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 Permanent Link:
 http://ufdc.ufl.edu/AA00030025/00001
Material Information
 Title:
 Grand unified supersymmetric inflationary cosmology
 Creator:
 RuizAltaba, MartÃ
 Publication Date:
 1987
 Language:
 English
 Physical Description:
 v, 56 leaves : ; 28 cm.
Subjects
 Subjects / Keywords:
 Asymmetry ( jstor )
Average linear density ( jstor ) Baryons ( jstor ) Fermions ( jstor ) Gravitinos ( jstor ) Mass ( jstor ) Protons ( jstor ) Quarks ( jstor ) Sine function ( jstor ) Symmetry ( jstor ) Cosmology  Mathematical models ( lcsh ) Dissertations, Academic  Physics  UF Physics thesis Ph. D
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 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1987.
 Bibliography:
 Bibliography: leaves 5054.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by MartÃ RuizAltaba.
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 University of Florida
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 University of Florida
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Full Text 
GRAND UNIFIED SUPERSYMMETRIC
INFLATIONARY COSMOLOGY
By
MARTI RUIZALTABA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1987
Per al Padri
To all those who have died in the struggle for justice while I wondered about the mathematical consistency
of observable reality.
ACKNOWLEDGEMENTS
I wish to thank all those who talked, discussed, and enjoyed physics with me during my student years. My advisor Pierre Ramond led me through many mazes with that characteristic serenity and insight of his, which only a grand master of knowledge commands. We had much fun together, even though we never played tennis. Were I to be fair, the list of people who have played a crucial role in the learning process of which this thesis is an official landmark would be very long, starting with my mother and growing every day. A detailed multilingual appendix for this purpose does not seem in order. Let me instead collectively acknowledge the good friends, the good lovers, the good teachers, the good students, the good neighbors, and the good fighters with whom I have participated in this mysterious process of life. Let me, however, acknowledge as well the unreliable friends, the poor lovers, the boring teachers, the lazy students, the gossipy neighbors and the cowardly traitors, for the wisdom I have gained in dealing with them and their personal irrelevance. This research was supported in part by the United States Department of Energy under contracts No. DEAS0581ER40008 and No. FG0586ER40272.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS . . . . . . . . .
ABSTRACT..... ................
INFLATION . . . . . . . . . . . . . . . .
SUPERSYMMETRY BREAKING . . . . . .
GRAND UNIFIED SECTOR . . . . . . . .
Fermion Masses . . . . . . . . . . . . .
The Full Superpotential . . . . . . . . . . PROTON DECAY . . . . . . . . . . . . .
Proton Decay Lagrangian . . . . . . . . . Renormalization of Proton Decay Operators . Proton Decay Rates . . . . . . . . . . .
BARYOGENESIS . . . . . . . . . . . . .
CONCLUSIONS . . . . . . . . . . . . . .
REFERENCES ...................
BIOGRAPHICAL SKETCH . . . . . . . . .
. . . . . . . . . . . iii
. . . . . . . . . . . v
. . . . . . . . . . . 5
. . . . . . . . . . . 10
. . . . . . . . . . . 13
. . . . . . . . . . . 15
. . . . . . . . . . . 26
. . . . . . . . . . . 29
. . . . . . . . . . . 31
. . . . . . . . . . . 35
. . . . . . . . . . . 39
. . . . . . . . . . . 44
. . . . . . . . . . . 49
. . . . . . . . . . .50
. . . . . . . . . . . 55
iv
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
GRAND UNIFIED SUPERSYMMETRIC INFLATIONARY COSMOLOGY
By
Marti RuizAltaba
May 1987
Chairman: Pierre Ramond
Major Department: Physics
A theoretical laboratory is presented, within the framework of supergravity unification, which satisfies all experimental and cosmological constraints below the Planck scale, notably those from standard big bang nucleosynthesis, baryogenesis, and the fermionic spectrum. An intermediate scale, around 1010  1011 GeV, arises from the examination of inflation, supersymmetry breaking, the PecceiQuinn symmetry, fermion masses, proton decay, and the baryon asymmetry of the universe. Careful attention is paid to the renormalization of proton decay operators. Although a realistic low energy spectrum follows from an isospinlike global symmetry (familiarity), a rather baroque higgs sector is needed to cancel anomalies and implement the missing multiplet mechanism. A low reheat temperature and baryogenesis from outofequilibrium decays are perhaps the most noteworthy aspects of the model, whose general success is not yet matched by any models derived from strings rather than point fields.
v
!~ r
INTRODUCTION
One of the most astounding successes of the grand unification program,1 initiated about ten years ago, was the understanding of the origin of the baryon asymmetry of the universe.2 The program suffered an early setback when minimal SU(5) failed to explain quantitatively this baryon asymmetry. It also predicted a fast proton decay rate which was soon ruled out by experiment. Supersymmetry3,4,5 generally extends the proton's lifetime and prefers the strange and muonic decay modes. The experimental limits on the proton's lifetime6 and the requirements of efficient baryogenesis tightly constrain the parameters in the Lagrangian of candidate unification models. An excellent theoretical laboratory arises from the locally supersymmetric grand unified model below,7,8,9,10,11,12,13 which is consistent a) internally below the Planck scale (anomalyfree), b) with the standard SU(3) x SU(2) x U(1) model at accelerator energies, c) with big bang inflationary cosmology14,15,16 at very early times, and d) with underground detector limits on proton decay.
The general traits of this supersymmetric inflationary cosmology (SIC) are presented below. The seminal papers for SIC are Refs. 7 and 8. Further work appeared in Refs. 11 and 12. Although the model is not particularly aesthetic, it serves to prove, by construction, the consistency of all known constraints with simple supergravity unification.
Supersymmetry is spontaneously broken by the O'Raifeartaigh mechanism1718 and its breaking is related to the end of the inflationary period in the evolution of our universe. The cosmological constant can be set to zero
1
2
while supersymmetry is broken, but a contribution of O(M2M~1auck) cannot be avoided after the observable sector finds out about supersymmetry breaking. No mechanism is known in any model to solve the cosmological constant problem, so we proceed with the rest of the program without much uneasiness.
A specific model with SU(5) invariance is written down, relying freely on insights gathered from explorations with larger groups. The mass spectrum of leptons and quarks is incorporated at the tree level with the aid of couplings between matter and a nonminimal higgs sector including19 a 45. The element KMcb of the quark mixing matrix has been measured from the (long) Bmeson lifetime.20 We incorporate it and a low topquark mass (claimed and subsequently disclaimed by the UA1 group21), obtaining reasonable fermion mass relationships and adequate CP violation.
Since gravitinos are created during reheat,22 the temperature TR of the heat bath generating them is bounded above.23 If reheated fields are thermalized immediately, the products of the reheating process are restricted to having masses < TR, lest their abundance be strongly suppressed,  exp(TR/m). However, if the reheat is not an equilibrium process,24 then no such strict bound can be derived for the reheated products, which could be more massive than TR. In the SIC theory presented here, reheat is caused by the decay of inflatons, of mass around the intermediate scale, 1010  1011 GeV, and yet the reheat temperature is only about 106 GeV.
Higgs triplets of masses around the intermediate scale can be used for generating the baryonnumber asymmetry25,26,27 without causing fast proton decay via the dimension five higgsinomediated operators.28,29,30 The method presented here calls for righthanded neutrinos with masses around, again, the intermediate scale.
3
Proton decay rates, with largest branching ratios into strange and muonic modes, put a lower bound on the masses of the higgs triplets whose interactions violate baryon number. The requirement that these same triplets be produced from inflaton decay to generate the baryon asymmetry of the universe sets an upper bound on their masses. A narrow window satisfying these two bounds is found at the intermediate scale.
This dissertation is organized as follows. First, the SIC superpotential is presented and the hidden sectors are glossed over. The grandunified (GUT) sector is then presented in detail along with its implications for fermion masses, CP violation, and neutrino oscillations,31 in terms of the Yukawa couplings of the superpotential and the various symmetrybreaking scales. We then compute the decay rates for the proton, under flavorhelicity SU(6), paying careful attention to the renormalization of the effective Bviolating lowenergy operators. Finally, we study baryogenesis and present a mechanism involving heavy righthanded neutrinos, whose masses are also around the intermediate scale.
The field content of supergravity32,33 is as follows. Chiral superfields (Dai = 0) contain matter, in spinorial and scalar varieties. Gauge superfields (Wa = D2 DV, V = Vt) contain gauge vector bosons and the spinorial gauginos. In addition, the spectrum contains the gravity supermultiplet, with the graviton, the gravitino and the KalbRamond field. Supergravity theories are never torsionfree due to the interdependence of the fermionic and bosonic coordinates of superspace.34
Minimal N = 1 supergravity in four dimensions has two arbitrary independent functions, which reduce to only one superpotential P in the "minimal"
4
case. Kinetic terms for all fields are in this case canonical, and the only requirement on P is that it be analytic, i.e., function only of the superfields and not of their conjugate partners.
The superpotential for the SIC model consists of three sectors,7
P = I + G + S (1.1)
where fields in one sector do not appear in the other two. Both I and S contain only gauge singlets, responsible for inflationary expansion and supersymmetry breaking respectively. The sector G contains all the usual matter, the higgs sector, and more.
The purely scalar potential for a minimal supergravity theory with superpotential P is given by
Vscalar = exp {M2I 1,12 ap + 2 3 2 + D  terms
(1.2)
where the sums run over all scalar (super)fields in the theory, denoted symbolically by 4, and M is the reduced Planck mass M = MPlanck = 2.4 x 1018 GeV (1.3)
Crosstalk between the sectors is always suppressed by powers of M. Each of the three sectors interacts with the other two only gravitationally, so thermal equilibrium is possible only within each sector or as a consequence of finetuned initial conditions.
INFLATION
The inflation sector I of the superpotential can be constructed very simply7 by assuming it contains only one chiral superfield, . Scaling all fields by M and considering the scalar potential due to I alone, VI = e'J'2 [I + &tI  31I12 (2.1)
At the minimum of this potential (the true vacuum up to interference with other sectors), the cosmological constant should vanish, so Vj( o) = 0. Requiring further supersymmetry to be unbroken at this minimum implies that the gravitino mass must be zero, which imples that
+ 4iI = 0 (2.2)
Hence
I(o) = (io) = 0 (2.3)
A very simple I satisfying the above requirements is7 a quadratic superpotential
I = A2(, _ o)2 (2.4)
where A is a mass parameter setting the scale for inflation. This is the small parameter which appears in all succesful inflationary models,2 fixed around 104M to avoid the supersymmetry entropy crisis or Polonyi problem.8,22 Without loss of generality, we can take the vacuum value of the inflaton,
(4) = 4O, to be real.
5
6
Requiring the scalar potential to be flat at the origin,
(0) = 0 (2.5)
so that (new) inflation can occur with a slow rollover, sets 4o = M = 1. Letting the scalar component of the chiral superfield 4 be 4eiO, the scalar potential from this sector only is V, = Ae2 [06  405cosO + (3 + 4cos2O)04  4cos 003+ (3  4cos20)02+1] (2.6)
For any value of 4 ("the radius") this potential has a minimum at 6 = 0. It is the only local minimum, so there is no 0degeneracy and no domain walls may arise.
VI(O) = Vj(,O = 0) = A4 e2 (6 4 5 + 704  403 _ 2 + 1)
= A4 (I  443 + 2  845 + 3 +O(7) = 4e A4(4  1)2 + 12e A4(0  1)3 + O((0  1)4) (2.7)
whence the mass of the inflaton is mo = 2ve/A2 (2.8)
Similarly,
V1(0) = VI(9,4 = 1) = 8e A4(1 cos O) (2.9)
so the mass of the angular field 0 is me = m = 2v/ A2 (2.10)
The potential (2.7) is a standard inflationary potential, flat at the origin, with a slow descent towards the minimum at M, and a steep divergence for values of the field larger than M. The inflaton 4 has cubic selfinteraction
7
of strength 12A2, and linear (gravitational) interactions with other sectors of strength  A2 as well. The inflaton will couple to fields in S and G, with a preference for the heaviest ones, and a decay rate ro ~ (A2)2 mO ~ A6 (2.11)
At Planck time all particle species in the universe are maintained in thermal equilibrium by gravity. Regardless, however, of the initial conditions, each of the three sectors decouples from the others while gravity becomes weaker as the universe expands. Fields within each sector thermalize through Yukawa and gauge interactions, so the inflaton superfield effectively evolves alone in a thermal bath provided by S and G.
The inflaton will roll away35 from its initial configuration36,37
4 < HO , 4  0 (2.12)
destabilized by the (otherwise negligible) Hawking radiation term
AVH  A8O (2.13)
As the universe expands, the energy density becomes dominated by the potential energy of the inflaton, which can reasonably be assumed38 to evolve according to the classical equations of motion 4 + HqS+dV1
+ 3H + = 0 (2.14)
d4
H2 32 (2 + VI(0) (2.15)
where H is the Hubble constant (parameter, rather) with its value in antideSitter spacetime given by V(0) A2
HO = = (2.16)
3M2v/3M
8
Inflation occurs with 4 << 3HO 4 from a time t. when 4 ~ Ho until a time tj ~ M . The inflaton then starts evolving like matter, until a time tR ~ F when it decays into radiation and reheats the universe.7>8 Using conservation of energy, the increase in radiation energy density is determined by
PO(tR) = P4(t4)
(2.17)
~ A12 I l
15
with g* the effective number of light degrees of freedom at the time of reheat tR. The reheat temperature TR for the heat bath in the causal bubble is, by definition,
TR ~ A3 (2.18)
To avoid the monopole problem,14,39 mo must be less than the unification scale= Mx; hence A2 < 102. Below we shall find that actually A ~ 104, so the reheat temperature TR is only about 106 GeV. Despite this low reheat temperature (which does not, by far, produce any problems with nucleosynthesis), inflatons are produced copiously and inflaton decay does generate enough baryon asymmetry, due to the lack of equilibrium among the various hidden sectors.
The number NJ of efolds of inflationary expansion (RR ' eNIR) will be9 heatt
NJ=J Hodt (2.19)
Initial
This is easily much larger than the observationally required NJ ;> 65. NJ is around 108 if 41 < HO , r M/10, so there is a large set of initial conditions leading to sufficient inflation.
9
In the deSitter (inflationary) stage, the quantum fluctuations40,41,42,43 of the inflaton are of order HO. Their comoving size leaves the horizon during inflation (Hubble radius is constant), and they reenter the horizon after inflation, thus generating a spectrum of density fluctuations at scale A: 6p H 3 2
(p ~ 0 1 + log(AHo) (2.20)
p #2(t1) L 2
Approximating VI by its truncated Taylor series around the origin, Eqs. (2.14) and (2.15) can be solved for 4(t1) to give7 p 104A2 ~ 10~4 (2.21)
p
which is the preferred value for galaxy formation with adiabatic density fluctuations, known in the trade as the "cold dark matter scenario." Interestingly, Hawking has shown44 that if < 104, then the mass of the inflaton must
p
be less than 1015 GeV, or A < 101.5. Let us emphasize that, given the value of A ~ 104 required from the discussion of the gravitino problem,8 the magnitude of the density fluctuations comes out as a prediction of the theory.
