Available at: http://www. ictp.trieste. it/~pubof f IC/98/106

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THRESHOLD CORRECTIONS TO THE MISSING DOUBLETSUPERSYMMETRIC SU(5) GRAND UNIFIED THEORY

A.C. Wimal Lalith De Alwis1

Department of Mathematics, University of Colombo, Colombo, Sri Lankaand

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

Threshold corrections have been done to the missing doublet supersymmetric SU(5) grandunified theory.

MIRAMARE - TRIESTE

August 1998

1 Junior Associate of the ICTP.E-mail: alwis@ictp.trieste.it and alwis@mail.cmb.ac.lk

1

1 Introduction

Unifying all elementary particles and their fundamental interactions of nature [1] in a one solid

undeniable theoretical framework is the ultimate goal of theoretical high energy physics. The

minimal SU(5) grand unified theory [2,3] which was originally proposed as a candidate to unify

strong, weak and electromagnetic interactions has been ruled out [4] by proton decay experi-

ments [5] and precise measurements of the Weinberg angle [6,7] of electroweak theory. On the

other hand, supersymmetric SU(5) grand unified theory [8] emerged as a promising candidate

in the hope of approaching closer to the desired target. The basic fact is that the theoretical

predictions of the latter are consistent with the latest experimental results [9]. This is a remark-

able achievement for which many great intellectuals [1-4],[8-10] have contributed. This can be

regarded as an indirect evidence for supersymmetry [10] as a symmetry of nature. But one can

still ask, is that just a coincidence or a reality. The reason is that so far no supersymmetric par-

ticles have been discovered within the current experimental energy range even though physicists

have speculated that such particles may well exist within one TeV and can be detected by using

a future generation of particle accelerators. Once the threshold corrections [11,20] will be done

at low and high energy scales to the theoretical predictions of SUSY SU(5) GUT it is possible

to derive more refined theoretical values which are extremely important while comparing with

values extracted from high energy physics experiments. But there are some theoretical problems

remaining in the minimal SUSY SU(5) GUT because there is no natural doublet-triplet splitting

in the Higgs representation of this theory. The missing doublet SUSY SU(5) GUT model [12,18]

has a larger Higgs representation in which the doublet-triplet splitting can be achieved naturally

without fine-tuning of parameters. The aim of this paper is to do the threshold corrections to

the later version of SUSY SU(5) GUT model which includes the top quark threshold corrections

as well for which the latest CDF [13,14] and D0 [15] data of the Fermilab in Batavia, Illinois,

USA has been used where both groups have concluded the observations of top quark.

While including full exact threshold corrections into the two-loop predictions by using the

missing doublet SUSY SU(5) grand unified theory we could be able to derive a theoretically

consistent, phenomenologically viable solution for super-heavy masses, the right selection of

values for the effective supersymmetric particle masses, and for the parameter which determines

the threshold corrections due to non-renormalizable operators using which Weinberg angle could

be predicted up to four decimals correctly that is 0.2319 and the strong coupling constant up to

three decimals correctly that is 0.120 the exact latest experimental values.

These exact values could be derived from the theory only when top quark mass is 176 GeV/C2

as claimed by CDF group of Fermilab in 1995. Therefore our calculation further confirms the

existence of top quark at 176 GeV/C2. Further the effective masses which were introduced to

handle the thresholds due to supersymmetric particle mass spectrum played a crucial role while

deriving these values theoretically. Therefore supersymmetric particles should necessarily exist in

between MZ and 1 TeV which may be detected in large hadron collider in future. Now onwards

missing doublet SUSY SU(5) grand unified theory is emerging as one of the most promising

candidates for unifying strong, weak and electromagnetic interactions. Moreover this specific

GUT model does not possess any theoretical shortcomings either. Hence, we can conclude that

the missing doublet model as one of the noble theory for unification which has a remarkable

predictive power as its own merits.

Let us first remember the doublet-triplet splitting problem in the minimal SUSY SU(5)

theory. The relevant part of the superpotential is W ~ A0£0 + M^cfxp with three chiral super-

multiplets: Σ(24), φ(5) and 0(5).

The breaking of SU(5) in the SU(3)cx SU(2)Lx U(1)Y direction via A(E(24)) = \M diag

(2,2,2,-3,-3) gives us W ~ 0(3)0(3)(M5 + 23M) + 0(2) • 0(2) • (M5 - M).

Selecting M5 = M we will end up with massless Higgs doublet. But fine tuning of parameters

in this way is not natural. If we have omitted the direct mass term the Higgs doublet would

obtain a superheavy mass through the expectation value of the adjoint Higgs. The procedure

that we have to follow is to introduce a representation which contains a Higgs triplet but no

doublet. Then, if we do not put a direct mass term for 5 and 5 there is no doublet mass

term in the superpotential. The interesting observations that has been made is that the 50

representation of SU(5), under SU(3)C x SU(2)L X U(1)Y does not contain any SU(3) singlet -

3

SU(2) doublet.

50 = (8, 2) + (6, 3) + (6, 1) + (3, 2) + (3, 1) + (1, 1) (1)

An expectation value that conserves color and charge is not possible for 50. An anomaly

free supersymmetric SU(5) requires a 50, since it is a non-real representation. In order to write

mixing terms between 5,5 and the 50, 50, we have to use the 75 instead of the 24 representation

to break SU(5). This is known as missing partner mechanism.

The minimal model is not satisfactory since the fine tuning has to be arranged at a high

level of accuracy for no particular reason. The missing doublet model (MDM) explains the

doublet-triplet splitting in the multiplets φ and (f) by coupling them to Φ(50) and Φ(50) which

do not contain doublet components. The required part of the superpotential is

WMDM = m 7 5 Tr(Σ 2 ) + λ1Tr(Σ3) + λ 2ΦΣφ + A 3 $E^ + m50ΦΦ (2)

Where Σ is now a 75 chiral supermultiplet. A non-zero vacuum expectation value of Σ(75)

breaks SU(5) in a unique way to SU(3)C x SU(2)L X U(1)Y then only the triplet components of

(f) and (f) gain masses of the grand unified energy scale. The (3,2, ±56) components of Σ combine

with the (x, y) gauge multiplet and makes a massive gauge - Higgs supermultiplet as in the

minimal theory (see Table I).

We find m2

x = 24g52V752 with V75 = m75/ 4λ1. The remaining chiral supermultiplet of Σ

acquire masses proportional to m75 (see Table II). The (8,3,0) component has the biggtest mass

20m75 which will be denoted by mΣ. The supermultiplet Σ(75) splits into several multiplets

with different masses and nontrivial SU(3)C x SU(2)L X U(1)Y quantum numbers which leads to

threshold effects which are independent of the parameters of the theory. All the supermultiplets

of Φ(50) and $(50) obtain the common mass mΦ = m50, except for their (3,1, ±31) to form two

Dirac supermultiplets D1 and D2 of masses.

(m250 + ( λ 2 + λ 3)2V 75 2) 1 / 2 ± (m250 + ( λ 2 - λ3)2V752)1/2

(3)

All the five parameters of the superpotential are independent, and we can take the five physical

masses mx, mΣ, mD1, mD2

a n d mΦ as the parameters of the theory.

The two-loop renormalization group equation is given by

where bi, bij are the one and two-loop β function coefficients respectively. The αi, i = 1,2,3 are

the normalized coupling constants of electromagnetic, weak and strong interactions respectively.

