new supersymmetric contributions to

New supersymmetric contributions to t˜cV

Jorge L. Lopez*Bonner Nuclear Lab, Department of Physics, Rice University, 6100 Main Street, Houston, Texas 77005

D. V. Nanopoulos†

Center for Theoretical Physics, Department of Physics, Texas A&M University, College Station, Texas 77843-4242and Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Mitchell Campus, The Woodlands, Texas 77381

Raghavan Rangarajan‡

Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Mitchell Campus, The Woodlands, Texas 77381~Received 18 February 1997; revised manuscript received 2 May 1997!

We calculate the electroweaklike one-loop supersymmetric contributions to the rare and flavor-violatingdecay of the top quark into a charm quark and a gauge boson:t→cV, with V5g,Z,g. We consider loops ofboth charginos and downlike squarks~where we identify and correct an error in the literature! and neutralinosand uplike squarks~which have not been calculated before!. We also account for left-right and generationalsquark mixing. Our numerical results indicate that supersymmetric contributions tot→cV can be up to 5orders of magnitude larger than their standard model counterparts. However, they still fall short of the sensi-tivity expected at the next-generation top-quark factories.@S0556-2821~97!01517-8#

PACS number~s!: 14.65.Ha, 12.15.Lk, 12.38.Bx, 12.60.Jv

I. INTRODUCTION

The discovery of the top quark by the Collider Detector atFermilab~CDF! and D0 Collaborations@1# at Fermilab andits subsequent mass determination (mt517566 GeV) haveinitiated a new era in particle spectroscopy. However, unlikethe lighter quarks, the top quark is not expected to form anybound states, and therefore its mass and decay branchingratios may be determined more precisely, both theoreticallyand experimentally. Upcoming~run II, Main Injector! andproposed~run III, TeV33! runs at the Tevatron will yieldlarge numbers of top quarks, as will be the case at the CERNLarge Hadron Collider~LHC!, turning these machines effec-tively into ‘‘top-quark factories.’’ Even though higher preci-sion in the determination ofmt is expected~it is alreadyknown to 3%!, more valuable information should come fromthe precise determination of its branching fractions into tree-level and rare~and perhaps even ‘‘forbidden’’! decay chan-nels.

The purpose of this paper is to study one class of suchrare decay modes:t→cV, with V5g,Z,g. The particularcase oft→cg has received some phenomenological attentionrecently as a means to probe the scale at which such new andunspecified interactions might turn on@2#. Our purpose hereis to consider an explicit realization of this coupling withinthe framework of low-energy supersymmetry. This is differ-ent in spirit from the line of work in Ref.@2#, as the effectivemass scale at which such vertices ‘‘turn on’’ is determinedhere by the interactions of presumably rather light sparticles.

Within supersymmetry, thet→cV vertex was first contem-plated in Ref.@3#, where the one-loop QCD-like~loops ofgluinos and squarks! and electroweaklike~loops of charginosand downlike squarks! contributions were calculated. TheQCD-like supersymmetric corrections were subsequently re-evaluated and generalized in Ref.@4#, which pointed out aninconsistency in the corresponding results of Ref.@3#. Herewe study the electroweaklike supersymmetric contributionsto t→cV. We reconsider the chargino–downlike-squarkloops and point out and correct an inconsistency, essentiallya lack of gauge invariance because of the apparent omissionof a term, in the corresponding results of Ref.@3#. We alsoconsider for the first time the neutralino–uplike-squarkloops, and include the effects of left-right and generationalsquark mixing.

Our numerical results indicate that for typical values ofthe parameters one gets a large enhancement over standardmodel predictions of top-quark decays to gauge bosons@5#.For the most optimistic values of the parameters the en-hancement can be as large as five orders of magnitude. How-ever, even for the most optimistic values of the parameters,such rare decay channels fall short of the expected sensitivityof the next-generation top-quark factories.

II. ANALYTICAL RESULTS

In this section we obtain the one-loop electroweaklike su-persymmetric effective top-quark–charm-quark–gauge-boson vertex by considering loops involving charginos andneutralinos, including the effects from left-right and genera-tional squark mixing. We then present the decay rates of thetop quark to the charm quark and a gauge boson.

