The Strong Coupling Constant in Grand Uni Ed Theories

Home , Coupling constant

The Strong Coupling Constant in Grand Uni ed Theories

Damien M. Pierce

Stanford Linear Accelerator Center

Stanford University

Stanford, California 94309

ABSTRACT

The prediction for the strong coupling constant in grand uni ed the-

ories is reviewed, rst in the Standard Mo del, then in the sup ersym-

metric version. Various corrections are considered. The predictions in

b oth sup ergravity-induced and gauge-mediated sup ersymmetry break-

ing mo dels are discussed. In the region of parameter space without

large ne tuning, the strong coupling is predicted to b e M 0:13.

s Z

Imp osing M = 0:118, we require a uni cation scale threshold

s Z

correction of typically 2, which is accommo dated by some GUT

mo dels but in con ict with others.

Work supp orted by Department of Energy Contract DE{AC03{76SF00515.

1 bf Intro duction

In the Standard Mo del, given values of the three gauge couplings, the Yukawa

couplings, the Higgs b oson self-coupling, and a dimensionful observable e.g., the

muon lifetime, a gauge b oson mass, or a quark mass, any other observable can

then b e computed. In a grand-uni ed theory GUT, one need only input two

of the gauge couplings, then the third one can b e predicted, as well as other

observables.

Thus, GUT theories are more predictive. Because the SU2 and U1 cou-

plings are measured quite precisely,we will examine the prediction of the strong

coupling constant in grand-uni ed theories, b oth in the Standard Mo del and in

sup ersymmetric mo dels. We start with the one-lo op results and then pro ceed to

discuss the next higher-order corrections. The inclusion of these corrections leads

to a precise prediction of the strong coupling constant.

2 bf Renormalization Group Equations

We are interested in measuring gauge couplings at the weak-scale or b elow, and

running the gauge couplings up to higher scales. The solution of the renormaliza-

tion group equations RGE's accurately describ es the evolution of the couplings,

even as they are evolved over many decades. At one-lo op, the renormalization

group equations for the gauge couplings are

b Q dg

i i

= g ; t =ln ; 1

dt 16 Q

where the b are the one-lo op b eta-constants. These constants receive contribu-

tions from every particle which circulates in the one-lo op gauge-b oson self-energy

diagram. For example, for a fermion doublet ; e, b =1=3.

The one-lo op renormalization group equation is easily solved. The inverse of

the gauge coupling evolves linearly with the log of the scale,

Q b

1 1

ln : 2 Q= Q

i i

2 Q

If we assume that the couplings do unify at some scale, wehave that

b b b b b b

1 2 2 3 3 1

+ + =0 : 3

Q Q Q

3 1 2

This equation is valid for any scale Q between the weak scale and the uni cation

MS values of the U1 and SU2 gauge couplings from scale. Next, we deduce the

1 2 1 2

the quantities ^ = 127:90 0:09 ands ^ =0:2315 0:0004 Ref. ^ ands ^

are the MS values of the electromagnetic coupling and sine-squared weak mixing

angle evaluated at M

1 1

M =58:97 0:05 ; M =29:61 0:05 : 4

Z Z

1 2

Given the Standard Mo del values of the one-lo op b eta-constants

19 41

; b = ; b = 7 ; 5 b =

2 3 1

10 6

we determine the prediction of the strong coupling constant in the Standard Mo del

M =0:07 Standard Mo del, one-lo op 6

s Z

with negligible error. Comparing with the measured value as quoted by the Par-

ticle Data Group, M =0:118 0:003, we see that the Standard Mo del pre-

s Z

diction of the strong coupling constant is ab out 16 standard deviations to o small.

Including higher order corrections cannot help repair this situation by more than

a few standard deviations. In order to remedy this huge discrepancy,we need to

go b eyond the Standard Mo del, adding matter to change the b eta-constants b .

We also need to sp ecify the mass scale of the new matter.

