The Superpartner Spectrum of Gaugino Mediation

SLAC-PUB-8442 April 2000 UCSD/PTH-00-10

Martin Schmaltz x and Witold Skiba y

xStanford Linear Accelerator Center Stanford University, Stanford, CA 94309

yDepartment of Physics, University of California at San Diego, La Jolla, CA 92093

Submitted to Physical Review D

Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 Work supported by Department of Energy contract DEÐAC03Ð76SF00515.

SLAC-PUB-8442

UCSD/PTH-00-10

hep-ph/yymmnnn

The Sup erpartner Sp ectrum of Gaugino Mediation

x y

Martin Schmaltz and Witold Skiba

SLAC, Stanford University, Stanford, CA 94309

[email protected]

Department of Physics, University of California at San Diego, La Jol la, CA 92093

[email protected]

Abstract

We compute the sup erpartner masses in a class of mo dels with gaugino

mediation or no-scale b oundary conditions at a scale b etween the GUT and

Planck scales. These mo dels are comp elling b ecause they are simple, solve the

sup ersymmetric avor and CP problems, satisfy all constraints from colliders

and cosmology, and predict the sup erpartner masses in terms of very few

parameters. Our analysis includes the renormalization group evolution of the

soft-breaking terms ab ove the GUT scale. We show that the running ab ove the

GUT scale is largely mo del indep endent and nd that a phenomenologically

viable sp ectrum is obtained. 1

I. INTRODUCTION

The soft-breaking terms in the Minimal Sup ersymmetric Standard Mo del MSSM need

to have a very sp ecial form for the mo del to be viable. Generic mass matrices for the

squarks and sleptons lead to unacceptably rapid avor-changing and lepton-numb er-violating

pro cesses. Similarly, the tri-linear soft-breaking breaking A-terms also require ne tuning

for the MSSM to agree with exp eriment. It is imp ortant to explore mo dels which naturally

solve the avor ne-tuning problem.

The rates for avor-changing pro cesses induced by the soft masses are prop ortional to

the ratios of the o -diagonal mass squares to the diagonal ones. In order to make such ratios

small one needs to minimize the o -diagonal terms or increase the avor-preserving masses.

Examples of natural scenarios are gauge mediation [1,2] and anomaly mediation [3], which

pro duce diagonal mass matrices due to the universality of the gauge coupling, and e ective

sup ersymmetry [4], which p ostulates large soft masses for the rst two generations, for which

the exp erimental constraints are most stringent. The p opular minimal sup ergravity [5]

mo del do es not contain a solution to the avor problem, instead one simply assumes that

the higher-dimensional op erators which pro duce the scalar masses are avor preserving.

It has b een noticed that in \gaugino-dominated" mo dels the avor problem is less se-

vere [6]. If at a high scale gaugino masses are larger than other soft parameters at that scale,

then at low energies the soft masses consist of large avor-conserving masses with smaller

avor-violating comp onents. This is b ecause the renormalization group running induces

p ositive! universal soft scalar masses prop ortional to the gaugino masses. The generated

scalar masses are avor universal b ecause couplings to gauginos are generation-indep endent.

The most app ealing gaugino-dominated scenarios have no soft masses for squarks and slep-

tons and no A-terms at a high scale M . Since at the high scale there are no masses

and no A-terms, the only sources of avor violation are the Yukawa matrices, thus the

sup ersymmetric avor problem is solved by a \sup er-GIM" mechanism [7].

Gaugino domination also alleviates the sup ersymmetric CP problem b ecause the only

sources for new phases are the gaugino masses, , and B . Two phases can b e rotated away

by eld re-de nitions, leaving only one p ossible new phase. For the sp ecial case B =0 the

sup ersymmetric CP problem is solved automatically.