The invariant amplitude for top-quark decay to a charmquark and a gauge boson can be written as

*Electronic address: [email protected]†Electronic address:[email protected]‡Electronic address: [email protected]

PHYSICAL REVIEW D 1 SEPTEMBER 1997VOLUME 56, NUMBER 5

560556-2821/97/56~5!/3100~7!/$10.00 3100 © 1997 The American Physical Society

M5M01dM , ~1!

where M0 is the tree-level amplitude anddM is the first-order supersymmetric correction. As there are no explicitflavor-violating tcV couplings in the LagrangianM050,whereasdM is given by

idM5 u~p2! Vmu~p1! em~k,l!, ~2!

where p1, p2, and k are the momenta of the incoming topquark, outgoing charm quark, and outgoing gauge boson,respectively, andem(k,l) is the polarization vector for theoutgoing gauge boson. The verticesVm may be written as

Vm~ tcZ!52 igm~PLFZ11PRFZ18 !1knsmn~PRFZ2

1PLFZ28 !, ~3!

Vm~ tcg!52 igm~PLFg11PRFg18 !1knsmn~PRFg2

1PLFg28 !, ~4!

Vm~ tcg!52 iTagm~PLFg11PRFg18 !1Taknsmn~PRFg2

1PLFg28 !, ~5!

where as usual we have definedPR,L51/2(16g5) andsmn5( i /2)@gm,gn#. Ta are the generators of SU(3)C . The

form factors F1,2 and F1,28 encode the loop functions anddepend on the various masses in the theory. The Feynmanrules used to obtain them are given in Refs.@6,7# and thecorresponding Feynman diagrams are shown in Fig. 1. Thevertices are derived assumingp12p22k50.

The form factors for the electroweaklike corrections dueto loops involving charginos and strange and bottom squarksare given by

Fi1c 5

1

4p2 (j 51

2

(r51

2

(l 51

2

(e51

2

(m51

2

$Acj ,r,lDc

j ,e,mEicr,e @mt

2~c121c232c112c21!22c241mc2~c232c12!#

1Bcj ,r,lCc

j ,e,mEicr,emcmt ~2c232c112c21!1Ac

j ,r,lCcj ,e,mEic

r,emtmxj6~c01c11!1Bc

j ,r,lDcj ,e,mEic

r,emcmxj6

3~c01c11!%~2p1 ,k,x j

6 , e m , r l !11

2p2 (j 51

2

(k51

2

(r51

2

(l 51

2

$Bcj ,r,lCc

k,r,lFicj ,kmcmt ~c211c2222c23!1Ac

j ,r,lDck,r,lEic8

j ,k

3@mc2c222mi

2~c121c23!22c24112 #1Bc

j ,r,lDck,r,lFic

j ,kmcmxk6~c122c11!1Ac

j ,r,lCck,r,lFic

j ,kmtmxj6~c122c11!

1Acj ,r,lDc

k,r,lFicj ,kmx

j6mx

k6c0%

~2p1 ,p2 ,xk6 , r l ,x j

6!1

1

2p2

1

mt22mc

2 (j 51

2

(r51

2

(l 51

2

$Acj ,r,lDc

j ,r,lHicmt2~2B1!

1Acj ,r,lCc

j ,r,lHicmtmxj6B01Bc

j ,r,lCcj ,r,lHicmcmt~2B1!1Bc

j ,r,lDcj ,r,lHicmcmx

j6B0%~2p1 ,x j

6 , r l !

11

2p2

1

mt22mc

2 (j 51

2

(r51

2

(l 51

2

$Acj ,r,lDc

j ,r,lHicmc2B11Ac

j ,r,lCcj ,r,lHicmtmx

j6~2B0!1Bc

j ,r,lCcj ,r,lHicmcmtB1

1Bcj ,r,lDc

j ,r,lHicmcmxj6~2B0!%~2p2 ,x j

6 , r l !, ~6!

Fi18c5Fi1

c ~A,B,C,D,E8,F,H→B,A,D,C,F,E8,G!, ~7!

FIG. 1. Feynman diagrams for one-loop electroweaklike super-symmetric contributions to thet-c-V (V5g,Z,g) vertex. Herex

represents the chargino or the neutralino andq represents thedown-type or up-type squarks, respectively. The subscripts are ex-plained in the text. The arrows on the squark lines indicate thedirection of flow of flavor; the arrows on the gauge bosons indicatethe direction of momentum flow. Diagram~b! is absent forV5g,and forV5g whenx5x0.