Rather than determining what additional matter content will lead to successful

gauge coupling uni cation in an ad ho c fashion, we will examine the implications

of a motivated mo del. It is widely accepted that the Standard Mo del is an e ective

theory. Some new physics must b ecome manifest at scales b elow the multi-TeV

scale. A promising mo del of new physics is sup ersymmetry, which, among other

virtues, explains how a theory with a weak-scale elementary Higgs b oson is stable

with resp ect to the Planck scale. Sup ersymmetry also naturally explains the

breaking of electroweak symmetry, and it typically contains a natural dark matter

candidate.

In order to make the Standard Mo del sup ersymmetric, we are required to

add a sp eci c set of new particles, the sup erpartners. For every Standard Mo del

gauge b oson, wemust add a spin-1/2 ma jorana particle, a gaugino. And for every

Standard Mo del fermion wemust add a spin-0 partner, a squark or a slepton. In

order to give b oth up- and down-typ e fermions a mass, and to ensure anomaly

cancellation, wemust also add an additional Higgs doublet. We refer to the two

Higgs doublets as \up-typ e" and \down-typ e." This additional particle content

de nes the minimal sup ersymmetric standard mo del MSSM. In the simplest and

most often considered versions of the MSSM with high-energy inputs, the entire

sup ersymmetric sp ectrum scales with one or two parameters whichmust b e near

the weak scale by naturalness considerations.

Given this new matter content and the sup erpartner mass scale, whichwe

taketobeM for the moment, we can take a rst-order lo ok at the prediction

of the strong coupling constant in the MSSM. As b efore, we use Eq. 3 to solve

for M as a function of M and M . Now the b eta-constants are

s Z 1 Z 2 Z

di erent. Because of the new matter content, wehave the following changes from

the Standard Mo del values

b =4:1! 6:6 ; b =3:2! 1 ; b =7!3 : 7

1 2 3

We use the central values and errors in Eq. 4 to nd

M =0:116 0:001 MSSM, one-lo op. 8

s Z

The new matter content mandated by sup ersymmetry brings the prediction for

the strong coupling constant to the measured value 0:118 0:003 within one

standard deviation! This could b e a coincidence, or it could b e a hint that we are

on the right trackby considering minimal low-energy sup ersymmetry. Next we

will consider corrections to the prediction of in the MSSM.

3 bf Corrections to the prediction of

In this section, we consider two sources of corrections to the prediction of the

strong coupling constant in the MSSM. The rst involves improving the evolution

of the couplings from their initial values at the grand uni cation scale to the weak

scale or, the sup ersymmetry breaking scale. We simply extend the renormal-

ization group equations to one higher lo op order. This means that in addition to

resumming logarithms of the form [ =4 lnM =M ] from the one-lo op RGE,

GU T Z

2 n

wenow resum logarithms of the form [ =4 lnM =M ] from the two-lo op

GU T Z

terms. Attwo lo ops, it is necessary to run b oth the gauge and Yukawa couplings

together, as they form a coupled set of di erential equations. The general form

for either a gauge or Yukawa coupling RGE at two-lo ops is

b b dg

ij k ij i

2 2 2

: 9 + g = g g g

k j j

2 2 2

dt 16 16

Since we can safely ignore the small Yukawa couplings of the rst two generations,

the set g includes fg ; g ; g ; ; ; g. The equations are readily solved

i 1 2 3 t b

numerically. The correction due to the two-lo op RGE's is large and p ositive. The

prediction of M increases by ab out 0:01.

s Z

The second source of corrections we will consider in this section are the sup er-

symmetric threshold corrections. These are divided into twotyp es, the logarithmic

corrections and the nite i.e., nonlogarithmic corrections. We will discuss these

in turn.