In fact, the sp ecial b oundary conditions with vanishing sup erpartner masses and A-

terms are theoretically well motivated. They arise, for example, in no-scale sup ergravity [8]

or in the recently constructed higher-dimensional \gaugino mediation" mo dels [9{11]. In

gaugino mediation the MSSM matter elds are con ned to a brane in higher dimensions.

Sup ersymmetry is assumed to break on a distant parallel brane. Extra dimensional lo cality

forbids direct couplings between the two branes and thereby suppresses all soft masses

which involve MSSM matter elds squark and slepton masses, A-terms. Gauge elds and

gauginos propagate in the bulk and couple directly to sup ersymmetry breaking, allowing for

the generation of gaugino masses. 2

Extra-dimensional lo cality only forbids scalar masses at energies large compared to the

compacti cation scale M . At long distances the theory is four-dimensional and masses are

generated from renormalization as usual. The compacti cation scale in gaugino mediation

corresp onds to a free parameter. Gauge coupling uni cation motivates us to cho ose M >

M , and an upp er limit on M is given by the length scale at which Nature b ecomes

GU T BC

non-lo cal, presumably the string scale or Planck scale. In no-scale sup ergravity the scale

M at which soft scalar masses vanish is related to the string scale. In the following we will

treat M as a free parameter sub ject to the constraint M M M , and refer

BC GU T BC P l anck

to the b oundary condition of vanishing scalar masses and A terms as gaugino mediation.

Clearly this scenario is very app ealing and it is crucial to ask if it is phenomenologically

viable. For the particular choice of M = M one nds that the stau is the lightest

BC GU T

sup erpartner LSP which is problematic b ecause the calculated relic abundance of stable

staus exceeds exp erimental limits by many orders of magnitude. This observation is often

conceived as a failure of no-scale mo dels and has motivated construction of mo dels with

new elds at intermediate scales to mo dify the renormalization group equations. In this

pap er we rep eat the analysis for general M M and nd the good news that the

BC GU T

problematic stau-LSP is very sp ecial to M M . For compacti cation scales slightly

BC GU T

higher than M the stau mass gets a large new contribution from running in the uni ed

GU T

theory ab ove M which lifts its mass ab ove the mass of the Bino. This results in a very

GU T

satisfying cosmological picture with a Bino-LSP.

Usually, the renormalization ab ove the GUT scale is ignored see, however, Refs. [12{15].

There are two seemingly go o d reasons for this negligence, we nd that b oth are invalid. The

rst reason given is that logM =M is negligibly small compared to logM =M .

BC GU T GU T w eak

However, we nd that the smallness of the logarithm is comp ensated for bymuch larger group

theory factors which arise in GUT theories. In particular, the right-handed sleptons receive

only very small contributions from running b elow the GUT scale b ecause they carry only

hyp ercharge, but ab ove M they are uni ed into a much larger representation with large

GU T

corresp onding group theory factors. The second reason is that the renormalization group

equations ab ove M are necessarily mo del-dep endent. Obviously, GUTs have more elds

GU T

than just the MSSM multiplets and the adjoint eld needed to break the GUT symmetry

to the Standard Mo del gauge group. For example, there must b e additional multiplets that

guarantee the splitting of the doublet and triplet Higgs elds. All such new GUT-scale elds

app ear with mo del-dep endent SUSY couplings. However, this do es not necessarily imply

that the running of all soft parameters ab ove the GUT scale is mo del dep endent. As we

will demonstrate in gaugino mediation most of the soft masses decouple from the unknown

physics and many predictions can be made in a mo del-indep endent fashion.

In the next two sections we describ e the renormalization group analysis ab ove Section I I

and b elow Section III M in detail. We nd that the number of mo del-indep endent

GU T

predictions that can b e obtained is related to the form of the b oundary conditions at M .

General gaugino mediation b oundary conditions allow arbitrary soft Higgs masses [10,16], 3

a B-term and gaugino masses at the scale M . We show that in this case the sp ectrum

of the rst two generations and the ratios of gaugino masses can be predicted. When

only M and B are non-zero at M one can also predict third generation scalar masses.