56 3101NEW SUPERSYMMETRIC CONTRIBUTIONS TOt→cV

Fi2c 5

1

4p2 (j 51

2

(r51

2

(l 51

2

(e51

2

(m51

2

$Acj ,r,lDc

j ,e,mEicr,emt~c121c232c112c21!1Ac

j ,r,lCcj ,e,mEic

r,emxj6~c01c11!

1Bcj ,r,lCc

j ,e,mEicr,emc~c232c12!%

~2p1 ,k,x j6 , e m , r l !1

1

2p2 (j 51

2

(k51

2

(r51

2

(l 51

2

$Acj ,r,lDc

k,r,lEic8j ,kmt~c111c212c122c23!

1Bcj ,r,lCc

k,r,lFicj ,kmc~c222c23!1Ac

j ,r,lCck,r,lEic8

j ,kmxk6~2c112c0!1Ac

j ,r,lCck,r,lFic

j ,kmxj6c12%

~2p1 ,p2 ,xk6 , r l ,x j

6!. ~8!

Fi28c5Fi2

c ~A,B,C,D,E8,F→B,A,D,C,F,E8!. ~9!

In these expressionsi 5Z,g,g, the sums overj ,k51,2 runover the two chargino mass eigenstates,r,e52,3 representstrange and bottom squarks~we ignore the mixing with thedown squark!, and l ,m51,2 represent a sum over squarkmass eigenstates which are obtained from theqL,R

gauge eigenstates viaqr15cosurqrL1sinurqrR andqr252sinurqrL1cosurqrR. The variousB andc functionsin the above expressions are the well-documented Passarino-Veltman functions @8# ~adapted to our metric wherepi

25mi2); the arguments of theB and c functions are indi-

cated by the superscripts on the braces in Eqs.~6!–~9!. Forexample, the arguments of thec functions that appear withinthe first set of braces of Eq. ~6! are(2p1 ,k,mx

k6,mqem

,mqr l) while the arguments of theB

functions that appear in the third set of braces of Eq.~6! are(2p1 ,mx

k6,mqr l

). ~Note that the Passarino-Veltman func-

tions depend only on the square of their arguments.! ThePassarino-Veltman functions contain infinities which canceleach other out, as they should since there is not-c-V vertexin the Lagrangian. The coefficient functions are given by

Acj ,r,l5

g

2F2U j 1~KG2†!2rH cosur ~ l 51!

2sin ur ~ l 52!J

11

A2mWcosbU j 2~KMdB2

†!2rH sin ur ~ l 51!

cosur ~ l 52!J G ,

~10!

Bcj ,r,l5

g

2F mc

A2mWsin bVj 2* ~KG2

†!2rH cosur ~ l 51!

2sin ur ~ l 52!J G ,

~11!

Ccj ,r,l5

g

2F mt

A2mWsin bVj 2~G2K†!r3H cosur ~ l 51!

2sin ur ~ l 52!J G ,

~12!

Dcj ,r,l5

g

2F2U j 1* ~G2K†!r3H cosur ~ l 51!

2sin ur ~ l 52!J

11

A2mWcosbU j 2* ~B2MdK†!r3H sin ur ~ l 51!

cosur ~ l 52!J G ,

~13!

EZcr,e52

g

cosuWF2

1

2 (p52

3

GQLrp GQL* ep1

1

3sin2uWdreG ,

~14!

Egcr,e5

e

3dre, ~15!

Egcr,e52gsd

re, ~16!

EZc8 j ,k5g

2cosuWF2U j 1* Uk12

1

2U j 2* Uk21sin2uWd jkG ,

~17!

Egc8 j ,k52e

2d jk, ~18!

Egc8 j ,k50, ~19!

FZcj ,k5

g

2cosuWF2Vj 1Vk1* 2

1

2Vj 2Vk2* 1sin2uWd jkG , ~20!

Fgcj ,k52

e

2d jk, ~21!

Fgcj ,k50, ~22!

GZc52g

2cosuWS 2

2

3sin2uWD , ~23!

Ggc52e

3, ~24!

Ggc52gs

2, ~25!

HZc52g

2cosuWS 1

22

2

3sin2uWD , ~26!

Hgc52e

3, ~27!

Hgc52gs

2. ~28!