When we arrived at prediction 8, we assumed that the masses of the sup er-

partners were equal to M . However, we know from collider searches that many

of the sup erpartner masses must b e heavier than M , and in general, they could

b e an order of magnitude heavier without straining naturalness considerations to o

much. As weevolve the gauge couplings down from high scales, wemust decouple

each sup erpartner contribution in turn as we cross the mass threshold. Hence,

we arrive at logarithmic corrections of the form b lnM =M . From these

i susy Z

corrections, we determine the shift in the prediction of the strong coupling and

nd, in general, for a particle of mass M >M with b contributions to the

Z i

b eta-constants

M 1

: 10 b b b +b b b +b b b ln M =

2 3 1 3 1 2 1 2 3 Z

2b b M

1 2 Z

Note that the b eta-constants b in this equation include the entire sup erpartner

sp ectrum. Plugging in all the sup erpartners, we arrive at the following sup ersym-

metric threshold correction

3 7

M M

1 M M

2 3

L Q

M = 3ln +32ln 28 ln 11

3 2 5

28 M M M M M

Z Z

D E U

jj M

: +12ln + 3ln

M M

Z Z

In the rst term, M and M M , M , and M denote the left-handed right-

Q L U D E

handed squark and slepton masses. This term is easily mo di ed to account for

generation-dep endent masses.

There are a few asp ects of this equation worth p ointing out. First, notice that

the rst term do es not contain M . This is b ecause the squarks and sleptons are

in complete SU5 multiplets, and degenerate complete multiplets do not a ect

at one lo op. It happ ens that this particular combination of soft squark and

slepton masses is near unity in the entire parameter space of b oth of the mo dels

that we consider in the following section. Hence, the squarks and sleptons do not

signi cantly a ect the prediction of M .

s Z

Another p oint is that the SU2 and SU3 gauginos contribute corrections

which largely combine into the term 1= lnM =M . In GUT mo dels, the gaug-

2 3

ino masses are degenerate ab ove the uni cation scale. At one-lo op, they are

renormalized prop ortional to the corresp onding gauge couplings, so the ratio

M =M = = . At the weak scale, this ratio is ab out 8=30 ' 0:27. Hence,

2 3 2 3

this term contributes ab out +0:006 to M , indep endent of parameter space.

s Z

The remaining terms include the residual term from the SU2 gaugino mass,

and the contributions of the heavy Higgs b osons and Higgsinos. If we take these

three masses to b e characterized by a single scale M , then these terms combine

susy

to give

19 M

susy

: 12 M = ln

28 M

Hence, as we increase the sup ersymmetric mass scale, the prediction of the strong

coupling constant decreases. This makes sense, since in the limit that the entire

sup ersymmetric sp ectrum is raised ab ove M ,we should recover the prediction

GU T

1 1

of the Standard Mo del. ]In fact 0:116 + 19=28 lnM =M =0:064 .]

GU T Z

If M = 1TeV, this correction yields M = 0:007. Note that the

susy s Z

largest contribution to Eq. 12 is due to the lnjj=M term. Because we imp ose

electroweak symmetry breaking, jj=M is a measure of the ne tuning necessary

to obtain the Z -mass from the input mass parameters. The predicted strong

coupling is larger in the region of no ne tuning jj=M 2 than in the region

of appreciable ne tuning jj=M 10.

There is one last p oint ab out Eq. 11. Because of the particle contentof

the sup ersymmetric standard mo del, the particles which are charged under SU2

always come into the expression for with a p ositive co ecient in front

of the logarithm of the mass. The SU2 singlets always come with a negative

~ ~

co ecient. Hence, heavy SU2 doublets, for example, W , H ,orH, will decrease

, and heavy SU2 singlets, e.g., the gluino, will increase the prediction of the

strong coupling.

Besides the logarithmic corrections of Eq. 11, there are also nite corrections.

These arise when the full one-lo op correction tos ^ is taken into account. Taking

the electromagnetic constant, the Z -b oson mass, and the muon lifetime as inputs,

6 7

DR Ref. renormalized weak mixing angle, we determine the

^ ^

M ^ 0

ZZ WW

2 2

+ : 13 s^ c^ = ; ^r =

2 2

M M

2G M 1 ^r

W Z

The correction r ^ is comprised of the real and transverse DR gauge-b oson self-

energy contribution the oblique correction, and the vertex and b ox diagram

contributions, the non-universal part. The oblique correction contains b oth

logarithmic and nite contributions, while is purely nite. The nite contri-

2 2

butions decouple as M =M for large sup ersymmetric masses.

