

SUSY +(AR) = -iVgT
ij (D,)(oj)
Y-(AR) = +i RgT
(o0 )(3D)
The o's a's, o's, etc. represent field-types (i.e., Q, u, d, etc.). Note that the

Y's which represent TR's are diagonal in type (i.e., both indices are o's as

exemplified above). The + and indicate that the scalar index is complex

conjugated (equivalently, is the left subscript on Y's) or not complex conju-

gated (equivalently, is the right subscript on Y's), respectively. Written this

way, the i above represents the ith group component of the scalar, and the

j represents the group component of the fermion. Contracting two Y's (i.e.,

using underscores) implies in their case contracting the superscript, group gen-

erator indices (the R's).

In order to construct the one loop SUSY beta function formula for the
Yukawa matrices the following must be supersymmetrized
yytya SUSY (.r(AR)-r:t(AR)y+(7))[3.]&
-(AR) +(AR)y+(O) + 1r+(AR),r-(AR)y+(O)
= 0Oo 0OI []o.& 10 -[o.],a

=(C2(0)+ C2))Y )
-2g2C2(R)Y(O) (B.3.3)

yayty SUS (R))
y +(0) -,-(AR)m^+(AR)

=Y+() C2(a)
2g2Y() C2(R)
Furthermore, in obtaining the formulas for the beta functions of the different
Yukawas the "external subscripts" are generally two fermions (not scalarinos)
which are represent by o's, a's, o's, etc., and the "external superscript" is a
scalar (a Higgs not an sfermion) which is represented by a (0). Hence, it should
be clear from inspection that the following terms cannot be constructed for any
choice of "internal subscripts"

YTr{Ytya SUSY (Y(AR)Tr{F(R)y()})o 0 ,
yyaty SUSY (( AR)y()Y:F(AR)) 0 .

Application of Eq. (B.3.3) to the up-Yukawa (Y ) ; see section where
the SUSY Y's are defined) yields

C2(R)Y)- C Y)u + =B) + C[ 2PoYO
C(R .](QB) = C2u [o ) [u](QB) (B
= (C2() + +0 U (( B(B.3.5)
---- (62() + C2(BU,]--[uo](QB) ,

where use was made of the diagonal nature of C20S ~ 6&AC2(0). Note that
C2 on the left can act on both left subscripts. Likewise

(YC2(R) Y+(u)o) +U(C)
[u](QB) [u C2(QB) (B.3.6)
-= WC2 *[u-](QB) '
The standard one loop beta function for the Yukawas is
(4r)2/a 1 [yytya + yayty] + 2YYatY + YTr{Ytya}
2L (B.3.7)
3g2 C2(F), ya
Supersymmetrizing according to the results derived above gives

(47)2a = 1 [yytya + yayty] + 2yyatY + YTr{YtYa}
Sr (B.3.8)
(3 g)g2 C2(R), Ya
where 6g equals 1 or 0 depending on whether one is above or below, respectively,
the mass threshold of the gaugino in question. The labels Ti, i = 1,..., 5, will
be used to denote the five terms appearing in Eq. (B.3.8) when computing the
supersymmetric beta function for the up-Yukawa

T = (yty+u(.))[u](QB)
[Y+u(OD) V-u(OD) v+u(QD) -u(QD)] +u(OA)
[ [u.](QC) (QC)[.u] + [u(C)]-* -[(OC)u] Y[u.](QB)
= [(ia2)DC(--i2)CD + (iu2)CD(--i2)Dc ](ia2)ABUUtU

= (1i2)ABU[4UtU] ,

T = (y+u()YY)-Y [u.](QB)
= y+u(OA) [r-u(D) v+u(OD) -+Y-(u) v+U(u)
[u.](QC) [ (QC)[.u] [u.](QB) (QC)[(4D).] I[.(D)](QB)
+ -d()d(D) Y+d(D) + (-d(d) v+d(d)
(QC)[.dl [d.](QB) (QC)[(OD).] [.(OD)](QB)(
= (io2)ACU [(-iD2)CD(i 2)DBUtU + (--i2)CD(i02)DBUtU

+ (-ia2)CD(ia2)DBDtD + (-io2)CD(ia2)DBDD]

= (ia2)ABU[2UtU + 2DtD ,


T3 = (YY-^+uY_) )
SY [nu.](QB) (B.3.11)
T4 = (YTr{YtY-u()}) [u.](B)
= v+u((C) Tv -u( Y C) v+u(A)
-[u.](QB) (QD)[u] [u.](QD) (B.3.12)
= (ia2)CBUTr{3UtU}(--iO2)DC(iO2)AD