3102 56LOPEZ, NANOPOULOS, AND RANGARAJAN

The chargino mixing matricesUi j and Vi j and the genera-tional mixing matricesK, G2 , andB2 which appear in theseexpressions are defined in Ref.@6#; Md is the diagonal matrix(md ,ms ,mb) andGQL is the squark mixing matrix defined inRef. @7#.

In deriving the above form factors, we have used the re-lations

(l

kmknem~k,l!en~k,l!50, ~29!

u~p2!p1mPR,Lu~p1!5 u~p2!@mtg

mPL,R

1 ip1nsmnPR,L#u~p1!, ~30!

u~p2!p2mPR,Lu~p1!5 u~p2!@mcg

mPR,L

2 ip2nsmnPR,L#u~p1!, ~31!

u~p2!~p11p2!mPR,Lu~p1!em~k,l!

5 u~p2!2p1mPR,Lu~p1!em~k,l!

5 u~p2!2p2mPR,Lu~p1!em~k,l!

5 u~p2!@mtgmPL,R1mcg

mPR,L

1 iknsmnPR,L#u~p1!em~k,l!. ~32!

@The first two equalities in Eq.~32! are only true when onetakes the absolute square of both sides of the equation anduses Eq.~29!.#

Our results above for the chargino-squark loops disagreewith those of Ref.@3# in the limit of mc50. We have anadditional term

1

2p2 (j 51

2

(r51

2

(l 51

2

(e51

2

(m51

2

3$Acj ,r,lDc

k,r,lFicj ,kmx

j6mx

k6c0%

~2p1 ,p2 ,xk6 , r l ,x j

6!

~33!

in Fi1c . This term is required by gauge invariance; i.e., it is

needed to ensure that the coefficient ofgm in Eqs.~4! and~5!vanish@9# for the massless gauge bosons.

The form factors for the electroweaklike corrections dueto loops involving neutralinos and top and charm squarks aregiven by Eqs.~6!–~9! with mx6 replaced bymx0 and withthe coefficientsAc–Fc replaced by

Anj ,r,l52

1

A2F2

3eNj 18 1

g

cosuWS 1

22

2

3sin2uWDNj 28 G

3~G1†!2rH cosur ~ l 51!

2sin ur ~ l 52!J 2

1

A2F g

2mWsin bNj 4G

3~MuB1†!2rH sin ur ~ l 51!

cosur ~ l 52!J , ~34!

Bnj ,r,l52

1

A2F g

2mWsin bNj 4* G~MuG1

†!2rH cosur ~ l 51!

2sin ur ~ l 52!J

11

A2F2

3eNj 18* 2

g

cosuWS 2

3sin2uWDNj 28* G

3~B1†!2rH sin ur ~ l 51!

cosur ~ l 52!J , ~35!

Cnj ,r,l52

1

A2F g

2mWsin bNj 4G~G1Mu!r3H cosur ~ l 51!

2sin ur ~ l 52!J

11

A2F2

3eNj 18 2

g

cosuWS 2

3sin2uWDNj 28 G

3~B1!r3H sin ur ~ l 51!

cosur ~ l 52!J , ~36!

Dnj ,r,l52

1

A2F2

3eNj 18* 1

g

cosuWS 1

22

2

3sin2uWDNj 28* G

3~G1!r3H cosur ~ l 51!

2sin ur ~ l 52!J 2

1

A2F g

2mWsin bNj 4* G

3~B1Mu!r3H sin ur ~ l 51!

cosur ~ l 52!J , ~37!

EZnr,e52

g

cosuWF1

2 (p52

3

GQLrp GQL* ep2

2

3sin2uWdreG ,

~38!

Egnr,e52

2

3edre, ~39!

Egnr,e52gsd

re , ~40!

EZn8 j ,k5g

2cosuWF1

2Nj 38* Nk38 2

1

2Nj 48* Nk48 G , ~41!

Egn8 j ,k50, ~42!

Egn8 j ,k50, ~43!

FZnj ,k5

g

2cosuWF2

1

2Nj 38 Nk38* 1

1

2Nj 48 Nk48* G , ~44!

Fgnj ,k50, ~45!

Fgnj ,k50. ~46!