Z susy

In the regions of parameter space where some SU2 nonsinglet particles are

of order M , the nite corrections to the weak mixing angle can substantially

9,10

increase the prediction of the strong coupling constant. We illustrate this in

Fig. 1. We show the prediction of the strong coupling with and without taking

the nite corrections into account in the sup ergravity mo del describ ed b elow. If

all the sup ersymmetric particles are ab ove a few hundred GeV, the nite cor-

Fig. 1. The prediction of M with solid and without dot-dashed including

s Z

the nite corrections. From Ref..

rections are negligible. In that case, the low-energy threshold corrections are

well-approximated by the logarithms in Eq. 11.

4 bf M at Two Lo ops in Two Sup ersymmetric Mo d-

s Z

els

Utilizing the two-lo op renormalization group equations and the weak-scale thresh-

old corrections, we are now in a p osition to present the improved prediction of

the strong coupling constant in the sup ersymmetric case. We still need to sp ecify

the sup ersymmetric particle sp ectrum. In what follows, we consider two mo dels

in turn, rst a minimal sup ergravity mo del, then a simple mo del with gauge-

mediated sup ersymmetry breaking.

4.1 Minimal Sup ergravity Mo del

In the minimal sup ergravity mo del, there are universal soft-sup ersymmetry break-

ing parameters induced at the grand uni cation scale. These include a gaugino

mass M , a scalar mass M , and a trilinear scalar coupling A . Also, we re-

1=2 0 0

quire that electroweak symmetry is broken radiatively. This happ ens naturally,

as the large top-quark Yukawa coupling drives the up-typ e Higgs mass-squared

negative near the weak scale. Given the Z -b oson mass and tan [the ratio of

Fig. 2. The prediction of M in the M , M plane, with tan = 2 and A =0.

s Z 0 1=2 0

From Ref..

up-typ e Higgs b oson vacuum exp ectation value vev to down-typ e], we imp ose

electroweak symmetry breaking, and this allows us to solve for the masses of the

heavy Higgs b osons and their sup erpartners, M and jj.

In Fig. 2, we show the prediction of the strong coupling in the M , M plane.

1=2

After including the corrections, the sup ersymmetric prediction is not as consistent

with the measured value 0:118 0:003. We nd that for squark masses b elow ab out

1TeV where the mo del is more natural, is predicted to b e greater than 0.127.

For the smallest sup ersymmetric masses, we nd numb ers larger than 0.140. The

predicted value is three to seven standard deviations larger than the measured

value.

The prediction of dep ends slightly on m .Itchanges by ab out 0.001 if m

s t t

changes by 10 GeV. Of the three input parameters M ;G ;and ^ , only ^ has an

appreciable error. Changing ^ by1-changes by ab out 0.001. The prediction

weakly dep ends on tan b ecause at small 1 or large 30 tan , the top,

b ottom, and/or tau-Yukawa couplings b ecome large, and they enter into the two-

lo op renormalization group equations of the gauge couplings. The prediction for

M can b e lowered by ab out 0.002 at the extreme values of tan . The strong

s Z

coupling is also insensitive to the sign of the Higgsino mass term . In Fig. 2 and

the following gures, we set >0.

Hence, we nd that we cannot avoid the large values of shown in Fig. 2.

The change 0:002 is due solely to the Yukawa couplings entering into the two-lo op RGE's.

There may b e an additional dep endence b ecause the particle sp ectrum dep ends on tan .

We will giveaninterpretation of these large numb ers after discussing the gauge-

mediated case.

4.2 Minimal Gauge-Mediated Mo del

Mo dels with gauge-mediated sup ersymmetry breaking are an attractive alterna-

tive to the sup ergravity mo dels. In the sup ergravity mo del we considered, wechose

universal b oundary conditions, which suppress dangerous squark- and slepton-

mediated avor changing neutral currents. However, there is no symmetry which

protects this choice of b oundary conditions, and hence it is an arti cial imp osition.

The advantage of the gauge-mediated mo dels is that avor changing neutral cur-

rents are automatically suppressed, since the soft sup ersymmetry breaking masses

which are induced by the gauge interactions are avor diagonal and generation

indep endent.