10
SUPERSYMMETRY BREAKING
Consider now the supersymmetrybreaking piece S of the superpotential P. It cannot be22 of the Polonyi45 type (linear in the superfield). Take S to be of the O'Raifeartaigh type,8,17
S = A(AB2 + [2) + mBC + n (3.1)
where A, B, C are chiral superfields and A, pt, m, and n are parameters. We can set m ~ A and adjust n to ensure that the cosmological constant vanishes at the minimum of the full potential, taking into account the inflation and gauge sectors.
The F term S breaks supersymmetry17'18 so that the spinor component of A is the Goldstone fermion which provides the gravitino with the helicity i states it needs to become massive. The scalar component of A, the O'Raifearton, couples weakly and stores potential energy during inflation. Then it evolves like matter until it decays preferentially into a gravitino pair through a oneloop diagram with three internal B propagators, with rate
3
A, 16 2mA (3.2)
where aA = A2/47r, and mA ~ aAy is the oneloop O'Raifearton mass.
The gravitino mass is m3 = A2/M, and the B, C fields have a mass of order
2
it. For supersymmetry to solve the gauge hierarchy problem46,47,48,49,50,51 (its precise raison d'itre), m must be at most around a TeV so 1i  101011 GeV
(107  108)M.
11
Now gravitinos couple to matter very weakly (with gravitational strength), so they are longlived particles. They may thus decay into photons and photinos (lightest oddR particles, R = (1)3B+L+2S) after the onset of nucleosynthesis. Their number density at decay is severely constrained52,53,54 by the success of conventional nucleosynthesis55 in explaining the primeval abundances of light element. The decay of gravitinos into gluongluino pairs, for instance, can dramatically alter the deuterium abundance.56,57
The abundance of gravitinos at the time of nucleosynthesis is constrained, whence the reheating temperature is bounded24
TR < 2.5 x 10 GeV (3.3)
Using TR ~ A3 and m3 ~ 2, this implies
A3A2 < 4.2 x 1027 (3.4)
which entails, under the requirement (for the solution of the hierarchy problem) that u ~ 108, that8
A < 10~4 (3.5) If gravitinos happened to be stable, then the bound (3.3) is less restrictive,24 TR < 1.3 x 1011 GeV (3.6)
implying
A < 102.5 (3.7)
To make sure that inflaton production of gravitinos is small, since the inflaton couples most strongly (~ A2) to the heaviest fields, we must kinematically forbid the inflaton from decaying into O'Raifeartons, as well as B and C fields,
mn < mA, mB, mC
(3.8)
12
and require also aA to be small, aA < 103, in order to suppress oneloop effects.8
A careful analysis8 of these oneloop effects reveals a lower bound for the inflation scale
A < 104.2 (3.9)
whence A is rather tightly constrained, much to nature's credit.
13
GRAND UNIFIED SECTOR We write the superpotential for matter fields asil
( 0 a'H 0 (do dK 0
G1 =FI aH cL 0 T + T dK eK' fK" T
0 0 bH 0 fK" gK
('0 d1K 0 (4.1)
+ ST 2K K' f1K" F + (1S1 S2 + j2S3S3) 41
0 12K" K
where TT = (T1, T2, T3), F = (Fl, F2, F3), and ST = (S1, S2S3) are the N = 1 chiral superfields containing matter in three generations or families, and H, L, K, K' and K" are higgs superfields, and 41 is a group singlet whose vacuum expectation value (41) = Mo = 1010 GeV breaks the PecceiQuinn symmetry58.
The explicit field content of F, T,, and S, is shown below, where Greek indices run from one to three and denote color. All fields are chiral superfields with lefthanded Weyl spinor components. The fields F are vectors, T" antisymmetric tensors, and Sn singlets:
Fn ~(dn) a
F(d= en
1 afi(Uc)" una dn, (4.2)
Tn = (2_ 0 e(
0
Sn = V
The representations and global symmetry charges of the various fields in GI are shown in Table I. The only unusual feature is the higgs field L, which
14
transforms like a 45. The first two terms in (4.1) constitute the GeorgiJarlskog model19 with an extra eK'T2T2 coupling, needed to correct the prediction KMcb eA , incompatible with the value inferred from measurements of the
Bmeson lifetime20 (KMcb = 0.0435 0.0003) and the small topquark mass (< 150 GeV) required by unitarity.
TABLE I. Field content and symmetries (vectorial, PecceiQuinn, familiarity) of G1.
Superfield SU(5) irrep V X 7
F, Ti
Si F2
T2 S2 F3 T3 S3
H
L
K K' K"
01
10 10
1
10
10
5
5
5
1
3
1
5
3
1
5
3
1
5
2
2
2
2
2
10
The four phase symmetries can be understood as follows."1 First, there is the vectorial symmetry V, which becomes BL below the electroweak breaking
15
scale. It forbids unwanted terms like FK and FFT and has no anomalies. Second, we have the anomalous PecceiQuinn symmetry X, needed to solve the strong CP problem.59 Third is the chiral R symmetry of cubic superpotentials, broken by gravitational couplings to the hidden sectors I and S. Lastly, we have the 7 symmetry, familiarity, which is the only one to distinguish among the three families or generations. Familiarity 7 is like family isospin, traceless in family space.
Crucially, we forbid"1 mixing terms between the two sets of higgs fields with different V, such as HK, HK', HK" or LEK, LEK', LEK", where E ~ 24 or 75. Such mass terms would lead to disastrously fast proton decay rates, through dimensionfive operators (see the discussion below on proton decay).
As a further and rather innocent simplification, we shall assume that the matrices are symmetric, i.e., that a = a , = d2, and fi = f2.
Fermion Masses
The higgs superfields produce after SU(5) breaking light ( MW) doublets and heavy (~ M2/M) triplets, via the missing multiplet mechanism.60,61 Our purpose now is to determine the fermion masses, in order to constrain the various Yukawa parameters appearing in the superpotential. The AIw = I masses come from the SU(2)L x U(1)y  U(1)e phase transition, when
neutral components of higgs SU(2)L doublets acquire a vacuum expectation value (vev) a. The AIw = 1 masses come from the breaking of the PecceiQuinn phase symmetry, when some singlets acquire vev's Mo. Rewriting (4.1) in terms of the higgs (3,1) triplets and (1,2) doublets, we arrive at the
16
SU(3) x SU(2)L x U(1)y superpotential for this sector, with due care for the normalization of the kinetic terms of the component fields.
The notation $x is shorthand for TCX, with C the chargeconjugation matrix and T the transposition in Dirac space. A useful identity is
( c )T = IR (4.3)
Since all spinors are lefthanded,
$TCx = $TLCXL = IPRXL = xTC L = XR OL (4.4)
In the more convenient twocomponent notation, in which
)R  *2kR (4.5)
the various products are of one of the forms below:
O OT
'X bLXL
$'CxC (u2$*9)T(u2x*R =x (1kRXR)*
T f (4.6)
O'X (O2W*) XL = Ofr2XL = 'RXL
Pxc _ TO2X) = (OLtr2XR)* = (?LXR)*
corresponding to the left Majorana mass term, the right Majorana mass term, and the two pieces of the Dirac mass term. The mass Lagrangian for matter fermions is thus
= (dcscbc)M_ () + (UC, cC, tC)M2 ( + (ec s, Tc)M ()
+ + C.
Ve vc + (Ve',V ,1Vr)MD )p + (~e, vg 4) MR VAV"C + h.c.
(4.7)
The mass matrices are given below in terms of the parameters in the superpotential and the vev's of the various doublets. It is clear that the first four
17
terms in (4.7) are Dirac masses of the form IPRIL + h.c. In contrast, the last term is a "righthanded" Majorana mass of the form ORL.
The quark mass matrices are
0 a (H2) 0
M I a (H2) c (L2) 0 (4.8)
3 F2 0 N/0 b (H2)
0 d(K2) 0 M2 = 4p d (K2) e (Kj) f (K1) (4.9)
3 ( f (K') g (K2)
The parameter p represents an SU(3) renormalization group enhancement factor between the GUT scale and low accelerator energies. The parameter p is determined by any one wellknown ratio of eigenvalues of M_ and M at low energies (see below), neglecting differences in the SU(2) and U(1) renormalization of the fermion masses.
The lepton mass matrices are
0 a (H2) 0
Me = a (H2) 3c(L2) 0 (4.10)
0 0 b(H2)
0 Z(K2) 0
MD= (K2) (K) fK) (4.11)
0 (K2") g(K2)
0 jIMO 0
MR= (i1o 0 0 (4.12)
0 0 j2MO)
Now the two doublets from H and L (coupling to the 5x 10 sectors) will mix with doublets from other heavy fields, leaving light only the linear combination cos a H2 + sin a L2. Similarly, the only doublet from higgses coupling to the 10 x 10 sector which remains light is
alK + a2K2 + a(4K1
(4.13)
18
Inverting the unitary mixing matrices, we can express the vev's in the mass matrices (4.8)(4.12) in terms of the vev's aFT and UTT of the light higgs doublets:
(H2) = al = cosa aFT
(L2) = a2 = sina aFT
t
K2) = = a IOTT (4.14)
(K') = c74 = a2 atT
(K) = O75 = a3 aTT with
5 M2
1;,2 = = (188 GeV)2 (4.15)
~.iig2 (4.15)4g
t=1
We have enough freedom to redefine the righjt and lefthanded fields so that we can make the a1 real by rotating their phases away.
Proceed now to find the eigenvalues of the above mass matrices and the mass eigenstates V, as a linear combination for the current eigenstates 0 which we have dealt with so far,
0 = V Om (4.16)
with V unitary.
For charged fermions, all mass terms are Dirac, of the form (+h.c.)
ORMOL = tPRVV MVV1PL = ;mRAPmL (4.17)
where A = V1MV is diagonal. These mass eigenvalues can be made real and positive by further rotating the chargeconjugate ("righthanded") fields by some phases. Indeed, righthanded and lefthanded fields mix differently in general.
19
For charge( ) fermions gettting their mass from uFT, cos O sinG 0 Ve = sin O cose 0 (4.18)
0 0 1/
and
cos 0 sinO0 0 V_ = (ine cos0, 0 (4.19)
S0 0 1
so that
At = diag(me,mu,mr)
1 (au,)2 3 1 (4.20)
= diag , cfaCU 12, V2 bai
cu2 v'2
and
A_ = diag(md, m9, mb)
diag p 3(a1)2 p pL2, (4.21)
c2 Nag , C2,with
cos 20c = mn , .e., Oc ~ (4.22)
ms + md M.
and
cos 20, = m1  me ,.e., e me (4.23)
m + me M
The GeorgiJarlsksg relationships9 are
me= lmd (4.24)
MIL 9 M,
m = 3 (4.25)
mr mb
and we can identify the Yukawa couplings a, b, c, as well as the renormalization factor p:
1
a =  2memp Orl
b =  mn3 (4.26)
Orl
2
P= b = 2.81 (4.27)
20
To make masses positive, let
VeL=V, V L=V (4.28)
3 3
and redefine the chargeconjugate fields such that
1 1
VtR = Vt 1 V !R = V_! 1 (4.29)
For the chargej quarks, which get their masses from aTT, first disregard entries in M2 proportional to d (<< e, f,g), which means disregard the up3
quark mass. There is then enough freedom to absorb all phases in M2 by
3
redefining the c and tquark fields (and their chargeconjugates). Assume, equivalently, that eW4, du5, ga3 are real. Then (1 0 0 m
V2 cos sin) + O(mu) (4.30)
0 sinr7 cosr/ q with
9~ 3 
so that
A2 ~diag[0, mC, mt]
3 (4.32)
 4p diag [0, eu4  2rl f a5, g03 + 217fu5 The Yukawa couplings are thus
 mc
e=
4pu4
_ r,(mt  mc) (4.33)
4por5 U5
mt + mC
g 4
Reintroducing now the entry do3, complex with phase 1b, we can find CP violation in the full mixing matrix V2. First examine the determinant of the mass matrix
det M2 = ej mumcmt = (dC3)2(go3) (4.34)
21
where e'6 is the determinant of V2L. The full mixing matrix is then
cos 0. V2  sin 0
77 sin 0,,
sin 60 e 2 r7 sinu, e 2 cos 0, e 2 T7 + e 2
 lt i1
where sin 0 ~ t ~  mu mc
and A 2 = diag[mu,mc,mt]
3
with mc and mt given by (4.32) and
m, = 2p Idu32 c4
jdr31 =  MmU mc
4p
V2L VL = V2
33
V2R = V ( 3 3
and recalling that the hadronic weak current is
JA = p;qYU Y (KM) Om 1
3 2 3(4.41)
39 3
one can check that the KobayashiMaskawa matrix,62 KM, ends up being
KM=V R  L
3 V 3
(cos c sin c Cos sin Oc sin ql
sin Oc
cos Oc cos 7
 cos 0, sin q
up to CPviolating effects, better studied with the help of the determinant
(4.34).
(4.35)
(4.36)
(4.37)
so that
(4.38)
Letting
(4.39)
1
(4.40)
0
sin r7 cos r
(4.42)
22
This prediction is to be compared with the KM matrix obtained from an analysis of experimental data, which yields63 d S b
u 0.9733 0.0024 0.231 0.003 < 0.0052
KM = C 0.24 0.03 0.97 i 0.007 0.0435 0.003 (4.43)
t  ~ 0 < 0.076  0.999 1
The best fit of (4.42) to (4.43) providesil
Oc = (13.33 0.16)0
(4.44)
Y = (2.49 0.01)'
so that we can establish (in disagreement with the usual Fritzsch mass matrix result,64'65 and believing the top mass to lie21 around 45 GeV) that
7 ~ (4.45)
mt
We can now bound the Yukawa couplings involved, because it follows from (4.15) that JaiI < 188 GeV. We use the values of p and 7 given by (4.27) and (4.31), the standard masses for charged leptons, and masses66 of (5 Mev, 1.84 GeV, 45t GeV), where t is a dimensionless parameter assumed by certain wishful propagandists close to one.
jal > 5.5 x 10'
IbI > 1.3 x 102
Icf > 6.5 x 104
(4.46)
Idl > 4.5 x 105
e, IfI > 8.7 x 104
g > 2.1 x 102 t2
To find the eigenvalues and eigenmasses for neutrinos ( lefthanded ones are naturally light due to the seesaw mechanism67,68), we must diagonalize
23
a 6 x 6 matrix which takes a particular form, with the upperleft 3 x 3 block equal to zero. Following the construction below,69 we can reduce the problem to the diagonalization of two 3 x 3 matrices instead.
In the basis (ve, i,, v, v', v,), the full mass matrix is
M = T MR) (4.47)
There exist
H =)(t (4.48)
and
U = U, U (4.49)
with U, and U2 unitary, such that VTMV (L 0) (4.50)
with AL and AR diagonal, real and positive, with V = e U (4.51)
Indeed, let
S = iM,(M )1 (4.52)
and let U, and U2 be the unitary matrices such that AL = U1miU1
mi =  MDM1 MT (4.53)
and
AR = U2m2U2
M R (4.54)
M2 = MR + U(MR)MDMD + DMDMb(Mr)5
One can then predict neutrino masses and oscillation lengths from the parameters in the mass matrices, which are more or less known for MD and
24
open to speculation for MR. The flavor of these speculations, rather than a solid result, can be obtained with a couple of simplifications.
