Rb� and new physics: A comprehensive analysis

P.�

Bamert,1 C.�

P. Burgess,1 J.�

M. Cline,1 D.�

London,1,2 and� E. Nardi3�

1Physics Department, McGill University, 3600 University Street, Montreal, Quebec, Canada H3A 2T82Laboratoire�

de Physique Nucleaire,� Universite de

Montreal,� C. P. 6128, succ. centre-ville, Montreal, Quebec,

Canada H3C 3J73�Department�

of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel Received1 March 1996�

We�

surveytheimplicationsfor newphysicsof thediscrepancybetweenthemeasurementof Rb� at� theCERN

e� � e � collider� LEP andits standardmodelprediction.Two broadclassesof modelsareconsidered: � i� � thosein�

which new Zb� ¯b couplings� ariseat the treelevel, throughZ

�or� b-quarkmixing with new particles,and � ii �

thosein which newscalarsandfermionsalter the Zbb

vertexat oneloop. We keepour analysisasgeneralaspossible� in orderto systematicallydeterminewhatkindsof featurescanproducecorrectionsto Rb

� of� the rightsign� andmagnitude.We areableto identify severalsuccessfulmechanisms,which includemostof thosewhichhave�

recentlybeenproposedin the literature,aswell assomeearlierproposals e.g.,! supersymmetricmodels" .By seeinghow suchmodelsappearasspecialcasesof our generaltreatmentwe areableto shedlight on thereasonfor, andthe robustnessof, their ability to explainRb

� . # S0556-2821$ 96% 05617-2& '

PACSnumber( s� ) :* 13.38.Dg,13.65.+ i, 14.40.Nd

I.,

INTRODUCTION

The standardmodel - SM. /

of0 electroweakinteractionshasbeen1

testedandconfirmedwith unprecedentedprecisionoverthe2

pastfew yearsusingmeasurementsof e3 4 e3 5 scattering6 atthe2

Z7

resonance8 at theCERNe3 9 e3 : collider; LEP < 1= and� theSLAC.

linear collider > SLC. ?A@

2B . A particularly striking ex-ample� of the impressiveSM synthesisof thedatacamewiththe2

discovery,at the collider detectorat fermilab C CDF� D

and�D0�

collaborationsE 3F G ,H of the top quarkwith a masswhich isin excellentagreementwith the value implied by the mea-surements6 at LEP.

TheI

biggest, and only statistically important, fly to befoundJ

sofar in theproverbialSM ointmentis theexperimen-tal2

surplusof bottomquarksproducedin Z decays,K

relativetothe2

SM prediction. With the analysisof the 1994 data asdescribedK

at last summer’sconferencesL 1,2M ,H this discrep-ancy� hasbecomealmosta 4N deviation

Kbetweenexperiment

and� SM theory.The numbersare

RbO PRQ

bO /S T

hadU 0.2219V W

0.0017,V

while RbO X SM. Y[Z

0.2156.V \

1]The SM prediction assumesa top quark massof m^ t_ ` 180GeVa

andthe strongcouplingconstantb sc (d M Ze )f g 0.123,

Vasis

obtained0 by optimizing the fit to the data.ThereI

areothermeasurementswhich differ from their SMpredictionsh at the i 2j level: Rck l 2.5mon ,H A FB

p0q rtstuvr2.0wox ,H and

the2

inconsistencyy 2.4zo{ between1

A e|0q as� measuredat LEPwith} that obtainedfrom A

~LR0q

as� determinedat SLC � 2� � . Infact,J

sincethe R�

ck and� R�

bO measurements� are correlated,and

because1

theywereannouncedtogether,someauthorsrefer tothis2

as the ‘‘ RbO � Rck crisis.’’; One of the points we wish to

make� in this paper is that there is no R�

ck crisis.; If the R�

bO

discrepancyK

canbe resolvedby the additionof new physics,one0 thenobtainsanacceptablefit to thedata.In otherwords,Rck ,H as well as A FB

p0q �����and� A LR

�0q ,H can reasonablybe viewedsimply6 asstatisticalfluctuations.

On�

the other hand, it is difficult to treat the measuredvalue� of R

�bO as� a statisticalfluctuation. Indeed,largely be-

cause; of RbO ,H the dataat face value now exclude3 the

2SM at

the2

98.8%confidencelevel. If we supposethat this disagree-ment� is not an experimentalartifact, then the burningques-tion2

is the following: What doesit mean?Our�

main intentionin this paperis to surveya broadclassof0 modelsto determinewhatkindsof newphysicscanbringtheory2

backinto agreementwith experiment.SinceR�

bO is�

themain� culprit we focuson explainingboth its sign andmag-nitude.This is nontrivial,but not impossibleto do,giventhatthe2

discrepancyis roughly the samesize as, though in theopposite0 direction to, the large m^ t_ -dependentSM radiativecorrection.; The result is thereforejust within the reachofone-loop0 perturbationtheory.

Our�

purpose is to survey the theoretical possibilitieswithin} a reasonablybroadframework,andwe thereforekeepour0 analysisquitegeneral,ratherthanfocusingon individualmodels.This approachhasthe virtue of exhibiting featuresthat2

aregenericto sundryexplanationsof the Z7 �

bb� ¯ width,}

and� manyof theproposalsof the literatureemergeasspecialcases; of the alternativeswhich we consider.

In the end we find a numberof possibleexplanationsofthe2

effect, eachof which would haveits own potentialsig-nature� in future experiments.Thesedivide roughly into twocategories:; thosewhich introducenew physicsinto Rb

O at�tree2

level, and thosewhich do so starting at the one-looplevel.�

TheI

possibilitiesareexploredin detail in theremainderofthe2

article, which has the following organization.The nextsection6 discusseswhy Rb

O is the only statisticallysignificantdiscrepancyK

betweentheoryandexperiment,andsummarizesthe2

kinds of interactionsto which the data points. This isfollowed by severalsections,eachof which examinesa dif-ferentclassof models.SectionIII studiesthe tree-levelpos-sibilities,6 consistingof modelsin which theZ

7boson1

or theb�

quark� mixes with a hitherto undiscoveredparticle.We findseveral6 viable models,someof which imply comparativelylarge modifications to the right-handedb

�-quark neutral-

PHYSICAL REVIEW D 1 OCTOBER1996VOLUME�

54, NUMBER 7

54�

0556-2821/96/54� 7� � /4275� �

26� �

/$10.00�

4275 © 1996The AmericanPhysicalSociety

current; couplings.SectionsIV andV thenconsiderloop con-tributions2

to RbO . SectionIV concernsmodificationsto the

t� -quark sector of the SM. Although we find that we canreduce8 the discrepancyin R

�bO to2 �

2� �

,H we do not regardthisas� sufficientto claim successfor modelsof this type.SectionV�

then considersthe generalform for loop-level modifica-tions2

of the Zb7 ¯b

�vertex� which arisefrom modelswith new

scalars6 andfermions.The generalresultsarethenappliedtoa� numberof illustrative examples.We are able to seewhysimple6 models,like multi-Higgs-doubletandZee-typemod-els  fail to reproducethe data, as well as to examinetherobustness8 of the difficulties of a supersymmetricexplana-tion2

of RbO . Finally, our generalexpressionsguideusto some

examples  which do¡

makeexperimentallysuccessfulpredic-tions.2

SectionVI discussessomefutureexperimentaltestsofvarious� explanationsof theRb

O problem.h Our conclusionsaresummarized6 in Sec.VII.

II.,

THE DATA SPEAKS

Takenat face value, the currentLEP/SLC dataexcludesthe2

SM at the 98.8% confidencelevel. It is natural to askwhat} new physicswould be requiredto reconciletheoryandexperiment  in the eventthat this disagreementsurvivesfur-ther2

experimentalscrutiny. Before digging through one’stheoretical2

repertoirefor candidatemodels,it behoovesthetheorist2

first to askwhich featuresarepreferredin a success-ful explanationof the data.

An efficientway to dosois to specializeto thecasewhereall� new particlesare heavyenoughto influenceZ

7-pole ob-

servables6 primarily through their lowest-dimensioninterac-tions2

in an effectiveLagrangian.Then the variouseffectivecouplings; may be fit to the data,allowing a quantitativesta-tistical2

comparisonof which onesgive thebestfit. Althoughnot� all of thescenarioswhich we shalldescribeinvolve onlyheavy particles,many of them do and the conclusionswedrawK

usinganeffectiveLagrangianoftenhavea muchwiderapplicability� thanonemight at first assume.Applicationsofthis2

type of analysisto earlierdata ¢ 4,5£ ¤

have¥

beenrecentlyupdated¦ to includelastsummer’sdata § 6¨ © ,H andthepurposeofthis2

sectionis to summarizethe resultsthat werefound.ThereI

are two main typesof effective interactionswhichplayh an importantrole in the analysisof Z

7-resonancephys-

ics, and we pausefirst to enumeratebriefly what theseare.ªFor moredetailsseeRef. « 4¬®­ . The first kind of interaction

consists; of the lowest-dimensiondeviations to the elec-troweak2

bosonself-energies,andcanbeparameterizedusingthe2

well-known Peskin-Takeuchiparameters1 S¯

and� T ° 7± ² .The secondclass of interactionsconsistsof nonstandarddimension-fourK

effective neutral-currentfermion couplings,which} may be definedasfollows:2

³eff´NCµ ¶ e3

s· w¸ c¹ w¸ Z º f» ¼o½¿¾ÁÀ g L

� fà ÄÆÅg L� fà ÇÉÈ

L� ÊÌË g R

Í fà ÎÆÏg RÍ fà ÐÉÑ

RÍ Ò f»

. Ó2� Ô

In this expressiong L� fà and� g R

Í fà denoteK

the SM couplings,which} are normalized so that g L

� fÃ Õ I 3� fÃ Ö Q

× fÃs· w¸2Ø and�

g Rfà ÙÛÚ

Q× fÃs· w¸2Ø ,H where I

Ü3� fà and� Q

× fÃ

are� the third componentofweak} isospin and the electric chargeof the correspondingfermion, f

». s· w¸ Ý sin6 Þ

w¸ denotesK

thesineof theweakmixingangle,� and ß L

�(àRÍ

)á âRã 1äæå 5

ç è /2.S

Fittingé

theseeffective couplingsto the data leadsto thefollowingJ

conclusions.(1)ê

What must be explained. Although the measuredval-ues¦ for severalobservablesdepartfrom SM predictionsat the2� ë

level�

and more,at the presentlevel of experimentalac-curacy; it is only the R

�bO measurement� which really must be

theoretically2

explained.After all, some2ì fluctuationsarenot surprisingin any sampleof twenty or moreindependentmeasurements.� í Indeed,

îit would be disturbing,statistically

speaking,6 if all measurementsagreedwith theory to within1ï .ð This observationis reflectedquantitativelyin the fits ofRef. ñ 6¨ ò ,H for which the minimal modification which is re-quired� to accommodatetheR

�bO measurement,� namelythead-

ditionK

of only new effective Zb7 ¯b

�couplings,; alreadyraises

the2

confidence level of the fit to acceptable levelsóõôminö2Ø /SN÷

DFø ù 15.5/11ascomparedto 27.2/13for a SM fitú . We

therefore2

regardtheevidencefor otherdiscrepancieswith theSM,.

suchasthevalueof R�

ck ,H as beinginconclusiveat presentand� focusinsteadon modelswhich predictlargeenoughval-ues¦ for Rb

O .(2)ê

The significance of Rck . Sincethe 1995summercon-ferencesJ

havehighlightedthe nonstandardmeasuredvaluesfor theZ branching

1ratio into both

�c and� b

�quarks,� it is worth

makingtheabovepoint morequantitativelyfor theparticularcase; of thediscrepancyin R

�ck . This wasaddressedin Ref. û 6¨ ü

by1

introducingeffectivecouplingsof the Z7

to2

both b�

and� c¹quarks,� andtestinghow muchbettertheresultingpredictionsfit the observations.Although the goodnessof fit to Z-poleobservables0 does

¡improve�

somewhatý with} þmin2 /SN÷

DF ÿ 9.8/9� �

,H

1Thethird parameterU also� appearsbut doesnot play a role in theZ-pole observables.

2�Herewe introducea slight notationchangerelativeto Ref. � 4� in

that our couplings � g L�

,� R�f

correspond� to g L�

,� R�f

of Ref. � 4� .�

FIG. 1. A fit of the Zb� ¯b couplings� � g L

�,� R�b�

to�

Z�

-pole datafromthe 1995 SummerConferences.The four solid lines respectivelydenotethe 1� , 2� , 3� � , and4� error! ellipsoids.The SM predictionlies at the origin � 0,0

& �.� This fit yields � s� (M Z)

� �0.101& �

0.007.&

4276 54BAMERT,�

BURGESS,CLINE, LONDON, AND NARDI

it doessoat theexpenseof driving thepreferredvaluefor thestrong6 coupling constantup to � sc (d M� Z)

f 0.180V !

0.035,V

indisagreementK

at the level of 2" with} low-energydetermina-tions,2

which lie in therange0.112# 0.003V $

8% &

. This changeinthe2

fit value for ' sc (d M z( )f is given by the experimentalcon-straint6 that the total Z

7width} not changewith the additionof

the2

new Zc7 ¯c¹ couplings.; 3

�Once�

the low-energydetermina-tions2

of ) sc (d M z( )f arealso included, * minö2Ø /SN÷

DFø not only drops

back1

to the levelstakenin the fit only to effectiveZbb�

cou-;plings,h but the best-fitpredictionfor R

�ck again� movesinto a

roughly8 2+ discrepancyK

with experiment.It is neverthelesstheoreticallypossibleto introducenew

physicsh to accountfor Rck in a way which doesnot drive upthe2

value of the strong coupling constant.As arguedonmodel-independent� groundsin Ref. , 6¨ - ,H and more recentlywithin} the contextof specificmodels . 9,10

� /,H an alterationof

the2

c¹ -quark neutral-currentcouplings can be compensatedforJ

in the total Z7

width} by also altering the neutral-currentcouplings; of light quarks,suchasthe s· . We put thesetypesof0 modelsasidein the presentpaper,consideringthemto beinsufficiently motivatedby the experimentaldata.

(3)ê

LH vs RH couplings. TheI

data do not yet permit adeterminationK

of whetherit is preferableto modify the left-handed0 LH 1 or0 right-handed2 RH3 Zbb

�coupling.; Themini-

mum valuesfor 4 2Ø

found in Ref. 5 6¨ 6 for a fit involving LH,RH,7

or both couplings are, respectively, 8 min2 /SN÷

DF 9 LH: ;

< 17.0/12, = min2 /SN÷

DF > RH7 ?A@

16.1/12, or B min2 /SN÷

DF C both1 D

E 15.5/11.(4)ê

The size required to explain RbO . The analysisof Ref.F

6¨ G

also� indicatesthesizeof thechangein theneutral-currentb�

-quark couplingsthat is requiredif theseare to properlydescribeK

the data.The bestfit valueswhich arerequiredaredisplayedK

in Fig. 1, and are listed in Table I. Table I alsoincludes�

for comparisonthe correspondingtree-level SMcouplings,; aswell asthe largestSM one-loopvertexcorrec-tions2 H

those2

which dependquadraticallyor logarithmicallyon0 the t� -quarkmass4

Im^ t_ J ,H evaluatedat s· w¸2Ø K 0.23.

VFor making

comparisons; we takem^ t_ L 180 GeV.AsM

we now describe,the implicationsof the numbersap-pearingh in TableI dependon the handednessN LH vs RHO of0effective  new-physicsZbb

�couplings.;

(4a)ê

LH couplings. TableI showsthattherequiredchangein�

the LH Zb7 ¯b

�couplings; mustbe negativeandcomparable

in�

magnitudeto them^ t_ -dependentloop correctionswithin theSM..

The sign must be negativesincethe predictionfor theZ P bb

� ¯ width} mustbe increasedrelativeto the SM result inorder0 to agreewith experiment.This requires Q g L

bO

to2

havethe2

samesign asthe tree-levelvaluefor g LbO,H which is nega-

tive.2

As we shall see,this sign limits the kinds of modelswhich} can producethe desiredeffect. Comparisonwith theSM.

loop contributionshowsthat the magnituderequiredforRg L

bO

is�

reasonablefor a one-loopcalculation.Sincethe sizeof0 the m^ t_ -dependentpart of the SM loop is enhancedby afactor of m^ t_2Ø /S M W

2Ø

,H the requirednew-physicseffect must belargerS

than2

a genericelectroweakloop correction.(4b)ê

RH couplings. Since.

the SM tree-levelRH couplingis oppositein sign to its LH counterpartand is somefivetimes2

smaller,the new-physicscontributionrequiredby thedata,K T

g RbO,H is positiveandcomparablein sizeto thetree-level

coupling.; This makesit likely thatanynew-physicsexplana-tion2

of the datawhich relies on changingg RÍbO must ariseat

tree2

level, ratherthanthroughloops.(5)ê

Absence of oblique corrections. AM

final provisois thatany� contributionto g L

bO

or0 g RbO

should6 not be accompaniedbylargecontributionsto otherphysicalquantities.For example,Ref. U 6¨ V finds that the best-fit valuesfor the obliqueparam-eters  S

¯and� T

Ware�

S¯ XZY

0.25V [

0.19,V

T \Z] 0.12V ^

0.21V _

3F `

awith} a relativecorrelationof 0.86b even  when c g L

�,RÍb

Oare� free

to2

float in the fit. SinceT often0 getscontributionssimilar insize6 to d g L

bO

these2

boundscanbe quite restrictive.Noticee

that we neednot worry about the possibility ofhaving¥

large cancellationsbetweenthe new-physicscontri-butions1

to the obliqueparametersand f g L�bO in Rb

O . It is truethat2

sucha partialcancellationactuallyhappensfor g bO in the

SM,.

where the loop contributionsproportionalto m^ t_2 in�

TW

and� h g L�bO exactly  cancelin thelimit thats· w¸2 i 1

4,H andsoendupbeing1

suppressedby a factor s· w¸2Ø j 14. We neverthelessneed

not� considersucha cancellationin R�

bO since6 the obliquepa-

rameters8 k especially  TW l

almost� completelycancelbetweenm bO

and� nhado . Quantitatively,we have p 4q

rbO sut

bOSMv w

1 x 4.57y g L�bO z 0.828

V {g RÍbO | 0.00452

VS¯ }

0.0110V

T ~ ,H�

had�u� hadSMv �

1 � 1.01� g LbO �

0.183V �

g RbO �

0.00518V

S¯ �

0.0114V

TW �

,H

3�Introducingeffective b-quark couplingshaspreciselythe oppo-

site effect, sincethe SM predictionfor � b� is�

low andthat for � c� ishigh relativeto observations,lowering thestrongcouplingconstantto � s� (M Z) � 0.103

& �0.007.&

4�More precisely � 11� , we use � g� L

b� �

(� �

w� /16� �

) � r � 2.88lnr � ,�wherer � m� t 2� /� M¡ W

2�

.