The neutralino mixing matricesNi j andNi j8 and the genera-tional mixing matricesG1 and B1, are defined in Ref.@6#,andMu is the diagonal matrix (mu ,mc ,mt). In Eqs.~6!–~9!the sums overj ,k now run from 1 to 4 over the four neu-tralino mass eigenstates, andr,e,p52,3 represent charm andtop squarks~we ignore the mixing with the up squark!;l ,m51,2 represent a sum over squark mass eigenstates, asearlier. The coefficientsG andH are unaltered.


While we have included the charm-quark mass in theform factors above for completeness, we setmc50 hereafter.The supersymmetric electroweaklike contribution to the de-cay rates is then given in terms of the form factors obtainedabove, as

G~ t→cZ!51

32pmt3 ~mt

22mZ2!2F S 21

mt2

mZ2D ~FZ1

2 1FZ182!

26mt ~FZ1FZ21FZ18 FZ28 !

1~2mt21mZ

2!~FZ22 1FZ282!G , ~47!

G~ t→cg!5mt

32p@~2~Fg1

2 1Fg182!26mt ~Fg1Fg21Fg18 Fg28 !

12mt2~Fg2

2 1Fg282!], ~48!

G~ t→cg!5mt

24p@2~Fg1

2 1Fg182!26mt ~Fg1Fg21Fg18 Fg28 !

12mt2~Fg2

2 1Fg282!]. ~49!

In these expressions each form factor receives contributionsfrom both chargino-squark and neutralino-squark form fac-tors. It is not hard to verify that formc50, the charginocontributions toF1,28 vanish.@In Ref. @3# there is a factor ofpmissing in the expression forG(t→cZ) and a factor of 1/3missing inG(t→cg).#

III. NUMERICAL RESULTS

Before we attempt to evaluate the rather lengthy expres-sions given above, we would like to consider qualitativelythe possibility of dynamical enhancements of the loop am-plitudes. Experience with similar diagrams contributing tothe self-energy of the top quark in supersymmetric theories@10# indicates possible large corrections when the mass ofthe top quark equals the sum of the masses of the otherparticles leaving the vertex involving the top quark.1 In thepresent case we have vertices with top quarks and~i! gluinosand top squarks in the case of QCD-like contributions~cal-culated in Refs.@3,4#!, ~ii ! charginos and downlike squarksin the case of ‘‘charged’’ electroweaklike corrections; and~iii ! neutralinos and uplike squarks in the case of ‘‘neutral’’electroweaklike corrections. Given the presently knownlower bounds on the squark~excluding t ) and gluino masses~i.e.,mq ,mg.175 GeV,mq 'mg.230 GeV@11#!, this typeof enhancement might only be present in the third type ofcontribution whenmt'mx1mt 1

, which requires a light topsquark whose mass is constrained experimentally tomt 1

*60 GeV@12#. The ‘‘neutral’’ electroweaklike contribu-tions might also be enhanced by large Glashow-Iliopoulos-Maiani- ~GIM! violating top-squark–charm-squark mass

splittings. This latter enhancement is also present in the‘‘charged’’ electroweaklike contributions. However, the‘‘charged’’ contributions fall short of the ‘‘neutral’’ elec-troweaklike corrections for the gluon and photon cases. In-terestingly, the ‘‘charged’’ contribution for theZ is higherthan in the ‘‘neutral’’ case. We first address the neutral elec-troweaklike contributions and comment on the charged con-tributions afterwards.

The neutral electroweaklike contributions might be en-hanced as discussed above, but this is subject to other mixingfactors in theAn–Fn coupling functions in Eqs.~34!–~46!being unsuppressed. At the root of this question is whetherthe quark-squark-neutralino couplings might be flavor non-diagonal as a result of their evolution from the unificationscale down to the electroweak scale. This question might beexplored by considering the squared squark mass matrices atthe electroweak scale that are obtained by renormalizationgroup evolution of a universal scalar mass at the unificationscale@6,13#:

XiR2 5MW

2 m iR~0!I 1m iR

~1!XiXi† ~ i 51,2!, ~50!

X1L2 5MW

2 m1L~0!I 1m1L

~1!X1X1†1m1L

~2!X2X2† , ~51!