In the model at hand, asume first that d/d = i/e = f/f = /g = e. This assumption would follow from an SO(10) symmetry broken at a higher scale than the SU(5) unification we are considering. Such further unification must be rejected in the light of the discussion on baryogenesis (see below). Let x; = r = (jiMo)1, and let a be any of the a1.
The eigenmasses are then, in terms of the quark masses,
[mumUmC 2 e2 , r,  E1, rl + El, r2 + E2 (4.55)
2r ' r, r2
with
El (mc) (4.56)
2r,
and
2= (mte)2 (4.57)
The 90% confidencelevel limits on neutrino masses70,71 improve with every new analysis of e+e runs and yet, given their order of magnitude mV, < 250 KeV
(4.58)
mv, < 125 MeV
they provide only very weak constraints. Tighter ones can be obtained from Zeldovich's cosmological limit72,73
mv, + m,, + my, < 100 eV (4.59)
implying
r, < 10 4GeV
(4.60)
r2 < 108 GeV
25
so that 1 0 (ol)
.72 < 10 01 (.1
MO (4.61)
j2 < 10 3 O
where MO is measured in GeV (109 < MO < 1012).59
The mixing scheme depends very heavily on the relative magnitudes of the various parameters. Let us assume that ri = r2 = x 1, and neglect d (i.e., neglect the upquark mass) from all expressions. Then
Ul ~ o0 1 1(4.62)
where we have taken a4 ~ 05 (=> f ~ e). The estimate
V22 1 2x2(e4)2 (4.63)
2
can be identified with the cosine of the effective mixing angle for muon neutrino disappearance,
a, ~ ( xeU4  < 108 (4.64)
r
The other parameter relevant for neutrino oscillations is AM2 = M  m2 1
~ 2 m__ (4.65)
< 3 x 106 eV2
where we have used mt ~ 45 GeV. Given that the bound on j, is weaker, the mixing angle could be brought up to about 104. An increase in the mixing angle, however, would probably imply an increase in Am2, so the outlook is rather grim: neutrino oscillations seem to be at least two orders of magnitude away from currently imaginable detectability.74
26
The Full Superpotential
In order to spontaneously break SU(5) at a scale MX, and the anomalous PecceiQuinn symmetry at a scale MO (109 < MO < 1012 GeV, from the existence of red giants and from axions not closing the universe59), we add a term to the superpotential11,12
2= 01(E'M)3 + 72(0102  M2)04 + 732 + 774E3 (4.66)
The SU(5) content of the various fields introduced, as well as their charges under the global U(1)'s, are shown in Table II.
In order to give mass to color triplets while leaving the isodoublets light,60'01 we introduce the term11,12
G3 = (A1K + A2K' + A3K")MU + (#1K + #2K' +#3K")MU'
(4.67)
+ (11H + 2L)EV
which, taking M >> (E) = MX, (a) gives mass to two combinations of the quintets K, K', K" pairing them with the 5 fields U and U', leaving one light quintet, and (b) gives mass to one quintet (or rather, a doublet and a triplet) formed with L and H.
Adding furthermore a term11,12
G4 = (61H91 + 62LO1 + 63J02)E + 64MO102
+ (pKO' + P2K'E' + P3K"E'2 + p4J'('.8' + P5M)9
masses can be given to the colortriplet components of the two light quintets formed with G3, pairing them with the color triplets in J and J'. We thus end up with four triplets with mass around the intermediate scales 6M2/M and pM2/M (two each), and four massless doublets, which will acquire a mass
MW from spontaneous electroweak symmetry breaking. Notice that only two of the doublets (triplets) couple to matter at the tree level. From the
27
discussion on the renormalization group equations below, we shall obtain an expression for MX. The upper and lower bounds on the triplet masses translate into the bounds
105 < pt ,b; < 101 (4.69)
The anomalous PecceiQuinn symmetry X is not broken explicitly; hence, dimension five operators are suppressed.11,12 The full matter superpotential G = G1 + G2 + G3 + G4 (4.70)
is anomalyfree.
The Z3 discrete symmetry D under which
D 2i
E  e 3 E (4.71)
prevents terms like KJ'E or HJJE from appearing in G4. It effectively forces E to be a 75 instead of a 24, thus allowing the missing multiplet mechanism implemented here to work. Charges under D are displayed in Table II. Familiarity is unfortunately not preserved in the full G, so it is perhaps useful to think of it as a lowenergy quasisymmetry. No familons are to be expected in this model, although one could imagine more complicated versions of G2 + G3 + G4 that would require them at some intermediate scale.
28
TABLE II. Field content and symmetries (vectorial, PecceiQuinn, discrete Z3) of G. Superfield SU(5) irrep V X D
H 5 2 2 2
L 45 2 2 2
K 5 2 2 2
K' 5 2 2 2
K" 5 2 2 2
01 1 10 2 0
02 1 10 2 0
03 1 0 0 0
04 1 0 0 0
E 75 0 0 1
E' 75 0 0 2
U 5 2 2 1
U' 5 2 2 1
V 50 2 2 0
J 5 2 2 2
J' 5 2 2 0
81 50 2 2 0
50 2 2 1
E50 2 2 0
25 2 2 2
PROTON DECAY
We know 75,76,77 that all possible protondecay78 operators are of one of the forms below. Greek indices run from one to three and represent color, Latin indices from the middle of the alphabet run from one to two and represent weak isospin and Latin indices from the beginning of the alphabet run from one to three and represent family or generation. Chirality is indicated for definiteness:
0abcd (daaLUIbR)( %cRqjdL)fafs ij 2bcd ( ZaaRqjbL)(cLuydR) Ecqpy Ej(
Oabcd = (daaLUfibR)(iLuydR)Ea#30abcd (4iaaRqjpbL) (1mcR nqydL)Ef.y Eij Emn
Recall that, as usual, ?PLXR = ZLOR, and notice that
0 2 o2 (5.2)
abcd  bacd
The form of these operators is constrained by the requirement of gauge invariance under the standard SU(3) x SU(2) x U(1). They arise dynamically from integrating away heavy particles (higgs fields or gauge bosons of broken symmetries). The effective Lagrangian for the fermionic components of matter fields contains, in our case, 01 from (FT)2 terms and 02 from (TT)2, with the family structure dictated by familiarity (7) invariance. There are no 03 or 04 operators, though, because we do not allow H  K nor L  K mixing terms; i.e., we suppress protondecay operators involving higgsinos28,29,30 with a chiralityflipping mass insertion in their propagator and gluinos.
29
30
To illustrate the point, consider a onefamily toymodel with a Yukawa superpotential of the form
a FTH + b TTK (5.3)
where F, T, H and K transform as 5, 10, 5 and 5 of SU(5). Both H and K contain color (anti)triplet, isosinglet pieces. These couple both to leptoquark and to diquark superfields, causing proton decay in two different channels. FF can go into TT via H exchange, and TT can go to TT via K exchange.
Because of the helicity structure required by the structure of chiral vertices, these processes correspond to "dimensionsix" protondecay operators, in which all four external lines are fermionic and the bosonic component of the triplet (from H or K) propagates through the internal line.
Were we to further allow the mixing term
kt HK (5.4)
then tripletinos (but not bosonic triplets) would mix, inducing the process squark + squark 4 antiquark + lepton in the channel FT  T T. The engineering dimension of such a diagram, with two scalar and two fermionic external legs, is five. The initial state can be virtually produced from a quarkquark pair (in the nucleon) via gluino exchange. Since gluinos couple strongly, the amplitude for the whole process (gluino exchange and higgsino exchange) is essentially the same as for the dimensionfive higgsino exchange subprocess. This amplitude crucially differs from the one for quark + quark f antiquark + lepton via higgs boson exchange, in that it involves a triplet fermion propagator (rather than a triplet bosonic one), so it is suppressed by the mass (rather than the mass squared) of the triplet superfield.
31
Proton decay thus proceeds faster through these higgsinomediated diagrams (dimension five) than through higgsboson mediated ones (dimension six) roughly by a factor of ( Hi$ )2. If the theory contains dimension five opproton
erators, the mass of the supertriplet must be boosted above ~ 1014 GeV (rather than above ~ 1010 GeV) in order to prevent fast proton decay rates.60,79,80,81,82 We can suppress the mixing term uHK at the cost of losing the usual baryogenesis mechanism, from the interference between H + TT at tree level after tmixing of H into K, and H  TT at one loop with T, F and H  K in the internal propagators. Note that in this onefamily toy model this interference is zero even with the mixing term because the product of Yukawa couplings is necessarily real.83 (The chiral structure of the processes is what interests us now.) Baryogenesis seems to require an interaction among, and hence the existence of, different families!
The baryogenesis problem will be solved shortly, but we can foresee that since the higgs triplets will have to be rather light, the dangerous mixings must be suppressed. The action presented above does so naturally with two of the global U(1)'s, PecceiQuinn and familiarity. Around the electroweak scale, nevertheless, supersymmetryviolating effects will induce such mixing, safely suppressed by 0( ).
Proton Decay Lagrangian
The superpotential arising from G1 (Eq.(4.1)) involving matter and colortriplet higgses is given below, where family is indicated explicity but color and Dirac indices have been dropped.
In fourcomponent notation, V)x stands for
POCa X = Vi)Cxi = XCPL (5.5)
32
For calculational purposes, it is much easier to use twocomponent notation and the physically relevant left and righthanded fields, substituting in all the previous expressions
(5.6)
and transposing the first entry in VX after the substitution:
a
GtripLeta = H3 '+  1iq2i  12iq1i
b ij c
+ H3 Ucdc e3.q3ij + L3 (  2
3,2 ( (U3 2 F 2 q(5.7)
 4dK3 ucec + ucec + qlq23Etj)  4gK3 (ucec + u3d3)  4eK3 (uec + u2d2)  4fK (ue + ucec + q2iq3j)
where = , q = (), all the triplets have canonical kinetic terms, and
the Yukawa couplings a, b, c, d, e, f, g were given in terms of measured fermion masses and constrained doublet vev's above.
Triplets coupling to the same matter in Eq. (5.7) mix,
(H3 _ cos a sina EFT cosa 'FT +L3I sina cosa) X sina 3FT + J
(Kf at 11 TT ETT+ (5.8)
Kj3 =; a2 p ~ X) = a rETT+ +1~a ETT a4 ET
K 2 2 2 X "I2 T + Only EFT and STT remain light, with masses MFT, MTT around the intermediate scale, light by unification standards. The a. can be chosen to be all real, just as we did with the doublets when we were dealing with the fermion masses. Although we have used the same notation (a;) for the entries of the mixing matrices of doublets and triplets, there is no reason why they should be the same.
33
The superpotential (5.7) thus becomes effectively
Gtri pets = CFT cos a = (uld' + u'd'  tj 2je6  2iq1jfi3
b
+ cosa Ud  t3 qj j
 sin a ucd  eq2'ii)] (5.9)
 42TT 1dai (uie + u2ei + glq2jE) + gal (u ej + u3d3
+ ea2 (ueM + u2d2) + fa3 (u~ei + e + q2333jE2)
This is the expression from which we will compute the baryon asymmetry of the universe. Since 2FT and PTT do not mix, we need not consider higgsinos. The interaction term in the Lagrangian involving the scalar components of the higgs superfields E and the fermionic components of matter in (5.9) is exactly the same expression (5.9), with the usual evenR fields instead of superfields.
At energies much lower than the masses of EFT or =TT we can integrate away PFT and 2TT. Keeping only terms with only one lepton field, we arrive at the following lowenergy Lagrangian for proton decay, where all fields are still current and not mass eigenstates:
L = 1 (Labcd < abcd >v Labcd < abcd >e +Rabcd[abcdl) (5.10)
The L's and R's are coefficients with indices in family space (a, b, c, d E {1,2,3}). The various proton decay operators are, in the notation of (5.1), abd = < abcd >v  < abcd >e
Obcd = [abcd] + [bacd] (5.11)
so we are breaking up the SU(2) doublets q and I appearing in (5.1), since proton decay takes place at energies below the SU(2) breaking scale.
34
Explicitly,
< a
< a
bed >v = (d'aucl (cdd )
= (4LubR) (cRddL) bcd >e = (d'uc)t (ecud)
= (LubR) (RudL) abedd] = (u'd') (ecud)t
= (aRdbL) (ELudR)
or, in twocomponent notation,
< abcd >v =  (UTRdaR) (vcLdL) < abcd >e =  U TRdaR eLudL)
[abcd] =  dTRUaR (4TudL
TABLE III. Coefficients of baryonnumberviolating fourFermi operators in effective Lagrangian before electroweak breaking. abed 2M'TLabcd 16'MT Rabcd
a2 cos2 a ac sin a cos a
ab cos2 a
c2 sin2 a bc sin a cos a
b2 cos2 a
(5.12)
(5.13)
1212 1222 1223 1233
2222 2223 2233 2323 2333 3333
d2 a2 de ala2 df a1a3 dg ac e2a efa2a3 ge ala2 f2 a2 gf al a3 92a2 g1
35
The coefficients Labcd and Rabcd satisfy the identities Labcd = L(ab)(cd) = L(cd)(ab)
(5.14)
Rabcd = R(ab)(cd) = R(cd)(ab)
and they are all zero except the ones in Table III and their symmetrizations according to (5.14).
Rotate now all the quark and lepton fields into the mass eigenstates, denoted by the additional subscript m, and related to the current eigenstates by a unitary transformation 0 = V,m. The new proton decay operators have the same form as the old ones, but with massive fields in them. They are related to the old ones by four mixings, one for each field. Summing over the various neutrino flavors, indistinguishable in protondecay experiments, and neglecting the seesaw coupling between ordinary neutrinos and massive righthanded ones, we can express the Lagrangian (5.10) in terms of mass eigenstates:
L = 1 (Labd < abd > + Labcd < abcd > +RebcdjabcdI) (5.15)
The coefficients
Lvb Lefgh(t 1i ) b 2 ) bd
Labd d
e,f,gh 3 3
Lbcd =  5 Lefgh C VtlR a(VIR)f (VtL)gc(V !L)h (5.16)
e,f,g,h
Rabed= RefghM2Rg)'a(VR b(VtLicg( d
e,f,g,h
can be evaluated using symbolic manipulations on a computer, without much problem. The full formal expressions are long and unilluminating.
Renormalization of Proton Decay Operators
One last step must be taken to obtain the effective proton decay Lagrangian, useful to compute proton decay rates: we must renormalize84 all
36
the (dimension six) operators down from unification to our energies.75,77,80 Indeed, the expressions for the coefficients L', Le, R' of the various operators are valid at the unification scale. The renormalization of the Yukawa couplings themselves is taken into account by using lowenergy fermion masses, but the full operators undergo further renormalization at lowenergies, by a multiplicative enhancement factor which can be calculated using dimensional regularization. We neglect U(1) renormalization, assume exact supersymmetry above MW, and renormalize with oneloop equations down to the charm mass. Below MW, we consider only SU(3) renormalization. These are all reasonable assumptions in light of the requisite accuracy.