1=2

Finally, when B is also set to zero one obtains the b oundary conditions of Minimal Gaugino

Mediation [11]. In this case the entire sup erpartner sp ectrum can be predicted in terms of

just two parameters: M and M . We conclude in Section IV.

1=2

II. RENORMALIZATION ABOVE THE GUT SCALE

In this section we discuss the renormalization of soft mass parameters ab ove the grand

uni cation scale, which we take to be M =2 10 GeV. We rst brie y summarize the

GU T

situation in a general SUSY-GUT with soft masses and then turn to gaugino dominated

scenarios such as gaugino mediation or no-scale sup ergravity. We assume that there is a

direct coupling of sup ersymmetry breaking to the gauginos. Then the one-lo op \anomaly

mediation" [3] contributions to sup erpartner masses are always negligible.

A. General case

A realistic GUT theory requires a numb er of new elds ab ove the GUT scale for breaking

the GUT symmetry, splitting the Higgs doublets and triplets, and p ossibly also for generating

avor. There exists a large number of di erent prop osals for addressing all these problems,

but unfortunately present day exp eriments do not allow us to single out a unique \GUT

Standard Mo del". Not knowing the exact sp ectrum and couplings ab ove the GUT scale

makes it imp ossible to p erform a reliable renormalization group calculation of all sup erpart-

ner masses ab ove the GUT scale. A conservative approach would then be to parameterize

our ignorance by assuming general non-universal but GUT-symmetric sup erpartner masses

at M . To simplify and to avoid con icts with exp erimental b ounds on avor viola-

GU T

tion one often assumes that the soft parameters are approximately avor symmetric. This

approximation can be poor for third generation scalar masses which can be signi cantly

mo di ed b ecause of the large Yukawa couplings. However, the rst and second generation

Yukawa couplings are presumably small also ab ove M and can therefore b e neglected in

GU T

the running. Then the one-lo op renormalization of the soft scalar masses for the rst and

second generation dep ends only on the uni ed gauge coupling and is avor-universal. It is

therefore p ossible to compute the running of these soft masses and one can make predictions

if they are known at some high scale. The results of this running are well-known [17].

B. Gaugino domination

In gaugino-dominated scenarios the situation is di erent. In gaugino domination one

assumes that the scalar masses as well as the tri-linear soft terms vanish at some high scale 4

M , and their low-energy values are generated from the renormalization group. Thus at

the scale M M the non-vanishing mass parameters are the universal gaugino mass

BC GU T

M and the sup ersymmetry preserving and violating Higgs mass parameters and B . In

1=2

addition, one could also have soft masses for the Higgs elds, but for the time b eing let us

assume that m and m are zero at M .

H H BC

As we will see b elow, do es not enter any renormalization group equation for the soft

terms neither ab ove nor b elow M . Since we do not have aphysical principle which tells

GU T

us the value of at the high scale, this means that there is no need to run the value of .

In the phenomenological analysis we simply work with its low-energy value. B also do es not

enter any renormalization group equations. However, Minimal Gaugino Mediation predicts

B = 0 at the high scale, it is therefore imp ortant to predict the low-energy value of B from

the renormalization group running b etween M to M .

BC w eak

The evolution of the uni ed gaugino mass b etween M and M is easily determined

BC GU T

b ecause at one lo op M simply tracks the evolution of the uni ed gauge coupling

1=2

1 d d

1=2

= 2b =0: 2.1

GU T

2 2

dt g dt g

Here we de ned t = 1=16 logM=M . The renormalization group equations for all

GU T

other soft masses ab ove M are

GU T

d 3 d 3 d 144

2 2 2 2 2

m = m = m = g M ; 2.2

10 5 5 1=2

dt 2 dt 2 dt 5

d 8 d d 192

A = A =2 B = g M ; 2.3

u d

1=2

dt 7 dt dt 5

for the case of SU 5 and

5 d d

2 2 2 2

m = m = 45g M ; 2.4

16 10 1=2

dt 4 dt

d 7 d

A = B =63g M ; 2.5

u;d 1=2

dt 4 dt

for SO10. On the right-hand side of these equations we assumed that all soft masses except

M are negligible. This is a very go o d approximation at energies near M where they all

1=2 BC

vanish but b ecomes worse if signi cant scalar masses are generated from the renormalization

group running.