In the simplest mo dels with gauge-mediated sup ersymmetry breaking, there

is a messenger sector with the sup erp otential interaction

M; 14 W = S M

where S is a Standard Mo del singlet sup er eld, M and M are a pair of messenger

elds which are vector-like under the Standard Mo del gauge group, and is the

messenger Yukawa coupling. In a grand uni ed theory, M and M come in full

SU5 representations. We will consider n 5+5 pairs and n 10+10 representa-

5 10

tions. Perturbative uni cation of the gauge couplings is ensured if we allowat

most n ;n = 1,1 or 4,0.

5 10

In these mo dels, the singlet sup er eld S couples to a sector in which sup ersym-

metry is dynamically broken, and as a result, it acquires b oth a vev S and an

F-term F . This in turn generates sup ersymmetry-conserving diagonal entries

and sup ersymmetry-violating o diagonal entries in the M; M scalar mass ma-

trix. The M and M elds enter into lo op diagrams with Standard Mo del elds on

the external legs, thereby generating soft sup ersymmetry breaking masses for the

sup erpartners and the Higgs b osons. The gaugino and scalar masses are generated

2 11

at one- and two-lo ops, resp ectively, and in the limit F S are given by

; 15 M M = n +3n

i 5 10

2 2

m~ M = 2n +3n C ; 16

5 10 i

i=1

where M = S is the messenger mass scale, = F=S, and C =3Y =5;3=4;4=3

for fundamental representations and zero for singlet representations.

The mass parameters 15,16 serve as b oundary conditions for the renormal-

ization group equations at the messenger scale. In order to determine the su-

p erparticle sp ectrum, we run these parameters from the messenger scale to the

weak scale using the two-lo op renormalization group equations. As b efore, we

imp ose radiative electroweak symmetry breaking and determine the Higgs b oson

and Higgsino masses M and jj for a given tan . Hence, the parameter space

of the mo del under consideration is

tan ; M; ; n ; n ; ; sg n;

5 10

where is the value of the messenger Yukawa coupling at the grand uni cation

scale we assume a single Yukawa coupling. Note that a physical messenger

sp ectrum requires M>.

After we determine the sup erpartner sp ectrum at the weak scale, we apply

the same weak-scale gauge coupling threshold corrections as in the sup ergravity

mo del. However, the messenger sector gives rise to additional corrections. Not

only are the one- and two-lo op renormalization group equations altered ab ove the

messenger scale, but there are also new threshold corrections at the scale M .

A degenerate SU5 multiplet do es not a ect the prediction of at one lo op.

However, the evolution of the messenger Yukawa couplings from the grand uni-

cation scale down to the messenger scale splits the messenger multiplets. If we

5 pair of messenger elds into their doublet and triplet comp o- decomp ose a 5+

nents, resp ectively denoted L and D , then the messenger sector sup erp otential

b elow the grand uni cation scale b ecomes

W = SM M + SM M : 17

L L L D D D

Thus, the messenger doublet triplet ends up with mass S S . According

L D

to Eq. 10, this mass splitting leads to the threshold correction the 10 + 10 elds

decomp ose into SU2,SU3 representations Q 2,3, U 1,3, and E 1,1

9n n

Q 5 L 10 Q

15 ln : 18 M = ln + +6ln

14 14

D U E

The Yukawa couplings are evaluated at the messenger scale, either S ' S or

L D

1 1

S ' S ' S . Note that at one lo op order M = M.

Q U E Z

s s

To determine the messenger Yukawa couplings at the messenger scale, we solve

the renormalization group equations which are of the form

1 0

X X

i i

2 2

A @

+ ; 19 = a g c

k j

dt 16

There is an additional negligible correction due to the splitting within each U1, SU2, or

SU3 multiplet. Each logarithm in Eq. 18 is replaced according to ln = ! ln = +

1 2 1 2

2 2 2 2

1=12 ln[1 F= S ]=[1 F= S ].