Both operators 01 (mediated by scalar H3 and L3) and 02 (mediated by scalar K3, K', Kg) could mix under SU(3) renormalization with their supersymmetric partners, the dimension five operators mediated by higgsinos with the same quantum numberswere those operators allowed by the theory at all. They cannot, however, mix with any other operators under SU(2) renormalization because under exact supersymmetry the SU(2) gaugino is Majorana massless: were it to be exchanged between two external legs with SU(2) quantum numbers, it would undergo a helicity flip to preserve the chiral structure of the vertices, which it cannot do because (as advertised) it is massless.85
Let us consider the spectrumil contributing to the renormalizationgroup equations between MX (unification scale) and the lowenergy world. We have three light fermion families (the singlets are heavy but irrelevant for group behavior), with the topquark mass around 45 GeV, four light isodoublets, with masses around Mw, and four light color triplets with masses around Mo _ 1010 GeV. We assume all other higgs superfields have masses of at least O(M)
37
If the coupling constant for the ith simple factor of SU(5) is, to one loop, of the form
4ir
Cti(6) =7 (5.17)
b, log (p2/A2)
we can find the coefficients b; from the following expressions, where nq (ne, nd, nt) is the number of quark flavors (lepton flavors, isodoublets, color triplets) below the energy scale it,
For nonsupersymmetric QCD, (SU(3) below Mw),
b3 = 11  2nq (5.18)
For supersymmetric QCD, (SU(3) above Mw),
1 1
b3 = 3 x 3  (2nq + nt) = 3  nt (5.19)
2 2
where the 2 comes from counting quarks and antiquarks.
For supersymmetric SU(2),
1 1
2 3 x 2  (nq+fnl+nd) = nd (5.20)
Finally, for supersymmetric U(1),
1 3
b=  (nq + nt)  frac3l0nd = 6  3 nd (5.21)
As boundary conditions for the renormalizationgroup equations we use a'(Mw) = 128.5 a '(Mw) = 0.12~1 = 8.3( 0.4) (5.22)
Recall also that a~(MW ) = a(Mw) sin20W = ae (Mw)xw a (Mw) =3 a1(M)cos2O0 (5.23)
5 M
3
= a I(MW) (1  XW)
5 C
w
38
whereas the unification scale is defined by
a = ai(MX) = a2(MX) = a3(MX) (5.24)
The oneloop renormalization equations are thus
a_  a2'(Mw) = 1 (2) log
a1  a'(Mw) = (7.2) log
27r (Mw) (5.25)
a~1  a31(M) = (1) log
3 ~27r MX
a3~I(M)  a3 (Mw) = (3) log
27r \MW/
Solving them, one easily finds11,86
MX = 5.6 x 1016 GeV a = a(MX) = (19.60)' (5.26)
xw = sin2 Ow(Mw) = 0.237 with errors of about 5% arising mostly from uncertainites in Ag, where MS is the modified minimal substraction scheme in QCD. The rather high value for xw is in excellent agreement with the most recent measurements of xw in deepinelastic neutrino scattering.87,88
Now for the actual renormalization of the qqqe operators. Since we are neglecting U(1) effects, all operators at hand get renormalized in the same way. Letting A be any of the coefficients L', L', R' in the effective Lagrangian, the enhancement factors relate A at one scale to A (and all other B with the same quantum numbers) at some other scale. Specifically,
A(mp) ~ A(m,) ~ E2E3nsE3ssA(MX)
(5.27)
w
39
where we have neglected El and the enhancement factors are
E2 = a2(MW) ]'12/b2
a
[ a(mt l/b(G lb5 [a( 1~/33
E3  a3(Mt) , (a(mb) 3 a3 (mb) (5.28)
E38  a3(M W ) 3da3)) J
a3(MO) I/ a(o
The y's are the relevant eigenvalues77 of the anomalous dimension matrices, which happen to be the same as those recently considered for the case of dimension five operators:85
3
=2
ns = 2 (5.29)
Y3
The 0function coefficients can be read of from (5.18), with nd = 4. The superindex in E3n, denotes nq, and nt in E388.
With a topquark mass of 45 GeV [and a3(MO) ~ 171, MO ~ 1010.5 GeV], the enhancement factor is
E = E2 x E3na x E388 = 1.39 x 1.26 x 1.65 = 2.89 0.2 (5.30)
Proton Decay Rates
Armed with this lowenergy potential, we can calculate the proton (and neutron) decay amplituyes into various channels. However, a nucleon at rest is hardly a system made up of three bound pointlike quarks, and although much effort has been made, using various techniques,78 to produce reliable amplitudes in terms of the qqqe operators, large uncertainties remain. We follow here the most straightforward approach.89 For the phase space, we assume the
40
quarks to be nonrelativistic and the antilepton extremely relativistic.90 All nuclear effects are disregarded, and we concentrate on twobody p + mi decays; that is, we ignore the pion pole contributions and all threebody decays. Proton and neutron wave functions are taken to be SU(6) symmetric,91,92 where SU(6) D SU(2)s,, x SU(3)flavor is the lightcone spinflavor symmetry.
Following Ref. 89, the lifetimes of each mode can be conveniently written in the generic formal
= 101(5.31) where m is the mass (in GeV) of the color triplet mediating the decay, MFT if the outgoing antilepton is righthanded, and MTT if it is lefthanded. The y coefficients (in years) for the main decay modes are shown in Table IV. Their explicit dependence on the triplet mixing parameters is also shown, and since we are looking for an limit on MFT and MTT, the upper bound (188 GeV) for all doublet vev's has been used, i.e., the bounds in (4.46) have been saturated.
At 90% confidence level, the latest limits on the proton lifetime into strange modes are6
r (P __ A+ KO > 4 x 10 31 years (.2
>4x 1031(5.32)
r (p + K+) > 5 x 1031 years
These limits can be translated into lower bounds on the masses of the color triplets mediating proton decay:
mFT > 0.75 x 115c2  2.8s2  0.055sc x 1010 GeV
mFT > 0.53 x V13.2c2  12.2s2  20sc x 1010 GeV
mTT > 0.32 x 1010 GeV
x  319a2 + 319a2 + 0.6a  2.7ala2  0.17a1a3  27.8a2a3
(5.33)
41
where c, s stand for cosa, sina, and a1, a2, and a3 stand, respectively, for cos a', sin a'cos a", and sin a' sin a". For any values of the mixing angles, the above expressions can only vary so much, so that a perhaps more useful result is
mFT (4  10 X 1010 GeV
mF4 x 10)4
4 xO1 years (5.34)
mTT = (5  18) T K 4 x 1010 GeV
5 x 1031years]
where the lifetimes are normalized by the current limits, for years of use.
It follows that masses of the color triplets are bound below by
mFT, mTT > 2 x 1010 GeV (5.35)
This lower bound on the triplet masses translates into a lower bound for the mass of the inflaton (A2/M) from which triplets decay,7,8,9,10,11,12,13 so we can extract a bound on A purely from underground detector results: A > 104.25 (5.36)
Remarkably, proton decay experiments set a lower limit on A which is essentially the same as the lower limit (3.9). The upper limit (3.5) on A arising from cosmological considerations on gravitino abundance at the time of nucleosynthesis is very close, so A is (as repeatedly advertised all along) very tightly constrained.
It is perhaps amusing to extract more mileage from the relationship between the reheat temperature and the gravitino mass, Eq.(3.4). The bound (5.35) translates into the bound (5.36) because inflatons must decay into triplets in order to produce the baryon asymmetric universe we live in. The two inequalities (3.4) and (5.36) result in
A < 1.54 x 107 = 3.7 x 1011 GeV
(5.37)
42
Proton decay experiments have thus obtained an upper bound on the gravitino mass! In numbers the result is not so impressive: m3 < 57 TeV (5.38)
2
Along the same lines, the lack of experimental confirmation for supersymmetry in accelerator experiments allows us to expect m3 > 100 GeV (5.39)
from which it follows that A < 103.3 (5.40)
hence
m3 < 1.9 x 101 GeV ==> TK < 1.3 x 1040years (5.41)
On the other hand, assuming supersymmetry is not found at the SSC, then the gravitino mass will be narrowly constrained between lower bounds from the SSC and upper bounds from upgraded proton decay experiments. If the largest conceivable Earthbased detectors6 are built and work, the proton lifetime limit will improve to TyK > 5 x 1034years (5.42)
Then
A > 103.9 (5.43)
and
m3 < 7 TeV (5.44)
2
which could be a useful constraint.
43
TABLE IV. Decay rates for twobody final states Mode F' Branching Rates (yr1)
e+,r0 1.02 x 1027E x1111
R1.02 x 10271E x L'
e+RKO 1.39 x 1O27jE x L11
e +K0 1.39 x 1027E x1111
p+R,,0 1.00 x 10271E x L11
L1.00 x 10271E x R1 p+KO 1.34 x 10271E x Le12
e4K0 1.34 x 10271E x L 2
A R7r 2.05 x 10271E x L 1
ARK+ 1.39 x 1027JE x LI12
VRK+ 0.012 x 1027.E x (312  L2112
e~r; 0.20 x [ e+70 ep+0 0.057 x 1 e+7r01
e+w 0.49 x [e+r0 1
e+K*O 0.0088 x T[e+ K
e+7 0.20 x F[p+7rO
ep0 0.045 x F[p+7r0J
,+77 0.20 x [0.+7ro
T p+ 0.20 x ]F[Vr+]
BARYOGENESIS
The observed baryon asymmetry2,93 of the universe,
nBB  1010 (6.1)
is a quantification of two surprising facts. One, that no significant amounts of antimatter exist. Two, that in fact there is much more matter per unit entropy (per photon) than one would expect from an equilibrium evolution of the primeval "singularity." The generation of this baryon asymmetry, a process known as baryogenesis, requires three fundamental conditions.94 First, baryon number must obvioulsy be violated. Next, thermal equilibrium must also be violated. Finally, C and CP must also not be conserved.
Baryogenesis may proceed after the inflationary phase of the evolution fo the universe, by allowing the matter sector of the superpotential G to contain higgs triplets which produce a nonzero baryonnumber asymmetry 6B per tripletantitriplet decay, and such that the inflaton can decay into them preferentially (i.e., m3 < mo).
Since the inflaton is never in equilibrium with other fields, we do not have to worry about producing enough triplets via a Boltzmann distribution.24 Indeed, the inflaton oscillates into the heaviest fields around, and we can simply estimate7,8,11,12 that all of the inflaton's energy is released into heavy triplets which quickly decay into radiation, reheating the bubble to a temperature TR:
44
45
nBB ~ BBn~6B n 6B TR (6.2)
Pp/TR m
bBA
~6B
M
In this expression 6B is the baryon number asymmetry produced per decay of an inflaton and an antiinflaton. Given that the inflaton's branching ratio into color triplets is practically one, then 6B is essentially the baryonnumber asymmetry produced per tripletantitriplet decay, which we calculate from the matter sector of the GUT superpotential. Clearly, G must contain triplets of mass a little lower than the inflaton's, with a rather high decay asymmetry 6B  106 (6.3)
At tree level, all jamplitudes12 are real, so CP violation can be produced perturbatively only at the oneloop level. We need oneloop decays which can interfere with treelevel decays to produce a net baryon asymmetry.25,26,27,95,2
Indeed, if a generic particle X (colored higgs, say) can decay into two channels with different baryon number, for instance
2
X 44q (Bfinal =2)
X 1~ (Bfinal 3 (6.4)
3
where q is a quark and t a lepton (superfields), then the baryon asymmetry 6B produced per decay of a pair of X and X is
1 (X X)  2P(X 4(X e) + !(X  qq) bB = 3+   3
IF (X +tg)+ + 4)T X i + r (X + qq)
= F(X + qq)  J(X v) (6.5)
r(X + anything)
1(X  qt  (y i+ g)
r(X  anything)
46
where we have used CPT invariance.
In general,
6B= Bi(r  Y) (6.6)
where the sum extends over all decay channels i of X, each with a net baryon number B;.
Let
F(X + qt) = gTfT + 9LfL2 (6.7)
where the amplitude is the sum of a treelevel graph and a oneloop graph (disregarding higher orders in h), with g the (product of) coupling constant(s) and f the dynamical factor, which depends on masses and external momenta and is the same, by CPT invariance, for X as for X decays. Then F(X *i ) = g'fT + g*fL2 (6.8)
so that (6.5) becomes simply 4Im(fjfL)Im(g*g L)
bB = (6.9)
L(X  anything)
or, in the general case with more than two decay channels,
4 E B;m(f fLi)Im(g* 9LJ
E IgTifT + gLi 12 (6.10)
4 E B;Im(f f(i)Im)(g gL; E IgTn fTi IT
The expression Im(f fL) can be calculated using the LandauCutkosky rules for the interference diagram fjfL. Since at tree level all the phases disappear, we may choose couplings so that f = fT. Then Im(fjfL) = fTImfL. To evaluate this, all one has to do is modify the internal propagators in fL separating initial from final states by putting them on shell.
47
All said and done, only the innermost line in the diagrams, where a higgs triplet is usually propagated, is allowed to be offshell. We could add gauge interactions to these diagrams but (contrary to lowenergy processes, such as proton decay) at the energies relevant for our dicussion, all gauge couplings are weak, so that renormalization of these operators can be neglected. All vertices are chiral, and even cubic if we are willing to introduce spurious fields to take into account mass terms.
The only way arrows (chirality) can arrange themselves in the interference diagrams is with mass insertions (chirality flips) in either the external line only, or else in two of the internal nonhiggs lines. The origin of these constraints is, in supergraph language, the fact that interactions in the superpotential involve only chiral fields, so all three arrows point either into or out of any given vertex.
All the baryon asymmetry is generated from the higgs fields coupling to the 10  10 sector, for otherwise the product of couplings gTgL is real. More particularly, 6B $ 0 arises from the interplay between the only two higgs superfields coupling to the same sector, both with nondiagonal (in family space) couplings. Remarkably, both higgs bosons and higgsinos can produce a baryon asymmetry upon decay.
Taking into account that K" = a tT, K3 = at , one can evaluate the interference and obtain the following asymmetry per decay:11,12
b1 a I2a32Im(gff21j2) MO (6.11)
327r2 11,12 + 11212 + 1j,12 + II2 + Il2 [MTT
Given that the triplet must decay into neutrinos (righthanded ones), with masses given by (4.55), it is clear that
jMO, j2MO < MTT
(6.12)
48
so we can write, assuming a hierarchy of Yukawa couplings and all phases to be of order unity,
6B < 1 a1! a31 V11 1f21 (6.13)
327r2 I0I
To estimate this, take a,  a3 ~ 1, disregard all doublet mixings, and assume proportionality between tilded and untilded Yukawa couplings [via SO(10)]. The value thus obtained,
2
6B < 103 ~mmm 1010 (6.14)
is much too low, so the presumed SO(10)inpired proportionality between tilded and untilded Yukawa couplings is to be rejected. Indeed taking all Yukawa couplings in (6.13) to be of order 101, 6B is about 105, an order of magnitude above the required value.
It is thus rather easy to produce the baryon asymmetry of the universe from triplet decays into righthanded neutrinos and ordinary quarks. The parameters to play with are those in the neutrino Dirac mass matrix, while making the righthanded Majorana masses as large as allowed, i.e., smaller than the triplet mass MTT. The rather large Yukawa couplings between SU(5) singlets and quarks could be ruled out if neutrino oscillations were to be observed, as pointed out above.