The [i is the scale parameter.

The solution to the two-loop renormalization group equation takes the following form:

, „ . 1 + b i t + θi - Ai for i = 1,2,3(Xi{Mz) (XQ

where

and

The MG is the grand unification scale and ΑG is the coupling at that point. A» are threshold

and other corrections, which should be calculated to a precision consistent with the θi.

At the Z threshold we have

1 3 1 - S2(MZ) S2(MZ) 1 ." = 5 —^TT^> ^77^V> TTTT^T f o r * = 1, 2, 3, . . . (6)

respectively, S2(MZ), α(MZ) and αS(MZ) are the weak angle, electromagnetic coupling and

strong coupling constants respectively; the three low scale parameters which are defined in the

modified minimal substraction scheme (MS) [19] and evaluated at the Z pole. Here, mG serves

as the high scale boundary of the desert whle MZ serves as the low scale boundary of the desert.

The two-loop terms can be rewritten using the lowest-order solution for the couplings, i.e.

1 1 α OLG(= αi

at(Mz) aG

+ • a , ( M z )

Therefore

1 ^ 3 bθi = 1 b —^-£n(l + bjαGt) for i = 1,2,3 (7)

where the one-loop expressions for ΑG and t are to be substituted.

The one-loop and two-loop β-function coefficients [16] are given by

/ f \ / 7.96 5.4 17.6 \

bi=\ 1 and bij = I 1.8 25 24 | (8)

The correction terms Aj for i = 1, 2, 3 are given by

A • = /\conversion + y ^ y ^ b i £n

boundary C ^ V V M b o u n d a r y

+ A f p + Afu k a w a + A,N R O (9)

The first term is a constant, which depends only on the gauge group Gi:

A conversion C 2 ( G i )

* ~ 12vr K '

where C2(Gi) is the quadratic Casimir operator for the adjoint representation

C2(Gi) = N for d = SU(N)

= 0 for d = U(1)

^conversion i r e s u l t s from the n e e d t o u s e the dimensional-reduction (DR) scheme in the MSSM,

so that the algebra is kept in four dimensions. Thus, we convert the MS couplings above M

conversion-*- -*- A

aMS-aDR- *

The second term in the expression for Aj sums over the one-loop threshold corrections, bζ

is the (decoupled) contribution of a heavy field to the β function coefficient bi between Mζ

and MboundaryCJζ is a mass-independent number, which depends on the spin Jζ of ζ and on

the regularization scheme used. In MS (using dimensional regularization) one has

C1/2 = C0 = 0. These are to be used at the low-scale boundary while at the other boundary1V1 o 1V1 o

(using dimensional reduction) we have Cj^ = 0.

The above summation has to be done at the low-scale boundary in the minimal supersym-

metric standard model (MSSM) (which is embedded in the missing doublet SUSY SU(5) grand

unified model) for the sparticles and heavy Higgs doublet. Instead of considering the individual

masses of these particles which can be calculated given a small number of high-scale parameters

- i.e., a universal gaugino mass M1/2: a universal scalar mass M0: the Higgs mixing parame-

ter /imixing: a universal trilinear coupling A: and the top Yukawa coupling ht (here we omit

all other Yukawa couplings) - by solving a set of coupled renormalization - group equations

(RGE's) (other mass parameters, such as the universal bilinear coupling B, are related to the

parameters above by boundary conditions and the constraint setting the weak breaking scale.);

we use a parametrization in terms of three low-energy effective parameters defined by [1 7]

In our paper we use the following data [7] from principal LEP and other recent observables.

MZ = 91.187 ± 0.007 GeV, 60GeV < MH < 1TeV, the global best fit values: mt =

169+ig -20 G e V ,

S\ = 0.2319 ± 0.0005 ± 0.0002 ,

αs(MZ) = 0.120 ±0.007 ±0.002 (13)

where the central values are for a Higgs mass of 300 GeV, and the second error bars are for

mH -> 1000(+) or 60(-).

In the modified minimum substraction (MS) scheme [19], [23],

a{Mz)~l = 127.9 ± 0 . 1 (14)

We will now discuss the threshold corrections due to heavy top quark. In the MS scheme to

account for mt > mZ one can define threshold corrections [20] to α(MZ) and αs(MZ), i.e.

top / r> \ /] / 1 / 7 top / r, \ /j / _ L - 1

2ir j n m t mZ and I o3 3 2vr I n m t mZ respectively ,

where bnP and b3top are the top contributions to the relevant one-loop β function slope. In MS

definition for our central value of mt = 169GeV, our value of α(MZ) already includes the top

threshold correction, and we have to further correct α(MZ) only for different values of mt. Thus

mmSimilarly, the mt threshold corrections are already included in S2(MZ) definition. However, the

input value of S2(MZ) extracted from the data depends both quadratically and logarithmically

on mt. In particular the value S02(MZ) = 0.2319 ± 0.0005 is for the best fit value mt = mt0 =

169GeV. For other mt the corresponding S2(MZ) is [19]

where GF = 1.166392 x 10 5 GeV 2 is the Fermi coupling, and we have neglected logarithmic

dependences on mt.

We then have S2(MZ) = S02(MZ) + A^T where

A^T = -1.041 x 10"7 [m2 - (169)2] (18)

Therefore

S2(MZ) = 0.2319 - 1.041 x 10"7[m2 - 28561] (19)

The mt dependence of the "true" S2(MZ) is A ^ 1 will be included together with the threshold

corrections in A-op. Thus

to p 8

p 1 =

-S2(MZ)} mt 1 3

A t o P = 8S02(MZ) ( mtA 2 - 9vr in{l69G

5 a(Mz)

A top

(20)

) (22)A^ £ n ( )3 3vr \91.187GeVj

Another issue that is related to the heavy top is the contribution of the top Yukawa coupling

ht to the two loop β function. If ht ~ 1, we have to reintroduce the relevant term (that was

neglected above) in the β function, i.e.

Qoti bi bij h2 a2

- (23)

where bi:top = ^§, 6, 4 for i = 1,2,3 in the minimum supersymmetric standard model. ht is

running and is coupled to ΑI at the one-loop order. A^ u k a w a are functions of the couplings ht and

(IQ at the unification point, and of the unification point parameter t, and have to be calculated

numerically.

Here we use an approximation in which ht is constant. Then the new term in the above

equation is realized as a negative correction to bi, and

AYukawa

8

• = ^ % 3 ( 2 4 )

ht ~ 1 ~ hfixed is a reasonable approximation (hfixed is the fixed point of one-loop top Yukawa

Renormalization Group equation).

Finally, we consider contributions from nonrenormalizable operators at the high scale, which

may be induced by the physics between mG and MPlanck = 1.22 x 1019GeV/C2. We consider

only dimension-five operators,

* MPlanck

where η is a dimensionless parameter and F^v is the field strength tensor. In the missing doublet

SUSY SU(5) model Σ is the 75 chiral supermultiplet (contributions from higher dimension

operators are suppressed by power of Mp^nck). When Σ acquires an expectation value the effect

is to renormalize the gauge fields, which can be absorbed into a redefinition of the couplings.