Note that all dep endence on unknown couplings ab ove the GUT scale has dropp ed out

of Eqs. 2.3 and 2.5. The remaining mo del dep endence lies in the choice of grand uni ed

group and in the evolution of the GUT gauge coupling sp eci cally b .

GU T

We de ne the soft mass parameters A and B such that they multiply the Yukawa couplings and

in the Lagrangian, resp ectively. For example, L A Y 10 10 5 + B 5 5 .

t t H H H

sof t

U U D 5

The solutions to these renormalization group equations are most easily written by using

Eq. 2.1 to replace M ! M M = and de ning

GU T GU T

1=2 1=2

1 M

4 2

I = ; 2.6 g dt = log

GU T

M b

BC GU T GU T

log 1

GU T

2 M

GU T

b M

GU T GU T BC

1 log

4 M

6 3

GU T

I = g dt =4 log : 2.7

GU T

b M

GU T GU T BC

1 log

GU T

2 M

GU T

We nd for SU 5

3 3 144

2 2 2 2

m = m = m = M I ; 2.8

10 5 5 1=2

2 2 5

192 8

A =2B = M I ; 2.9 A =

d 1=2 4 u

7 5

and for SO10

2 2 2

m =45 M I ; 2.10 m =

10 1=2 16

A = B = 63 M I ; 2.11

u;d 1=2 4

where from now on M stands for the uni ed gaugino mass evaluated at M .

1=2 GU T

Now all soft sup ersymmetry breaking at the GUT scale is determined in terms of the ve

parameters M , , B , I and I . I and I are given in terms of the two parameters b

1=2 4 6 4 6 GU T

and logM =M . Note that I and I are small enough for our approximation ignoring

BC GU T 4 6

all soft masses except for the gaugino mass to b e valid unless the denominators in Eqs. 2.6

and 2.7 go to zero. But the same vanishing denominators also app ear in the evolution of

the gauge coupling. Therefore our approximation is good if and only if the theory stays

p erturbative up to the mass scale M . For example, for =10 p erturbativity allows

GU T

a b eta function co ecient b as large as 50. Note that this leaves sucient ro om for non-

GU T

minimal Higgs sectors ab ove M b ecause the minimal SU 5 theory only has b = 3,

GU T GU T

an extra generation adds 2 and extra adjoint sup er elds contribute 5. If the theory ab ove

the GUT scale has large new Yukawa couplings, for example a coupling of H and H to

u d

a GUT adjoint sup er eld, then terms prop ortional to scalar masses on the right hand side

can make contributions to scalar masses which b ecome imp ortant for large M M .

BC P l anck

This can b e seen from the numerical solutions to the renormalization group equations of the

minimal SU 5 mo del with gaugino mediation b oundary condition presented in Ref. [15].

This discussion changes if we allow non-zero sup ersymmetry violating masses m and

m at M . Not only is the running of these masses very sensitive to new physics ab ove

the GUT scale such as the doublet-triplet splitting mechanism or a non-minimal GUT

Higgs sector, but also the third generation scalar masses are now mo del dep endent b ecause

of contributions prop ortional to the large Yukawa couplings. Thus, in this case mo del-

indep endent predictions are only p ossible for the rst and second generation scalar masses. 6

III. THE SPECTRUM

In this section we discuss the sup erpartner masses which result from running from the

compacti cation scale all the way down to the weak scale. Throughout the discussion we

assume that the gravitino mass is larger than the gaugino masses. This assumption is

imp ortant for the phenomenology of the mo del. Parametrically, in gaugino mediation one