Baryon number violation has been searched for with greatest care in underground detectors, to no avail. The rather strong limits on proton decay rates provide us with a lower limit on the mass of the baryonnumberviolating triplets. The upper limit on this mass (assuming Yukawa couplings to be perturbative, i.e., smaller than one) sets an upper limit on the lifetime of the proton, limit which unfortunately falls beyond the neutrinoinduced background on Earth.
CONCLUSIONS
We have achieved to construct a model in which the two bounds on the mass of the color triplets (lower from protondecay experiments, upper from the requirement of baryogenesis) are met. Good relationships between fermion masses and a reasonable KobayashiMaskawa matrix (including CP violation) have been incorporated into an anomalyfree grandunified model.
We have also established a hierarchy of masses
Mrightneutrino
at the intermediate scale, 1010 GeV This intermediate scale arises from a plurality of physical considerations associated with supersymmetry breaking, inflation, baryogenesis, and the breaking of the PecceiQuinn symmetry.
49
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(1984).
BIOGRAPHICAL SKETCH
Born a month into the decade of the sixties in Mexico, the candidate developed an early interest in physics while riding the giant roller coaster in the park of Chapultepec. A total solar eclipse the same year of the massacre of Tlatelolco further kindled a desire to understand nature, without necessarily pretending to dominate it.
His discovery in 1970 of the Mediterranean sea, its waves, and its peoples did not distract him from academic endeavors until 1976, the first year after fascism (and after highschool), during which he decided to join the Universitat de Barcelona as a freshman in physics. That tumultuous and exciting first taste of academia ended when the candidate moved to New York, to witness the last year of the "humanrights" presidency.
Disoriented by the contradictions apparent in the streets of New York, the candidate joined, along with most of his close friends, the CCFRR neutrino collaboration. After a rather long winter at Fermilab, he quit experimental high energy physics, secured a Master of Arts from Columbia University, and joined the Particle Theory group at the University of Florida, under the academic guidance of Professor Pierre Ramond.
Since then (1983), his research interests have meandered from the basic mechanics of grand unification, to the interface of particle physics and astrophysics, to the more formal issues associated with string theories and their compactifications, paying nevertheless close attention to the possible phenomenological implications of mathematical ideas.
55
56
The charming rural setting has proven most conducive towards research activities in a field somewhat unrelated to social conditions. The candidate appreciates some sports but particularly dislikes football, and deeply deplores the racist atmosphere of the University of Florida, the most obvious exponent of which has perhaps been the obstinately reactionary stance taken by its Administration on the apartheid regime and the divestment issue.
The candidate expects to continue research on the forefront of particle physics at the interEuropean research center CERN. Two years there seem to constitute the only certainty of his personal future, which nevertheless looks brighter than that of the world at large. He hopes that the mathematical study of nature can contribute to a greater understanding of all the peoples of the world in tolerance, peace, and justice.
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Pierre Ramond, chairman
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Ric ar . Field, CoChairman
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
60? ['Ai
I
1~
J 1es of PysU s P fessor of Physics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
ra ee . umar
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
Pierre Sikivie
Professor of Physics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.
ThmsW. Smo
Professor of Philosophy
This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy.
May 1987
Dean, Graduate School

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GRAND UNIFIED SUPERSYMMETRIC INFLATIONARY COSMOLOGY By MARTI RUIZALTABA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1987
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Per al Padri To all those who have died in the struggle for justice while I wondered about the mathematical consistency of observable reality.
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ACKNOWLEDGEMENTS I wish to thank all those who talked, discussed, and enjoyed physics with me during my student years. My advisor Pierre Ramond led me through many mazes with that characteristic serenity and insight of his, which only a grand master of knowledge commands. We had much fun together, even though we never played tennis. Were I to be fair, the list of people who have played a crucial role in the learning process of which this thesis is an official landmark would be very long, starting with my mother and growing every day. A detailed multilingual appendix for this purpose does not seem in order. Let me instead collectively acknowledge the good friends, the good lovers, the good teachers, the good students, the good neighbors, and the good fighters with whom I have participated in this mysterious process of life. Let me, however, acknowledge as well the unreliable friends, the poor lovers, the boring teachers, the lazy students, the gossipy neighbors and the cowardly traitors, for the wisdom I have gained in dealing with them and their personal irrelevance. This research was supported in part by the United States Department of Energy under contracts No. DEAS0581ER40008 and No. FG0586ER40272. Â• Â• in
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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS Hi ABSTRACT v INFLATION 5 SUPERSYMMETRY BREAKING 10 GRAND UNIFIED SECTOR 13 Fermion Masses 15 The Full Superpotential 26 PROTON DECAY 29 Proton Decay Lagrangian 31 Renormalization of Proton Decay Operators 35 Proton Decay Rates 39 BARYOGENESIS 44 CONCLUSIONS 49 REFERENCES 50 BIOGRAPHICAL SKETCH 55 iv
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy GRAND UNIFIED SUPERSYMMETRIC INFLATIONARY COSMOLOGY By Marti RuizAltaba May 1987 Chairman: Pierre Ramond Major Department: Physics A theoretical laboratory is presented, within the framework of supergravity unification, which satisfies all experimental and cosmological constraints below the Planck scale, notably those from standard big bang nucleosynthesis, baryogenesis, and the fermionic spectrum. An intermediate scale, around 10 10 10 n GeV, arises from the examination of inflation, supersymmetry breaking, the PecceiQuinn symmetry, fermion masses, proton decay, and the baryon asymmetry of the universe. Careful attention is paid to the renormalization of proton decay operators. Although a realistic low energy spectrum follows from an isospinlike global symmetry (familiarity), a rather baroque higgs sector is needed to cancel anomalies and implement the missing multiplet mechanism. A low reheat temperature and baryogenesis from outofequilibrium decays are perhaps the most noteworthy aspects of the model, whose general success is not yet matched by any models derived from strings rather than point fields. v
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INTRODUCTION One of the most astounding successes of the grand unification program, 1 initiated about ten years ago, was the understanding of the origin of the baryon asymmetry of the universe. 2 The program suffered an early setback when minimal SU(5) failed to explain quantitatively this baryon asymmetry. It also predicted a fast proton decay rate which was soon ruled out by experiment. Supersymmetry 3,4,5 generally extends the proton's lifetime and prefers the strange and muonic decay modes. The experimental limits on the proton's lifetime 6 and the requirements of efficient baryogenesis tightly constrain the parameters in the Lagrangian of candidate unification models. An excellent theoretical laboratory arises from the locally supersymmetric grand unified model below, 7 ' 8 ' 9,10 ' 11, 12,13 which is consistent a) internally below the Planck scale (anomalyfree), b) with the standard SU(3) x SU(2) x U(l) model at accelerator energies, c) with big bang inflationary cosmology 14,15 ' 16 at very early times, and d) with underground detector limits on proton decay. The general traits of this supersymmetric inflationary cosmology (SIC) are presented below. The seminal papers for SIC are Refs. 7 and 8. Further work appeared in Refs. 11 and 12. Although the model is not particularly aesthetic, it serves to prove, by construction, the consistency of all known constraints with simple supergravity unification. Supersymmetry is spontaneously broken by the O'Raifeartaigh mechanism 17 ' 18 and its breaking is related to the end of the inflationary period in the evolution of our universe. The cosmological constant can be set to zero
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while supersymmetry is broken, but a contribution of 0{M^Mp l&nciL ) cannot be avoided after the observable sector finds out about supersymmetry breaking. No mechanism is known in any model to solve the cosmological constant problem, so we proceed with the rest of the program without much uneasiness. A specific model with SU(5) invariance is written down, relying freely on insights gathered from explorations with larger groups. The mass spectrum of leptons and quarks is incorporated at the tree level with the aid of couplings between matter and a nonminimal higgs sector including 19 a 45. The element KM c b of the quark mixing matrix has been measured from the (long) Bmeson lifetime. 20 We incorporate it and a low topquark mass (claimed and subsequently disclaimed by the UA1 group 21 ), obtaining reasonable fermion mass relationships and adequate CP violation. Since gravitinos are created during reheat, 22 the temperature Tr of the heat bath generating them is bounded above. 23 If reheated fields are thermalized immediately, the products of the reheating process are restricted to having masses < Tr, lest their abundance be strongly suppressed, ~ exp(Â— Tji/m). However, if the reheat is not an equilibrium process, 24 then no such strict bound can be derived for the reheated products, which could be more massive than Tr. In the SIC theory presented here, reheat is caused by the decay of inflatons, of mass around the intermediate scale, 10 10 Â— 10 11 GeV, and yet the reheat temperature is only about 10 6 GeV. Higgs triplets of masses around the intermediate scale can be used for generating the baryonnumber asymmetry 25 ' 26 ' 27 without causing fast proton decay via the dimension five higgsinomediated operators. 28 ' 29 ' 30 The method presented here calls for righthanded neutrinos with masses around, again, the intermediate scale.
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Proton decay rates, with largest branching ratios into strange and muonic modes, put a lower bound on the masses of the higgs triplets whose interactions violate baryon number. The requirement that these same triplets be produced from inflaton decay to generate the baryon asymmetry of the universe sets an upper bound on their masses. A narrow window satisfying these two bounds is found at the intermediate scale. This dissertation is organized as follows. First, the SIC superpotential is presented and the hidden sectors are glossed over. The grandunified (GUT) sector is then presented in detail along with its implications for fermion masses, CP violation, and neutrino oscillations, 31 in terms of the Yukawa couplings of the superpotential and the various symmetrybreaking scales. We then compute the decay rates for the proton, under flavorhelicity SU (6), paying careful attention to the renormalization of the effective Bviolating lowenergy operators. Finally, we study baryogenesis and present a mechanism involving heavy righthanded neutrinos, whose masses are also around the intermediate scale. The field content of supergravity 32,33 is as follows. Chiral superfields (D a { = 0) contain matter, in spinorial and scalar varieties. Gauge superfields (W a Â— D D a V, V = V>) contain gauge vector bosons and the spinorial gauginos. In addition, the spectrum contains the gravity supermultiplet, with the graviton, the gravitino and the KalbRamond field. Supergravity theories are never torsionfree due to the interdependence of the fermionic and bosonic coordinates of superspace. 34 Minimal N = 1 supergravity in four dimensions has two arbitrary independent functions, which reduce to only one superpotential P in the "minimal"
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case. Kinetic terms for all fields are in this case canonical, and the only requirement on P is that it be analytic, i.e., function only of the superfields and not of their conjugate partners. The superpotential for the SIC model consists of three sectors, P = I + G + S (1.1) where fields in one sector do not appear in the other two. Both I and S contain only gauge singlets, responsible for inflationary expansion and supersymmetry breaking respectively. The sector G contains all the usual matter, the higgs sector, and more. The purely scalar potential for a minimal supergravity theory with superpotential P is given by 2 scalar ex P Â— ipi 2 + D Â— terms (1.2) where the sums run over all scalar (super) fields in the theory, denoted symbolically by fa, and M is the reduced Planck mass M = Â±=M Planck = 2.4xl0 18 GeV (1.3) V87T Crosstalk between the sectors is always suppressed by powers of M. Each of the three sectors interacts with the other two only gravitationally, so thermal equilibrium is possible only within each sector or as a consequence of finetuned initial conditions.
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INFLATION The inflation sector J of the superpotential can be constructed very simply 7 by assuming it contains only one chiral superfield, $. Scaling all fields by M and considering the scalar potential due to / alone, Vj 1*1' 31/1 (2.1) At the minimum of this potential (the true vacuum up to interference with other sectors), the cosmological constant should vanish, so Vj($q) = 0. Requiring further supersymmetry to be unbroken at this minimum implies that the gravitino mass must be zero, which imples that 91 *tr ^ + $ 1 = 0 $=$o Hence mo) = (*o) = 0 (2.2) (2.3) A very simple J satisfying the above requirements is a quadratic superpotential J = A 2 ($$ 0 ) 2 (2.4) where A is a mass parameter setting the scale for inflation. This is the small parameter which appears in all succesful inflationary models, 2 fixed around 10 4 M to avoid the supersymmetry entropy crisis or Polonyi problem. 8 ' 22 Without loss of generality, we can take the vacuum value of the inflaton, (<Â£) = $o> to be real. 5
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6 Requiring the scalar potential to be flat at the origin, ( d. (0) = 0 (2.5) so that (new) inflation can occur with a slow rollover, sets $o = M = 1. Letting the scalar component of the chiral superfield $ be e %6 the scalar potential from this sector only is Vj = A V 2 6 4(f> 5 cos 0 + (3 + 4 cos 2 0) 4 4 cos 0 3 + (3 4 cos 2 0) 2 + 1 (2.6) For any value of 5 + 7 4 4 3 2 + l) = A 4 (l 4* 3 + j 4 * 5 +  ** + l) 2 + 12eA 4 (<Â£ l) 3 + 0{{ l) 4 ) (2.7) whence the mass of the inflaton is = 2v/eA 2 (2.8) Similarly, Vj{0) = V I {0, = l) = 8eA 4 (lcos0) (2.9) so the mass of the angular field 0 is m e y/im^ = 2\/2eA 2 (2.10) The potential (2.7) is a standard inflationary potential, flat at the origin, with a slow descent towards the minimum at M, and a steep divergence for values of the field larger than M . The inflaton has cubic selfinteraction
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7 of strength 12A 2 , and linear (gravitational) interactions with other sectors of strength ~ A 2 as well. The inflaton will couple to fields in S and G, with a preference for the heaviest ones, and a decay rate I> Â« (A 2 ) 2 m^ cA 6 (2.11) At Planck time all particle species in the universe are maintained in thermal equilibrium by gravity. Regardless, however, of the initial conditions, each of the three sectors decouples from the others while gravity becomes weaker as the universe expands. Fields within each sector thermalize through Yukawa and gauge interactions, so the inflaton superfield effectively evolves alone in a thermal bath provided by S and G. The inflaton will roll away 35 from its initial configuration 36 ' 37 ~ 0 (2.12) destabilized by the (otherwise negligible) Hawking radiation term AV H ~ AV (2.13) As the universe expands, the energy density becomes dominated by the potential energy of the inflaton, which can reasonably be assumed 38 to evolve according to the classical equations of motion j> + 3H+ = 0 (2.14) a ) < 215 > where H is the Hubble constant (parameter, rather) with its value in antideSitter spacetime given by
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8 Inflation occurs with $ Â« 3H 0 <$> from a time t* when ~ H 0 until a time tj ~ mT 1 . The inflaton then starts evolving like matter, until a time i# ~ T^ 1 when it decays into radiation and reheats the universe. 7,8 Using conservation of energy, the increase in radiation energy density is determined by P{tR) = P+{tj) 2 = A 4 ~ A 12 TT 2 *T R (2.17) with g* the effective number of light degrees of freedom at the time of reheat tft. The reheat temperature Tr for the heat bath in the causal bubble is, by definition, T R Â« A 3 (2.18) To avoid the monopole problem, 14,39 must be less than the unification scale= Mx\ hence A 2 < 10~ 2 . Below we shall find that actually A ~ 10 4 , so the reheat temperature Tr is only about 10 6 GeV. Despite this low reheat temperature (which does not, by far, produce any problems with nucleosynthesis), inflatons are produced copiously and inflaton decay does generate enough baryon asymmetry, due to the lack of equilibrium among the various hidden sectors. The number Nj of efolds of inflationary expansion (Rr ~ e Nl Rj) will be 9 freheat Nj = / H 0 dt (2.19) JfcnitiaX This is easily much larger than the observationally required Nj > 65. Nj is around 10 8 if fa < Hq , r ~ M/10, so there is a large set of initial conditions leading to sufficient inflation.