TABLE I. Requiredneutral-currentb

-quark couplings: The last two columnsdisplay the size of theeffective! correctionto the left- andright-handedSM Zbb couplingswhich bestfit thedata.The ‘‘individualfit’’¢

is obtainedusingonly oneeffectivechiral coupling in addition to the SM parametersm t  and� £s� (� M¡ Z

¤ ).�The¥

‘‘fit to both’’ includesboth couplings.Also shown for comparisonare the SM predictionsfor thesecouplings,� both the tree-levelcontribution ¦ ‘‘SM tree’’§ ,� and the dominantm t  -dependentone-loopvertexcorrection,� evaluatedat s¨ w�2� © 0.23

& ª‘‘SM top loop’’ « .

Coupling¬

g ­ SM tree® ¯ g� ° SM±

top loop² ³ g� ´ individual�

fit µ ¶ g� · fit¢

to both

g� Lb� ¹

0.4230 0.0065 º 0.0067» 0.0021& ¼

0.0029½ 0.0037g� R¾b� 0.0770 0 0.034¿ 0.010

&0.022À 0.018

54 4277RÁ

b� AND NEW PHYSICS:A COMPREHENSIVEANALYSIS

so6 R�

bO Â R

�bOSMv Ã

1 Ä 3.56F Å

g LbO Æ

0.645V Ç

g RbO È

0.00066V

S¯

É 0.0004V

T Ê . Ë 4ÌWeÍ

now turn to a discussionof the circumstancesunderwhich} theaboveconditionsmaybeachievedin a broadclassof0 models.

III.,

TREE-LEVEL EFFECTS: MIXING

At treelevel theZbb�

couplings; canbemodifiedif thereismixing� amongstthechargeÎ 1

3Ï quarks,� or theneutral,colour-

less�

vector bosons.Being a tree-leveleffect it is relativelyeasy  and straightforwardto analyzeand comparedifferentscenarios.6 Also, since mixing effects can be large, mixingcan; providecomparativelylargecorrectionsto the Zb

7 ¯b�

cou-;pling,h suchasis neededto modify R

�bO through2

changesto g RbO.

Note

surprisingly, a numberof recentmodels Ð 9,10,12�

–14Ñuse¦ mixing to try to resolvetheRb

O Ò and� Rck Ó discrepancy.K

Ouraim� hereis to surveythepossibilitiesin a reasonablygeneralway.} We thereforepostponefor themomenta moredetailedphenomenologicalh analysisof the variousoptions.

In generalwe imaginethat all particleshaving the samespin,6 color, andelectricchargecanbe relatedto oneanotherthrough2

massmatricesÔ some6 of whoseentriesmight becon-strained6 to be zero in particularmodelsdue to gaugesym-metries or restrictionson the Higgs-field representationsÕ .WeÍ

denotethe color-triplet, chargeQ× ÖØ× 1

3Ï ,H quarks in the

flavorÙ

basis by BÚ Û

,H and label the correspondingmasseigenstates  5

çby1

b� i. The mass-eigenstatequarks,b

� i,H are ob-tained2

from the B Ü by1

performingindependentunitary rota-tions,2 ÝßÞ

L,R† )f à i amongst� the left- andright-handedfields.The

b�

quark� thathasbeenobservedin experimentsis the lightestof0 themasseigenstates,b

� áb� 1,H andall othersarenecessarily

muchheavierthanthis state.Similar.

considerations also apply for colorless,electrically-neutral  spin-oneparticles.In this casewe imag-ine theweakeigenstates,Z âw¸ ,H to be relatedto themasseigen-states,6 Z ãmä ,H by an orthogonalmatrix, å wm¸ . We take thephysicalh Z

7,H whosepropertiesaremeasuredin suchexquisite

detailK

at LEP andSLC, to be the lightestof the masseigen-states:6 Z æèç Z é1 .

AssumingM

thatall of theb� i and� Zmä ê except  for the lightest

ones,0 the familiar b�

and� Z7

particlesh ë are� too heavy to bedirectlyK

producedat Z-resonanceenergies,we find that theflavor-diagonaleffective neutral-currentcouplings relevantforJ

R�

bO are�

g L�

,RÍb

O ìîíg mä ï 1 ð L

�,RÍ11 ñóòôöõ w¸ ÷ g w¸ ø L

�,RÍùûúýü

L�

,RÍþ 1* ÿ L

�,RÍ� 1 � w¸ 1

��� �w¸ � g w¸ �� L,R � L,R

� 1 2Ø � w¸ 1,H � 5� �

where} the neutral-currentcouplingsaretakento be diagonalin the flavor B � basis.

1 6�

ThisI

expressionbecomesreasonablysimple in the com-mon� situation for which only two particlesare involved inthe2

mixing. In this casewe may write BÚ ���

(d

B �B ),f

b� i � (

dbO �bO

),f

and� Z7 w¸ � (

dZ �Z )f, and take � L

� ,H � RÍ ,H and � to

2be two-by-two

rotationmatricesparametrizedby the mixing angles� L ,H R ,Hand� !

Z . In this caseEq. " 5� # reduces8 to

g L,RbO $&%('

g ZB )

L�

,RÍ c¹ L,R

2 *&+ g ZB ,.-

L�

,RÍ s· L,R

2 / c¹ Ze 02143 g Z 5B 6

L�

,RÍ c¹ L,R

2

7&8g Z 9B :.;

L,Rs· L,R2 < s· Z ,H = 6¨ >

where} s· L denotesK

sin ? L ,H etc. IncreasingR�

bO requires8 increas-

ing�

the combination(g L�bO )f 2 @ (

dg RÍbO )f 2. To seehow this works

we} now specializeto morespecificalternatives.

A.A

ZB

mixing

First considerthecasewheretwo gaugebosonsmix. ThenEq. C 6¨ D reducesto

g L,RbO E&F

g ZBG H

L�

,RÍ c¹ Ze I&J g Z

e KBG L

L�

,RÍ s· Ze ,H M 7± N

where} (g ZeBG )f

L�

,RÍ is the SM couplingin the absenceof Z mix-

ing, and (g Ze OBG

)f

L,R is the b�

-quark coupling to the new fieldZ P Q�R which} might itself be generatedthrough b

�-quark mix-

ing� S

. It is clearthatso long astheZ7 T

b�

b�

coupling; is nonzero,then2

it is alwayspossibleto choosethe angle U Ze to2

ensurethat2

the total effectivecoupling is greaterthan the SM one,(dg Z

B)f

L,R . This is becausethe magnitudeof any function ofthe2

form f»

(d V

Z)f W

Ac~

Z X BsÚ

Z is�

maximized by the angletan2 Y

Ze Z B/

SA,H for which [ f» \ maxö ]_^ A/

Sc¹ Ze `ba&c A d .

The model-buildingchallengeis to ensurethat the sametype2

of modificationsdo not appearin an unacceptablewayin�

theeffectiveZ7

couplings; to otherfermions,or in too largean� M Z

e shift6 due to the mixing. This can be ensuredusingappropriate� choicesfor the transformationpropertiesof thefieldse

underthe new gaugesymmetry,andsufficiently smallZ7

-Z7 f

mixing� angles.Models along theselines have beenrecentlydiscussedin Refs. g 9,

�16h .

B. bi

-quark mixing

The secondnatural choice to consideris pure b�

-quarkmixing, with no newneutralgaugebosons.We consideronlythe2

simplecaseof 2j 2�

mixings,sincewith only onenewBÚ k

quark� mixing with the SM bottom quark,Eq. l 6¨ m simplifies6considerably.; As we will discussbelow,we believethis to besufficient6 to elucidatemost of the featuresof the possibleb�

-mixing solutionsto the R�

bO problem.h

Let:

us first establishsomenotation.We denotethe weakSU. n

2� o

representations8 of the SM BÚ

L,R and� of the BÚ

L,R

pas�

5qWe imaginehavingalreadydiagonalizedthe SM massmatrices

so that in the absenceof this non-standardmixing one of the B rreducesto the usualb

quark,with a diagonalmassmatrix with the

d ands¨ quarks.s

6tEquation u 5� v describesthe mostrelevanteffectsfor the Rb

� prob-�lem, namelythe mixing of Z

�and b

with new states.However,in

generalotherindirecteffectsarealsopresent,suchas,for example,a shift in M Z dueto themixing with theZ w .� For a detailedanalysisof the simultaneouseffectsof mixing with a Z x and� new fermions,seeRef. y 15z .

4278 54BAMERT,�

BURGESS,CLINE, LONDON, AND NARDI

R�

L,R and� R�

L,R

{,H respectively,where R

� |(dIÜ,H IÜ 3� )f . The SM

B-quark assignmentsare RL }�~ 12� ,H � 1

2� � ,H and R

�R �_� 0,0

V �. By

definition,K

a BÚ �

quark� must haveelectric chargeQ× �_�

1/3,but1

may in principle have arbitrary weak isospin R�

L,R

�� (dIL�

,R� ,H I3

�L�

,R�).f

In terms of the eigenvaluesI3�

L�� and� I3

�R� of0 the weak-

isospin�

generatorIÜ

3� acting� on B

ÚL� and� B

ÚR� ,H the combination

of0 couplingswhich controls � bO becomes1

�bO ��� g L

�bO � 2 �&� g RÍbO � 2 � � c¹ L

�2Ø2 � s· w¸2Ø

3F � s· L

�2Ø I3�

L�� 2Ø �

s· w¸2Ø3F   s· R

Í2Ø I3�

RÍ¡ 2Ø.¢

8% £

In orderto increase¤ bO using¦ this expression,¥ L

� and� ¦RÍ must

be1

suchasto makeg L�bO morenegative,g R

ÍbO morepositive,orboth.1

Two waysto ensurethis areto choose

I3�

L�§_¨2© 1

2or0 I3

�Rͪ¬« 0.

V ­9� ®

ThereI

arealsotwo otheralternatives,involving largemixingangles� or large B

Ú ¯representations:8 I

Ü3�

L

°_±0V

, with s· L2(dIÜ

3�

L

²´³12� )f µ 1 ¶ 2

�s· w¸2 /3S ·

0.85,V

and IÜ

3�

R

¸¬¹0V

, with s· R2 º IÜ 3�

R

»½¼b¾2�

s· w¸2 /3S¿ 0.15.

VNote that, in the presenceof LH mixing, the

Cabibbo-Kobayoshi-Maskawa�

elementsVqbÀ (dqÁ  uà ,H c¹ ,H t� )f get

rescaledasVqbÀ Ä c¹ L� VqbÀ ,H thus leadingto a decreasein rates

forJ

processesin which theb�

quark� couplesto a W. Thereforecharged-current; datacanin principleput constraintson largeLH mixing. For example,futuremeasurementsof thevarioust� -quark decaysat the Tevatronwill allow the extractionofV tb_ in

�a model-independentway, thus providing a lower

limit�

on c¹ L . At present,however,when the assumptionofthree-generation2

unitarity is relaxed Å as� is implicit in ourcases; Æ the

2current measurementof B(

dt� Ç Wb)/

fB(dt� È Wq )

fimplies�

only the very weaklimit É V tb_ ÊbË 0.022V Ì

at� 95% C.L.ÍÎ17Ï . Henceto date thereare still no strongconstraintson

large LH mixing solutions.Regardingthe RH mixings, asdiscussedK

below there is no correspondingway to deriveconstraints; on c¹ R ,H andso larges· R solutions6 arealwayspos-sible.6

WeÍ

proceednow to classify the modelsin which the SMbottom1

quarkmixeswith othernew Q× Ð_Ñ

1/3 fermions.Al-though2

thereare endlesspossibilitiesfor the kind of exoticquark� onecould consider,the numberof possibilitiescanbedrasticallyK

reduced,and a completeclassificationbecomespossible,h after the following two assumptionsare made: Ò i ÓThereI

areno new Higgs-bosonrepresentationsbeyonddou-blets1

andsinglets; Ô ii� Õ the2

usualBÚ

quark� mixeswith a singleB Ö ,H producingthemasseigenstatesb

�and� b

� ×. This constrains

the2

massmatrix to be 2Ø 2:

ÙB B ÚÜÛ L

M 11

M�

21

M 12

M�

22

BBÚ Ý

R

. Þ 10ßWeÍ

will examineall of the alternativesconsistentwiththese2

assumptions,both of which we believeto be well mo-tivated,2

andindeednot very restrictive.Theresultingmodelsincludethe‘‘standard’’ exoticfermionscenariosà 18áãâ vector�singlets,6 vectordoublets,mirror fermionsä ,H as well asa num-ber1

of others.

Let us first discussassumptionå i æ . From Table I andEq.ç8% è

one0 seesthat the mixing anglesmustbe at leastaslargeas� 10% to explainR

�bO ,H implying that the off-diagonalentries

in�

themassmatrix Eq. é 10ê which} give riseto themixing areof0 orders· L

�,RÍ M 22

Ø ë 10 GeV. If theseentriesaregeneratedbyHiggs fields in higher than doublet representations,suchlarge�

vacuumexpectationvalves ì VEV’s� í

would} badly un-dermineK

the agreementbetweentheory and experimentforthe2

M W/SM Ze massratio.7

îAccording to assumptionï i ð ,H the permittedHiggs repre-

sentations6 are R�

H ñ_ò 12� ,H ó 1

2� ô and� õ 0,0

V ö. It is then possibleto

specify6 which representationsR�

L,R

÷allow� the B

Ú øto2

mix withthe2

B quark� of the SM.ù1ú Since

.the B

Ú ûshould6 be relatively heavy,we require

that2

M�

22ü 0.V

Then the restriction ý i� þ on0 the possibleHiggsrepresentationsimplies that

ÿIÜ

L� � IÜ

R

�����0,V 1

2� � 11�

and�IÜ

3�

L

��IÜ

3�

R

����0,V 1

2� . � 12�

�2� To haveb

�-b� �

mixing, at leastoneof the off-diagonalentries,  M 12 or0 M 21

Ø ,H must be nonzero.Theseterms ariserespectively from the gauge-invariantproducts RH � R

�L�

R�

R� and� R�

H � R�

R � R�

L� so6 that R�

L(àR)á�

must� transformas theconjugate; of the tensorproductRH

� � RRÍ

(àL�

)á :

R�

L�! R�

H� " R

�RÍ #%$ 0,0

V &,H 1

2� ,H ' 1

2� ,H ( 13)

or0

R�

R*,+ R�

H� - R

�L� .%/ 0,0

V 0,H 1

2� ,H 1 1

2� ,H 2 1,3 1 4 ,H 5 1,06 . 7 148

Thus the only possiblerepresentationsfor the B 9 are� thosewith} IR

Í :,; 0,V 1

2,H 1 andIL� <!= 0,

V 12, 1H , 3

Ï2,H subjectto therestrictions>

11? – @ 14A .AsM

for assumptionB ii� C ,H it is of coursepossiblethatseveralspecies6 of B D quarks� mix with theB,H giving rise to anN

÷ EN÷

massmatrix, but it seemsreasonableto study the allowedtypes2

of mixing one at a time. After doing so it is easytoextend  the analysisto the combinedeffectsof simultaneousmixing with multiple B F quarks.� Thus G ii H appears� to be arather8 mild assumption.

7IThe¥

contribution of theserelatively large nonstandardVEV’scannot be effectively compensatedby new loop effects. On theotherhand,beyondHiggs doublets,the next caseof a Higgs mul-tiplet preservingthetree-levelratio is thatof I3

JH K 3,

LY H M 2. We do

not considersuchpossibilities,which would alsorequirethemixedBN O

to�

belongto similarly high-dimensionalrepresentations.We alsoneglect alternativescenariosinvoking, for example,more HiggstripletsandcancellationsbetweendifferentVEV’s, sincethesesuf-fer from severefine-tuningproblems.

54 4279RÁ

b� AND NEW PHYSICS:A COMPREHENSIVEANALYSIS

Thereis one sensein which P ii Q might appearto restrictthe2

classof phenomenawe look at in a qualitativeway: itis�

possibleto obtainmixing betweentheBÚ

and� a BÚ R

in�

oneofthe2

higher representationswe haveexcludedby ‘‘bootstrap-ping,’’h thatis, by intermediatemixing with a B

Ú1S in�

oneof theallowed� representations.The ideais that, if the SM B mixeswith} such a B1T ,H but in turn the latter mixes with a B2

Ø U of0larger�

isospin,this would effectively inducea BÚ

-BÚ

2V mixing,�which} is not consideredhere.However,sincemassentriesdirectlyK

coupling B to2

B2Ø W are� forbiddenby assumptionX i Y ,H

the2

resultingBÚ

-BÚ

2Z mixing� will in generalbe proportionaltothe2

BÚ

-BÚ

1[ mixing,� implying that theseadditionaleffectsaresubleading,6 i.e., of higher order in the mixing angles.Thismeansthat if the dominantB-B1\ mixing effectsare insuffi-cient; to accountfor the measuredvalueof R

�bO ,H addingmore

B ] quarks� with larger isospinwill not qualitatively changethis2

situation.ThereI

is, however,a loophole to this argument.If themassmatrix hassomesymmetrywhich givesriseto a special‘‘texture,’’ thenit is possibleto havelargemixing anglesandthus2

evadethe suppressiondue to productsof small mixingangles� alludedto above.Indeed,we haveconstructedseveralexamples  of 3 3

Fquasidegeneratematriceswith three and

four texture zeros,for which the B-B2Ø _ mixing is not sup-

pressedh and,dueto the degeneracy,canbe maximally large.Foré

example, let us chooseBÚ

1 in�

a vector doublet withIÜ

3�

L,R a�b 1/2 and BÚ

2c in�

a vector triplet with IÜ

3�

L,R d�e 1. Be-cause; of our assumptionof no Higgstriplets,directB mixingwith} sucha B2

Ø f is forbidden,and M 13g M 31� h M 12i 0

V. It is

easy  to checkthat for a genericvaluesof the nonvanishingmassmatrix elements,the induceds· L

�,RÍ13 mixings are indeed

subleading6 with respectto s· L�

,RÍ12 . However,if we insteadsup-

poseh that all the nonzeroelementsare equal to somelargemass� j ,H thentherearetwo nonzeroeigenvaluesm^ b

O kmlon and�m^ bO p!q 2r while} the B-B2

Ø s mixing angless· L�13tvu 1/3 and s· R

Í13

wvx 3/8F

are unsuppressedrelative to s· L�

,RÍ12 .8 Although it may

be1

unnaturalto havenearequalityof themassentriesgener-ated� by singletanddoubletHiggsVEV’s, asis neededin thiscase; and in most of the otherexampleswe found, it is stillpossibleh that someinterestingsolutionscouldbeconstructedalong� theselines.

ApartM

from somespecialcasesanalogousto the oneout-lined�

above,we canthereforeconcludethat neitherdoesas-sumption6 y ii z seriously6 limit the generalityof our results.

WeÍ

can now enumerateall the possibilitiesallowed byassumptions� { i� | and� } ii� ~ .

WithÍ

the permittedvaluesof IÜ

R� and� IÜ

L� listed�

above,andthe2

requirementthat at leastone of the two conditions � 13�and� � 14� is satisfied,thereare19 possibilities,listed in TableII.î

Althoughnot all of themareanomalyfree, theanomaliescan; always be canceledby adding other exotic fermionswhich} haveno effecton R

�bO . Sinceonly thevaluesof I

Ü3�

L

�and�

I3�

R� are� importantfor theb

�neutralcurrentcouplings,for our

purposeh modelswith thesameI3�

L�

,R�

assignments� areequiva-lent,�

regardlessof IÜ

L,R

�or0 differencesin the massmatrix or

mixing pattern.Altogether thereare 12 inequivalentpossi-bilities.1

Equivalentmodelsareindicatedby a prime ����� in�

the‘‘Model’’ columnin TableII.