Vm where V > 8 is the volume of the extra dimensions in fundamental nds m =

1=2 3=2

Planck units [9,10]. We pro ceed by presenting three qualitatively di erent scenarios. All

three scenarios have vanishing squark and slepton masses and A-terms at the high scale

but they di er in the assumptions made ab out the Higgs sector. In scenario A we allow

2 2

general soft Higgs mass parameters m , m , B , and . In B we sp ecialize to mo dels with

H H

2 2

vanishing non-holomorphic masses m = m = 0. And in C we also assume B = 0, so

H H

that the only remaining mass parameters of the mo del are M ;;M and M . In the

1=2 BC GU T

last section we also give three example sup erpartner sp ectra corresp onding to representative

sets of mo del input parameters which satisfy the M~gM b oundary conditions.

We solve the one-lo op renormalization group equations of the MSSM [18] b elow the GUT

scale numerically. We include the e ects of third generation Yukawa couplings and neglect

the smaller Yukawa couplings. The GUT scale b oundary conditions for all couplings follow

from the analysis in Section II. Note that there are no threshold contributions to the su-

DR renormalization scheme from integrating out p ersymmetry breaking parameters in the

the heavy GUT gauge b osons and gauginos. Explicitly, this follows b ecause diagrams renor-

malizing the scalar masses with heavy GUT gauginos in a lo op have vanishing nite pieces

in DR. At the weak scale we use a one-lo op improvement for the Higgs p otential [19,20]

which captures the e ect of top lo ops b elow the stop mass threshold. The top lo ops mo dify

y 2

the co ecient of the H H quartic term and represent the dominant correction to the

mass of the lightest Higgs particle. The accuracy of this approximation is to b etter than 10

GeV [21], when the running top quark mass m m is used for the calculation.

top top

A. General Higgs mass parameters

2 2

When we allow Higgs masses m and m at the scale M then only the soft masses of

H H

the rst two generations and gaugino masses can b e predicted without knowledge of details

2 2

of the GUT physics. It is convenient to use the average soft Higgs mass m + m =2 and

H H

2 2

the di erence m m as input parameters. The average Higgs mass do es not contribute

H H

directly to rst and second generation scalar masses. Indirectly,itdoescontribute to scalar

masses through weak scale D-terms, but for large enough tan these D-terms can b e written

universally in terms of the W and Z masses. The di erence do es contribute b ecause it

generates a D-term for hyp ercharge. This D-term is prop ortional to the renormalization

group invariant quantity S de ned as

2 2 2 2 2 2 2 2 2

: 3.1 S = m m +Trm 2m + m + m m = m m

H H Q U E D L H H

u u

d d

GU T 7

where the second equality is valid only at the GUT scale b ecause the squark and slepton

masses in the trace are GUT symmetric and therefore drop out of the equation. The D-term

mass shift for each scalar at the weak scale is simply

2 2

y S g dt ':078 y S : 3.2 m =

i i

1 i

w eak

2 2

We rst sp ecialize to the case with no D-term for hyp ercharge, i.e. m = m at

H H

M . Figure 1 illustrates the evolution of soft masses for the rst two generations and

GU T

the gauginos from M > M to the weak scale. At M the soft masses vanish and

BC GU T BC

evolve according to SU 5 RGEs down to M . For the purp ose of this plot we assumed

GU T

SU 5 uni cation therefore the 5 and 10 evolve at di erent rates. The gaugino masses are

uni ed between M and M . Below M the RGEs resp ect only the symmetries of

BC GU T GU T

the Standard Mo del. The evolution dep ends on the gauge charges of the elds. Gaugino

masses and scalar masses are prop ortional to the squares of the gauge couplings. As a result,

colored elds are always heaviest and have masses ab out four times larger than elds with

only hyp ercharge.