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9 In the deSitter (inflationary) stage, the quantum fluctuations 40,41,42,43 of the inflaton are of order Hq. Their comoving size leaves the horizon during inflation (Hubble radius is constant), and they reenter the horizon after inflation, thus generating a spectrum of density fluctuations at scale A: 2 (2.20) Approximating Vj by its truncated Taylor series around the origin, Eqs. (2.14) and (2.15) can be solved for {tj) to give 7 ^ 10 4 A 2 ~ 1(T 4 (2.21) P which is the preferred value for galaxy formation with adiabatic density fluctuations, known in the trade as the "cold dark matter scenario." Interestingly, Hawking has shown 44 that if < 10 4 , then the mass of the inflaton must be less than 10 15 GeV, or A < 10~ 1,5 . Let us emphasize that, given the value of A ~ 10 4 required from the discussion of the gravitino problem, 8 the magnitude of the density fluctuations comes out as a prediction of the theory. 4>Htr) 1 + log(Atfo)
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10 SUPERSYMMETRY BREAKING Consider now the supersymmetrybreaking piece S of the superpotential P. It cannot be 22 of the Polonyi 45 type (linear in the superfield). Take S to be of the O'Raifeartaigh type, 8 ' 17 S = A{\B 2 + fi 2 ) + mBC + n (3.1) where A, B, C are chiral superfields and A, /x, m, and n are parameters. We can set m ~ \i and adjust n to ensure that the cosmological constant vanishes at the minimum of the full potential, taking into account the inflation and gauge sectors. The F term Â§4 breaks supersymmetry 17 ' 18 so that the spinor component of A is the Goldstone fermion which provides the gravitino with the helicity ^ states it needs to become massive. The scalar component of A, the O'Raifearton, couples weakly and stores potential energy during inflation. Then it evolves like matter until it decays preferentially into a gravitino pair through a oneloop diagram with three internal B propagators, with rate 1^3 3 m TtVm (3.2) where = A 2 /47r, and a\n is the oneloop O'Raifearton mass. The gravitino mass is mz = H 2 /M, and the B, C fields have a mass of order 3 fj,. For supersymmetry to solve the gauge hierarchy problem 46 ' 47,48,49 ' 50,51 (its precise raxson d'etre), mz must be at most around a TeV so /x ~ 10 1011 GeV 2 ~ (10 7 10 _8 )M.
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11 Now gravitinos couple to matter very weakly (with gravitational strength), so they are longlived particles. They may thus decay into photons and photinos (lightest oddfl particles, R = (_i)3^+^+ 25 ) after the onset of nucleosynthesis. Their number density at decay is severely constrained 52,53,54 by the success of conventional nucleosynthesis 55 in explaining the primeval abundances of light element. The decay of gravitinos into gluongluino pairs, for instance, can dramatically alter the deuterium abundance. 56,57 The abundance of gravitinos at the time of nucleosynthesis is constrained, whence the reheating temperature is bounded 24 1 TeV' T R < 2.5 x 10 7 GeV Using Tr ~ A 3 and mz ~ /z 2 , this implies f/l3 2 (3.3) A V < 4.2 x 10" 27 (3.4) which entails, under the requirement (for the solution of the hierarchy problem) that n ~ 10~ 8 , that 8 A < 1(T 4 (3.5) If gravitinos happened to be stable, then the bound (3.3) is less restrictive, 24 T R < 1.3 x 10 11 GeV 1 TeV T7l3 2 implying A < 10 2.5 (3.6) (3.7) To make sure that infiaton production of gravitinos is small, since the inflaton couples most strongly (~ A 2 ) to the heaviest fields, we must kinematically forbid the infiaton from decaying into O'Raifeartons, as well as B and C fields, m < m A> m B > m C (3.8)
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12 and require also to be small, < 10 3 , in order to suppress oneloop effects. 8 A careful analysis 8 of these oneloop effects reveals a lower bound for the inflation scale A < 10" 4 2 (3.9) whence A is rather tightly constrained, much to nature's credit.
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13 GRAND UNIFIED SECTOR 11 (4.1) We write the superpotential for matter fields as Â„ / 0 a'H 0 \ Â„ / 0 dK 0 \ G y = F r oH cL 0 J T + T I dK eK' fK" T V 0 0 bHj V 0 fK" gK J ( 0 d x K 0 \ + S T d 2 K eK' hK" F + {j^S^ + y 2 5 3 5 3 ) V o / 2 x" sir 7 where T T = (ri,r 2 ,r 3 ), F = {F l ,F 2 ,F 3 ), and S r = (S U S 2 S 3 ) are the N = 1 chiral superfields containing matter in three generations or families, and H, L, K, K' and K" are higgs superfields, and 4>\ is a group singlet whose vacuum expectation value (<Â£i) = Mq = 10 10 GeV breaks the PecceiQuinn symmetry 58 . The explicit field content of F n , T n , and S n is shown below, where Greek indices run from one to three and denote color. All fields are chiral superfields with lefthanded Weyl spinor components. The fields F n are vectors, T n antisymmetric tensors, and S n singlets: F n = t n 1 (ZafaiO 1 Â«na ~4 Tn = ~E [ 0 v/2 na c 0 (4.2) Sn =V n The representations and global symmetry charges of the various fields in G\ are shown in Table I. The only unusual feature is the higgs field L, which
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14 transforms like a 45. The first two terms in (4.1) constitute the GeorgiJarlskog model 19 with an extra eK'T 2 T2 coupling, needed to correct the prediction KM c i =52 incompatible with the value inferred from measurements of the Bmeson lifetime 20 [KM cb = 0.0435 Â± 0.0003) and the small topquark mass (< 150 GeV) required by unitarity. TABLE I. Field content and symmetries orial, PecceiQuinn, familiarity) of G\. (vectSuperfield SU(5) irrep V X 7 5 3 1 ri 10 1 1 Si 1 5 1 F 2 5 3 1 T 2 10 1 1 S 2 1 5 1 1 3 ^ o xJ 0 r 3 10 1 0 S3 1 5 0 H 5 2 2 0 L 45 2 2 2 K 5 2 2 0 K' 5 2 2 2 K" 5 2 2 1 4>1 1 10 2 0 The four phase symmetries can be understood as follows. 11 First, there is the vectorial symmetry V, which becomes BÂ—L below the electroweak breaking
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15 scale. It forbids unwanted terms like FK and FFT and has no anomalies. Second, we have the anomalous PecceiQuinn symmetry X, needed to solve the strong CP problem. 59 Third is the chiral R symmetry of cubic superpotentials, broken by gravitational couplings to the hidden sectors / and S. Lastly, we have the J symmetry, familiarity, which is the only one to distinguish among the three families or generations. Familiarity 7 is like family isospin, traceless in family space. Crucially, we forbid 11 mixing terms between the two sets of higgs fields with different V, such as HK, HK', HK" or LEff, LEif', LEK", where E ~ 24 or 75. Such mass terms would lead to disastrously fast proton decay rates, through dimensionfive operators (see the discussion below on proton decay) . As a further and rather innocent simplification, we shall assume that the matrices are symmetric, i.e., that o a', d\ Â— d>2, and f\ = fa. Fermion Masses The higgs superfields produce after SU(5) breaking light (~ M\y) doublets and heavy (~ M^/M) triplets, via the missing multiplet mechanism. 60 ' 61 Our purpose now is to determine the fermion masses, in order to constrain the various Yukawa parameters appearing in the superpotential. The A/jy =  masses come from the 5^(2)^ x U(l)y Â— > ^(l)em phase transition, when neutral components of higgs SU{2)i doublets acquire a vacuum expectation value (vev) a. The Alyy = 1 masses come from the breaking of the PecceiQuinn phase symmetry, when some singlets acquire vet/'s Mq. Rewriting (4.1) in terms of the higgs (3,1) triplets and (1,2) doublets, we arrive at the
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16 SU(3) x SU(2)i x U(l)y superpotential for this sector, with due care for the normalization of the kinetic terms of the component fields. The notation ipx is shorthand for tp T Cx, with C the chargeconjugation matrix and T the transposition in Dirac space. A useful identity is ($L) C = rP R (4.3) Since all spinors are lefthanded, rf> T C X = rPlCxL = *RXL = Xl C 1>L = XÂ°R^L (44) In the more convenient twocomponent notation, in which V c o 2 tf R (4.5) the various products are of one of the forms below: V>V M) T (*2X R ) =  R x R = Wrxr)* (4,6) VÂ» C X ~* {^*r) XL = 1>R?2XL = $RXL <1>X C ~* ^l(Â°2Xr) = {1>\Â°2XRY = $LXR)* corresponding to the left Majorana mass term, the right Majorana mass term, and the two pieces of the Dirac mass term. The mass Lagrangian for matter fermions is thus (4.7) The mass matrices are given below in terms of the parameters in the superpotential and the veto's of the various doublets. It is clear that the first four
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17 terms in (4.7) are Dirac masses of the form + ^c m contrast, the last term is a "righthanded" Majorana mass of the form ipR^LThe quark mass matrices are 0 a (H 2 ) 0 M : = 4 I f 6 (L 2 ) 0 {4 . 8) 0 0 b(H 2 ) \/2 / 0 d(K 2 ) 0 \ M 2 = 4p d(K 2 ) e(K' 2 ) f(KÂ») (4.9) V 0 /(*?> Â»(^)/ The parameter p represents an SÂ£/(3) renormalization group enhancement factor between the GUT scale and low accelerator energies. The parameter p is determined by any one wellknown ratio of eigenvalues of M_i and Mi at 3 low energies (see below), neglecting differences in the SU(2) and U(l) renormalization of the fermion masses. The lepton mass matrices are 3c (L 2 ) 0 (4.10) 0 d(K 2 ) 0 d(K 2 ) e(K' 2 ) f(KÂ»)  (4.11) ^ 0 i{KÂ«) g(K 2 ) ( 0 ixMo 0 \ Mr = iiM 0 0 0 (4.12) V 0 0 j 2 M 0 J Now the two doublets from H and L (coupling to the 5 x 10 sectors) will mix with doublets from other heavy fields, leaving light only the linear combination cos a H 2 + sinaL2Similarly, the only doublet from higgses coupling to the 10 x 10 sector which remains light is a x K + a 2 K' 2 + a 3 K' 2 ' (4.13)
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18 Inverting the unitary mixing matrices, we can express the ueu's in the mass matrices (4.8)(4.12) in terms of the vev's afp and oj (K 2 ) = a 3 = a{ a TT (4.14) (K J) = a 4 = Â°TT {K'i) = a 5 = a* a rr with El". 2 = "2 = = (188 GeV) 2 (4.15) i=l 9 We have enough freedom to redefine the righjtand lefthanded fields so that we can make the e*j real by rotating their phases away. Proceed now to find the eigenvalues of the above mass matrices and the mass eigenstates as a linear combination for the current eigenstates which we have dealt with so far, V> = Vxjjm (4.16) with V unitary. For charged fermions, all mass terms are Dirac, of the form (+h.c.) t R Mxl> L = 4> R VVl MVVl j> L = VwAVw, (4.17) where A = V~ 1 MV is diagonal. These mass eigenvalues can be made real and positive by further rotating the chargeconjugate ("righthanded") fields by some phases. Indeed, righthanded and lefthanded fields mix differently in general.