8�A�

small perturbationof the orderof a few GeV canbe addedtosomeof the nonzeromassentriesto lift the degeneracyandgive anonzerovaluefor mb

� .�

TABLE II. Modelsandchargeassignments.All thepossiblemodelsfor B-B � mixing allowedby theassumptionsthat � i � hereareno newHiggsrepresentationsbeyondsingletsanddoublets,and � ii � only mixing with a singleB � is considered.Thepresenceof LH or RH mixingswhich canaffect the b

neutralcurrentcouplingsis indicatedunder‘‘Mixing.’’ Subleadingmixings, quadraticallysuppressed,aregiven in

parentheses.Equivalentmodels,for the purposesof RÁ

b� , areindicatedby a prime ��� � in

�the ‘‘Model’’ column,while modelssatisfyingEq.�

9� and� which canaccountfor thedeviationsin Rb� with smallmixing angles,arelabeledby anasterisk� * � . LargeRH mixing solutionsare

labeledby a doubleasterisk� **   , while models7, 7 ¡ , and7¢ allow for a solutionwith largeLH mixing.

IL£ ¤ I3

�L£¥ IR

¦ § I3�

R¦¨ Model

©Mixing

0 0 0 0 1 Vector singlet L�

1/2 ª 1/2 2« ** ¬ Mirror family L�

,RÁ

­1/2 3® * ¯ (L),

�R

1/2 ° 1/2 0 0 4 Fourthfamily1/2 ± 1/2 5² ** ³ Vector doublet ´ Iµ ¶ R

Á1 · 1 6 ** ¹ R

0 4º»1/2 0 0 7 L

�1/2 ¼ 1/2 8½ * ¾ Vector doublet ¿ IIµ À (L

�),RÁ

1 0 7 Á LÂ1 9Ã * Ä (L),

�R

1 Å 1 1 Æ 1 10Ç * È Vector triplet É Iµ Ê L�

,(RÁ

)1/2 Ë 1/2 11Ì * Í L

�,(RÁ

)0 1 0&

1 Î Vector triplet Ï II Ð L1/2 Ñ 1/2 2 Ò L,(R)

3/2 Ó 3/2 1 Ô 1 12Õ * Ö L�

,(RÁ

)× 1/2 1 Ø 1 6Ù (�RÁ

)�

0 4ÚÛ1/2 1 0 7Ü L

4280 54BAMERT,�

BURGESS,CLINE, LONDON, AND NARDI

Due to gaugeinvarianceand to the restriction Ý i Þ on0 theHiggssector,in severalcasesoneof theoff-diagonalentriesM�

12 or0 M�

21 in�

Eq. ß 10à vanishes,� leadingto a hierarchybe-tween2

the LH and the RH mixing angles.If the b� á

is muchheavierthan the b

�,H M 12â 0

Vyields s· R

Í ã M 21Ø /SM 22Ø ,H while the

LH:

mixing is suppressedby M�

22ä 2Ø. If on the other hand

M�

21å 0,V

thenthe suppressionfor s· R is�

quadratic,leavings· L

as� thedominantmixing angle.For thesecases,thesubdomi-nantmixingsareshownin parenthesesin the ‘‘Mixing’’ col-umn¦ in TableII. Notice that while models2 and6 allow fora� large right-handed mixing angle solution of the R

�bO

anomaly,� the ‘‘equivalent’’ models2 æ and� 6 ç doK

not, pre-cisely; becauseof sucha suppression.

Six.

choicessatisfy one of the two conditionsin Eq. è 9� é ,Hand� hencecan solve the R

�bO problemh using small mixing

angles.� Theyarelabeledby anasteriskê * ë in TableII. Threeof0 thesemodels ì 10,11,12í satisfy6 the first condition for so-lutions�

using small LH mixings. Since for all thesecasesIÜ

3�

R

îðï0V

, a largeRH mixing could alternativelyyield a solu-tion2

but becauses· RÍ is alwayssuppressedwith respectto s· L

� ,Hthis2

latter possibility is theoreticallydisfavored.The otherthree2

choicesñ models� 3,8,9ò ó

satisfy6 the secondconditionforsolutions6 usingsmall RH mixing. It is noteworthythat in allsix6 modelsthe relevantmixing neededto explain Rb

O is au-tomatically2

the dominantone,while the other,which wouldexacerbate  the problem, is quadratically suppressedandhence¥

negligiblein thelargem^ bO ô limit.�

Therearetwo choicesõmodels� 5,6

ö ÷forJ

which IÜ

3�

R

øðù0V

andthereis only RH mixing,and� one ú model2û for which I3

�RÍüðý

0V

ands· RÍ is unsuppressed

with} respectto s· L . Thesethreecasesallow for solutionswithlargeRH mixings,andarelabeledby a doubleasteriskþ ** ÿ .Finally, a solutionwith largeLH mixing is possible� models7,H 7 � ,H and 7��� in which I3

�L����� 1/2, and I3

�R�� 0

Vimplies no

RH7

mixing effects.In the light of TableII we now discussin moredetail the

most popular models,as well as some other more exoticpossibilities.h

Vector singlet. Vector�

fermionsby definition haveidenti-cal; left- andright-handedgaugequantumnumbers.A vectorsinglet6 model1� is one for which IL

� ��� IRÍ ��� 0

V. Inspectionof

Eq.� �

8% �

shows6 that mixing with sucha vector-singletquarkalways� actsto reduceR

�bO .9�

Mirror family. A mirror family � model2� is a fourth fam-ily�

but with the chiralities of the representationsinter-changed.; BecauseI3

�L�� vanishes,� LH mixing actsto reducethe

magnitudeof g L�bO ,H andso tendsto makethepredictionfor Rb

Oworse} than in the SM. For sufficiently large RH mixingangles,� however,this tendencymaybereversed.As wasdis-cussed; immediatelybelow Eq. � 9� � ,H since I

Ü3�

R

�is�

negativeacomparatively; large mixing angle of s· R

Í2Ø � 1/3 is neededtosufficiently6 increaseRb

O . Such a large RH mixing angle isphenomenologicallyh permittedby all off-resonancedetermi-nations� of g R

bO

19! . In fact, the b�

-quark production crosssection6 andasymmetry,asmeasuredin the " -Z interference

region # 21,22$ ,H cannotdistinguishbetweenthe two valuess· RÍ2Ø % 0

Vand 4s· w¸ /3,

Swhich yield exactly the same rates.10

Hence&

this kind of modelcansolvethe R�

bO problem,h though

perhapsh not in the most aestheticallypleasingway. As isshown6 in Fig. 2, the allowedrangeof mixing anglesis lim-ited to a narrowstrip in the s· L

�2Ø -s· RÍ2Ø plane.h

Fourth'

family. AM

fourth family ( model� 4) cannot; resolveR�

bO via� tree-leveleffectsbecausethe new B

Ú *quark� hasthe

same6 isospinassignmentsastheSM b�

quark,� andsotheydonot mix in the neutral current.11 Two other possibilities+models4 , and� 4-�. yield/ the sameI3

�L�

,RÍ0

assignments� as thefourthJ

family model, and are similarly unsuccessfulin ex-plainingh Rb

O since6 they do not modify the b�

quark� neutralcurrent; couplings.

Vector doublets. ThereI

aretwo possibilitieswhich permita� Q× 1�2 1

3Ï quark� to transformas a weak isodoublet,and in

both1

casesmixing with the SM b�

is allowed. They can belabeledby the different hyperchargevalue using the usualconvention; Q

× 3IÜ

3� 4 Y .

WithÍ

the straightforwardchoice IÜ

3�

L

5�6IÜ

3�

R

798:1/2 ; model�

5� <

,H we haveY L=�> Y R?�@ 1/6.This typeof modelis discussedinRef. A 13B ,H wheretheisopartnerof theB C is

�a toplike quarkT

W D

9EA Q FHG 2/3

�vector singlet can howeverbe usedto reduceR

ÁcIJ

10,12,14K , provided that stepsare taken,as suggestedin Sec. IIabove,to avoid the resultingpreferencefor an unacceptablylargevaluefor L s� (� M¡ Z

¤ ).

10The current90% C.L. upperbounds¨ R2� M

0.010 N 20O holdsin thesmall mixing angleregions¨ R

P2 Q 1/3.11Thesemodelshavethe further difficulty that, exceptin certain

cornersof parameterspaceR 23S ,� they producetoo largea contribu-tion to the oblique parameters,S and T,� to be consistentwith thedata.

FIG. 2. The experimentallyallowedmixing anglesfor a mirrorfamily. Thethick line coverstheentireareaof valuesfor s¨ L

� and� sRP

which areneededto agreewith theexperimentalvaluefor RÁ

b� to the

2T levelU

or better.Thethin line representstheone-parameterfamilyof mixing angleswhich reproducethe SM prediction.Notice thatthe small-mixing solution, which passesthrough s¨ L

� V sRP W 0,

&is

ruled out sinceILXZY 0&

implies that any LH mixing will reduce g Lb�

andthusincreasesthe discrepancywith experiment.

54 4281RÁ

b� AND NEW PHYSICS:A COMPREHENSIVEANALYSIS

having charge [ 23Ï . Sincetheseare the samechargeassign-

mentsasfor thestandardLH b�

quark,� this leadsto no mixingin�

the neutralcurrentamongstthe LH fields, and thereforeonly0 the right-handedmixing angle s· R

Í is�

relevant for R�

bO .

Since.

IÜ

3�

R

\is�

negativea comparativelylargemixing angleofs· R

2Ø ]

1/3 is neededto sufficiently increaseR�

bO ,H in much the

same6 way as we found for the mirror-family scenariodis-cussed; above.The requiredmixing anglethat gives the ex-perimentalh value,Rb

O ^ 0.2219V _

0.0017,V

is

s· RÍ2Ø ` 0.367

V a0.014qb 0.013q

. c 15dThe otherway to fit a Q

× e�f1/3 quark into a vectordou-

blet1

correspondsto I3�

L�g�h I3

�RÍi9jk 1/2 l model 8m and� so

Y n�o�p 5/6� q

10r . The partnerof the BÚ s

in�

the doubletis thenan� exoticquark,R

�,H havingQ

× t�u4/3.£

HereIÜ

3�

L

vhas¥

thewrongsign6 for satisfyingEq. w 9� x and� somixing decreases

¡the2

mag-nitude of g L

�bO . On the other hand, I3�

RÍy has the right sign to

increase�

g RbO. Whetherthis type of modelcanwork therefore

dependsK

on which of thetwo competingeffectsin RbO wins.} It

is easyto seethat in this model the M 21 entry  in the B-B zmass� matrix Eq. { 10| vanishes,� which as discussedaboveresults8 in a suppressionof s· L quadratic� in the largemass,butonly0 a linear suppressionfor s· R

Í . Hences· L� becomes1

negli-gible} in the largem^ b

O ~ limit, leavings· R as� thedominantmix-ing�

angle in R�

bO . The mixing angle which reproducesthe

experimental  valuefor R�

bO then2

is

s· RÍ2Ø � 0.059

V �0.015q� 0.013q

. � 16�However,&

in order to accountfor sucha large value of themixing anglein anaturalway, theb

� �cannot; bemuchheavier

than2 �

100 GeV.Similarly.

to the Y ����� 5/6�

vectordoubletcase,models3and� 9 alsoprovidea solutionthroughRH mixings. In model3,F

the subdominantcompetingeffect of s· L� is further sup-

pressedh by a smallerI3�

L�� ,H while in model9 theeffectof s· R

Í isenhanced  by I3

�R

�9��1, andhencea mixing anglea factorof

4 smallerthat in Eq. � 16� is sufficient to explainRbO .

Vector triplets. Thereare threepossibilitiesfor placingavector� B � quark� in an isotriplet representation: I3

�L���� I3

�R���� 1,0,� 1. The last does not allow for b

�mixing, if only

Higgs&

doubletsand singlets are present,and for our pur-poses,h I

Ü3�

L

���I3�

R

�9�0V �

model 1 ��� is equivalentto the vectorsinglet6 case already discussed.Only the assignmentI

Ü3�

L

�  I3

�RÍ¡9¢£ 1 ¤ model 10¥ allows¦ for a resolution of the Rb

§problem,¨ and it was proposedin Ref. © 12ª . If B

Ú «is¬

thelowest-isospin­

memberof the triplet thereis an exotic quarkof® chargeQ

¯ °�±5/3²

in the model.Again in the limit of largeb³ ´

massonecombinationof mixing anglesµ in this cases¶ R · isnegligible,¸ due to the vanishingof M

¹12 in¬

Eq. º 10» . As aresult,¼ s¶ L

½ plays¨ themain role in R¾

b§ . Agreementwith experi-

mentrequires

s¶ L½2¿ À 0.0127

Á Â0.0034.Á Ã

17ÄSinceÅ

the resultingchangeto gÆ L½b§ is so small, sucha slight

mixing anglewould haveescapeddetectionin all other ex-periments¨ to date.

SimilarlyÅ

to this case,models11Ç

and¦ 12Ç

also¦ provide asolutionÈ throughLH mixings. In model11 the

Éunwantedef-

fectsof s¶ RÊ are¦ further suppressed,while for model12 a L¦ H

mixing somewhatsmaller than in Eq. Ë 17Ì is sufficient toexplainÍ the data.

OurÎ

analysisof tree level effectsshowsthat both ZÏ

mix-Ðing and b

³mixing can resolvethe Rb

§ discrepancy.Ñ

b³

-quarkmixing solutionssatisfyingthetwo assumptionsthat Ò i Ó there

Éare¦ no new Higgs representationsbeyondsingletsand dou-blets,Ô

and Õ ii¬ Ö only® mixing with a singleBÚ ×

is¬

relevant,havebeenÔ

completely classified.The list of the exotic new B ØquarksÙ with the right electroweakquantumnumbersis givenin¬

TableII. Solutionswith smalls¶ R and¦ s¶ L mixingÐ anglesarepossible¨ when the B

ÚRÛ is¬

the memberwith highestIÜ

3Ý

R

Þin¬

anisodoublet¬

or isotriplet,or whenBÚ

L½ ß is thememberwith low-

estÍ I3Ý

L½à in an isotriplet or isoquartet.In all thesecases,new

quarksÙ with exotic electric chargesare also present.Someother® possiblesolutionscorrespondto I

Ü3Ý

R

á9â0Á

andareduetomixing amongsttheRH b

³quarksÙ involving ratherlargemix-

ing angles,while for I3Ý

L½ã�äå 1/2 we find anothersolution

requiring¼ even larger LH mixing. It is intriguing that suchlarge mixing anglesare consistentwith all other b

³-quark

phenomenology.¨ We havenot attemptedto classify modelsin¬

which mixing with new stateswith very large valuesofIÜ

3Ý

LR

æcanç ariseasa resultof bootstrappingthroughsomein-

termediateÉ

B è mixing. Under special circumstances,theycouldç allow for additionalsolutions.

Foré

someof the modelsconsidered,the contributionstotheÉ

obliqueparameterscould be problematic,yielding addi-tionalÉ

constraints.However,for the particularclassof vec-torlikeÉ

modelsê whichë includestwo of thesmallmixing anglesolutionsÈ ì loop

­effects are sufficiently small to remain

acceptable.¦ 12 This is because,unlike the top quark whichbelongsÔ

to a chiral multiplet, vectorlikeheavyb³ í

quarksÙ tendtoÉ

decouplein the limit that their massesget large.Introduc-ing¬

mixing with other fermions doesproducenonzeroob-lique corrections,but theseremain small enoughto haveevadedÍ detection.Exceptionsto this statementare modelsinvolving¬

a largenumberof new fields, like entirenew gen-erations,Í sincethesetend to accumulatelarge contributionstoÉ

Sî

and¦ T.

IV. ONE-LOOP EFFECTS: tï-QUARK MIXING

Weð

now turn to the modificationsto the Zbb³

couplingsçwhichë canariseat oneloop. Recall that this option canonlyexplainÍ R

¾b§ if¬

the LH b³

-quarkcoupling,gÆ Lb§,ñ receivesa nega-

tiveÉ

correctioncomparablein size to the SM mò tó -dependentcontributions.ç As wasarguedin Sec.II, it is theLH couplingweë areinterestedin becausea loop-levelchangein gÆ R

b§

is¬

toosmallÈ to fix thediscrepancybetweentheSM andexperiment.

The fact that the Rb§ problem¨ could be explainedif the

mò tó -dependentone-loopcontributionsof the SM wereabsentnaturally¸ leadsto the idea that perhapsthe tô -quark couplesdifferentlyÑ

to the b³

-quarkthanis supposedin the SM. If thetô quarkÙ mixessignificantlywith a new tô õ quarkÙ onemight beable¦ to significantly reducethe relevantcontributionsbelowtheirÉ

SM values.In this sectionwe show that it is at best

12Vectorlike modelshavethe additionaladvantageof beingauto-matically anomalyfree.

4282 54BAMERT,ö

BURGESS,CLINE, LONDON, AND NARDI

possible¨ to reducethe discrepancyto ÷ 2ø in modelsof thistype,É

andso they cannotclaim to completelyexplainthe Rb§

data.Ñ

OurÎ

surveyof tô -quarkmixing is organizedasfollows. Wefirst describethe frameworkof modelswithin which we sys-tematicallyÉ

search,andwe identify all of the possibleexotictô -quarkquantumnumberswhich canpotentiallywork. ThisstudyÈ is carriedout much in the spirit of the analysisof b

³mixing presentedin Sec.III. We thendescribethepossibletô ùloop contributionsto the neutral-currentb

³couplings.ç Since

thisÉ

calculation is very similar to computing themò tó -dependenteffectswithin the SM, we briefly review thelatter. Besidesproviding a usefulcheckon our final expres-sions,È we find that the SM calculationalso hasseveralles-sonsÈ for the moregeneraltô -quarkmixing models.