= (i2)ABU[Tr{3UtU}] ,

T5 = (C2), +u(}) ) (
= (ia2)ABU Z [C+'i) + C2(Q) + C2k(+)]
.(4 1 1 2 3 3 2 (B.3.13)
= (ia2)ABU[( + + + (0 + + )g
+( + + +0)
[13 3 2 8 2]
= (i2)ABU [ + 2 + 3
In T4 the 3 comes from the trace over the suppressed color indices. Also, in
T5 use was made of the fact that if the gauge group is not simple but a direct
product G1 x ... x Gn with couplings gl,... ,g9n, then g2C2 -- k =1 gC2k

B.4 Thresholds
To implement the super particle thresholds in the minimal low energy super
gravity model, the renormalization group P function must be calculated in a
form that has not appeared in the literature. In the following example, the one
loop p function of g3 is considered. Starting from Eq. (B.1.1.12)

(47r)2/31) /3 = 2T2(F3) + T2(S3) + C2(G3)g- C2(G3) (B.4.1)
3 3 3 3
where F3, S3 refer to the fermion and scalar representations, and G3 = SU(3).

01= .< Mg9 (B.4.2)

where Mg is the mass of the gluino. When dealing with a direct product group,
like SU(3) x SU(2) x U(1),

T2(R3) 2(R3)d(R)d(R2) (B.4.3)

From the definition of T2(R) in Eq. (B.1.1.2), one obtains the following result
in the SU(3) case

T2(R3) = 2( )NQ + ( )N- + ( )N (B.4.4)

where Np equals the number of generations of particle p. This result is valid
for both fermion (R = F) and scalar (R = S) representations. The notation
will be such that N = n for the particle and N = n for the SUSY partner.
Equations (B.4.1) and (B.4.4) lead to
(420(1)/3 2 12 1_
(43)2 1)/g3 = (nu + nd) + -nQ + 1- + -6- + 26g 11 (B.4.5)

It has been assumed that nQ = (nu + nd)/2. Also the fact that left-handed
and right-handed quarks of a given flavor have the same mass implies nu = nU
and nd = n-. Note that this reduces to the right standard model result when
h = Sg = 0, and to the right supersymmetric result (J. > MSUSY) when h = 3,
g = 1. Similar formulas are calculated for gl and g2

(4.)2(l)/g 2.17 5 5 1 1 4 1
(47r)2 3 (-nu + -" d + ne + -nv) + i-iQ + +- -- + -n-d
512 12 4 4 3015 15
1 1- 1 1
+ 10 L 5h + 1 ni + 1 "nu + ) + 10 (no + n d) '
(47r2l)/g3 22 1 1 1 1
(4 -- + ("u + nd) + (ne + nv) + 2nQ + niL
1 1 42
+ 3(" + i)+ ) + (n + n ) + W w.
For the gauge couplings, the two loop contributions were also calculated in this

manner. For the Yukawa couplings, this form of the j function was calculated
to one loop.

Table 7
G C2(G) d(G)
U1 0 1
SU2 2 3
SU3 3 8

Table 6: Group Theory Factors
T2(R) C2(R) d(R)
U1 SU2 SU3 Ui SU2 SU3 UI SU2 SU3
Q_ (+1)2 +)2 1 4 1 2 3
yT (-2)2 0 1 (-2)2 0 1 1 3
d (+3)2 0- 1 ()2 02 1 1 3
L (-2)2 0 (-)2 T 0 1 2 1
e (+1)2 0 0 (+1)2 0 0 1 1 1
_, (+0)2 1 0 (+ )2 3 0 1 2 1
d (_1)2 1 0 (-4)2 3 0 1 2 1
-l~ -- 1 : --- 1 -


Using some of the notation of Falck [80], the superpotential and soft sym-

metry breaking potential are as follows

W = YvQU + dYd dQ + eYe dL + Apiud + h.c.,

V'oft = M2 I + m d + B(ud d+ h.c.)
+ m tQ + m2 Lu t + m + 2f 2

+ ( AY~Y~i Qj + A d didQj + AYeidLj + h.c. ),
1 3
Vgaugino = 2 MiAA + h.c..

Various a2's have been omitted, and a sum over the number of generations is

implied in the squark and slepton mass terms. Also, hats imply superfields

and tildes the superpartners of the given fields.

First the gauge couplings

dg, 1 3
dt 16-b,2 I

(16r2)2 [Y 1 Tr{CluYutYu + CldYdYd + CleYeYe}]
where t = lnp 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
bl = -e 2ng ,

b2 = 5 2ng (C.3)

b3 = 9 2ng ,
38 6 88 9 9
17 5 IT M 5 0
(blk)= 14 8 ng 3 17 (C.4)
11 3 68 0 -54

26 1r 14 8
(Cl) = with f = nu, d, e, (C.5)
4 4 0
with ng= =nfl.
In the following, the beta functions for the parameters of the superpotential
are listed.