The running couplings at mG are related to the underlying gauge coupling ΑG(MG) by [21,22]

G) = (1+ &i)/otG, where

ff(p) (26)

In this model r = ^ * and ki = \, | , —1 for i = 1,2,3 respectively. We treat these operators

perturbatively (i.e. for |η| < 10), by defining

/ \ 1 / 2

(27)

where it is sufficient to use the one-loop expressions for ΑG and mG = MZ • e 2 π t

/ 2 V/2

AfRO = -7]ki 2 5 - • (0.74743443) x 1 0 " 1 7 • e2πt (28)

Now substituting values for b1, b2 and b3 we obtain the following two sets of equations for t,

CXQ1, S2(MZ) and t, CXQ1, αS(MZ) respectively

*Here an extra factor has been absorbed into η. See Mathematical Appendix.

(A) t = 1 ' 3

60 α(MZ)

αG = 20 α(MZ) + α

s ( M Z )

S2(MZ) = 0 2 + — •0. 2+ 1 5

θ2 -289s -209,)

60 (20Ai - 48A2 + 28A3)

(B)

1αG

3-36gg(M z) (5gi-33e2) (33A 2-5Ai)28α(0MZ) + 28 + 28

-28(93)

where

1 = 7.7222459 - -In5 ( mx\ 10

in H IvrH

ir— ^n5vr

1 , A m D 2 \ 173

5vr V

where

11,2 = 2

1/2

m502 + (λ2 + λ 3) 2

16λ2

(A2 -

A 2 = 9.614596 + 1 1 I n + —invr \rri J 2vr

m%z,

16λ21

3̂ /mx\in inM

2\—In + in in + in

vr \rriG J 2vr \niG J vr \niG J 12vr \MZ J

A3 = 8.7037823 + 2^i2 2vr 2vr

mG

Further,

- A2) = -1.892044 in m X .„.„ . . „.„ m

vr \niG J vr \rriG J 5vr \ m c .1 , /mfll+1in 1

5vr V V m

(33A2 - 5A l } = 278.66033 - ™ln

7T

m G y vr V m G /

127r M "

\mG J

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

10

(20Ai - 48A2 + 28A3) = -63.33509f - —In ( — X - -In (—IT \mGJ vr \mG

1 8o m 5 0 , 1 8 . fmDl\ 18 (mD2\ 25 / M A^n H in L H in H in —— (40)

vr Vmcy vr \mG J TT \mGJ vr \MZ j100 / M 2 \ 56 /M

3A2 - 8A3) = -2.f7524 - £„ ( ) ln (IT \rriGj vr \rriG

— £ n £n £n + —-£n —— (41)IT \mG) IT \mG ) IT \mG + 4TT \MZ J

, 25 /M2\ 16. fM3\+T/n {Wz) ~ Tln [Wj

(12A3 + 5A1 + 3A2) = 171.90041 - 5 ( ) + n̂ f ^ + ̂ n fvr \mG J IT \mG J IT \mG

H7^n + -in m D H £n

IT V mG J IT V me / vr

25 (M2\ 24 / M 3 \

4vr V M Z ; TT V M Z ;

Now we have to do the numerical calculations.

We use the recent data in Table III from Fermilab in USA.

We will evaluate how this higher values of top quark mass effect on the value of the Weinberg

angle using the formulas (18) and (19) (see Table IV).

Now let us see how the top quark mass effect on the strong coupling constant αs using he

formula (16) and on the electromagnetic coupling α using the formula (15) (see Table V).

Finally we evaluate the effect of top mass on α1 and α 2 using the formulas (20) and (21)

(see Table VI).

Now we have to evaluate the numerically predictable values of t (hence MG), aG

l (hence

αG), S2(MZ) and αs(MZ) using the expressions which come under (A) and (B) given earlier up

to one-loop level at first (see Table VII).

We can enumerate the two-loop corrections by using the expression (7) where we have to use

the one-loop values of αG and t derived in Table VII (see Table VIII).

We will now calculate the two loop predictions (see Table IX).

Using the expression:

1 3 ft-E -in0- + bjαGt + θjαG)

3

for i = 1,2,3

11

we will enumerte the two-loop terms up to two-loop level (see Table X).

Now we have to evaluate the heavy top Yukawa coupling threshold corrections by taking

ht ~ 1 in the expression (24) (see Table XI).

Finally we have to calculate the threshold corrections due to nonrenormalizable operators

using the expression (28) (see Table XII).

Let us now evaluate the A™nvers ion factor (see Table XIII).

Let us evaluate δi = Afn v e r s i o n + A?p + Ajakawa i A t

N R O for i = 1,2,3 (see Table XIV).

The values of δ1, δ2 and δ3 will be added to the expressions for Ai, A2 and A3 respectively.

Further, the value of (δ1 - δ2), (33δ2 - 5δ1), (20δ1 - 48δ2 + 28δ3), (5δ1 + 3δ2 - 8δ3) and (12δ3 +

5δ1+3δ2) will be added to the expressions for ( A i - A 2 ) , (33A 2 -5Ai) , ( 2 0 A i - 4 8 A 2 + 28A3),

(5Ai + 3A2 - 8A3) and (12A3 + 5Ai + 3A2) respectively. Now take h = A*°p + Afukawa (for

i = 1,2,3) which contains the heavy top quark and the heavy top Yukawa coupling threshold

corrections. Since λi terms can be evaluated we can easily see how recently observed heavy

top quark effects explicitly on the predictions of the missing doublet SUSY SU(5) grand unified

theory. These heavy top quark effects are quantitatively the same for the minimal SUSY SU(5)

GUT predictions as well. Let us now evaluate these effects. For this purpose we use the two

loop values that come under (A) and (B) (see Table XV).

Using the above table finally we evaluate the threshold corrections due to heavy top quark

plus heavy top Yukawa Coupling on αS(MZ), S2(MZ), t and CXQ1 (see Table XVI).

In Table XVI, the explicit effect of the heavy top quark plus heavy top Yukawa coupling

threshold corrections on the predictions of the missing doublet SUSY SU(5) GUT model have

been evaluated which are valid for the minimal SUSY SU(5) GUT model as well. According

to the results that we have obtained here, these threshold corrections will have an effect at the

third decimal of the value of t predicted by the set of equations that come under (A). It will

have an effect at the first decimal of the predicted value of CXQ1 by the same set of equations,

this effect will be more if the top quark mass is around 199 GeV/C2 as concluded by the D0

group of Fermilab recently. These corrections will have an effect at the fourth decimal of the

predicted value of the Weinberg angle S2(MZ) by the set of (A) equations. It can change the

12

third decimal if the top quark mass is about 199 GeV/C2. These corrections will change the

third decimal of the value of t predicted by the set of (B) equations and will have an effect at

the second decimal if the top quark mass is around 199 GeV/C2.

The same corrections will change the first decimal of the predicted value of aG

l by the set

of (B) equations but the effect will decrease if the top quark mass increases further. Hence, if

the top quark mass is about 199 GeV/C2 the effect will be at the second decimal. Finally we

can see that, these corrections will have an effect at the fourth decimal of the prediction of the

value of the strong coupling constant αs(MZ) by the set of (B) equations and when the top

quark mass increases the effect will also increase, hence it will be at the third decimal place for

the top quark mass of 199 GeV/C2 or the average value of 183 GeV/C2.