A. Enumerating the models

Inú

this section we identify a broad class of models inwhichë the SM top quarkmixeswith otherexotic top-quark-like fermions.As in theprevioussectionconcerningb

³-quark

mixing, we denotethe electroweakeigenstatesby capitals,Tû ü

,ñ and the masseigenstatesby lower-caseletters, tý i. Toavoid¦ confusion,quantitieswhich specificallyrefer to the b

³sectorÈ will be labeledwith thesuperscriptB. By definition,aT þ quarkÙ musthaveelectricchargeQ

¯ ÿ2/3, but may in prin-

cipleç have arbitrary weak isospin RL½

,R���

(�IL½

,RÊ� ,ñ I3

ÝL½

,R�)�. Fol-

lowing closely the discussionin the previous section,wemakethreeassumptionswhich allow for a drasticsimplifi-cationç in the analysis,without muchlossof generality.�

i¬ �

First,é

the usualTû

quarkÙ is only allowedto mix with asingleÈ T quarkÙ at a time, producingthe masseigenstatestýand¦ tý .�

ii¬ �

Second,Å

for the Higgs-bosonrepresentations,we as-sumeÈ only one doublet and singlets. Additional doubletswouldë complicatethe analysisof the radiativecorrectionsina¦ model-dependentway dueto the extradiagramsinvolvingchargedç Higgs bosons.

iii¬ �

Finally,é

certainTû �

-quarkrepresentationsalsocontainnew¸ B

Ú �quarks.Ù We denotetheB

ÚL� and¦ B

ÚR� as¦ ‘‘exotic’’ when-

everÍ theyhavenonstandardweakisospinassignments,that isI3Ý

L½� B� ��� 1

2or® I3Ý

R� B� � 0

Á. As we havealreadydiscussed,for exotic

BÚ �

quarksÙ b³

-b³ �

mixingÐ will modify the b³

neutral-current¸couplingsç at tree level, overwhelmingthe loop-suppressedtý -tý � mixing effectsin Rb

§ . We thereforecarry out our analy-sisÈ underthe requirementthatany b

³-b³ �

mixingÐ affectingtheb³

neutral-current¸ couplingsbe absent.OurÎ

purposeis now to examineall of the alternativeswhichë canarisesubjectto thesethreeassumptions.Accord-ing¬

to � i¬ � ,ñ the Tû

-Tû

massÐ matriceswe considerare2! 2,"

andcanç be written in the generalform

#T T $&% L

½ M¹

11

M 21¿ M

¹12

M 22¿ T

ûT '

R

. ( 18)Due*

to our restriction + ii¬ , on® the Higgs sector,certainele-mentsof this massmatrix are nonzeroonly for particularvalues- of the T . weakë isospin.Moreover,wheneverTR

Ê / be-Ô

longs to a multiplet which also contains a Q¯ 021

1/3 BRÊ 3

quark,Ù the M¹

12B and¦ M

¹12 entriesÍ of the B

Ú-BÚ 4

and¦ Tû

-Tû 5

massÐmatricesarethe same.In thosecasesin which the B 6 quarkÙ

is exotic, assumption7 iii 8 thenÉ

forcesus to set M 129 0.Á

Incontrast,ç the M 21 entriesÍ are unrelated—forexample, thechoiceç M

¹21B : 0

Áis alwayspossibleevenif M

¹21; 0

Áfor the T

ûand¦ T

û <quarks.Ù

Inú

order to selectthoserepresentations,R¾

L,R

=,ñ which can

mix with the SM T quark,Ù we requirethe following condi-tionsÉ

to be satisfied.>1? Inú

order to ensurea largemassfor the tý @ ,ñ we requireM 22¿ A 0.

ÁAnalogouslyto Eqs. B 11C and¦ D 12E ,ñ this implies

FIÜ

LGIH IÜ

RJLKNM 0,Á 1

2" O 19P

and¦QIÜ

3Ý

L

R2SIÜ

3Ý

R

TVUNW0,Á 1

2" . X 20

" YZ2[ To ensurea nonvanishingtý -tý \ mixing we requireat

leastoneof the two off-diagonalentries,M 12 or® M 21¿ , tñ ob e

nonvanishing.¸ This translatesinto the following conditionson® R

¾L] and¦ R

¾R :

RL½ _I` RH

� a RRÊ bdc 0,0

Á e,ñ 1

2,ñ f 1

2,ñ g 21h

or®

RRÊ ikj RH

� l RL½ mdn 0,0

Á o,ñ 1

2,ñ p 1

2,ñ q 1,0r ,ñ s 1,t 1 u . v 22w

x3y z

Wheneverð

R¾

R{ containsç a Q¯ |2}

1/3 quark, and eitherBÚ

L~ or® BÚ

R� have�

nonstandardisospinassignments,we requireM 12� 0.

ÁThis ensuresthat at tree level the neutralcurrentb

³couplingsç are identical to thoseof the SM. Clearly, in thecasesç in which theparticularR

¾L� representation¼ impliesavan-

ishing M 21¿ element,Í imposing the condition M 12� 0

Ácom-

pletely¨ removesall tý -tý � mixing.ÐWeð

now may enumerateall the possibilities.From Eqs.�19� – � 22� ,ñ it is apparentthat as in the B � caseç the only al-

lowed representationsmusthaveIRÊ �k� 0,

Á 12,1ñ and IL

½ ��� 0,Á 1

2,1,ñ 3�2.

Consider�

first IL½ �I� 1 or 3

�2. In this case,from Eq. � 21� ,ñ

M¹

21� 0.Á

Thus we need M¹

12� 0Á

if there is to be any tý -tý �mixing.Ð The four possibilitiesfor R

¾R� are¦ shownin Eq. � 22

" �.

OfÎ

these,RRÊ �k� (

�0,0) is not allowed since Eq. � 19� is not

satisfied.È In addition,RR�¡  (� 1

2¢ ,ñ 1

2¢ )� and £ 1,0¤ both

Ôcontainexotic

BÚ ¥

quarksÙ ¦ IÜ 3Ý

R

§ B ¨�© 12¢ or® ª 1« and¦ so M

¹12 is¬

forcedto vanish,leadingto no tý -tý ¬ mixing. This leavesRR

Ê ­k® (�1,1) asa possi-

bility,Ô

sincethe BRÊ ¯ is not exotic (I3

ÝRÊ° B� ± 0

Á). If we chooseRL

½ ²suchÈ that I

Ü3Ý

L

³ B ´�µ 12¢ ,ñ then both B

ÚL¶ and¦ B

ÚR· are¦ SM-like, and

b³

-b³ ¸

mixing is not prohibitedsinceit doesnot affect the b³

neutralcurrentcouplings.Thus the combinationRL½ ¹Iº (

� 3�2,ñ 1

2),�

R¾

R»k¼ (�1,1) is allowed.

Next½

considerIÜ

L¾�¿ 0 oÁ

r 12¢ . Here,regardlessof thevalueof

I3Ý

L½À ,ñ M 21

¿ canç benonzero.ThusanyRRÊ Á representationwhich

satisfiesÈ Eqs. Â 19Ã and¦ Ä 20" Å

is¬

permitted.It is straight-forwardtoÉ

showthat thereare11 possibilities.TheÆ

list of theallowedvaluesof IÜ

3Ý

L

Çand¦ I

Ü3Ý

R

Èwhichë under

our® assumptionslead to tý -tý É mixingÐ is shownin Table III.ThereÆ

are twelve possiblecombinations,including fourth-

54 4283RÊ

bË AND NEW PHYSICS:A COMPREHENSIVEANALYSIS

generationÌ fermions, vector singlets, vector doublets,andmirror fermions.Not all of thesepossibilitiesare anomalyfree,Í

but asalreadynotedonecouldalwayscancelanomaliesbyÔ

adding other exotic fermions which give no additionaleffectsÍ in Rb

§ .It is useful to group the twelve possibilities into three

differentÑ

classes,accordingto the particularconstraints,ontheÉ

form of the Tû

-Tû Î

massÐ matrix in Eq. Ï 18Ð .Thefirst two entriesin TableIII, which we haveassigned

toÉ

groupA,ñ correspondto thespecialcasein which the BL½

,RÊ

and¦ BL½

,RÊÑ have the samethird componentof weak isospin,

hence�

leaving the b³

neutral¸ current unaffectedby mixing.BecauseÒ

both BÚ

LÓ and¦ BÚ

RÔ appear¦ in the samemultipletswithTL½ Õ and¦ TR

Ê Ö ,ñ two elementsof the B-quarkandT-quarkmassmatricesÐ areequal:

M¹

12× M¹

12B ,ñ M

¹22Ø M

¹22B . Ù 23

" ÚAsÛ

we will see,this conditionis importantsinceit implies arelation¼ betweenthemixingsandthe mÜ tÝ ,ñ mÜ tÝ Þ massÐ eigenval-ues.ß Althoughoutsidethesubjectof this paper,it is notewor-thyÉ

that for thesemodelsthe simultaneouspresenceof bothb³

-b³ à

and¦ tý -tý á RHâ

mixing generatesnew effects in thechargedç currents: right-handedWtb

³chargedç currentsget

induced,proportionalto the productof the T and¦ B quarkÙmixings s¶ Rs¶ R

B�. Comparedto the modificationsin the neutral

currentsç andin the LH chargedcurrents,theseeffectsareofhigher�

order in the mixing angles ã 18,19ä and,¦ most impor-tantly,É

theycanonly changetheRH b³

coupling.ç But asnotedabove,¦ gÆ R

b§

is far too small to accountfor the measuredRb§

value- using loop effectsof this kind. Thereforethe mixing-inducedRH currentsallowedin modelsA1 and¦ A2

¿ are¦ inef-fective for fixing theRb

§ discrepancy,Ñ

andwill not beconsid-eredÍ in the remainderof this paper.

Foré

the modelsin groupBÚ

,ñ the condition

M¹

12å 0Á æ

24" ç

holds.�

In the four casescorrespondingto R¾

Rèké (�1,0) ê modelsÐ

B1,ñ B3Ý ë and¦ RR

Ê ì¡í (�1/2,1/2) î modelsB2

¿ ,ñ B6ï ð ,ñ an exotic BR

Ê ñquarkÙ is presentin the sameTR

Ê ò multiplet. HenceM 12 hastobeÔ

set to zero in order to forbid the unwantedtree-levelb³

mixing effects.In theotherthreecasesbelongingto groupB,ñTRÊ ó correspondsç to the lowest componentof nontrivial mul-

tiplets:É

RRÊ ô¡õ (1

�, ö 1) ÷ model B4

ø ù and¦ RRÊ úkû (

�1/2,ü 1/2)ý

modelsÐ BÚ

5þ ,ñ BÚ

7ÿ � . For thesevaluesof I

Ü3Ý

R

�,ñ M¹

12� 0Á

is auto-matically ensured,dueto our restrictionto Higgs singletsordoublets.Ñ

Furthermore,theserepresentationsdo not containaBRÊ � quark,Ù and no BL

½ � quarkÙ appearsin the correspondingR¾

L� . Thereis thereforeno b³

-b³ �

mixing.ÐWeð

should also remark that in model B3Ý no BL

½ � quarkÙappears¦ in RL

½ � . However,a BL½ is neededasthehelicity part-

ner¸ of the BÚ

R present¨ in R¾

R� � (�1,0). Becauseof our restric-

tionÉ

on theallowedHiggsrepresentations,BÚ

L� mustÐ belongtoRL½ ��� (

�1,0) or RL

½ � (� 1/2,1/2), which in turn containa new TL½ ��

Tû

L� . While the first choice correspondsto a type of Tû

L

�mixing which we havealreadyexcludedfrom our analysis,theÉ

secondchoiceis allowedandcorrespondsto model B1.Followingé

assumption� i¬ � ,ñ evenin this casewe neglectpos-sibleÈ T

ûL

�mixingsÐ of type B

Ú1,ñ whenanalyzingB

Ú3Ý .

Finally, the remainingthreemodelsconstitutegroup C,ñcorrespondingç to RR

Ê � � (�0,0). In In this group,TR

Ê � is an isos-inglet,¬

asis theSM Tû

R ,ñ implying thatonly LH tý -tý � mixingÐ isrelevant.¼ For C2 and¦ C3

Ý ,ñ R¾

L� doesÑ

not containa BÚ

L� ,ñ while forC1 the

ÉBL½ is not exotic. Hencein all the threecasesthe b

³neutral-current¸ couplingsareunchangedrelative to the SM,and¦ we neednot worry abouttree-levelb

³-mixing effects.

B. t!-quark loops within the standard model

Beforeexaminingthe effect of tý -tý " mixing on the radia-tiveÉ

correctionto Zbb³

,ñ we first review the SM computation.Weð

follow the notationand calculationof Bernabe´u,ß Pich,and¦ Santamarı´a¦ # 11$&% BPS

Ò '. Thecorrectionsaredueto the10

diagramsÑ

of Fig. 3. All diagramsarecalculatedin ’ tý Hooft–Feynmangauge,andwe neglecttheb

³-quarkmassaswell as

theÉ

difference ( V tbÝ ) 2¿ * 1.Due*

to theneglectof theb³

-quarkmass,anddueto theLHcharacterç of the charged-currentcouplings,the tý -quarkcon-tributionÉ

to theZbb³

vertex- correctionpreserveshelicity. Fol-lowing­

BPS we write the helicity-preservingpart of theZÏ +

bb³ ¯ scatteringÈ amplitudeas

TABLE,

III. Modelsandchargeassignments.Valuesof the weakisospinof TL- . andTR

/ 0 which,1 underthe only restrictionsof singletanddoubletHiggs representations,leadto nonzerot2 -t 3 neutralcurrentmixing. The ‘‘Model’’ columnlabelsthe morefamiliar possibilitiesforthe T 4 quarks: vectorsinglets,mirror fermions,fourth family, andvectordoublets.The othermodelsaremoreexotic.

IL5 I36

L

7IR8 I3

6R

9Model:

Group

3/2 ; 1/2 1 < 1 A1

1/2 = 1/2 1 > 1 A?

2@

0 BA

1

1/2 B 1/2 Vector doublet C I D B2

0 0 Fourthfamily C1

1/2 E 1/2 1 0 BA

3FG 1 B4

1/2 H 1/2 Vector doublet I III J B5K

0 0 C2@

0 0 1/2 L 1/2 Mirror fermions BA

6MN 1/2 B7O

0 0 Vector singlet C3F

4284 54BAMERT,P

BURGESS,CLINE, LONDON, AND NARDI

QSRUT eVs¶ wW cX wW b

³ Y pZ 1 ,ñ [ 1 \^]`_ b³ a

pZ 2 ,ñ b 2 cedgfih qj ,ñ kml ,ñ n 25o

withëp`qsrut

0v wsxzy|{`} ,ñ ~|�`�s���

2 ����� LI � s¶ ,ñ r � ,ñ � 26�

whereë ����� represents¼ theloop-inducedcorrectionto theZbÏ ¯b³

vertex.- I(�s¶ ,ñ r)

�is a dimensionlessandLorentz-invariantform

factor which depends,a� priori,ñ on the threeindependentra-tios:É

r� � mÜ tÝ2¿ /� M¹ W2¿

,ñ s¶ � M¹

Z2¿/�M¹

W2¿

,ñ andqj 2¿/�M¹

W2¿

. For applicationsat¦ the Z

Ïresonance¼ only two of theseareindependentdueto

theÉ

mass-shellconditionqj 2 � M Z�2. Moreover,for an on-shell

Z,ñ nonresonantbox-diagramcontributionsto eV � eV ��� bb³ ¯ are¦

unimportant,ß and IÜ(�s¶ ,ñ r� )� can be treated as an effectively

gauge-invariantÌ quantity.Thecontributionsdueto the tý quarkÙ maybeisolatedfrom

other® radiativecorrectionsby keepingonly the r-dependentpart¨ of I

Ü(�s¶ ,ñ r� )� . BPSthereforedefinethe difference

F� �

s¶ ,ñ r �g� I   s¶ ,ñ r ¡g¢ I £ s¶ ,0ñ ¤ . ¥ 27¦Given§

this function, the mÜ tÝ dependenceÑ

of the width Z ¨ bb³ ¯

is obtainedusing

©b§SMª «

r� ¬g­u® b§SMª ¯

r� ° 0Á ±

1 ²´³µ gÆ Lb§

¶gÆ L

b§ ·

2¿ ¸º¹

gÆ Rb§ »

2¿ F� SMª ¼

s¶ ,ñ r� ½

¾VP¿ À

s¶ ,ñ r� Á . Â 28" Ã

In this lastequationVP (s¶ ,ñ r)�

denotesthemÜ tÝ -dependentcon-tributionsÉ

which enterÄ b§ throughÉ

the loop correctionsto thegauge-bosonÌ vacuumpolarizations.

TheÆ

functionF� SMª

(�s¶ ,ñ r� )� is straightforwardto compute.Al-

thoughÉ

the resultingexpressionsare somewhatobscure,thespecialÈ cases¶ Å 0

Árevealssomeinterestingfeatureswhich are

also¦ presentin our new-physicscalculations,andsowe showtheÉ

s¶ Æ 0Á

limit explicitly here.For s¶ Ç 0,Á

an evaluationof thegraphsÌ of Fig. 3 givesthe expressions

F1È aÉ ÊÌËÎÍ 1

2"

s¶ wW2gÆ L½tÝ

2

r� Ï r� Ð 2" Ñ

Òr Ó 1 Ô 2 lnr Õ r�

r Ö 1 × gÆ RÊtÝ r�Ø

r Ù 1 Ú 2 ln r

Û r

r� Ü 1,ñ Ý 29

" Þ

F1ß bà áÌâ 3y

cX wW24s¶ wW2¿

r2ãr ä 1 å 2

¿ ln r æ r

r ç 1,ñ è 30

y é

F1ê cë ìÌí 1î dï ðÌñ 1

121 ò 3

y2"

s¶ wW2¿r� 2ó

r ô 1 õ 2¿ ln r ö r�

r ÷ 1,ñ ø 31y ù

F� 1ú eû üÌý 1þ f ÿ�� r�

2" r��

r� � 1 � 2 ln­

r� � 1

r� � 1,ñ � 32

y �

F� 2 aÉ ��� r

4s¶ wW2gÆ R

tÝ2" ��� r � r � 2 ��

r� � 1 � 2 ln­

r� � 2r � 1

r� � 1

�gÆ L½tÝ r�

r � 1 � 2¿ ln r � r

r 1,ñ ! 33

y "

F� 2# bà $�%'& 1

8( 1 ) 1

2s¶ wW2 r� *�+ r� 2¿,

r� - 1 . 2 ln­

r� / 1

r� 0 1,ñ 1

34y 2

F2¿ 3

cë 4�5 2¿ 6

dï 7�8 1

241 9 3

y2s¶ wW2¿ r :�; r2<

r = 1 > 2¿ ln r ? 1

r @ 1,ñ A35y B

withëC�D 2

nE F 4G�HJI

ln K M W2 /4� LNM 2

¿ OQP 3y2

,ñ R 36y S

whereë nE is¬

the spacetimedimensionarising in dimensionalregularization,¼ and

gÆ L½tÝ T 1

2 U 2"3y s¶ wW2¿ ,ñ gÆ R

ÊtÝ V'W 2"3y s¶ wW2¿ . X 37

y YTheÆ

picture becomesmuch simpler after summingthe dia-gramsÌ to obtainthe total SM contribution:

F� SMª Z

s¶ [ 0,Á

r� \Q]_^i ` 1a aÉ b

2¿ c

dd e

F� i f 1

8(

s¶ wW2¿r2¿

r� g 1 h 6i r

r� j 1

k r l 3y r m 2 nor p 1 q 2

¿ ln r . r 38y s

FIG. 3. The Feynmandiagramsthrough which the top quarkcontributesto the Zb

t ¯bu

vertexv within the standardmodel.

54 4285Rw

bx AND NEW PHYSICS:A COMPREHENSIVEANALYSIS

Therearetwo pointsof interestin this sum.First, it is ultra-violet- finite sinceall of thedivergencesy 1/z nE { 4| havecan-celled.ç This is requiredon generalgroundssincetherecanbeno¸ r� -dependentdivergencesin I

Ü SMª

(�s¶ ,ñ r� )� , and so thesemust

cancelç in FSMª

(�s¶ ,ñ r)

�. A similar cancellationalsooccurswhen

new physics is included, provided that it respects theSUÅ

L } 2" ~�� U�

Y � 1� gaugeÌ symmetryandthat thecompletesetofnew¸ contributionsis carefully included.