X2L2 5MW

2 m2L~0!I 1m2L

~1!X1X1†1m2L

~2!X2X2† , ~52!

where i 51 (i 52) corresponds to up-type~down-type! fla-vors, them (0,1,2) are renormalization-group-equation-~RGE-!dependent coefficients, andX1 (X2) are the up-type~down-type! Yukawa matrices. The matricesBi[U i* Ui

T , appearing

in the equations in Sec. II above, are obtained from theU i

matrices that diagonalizeXiR2 , and theUi matrices that diag-

onalize the right-handed quark mass matrices. Because of thesimple form for XiR

2 in Eq. ~50!, it can be shown that

U i5Ui and thereforeBi5I @6#. ~Note that the quark mixingmatricesUi and Vi mentioned in this section are differentfrom the chargino mixing matricesUi j andVi j mentioned inSec. II.!

The other relevant set of matrices areG i[ViVi† , obtained

from theVi matrices that diagonalizeXiL2 and theVi matrices

that diagonalize the left-handed quark mass matrices. In thecase ofl t@lb , which requires tanb;1, theX2X2

†}lb2 term

in Eqs. ~51! and ~52! is small compared to theX1X1†}l t

2

term, and therefore the former may be neglected. This im-plies that bothX1L

2 and X2L2 are diagonalized by the same

matrix V15V25V1, and thereforeG15V1V1†5I , whereas

G25V2V2†5V1V2

†5K reduces to the regular Cabibbo-Kobayashi-Maskawa~CKM! matrix @6#. As the quark-squark-neutralino couplings in flavor space are proportionalto G i , we see that forl t@lb there are no flavor off-diagonalcouplings in the up-quark sector, as required for an unsup-pressed contribution to the ‘‘neutral’’ electroweaklike contri-butions tot→cV.

One might consider instead a scenario wherel t;lb , aswould be consistent with tanb@1. In this case theX2X2

†

}lb2 term in Eqs.~51! and ~52! is no longer negligible and

G1ÞI is expected. The precise form ofG1 requires a com-

1The sign of these corrections depends on the observable beingcalculated: In the case ofB(t→cV) they are positive, whereas inone-loop supersymmetric corrections tos(pp→t t) @10# they werenegative.


plicated calculation, essentially solving the matrix renormal-ization group equations. For our purposes here it suffices toconsider the effective form

G15S 1 0 0

0 1 e

0 2e 1D , ~53!

wheree parametrizes the size of the ratiolb /l t . For mod-erate values of tanb this form should be adequate~i.e., e nottoo close to 1!. We still expectG2'K. We assume that thelower (232) right corner ofV1 is approximately the identityto relateGQL to G1.

The above forms forG1,2 plus the resultB1,25I aboveallow us to evaluate numerically the branching ratios of Sec.II. Perhaps the most optimistic top-quark factory being con-templated at the moment is a high-luminosity upgrade of theTevatron, where studies show that one might be sensitive toB(t→cg)'431024 (831025) @14#, B(t→cZ)'431023 (631024) @14#, and B(t→cg)'531023 (131023) @2# with an integrated luminosity of10 (100) fb21, where the branching ratios are with respect toG(t→bW). These expected sensitivities will not allow directtests of the standard model predictions for these processes,B(t→cg)SM;10212, B(t→cZ)SM;10212, andB(t→cg)SM;10210 @5#, but might uncover virtual newphysics effects that enhance these rates over standard modelexpectations.

Indeed, we generally find thatB(t→cV) greatly exceedsthe corresponding standard model contribution, but unfortu-nately falls below the expected experimental sensitivities, aswas observed also in previous studies of the QCD-like cor-rections @4#. Specifically, concentrating on the ‘‘neutral’’electroweaklike corrections, we have as the dominant inputsthe masses of the charm squark and top squark, the top-quarkmixing angle, the mass of the neutralino~s!, and the neu-tralino composition. The results forV5g,g scale withe2 asdefined in Eq.~53!; we takee50.5 for concreteness.~ForV5Z, the cross section increases montonically withe fore<0.5.! Numerically2 we find that whenmt'mx1mt 1