Now let us evaluate the total threshold effects quantitatively. Using the following expressions

for mD1 and mD2:

•"* = \ ((•»*> + < * » + * . > a i j | ) V 2 + (•»*> + <*» - * .> a i j | )

250 + (λ2 + λ3)2m752 - m5

02 + (λ2 - λ3)

We can easily prove that m75 = rrh~ (mD1 m D 2 ) 1 / 2 a n d

77150 ~

Let us take λ2 = λ3 = λ: then m50 = (mD1 — MD2) a n d

m 7 5 = 4λλ1(mD1 • mD2)1/2.

If we select mD1 = mG and mD2 = 21mG = 0.5mG then

m 5 0 = (mD1 - mD2) = 1mG = 0.5mG

If we select λ2 = λ3 = λ = 4 and λ1 = \/2 then by

4λ1 1 / 2 4^/2 mG 1 / 2m75 = 4— (mDl -mot) ' = 4 2(mGm-) ' = mGA 4 G

(mDl mot)A 4

Therefore

= m75 = mG and

13

Further we take mX = 0.3mG ** M1 = 4MZ, M2 = MZ, M3 = 6MZ (see Table XVII).

Table XVIII gives the other threshold contributions which are valid for both the models.

But the value of η has to be selected according to the specific requirement of each model such

that |η| < 10

Now we will evaluate the predictions of the missing doublet model by adding the threshold

corrections due to superheavy masses and supersymmetric particle masses with other threshold

corrections to the two-loop predictions (see Table XIX).

Now we select η to be one, i.e. η = 1 (see Table XX).

At last we calculate the grand unified energy and proton lifetime after full threshold correc-

tions due to the missing doublet model (see Table XXI).

Unified Coupling constant after full threshold corrections due to the missing doublet model

(Table XXII).

Conclusions

Using the solution we found theoretically for the superheavy masses in the missing doublet model,

selecting right values for the supersymmetric particle effective masses in between MZ and 1TeV

and selecting the right value for the parameter which determines the threshold corrections due

to non-renormalizable operators together with the threshold corrections due to the massive top

quark and top Yukawa coupling and conversion factors we were successful in deriving the latest

experimental values by the theory for the Weinberg angle which is 0.2319 and the strong coupling

constant which is 0.120.

When the top quark mass is 176 GeV/C2 these values could have been derived. This fur-

ther confirms the existence of top quark exactly at 176 GeV/C2, neither less nor more. Since

the supersymmetric particle effective masses played a crucial role while deriving these values

theoretically they should necessarily exist in between MZ and 1 TeV hopefully may observe

in the large hadron collider (LHC) in the future. The set of (A) equations predict the unified

coupling constant to be 0.032 and the set of (B) equations predict it to be 0.038 up to first three

"Here mX = 0.3MG a lower value has been used but for the two-loop value mX = 0.6mG withdifferent Mi's the calculation is explained in the mathematical appendix. See the appendix.

14

decimals. For the top quark mass of 176 GeV/C2 under set of (A) equations using two-loop

values under (A)/(B), the grand unified energy is predicted to be 1.925 x 1016GeV/2.538 x 1016

GeV and under the set of (B) equations using two-loop values under (A)/(B) it is predicted to

be 1.135 x 1016GeV/ 1.616 x 1016GeV. While the top quark exists at 176GeV/C2 under set of (A)

equations using two-loop values under (A)/(B) the proton lifetime is predicted to be 2.369 x 1034

years/7.032 x 1034 years and under set of (B) equations using two-loop values under (A)/(B)

the proton lifetime is predicted to be 2.046 x 1033 years/8.257 x 1033 years that we leave for

experimentalists for further tests which are of course beyond the lower bound given by them.

Acknowledgments

The author would like to thank Dr. M.K. Parida for the important discussions, suggestions

and the hospitality at the North Eastern Hill University, India during part of this work. The

author would also like to thank Dr. R.M. Godbole of the Department of Physics, University of

Bombay, India for the valuable discussions and suggestions. The author acknowledges financial

support from the University of Colombo, Sri Lanka. This work was done within the framework

of the Associateship Scheme of the Abdus Salam International Centre for Theoretical Physics,

Trieste. Financial support from the Swedish International Development Cooperation Agency is

acknowledged.

15

Mathematical Appendix

1. Non renormalizable Operators The term — ̂ T r ^ i ^ E i 7 7 ^ ) M ^ n c k can occur in the

Lagrangian of the full unified theory for energies mG < E < MPlanck. That is

C = Co- 1 η T r ( F ^ E F ^ ) where F^ = F*^-* ηPlanck *

is the field strength. The dimensionless constant η may arise with order of unity in quantum

gravity. The above term is of dimension five therefore it is a non-renormalizable operator.

When universe cooled from Planck energy gravity decoupled. There may be d > 5 operators

induced by gravity and enter the Lagrangian scaled by factors of (MP l a n c k)~(d~4) with order unity

coefficients which are subject only to the constraints of symmetries such as gauge invariance and

supersymmetry of the energy theory. When considered within a Kaluza-Klein framework, any

GUT Lagrangian is expected to be modified through the addition of nonrenormalizable terms

whose form is dictated by the appropriate local and global symmetries present. Let us discuss

the SU(5) case in detail. The standard Lagrangian contains the pure gauge boson term

Tr(F^F^) where F^ = d^Av - 8VA^ - ig5[A^, Aν]

g5 being the SU(5) gauge coupling, {A^)b

a = A^(Xi)^, and λi are SU(5) generators, normalized

such that

•" 2 J

We now introduce the following SU(5)-invariant nonrenormalizable (NR) (dimension-five)

interaction term

LNR = η/MPlanck ( - i ) T r ( F ^ S F ^ )

where Σ denotes Higgs 24-plet (Σ(24)) or 75 chiral supermultiplet (Σ(75)) in the minimal model

and missing doublet model respetively.

Now let Σ acquire a nonzero vaccum expectation value

/ 1 \ 1/2 / 1 \ 1/2 3 3

(£(24)) = 1 •F24diag(-2,-2,-2,3,3)= 1 • 2F2 4diag(-l,-1,-1,3, 3)\_LO/ \ X o / Zi Zi

(£(75)> = (J^j F75diag(-2, -2, -2,3,3) = (J^j 2F75diag(-l, - 1 , - 1 , | ^)

16

The SU(5) gauge symmetry breaks to SU(3)C x SU(2)L X U(1)Y and the gauge bosons that me-

diate baryon-number-nonconserving processes acquire masses mX2 = 56g52V242 and mX2 = 24g52V7

5

respectively in the minimal and missing doublet model.