The secondinterestingfeatureof Eq. � 38y �

lies in its de-pendence¨ on the weakmixing angle,s¶ wW . Eachof the contri-butionsÔ

listed in Eqs. � 29" �

– � 35y �

has�

the formF� i � (

�x� i � y� is¶ wW2 )/

�s¶ wW2 ; however,all of the terms involving y� i

havecancelledin thesum,Eq. � 38y �

. This very generalresultalso¦ appliesto all of the new-physicsmodelswe considerinsubsequentÈ sections.As will be provedin Sec.V, the can-cellationç is guaranteedby electromagneticgaugeinvariance,becauseÔ

the termssubleadingin s¶ wW2¿ are¦ proportionalto theelectromagneticV b-quarkvertexat qj 2

¿ �0,Á

which mustvanish.ThisÆ

givesa powerful checkon all of our calculations.Rather than using completeexpressionsfor F(

�s¶ ,ñ r),

�we

find it moreinstructiveto quoteour resultsin the limit r � 1,whereë powersof 1/r� and¦ s¶ /� r� may� be neglected.We do thesameÈ for the ratio of massesof other new particlesto M

¹W2

whenë thesearise in later sections.Besidespermitting com-pact¨ formulas,this approximationalsogivesnumericallyac-curateç expressionsfor mostof the models’parameterrange,as¦ is alreadytrue for the SM, eventhoughr� in

¬this caseis

only� � 4. In the large-r limit FSMª

(�s¶ ,ñ r)

�becomes

F� SMª �

r� �Q� 1

8(

s¶ wW2 r� � 3y � s¶

6i � 1 � 2

"s¶ wW2 � ln

­r� �'����� ,ñ �

39y  

whereë the ellipsis denotesterms which are finite as r ¡�¢ .SeveralÅ

pointsarenoteworthyin this expression.£1¤ TheÆ

s¶ -dependenttermappearingin Eq. ¥ 39y ¦

is¬

numeri-callyç very small, changingthe coefficientof ln r from 3 to2.88.This typeof s¶ dependence

§is of evenlessinterestwhen

weë considernew physics,sinceour goal is then to examinewhetherë the new physics can explain the discrepancybe-tweenÉ

theoryandexperimentin Rb§ . That is, we want to see

if the radiativecorrectionscanhavethe right sign andmag-nitude¸ to change b

§ byÔ

the correctamount.For thesepur-poses,¨ so long as the inclusion of qj 2-dependenttermsonlychangesç thenumericalanalysisby factors © 25% ª as¦ opposedtoÉ

changingits overall sign« theyÉ

may be neglected.¬2" ­

TheÆ

above-mentionedcancellationof the terms pro-portional¨ to s¶ wW2 when® s¶ ¯ 0

Áno longer occursoncethe s¶ de-

§pendence¨ is included.This is as expectedsincethe electro-magnetic Ward identity only enforcesthe cancellationatqj 2 ° 0,

Ácorrespondingto s¶ ± 0

Áin thepresentcase.Notice that

theÉ

leading term, proportional to r� , iñ s s¶ independent,¬

andbecauseÔ

of the cancellationit is completelyattributabletographÌ ² 2a³ of� Fig. 3. All of the other graphscancelin theleading´

term. Due to its intrinsic relationwith the cancella-tionÉ

of the s¶ wW2 -dependentterms,the fact that only onegraphis responsiblefor the leadingcontributionto µ gÆ L

½b§ still¶ holdsonce� new physics is included. This will prove useful foridentifying¬

which featuresof a givenmodelcontrol theover-all· sign of the new contributionto ¸ gÆ L

b¹.

º3y »

SinceÅ

the large-r limit correspondsto particlemasses¼in this casemÜ t

Ý ½ that¾

arelargecomparedto M W and· M Z ,¿ thisisÀ

the limit where the effective-Lagrangiananalysis de-scribed¶ in Sec.II directly applies.ThenthefunctionF

�canç be

interpretedastheeffectiveZbb³

couplingç generatedwhentheheavy particle is integratedout. Quantitatively, Á gÆ L

b¹

is re-lated´

to F�

byÂ

ÃgÆ L½b¹ Ä Å

2 Æ F. Ç 40ÈÉ4Ê Ë

TheÆ

vacuumpolarizationcontributionsto Ì b¹ of� Eq.Í

28Î Ï

haveÐ

a similar interpretationin the heavy-particlelimit.In this casethe removalof the heavyparticlescangenerateoblique� parameters,which also contribute to Ñ b

¹ . In theheavy-particleÐ

limit Eq. Ò 28Î Ó

therefore¾

reducesto the first ofEqs.Ô Õ

4Ê Ö

.

C. × gØ LÙbÚ in the t

!-quark mixing models

Weð

maynow computehow mixing in thetop-quarksectorcanÛ affecttheloop contributionsto theprocessZ Ü bb

Ý ¯. As inthe¾

SM analysis,we set mÜ b¹ Þ 0.

ÁIn addition, following the

discussion§

in the previoussubsection,we neglectthe sß de-§

pendenceà in all our expressions.We alsoignoreall vacuum-polarizationà effects,knowing that they essentiallycancelinRb¹ . Finally, in the CKM matrix, we set á V id âäãæå V is çäè 0,

éwhere® i

ê ëtý ,¿ tý ì . Thusthecharged-currentcouplingsof interest

to¾

us are describedby a 2í 2Î

mixing matrix, just as in theneutral-currentsector.In theabsenceof tý -tý î mixing this con-dition§

implies ï V tbÝ ðäñ 1.Foré

tý -tý ò mixing,� independentof the weak isospinof theTû ó

,¿ we write

TûT ô

L,R õö

L÷

,Rø tý

tý ùL,R

,¿ ú L÷ û cX L

÷ü sß L

sß L÷

cX L,¿

ýRø þ cX R

sß Rø ÿ sß R

cX Rø ,¿ � 41

� �where® cX L

÷ � cosÛ �L÷ ,¿ etc. The matrices� L

÷,Rø are· analogousto

the¾

bÝ

-bÝ �

mixing matricesdefinedin Eq. � 5� in our tree-levelanalysis· of b

Ýmixing.�

In

the presenceof tý -tý � mixing,� the diagonal neutral-currentÛ couplingsaremodified:

g� L,Ri �

a� � T,T � g� L,Ra� ���

L,Rai� � 2 � g� L,R

tÝ ,SM � g� L,Ri ,¿ � 42

� �

where® iê �

tý ,¿ tý � ,¿ andg� L,RtÝ ,SM are· theSM couplingsdefinedin Eq.�

37� �

. The new termsg� L÷

,Røi explicitly� read

g� LtÝ I

!3"

L

#%$ 1

2Î sß L

2,¿ g� RtÝ & I!

3"

R

'sß R

2 ,¿ ( 43� )

g� L÷ tÝ *�+ I3

"L÷,%- 1

2cX L÷2. ,¿ g� R

ø tÝ /�0 I3"

Rø1 cX Rø2. . 2 443

In

addition,wheneverthe Tû

L,R

4hasÐ

nonstandardisospinas-signments,¶ I

!3"

L

5%61/2 or I

!3"

R

7980é

, flavor-changing neutral-currentÛ : F.C.N.C.; couplingsÛ arealso induced:

4286 54BAMERT,P

BURGESS,CLINE, LONDON, AND NARDI

g� L÷

,Røi j < =

a� > T,T ? g� L÷

,Røa� @

L÷

,Røai� A

L÷

,Røa j� B

g� L÷

,Røij ,¿ C 45D

where® iê,¿ jE F tý ,¿ tý G ,¿ and i

ê HjE. Here,

g� L÷ ttÝ IKJ 1

2 L I3"

L÷M sß LcX L ,¿ g� R

ø ttÝ N�O I3"

RøP sß RcX R . Q 46R

EquationS T

41� U

determinesV

the effective tý and· tý W neutral-XcurrentÛ couplings Y Eqs.

S Z42� [

– \ 46� ]K^

. However, the charged-currentÛ couplingsdependon thematrix _a`%b L

† cLB . Hencewe

needto consideralso bÝ

mixing, since,as discussedin Sec.IV

A, in thosecasesin which the Bd e

quarkf is not exotic(�I!

3"

L

g B hji 1/2, I!

3"

R

k B l 0é m

,¿ we haveno reasonto require n LB o I

!pi.e., no b

Ý-bÝ q

mixingr . We thendefinethe2s 2 chargedcur-rentt mixing matrix

uavxwL† y

LB ,z { tbÝ | cX L

÷ cX LB } sß L

÷ sß LB ,z ~ tÝ � b� � sß L

÷ cX LB � cX L

÷ sß LB ,z �

47�which� trivially satisfies the orthogonality conditions��� † ��� †�a� I. In the absence of b

�-b� �

mixing, clearly�tbÝ � cX L

÷ ,z � tÝ � b� � s� L÷ . We also note that, by assumption,

whenever� �a���L÷ we� necessarilyhave I3

"L÷�%�j� 1/2 � so� that

I!

3"

L

� B� �j� 1/2  in¡

order to guaranteethat the Bd

L¢ is¡

not exotic.

From£

Eqs. ¤ 43¥ ¦

,z § 44¥ ¨

,z and © 46¥ ª

,z this implies that g� LtÝ « g� L

tÝ ¬­ g� L

® ttÝ ¯K° 0±

, that is, the mixing effects on the LH tý and² tý ³neutral-current´ couplingsvanish.

Theµ

Feynmanrules of relevancefor computingthe Zb¶ ¯b�

vertex· loop correctionsin the presenceof a mixing in thetop-quark¸

sectorcannow be easilywritten down:

Wt ib�

:igê¹»º tÝ i¼ b� ½¿¾À½ L

® ,z

Átý ib� :

igê

ÂM W

ÃtÝ i¼ b� mÜ i Ä L ,z

Zt¶

itý

i :igêcX wW Å¿ÆÈÇÊÉ gË L

tÝ ,SM ÌL® Í gË R

tÝ ,SM ÎRÏ ÐÒÑÔÓ tý LtÝ i¼ Õ

L® Ö gË R

tÝ i¼ ×RÏ ØÚÙ ,z

Zt tÛ Ü : igêcX wW Ý¿ÞÈß gË L

® ttÝ à�áL â gË R

Ï ttÝ ãåäR æ ,z ç 48è

where� é are² theunphysicalchargedscalars,andtÛ i ê tÛ ,z tÛ ë . Thevertices· listedin Eq. ì 48

¥ íreducet to theSM Feynmanrulesin

the¸

limit of no mixing.As pointedout at theendof Sec.IV A, in somegroupsof

modelsî equalitiescan be found betweensomeelementsofthe¸

Tï

-Tï ð

and² Bd

-Bd ñ

massî matrices.Thesehave importantconsequences.ò In particular,onceexpressedin termsof thephysicaló massesandmixing angles,theequalitiesof Eq. ô 23õöwhich� hold in the modelsof groupA

÷ øcanò be written

ù�úLMû

diagd ü

R† ý

aþ 2 ÿ���� LBMû

diagdB �

RB†�

aþ 2 ���� LB

aþ 2mÜ b� � cX R

B

�a� 1,2� ,z � 49�

where� Mû

diagd � diag[

VmÜ tÝ ,z mÜ tÝ � ]� , and we have used M

ûdiagdB���

i2� mÜ b� ��� recallt thatwe takemÜ b

� � 0± �

. Multiplying now on theleft�

by ( � LB†)�

1aþ and² summingover a� we� obtain

�! †M diagd "

Rφ #

12$ mÜ tÝ % tbÝ s� R & mÜ tÝ ')( tÝ * b� cX R + 0.± ,

50- .

For/

the modelsin groupBd

,z the vanishingof Mû

12 implies¡

nob�

mixing. Then 02143 L® ,z andEq. 5 496 still� holds in the limit7

tbÝ 8 cX L ,z 9 t

Ý : b� ; s� L . For themodelsin groupC no particularrelationt betweenmassesand mixing anglescan be derived.For/

example,it is clear that in the fourth family model C1,zEq.< =

50- >

doesV

not hold. However, for all thesemodelsI?

3@

R

AB 0±

. Hence,noting that all the gË R couplingsò in Eqs. C 43¥ D

,zE44¥ F

,z and G 46¥ H

are² proportional to I?

3@

R

I,z and defining r� JK mÜ tÝ L2 /

MM W

2 ,z squaringEq. N 50- O

yieldsP a relation which holdsforQ

all modelsin TableIII:

RtbÝ 2gË R

tÝr� SUT t

Ý Vb�2 gË R

tÝ W

r� XZY�[U\tbÝ ] tÝ ^ b� gË R

ttÝ _�`

rr� a . b 51c d

This relation is used extensivelyin the calculation whichfollows.Q

Howe

do we generalizethe SM radiativecorrectionto in-cludeò tf -tf g mixing?First notethat for eachof thediagramsinFig. 3, thereis alsoa diagramin which all the tf quarksh arereplacedt by tf i quarks.h Second,thereare two new diagramsjFig.k

4l dueV

to theFCNCcouplingof theZ¶

to¸

thetf and² tf m . Soto¸

generalizethe SM result to the caseof mixing, threethings¸

haveto be done: n i o multiply Eqs. p 29q – r 35s t

byu v

tbw2xfor the tf contributionò and y tw z b�2x for tf {}| with� r ~ r �)� ,z � ii � replacegË L,R

tw byu

themodifiedcouplingsin Eq. � 42¥ �

,z addingEqs. � 43¥ �

and² � 44� respectivelyfor tf and² tf � ,z and, � iii � includediagrams3s �

a² � and² 3� bu ��� Fig. 4� correspondingò to the FCNC couplings�Eqs.� �

45¥ �

and² � 46¥ ���

.A�

glanceat the Feynmanrules in Eq. � 48¥ �

shows� that inthe¸

first step � i¡ � ,z a correctionproportionalto gË L,Rtw ,SM ,z and in-

dependentV

of the gË L®

,R  couplings,ò is generated.This correc-

tion¸

is commonto all modelsin TableIII—it appearseveninthe¸

casein which the tf NC¡

couplingsarenot affected¢ fourthQ

family£ . In contrast,steps ¤ ii ¥ and² ¦ iii § generate¨ correctionswhich� differ for different models.It is useful to recasttheminto¡

two types,one proportionalto the LH neutral currentcouplingsò ©�ª�«

ib ¬ jb­ gË L ® ,z andthe otherproportionalto the RH

neutralcurrentcouplings ¯�°�± ib ² jb­ gË R

  ³ . The LH andRH cor-

FIG. 4. The additionalFeynmandiagramswhich are requiredfor models in which the t quark´ mixes with an exotic, heavy tµ ¶quark.

54 4287R·

b¸ AND NEW PHYSICS:A COMPREHENSIVEANALYSIS

rectionst vanishrespectivelyfor I¹

3º

L

»U¼�½1/2 andI

¹3º

R

¾À¿0±

, whenthe¸

correspondingneutral-currentcouplingsarenot affectedbyu

the mixing.InÁ

the presenceof mixing, the correctiondue to the dia-grams¨ of Fig. 3 involving internal tf quarksh becomes

Âi à 1Ä aÉ Å

2x Æ

dd Ç

F i ÈUÉtbw 2 Ê FSM

Ë Ìr ÍÏÎ F Ð gË L

®,R tw ,z r ÑÓÒ ,z Ô 52

c Õ

whereÖ F× SMË

(ØrÙ )Ú is given by Eq. Û 38

s ÜandÝ

F Þ gË L®

,R tw ,z r ßÏà 1

8á

s� wâ2x gË L® tw r 2 ã 4

¥r ä 1

ln r å gË R  tw r æUç 2

èrÙ é 5

cr ê 1

ë rÙ 2 ì 2è

rÙ í 4¥

îr ï 1 ð 2 ln r . ñ 53

c ò

Theµ

third step ó iii¡ ô gives¨ rise to a new contribution

F× 3º õ

aÉ öø÷ F× 3º ù

bú ûýüUþ

tbw ÿ

tw �

b� Fט � gË L,R

ttw � ,� r٠,� r٠��� . � 54c �

Evaluating�

diagrams3 aÝ andÝ 3� bu �� Fig.k

4� weÖ find

F3º �

aÉ ����� 1

s� wâ2x � tbw � tw � b� 1

2gË L® ttw � 1

r ��� r

r � 2r ��� 1

ln r ! rÙ 2

r " 1ln r # gË R

  ttw $�% rr & 1

r '�( r

rÙ )r *,+ 1

ln r -. r

rÙ / 1ln0

rÙ ,� 1 55c 2

F3º 3

bú 4�5 1

4¥

s� wâ2x 6 tbw 7 tw 8 b9 2gË L® ttw : rrÙ ;

r <�= r

rÙ >r ?,@ 1

ln r A�B rÙr C 1

ln r

D gË R  ttw EGF rr H IKJ 1 L 1

r M�N r

rÙ O 2r P�Q 1

ln r RS r2

xrÙ T 1

ln0

rÙ . U 56c V

Puttingall thecontributionstogether,for thegeneralcaseweÖ find

F WYXi Z 1[ aÉ \

3º ]

bú ^

F i _a`j­ b

1,2 c tw jd b92x e

FSMË f

r j­ gih F j gË L

k,R tw jd ,� r j

­ l�mnpo

tbw q tw r b9 F s gË L,Rttw t ,� r,� r u�v ,� w 57

c xwhereÖ tf j

­ y tf ,� tf z andÝ r j­ { r,� r | . We notethatdueto Eq. } 51

c ~allÝ

the�

divergenttermsproportionalto g� R � cancel� in the sum.Now,¡

the correction� � g� Lb9 �

(Ø �

/2M �

)ÚX�

corr� to�

theSM resultcanbe�

explicitly extractedfrom Eq. � 57c �

by�

meansof therelation�tbw 2 � 1 ��� tw � b92 .