= 162[ Tr{3YutYu + 3YdYd + YeYe} 3( 2 +2) g (C.6)

In the Yukawa sector the 3 functions are

dYue 1 (1) 1 (2) (C.7)
dt Yu,d,e( 162 (ud,e + 16ir2)2 u,d,e)

where the one-loop contributions are given by

,31) = 3YutYu + YdtYd + 3Tr{YutYu} (13 2 + 16 2
(1) = 3YdtYd + Y tYu + Tr{3YdtYd + YetYe} (792 3g2 9 )

31) = 3YetYe + Tr{3YdtYd + YetYe} (9g2 + 3922)

and the two-loop contributions are given by

32) = -4(YutYu)2 2(YdtYd)2 YdtYdYutYu 9Tr{YtYu}YYu

Tr{3YdtYd + YetYelYdtYd 3Tr{3(Yu Yu)2 + YdtYdYutYu
2 2 2 + d 4 2 2)Trlyutyu,
+ ( 9 + 6g )YutY + (J2)Y d + ( + Y16 Yu
26 403 21 32 3044
+ (ng + 45)g + (6ng 32 92+- 9)g3
22 136 22 + 8 33
+ 9192 + -3 + g2g3
2) = -4(YdtYd)2 2(YutYu)2 2YutYuYdtYd 3Tr{YutYu}YutYu

3Tr{3YdtYd + YeYe}YdtYd

3Tr{3(YdtYd)2 + (Yete)2 + YdYdYU Y} + (4gu)yytYu

+( (g + 6g2)YdtYd + (- g + 16g )Tr{YdfY}
6 14 7 21
+(g )Tr{YeY} + ( -15n + +(6n- 2
+32 304, 4 22 8 22 22
+ ( g --)g3 + 9192 + 9g3 + 23
e2) = -4(YetYe)2 3Tr{3YdtYd + YetY }etYe

3Tr{3(YdtYd)2 + (Yetye)2 + YdtYdYutu}
+ (6g2)YetYe + ( 2)Tr{Yet} + + 2 + 16g )Tr{YtYd}
18 27 214 9 2 2
+ ( 1ng + 7)g + (6ng 2) + 9192
The evolution of the vacuum expectation values of the Higgs's is given by

dlnvbuOd 1 (1) 1 (2)
dt 1672 .,d + (167r2)2 ,,d (

where the one-loop contribution is given by
3 1 2 2)
= ( +g) 3Tr{YutYu}
7 = ( +2 2 3Tr{YdtYd) Tr{YeYe) ,
Od =(591+92

and the two-loop contribution is given by

(2) = Tr{3(YutYu)2 + 3YutYuYdtYd}
19 2 9 2+ 20rg3)Trjyutyu
(-g+ -g92 + 2093T
279 1803 4 207 357 4 27 9 22
(0+ n1g)g1 -( + ng)g2 ( + 9)g192 2
800 1600 32 64 80 80
( = 3Tr{3(Ydtd)2 + 3YdtYdYutY + (Yetye)2}
-(g2 + g + 20g)Tr{YtYd} (92 + 2)Tr{YeYe}
279 1803 4 207 357 4 27 9 2 2
-(8 + 1600ng)gl (2 + ng)g2 + Ong)g2
800 1600 32 64 80 80

The renormalization group equations for soft symmetry breaking terms are

dA= 1[ 4(YeYet)ik Ae y
dt -16x2 Y,--

+ 5Ak y (Y )kj 3 Ae(YeYetYe)
Ye Ye

+ 2(Akm Yekm2 + 3AdmIY~m) 6( 21M1 + gM2)] ,
kYtk yik
[ 4(YdYdt Ad + 5A~ (YYd)kj 3 d (YYdYd)i
16 d d d
i .. yikk__ 2)ikkuj
+ (Ak A')(YutYu)kj + 2(YdYut)ikAy
+ + dd d 3d dk 2
+ 2(Amlykm2 + 3Akm iym2) 1Mi 6g2M2 g32 23
15 3

dA" 1 4y"
Sd-t6 4(Y uYt)^kAkJ +
dt 1672 4YIU Yu

5Aik u(yty)kj 3 A(YuYtYu)
Y2 Yu.j

+ (A' Aj )(YdtYdj + 2(YuYdt)ikA d

+ 6Amiyrm2 2621M 692M2 M ]3 ,
15 3