600

500

400

300 [GeV]

200

100

4 6 8 10 12 14 16 18 µ

log 10 ( /GeV)

FIG. 1. Evolution of the soft masses of the rst two generations solid and the gauginos

dashed as a function of renormalization scale . The input parameters are M = 250 GeV,

1=2

M = 2 10 GeV, and vanishing hyp ercharge D-term S = 0. The scalar elds are, from the

lightest to heaviest at the weak scale, the right-handed selectron, the left-handed sleptons, the

right-handed down and up squarks, and the left-handed squarks. The gaugino masses start at a

nonzero value at M . At the weak scale the gluino is heaviest and the Bino lightest.

The e ects of the running ab ove the GUT scale are depicted in Figure 2. Scalar masses

receive additive GUT-symmetric contributions from the running ab ove M . This e ect

GU T

is most imp ortant for scalars which do not receive large masses from running b elow the

GUT scale. The mass shifts are prop ortional to the Casimirs of the corresp onding uni ed

representations, thus they are larger in SO10 compared to SU 5. 8 250

225

200

175 [GeV] 150

125

100 1 2 3 4

log (MBC /M GUT )

FIG. 2. Weak scale sup erpartner masses as a function of log M =M . The common

BC GU T

gaugino mass is 250 GeV at M and we take S =0. The lightest particle is the neutralino, its

GU T

mass is indep endentof M . The sleptons are, from the lightest to the heaviest, the right-handed

selectron, the left-handed sneutrino, and the left-handed selectron. The solid lines corresp ond to

the running in SU 5, dashed lines corresp ond to SO10.

A non-vanishing D-term intro duces the additional input parameter S . The S -dep endence

of the rst and second generation scalar masses is easily accounted for by using equa-

tion 3.2. The e ects of the D-term are largest for the lightest sup erpartners. In Figure 3

we show the slepton masses for the case of SU 5 GUT group and three di erent values

of S . The mass shifts from the hyp ercharge D-term are in opp osite directions for left- and

right-handed sleptons.

Note that the hyp ercharge D-terms also contribute to stau masses and may b e resp onsible

for lifting the right-handed stau mass ab ove the Bino mass in mo dels where M = M .

BC GU T

This has b een used in Refs. [10,16].

2 2

B. m = m =0

H H

When the Higgs masses are zero at M , the Yukawa couplings do not signi cantly con-

tribute to the running ab ove the GUT scale. Therefore the running for all three generations

can be computed mo del indep endently. For xed M and M , B and tan are related

1=2

at the minimum of the Higgs p otential. Thus, we can express B in terms of tan or vice

versa.

The relation b etween B at the high scale and tan is depicted in Figure 4 for di erent

values of M . It is clear that a value of B can b e picked for anyvalues of tan and M .

BC BC

It is therefore more convenient to treat tan as an input parameter, as is usually done in

analyzing sup ersymmetric theories.

Since the ratio of the b ottom and top Yukawa couplings dep ends on tan the masses

of the third generation particles vary with tan . In particular, the mixing between left- 9 240

220

200

180

160 [GeV] 140

120

100 1 2 3 4

log (MBC /M GUT )

FIG. 3. Weak scale sup erpartner masses as a function of log M =M for three di erent

BC GU T

2 2

values of S = M ; 0; +M . We take M = 250 GeV and GUT group SU 5. The lightest

1=2

1=2 1=2

particle is the neutralino, its mass is indep endent of M . The other solid lines corresp ond to,

from lightest to heaviest, the right-handed selectron, the left-handed sneutrino, and the left-handed

selectron for S = 0, as in Figure 2. Dashed lines corresp ond to S = M and dotted lines to

1=2

S =+M . Note that left- and right-handed slepton masses are shifted in opp osite directions.