Moreover,asanticipatedit is possibleto divide the vari-ous� contributionsto X

�corr� into�

threedifferent pieces: a uni-versal� correction,a correctiondueto LH mixing only, andacorrection� dueto the RH mixing. Hencewe write

Xcorr� � F � FSMË �

Xcorr�univ� �Xcorr�LH� �

Xcorr�RH�

,� � 58c �

whereÖXcorr�univ�  p¡

tw ¢ b92x £ FSMË ¤

r ¥�¦i§ FSMË ¨

r ©�ª ,� « 59c ¬

Xcorr�LH� ­p®

tbw 2x F ¯ g� Lk tw ,� r °i±p² tw ³ b92x F ´ g� L

k tw µ ,� r ¶�·i¸p¹ tbw º tw » b9 F ¼ g� L½ ttw ¾ ,� r,� r ¿�À ,�Á

60Â Ã

X�

corr�RH ÄpÅtbw 2Fט Æ gÇ R

tw ,� rÙ ÈiÉpÊ tw Ë b92 Fט Ì gÇ R

tw Í ,� rÙ Î�ÏiÐpÑ tbw Ò tw Ó b9 Fט Ô gÇ Rttw Õ ,� rÙ ,� rÙ Ö�× .Ø

61Â Ù

UsingÚ

theexplicit expressionsfor gÇ L,Rtw ,� gÇ L

½,RÛtw Ü ,� andgÇ L

½,RÛttw Ý asÝ

givenÞ in Eqs. ß 43à ,� á 44â ,� and ã 46ä above,Ý togetherwith rela-tion� å

51c æ

for the RH piece,theseread

Xcorrçunivè épêtw ë b92x fì

1corrç í

r,� r î�ï ,� ð 62 ñ

Xò

corrçLH óõô 1 ö 2÷

I¹

3º

L

øúù,ûtbw ü tw ý b9 sþ L

½ cÿ L½ fì

2corrç �

r٠,� r٠��� ,� � 63 �

XcorrçRH� ��

2I3º

R���

tbw 2x sþ RÛ2x fì

3ºcorrç �

r,� r ��� ,� � 64 �

withÖ

fì

1corrç �

r,� r ����� 1

8á

sþ wâ2rÙ ��� rÙ ��� 6

� �r �! 1

" rÙ #%$ 3s rÙ &�' 2÷ (

)r *�+ 1 , 2

x ln r -�. rÙ / rÙ 0 61 2

r 3 1 4rÙ 5 3s rÙ 6 2

÷ 78r 9 1 : 2

x ln r ,� ; 65< =

fì

2corrç >

rÙ ,� rÙ ?�@�A 1

8á

sþ wâ2xcÿ L B tw C b9sþ L D tb

w E rÙ F�G 2r HrÙ I�J 1

ln0

rÙ K L sþ L M tbwcN L O t

w Pb9 Q rÙ R 2r

rÙ S 1ln0

rÙ T 2r U 2x V r W 1 XYrÙ Z�[ 1 \^] rÙ _�` rÙ a ln

0rÙ b�c 2r2

x dr e�f 1 gh

rÙ i 1 j^k rÙ l�m rÙ n ln0

rÙ ,�o66< p

fq

3ºcorrr s

rÙ ,� rÙ t�u�v 1

8á

sw wâ2 rÙ x 1

2y 2y

rÙ z 5c

rÙ { 1| r2

x }2y

rÙ ~ 4�

�rÙ � 1 � 2 ln

0rÙ � 1

2y 2y

r٠��� 5c

r٠��� 1� r � 2x � 2

yr٠��� 4

��r٠��� 1 � 2 ln

0rÙ � � 4

� 1

rÙ ��� rÙrÙ �

r٠��� 1ln0

rÙ ��� rÙrÙ � 1

ln0

rÙ

�1 � 1

rÙ ��� rÙr � 2x

rÙ  �¡ 1ln0

rÙ ¢�£ r2x

rÙ ¤ 1ln0

rÙ . ¥ 67< ¦

4288 54BAMERT,§

BURGESS,CLINE, LONDON, AND NARDI

Note¡

that a value of V tbw differentV

from unity can be easilyaccountedÝ for by using the unitary condition¨ ©

tbw ª 2 «­¬ ® tw ¯ b9 ° 2 ±³² V tbw ´ 2 µ 1 ¶¸· V tsw ¹ 2 º¸» V tdw ¼ 2 in�

Eqs. ½ 62< ¾

– ¿ 67< À

.As�

we havealreadypointedout, becauseof our require-ment of no B-B Á mixing when the B Â is exotic, only whenI3º

LÃÄÆÅÈÇ 1/2 canwe havecN L

à ÉËÊtbw ,Ì sÍ L

à ÎÐÏtw Ñ bÒ . However,in this

caseÓ XÔ

corrÕLH vanishes.Ö Hence,without lossof generality,we canset× the LH neutralcurrentmixing equalto the chargedcur-rent mixing in X corrÕLH

Ø,Ì obtaining

XcorrÕLHØ Ù¸Ú

1 Û 2I3º

LÃÜÞÝ�ß

tbw 2x à tw á bâ2x

fã

2xcorrÕ ä

r,Ì r å�æ ,Ì ç 68< è

fã

2xcorrÕ é

r,Ì r ê�ë�ì 1

8á

sÍ wí2x î¸ï r ð r ñ�ò�ó 2rr ôr õ�ö r

lnr ÷r

. ø 69< ù

FromEqs. ú 62< û

,Ì ü 64< ý

,Ì and þ 68< ÿ

we� seethat thereareonly twoindependentmixing parametersrelevant for the completeanalysis� of our problem: theLH matrix element

�tbw and� the

RH�

mixing sÍ R� . Furthermore,notethat asr� ��� r� ,Ì all the cor-

rectionsin Eqs. 65<

,Ì � 67< �

,Ì and 69< �

vanish,Ö independentofthe�

mixing angles.This comesaboutbecauseof a GIM-likemechanismfor all the pieceswhich do not dependon I3

ºR�� .

The�

I¹

3º

R

�-dependentcontributionfrom the RH fermionscou-

pling� to the Z vanishesÖ in the limit r ��� r as� a consequenceof� Eq. � 50

c �.

In�

thelimit r� ,Ì r� ��� 1, for thefunctionsfã

icorrÕ

(�r� ,Ì r� � )� we obtain

fã

1corrÕ �

r,Ì r "!$# 1

8á

sÍ wí2 r %�& r ' 3 ls

nr� (r

,Ì ) 70* +

fã

2corrÕ ,

r� ,Ì r� -�.$/ 1

8á

sÍ wí2 021 r� 3 r� 4�5$6 2y

rr� 7r� 8�9 r� ln

0 r� :r� ,Ì ; 71

* <

fã

3ºcorrÕ =

r� ,Ì r� >"?$@ 1

8á

sÍ wí2x A r� B 1

2y 1 C r

r� Drr E

r� F�G r� ln0 r H

r�I 3

sr

r� JLK r� lnr Mr� N 3

s2y 1 O r

r� P . Q 72* R

Let usnow considerthenumericalvaluesof thesecorrec-tions�

in moredetail.UsingmS tw T 180GeV, MU

W V 80á

GeV,andsÍ wí2 W 0.23,

XEq. Y 38

s Zgives[ a SM radiativecorrectionof

F\ SM] ^

4.01.� _

73* `

The questionis whetherit is possibleto cancelthis correc-tion,�

thuseliminatingtheRbâ problem,� by choosingparticular

valuesÖ of mS tw a and� the mixing angles.For variousvaluesofmS tw b ,Ì the valueof X

ÔcorrÕ c Eq.

� d58c e�f

isg

shownin TableIV.Weh

seethat even for mS tw i�j mS tw , iÌ t isk

possible� to chooseI3º

LÃl ,Ì I3

ºR�m ,Ì and the LH and RH mixing anglessuchthat the

correctionÓ is negative.So the discrepancyin Rn

bâ betweeno

theory�

and experimentcan indeedbe reducedvia tf -tf p mix-qing.

Referring to the modelslisted in Table IV, the optimalchoiceÓ for the weak isospinof the T r is I3

ºLÃsutwv

1/2 and I3º

R�xywz 1, regardlessof thevalueof mS tw { . Furthermore,maximal

RH�

mixing, sÍ R2 | 1, is also preferred.However, even with

these�

choices,it is evidently impossibleto completely re-movetheRb

â problem.� Fromtheabovetable,thebestwe candoV

is to takemS tw }�~ 75*

GeV and � tw � bâ2 � sÍ LÃ2 � 0.6,

Xin which case

the�

total correctionis XcorrÕ �u� 3.68.s

This leavesa 1.5� dis-V

crepancyÓ in Rn

bâ ,Ì which would put it in the categoryof the

other� marginal disagreementsbetweenexperimentand theSM.�

However,sucha light tf � quark� hasother phenomeno-logical0

problems.In particular,CDF hasput a lower limit of91�

GeV on charge2/3 quarkswhich decayprimarily to Wb�24� . Unlessoneaddsothernew physicsto evadethis con-

straint,× the lightesttf � allowed� is aboutmS tw ��� 100GeV. In thiscase,Ó maximal LH mixing ( � tw � bâ2

x �sÍ LÃ2x � 1) gives the largest

effect:� XÔ

corrÕ �u� 2.7.y

The predictedvalueof Rn

bâ isg

thenstillsome× 2� below

othe measurednumber.

Anotherpossibility is that the charge2/3 quarkobservedbyo

CDF is in fact the tf � ,Ì while the real tf quark� is muchlighter,0

saymS tw � 100 GeV. Assumingsmall tf -tf � mixing,q andthat�

the tf � isg

the lightestmemberof thenewmultiplet, the tf �will� then decayto Wb,Ì as observedby CDF, but the SMradiativecorrectionwill be reduced.This situationis essen-tially�

identicalto that discussedabove,in which the LH tf -tf �mixingq is maximal, and mS tw �"� 100 GeV: the SM value ofRbâ will� still differ from the experimentalmeasurementby

about� 2� . The only way for such a scenarioto work is ifmS tw   M

UW . However,newphysicsis thenonceagainrequired

to�

evadethe constraintfrom Ref. ¡ 24y ¢

.For all the possibilitiesof this sectionour conclusionis

TABLE IV. Dependenceof the tµ -tµ £ mixing resultson m¤ t¥ ¦ :§ This table indicatesthe dependenceon themassof the tµ ¨ quarkof the correctionsto g L

b¸

dueto tµ -tµ © mixing, with the t massfixed at 180 GeV.

m¤ t¥ ª«GeV¬ ­

Xcorr®75 ¯ 3.31° t

± ²b³2´ µ 1.21(1 ¶ 2

·I¸

3¹

L

º)» ¼

t± ½

b³2´ ¾

tb± 2´ ¿ 1.39(2I

¸3¹

R

À)» Á

tb± 2´ sR

2´

100 Â 2.70Ã t± Ä b³2 Å 0.71(1Æ Ç

2·

I¸

3¹

LÈÉ )» Ê

t± Ë b³2 Ì tb± 2 Í 0.59(2I¸

3¹

RÎÏ )» Ð

tb± 2sRÎ 2

125 Ñ 1.97Ò t± Ó b³2´ Ô 0.34(1Æ Õ

2I3¹

L

Ö)» ×

t± Ø b³2´ Ù tb± 2´ Ú 0.22(2I3¹

R

Û)» Ü

tb± 2´ sR2´

150 Ý 1.14Þ t± ß b³2 à 0.10(1Æ á

2·

I¸

3¹

LÈâ )» ã

t± ä b³2 å tb± 2 æ 0.05(Æ

2I¸

3¹

RÎç )» è

tb± 2sRÎ 2

175 é 0.20ê t± ë b³2 ì 0.003(1Æ í

2I3¹

LÈî )» ï

t± ð b³2 ñ tb± 2´ ò 0.001(2I3¹

RÎó )» ô

tb± 2´ sRÎ 2´

200 0.84õ t± ö

b÷2ø ù 0.04(

ú1 û 2I

ü3ý

L

þ) ÿ t� �

b÷2ø �

tb� 2ø � 0.02(2Iü

3ý

R

�) � tb� 2ø s� R

2ø

225 1.97� t� � b÷2 0.23(ú

1 2Iü

3ý

L�� ) t� � b÷2 � tb� 2 � 0.07(2I

ü3ý

R�� ) � tb� 2s� R

� 2250 3.20� t� � b÷2ø � 0.55(1 � 2I3

ýL

�) � t� � b÷2ø � tb� 2ø � 0.15(

ú2I3ý

R

�) � tb� 2ø sR

2ø

275 4.52� t� b÷2 ! 1.01(1 " 2Iü

3ý

L

#) $ t� % b÷2 & tb

� 2 ' 0.24(ú

2Iü

3ý

R

() ) tb� 2sR

2

300 5.93* t� + b÷2 , 1.61(1 - 2I3ý

L�. ) / t� 0 b÷2 1 tb� 2 2 0.34(

ú2I3ý

R�3 ) 4 tb� 2sR

� 2

54 4289R·

b¸ AND NEW PHYSICS:A COMPREHENSIVEANALYSIS

therefore�

the same: it is not possibleto completelyexplainRbâ through�

tf -tf 5 mixing. The bestwe can do is reducethediscrepancyV

betweentheory and experimentto about 26 ,Ìwhich� might turn out to be sufficient, dependingon futuremeasurements.

V.7

ONE-LOOP EFFECTS: OTHER MODELS

Another way to changeg8 L9bâ at� the one-loop level is to

introduceexotic new particlesthat coupleto both the Z and�the�

bÒ

quark.� One-loopgraphsinvolving such particlescanthen�

modify the Zb: ¯bÒ

vertexÖ as measuredat LEP and SLC.Recall oncemore the conclusionfrom Sec.II: agreementwith� experimentrequiresthe LH b

Ò-quark coupling, g8 L

9bâ , tÌ oget[ a negative correction comparablein size to the SMm; tw -dependentcontributionssince loop-level changesto g8 R

bâ

are� too small to be detectable.In this sectionwe first exhibit the generalone-loopcor-

rection< dueto exoticnewscalarandspin-halfparticles,withthe�

goalof identifying thefeaturesresponsiblefor theoverallsign× and magnitudeof the result. We then usethis generalresult to investigatea numberof morespecificcases.

The=

answer is qualitatively different depending onwhether� or not the new scalarsand fermionscan mix, andthus�

haveoff-diagonalcouplingsto the Z boson.o

We there-fore treatthesetwo alternativesseparately.Thesimplestcaseisg

when all Z:

couplingsÓ are diagonal,so that the one-loopresults< dependonly upon two masses,thoseof the fermionand� the scalarin the loop. Then the correctionto the ZbbvertexÖ is given by a very simple analytic formula, whichenables� us to easilyexplainwhy a numberof modelsin thiscategoryÓ give the ‘‘wrong’’ sign, reducing > b

â rather thanincreasingit.

More?

generallyhowever,the new particlesin the loopshave@

couplingsto theZ:

which� arediagonalonly in theflavorbasiso

but not the masseigenstatebasis,so the expressionsbecomeo

significantly more complicated.This occursin su-persymmetric� extensionsof thestandardmodel,for example.After�

proposingseveralsamplemodelswhich can resolvethe�

Rbâ problem,� we useour resultsto identify which features

of� supersymmetricmodelsare instrumentalin so doing.

A. Diagonal couplings to the ZA

:B General results

WeC

now presentformulas for the correctionto the ZbbÒ

vertexÖ due to a loop involving genericscalarand spin-halfparticles.� In this sectionwe makethesimplifying assumptionthat�

all of the Z:

-bosoncouplingsare flavor diagonal.ThisconditionÓ is relaxedin later sectionswhere the completelygeneral[ expressionis derived.Theresultingformulasmakeitpossible� to seeat a glancewhethera given modelgives theright< signfor alleviatingthediscrepancybetweenexperimentand� the SM predictionfor Rb

â .The one-loopdiagramscontributingto the decayZ D bb

Ò ¯canÓ begroupedaccordingto whetherthe loop attachesto thebÒ

quark� E i.e., thevertexcorrectionandself-energygraphsofFig. 5F or� whether the loop appearsas part of the gaugebosono

vacuumpolarization G Fig.k

6H . For the typesof modelswe� considerthesetwo classesof graphsareseparatelygaugeinvariant and finite, and so they can be understoodsepa-rately.This is particularlyclearin the limit that the particleswithin� the loop are heavycomparedto M

IZ ,Ì sincethen the

vacuumÖ polarizationgraphsrepresentthe contributionof theoblique� parameters,S

Jand� T

K,Ì while the self-energyand

vertex-correctionÖ graphsdescribeloop-inducedshifts to thebÒ

-quarkneutralcurrentcouplings, L g8 L9

,RMb

â.

Furthermore,althoughwe must ensurethat the obliqueparameters� do not becomelarger thanthe boundof Eq. N 3s O ,ÌEq.� P

4� Q

shows× that they largely cancelin the ratio RR

bâ . We

therefore�

restrictour attentionin this sectionto thediagramsof� Fig. 5 by themselves.Thesumof thecontributionsof Fig.5c

is also finite as a result of the Ward identity which wasalluded� to in Sec.III. This Ward identity relatesthe vertex-part� graphsof Figs.5S a� T and� 5U bo V to

�theself-energygraphsof

Figs. 5W cÓ X and� 5Y dV Z . Since this cancellationis an importantcheckÓ of our results,let us explainhow it comesabout.

WeC

first consideran unbrokenU [ 1\ gauge[ bosonwith atree-level�

couplingof g8 bâ to�

thebÒ

quark.� This givesriseto thefamiliar Ward identity from quantumelectrodynamics:forexternal� fermionswith four-momentap] and� p] ^ ,Ì

_p] ` p] acbedgf dih g8 effj k SJ Fl 1 m p] npo S

JFq 1 r p] sctvu ,Ì w 74

* x

where� y{z isg

the one-particle-irreduciblevertex part andSJ

F| (}p] )~

is thefermionpropagator.If we denotethevertex-partcontributionsÓ � Figs. 5� a� � and� 5� bo �c� to

�the effectivevertexat

zero momentum transfer by � g8 bâ ,Ì and the self-energy-

inducedg

wavefunctionrenormalizationof thebÒ

quark� by Z:

bâ ,Ì

FIG. 5. The one-loopvertex correctionand self-energycontri-butionsto the Zbb

� ¯ vertexdueto fermion-scalarloops.

FIG. 6. Theone-loopcontributionsto theZbb�

vertexdueto thegauge-bosonvacuumpolarizations.

4290 54BAMERT,�

BURGESS,CLINE, LONDON, AND NARDI

then�

at one loop the Ward identity � 74* �

reduces tog8 bâ (1} �

Zbâ )(~ p] ��� p] ��� )~ � (

}g8 bâ ��� g8 b

â )(~ p] ��� p] ��� ),~ or

�g8 bâ � g8 b

â Zbâ � 0.

�  75* ¡

This=

lastequationis themoregeneralcontextfor thecancel-lation which we found in Sec. III; it statesthat the self-energy� graphs¢ Figs.5£ cÓ ¤ and� 5¥ dV ¦¨§ mustpreciselycancelthevertexÖ part © Figs.

k5ª a� « and� 5¬ bo ­c® in

gthe limit of zeromomen-

tum�

transfer.Another way of understandingEq. ¯ 75* °

isg

toimaginecomputingthe effectiveb

Ò-photonvertexdue to in-

tegrating�

out a heavyparticle.Equation ± 75* ²

is theconditionthat�

the two effectiveoperatorsbÒ ³ ´ bÒ and� b

ÒAµ ¶

bÒ

have@

the rightrelative< normalizationto begroupedinto thegauge-covariantderivative:V

bÒ

D· bÒ

.But for the externalZ boson,

othe Ward identity only ap-

plies� to thosepartsof the diagramswhich are insensitivetothe�

fact thattheU ¸ 1¹ symmetry× is now broken.Theseincludethe�

1/º nE » 4¼ poles� from dimensionalregularization,andalsothe�

contributionsto the bÒ

neutral-currentcoupling propor-tional�

to sÍ wí2 ,Ì sincethe latter ariseonly throughmixing fromthe�

couplingsof the photon.WeC

now returnto thediagramsof Fig. 5. Thefirst stepisto�

establishtheFeynmanrulesfor thevariousverticeswhichappear.� Sincewe careonly abouttheLH neutral-currentcou-plings,� it sufficesto considercouplingsof the new particlesto�

bÒ

L9 :

½scalar¾ y¿ f

À Ái fã Ã

LbÒ Ä

H.c. Å 76* Æ

and� we write the Z:

couplingÓ to fã

and� Ç as�È

NCÉ Ê eË

sÍ wí cÌ wí Z Í�Î fã ÏÑÐ�Ò g8 L9 fÀ Ó

L9 Ô g8 R

M fÀ ÕRM Ö fã ×

igØ

SÙ Ú † Û ÜÞÝißáà . â

77* ã

The couplings, g8 äæå g8 L9 fÀ ,Ì g8 R

M fÀ ,Ì g8 SÙ ç ,Ì are normalized so that

g8 è I3º é Qs

êwí2x for all fields, f

ãL9

,RMaë and� ì mí .