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19 For charge(j) fermions gettting their mass from Of?, (cos $i sin 6 1 0 \ sin^ cos0 Â£ 0 (4.18) 0 0 1/ and so that (cos 0 C sin 0 C 0 \ sin0 c cos0 c 0 (4.19) 0 0 1/ and with = diag(Â—m e ,mn,m T ) .. I 1 (q gl ) 2 3 1 A_i = diag(m d , m 3 ,m b ) 3 cos 20 c = , i.e., dents + m
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20 To make masses positive, let Va = V t , V_ kL = V_! (4.28) 3 3 and redefine the chargeconjugate fields such that V a = V t ^ 1 J , V_, R = 1 J (4.29) For the chargeÂ§ quarks, which get their masses from ayj, first disregard entries in M2 proportional to d (<< e,f,g), which means disregard the up3 quark mass. There is then enough freedom to absorb all phases in M2 by 3 redefining the cand iquark fields (and their chargeconjugates). Assume, equivalently, that ea 4 , do^, go$ are real. Then with so that 10 0 = ( 0 cost; sinr; + 0( Â— ) (4.30) 0 Â—sinr/ cosrj/ mc r/ Â— Â— << 1 (4.31) 0ct 3 ea 4 A 2 diag[0,m c ,m t ] 3 4p diag\0, ea 4 2r}fa 5 ,ga^ + 2r//a 5 ] The Yukawa couplings are thus m c e = 4pa 4 r?(m ( m c )
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21 where e tS is the determinant of Vg L . The full mixing matrix is then V 2 * 3 f cosO u sin0 u e*2 Â—T)s'mO u ^e l '< sin0 u cos^e^ ~ 1 1 + ^ e ~ t ^ 1 ,4\ V rj sin 0 U ^ + r/e ,5 where and sin0 u ~ 0 U ~ ./Â— V m c A 2 =
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22 KM u c t V (4.43) J This prediction is to be compared with the KM matrix obtained from an analysis of experimental data, which yields 63 d s b 0.9733 Â± 0.0024 0.231 Â± 0.003 < 0.0052 0.24 Â± 0.03 0.97 Â± 0.007 0.0435 Â± 0.003 ~ 0 < 0.076 ~ 0.999 The best fit of (4.42) to (4.43) provides 11 6 C = (13.33 Â±0.16)Â° t] = (2.49 Â±0.01)Â° so that we can establish (in disagreement with the usual Fritzsch mass matrix result, 64,65 and believing the top mass to lie 21 around 45 GeV) that m c (4.44) mt (4.45) We can now bound the Yukawa couplings involved, because it follows from (4.15) that a t  < 188 GeV. We use the values of p and tj given by (4.27) and (4.31), the standard masses for charged leptons, and masses 66 of (5 Mev, 1.84 GeV, 45t GeV), where t is a dimensionless parameter assumed by certain wishful propagandists close to one. \d W.I/ \9 > 5.5 x 10~ 5 > 1.3 x 10 2 > 6.5 x 10 4 ,5 (4.46) > 4.5 x 10" > 8.7 x 10" 4 > 2.1 x 10" 2 * 2 To find the eigenvalues and eigenmasses for neutrinos ( lefthanded ones are naturally light due to the seesaw mechanism 67 ' 68 ), we must diagonalize
PAGE 28
23 a 6 x 6 matrix which takes a particular form, with the upperleft 3x3 block equal to zero. Following the construction below, 69 we can reduce the problem to the diagonalization of two 3x3 matrices instead. In the basis {ue,!/^,^,!/^^^,^), the full mass matrix is There exist H = ( Â°, I) (4.48) and with U\ and tAj unitary, such that V T *>V = (% L A Â°J (<Â•Â«>) with A/, and A# diagonal, real and positive, with V = e iH U (4.51) Indeed, let S = iM* D {M* R )1 (4.52) and let U\ and i/2 be the unitary matrices such that mi = M D M~ l Ml and A* = U 2 T m 2 U 2 (4.53) 1 + 1 ^ ( 4 54 ) Â™ 2 = + (Af^)~ MpAfp + MlMiiMi)1 One can then predict neutrino masses and oscillation lengths from the parameters in the mass matrices, which are more or less known for Mj) and
PAGE 29
24 open to speculation for Mr. The flavor of these speculations, rather than a solid result, can be obtained with a couple of simplifications. In the model at hand, asume first that d/d=e/e Â— f/f = g/g= Â£Â• This assumption would follow from an S'O(lO) symmetry broken at a higher scale than the SU(5) unification we are considering. Such further unification must be rejected in the light of the discussion on baryogenesis (see below). Let x t = rr 1 = (jjMn) 1 , and let a be any of the a,. The eigenmasses are then, in terms of the quark masses, m u Jm u m c 2 m c Jm u m c 2 m 2 2 % C , Y 6 i Â— L Â£ i ri + ei, ^2 + ^2 2rj ri r 2 with KO 2 2ri and (4.55) (4.56) (2 = (4.57) r 2 The 90% confidencelevel limits on neutrino masses 70 ' 71 improve with every new analysis of e + e~ runs and yet, given their order of magnitude m Vfl < 250 KeV (4.58) m Vr < 125 MeV they provide only very weak constraints. Tighter ones can be obtained from Zeldovich's cosmological limit 72,73 m Ve + m U(t + mÂ„ T < 100 eV (4.59) implying r x < 10 4 GeV (4.60) r 2 < 10 8 GeV
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25 so that {M ?J Ml) where M 0 is measured in GeV (10 9 < M 0 < 10 12 ). 59 The mixing scheme depends very heavily on the relative magnitudes of the various parameters. Let us assume that = = Â£ 1 > and neglect d (i.e., neglect the upquark mass) from all expressions. Then /l 0 0\ U X Â« 0 1 I (4.62) vÂ° 1 V where we have taken (74 ~ as (=> / ~ Â— e). The estimate V22 Â« 1^ 2 *V 4 ) 2 (4.63) can be identified with the cosine of the effective mixing angle for muon neutrino disappearance, a VfL Â— fcre<7 4 Â» Â— < 10" 8 (4.64) The other parameter relevant for neutrino oscillations is AM 2 = 2 2 m m vx 2 2 Â„ (4.65) "V r < 3 x 10 6 eV 2 where we have used m t a 45 GeV. Given that the bound on j\ is weaker, the mixing angle could be brought up to about 10 4 . An increase in the mixing angle, however, would probably imply an increase in Am 2 , so the outlook is rather grim: neutrino oscillations seem to be at least two orders of magnitude away from currently imaginable detectability. 74
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26 The Full Superpotential In order to spontaneously break SU(5) at a scale M^, and the anomalous PecceiQuinn symmetry at a scale Mq (10 9 < Mo < 10 12 GeV, from the existence of red giants and from axions not closing the universe 59 ), we add a term to the superpotential 11, G 2 = i?i(EE ; M^3 + i7 2 (M2 AfÂ£)0 4 + r/ 3 E 3 + 77 4 Â£' 3 (4.66) The SU(5) content of the various fields introduced, as well as their charges under the global C/(l)'s, are shown in Table II. In order to give mass to color triplets while leaving the isodoublets light, 60 ' 61 we introduce the term 11 ' 12 G 3 = {XiK + \ 2 K' + \ 3 K")MU + faK + p 2 K' + foK")MU' (4.67) + {liH + l2 L)EV which, taking M Â» (Â£) = Mx, (a) gives mass to two combinations of the quintets K, K', K" pairing them with the 5 fields U and U', leaving one light quintet, and (b) gives mass to one quintet (or rather, a doublet and a triplet) formed with L and H. Adding furthermore a term 11 ' 12 G 4 = (SxHQi + 6 2 LÂ®i + <5 3 J0 2 )E + 6 A MÂ®iÂ® 2 (4.68) + (piKe' 2 + P2 K'G' 2 + p 3 K"Q' 2 + ptJ'O'jX' + PsMe'^ masses can be given to the colortriplet components of the two light quintets formed with G3, pairing them with the color triplets in J and J'. We thus end up with four triplets with mass around the intermediate scales 6M X /M and pM x /M (two each), and four massless doublets, which will acquire a mass ~ M\y from spontaneous electroweak symmetry breaking. Notice that only two of the doublets (triplets) couple to matter at the tree level. From the
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27 discussion on the renormalization group equations below, we shall obtain an expression for My. The upper and lower bounds on the triplet masses translate into the bounds 1(T 5
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28 TABLE II. Field content and symmetries (vectorial, PecceiQuinn, discrete Z3) of G. Superfield SU(5) irrep V X D H 5 2 2 2 L 45 2 2 2 K 5 2 2 2 K' 5 2 2 2 K" 5 2 2 2 *l 1 10 2 0 02 1 10 2 0 3 1 0 0 0 A 1 0 0 0 Â£ 75 0 0 1 E' 75 0 0 2 U 5 2 2 1 u' 5 2 2 1 V 50 2 2 0 J 5 2 2 2 J' 5 2 2 0 Â©1 50 2 2 0 e; 50 2 2 1 e 2 50 2 2 0 e' 2 50 2 2 2
PAGE 34
PROTON DECAY We know 75,76 ' 77 that all possible protondecay 78 operators are of one of the forms below. Greek indices run from one to three and represent color, Latin indices from the middle of the alphabet run from one to two and represent weak isospin and Latin indices from the beginning of the alphabet run from one to three and represent family or generation. Chirality is indicated for definiteness: Â°lbcd = {^aaL u fibR)^icR^ndL) Â€ ap 1 e ij Â°lbcd = {?iaaR?j0bL){e C cL u ldR)tafatij 3 ~c c (5l) Â°abcd = { d aaL u pbR){ eC cL u ldR)^ 1 Â°\bcd = tiiaaRQjpbLWmcRQn^dLjtapitijtrnn Recall that, as usual, xfriXR = Xi^pRi an d notice that Olbcd = Â°bacd ( 5 2 ) The form of these operators is constrained by the requirement of gauge invariance under the standard SU(3) x SU(2) x U(l). They arise dynamically from integrating away heavy particles (higgs fields or gauge bosons of broken symmetries). The effective Lagrangian for the fermionic components of matter fields contains, in our case, O 1 from (FT) 2 terms and O 2 from (TT) 2 , with the family structure dictated by familiarity [J) invariance. There are no O 3 or O 4 operators, though, because we do not allow H Â— K nor L Â— K mixing terms; i.e., we suppress protondecay operators involving higgsinos 28 ' 29 ' 30 with a chiralityflipping mass insertion in their propagatorand gluinos. 29
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30 To illustrate the point, consider a onefamily toymodel with a Yukawa superpotential of the form aFTH + bTTK (5.3) where F, T, H and K transform as 5, 10, 5 and 5 of SU(5). Both H and K contain color (anti)triplet, isosinglet pieces. These couple both to leptoquark and to diquark superfields, causing proton decay in two different channels. FF can go into TT via H exchange, and TT can go to TT via K exchange. Because of the helicity structure required by the structure of chiral vertices, these processes correspond to "dimensionsix" protondecay operators, in which all four external lines are fermionic and the bosonic component of the triplet (from H or K) propagates through the internal line. Were we to further allow the mixing term fiHK (5.4) then tripletinos (but not bosonic triplets) would mix, inducing the process squark + squark Â— antiquark + lepton in the channel FT Â— TT. The engineering dimension of such a diagram, with two scalar and two fermionic external legs, is five. The initial state can be virtually produced from a quarkquark pair (in the nucleon) via gluino exchange. Since gluinos couple strongly, the amplitude for the whole process (gluino exchange and higgsino exchange) is essentially the same as for the dimensionfive higgsino exchange subprocess. This amplitude crucially differs from the one for quark + quark Â— antiquark + lepton via higgs boson exchange, in that it involves a triplet fermion propagator (rather than a triplet bosonic one), so it is suppressed by the mass (rather than the mass squared) of the triplet superfield.
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31 Proton decay thus proceeds faster through these higgsinomediated diagrams (dimension five) than through higgsboson mediated ones (dimension six) roughly by a factor of ( w H TT at one loop with T, F and H K in the internal propagators. Note that in this onefamily toy model this interference is zero even with the mixing term because the product of Yukawa couplings is necessarily real. 83 (The chiral structure of the processes is what interests us now.) Baryogenesis seems to require an interaction among, and hence the existence of, different families! The baryogenesis problem will be solved shortly, but we can foresee that since the higgs triplets will have to be rather light, the dangerous mixings must be suppressed. The action presented above does so naturally with two of the global 1/(1) 's, PecceiQuinn and familiarity. Around the electroweak scale, nevertheless, supersymmetryviolating effects will induce such mixing, safely suppressed by O(^J^). Proton Decay Lagrangian The superpotential arising from (Eq.(4.l)) involving matter and colortriplet higgses is given below, where family is indicated explicity but color and Dirac indices have been dropped. In fourcomponent notation, rf>x stands for
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32 For calculational purposes, it is much easier to use twocomponent notation and the physically relevant leftand righthanded fields, substituting in all the previous expressions tj) Â— (5.6) 0 C Â°%$R and transposing the first entry in rpx after the substitution: GtripUu = + u c 2 d\ tug^ t 2iqij V) (5.7) 4d/f 3 (u^ + u\t\ + qutojt') ~ *9K 3 {u\e% + u 3 d 3 ) 4eK' 3 {u\t\ + u 2 d 2 ) AfK'i (u c 2 e c 3 + u%t\ + q 2i q^ ij ) where Â£ = (^J, q = ^ ^ , all the triplets have canonical kinetic terms, and the Yukawa couplings a, b, c, d, e, /, g were given in terms of measured fermion masses and constrained doublet vev's above. Triplets coupling to the same matter in Eq. (5.7) mix, (H 3 \ _ ( cos a sin a \ / 5 \ / cosa 5^ H \ L 3 J ~ y sin a cos a J \ X J \ sin a 3 ft H / K, K: K  i = l 4 A ij 2 "2 '2 t Alt J Â«2 JU) XT Â•7 (5.8) Only S/ry and Hjt remain light, with masses j, Mj^ around the intermediate scale, light by unification standards. The a, can be chosen to be all real, just as we did with the doublets when we were dealing with the fermion masses. Although we have used the same notation (o^) for the entries of the mixing matrices of doublets and triplets, there is no reason why they should be the same.
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33 The superpotential (5.7) thus becomes effectively (5.9) This is the expression from which we will compute the baryon asymmetry of the universe. Since 5 and Hyjdo not mix, we need not consider higgsinos. The interaction term in the Lagrangian involving the scalar components of the higgs superfields 5 and the fermionic components of matter in (5.9) is exactly the same expression (5.9), with the usual eveni? fields instead of superfields. At energies much lower than the masses of Ef t or &TT we can integrate away Sj^y and Ej>xKeeping only terms with only one lepton field, we arrive at the following lowenergy Lagrangian for proton decay, where all fields are still current and not mass eigenstates: C = ( L abcd < obcd > v L abcd < abed > e + R abcd [abcd}) (5.10) The L's and iE's are coefficients with indices in family space (a, b, c, d Â€ {1,2,3}). The various proton decay operators are, in the notation of (5.1), so we are breaking up the SU(2) doublets q and I appearing in (5.1), since proton decay takes place at energies below the SU{2) breaking scale. Â°lbcd = (Â°M + \ bacd \ (5.11)
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34 (5.12) Explicitly, < abed >Â„ = (d c a u c b y {u c d d ) = ( e = {d c a u c b y {t c u d ) = (daLHR) &R u dL) [abed] = (u c a d c b ) {e c u d )^ = (Kn d bL) {^cL u dR) or, in twocomponent notation, < abed >Â„ = (uf R d aR ^ \uf L d dL J < abed > e = (u^d a j?) {e^ L u dL J (5.13) [a&cd] = (dJ R u aR j (e1[ L u dL ) TABLE III. Coefficients of baryonnumberviolating fourFermi operators in effective Lagrangian before electroweak breaking. abed 2Mp T L abcd 1212 i i Â—a cos a d 2 a\ 1222 ac sin a cos a de
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35 The coefficients and R a b c d satisfy the identities L abcd = L (ab){cd) ~ L (cd)(ab) (5.14) R abcd = R {ab){cd) ~ R (cd)(ab) and they are all zero except the ones in Table III and their symmetrizations according to (5.14). Rotate now all the quark and lepton fields into the mass eigenstates, denoted by the additional subscript ra, and related to the current eigenstates by a unitary transformation ^ = V^VVnThe new proton decay operators have the same form as the old ones, but with massive fields in them. They are related to the old ones by four mixings, one for each field. Summing over the various neutrino flavors, indistinguishable in protondecay experiments, and neglecting the seesaw coupling between ordinary neutrinos and massive righthanded ones, we can express the Lagrangian (5.10) in terms of mass eigenstates: Â£ = Â£ (Kbd < abd > + Kbcd < abcd > +Kbcd\ abcd }) ( 5 15 ) The coefficients Kbd = Â£ W^^a e (^)/(^^ Kbcd = ~ Â£ L efgh i V ! I R}e a { V l^)\ [ V Il) 9 c {Vl^h ( 5 .16) e,f,9,h Kbcd = Â£ ^m^iji)*^^)^^)/^)/ e,f,g,h can be evaluated using symbolic manipulations on a computer, without much problem. The full formal expressions are long and unilluminating. Renormalization of Proton Decay Operators One last step must be taken to obtain the effective proton decay Lagrangian, useful to compute proton decay rates: we must renormalize 84 all
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36 Â• 75 77 80 the (dimension six) operators down from unification to our energies. 1 Â• Indeed, the expressions for the coefficients L v , L e , R e of the various operators are valid at the unification scale. The renormalization of the Yukawa couplings themselves is taken into account by using lowenergy fermion masses, but the full operators undergo further renormalization at lowenergies, by a multiplicative enhancement factor which can be calculated using dimensional regularization. We neglect U(l) renormalization, assume exact supersymmetry above M\y, and renormalize with oneloop equations down to the charm mass. Below M\\r, we consider only SU(3) renormalization. These are all reasonable assumptions in light of the requisite accuracy. Both operators O 1 (mediated by scalar #3 and L3) and O 2 (mediated by scalar K$, K' 3 , K'Â£) could mix under SU(3) renormalization with their supersymmetric partners, the dimension five operators mediated by higgsinos with the same quantum numberswere those operators allowed by the theory at all. They cannot, however, mix with any other operators under SU(2) renormalization because under exact supersymmetry the SU(2) gaugino is Majorana massless: were it to be exchanged between two external legs with SU(2) quantum numbers, it would undergo a helicity flip to preserve the chiral structure of the vertices, which it cannot do because (as advertised) it is massless. 85 Let us consider the spectrum 11 contributing to the renormalizationgroup equations between Mx (unification scale) and the lowenergy world. We have three light fermion families (the singlets are heavy but irrelevant for group behavior), with the topquark mass around 45GeV, four light isodoublets, with masses around M\y, and four light color triplets with masses around M 0 a 10 10 GeV. We assume all other higgs superfields have masses of at least 0(M X ).