1=2

200

100 tan β [GeV] 10 20 30 40 50

-100

FIG. 4. The dep endence of B , evaluated at M , on tan . We take M = 250 GeV, the

1=2

di erent curves corresp ond to log M =M =0;1;2;3;4 from left to right, resp ectively.

BC GU T

and right-handed sleptons increases with tan . As a result one of the mass eigenstates

b ecomes lighter with increasing tan . Figure 5 illustrates this strong dep endence in case of

the right-handed stau.

2 2

C. m = m =0, B =0

H H

In minimal gaugino mediation all soft terms except for M and vanish at M .

1=2 BC

The values of tan which corresp ond to B = 0 can be determined easily from Figure 4 10 150 ~e 140 R

130

120 ho

[GeV] 110

100 χο

90 ~τ R

5 10 15 20 25 30

tan β

FIG. 5. Masses of the right-handed selectron, stau, the lightest Higgs and neutralino as a

function of tan b eta. The common gaugino mass is 250 GeV at M and log M =M =2.

GU T BC GU T

as a function of M . The two remaining parameters are M and the scale M . With

BC 1=2 BC

only two free parameters the theory is highly predictive. To a good approximation M

1=2

sets the linear scale of all sup erpartner masses. The second parameter, logM =M ,

BC GU T

increases the slepton masses relative to gaugino masses. A more detailed discussion of M~gM

is contained in Refs. [11,15].

For demonstration, we present the sp ectra for three sample p oints of the parameter space

in Table I. The rst p oint with M = 200 GeV \lightM~gM" scenario is in the lower range

1=2

of exp erimentally allowed values of M . This p oint will be prob ed in the near future by

1=2

the ongoing Higgs search at LEP I I. The lighter chargino and the second lightest neutralino

can be observed at the Tevatron in the pp ! ! 3l channel. While the chargino and

neutralino in the \light" scenario might evade Run II they are certainly within the reach

of Run III [22]. Our second and third p oints, with M = 300; 500 GeV { \intermediate

1=2

M~gM" and \heavy M~gM" scenarios { cannot be tested at LEP II, but the lightest Higgs is

within the reach of Run I I at the Tevatron. The sup erpartners corresp onding to the second

and third study p oints are to o heavy to b e seen at LEP or the Tevatron, but they are easily

within the reach of b oth LHC and NLC.

One can set an upp er b ound on M 600 GeV by requiring that the relic abundance

1=2

of M~gM Bino-LSPs do es not contribute more than 0.1 to 0.3 to critical density [11]. We

nd that in our \light" scenario Binos are not abundant enough to account for all cold dark

matter, whereas the \intermediate" and \heavy" scenarios are in the preferred region.

Note that the lightest Higgs mass do es not scale linearly with M , b ecause at tree level the

1=2

Higgs mass is b ounded by M . Radiative corrections to the Higgs p otential from top and stop

lo ops intro duce a logarithmic dep endence on M .

1=2 11

I. \lightM~gM" I I. \intermediate M~gM" I I I. \heavy M~gM"

Field mass Field mass Field mass Field mass Field mass Field mass

e e e

g 520 g 780 g 1300

e e e e e e

147 317 235 457 406 750

1 2 1 2 1 2

0 0 0 0 0 0

e e e e e e

82 148 125 236 211 406

1 2 1 2 1 2

0 0 0 0 0 0

e e e e e e

441 456 739 750 294 315

3 4 3 4 3 4

e e e e e e

u 478 u 463 u 720 u 696 u 1188 u 1145

L R L R L R

e e e e e e

d 725 d 689 d 1191 d 1138 d 485 d 460

L R L R L R

e e e e e e

t 351 t 501 t 535 t 703 t 906 t 1117

1 2 1 2 1 2

e e e e e e

b 437 b 497 b 664 b 735 b 1121 b 1195

1 2 1 2 1 2

e e e e e e

e 167 e 126 e 245 e 183 e 373 e 235

L R L R L R

e e e e e e

147 147 231 231 364 364

e e e

e e e e e e

104 178 162 253 224 375

1 2 1 2 1 2

0 0 0 0 0 0

h 108 H 294 h 115 H 453 h 122 H 793

0 0 0

A 294 H 312 A 453 H 464 A 793 H 800

TABLE I. Masses of superpartners, in GeV, for M~gM study points I, II and III. They cor-