Inî

the exampleswhich follow, the field fã

canÓ representeither� anordinaryspinor ï e.g.,� tf ð or� a conjugateÌ spinor× ñ e.g.,�tf cò ó . This differencemustbe kept in mind wheninferring thecorrespondingÓ charge assignmentsfor the neutral-currentcouplingsÓ of the f

ã. For example,the left-handedtop quark

has@

I¹

3º

L ôöõ 12÷ , sÌ o g8 L

fÀ ø

12÷ ù 2

÷3ú sÍ wí2 and� I

¹3º

R û 0,�

so g8 RfÀ üþý

2÷3ú sÍ wí2 . If

the�

internal fermion were a top antiÿ quark,� however, wewould� insteadhaveg8 R

M fÀ � � 12� 2

3ú sÍ wí2x and� g8 L

9 fÀ ��� 23ú sÍ wí2x . Thelat-

ter�

couplingsfollow from the former using the transforma-tion�

of the neutral current under chargeconjugation:Ó ����

L9 ��� ��

RM .

WeC

quotetheresultsfor evaluatingthegraphsof Fig. 5 inthe�

limit whereMI

Z

�and� of coursem; b

â � are� negligiblecom-pared� to m; f

À and� MI �

,Ì sincetheyarequitesimpleandillumi-natingin this approximation.It will be shownthat the addi-tional�

correctionsdueto thenonzeromassof theZ bosono

aretypically�

lessthan10% of this leadingcontribution.WeC

find that

�g8 L9bâ � 1

32s � 2

�fÀ � nE cò � y¿ f

À ��� 2 � 2 � g8 L9 fÀ � g8 R

M fÀ "!$# r %&('*)

g8 RfÀ +

g8 Lbâ ,

g8 SÙ -/.10325476˜ 8 r� 9/:<; ,Ì = 78

* >

where� ? (}r� )~ and @˜(

}r� )~ are functions of the mass ratio

r� A m; fÀ2/BMI C2 ,Ì

D3Er� FHG r�I

r� J 1 K 2 L r� M 1 N ln0

r� O ,Ì P 79* Q

R˜ S r THU r�Vr W 1 X 2

Y Z r [ 1 \ r ln r ] . ^ 80á _

`badenotesV

the divergent combinationcbdfe2/g nE h 4ikjml�n ln o M p2Y /4B q5r 2

Y skt12,Ì and nE cò is a color factor

that�

dependson theSUcò u 3v w quantum� numbersof thefields xand� f

ã. For example,nE cò y 1 if z|{ 1

}or� fã ~

1} �

colorÓ singlets� ;nE cò � 2 i

�f fã �

3�

and� ��� 3�

or� 6�; nE cò � 16

3ú ifg

fã �

3�

and� �|� 8�.

The cancellationof divergenceswe expectedon generalgrounds[ is now evidentin thepresentexample,becauseelec-troweak�

gaugeinvarianceof the scalarinteraction � 76* �

im-g

plies� that the neutral-currentcouplingsarerelatedby

g8 SÙ � g8 L

9bâ � g8 RM fÀ � 0.

� �81� �

This forcesthe term proportionalto �˜ to�

vanishin Eq. � 78* �

.As advertisedthe remaining term is both ultraviolet finiteand� independentof sÍ wí2 ,Ì which cancelsin the combinationg8 L9 fÀ � g8 R

M fÀ .WeC

are left with the compactexpression

�g8 L

bâ � 1

16� 2Y �

fÀ � nE cò   y¿ f

À ¡�¢ 2Y £ g8 LfÀ ¤

g8 RfÀ ¥"¦3§

m; fÀ2/BMI 2 © . ª 82

� «

Interestingly,î

it dependsonly on theaxial-vectorcouplingofthe�

internal fermion to the gaugebosonW3º and� not on the

vectorÖ coupling.The functionof themasses¬ (}r)~

is positiveand� monotonically increasing,with ­ (

}r)~ ®

r as� r ¯ 0�

and°3±³²�´kµ1, ascanbe seenin Fig. 7.

Itî

is straightforwardto generalizeEq. ¶ 82� ·

to�

includetheeffect� of thenonzeroZ boson

omass.Expandingto first order

in M Z2Y,Ì oneobtainsan additionalcorrectionto the effective

vertex,Ö

FIG. 7. From top to bottom, the functions ¸º¹ m» f¼2/½M¾ ¿2 ),

ÀFRÁ (r) Â FS

à (r), andFLÄ (rÅ )À which appearin the loop contributionto

the left-handedZbb vertex,Æ Secs.V A andV C.

54 4291RÇ

bÈ AND NEW PHYSICS:A COMPREHENSIVEANALYSIS

ÉZg8 L

bâ ÊÌË

fÀ Í

Îy¿ fÀ Ï�Ð 2Y nE cò96Ñ Ò 2

Y MI

Z2

m; fÀ2Y 0

Ó1

dxÔ

Õ xÖ 3º ×

g8 Lbâ Ø

g8 RfÀ ÙHÚ

2� Û

1 Ü xÖ Ý 3ºg8 R

fÀ

xÖ Þ M ß2Y /Bm; fÀ2Y à 1 áHâ 1

ã ä 1 å xÖ æ 3ºg8 L

fÀ

çxÖ è M é2Y /

Bm; fÀ2Y ê 1 ëHì 1 í 2

Y . î 83ï ð

To=

seethat this is typically an unimportantcorrection,con-sider× the limit in which the scalarand fermion massesareequal,� r ñ 1. Thenthe total correction ò 82

ï ókô7õ83ï ö

is

÷g8 L

bâ øúù

Zû g8 L

bâ üÌý

fÀ þ

ÿy¿ fÀ ��� 2Y nE c�32� � 2 g8 L

fÀ �

g8 RfÀ � M

IZ2

12m; fÀ2 � g8 L

bâ

g8 LfÀ

g8 R

fÀ �

. � 84ï

Although�

the MI

Z2 correctionÓ can be significant if g8 L

fÀ �

g8 RfÀ

,Ìthe�

total correctionwould thenbetoo small to explaintheRbâ

discrepancy,V

andwould thusbe irrelevant.

B. Why many models do not work

WhatC

is importantfor applicationsis the relativesign be-tween�

the tree and one-loopcontributionsof Eq. � 82ï �

. Inorder� to increaseR

Rbâ so× as to agreewith the experimental

observation,� oneneedsfor themboth to havethe samesign,and� so � g8 L

9bâ � (}g8 L9 fÀ � g8 R

M fÀ )~ �

0�

in Eq. � 82ï �

. Thus an internalfermion�

with the quantum numbersof the bÒ

quark� hasg8 L

fÀ �

g8 RfÀ ���

12� and� would increaseR

Rbâ . Conversely,a fermion

like the t� -quarkhasg8 L9 fÀ � g8 R

M fÀ �! 12 and� so causesa decrease.

Moreover, becausethe combination(g8 L9 " g8 R

M )~

is invariantunder# chargeconjugation,the samestatementshold true forthe�

antiparticles: a bÒ

running< in theloop would increaseRR

bâ

whereas� a t� would� decreaseit.It thus becomesquite easyto understandwhich models

with� diagonalcouplingsto theZ:

bosono

canimprovethepre-dictionV

for RR

bâ . Multi-Higgs-doubletmodelshavea hardtime

explaining� an Rbâ excess� becausetypically it is the top quark

that�

makesthe dominantcontribution to the loop diagram,since× it has the largestYukawa coupling, y¿ f

À $&% 1, and thelargest0

mass,to which thefunction ' isg

very sensitive.How-ever,� for very largetan (*) the

�ratio of thetwo HiggsVEV’s + ,Ì

the�

Yukawa coupling of the t� quark� to the chargedHiggsbosono

can be madesmall and that of the bÒ

quark� can bemadeq large,asin Ref. , 25

� -. Figure7 showsthat, in fact, one

must go to extremevaluesof theseparameters,becauseinaddition� to needingto invert thenaturalhierarchybetweeny¿ t

.and� y¿ b

â ,Ì one must overcomethe big suppressionfor smallfermion�

massescomingfrom the function / .Preciselythe sameargumentappliesto a broadclassof

Zee-typemodels,where the SM is supplementedby scalarmultipletsq whose weak isospin and hyperchargepermit aYukawa0

coupling to the bÒ

quark� and one of the other SMfermions.So long asthescalarsdo not mix andtherearenonew fermionsto circulatein the loop, all suchmodelshavethe�

samedifficulty in explainingthe RR

bâ discrepancy.V

Belowwe� will give someexamplesof modelswhich, in contrast,are1 able� to explainRb

â .

C. Generalization to nondiagonal Z couplings2WeC

now turn to themorecomplicatedcasewheremixingintroducesg

off-diagonalcouplingsamongthe new particles.Becauseof mixing the couplingsof the fermions to the Zwill� be matricesin the massbasis.Similar to Eqs. 3 424 and�5456 7

we� write

8g8 L,R 9 f f

À :<;>=a? @BA<C L

9,RMa f? D

* E L9

,RMaf? F I3º

L9

,RMa? GIH f f

À JQK a? sÍ wí2 L ,Ì M 85

ï N

where� OL,Ra f? are� the mixing matrices.An analogousexpres-

sion× givesthe off-diagonalscalarZ:

couplingÓ in termsof thescalar× mixing matrix, P S

Qa? R . Of courseif all of the mixingparticles� sharethe samevalue for I3

º ,Ì then unitarity of themixingq matrices guaranteesthat the couplings retain thisform�

in any basis.This modificationof theneutral-currentcouplingshastwo

importanteffectson the calculationof S g8 L9bâ . One is that the

off-diagonal� Z:

couplingsÓ introducethe additionalgraphsofthe�

typeshownin Figs.5T a� U and� 5V bo W ,Ì wherethe fermionsorscalars× on eithersideof the Z vertexÖ havedifferent masses.The other is that the mixing matricesspoil the relationship,Eq.X Y

81ï Z

,Ì wherebythetermproportionalto [˜ canceledÓ in Eq.\78* ]

. But this is only becauseof the massdependenceof ^and� _a` . Thereforethe cancellationstill occursif all of theparticles� that mix with each other are degenerate,as onewould� expect. Moreover, the ultraviolet divergencesstillcancelÓ sincethey aremassindependent.

Evaluation of the graphsgives the following result atqb 2c d

M Z2c e

0:�f

g8 Lbâ g 1

32� h 2

c i Gj diagd k G

jf fÀ lnm G

j oporqts,Ì u 86

ï v

where� Gj

diagd /32B w 2

crepresentsthe contributioninvolving only

the�

diagonalZ:

couplings,Ó and so is identical to the previ-ously� derivedEq. x 78

* y. It is convenientto write it as

Gj

diagd z>{

fÀ | nE c� } y¿ f

À ~�� 2� 2 � g8 L9 � g8 R

M � f fÀ ���

r ��������� g8 RM � ffÀ �

g8 L9bâ

���g8 SQ ���p���������&���˜ � r�  ¢¡¤£ . ¥ 87

ï ¦Here and in the following expressionswe use the notationr § m; f

À2c /B M ¨2 and� r ©«ª m; fÀ ¬2c

/BM ­2 . As before®a¯ denotes

VtheUV-

divergentV

quantity °a±³² 2/� ´

nE µ 46 ¶B·¹¸»º

ln0 ¼

MI ½2/4

B ¾À¿ 2ÁB 12� .

The=

remaining terms in Eq. Ã 86ï Ä

comeÓ from the newgraphs[ of Figs. 5Å a� Æ and� 5Ç bo È ,Ì wherethe scalarsor fermionson� eithersideof the Z vertexÖ havedifferent masses,due tomixing:q

Gj

f fÀ ÉnÊÌËÍ

, fÀ Î

fÀ Ï nE c� y¿ f

À Ð y¿ fÀ Ñ�Ò* Ó 2 Ô g8 L Õ f f

À ÖØ×L Ù r,Ì r Ú<Û�Ü�Ý g8 R Þ f f

À ßáàáâ�ãä�å

RM æ r� ,Ì r� ç<è¢éëê ,Ì ì 88

ï í

Gî ïpïrð<ñ ò

fÀ

, ó&ôõórö nE c� y¿ fÀ ÷ y¿ f

À ørù* ú g8 Sû ü�ýþýrÿ���������

Sû xÖ ,Ì xÖ ���� ,Ì �

89ï �

where� �L(}r� ,Ì r� � ),~ �

R(}r� ,Ì r� � )~ , and � S

û (} xÖ ,Ì xÖ � )~ aregiven by

4292 54BAMERT,�

BURGESS,CLINE, LONDON, AND NARDI

�L � r,Ì r �����

�rr� �

r r !r�

r " 1ln r # r� $

r %'& 1ln r ( ,Ì ) 90

Ñ *

+RM , r� ,Ì r� -�.�/ 1

r� 0 r� 1r22

r� 3 1ln0

r� 4 r 5 22r� 6'7 1

ln0

r� 8 ,Ì 9 91Ñ :

;S< = xÖ ,Ì xÖ >�?�@ 1A

xÖ B 1 CED xÖ FHG 1 IKJ 1 L ln xÖ M�N xÖ O 2PxÖ Q'R 1 SUT xÖ VHW xÖ X

Y1 Z ln

0 xÖxÖ [ \ xÖ 2

2]xÖ ^ 1 _E` xÖ a xÖ b�c ,Ì d 92

Ñ e

and� xÖ ,Ì xÖ f are� the massratios xÖ g MI h2 /

Bm; fÀ2 and� xÖ iHj M

I kml2 /Bm; fÀ2.

Theseexpressionshave severalsalient featureswhich wenown discuss.First, Eqs. o 87

ï p,Ì q 88ï r

,Ì and s 89ï t

are� obviouslymuchq morecomplicatedthanEq. u 82

ï v. In particular,it is no

longerstraightforwardto simply readoff the sign of the re-sult.×

Second,w

thesumof theUV divergencesin Eqs. x 87ï y

,Ì z 88ï {

,Ìand� | 89

ï },Ì

Gî ~����

f fÀ �����m� y¿ f

À � y¿ fÀ ���m�* ����� g8 R

M � f fÀ �������m���

g8 L9bâ � f fÀ ���������

�� g8 S< ¡£¢�¢�¤�¥ f f

À ¦¨§,Ì © 93

Ñ ªis basisÒ

independent since× a unitary transformationof thefields«

cancelsbetweenthe Yukawaandneutral-currentcou-plings.� Thus it can be evaluatedin the electroweakbasiswhere� theneutral-currentcouplingsarediagonalandpropor-tional�

to ¬ g8 RM fÀ ­ g8 L

9bâ ® g8 S< ,Ì which vanishesdueto conservation

of� weak isospinand hyperchargeat the scalar-fermionver-tex.�

Wearethereforefreeto choosetherenormalizationscale¯ 22

in ln ° M ±22 /B ² 22 ³

to�

takeany convenientvalue.The M ´ de-V

pendence� of µ·¶ makesGî ¸�¸�¹

look unsymmetricunder theinterchangeg

of º and� »½¼ ,Ì but this is only an artifact of theway� it is expressed.For examplewhen thereare only twoscalars,× G

î ¾�¾�¿is indeedsymmetricunderthe interchangeof

their�

masses.Third,=

all the contributionsexceptÀ those�

of Gî Á�ÁÃÂ

are� sup-pressed� by powersof m; f

À /B MI Äing

the limit that thescalarsaremuchheavierthanthe fermions.Thusto get a largeenoughcorrectionÓ to g8 L

bâ

requiresthat Å i Æ not all of the scalarsbemuchq heavierthanthe fermionswhich circulatein the loop,or� Ç iig È the

�scalarsmix significantlyandhavetheright charges

so× that Gî ÉUÉÃÊ

is nonnegligibleand negative.We useoptionËii Ì in what follows to constructanothermechanismfor in-

creasingÓ RR

bâ .

Finally,Í

evenif thetwo fermionsaredegenerate,onedoesnot generallyrecoverthe previousexpressionÎ 78

* Ïthat�

ap-plied� in the absenceof mixing. This is becauseDirac massmatricesq are diagonalizedby a similarity transformation,MI ÐÒÑ

L†MI Ó

RM ,Ì not a unitary transformation.The left- and

right-handed mixing angles can differ even when thediagonalizedV

mass matrix is proportional to the identity.Thus,=

in contrast to Eq. Ô 93Ñ Õ

,Ì the expressionÖf fÀ ×ÙØÃØmÚ y¿ f

À Û y¿ fÀ Ü�ÝmÞ* ß (} g8 L)

~ f fÀ à�á

(}g8 R)~ f fÀ â¨ã

is not invariant undertransformations�

of the fields, becausey¿ fÀ ä is

grotatedby å Ræ

recall thaty¿ fÀ ç is theYukawacouplingonly for theRH f

ã’sè ,Ì

whereas� g8 L9 is rotatedby é L

9 .

WeC

cangetsomeinsightinto Eqs. ê 88ï ë

– ì 92Ñ í

byo

looking atspecial× valuesof the parameters.Let us assumethere is adominantV

Yukawa coupling y¿ betweeno

the left-handedbÒ

quark� anda singlespeciesof scalarandfermion, fã

1 and� î1 ing

the�

weakbasis,

ïscalarð y¿ ñ 1f

ã1 ò L9 bÒ ó

H.c. ô 94Ñ õ

In the massbasisthe couplingswill thereforebe

y¿ fÀ ö�÷ y¿ ø S

<1 ùûú�üRM1 fÀ ý

* . þ 95Ñ ÿ

Now�

gaugeinvarianceonly relatesthe � 1,1� elements� of theneutral-currentcouplingmatricesin the weakbasis:

�g8 S< � 11� g8 L

bâ ���

g8 RfÀ

11 0.� �

96Ñ �

Thereare three limiting casesin which the resultsbecomeeasier� to interpret.