1 2

Fig. 3. The ratios of the messenger masses at the messenger scale vs. the value

of the Yukawa coupling at the uni cation scale. Both the n = 1 dashed and

n =1;2;3 solid cases and are shown.

where the dots indicate that there may b e extra contributions due to interac-

tions of the singlet with the sup ersymmetry breaking sector elds. These extra

contributions are the same for all the messenger elds so they do not a ect the

ratios of Yukawa couplings. Note that for a large initial value of the messenger

Yukawa coupling, the renormalization group evolution is initially dominated by

the Yukawa term which leads to the same evolution for the various messenger

elds. Hence, there will b e less splitting if the initial value is large. We illustrate

this in Fig. 3 where we show the ratio of messenger Yukawa couplings at the mes-

senger scale vs. the starting value at the grand uni cation scale. In the 10+ 10

case, we see that = = varies from 1.2 to 1.3 2.3 to 2.6. Hence, the

Q U Q E

splitting is small and con ned to a narrow range of values. Plugging this splitting

into Eq. 18, we nd that the messenger threshold correction results in at most

M =0:003 for n =1,+0:001 for n = 1, and +0:005 for n =3.

s Z 10 5 5

Combining all the e ects together, we show the full result for the prediction

of M in the gauge-mediated mo del in the M=, plane in Fig. 4, with

s Z

tan =4,n = 1, and =3. We see slightly smaller numb ers than in the

sup ergravity case Fig. 2. Relative to other choices of n ; n ; and , this case

5 10

results in the smallest values of M . As b efore, the result is not very sensitive

s Z

to the value of tan . Changing the value of the messenger Yukawa coupling do es

not change the value of the strong coupling signi cantly. The dashed lines in Fig. 4 indicate that the region of parameter space with the

Fig. 4. Contours of M in the gauge mediated mo del with n = 1, tan =4,

s Z 5

and = 3. The dashed lines show contours of jj=M =2; 5, and 10.

jj < 10M jj < 2M

Z Z

=0:01 =3 =0:01 =3

n =1 > 0:125 > 0:124 > 0:137 > 0:136

Table 1. Summary of predictions for M inthe

s Z

n =3 > 0:126 > 0:125 > 0:139 > 0:136

n =1 > 0:127 > 0:128 > 0:137 > 0:140

least ne tuning jj=M ' 2 corresp onds to the largest value of M ' 0:137.

Z s Z

The strong coupling can b e reduced signi cantly at the cost of ne tuning. In the

ne tuned region jj=M ' 10, M ' 0:125.

Z s Z

We summarize the predicted values of in the gauge-mediated mo dels in

Table 1. Here we set m = 175 GeV, tan = 4, and > 0. We nd that

if jj < 10M , M is greater than 0.124. This is two standard deviations

Z s Z

larger than the measured value. Hence, we are faced with the fact that, like the

Standard Mo del, sup ersymmetric mo dels do not predict that the gauge couplings

meet. However, the discrepancy in the Standard Mo del case is enormous. The

small discrepancy in the MSSM can b e accounted for, and is required by, threshold corrections at the grand uni cation scale.

Fig. 5. The discrepancy " with M =0:118 vs. jj in a the sup ergravity

g s Z

mo del, and the messenger mo del with b n = 1 and c n = n =1.

5 5 10

5 GUT Threshold Corrections

At the GUT scale, there are incomplete SU5 multiplets most notably the color

triplet Higgs b osons and there can b e split multiplets as well. These elds give

rise to threshold corrections which are exp ected to b e of order a couple of p ercent

or larger if is larger.

GU T

If we take all three gauge couplings as input at the weak scale and run them

up to the grand uni cation scale M which is de ned to b e the scale where

GU T

g and g meet, we nd a discrepancy b etween the value of g and g = g . We

1 2 3 1 2

de ne the discrepancy " as

g M =g M 1 + " : 20

3 GU T 2 GU T g

If we x M =0:118, the discrepancy is negative. We show a scatter plot

s Z

of the discrepancy in Fig. 5 for three di erent mo dels: the sup ergravity mo del,

the messenger mo del with n = 1, and the messenger mo del with n = n =1

5 5 10

Ref. . For the sup ergravity mo del and the messenger mo del with n = 1,

" varies from ab out 3to1 as jj varies from 100 to 1000 GeV. For the

messenger mo del with n = n =1, " is larger b ecause is larger.