, thebranching ratios are enhanced compared to off-resonancevalues by a factor of 2–10. This factor depends on the spe-cific combination of neutralino and top-squark masses thatsatisfy this relation~all other parameters being kept fixed!;the enhancement decreases with increasing top-squark massand so is maximized whenmt 1

is 60 GeV. The off-resonance values themselves are larger than the standardmodel predictions for not-too-heavy sparticles. We alsoverify that largemc-mt 1

mass splitting enhances the results,because of its GIM-violating effect.~A useful test of ourcode is that the branching ratios go to zero due to the GIMmechanism, if we set squark masses equal.! With regards toneutralino composition, the largest branching ratios are ob-tained for neutralinos with comparableB-ino and Higgsinoadmixtures. Increasing the scale of the sparticle masses typi-

cally leads to a rapid decrease in the branching ratios forgand g. For theZ the decrease is very gradual, as has alsobeen noted in Ref.@4#. For g andg the cross section seemsto be maximized for top-squark mixing angles close to 0 orp. The effects of mixing forZ are very dependent on theother parameters chosen, such as the neutralino composition,etc. The effect of varying tanb is of order 1.

In varying the different parameters above we have workedin the most general framework of the minimal supersymmet-ric standard model~MSSM!, in which the various parameterscan be varied independently. In a more specific model, suchas one with universal scalar masses and radiative elec-troweak symmetry breaking, these parameters are not all in-dependent. Although our choice of mixing matrices was mo-tivated by certain specific scenarios, we vary our parametersfreely so as to look for the maximal supersymmetric contri-butions. Furthermore, we choosemc1,2

and mt 2to be ;1

TeV.For the most optimistic values of the parameters, i.e.,

when the above enhancing circumstances all simultaneouslyoccur, we findB(t→cg)&231027, B(t→cZ)&431027,andB(t→cg)&331025. We see thatB(t→cg) is the oneclosest to the level of experimental sensitivity expected at theTevatron, and so perhaps it would be the mode to be firstobserved at a future sufficiently sensitive machine. This hopeis further enlarged by recalling thatB(t→cg) receives com-parable contributions from the QCD-like supersymmetriccorrections@4#, which we have not evaluated here. Eventu-ally, such a process can be a possible test for supersymmetry.

We have also evaluated the charged electroweaklike cor-rections and have found them to be smaller~typically by afactor of 10 or more! than the neutral electroweaklike cor-rections discussed above forg and g, but a factor of 10higher for theZ. @Again we assume that the lower (232)right corner ofV2 is approximately the identity to relateGQLto G2.# In the case of universal squark masses at the unifica-tion scale, GIM-violating bottom-squark strange-squarkmass differences are generated by RGE evolution, resultingin shifts to the left-handed downlike squark mass matrices.The dominant term is from the second term of Eq.~52!

which may be rewritten as2ucuK†(mu)2K, wheremu5$mu ,mc ,mt% anducu<1 is an RGE-dependent constant.Inserting the values of the CKM matrix elements we find~approximately! msL

2 →msL

22ucu(mt/5)2 and mbL

2 →mbL

2

2ucumt2 . Choosing the maximal (ucu51) mass splittings, we

find B(t→cg)&1028, B(t→cZ)&231026, andB(t→cg)&1027 for squark masses as low as experimentally allowed.~These results are not much altered even if one drops theassumption of universal squark masses at the unificationscale.! The numerical results for the charged electroweaklikecorrections cannot be compared with the corresponding onesin Ref. @3# because, as we explained above, the formulaspresented in Ref.@3# are inconsistent with gauge invarianceconstraints.

We finally try to connect up with the recent literature inRef. @2#, where in addition to thet→cg decay mode, peoplehave considered hadronic processes likepp→t c, whichmight be more easily detectable. These works ignore anysubstructure that thet-c-g vertex might have, and replace it

2We used the software packageFF @15# to evaluate the Passarino-Veltman functions.


all by an effective scaleL, defined, for instance, by the re-lation G(t→cg)58asmt

3/3L2. A given branching ratio ob-tained in the supersymmetric theory then corresponds to ascaleL in the effective theory. Dividing this expression byG(t→bW) we find thatL'1 TeV/AB(t→cg). Therefore asupersymmetric prediction ofB(t→cg);1025 correspondsto L;300 TeV. The point of this exercise is to note howmisleading such estimates of new-physics scales might

be, as this actually corresponds in our case to sparticlemasses of a few hundred GeV.

ACKNOWLEDGMENTS

The work of R.R. has been supported by the World Labo-ratory. The work of J.L. has been supported in part by U.S.DOE Grant No. DE-FG05-93-ER-40717 and that of D.V.N.by U.S. DOE Grant No. DE-FG05-91-ER-40633.

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