The presence of the nonrenormalizable coupling modifies the usual kinetic energy terms of the

SU(3)C, SU(2)L and U(1)Y gauge bosons. To order Gj, the gauge boson part of the low-energy

Lagrangian is given by

~\ {(1+ e3)Tr(F^F(3)^) + (1+ e2)Tr(F$F^)} - ±(1+ ^)F$ F^v

where the superscripts 3, 2 and 1 refer to the gauge field strengths of SU(3)C, SU(2)L and

U(1)Y and Gj is defined by

1G,= 151 ηki2V24/MPlanck for i = 1,2,3

in the minimal model and

/ 1 \ 1/2Gi= 1 η ) ki2V75/MPlanck for i = 1, 2, 3

in the missing doublet model where k3 = —1, k2 = | , k1 = \ in both models. After appropriate

rescaling of the field variables, one finds that the SU(3)C, SU(2)L and U(1)Y gauge couplings

are related at the scale mG as follows

(1+ G3)#i(mG) = (1+ <E2)gl(mG) = (1+ ei)gl(mG) = g5

" 1 ^ = (1+ ei){gl}-Hir

1 1 (ql\= (1+ G j ) 1 where αG = — for i = 1,2,3

y a G \47r/

Therefore g5 = 4 π

Thus, the presence of the nonrenormalizable term modifies the boundary conditions usually

imposed on the gauge couplings which are g3(mG) = G2(MG) = G1(MG) at scale mG. Further

\l/2 mX (Q\l/2 mX ( 3 1/2

' m x

and

7 5 =

17

Therefore

/ 1 \ 1/2 / 3 \ I / 2<=• = I — 1 nh, • 9 I 1 m.v/M-r,, — -,. = I

η M P l a n c k

since mX ~ mG

A 2 \ 1 / 2 / mG \Gi= — M G for i = 1,2,3

\25iraGJ \ m Jin the minimal model.

In the missing doublet model

/ 1 \ 1/2 / 1 \i/2 / 1 \i/2 m X

Gi= 1 J/A* i 2 2 1 • m X / M P l a n c k = 1 k i ηMPlanck

since

/ 1 / 2

V3607TQ;G/

in the missing doublet model.

But

Rescaling; jy' = ^=r? or ^rf = η the Gj becomes

/ 2 \ 1 / 2

Si= 2 k i η ( m G / M P l a n c k ) for i = 1, 2, 3

in the missing doublet model.

Therefore we can use the same expression for G» in both the models:

1 (1+ £j) 1 + g t

2 V/2

k i ( m G / M P l a n c k ) for i = 1,2,3

Now we define the threshold corrections due to nonrenormalizable operator as /S.fRO = —

/ \ 1/2NRO _ h . [ l \ mG , • _ , 9 o

MPl a n c k

with ki = T;, | , —1 for i = 1,2,3 respectively.

Further G,= ki G where e = η ( 3 5 ^ ) 2 ' 7 3 ^ for i = 1, 2, 3 and

25

18

2. Missing Doublet Model In this model m2X = 24g52 • V7

52 and V75 = m75/4λ1; λ1 = \/2,

- Therefore

2m2X =

2 m275 \ 2 2 2

5 2;

\/3—

Therefore mX = \/3iTttG "^75; for ΑG = 0.042 (two loop value) and m75 = mG, mX ~ 0.6mG

But ΑG gets a lower value after full threshold corrections hence mX. For mX = 0.6mG and

rj = 1 with different values of Mi's the supersymmetric effective masses we can get the correct

values of S2(Mz) and α s(MZ) as in the text of this paper. Therefore in the missing doublet

model for mX = 0.6mG, m75 = mG, MD1 = mG, mD2 = 0.5mG

M1 = 4MZ, M2 = MZ, M3 = 7MZ and η = 1, mt = 176 GeV/c2

(20Ai - 48A2 + 28A3) = -63.335091

= -63.335091

The contribution to S2(mZ); ^ ^ (

The contribution to

{6.1299075 + 34.657359 + 108.97097}

= -15.665564

- 48A2 + 28A3) = -0.0020413.

2 o A i " 4 8 A a + 2 8 A s ) = " ° - ° ° 7 3 1 9 4

S2(M

Z) = 0.2334409 - 0.0020413 + 0.0002327 + 0.0003346 = 0.2319669

or

= 0.2334409 - 0.0020413 + 0.0002295 + 0.0004641 = 0.2320932

and

αs(m

Z) = 0.1249658 - 0.0073194 + 0.0008344 + 0.0011999 = 0.1196807

or

= 0.1249658 - 0.0073194 + 0.0008229 + 0.0016642 = 0.1201335

19

3. Derivations of Superheavy Mass Relations We start with the expressions

/ 2 \ 1/2 / 2 \ 1/2

1 = m 5

0 + (A2 + A 3 ) 2 ^ j λ ( m 2

0 + (λ 2 - λ3)216mλ2751 2 (a)

/ m2 V'2 ( m2 V / 2

2m D 2 = m 5

0 + (A2 + A 3 ) 2 ^ | J - (m2

50 + (λ 2 - λ3)216mλ2751 2 (b)

By/ 2

Therefore

(mD1 + mD2)2 = ( ml + (λ2 + λ3)

By

(a) - (b); 2(m D 1 - mD2) = 2 ( m250 + (λ2 - λ3

Therefore

(mD1 - mD2) = m 5 0 + (λ2 - λ3)

By

(A) - (B); (mD1 + m D 2 ) 2 - (m D 1 - m D 2 ) 2 = ((λ2 + λ 3 ) 2 - (λ2 -

(mD1 + mD2 - mD1 + mD2)(mD1

m2

(λ 2 + λ 3 — λ 2 + λ 3 ) (λ 2 + λ 3 + λ 2 — λ3)m75 2loA^

Therefore

•ml.2mD2 • 2mD1 = 2λ 3 • 2λ 2

Therefore

λ 3 λ 2 2mD1 •mD2 = ¥ ^ 2 - m75

Therefore

2 (4λ1) 2

m752 = mD1 • mD2λ2λ 3

m 7 5 = ~7== • (mD1 • mD2)1/2

By

(A) (mD1+mD2)2 m502 m2

_ _i_ "u'b (c)(λ2 + λ 3 ) 2 ; (λ2 + A3)

2 (A2 + λ 3 ) 2 + 1 6 λ 2 1 (

20

By

' ^ '""- ~~~ ^ ™ 2 " ' 7 5(B) (m D 1 - m p 2 ) 2 _ m502 m2

75

(A 2 -A 3 ) 2 ' (A 2-A 3) 2 (A 2 -A 3 ) 2 +16λ2

By

(c)-(Dy, ^ ; rrr - v ' ; r rrr = ^ 0 ' ^ x

(λ2 + A3)2 (A 2 -A 3 ) 2 = m 5 0 ( λ 2 + A3)2 (A 2 -A 3 ) 2 ,

- λ 3) 2 - (λ2 + λ3)2) = (m D 1 + m D 2 ) 2 (λ 2 - λ 3) 2 - (mD1 - m D 2 ) 2 (λ 2 + λ 3 ) 2

mgO(-4A2A3) = ((mD1 + m D 2 ) ( λ 2 - λ3) - (mD1 - mD2)(λ2 + λ3))

λ2 — λ3 — λ2 — λ3) + mD2 (λ2 — λ3 + λ2 + λ3))

D1 (λ 2 - λ3 + λ2 + λ 3) + mD2 (λ 2 - λ3 - λ2 - λ3))

—4m250λ2λ3 = (—

λ 2 λ 3

Therefore

2 ( λ 3 m D 1 -m250 = λ 2 λ 3

Therefore

_ λ 3 - A 2 m D 2 )(A 2 m D l - λ 3 mD 2 ) ) 1 / 2

Let us take

((λmD1 - λmD2)(λmD1 - λmD2))1/2

2

λ2 = λ3 = λ; then m 5 0 = 1 , " 1 / 2 D — — = ((m D 1 - M D 2

Therefore

m 5 0 =

and

4 λ 1 1/2 4 λ 1 1 / 2

m75 = l—-{mB1mB2)1 = —— (mD1mD2) 1

AA

21

4. Top Yukawa Coupling (ht) The top quark mass is given by mt = ht \~%); ht =

where v = 246.2212 GeV

Therefore

Therefore

mt mt

246.2212 = 174.1046812 2

Top quarkmass (mt)GeV/c2

174

176

199

183

Top Yukawacoupling (ht)

0.9993987

1.0108861

1.1429905

1.0510918

Therefore ht w 1 at mt = 176 GeV/c2

22

References

1. J.C. Pati and Abdus Salam, Phys. Rev. Lett. 31 (1973) 661 and Phys. Rev. D8 (1973)

1240, Phys. Rev. D10 (1974) 275.