5 10 g GU T

In a grand uni ed theory, there will b e a discrepancy due to the grand uni-

cation threshold corrections. . In any particular mo del of grand uni cation, we

can calculate " as a function of the GUT mo del parameter space. By varying the

parameters over their allowed range, we can see whether the mo del is consistent

There are p ossibly additional corrections due to higher dimension op erators suppressed by

M =M See Ref. for a discussion.

GU T P lanck

with the "measured" values of " shown in Fig. 5. Here we givetwo examples, the

14 15

minimal SU5 mo del and the missing doublet SU5 mo del.

10,16

In the minimal SU5 mo del, we nd the correction

3 M

GU T H

" = ; 21 ln

10 M

GU T

where M is the triplet Higgs b oson mass. It is constrained to b e larger than

16 17

ab out 10 GeV by the lower limits on the nucleon lifetimes. Both M and the

GU T

b ound on M are functions of the sup ersymmetric parameter space. The b ound

on M is typically such that M >M , so that in most of the minimal SU5

H H GU T

3 3

mo del parameter space, " is p ositive. From Fig. 5, we know that in order to

b e compatible with gauge coupling uni cation, " must b e negative. Hence, the

minimal SU5 mo del is not compatible with coupling constant uni cation.

5 of Higgs elds. The doublet parts The minimal SU5 mo del contains a 5+

are the MSSM Higgs elds with order M masses. The triplet parts mediate

nucleon decay via dimension ve op erators. In order to b e compatible with the

lower b ound on the nucleon lifetimes, the triplet Higgs particles must have GUT

scale masses. In general, it is problematic to nd a GUT mo del which naturally

yields light Higgs doublets and heavy triplets. In the minimal SU5 mo del, this

doublet-triplet splitting is imp osed by ne tuning sup erp otential parameters. The

missing doublet mo del elegantly solves the doublet-triplet splitting problem bya

50 representations are judicious choice of Higgs representations. The 75, 50, and

employed, and when the 75 gets a vev, the sup erp otential term 75 50 5 generates

a mass for the triplet, but not for the doublet.

10,19

In the missing doublet mo del, we nd

ef f

GU T

ln " = 9:72 : 22

10 M

GU T

ef f

Wehave de ned an e ective triplet Higgs mass M whichenters into the nu-

cleon decay amplitude in the same way as in the minimal SU5 mo del, so the

same b ounds apply. However, the splitting in the 75-dimensional representation

gives rise to a negative correction, such that in the missing doublet mo del, the

discrepancy " is negative, just as it must b e in order to b e consistent with the

measured values of g M ; g M , and g M . In fact, in each of the three

1 Z 2 Z 3 Z

mo dels shown in Fig. 5, the allowed range of " in the missing doublet mo del just

ab out overlaps the required values. Hence, the missing doublet mo del is consistent with gauge coupling uni cation.

6 Conclusion

In the end, what we really do when weinvestigate gauge coupling uni cation is

constrain the physics at the grand-uni cation scale. The weak-scale measurements

of the gauge couplings imply that " is negative. We can calculate " in various

g g

grand uni ed mo dels to see whether the grand uni ed mo del parameter space

can accommo date the required value. Here we showed that this is not p ossible

in the minimal SU5 mo del, but that it is p ossible in the missing doublet SU5

mo del. In other words, the missing doublet mo del requires that " is negative,

which corresp onds with the measured values of the gauge couplings. Similarly,

there are SO10 mo dels which can accommo date the "measured" " , and others

which cannot. Gauge coupling uni cation remains an e ective constraint on mo del

building.

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2 2

M +4M . In the mo del M 500 GeV, M 1000 GeV, and jA j3

1=2 0 0

1=2

, and 0:03 3. gauge-mediated mo dels, 300 TeV, 1:03 M= 10

We imp ose that the rst or second generation squark masses b e greater than

220 GeV, m > 65 GeV, m > 170 GeV, m > 45 GeV, and m > 60 GeV.

g~ h

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