respond to parameter values \light" M = 200 GeV, M =M = 10, tan = 17, \inter-

BC GU T

1=2

mediate" M = 300 GeV, M =M = 10, tan = 17, and \heavy" M = 500 GeV,

BC GU T

1=2 1=2

M =M =2, tan =12, respectively.

BC GU T

IV. CONCLUSIONS

We conclude that the MSSM with gaugino mediated sup ersymmetry breaking is not only

phenomenologically viable, but it also has anumber of very attractive features:

The mo del solves the sup ersymmetric avor problem b ecause the squark and slepton

masses which are generated from the renormalization group evolution are suciently

degenerate and aligned [23].

It is theoretically well motivated. The vanishing of the scalar masses and the A terms

at high scales is a natural prediction of mo dels with extra dimensions where MSSM

gauge elds are bulk elds, whereas the MSSM matter elds and the sup ersymmetry

breaking mechanism are lo calized on separate branes.

The Bino-LSP of these mo dels makes a good cold dark matter candidate. Cosmic

abundances in the range h = :1 :3 are obtained for right-handed selectron masses

in the range 150 250 GeV as shown in [11,24].

The mo del is very predictive b ecause sup erpartner masses dep end only on a small

number of input parameters. For example, in M~gM all sup erpartner masses can be 12

computed in terms of two parameters M and logM =M . In the more general

BC GU T

1=2

case with non-vanishing soft Higgs masses at M all gaugino masses and the rst and

second generation scalar masses are predicted in terms of the same two parameters

and p ossibly a hyp ercharge D-term S .

In our analysis of the mo del the renormalization group running ab ove the GUT scale was

essential for determining the masses of the lightest scalar sup erpartners. Its most imp ortant

e ect is that it raises the stau mass ab ove the mass of the Bino. Precise measurements

of the sup erpartner masses would allow a determination of the small contributions from

running ab ove the GUT scale [25]. Consistency with the gaugino mediation predictions

would constitute a decisive test of the scenario and allow an indirect measurement of the

GUT gauge group.

The framework is predictive b ecause at the one-lo op level ab ove-the-GUT-scale mo del

dep endence decouples from the soft sup ersymmetry breaking terms. This is a consequence

of vanishing scalar masses at M and would not be true for more general SUSY GUTs.

2 2

If the soft Higgs mass parameters m and m are non-vanishing at the GUT scale then

H H

the third generation scalar masses and Higgs masses b ecome mo del-dep endent, but rst and

second generation scalar masses as well as gaugino masses can still be predicted. In this

2 2

case the hyp ercharge D-term prop ortional to m m also plays an imp ortant role in

H H

determining the lightest sup erpartner masses.

We have left studies of the collider phenomenology of these mo dels for future work. In

particular, it would be interesting to determine the most promising signatures which allow

tests of this scenario and the implications for future colliders. As part of this analysis a more

accurate treatment of the renormalization group two-lo ops and weak scale threshold e ects

would allow sharp ened predictions for sup erpartner masses. This more accurate analysis is

necessary for comparison of our Higgs mass prediction with LEP I I b ounds. We also exp ect

interesting predictions for and constraints from avor violating transitions suchasb!s

and ! e .

ACKNOWLEDGMENTS

We thank Howard Baer, Markus Luty, Elazzar Kaplan, Konstantin Matchev, Michael

Peskin, Raman Sundrum, and Jim Wells for useful discussions. MS is supp orted by the

DOE under contract DE-AC03-76SF00515. WS is supp orted by the DOE under contract

DOE-FG03-97ER40506. 13

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