2. H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438.

3. H. Georgi, H.R. Quinn and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451.

4. A.J. Buras, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B135 (1978) 66.

5. S. Seidel et al., Phys. Rev. Lett. 61 (1988) 2522;

R. Becker-Szendy, et al., Phys. Rev. D42 (1990) 2974;

K.S. Hirata, et al., Phys. Lett. B220 (1989) 308;

C.Berger, et al., Z. Phys. C50 (1991) 385.

6. Langacker, P. and Mann, A., Physics Today 22 (December, 1989).

7. Particle Physics Booklet, July 1994 from the Review of Particle Properties Physical Review

D50 (1994) 1173.

8. Dimopoulos, S. and Georgi, H., Nucl. Phys. B193 (1981) 150;

Sakai, N., Z. Phys. C11 (1981) 153;

Witten, E., Nucl. Phys. B188 (1981) 513.

9. J. Ellis, S. Kelley, and D.V. Nanopoulos, Phys. Lett. B249 (1990) 441; Phys. Lett. B260

(1991) 131;

Amaldi, U., W. de Boar, and H. Furstenau, ibid. 260 (1991) 447;

F. Anselmo, L. Cifarelli, A. Peterman, and A. Zichichi, Nuovo Cimento A 104 (1991)

1817;

P. Langacker, and M. Luo, Phys. Rev. D44 (1991) 817.

10. J. Wess, and B. Zumino, Nucl. Phys. B70 (1974) 39.

11. Weinberg, S., Phys. Lett. B, Vol. 91, No. 1 (1980) 51.

23

12. A. Masiero, D.V. Nanopoulos, K. Tamvakis, and T. Yanagida, Phys. Lett. 115B (1982)

380.

13. F. Abe, et al., Phys. Rev. D50 (1994) 2966, Phys. Rev. Lett. 73 (1994) 225.

14. F. Abe, et al., Phys. Rev. Lett. 74 No. 14 (1995) 2625.

15. S. Abachi et al., Phys. Rev. Lett. 74 No. 14 (1995) 2632.

16. D.R.T. Jones, Phys. Rev. D25 No. 2 (1982) 581.

17. P. Langacker and N. Polonsky, Phys. Rev. D47 No. 9 (1993) 4028.

18. K. Hagiwara, and Y. Yamada, Phys. Rev. Lett., Vol. 70, No. 6 (1993) 709.

19. G. Degrassi, S. Fanchiotti and A. Sirlin, Nucl. Phys. B351 (1991) 49.

20. L. Hall, Nucl. Phys. B178 (1981) 75.

21. C.T. Hill, Phys. Lett. 135B (1984) 47.

22. Q. Shafi and C. Wetterich, Phys. Rev. Lett. 52 (1984) 875.

23. A. Sirlin, Comments Nucl. Part. Phys. 21 No. 5 (1994) 287.

24

Table I

SU(3)C x SU(2)L x U(1)Y quantum numbers, masses, β-function coefficients bi(j) of the

heavy multiplets j in the missing doublet SUSY SU(5) GUT gauge sector

j

X,Y

X,Y

HX,HY

sum

R

(3,2,±65)

(3,2,±56)

(3,2,±56)

mass

mx

mx

mx

mx

b1(j)

354

103

512

- 5

214

2

14

- 3

7~2

43

16

- 2

Table II

Higgs sector of the missing doublet model with 75, 50 50, 5 and 5

j

#(8,3), #(8,3)

#(3,1), #(3,1)

#(6,2), #(6,2)

#(1,1), #(1,1)

#(8,1), #(8,1)

D1,D1

D2,D2

#50, #50

R

(8,3,0)

(3,1,±35)

(6,2,±65)

(1,1,0)

(8,1,0)

(3,1,±31)

(3,1,±31)

Mass

m Σ

0.8mΣ

0.4mΣ

0.4mΣ

0.2mΣ

mD1

mD2

mΦ

0

5

5

0

0

15

15

17310

b2(j)

8

0

3

0

0

0

0

352

92

12

5

0

32

12

12

17

Comments

in 75

} in 5, 5, 50, 50

} 50 + 50-(3, l ,±| )

25

Table III

Year and experimentalgroup

CDF, Fermilab, 1994

CDF, Fermilab, 1995

D0, Fermilab, 1995

mt (mass of thetop quark)

174±10l^GeV/C2

176 ± 8 ± 10 GeV/C2

1991^ ± 22GeV/C2

Table IV

mt

174GeV/C2

176GeV/C2

199GeV/C2

The average value= 183GeV/C2

-0.0001785315

-0.0002514015

-0.001149264

-0.0005130048

S2(MZ) = S02 + AgP = 0.2319 + A^T

0.2317215

0.2316486

0.2307508

0.231387

26

Table V

mt

174 GeV/C2

176 GeV/C2

199 GeV/C2

The average value= 183 GeV/C2

0.0685578

0.0697705

0.0828022

0.0739087

ptop

0.0082496

0.0114832

0.0462344

0.0225186

Table VI

174

176

199

mt

GeV/C2

GeV/C2

GeV/C2

The average value=183 GeV/C2

0.

0.

0

0.

Atop

0174998

0245845

.109497

1004987

A £ O P

-0.0209171

-0.0294911

-0.1362609

-0.0603907

27

Table VII

The physicalquantity

t

mG

αG

S2(MZ)

αs(MZ)

The values up toone-loop level

using (A)

5.2838889

2.3897875 x 1016 GeV/C2

24.185

0.0413479

0.2304056

0.120

(The input value)

The values up toone-loop level

using (B)

5.2292829

1.6957156 xlO 1 6 GeV/C2

24.430727

0.040932

0.2319

(The input value)

0.1143788

Table VIII

θi (for (i = 1,2,3)

θ3

Using one-loopvalues derived

from (A)

0.6680172

1.0907898

0.5608773

Using one-loopvalues derived

from (B)

0.6476158

1.0591707

0.5438849

28

Table IX

The physicalquantity

t

mG

eg

αG

S2(MZ)

αs(MZ)

The two-loop valuesusing (A)

5.248465

1.9129165 x 1016 GeV/C2

23.517851

0.0425208

0.2334409

0.120 (The input value)

The two-loop valuesusing (B)

5.3027748

2.6908789 xlO 1 6 GeV/C2

23.298064

0.042922

0.2319

(The input value)

0.1249658

Table X

θi (for i = 1,2,3)

θ3

Using two-loopvalues derived

from (A)

0.6749068

1.1551888

0.580533

Using two-loopvalues derived

from (B)

0.6957552

1.1862567

0.5975848

29

Table XI

A Yukawa

(for i = 1,2,3)

A Yukawa

A Yukawa

A Yukawa

Using one-loopvalue of tunder(A)

0.1739952

0.2007637

0.1338424

Using one-loopvalue of tunder (B)

0.172197

0.1986889

0.1324592

Using two-loopvalue of tunder (A)

0.1728287

0.1994177

0.1329451

Using two-loopvalue of tunder (B)

0.1746171

0.2014812

0.1343208

Table XII

^NRO

(i=\,2,3)

ANRO

ANRO

NRO

Using one-loopvalues under (A)

-0.01858917?