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37 If the coupling constant for the ith simple factor of SU(5) is, to one loop, of the form Q ' (M) = wdpffl (5,I7) we can find the coefficients 6^ from the following expressions, where n q (n.Â£, n d , nt) is the number of quark flavors (lepton flavors, isodoublets, color triplets) below the energy scale //, For nonsupersymmetric QCD, (SU(3) below M\y), 6 3 = 11 (5.18) For supersymmetric QCD, (5(7(3) above M\y), 63 = 3 x 3\{2n q + n t ) = 3 \n t (5.19) where the 2 comes from counting quarks and antiquarks. For supersymmetric SU(2), 6 2 = 3x2^ + ^ + ^) = ^n d (5.20) Finally, for supersymmetric U(l), 1 3 h = {n q + n e )frac310n d = 6 Â—n d (5.21) As boundary conditions for the renormalizationgroup equations we use Â«7m{Mw) = 128.5 a^{M w ) = 0.12 1 = 8.3(Â±0.4) (5.22) Recall also that a 2 1 {M\y) = a er *(A%) sin 2 6 W w 31 ( 5 ' 23 ) a l {M\y) = g "cm i M w) cos 2 0 W 3
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(5.25) 38 whereas the unification scale is defined by a = a^Mx) = a 2 {M x ) = a 3 {M x ) (5.24) The oneloop renormalization equations are thus .fdM = s<0*(sÂ£) Solving them, one easily finds 11 ' 86 M x = 5.6 x 10 16 GeV a = a{M x ) = (19.60) 1 (5.26) x w = s'm 2 e w (M w ) = 0.237 with errors of about 5% arising mostly from uncertainites in Ajg^, where MS is the modified minimal substraction scheme in QCD. The rather high value for x w is in excellent agreement with the most recent measurements of x w in deepinelastic neutrino scattering. 87 ' 88 Now for the actual renormalization of the qqql operators. Since we are neglecting U(l) effects, all operators at hand get renormalized in the same way. Letting A be any of the coefficients L u , L e , R e in the effective Lagrangian, the enhancement factors relate A at one scale to A (and all other B with the same quantum numbers) at some other scale. Specifically, A{m p ) ~ A{m c ) ~ E 2 E 3ng E 333 A ( M x ) (5.27)
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39 where we have neglected E\ and the enhancement factors are x
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40 quarks to be nonrelativistic and the antilepton extremely relativistic. 90 All nuclear effects are disregarded, and we concentrate on twobody p > ml decays; that is, we ignore the pion pole contributions and all threebody decays. Proton and neutron wave functions are taken to be 51/(6) symmetric, 91 ' 92 where SU{6) D SU{2) spin x St/(3) flavor is the lightcone spinflavor symmetry. Following Ref. 89, the lifetimes of each mode can be conveniently written in the generic form 11 (5.31) r = y m LlOiOJ where m is the mass (in GeV) of the color triplet mediating the decay, MpT if the outgoing antilepton is righthanded, and M?t Â»* is lefthanded. The y coefficients (in years) for the main decay modes are shown in Table IV. Their explicit dependence on the triplet mixing parameters is also shown, and since we are looking for an limit on M F t and Mjr, the upper bound (188 GeV) for all doublet veu's has been used, i.e., the bounds in (4.46) have been saturated. At 90% confidence level, the latest limits on the proton lifetime into strange modes are 6 r (p ^ fJ. + KÂ°) > 4 x 10 31 years \ > (5.32) t (p Â» uK + ) > 5 x 10 31 years These limits can be translated into lower bounds on the masses of the color triplets mediating proton decay: m FT ^ 0.75 x Y/I15C22.8S2 0.0555CI x 10 10 GeV m FT > 0.53 x ^/3.2c 2 12.2s 2 20sc x 10 10 GeV m TT > 0.32 x 10 10 GeV x 319q2 + 319a 2 , + 0.6a 2.7aia 2 0.17aia 3 27.8a 2 a 3  (5.33)
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41 where c, s stand for cosa, sina, and cq, a 2 , and a 3 stand, respectively, for cos a', sin a' cos a", and sin a' sin a". For any values of the mixing angles, the above expressions can only vary so much, so that a perhaps more useful result is m fp = (410) 4 x 10 31 y ears . i x 10 10 GeV mj"T (518) TvK i x 10 lu GeV (5.34) 4 10 5 x 10 31 years where the lifetimes are normalized by the current limits, for years of use. It follows that masses of the color triplets are bound below by m FT ,m TT > 2xl0 10 GeV (5.35) This lower bound on the triplet masses translates into a lower bound for the mass of the infiaton {A 2 /M) from which triplets decay, 7 ' 8 ' 9 10 ' 11 ' 12 ' 13 so we can extract a bound on A purely from underground detector results: A > 1CT 4 25 (5.36) Remarkably, proton decay experiments set a lower limit on A which is essentially the same as the lower limit (3.9). The upper limit (3.5) on A arising from cosmological considerations on gravitino abundance at the time of nucleosynthesis is very close, so A is (as repeatedly advertised all along) very tightly constrained. It is perhaps amusing to extract more mileage from the relationship between the reheat temperature and the gravitino mass, Eq.(3.4). The bound (5.35) translates into the bound (5.36) because inflatons must decay into triplets in order to produce the baryon asymmetric universe we live in. The two inequalities (3.4) and (5.36) result in H < 1.54 xlO 7 = 3.7xlO n GeV (5.37)
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42 Proton decay experiments have thus obtained an upper bound on the gravitino mass! In numbers the result is not so impressive: m 2 < 57TeV (5.38) 2 Along the same lines, the lack of experimental confirmation for supersymmetry in accelerator experiments allows us to expect m 3 >100GeV (5.39) 2 from which it follows that A < 1(T 3 3 (5.40) hence ro 3 < 1.9 x 10 12 GeV => t^k < 1.3 x 10 40 years (5.41) On the other hand, assuming supersymmetry is not found at the SSC, then the gravitino mass will be narrowly constrained between lower bounds from the SSC and upper bounds from upgraded proton decay experiments. If the largest conceivable Earthbased detectors 6 are built and work, the proton lifetime limit will improve to T nK > 5 x 10 34 years (5.42) Then A > 10" 3 9 (5.43) and ms < 1 TeV (5.44) 2 which could be a useful constraint.
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43 TABLE IV. Decay rates for twobody final states Mode T : Branching Rates (yr *) 4Â° 1.02 x io 27 Â£ x L e nn \ 2 4*Â° 1.02 x 10 27 \E x i2i U1  2 4*Â° 1.39 x 10 27 Â£ x L e 2Ul \ 2 4*Â° 1.39 x 10 27 Â£ x L\ 2n \ 2 4^Â° 1.00 x 10 27 Â£ x L\ ni \ 2 4Â° i.oo x io 27 Â£ x flf m r 1.34 x 10 27 E x L 121  2 1.34 x 10 27 Â£7 x L\ 22l \ 2 2.05 x 10 27 Â£7 x L\ n \ 2 u R K + 1.39 x 10 27 Â£ x L v n2 \ 2 V R K*+ 0.012 x 10 27 Â£ x (3L^ 12 2L^ n ) 2 e + rj 0.20 x r[e + 7rÂ°] e+pÂ° 0.057 x r[c + 7rÂ°] e + u 0.49 x r[e + 7TÂ°] e+K*Â° 0 0088 x Tle+.K' 0 ! 0.20 X r[/X + 7TÂ°] 0.045 x I> + 7TÂ°] 0.38 x I> + 7rÂ°] I7p + 0.20 x T\u*+\
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BARYOGENESIS The observed baryon asymmetry 2,93 of the universe, "Bb/""! ~ 10 " 10 l 6 ' 1 ) is a quantification of two surprising facts. One, that no significant amounts of antimatter exist. Two, that in fact there is much more matter per unit entropy (per photon) than one would expect from an equilibrium evolution of the primeval "singularity." The generation of this baryon asymmetry, a process known as baryogenesis, requires three fundamental conditions. 94 First, baryon number must obvioulsy be violated. Next, thermal equilibrium must also be violated. Finally, C and CP must also not be conserved. Baryogenesis may proceed after the inflationary phase of the evolution fo the universe, by allowing the matter sector of the superpotential G to contain higgs triplets which produce a nonzero baryonnumber asymmetry SB per tripletantitriplet decay, and such that the inflaton can decay into them preferentially (i.e., m$ < m^). Since the inflaton is never in equilibrium with other fields, we do not have to worry about producing enough triplets via a Boltzmann distribution. 24 Indeed, the inflaton oscillates into the heaviest fields around, and we can simply estimate 7 ' 8 ' 11,12 that all of the inflaton's energy is released into heavy triplets which quickly decay into radiation, reheating the bubble to a temperature Tr: 44
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45 * 6B^ * SB^ n~f ~ 6B^=* 6B^ (6.2) ~ SBÂ— M In this expression S B is the baryon number asymmetry produced per decay of an inflaton and an antiinfiaton. Given that the inflaton's branching ratio into color triplets is practically one, then 6B is essentially the baryonnumber asymmetry produced per tripletantitriplet decay, which we calculate from the matter sector of the GUT superpotential. Clearly, G must contain triplets of mass a little lower than the inflaton's, with a rather high decay asymmetry SB ~ 10~ 6 (6.3) At tree level, all amplitudes 2 are real, so CP violation can be produced perturbatively only at the oneloop level. We need oneloop decays which can interfere with treelevel decays to produce a net baryon asymmetry/ 0 ^ 0 '^'' 5 ' 0 ^ Indeed, if a generic particle X (colored higgs, say) can decay into two channels with different baryon number, for instance 2 X Â— qq (Bfi na i = ), ! 3 (64) X Â— y ql (B final = ), where q is a quark and I a lepton (superfields) , then the baryon asymmetry SB produced per decay of a pair of X and X is = jT(X Â» ql) r(X > qq) ^(X Â» ql) + jfjX Â» qq) T{X+qe) + r{X+qq) + r{X+qi) + r{X+qq) T(X>qq)T(X^qq) (65) r(X + anything) T{X * ql) T(X > ql) T{X > anything)
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46 where we have used CPT invariance. In general, sb = ^^Bar.r.) (e.e) where the sum extends over all decay channels t of X, each with a net baryon number B{. Let T(X >
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47 All said and done, only the innermost line in the diagrams, where a higgs triplet is usually propagated, is allowed to be offshell. We could add gauge interactions to these diagrams but (contrary to lowenergy processes, such as proton decay) at the energies relevant for our dicussion, all gauge couplings are weak, so that renormalization of these operators can be neglected. All vertices are chiral, and even cubic if we are willing to introduce spurious fields to take into account mass terms. The only way arrows (chirality) can arrange themselves in the interference diagrams is with mass insertions (chirality flips) in either the external line only, or else in two of the internal nonhiggs lines. The origin of these constraints is, in supergraph language, the fact that interactions in the superpotential involve only chiral fields, so all three arrows point either into or out of any given vertex. All the baryon asymmetry is generated from the higgs fields coupling to the 10 Â• 10 sector, for otherwise the product of couplings g* T g L is real. More particularly, SB # 0 arises from the interplay between the only two higgs superfields coupling to the same sector, both with nondiagonal (in family space) couplings. Remarkably, both higgs bosons and higgsinos can produce a baryon asymmetry upon decay. Taking into account that K'J = ol\EtTi K s ajarr, one can evaluate the interference and obtain the following asymmetry per decay: 11 ' 12 SB _ i Â«ii 2 a3l 2 imKg/r/^Ty 2 ) 32 7 r 2 / 1 2 + / 2  2 + di 2 + eT + y2 Given that the triplet must decay into neutrinos (righthanded ones), with masses given by (4.55), it is clear that M 0 (6.11)
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48 so we can write, assuming a hierarchy of Yukawa couplings and all phases to be of order unity, Â„ < i MWillAllAl (6 . 13) 32tt 2 \g\ To estimate this, take aj ~ Â«s ~ 1, disregard all doublet mixings, and assume proportionality between tilded and untilded Yukawa couplings [via 50(10)]. The value thus obtained, SB < io3 V^ m ' ^ 10 io (6 14) m t a L is much too low, so the presumed 5O(10)inpired proportionality between tilded and untilded Yukawa couplings is to be rejected. Indeed taking all Yukawa couplings in (6.13) to be of order 10 1 , 6B is about 10~ 5 , an order of magnitude above the required value. It is thus rather easy to produce the baryon asymmetry of the universe from triplet decays into righthanded neutrinos and ordinary quarks. The parameters to play with are those in the neutrino Dirac mass matrix, while making the righthanded Majorana masses as large as allowed, i.e., smaller than the triplet mass Mj
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CONCLUSIONS We have achieved to construct a model in which the two bounds on the mass of the color triplets (lower from protondecay experiments, upper from the requirement of baryogenesis) are met. Good relationships between fermion masses and a reasonable KobayashiMaskawa matrix (including C P violation) have been incorporated into an anomalyfree grandunified model. We have also established a hierarchy of masses m rightneutrino < m triplet < m inflaton < m O'Raifearton at the intermediate scale, ~ 10 10 GeV This intermediate scale arises from a plurality of physical considerations associated with supersymmetry breaking, inflation, baryogenesis, and the breaking of the PecceiQuinn symmetry. 49
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BIOGRAPHICAL SKETCH Born a month into the decade of the sixties in Mexico, the candidate developed an early interest in physics while riding the giant roller coaster in the park of Chapultepec. A total solar eclipse the same year of the massacre of Tlatelolco further kindled a desire to understand nature, without necessarily pretending to dominate it. His discovery in 1970 of the Mediterranean sea, its waves, and its peoples did not distract him from academic endeavors until 1976, the first year after fascism (and after highschool) , during which he decided to join the Universitat de Barcelona as a freshman in physics. That tumultuous and exciting first taste of academia ended when the candidate moved to New York, to witness the last year of the "humanrights" presidency. Disoriented by the contradictions apparent in the streets of New York, the candidate joined, along with most of his close friends, the CCFRR neutrino collaboration. After a rather long winter at Fermilab, he quit experimental high energy physics, secured a Master of Arts from Columbia University, and joined the Particle Theory group at the University of Florida, under the academic guidance of Professor Pierre Ramond. Since then (1983), his research interests have meandered from the basic mechanics of grand unification, to the interface of particle physics and astrophysics, to the more formal issues associated with string theories and their compactifications, paying nevertheless close attention to the possible phenomenological implications of mathematical ideas. 55
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56 The charming rural setting has proven most conducive towards research activities in a field somewhat unrelated to social conditions. The candidate appreciates some sports but particularly dislikes football, and deeply deplores the racist atmosphere of the University of Florida, the most obvious exponent of which has perhaps been the obstinately reactionary stance taken by its Administration on the apartheid regime and the divestment issue. The candidate expects to continue research on the forefront of particle physics at the interEuropean research center CERN. Two years there seem to constitute the only certainty of his personal future, which nevertheless looks brighter than that of the world at large. He hopes that the mathematical study of nature can contribute to a greater understanding of all the peoples of the world in tolerance, peace, and justice.
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ^lerre Ramond, Chairman Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. "Field, CoChairman Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. fessor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 'tragi** L eep I Pradeep P. Kumar Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pierre Sikivie Professor of Physics
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Thomas W. Simon i Professor of Philosophy This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1987 Dean, Graduate School