-0.05576737?

0.0371782η

Using one-loopvalues under (B)

-0.0133917657?

-0.04017517?

0.0267834η

Using two-loopvalues under (A)

-0.0142683217?

-0.04280497?

0.0285366η

Using two-loopvalues under (B)

-0.0197903837?

-0.05937097?

0.0395806η

30

Table XIII

A conversion

(i = 1,2,3)

A conversionn 1

A conversionn 2

A conversion

The value

0

-± = -0.0530516

1 π = -0.0795774

31

Table XIV

mass of the top quarkwith the order of

the loop values

with one loop valuesunder (A)

mt = 174 GeV/C2

mt = 176GeV/C2

mt = 199GeV/C2

The average valuemt = 183GeV/C2

with one-loop valuesunder (B)

mt = 174GeV/C2

mt = 176GeV/C2

mt = 199GeV/C2

The average valuemt = 183GeV/C2

with two-loop valuesunder (A)

mt = 174GeV/C2

mt = 176GeV/C2

mt = 199GeV/C2

The average valuemt = 183GeV/C2

with two-loop valuesunder (B)

mt = 174GeV/C2

mt = 176GeV/C2

mt = 199GeV/C2

The average valuemt = 183GeV/C2

δ1

0.191495- (0.0185891)η

0.1985797 -(0.0185891)??

0.2834922-(0.0185891)??

0.2744939-(0.0185891)??

0.1896968 - (0.013391765)η

0.1967815-(0.013391765)??

0.281694- (0.013391765)η

0.2726957- (0.013391765)η

0.1903285 - (0.014268321)η

0.1974132- (0.014268321)η0.2823257- (0.014268321)η

0.2733274 - (0.014268321)η

0.1921169-(0.019790383)??

0.1992016- (0.019790383)η

0.2841141 - (0.019790383)η

0.2751158-(0.019790383)??

δ2

0.126795 - (0.0557673)η

0.118221 - (0.0557673)η

0.114512 -(0.0557673)??

0.0873214 - (0.0557673)η

0.1247202- (0.0401751)η

0.1161462- (0.0401751)η

0.0093764- (0.0401751)η

0.0852466- (0.0401751)η

0.125449 - (0.0428049)η

0.116875-(0.0428049)??0.0101052- (0.0428049)η

0.0859754 - (0.0428049)η

0.1275125- (0.0593709)η

0.1189385-0.0593709)??

0.0121687-0.0593709)??

0.0880389 - (0.0593709)η

δ3

0.1228228 + 0.0371782η

0.1240355+ (0.0371782)η

0.1370672+(0.0371782)η

0.1281737+(0.0371782)η

0.1214396+(0.0267834)η

0.1226523+(0.0267834)η

0.135684+(0.0267834)η

0.1267905+(0.0267834)η

0.1219255+(0.0285366)η

0.1231382+(0.0285366)η0.1361699+(0.0285366)η

0.1272764+(0.0285366)η

0.1233012+ (0.0395806)η

0.1245139+(0.0395806)η

0.1375456+(0.0395806)η

0.1286521+(0.0395806)η

32

Table XV

The mass of thetop quark

using two loop

values under (A)

mt = 174GeV/C2

mt = 176GeV/C2

mt = 199GeV/C2

The average value

mt = 183GeV/C2

using two-loop

values under (B)

mt = 174GeV/C2

mt = 176GeV/C2

mt = 199GeV/C2

The average value

The average value

mt = 183GeV/C2

λ1

0.1903285

0.1974132

0.2823257

0.2733274

0.1921169

0.1992016

0.2841141

0.2751158

λ2

0.1785006

0.1699266

0.0631568

0.139027

0.1805641

0.1719901

0.0652203

0.1410905

λ3

0.2015029

0.2027156

0.2157473

0.2068538

0.2028786

0.2040913

0.217123

0.2082295

33

Table XVI

The quantity

Using two loop valuesunder (A)/(B)

601(5λ1 + 3A2-8A3)The contribution

to t under (A)

1(12λ3 + 5λ1 + 3λ2)The contribution to

UQ1 under (A)

0Z)(20λ1 - 48λ2 + 28λ3)The contributions toS2(MZ) under (A)

i ( A i - A 2 )The contribution to

t under (B)

(33A2-5Ai)28

The contribution to

a.~Ql under (B)

28[a(Mz)]2(28A3+2OAi-48A2)( 6 0 S 2 ( M z ) - 1 2 ) 2

The contribution to

αs(MZ) under (B)

mt = 174GeV/C2

(A) -0.0020813

(B) -0.0020125

(A) 0.1952589

(B) 0.196841

(A) 0.0001147

(B) 0.0001115

(A) 0.0021111

(B) 0.002063

(A) 0.1785006

(B) 0.1785011

(A) 0.0004114

(B) 0.0003998

mt = 176GeV/C2

-0.0020813

-0.0018508

0.1964716

0.1980537

0.0001912

0.0001695

0.0049083

0.002063

0.1650182

0.1671309

0.0006857

0.0006079

mt = 199GeV/C2

-0.0020813

-0.0020125

0.2095033

0.2110853

0.0011278

0.0011246

0.0391373

0.0390881

0.0240194

0.0261321

0.004044

0.0040324

The average valuemt = 183GeV/C2

0.0021481

0.0022169

0.2132979

0.2148802

0.0005974

0.0005942

0.0239822

0.023933

0.1150447

0.1171574

0.0021421

0.0021306

34

Table XVII

The quantity

^ ( 5 A i + 3 A 2 - 8 A 3 )The contribution to t under (A)

^j(12A3 + 5Ai + 3A2)The contribution to CXQ1 under (A)

a (JJz )(20Ai 48A2 + 28A3)The contribution to S2(MZ) under (A)

— (Ai — A2)The contribution to t under (B)

(33A2-5Ai)28

The contribution to aG under (B)

2 8 ( α ( M Z ) ) 2

The contribution to

αS(MZ) under (B)

Value

-0.005343

7.081274

-0.0020544

-0.1024968

2.4547637

-0.0073662

35

Table XVIII

The quantityusing two-loop

values under (A)/(B)

^ ( 5 J i + 3 J 2 - 8 J 3 )

The contributionto t under (A)

201(12δ3 + 5δ1 + 3δ2)

The contributionto OQ1 under (A)

Z ) (20δ1 - 48δ2 + 28δ3)

The contributionto S2(MZ) under (A)

The contributionto t under (B)

(33δ2 - 5δ1)/28The contribution

O.Q1 under (B)

2%{a{mz)Y(60s2(mz)-12)2

•(28(53 + 20δ1 -48(52)The contribution

αs(MZ) under (B)

m t = 174GeV/C2

(A) 0.0058764+(0.0004756)η

(B) 0.0059452-(0.0098951)??(A) 0.1395547+(0.0071341)η

(B) 0.1411368+(0.0098951)η

(A) 0.0001562+(0.0003346)η

(B) 0.0001529+(0.0004641η

(A) +0.0115856+(0.0050958)η(B) 0.0115365+(0.0070679)η(A) 0.1138633-(0.0479007)?y

(B) 0.115976-(0.0664388)??

(A) 0.0005601

+(0.0011999)η

(B) 0.0005485+(0.0016642)η

mt = 176GeV/C2

0.0058764+(0.0004756)η

0.0059452-0.0098951)??

0.14076740.0071341)η

0.1423495+(0.0098951)η

0.00023270.0003346)η

0.0002295+(0.0004641)η

0.0143818+(0.0050958)η

0.0143326+(0.0070679)η

0.1024931-(0.0479007)??

0.1046058-(0.0664388)??

0.0008344

+(0.0011999)η

0.0008229+(0.0016642)η

mt = 199GeV/C2

0.0058764+ (0.0004756)η

0.0059451-(0.0098951)??

0.1537991+ (0.0071341)η

0.1553811+ (0.0098951)η

0.0011693+ (0.0003346)η

0.001161+ (0.0004641)η

0.0486108+ (0.0050958)η

0.0485616+ (0.0070679)η

-0.0385056-(0.0479007)?y

-0.0363929-(0.0664388)??

0.0041927

+ (0.0011999)η

0.0041631+ (0.0016642)η

The averagevalue

mt = 183GeV/C2

0.0101058+ (0.0004756)η

0.0101746-(0.0098951)??

0.157594+ (0.0071341)η

0.159176+ (0.0098951)η

0.0006389+ (0.0003346)η

0.0006356+ (0.0004641)η

0.0334557+ (0.0050958)η

0.0334065+ (0.0070679)η

0.0525196-(0.0479007)??

0.0546323-(0.0664388)??

0.0022909

+ (0.0011999)η

0.0022793(0.0016642)η

36

Table XIX

Prediction(using two-loop values

under (A)/(B))

t under (A)

U'Q1 under (A)

S2(MZ) under (A)

t under (B)

cig1 under

αs(MZ) under (B)

mt = 174GeV/C2

(A) 5.2489984+ (0.0004756)η

(B) 5.303377-(0.0098951)7?

30.73868+ (0.0071341)η

(B) 30.520475+ (0.0098951)η(A) 0.2315427(0.0003346)η

(B) 0.2315394+ (0.0004641)η(A) 5.1575538+ (0.0050958)η

(B) 5.2118145+ (0.0070679)η(A) 26.086478-(0.0479007)??

(B) 25.868804-(0.0664388)??(A) 0.1181597+ (0.0011999)η

(B) 0.1181481+ (0.0016642)η

mt = 176GeV/C2

5.2489984+ (0.0004756)η

5.303377-(0.0098951)??

30.739892+(0.071341)η

30.521688+ (0.0098951)η

0.2316192+ (0.0003346)η

0.231616+ (0.0004641)η

5.16035+ (0.0050958)η

5.2146106+ (0.0070679)η

26.075108-(0.0479007)?y

25.857434-(0.0664388)??

0.118434+ (0.0011999)η

0.1184225+ (0.0016642)η

mt = 199GeV/C2

5.2489984+(0.0004756)η

5.3033769-(0.0098951)??

30.752924+(0.0071341)η

30.534719+(0.0098951)η

0.2325558+(0.0003346)η

0.2325475+(0.0004641)η

5.194579+(0.0050958)η

5.2488396+(0.0070679)η

25.934109-(0.0479007)?y

25.716435-(0.0664388)??

0.1217923+(0.0011999)η

0.1217627+(0.0016642)η

The average valuemt = 183GeV/C2

5.2532278+ (0.004756)η

5.3076064-(0.0098951)??

30.756719+(0.0071341)η

30.538514(0.0098951)η

0.2320254+(0.0003346)η

0.2320221+(0.0004641)η

5.1794239+(0.0050958)η

5.2336845+(0.0070679)η

26.025134-(0.00479007)??

25.80746-(0.0664388)??

0.1198905+(0.0011999)η

0.1198789+(0.0016642)η

37

Table XX

Prediction(using two-loop values

under (A)/(B))

t under (A)

O.Q1 under (A)

S2(MZ) under (A)

t under (B)

a G ' under (B)

αs(MZ) under (B)

mt = 174GeV/C2

(A) 5.249474

(B) 5.2934819

(A) 30.745814

(B) 30.53037

(A) 0.2318773

(B) 0.2320035

(A) 5.1626496

(B) 5.2188824

(A) 26.038577

(B) 25.802365

(A) 0.1193596

(B) 0.1198123

mt = 176GeV/C2

5.249474

5.2934819

30.811233

30.531583

0.2319538

0.2320801

5.1654458

5.2216785

26.027207

25.790995

0.1196339

0.1200867

mt = 199GeV/C2

5.249474

5.2934818

30.760058

30.544614

0.2328904

0.2330116

5.1996748

5.2559075

25.886208

25.649996

0.1229922

0.1234269

The average valuemt = 183GeV/C2

5.2579838

5.2977113

30.763853

30.548409

0.23236

0.2324862

5.1845197

5.2407524

26.020344

25.741021

0.1210904

0.1215431

38

Table XXI

Prediction(using two-loop values

under (A)/(B))

mG (using valuesof t under (A)

mG (using valuesof t under (B))

Proton lifetimeTp (using values of

mG and a^1 under (A))

Proton lifetimeTp (using values

of ma&zctQ1

under (B))

mt = 174GeV/C2

(A) 1.9250824X1016GeV/C2

(B) 2.15382594X1016GeV/C2

(A) 1.1156475X1016GeV/C2

(B) 1.5884458X1016GeV/C2

(A) 2.3597328X1034 years

(B) 7.0323694X1034 years

(A) 1.9091323X1033 years

(B) 7.7037498X1033 years

mt = 176GeV/C2

1.9250824X1016GeV/C2

2.5382594X1016GeV/C2

1.1354216X1016GeV/C2

1.616589X1016GeV/C2

2.3697852X1034 years

7.0329283X1034 years

2.0463376 x1033 years

8.2573149X1033 years

mt = 199GeV/C2

1.9250824X1016GeV/C2

2.5382578X1016GeV/C2

1.4078605X1016GeV/C2

2.004494X1016GeV/C2

2.3619197X1034 years

7.0389152X1034 years

4.7848563X1033 years

1.9305772X1034 years

The average valuemt = 183GeV/C2

2.0308155 x 1016GeV/C2

2.6066156 x 1016GeV/C2

1.2799856 x 1016GeV/C2

1.8224274 x 1016GeV/C2

2.9258824 x 1034years

7.8303051 x 1034 years

3.3032386 x 1033 years

1.3284525 x 1034 years

39

Table XXII

Prediction(using two-loop valuesvalues under (A)/(B))

a.o under (A)

O.Q under (B)

mt = 174GeV/C2

(A) 0.0325247

(B) 0.0327542

(A) 0.0364045

(B) 0.0387561

mt = 176GeV/C2

0.0324556

0.0327529

0.0384213

0.0387732

mt = 199GeV/C2

0.0325096

0.0327389

0.0386306

0.0389863

The average valuemt = 183GeV/C2

0.0325056

0.0327349

0.0384314

0.0388484

40