1� Ifî

all the scalarsare degeneratewith eachother,andlikewise for the fermions,then the nonmixing result of Eq.�82ï �

holds,exceptonemustmakethe replacement

g8 LfÀ �

g8 RfÀ �����

RM SJ

m� � L†g8 L9 �

L9 SJ

m� � R† � g8 R

M � 11,Ì � 97Ñ �

where� SJ

m� isg

the diagonalmatrix of the signsof the fermionmasses.q �

2 If there are only two scalarsand if they are muchheavierthan all of the fermions,only the term G

î !"!$#is sig-

nificant.n Let % 1 and� &2 denoteV

the weak-eigenstatescalars,and� ' and� (*) the

�masseigenstates;then

+g8 L

bâ , y¿ 2

2n- c�

16. 2 / I¹ 3º 0 1 1 I

¹3º 2 23 4

c5 S<22 sÍ S<22 F6 S< 7 MI 822 /

BMI 9;:22 <

; = 98Ñ >

FS< ? r @"A r B 1

2 C r D 1 E ln r F 1, G 99Ñ H

where� c5 S< and� sÍ S

< are� thecosineandsineof thescalarmixingangle.� ThefunctionF

6S< (} r� )~ is positiveexceptat r� I 1 whereit

is zero,and so the sign of J g8 L9bâ is completelycontrolledby

the�

factor (I3º K 1 L I3

º M 23)~. We seethat to increaseRb

â it is neces-

sary× that I¹

3º N 1 O I

¹3º P 2.Q

3� R

WhenC

thereareonly two fermions,with weakeigen-states× f

ã1,Ì fã

2 and� masseigenstatesfã

,Ì fã S

botho

much heavierthan�

any of the scalars,then

Tg8 L9bâ U y¿ 2n- c�

16V 2 W g8 L911c5 LR

922 Xg8 L9222

sÍ LR922 Y

g8 RM11

Z�[g8 RM222 \

g8 RM11] c5 R

M22 sÍ RM22 FR

M ^ m; fÀ22 /B m; f

À _22 `

a 2� b

g8 L22c g8 L

11d c5 L9 sÍ L9 c5 RM sÍ RM F6

L9 e m; f

À2/Bm; fÀ f2 gih ; j 100k

where� sÍ LR and� c5 LR are� the sineandcosineof the differenceorl sumof theLH andRH mixing angles,m L n sÍ m� o R ,Ì depend-ing on the relative sign sÍ m� ofl the two fermion masseigen-values,p and

54 4293Rq

br AND NEW PHYSICS:A COMPREHENSIVEANALYSIS

FR s r t"u FS< v r w"x r y 1

2 z r { 1 | lnr } 1

and~ F6

L9 � r� �"�

�r�

r� � 1ln�

r� � 1. � 101�ThefunctionFL

9 hassomeof thesamepropertiesasFS< � FR

M ,�including invarianceunderr � 1/r,� being positive semidefi-nite� andvanishingat r� � 1. Plotsof thesefunctionsareshownin�

Fig. 7. Note that the first line of Eq. � 100� is�

the sameasEq. � 97

Ñ �.

To getsomeideaof theerrorwe havemadeby neglectingthe�

massof the Z:

boson�

one can computethe lowest ordercorrection� as in Sec.V C. The answeris more complicatedthan�

for the caseof diagonalZ couplings,� exceptwhen thefermionsaredegeneratewith eachotherandlikewise for thebosons.�

In that casethe answeris given againby Eq. � 84ï �

except� that g8 RfÀ �

(}g8 R

fÀ

)~ 11 and~ g8 L

fÀ �

(} �

RM SJ

m� � L†g8 L9 �

L9 SJ

m� � R†� g8 R

M )~ 11,� preciselyasin Eq.   97

Ñ ¡. Thuswe would still expect

it�

to be a small correctionevenwhenthereis mixing of theparticles¢ in the loop.

These simplifying assumptionscan be used to gain asemianalytic£ understandingof why certainregionsof param-eter� spacearefavoredin complicatedmodels,which is oftenmissingin analysesthattreattheresultsfor theloop integralsas~ a black box. The observationswe makeheremay be use-ful¤

whensearchingfor modificationsto a model that wouldhelp¥

to explainR¦

b§ . The next two sectionsexemplify this by

creating� somenew models that take advantageof our in-sights,£ and by elucidatingprevious findings in an alreadyexisting� model,supersymmetry.

D. Examples of models that work

In addition to ruling out certain classesof models,ourgeneral¨ considerationsalsosuggestwhat is

©requiredª in order

to�

explain R¦

b§ . Obviously new fermionsand scalarsare re-

quired,« whoseYukawacouplingsallow themto circulatein-side£ the loop. We give two examples,onewith diagonalandonel with nondiagonalcouplingsof thenewparticlesto theZ

¬boson.�

For our first examplewe introduceseveralexotic quarksF6

,� P­

,� and N®

,� and a new Higgs doublet ¯ ,� whosequantumnumbers� are listed in Table V. The unorthodox electriccharge� assignmentsdo not ensure cancellation of elec-troweak�

anomalies,but this canbefixed by addingadditionalfermions,¤

like mirrors of thosegiven, which do not contrib-ute° to R

¦b§ .

Thehyperchargesin TableV allow thefollowing Yukawainteractions:

±y² ³ yN¿ ¯RQ

´Li µ j¶ ·

i j ¸ g¹ pº P­

RF6

Li H» j¶ ¼

ij ½ g¹ n¾ N®¯RF6

Li H»˜ j¶ ¿

ij À H.c.,Á Â

102ÃwhereÄ Å

i j is the 2Æ 2 antisymmetrictensor,H is the usualSMÇ

Higgs doublet,andQ´

LÈ É (

Êb§

L

tË LÌ )Í

is the SM doubletof third

generation¨ LH quarks.When H gets¨ its VEV, Î H ÏÑÐÓÒ , w� efindÔ

two fermion masseigenstates,pÕ and~ n- ,� whosemassesare~ mÖ pº × g¹ pº Ø and~ mÖ n¾ Ù g¹ n¾ Ú and~ whoseelectric chargesareQ´

pº Û qÜ Ý 1 and Q´

n¾ Þ qÜ . Thereare also two new scalarmasseigenstates,� ß;à ,� whoseelectric chargesare Q

´ á;âqÜ ã 1

3ä and~

Q´ å;æ

qÜ ç 23ä .

Inè

themasseigenstatebasis,theYukawainteractionswiththe�

new scalarsare

éy² ê yn¿ ¯Rb

ëL ìîíðï yn¿ ¯Rtñ L òîóðô H.c.,

Á õ103ö

from which we seethatthen- couples� to thebë

quark« asin Eq.÷76ø ù

.Theú

weak isospinassignmentsof the n- are~ Iû

3ü

LNý þ ÿ

123 and~

I 3ü

R�n¾ �

0,�

so that g¹ LÈn¾ � g¹ R

�n¾ ��� 12. Therefore,from Eq. � 82

ï ,� one

obtainsl g¹ Lb§ �

0.�

Thecentralvalueof Rb§ can� bereproducedif�

g¹ Lb§ ��

0.0067,�

which is easilyobtainedby taking y� � 1 andr� � 1, so that � (

Êr� )Í � 1. The Yukawacouplingcould be made

smaller£ by putting the new scalarsin a higher color repre-sentation£ like the adjoint.

We�

havenot exploredthedetailedphenomenologyof thismodel,� but it is clearly not ruled out since we are free tomakethenewfermionsandscalarsasheavyaswewish.Andsince£ we canalwaystake mÖ pº � mÖ n¾ ,� thereis no contributionto�

the obliqueparameterT�

. The contributionto R¦

b§ does�

notvanish� evenas the massesbecomeinfinite, but this is con-sistent£ with decouplingin the sameway asa heavytñ quark,«since£ thenewfermionsget their massesthroughelectroweaksymmetry£ breaking.Thepricewe haveto pay for suchlargemassesis correspondinglylargecouplingconstants.

Next�

we build a modelthatusesour resultsfor nondiago-nal� couplingsto the Z

¬. It is a simplemodificationof theSM

that�

goesin the right directionfor fixing the R¦

b§ discrepancy�

but�

not quite far enoughin magnitude.Variations on thesame£ themecancompletelyexplainRb

§ at~ thecostof makingthe�

modelsomewhatmorebaroque.Our�

startingpoint is a two-Higgs-doubletextensionof theSM.Ç

We takethe two Higgs fields,

Hd� � Hd

�0ÓH»

d� � and~ Hu� � Hu�

H»

u�0Ó ,�to�

transformin theusualway undertheSM gaugesymmetry.Itè

was explainedearlier why this model doesnot by itselfproduce¢ the desiredeffect, but Eq. ! 98

Ñ "suggests£ how to fix

this�

problem by introducing a third scalar doublet, #$ (Ê %'&(')*)

)Í, which mixes with the other Higgs fields. The

charge� assignmentsof thesefields, listed in TableVI, ensurethat�

the two fields H u� + and~ ,.- can� mix even though theyhavedifferent eigenvaluesfor I3

ü .Inè

this model the new scalar field cannot have anyYukawa/

couplingsto ordinaryquarkssincetheseareforbid-den�

by hyperchargeconservation.The only Yukawa cou-plings¢ involving theLH b

ëquark« arethosewhich alsogener-

ate~ the massof the tñ quark:«

TABLE V. Field content and charge assignments: Elec-troweakquantumnumbersfor thenewfieldswhich areaddedto theSM to producethe observedvaluefor R

0b1 .2

Field Spin SUc3 4 35 6 SUL 7 28 UY 9 1:;0<

1 2 q = 16>

FL? 1

23 3 2 q @ 1

23

PR12 3 1 q A 1

NB

R12 3 1 qC

4294 54BAMERT,D

BURGESS,CLINE, LONDON, AND NARDI

EYukF G y� tË H tñ I L

È bHë

u� JLK H.c.,Á M

104NwhereÄ y� tË O mÖ tË /P Q u� is the conventionally-normalizedYukawacoupling.� We imagine R u� to

�be of the sameorder as the

single-Higgs£ SM value, and so we expecty� tË to�

be compa-rableto its SM size.

The scalarpotential for sucha model very naturally in-corporates� H

»u� S -TVU mixing.� Gaugeinvariancepermitsquartic

scalar£ interactionsof theform W (ÊH»

u�†X )(Í

H»

u�†H»

d� )Í Y H.c.,

Áwhich

generate¨ the desiredoff-diagonal terms: Z (Ê [�\

H»

u� ] * ^ u�* _ d�`�acb

Hd� dfe

u�22 * )Í g

H.c.

SinceÇ

the weak isospin assignmentsare I3ü hji.kml 1

2

and~ Iû

3üHn uo prqms 1

23 ,� the color factor is n- ct u 1, and the relevant

Yukawa/

coupling is y� v y� tË w ,� we seethat Eq. x 98y z

predicts¢the�

following contribution due to singly-chargedHiggsloops:

{g¹ LÈb§ |m} y� tË22

16~ 2 2c5 S<22 s� S<22 FS< � r � ,� � 105�

withÄ r being�

the ratio of the scalar mass eigenstates,r� M �2 /PM �f�22

. Taking optimistic values for the parameters13���S< ��� /4,

P2c5 S

<2s� S<2 � 1

23 ,� y� tË � 1, and M

� �/PM� �f���

10� , w� efi nd�g¹ L

b§ ���

0.0043,�

which is two-thirds of what isrequired: ( � g¹ L

Èb§ )Í expt� ��� 0.0067� �

0.0021.�

In additionto thecontributionof thesingly-chargedscalarloops,�

one should considerthoseof the other nonstandardscalar£ fields we introduced.Sinceall of the scalarsthat mixhavethe sameeigenvaluefor I3

ü ,� their contributionis givenby�

Eq.   82¡ ¢

,� which is small if the scalarsare much heavierthan�

thelight fermions.Thenonly the tñ -quarkcontributionisimportant.�

In this limit there are appreciablecontributionsonlyl from the three chargedscalarfields, one of which iseaten� by thephysicalW boson

�andsois incorporatedinto the

SMÇ

tñ -quarkcalculation,andtheothertwo of which we havejust£

computed.So,Ç

for an admittedlyspecialregion of parameterspace,this�

simple model considerablyamelioratesthe Rb§ discrep-�

ancy,~ reducingit to a 1¤ effect.� It is easyto adaptit soastofurther¤

increase¥ g¹ LÈb§ and~ alsoenlargethe allowedregionof

the�

model’sparameterspace.Thesimplestway is by increas-ing the size of the color factor n- ct orl the isospindifference

I3ü ¦f§�¨ I3

ü © . For instancethe new scalar,ª ,� could be put into a4«

ofl SUL ¬ 2­ ® ratherª thana doublet,andbegivenweakhyper-

charge� Y ¯�° 5±23 . Then the singly-chargedstate²´³ has

¥Iû

3ü µ'¶.·

¸ 3ä2,� making I

û3ü ¹fº�» I

û3ü ¼¾½m¿ 2

­, which is twice asbig as for the

doublet.�

More newscalarsmustbeaddedto generatemixingamongst~ the singly-chargedscalarstates.

A secondvariationwould be let the two new Higgs dou-blets�

be color octetssince this gives more than a fivefoldenhancement� of À g¹ L

b§

due�

to thecolor factor n- ct Á 163ä . It is still

possible¢ to write downquarticscalarinteractionswhich gen-erate� the desiredscalarmixings. Either of thesemodelshasmuch more room to relax the previouslytight requirementsfor¤

optimal scalarmassesandmixings.

E. The supersymmetric case

LetÂ

us now apply the aboveresultsto gain someinsightinto�

whatwould benecessaryto explainR¦

b§ in�

supersymmet-ric extensionsof the standardmodel.Therearetwo kinds ofcontributions� involving the top-quark Yukawa coupling,whichÄ oneexpectsto give thedominanteffect.Thesearethecouplings� of the left-handedb

ëquark« to the secondHiggs

doublet�

andthetop quark,or to thecorrespondingHiggsinosand~ top squarks,

y� tË bë LÈ hÃ

2,2

R�Ä tñ R� Å y� tË bë L

È tñ R� hÃ

22 Æ . Ç 106È

Of�

these,the secondone gives a loop contribution likethat�

of thetwo-Higgsdoubletmodelsdiscussedabove:it hasthe�

wrong sign for explaining R¦

b§ . Since the massof the

charged� Higgs bosonis a free parameterin supersymmetricmodels,we canimaginemakingit largeenoughcomparedtomÖ tË so£ that,accordingto Eq. É 82

¡ Ê,� it hasonly a smalleffecton

R¦

b§ . We thereforeconcentrateon the Higgsino-squarkpart.

The chargedHiggsinomixeswith the W-ino, and the right-handedtop squarkmixeswith its chiral counterpart,soin thenotation� of Eq. Ë 94

y Ì,� we havef

Í1 Î h

Ã2Ï ,� fÍ

2 Ð W Ñ ,� Ò 1 Ó tñ R ,� andÔ2 Õ tñ L . Thecorrespondingchargematricesfor thecouplings

to�

the W3ü are~

g¹ S< Ö 2

3ä s� w×22

0� 0

�12 Ø 2

3ä s� w×22 ;

g¹ LÈ Ù g¹ R

� Ú Û 123 Ü s� w×20� 0

�Ý

1 Þ s� w×22 . ß 107àBecauseá

there are two possiblecolor combinationsfor theinternal lines of the loops diagram,the color factor in Eqs.â87¡ ã

– ä 89¡ å

is n- ct æ 2.Beforeá

exploringthe full expressionfor ç g¹ Lb§

weÄ candis-cover� whatparameterrangesarethemostpromisingby look-ing at the limiting casesdescribedby Eqs. è 97

y é– ê 100ë . The

most important lessonsfrom theseapproximationsfollowfrom¤

the chargematricesì 107í . We do not want the squarksto�

bemuchheavierthanthecharginosbecausethenEq. î 98y ï

wouldÄ apply andgive the wrong sign for the correctiondueto�

thesignof theisospindifferencebetweenthesquarks.Theotherl two cases,wherethesquarksarenot muchheavierthanthe�

charginos,manifest a strong suppressionof the resultunless° the chargino mixing angles are such thatsin(£ ð

L ñ s� m� ò R)Í

is large,wheres� m� is the sign of the determi-nant� of the chargino mass matrix. If on the other handsin(£ ó

LÈ ô s� m� õ R

� )Í ö

0,�

there is exact cancellationbetweeng¹ LÈ

13Note÷

that the charged-scalarmixing in this modelis suppressedif one of the scalarmassesgetsvery large comparedto the weakscale.

TABLE VI. Field content and charge assignments: Elec-troweakquantumnumbersfor all of the scalars,including the SMHiggs doublet,of the three-doubletmodel.

Field Spin SUc3 ø 35 ù SUL ú 2û UY ü 1ýHþ

dÿ 0

<1 2 � 1

23

Hþ

u� 0<

12� 1

23�

0<

12 � 3ä2

54 4295R0

b1 AND NEW PHYSICS:A COMPREHENSIVEANALYSIS

and~ g¹ R� in�

theseequationsbecauseof thefact thatg¹ LÈ � g¹ R

� for¤

the�

charginos.In summary,our analytic formulas indicatethat�

thefavoredregionsof parameterspacefor increasingRb§

are~ where

tan� �

R� tan� �

LÈ � s� m� �� sgn£ mÖ f

� mÖ f� ��� ,� � 108�

and~ at leastoneof the squarksis not muchheavierthanthecharginos.�

In supersymmetricmodelstheYukawacouplingthatcon-trols�

the largestcontributionto R¦

b§ is�

that of the top quark,

and~ it dependson the ratio of the two Higgs VEV’s,tan� �����

2� /P � 1, b� y

y� f� ��� mÖ tË� sin£ � ,� 109!

whereÄ "$# (Ê %

12� &('

22�)Í 1/2) 174 GeV. Thereforeit is important

to�

find tan * in�

terms of the charginomassesand mixingangles.~ The charginomassmatrix is given by

+ g¹ , 2

g¹ - 1 M�

2� .0/ L

† mÖ f�

0� 0

�mÖ f� 1 2 R

� 3 c5 Lc5 RmÖ f� 4 s� Ls� RmÖ f

� 5c5 R� s� LÈ mÖ f� 6 s� R

� c5 LÈ mÖ f� 7 s� Rc5 LmÖ f

� 8 c5 Rs� LmÖ f� 9

s� LÈ s� R� mÖ f� : c5 L

È c5 R� mÖ f� ; ,� < 110=

whereÄ > is�

the coefficientof H»

1H»

2 in�

the superpotentialandM 2� is the soft-supersymmetry-breakingmassterm for the

W-ino. It follows that

tan� ?A@ mÖ f

� tan� B

R C mÖ f� D tan� E

L

mÖ f� tan� F

L G mÖ f� H tan� I

R. J 111K

Theaboveconsiderationsallow usto understandwhy val-ues° of tan L near� unity are necessaryfor a supersymmetricsolution£ to theR

¦b§ problem.¢ FromEq. M 111N and~ themaximi-

zation condition O 108P weÄ seethat tan Q is restrictedto liebetween� R

mÖ f� /P mÖ f

� SUT and~ V mÖ f� W /P mÖ f

� X . Equation Y 108Z together�

withÄ Eq. [ 110\ also~ implies

c5 L2 ] mÖ f

� ^`_ s� L2 a mÖ f

� bUc`dfe M�

W sin£ g ;

c5 LÈ2� h mÖ f

� ikjkl s� LÈ2� m mÖ f

� n`ofp M W cos� q . r 112s

Thisú

means that the averagevalue of the two charginomassescanbeno greaterthanM wt ,� so that theratio u mÖ f

� /P mÖ f� vkw

cannot� differ muchfrom unity unlessoneof thecharginosismuch� lighter than the W boson.

�Using the LEP 1.5 limit of

65x

GeV for thelightestcharginoy 26­ z

this�

would thenrequirethat�

tan {�| 1.5.In the casethat noneof our simplifying limits apply, we

have¥

searchedthe parameterspaceof the threeindependentratiosª betweenthe two scalarmassesand the two fermionmasses,andthe threemixing angles} R

� ,� ~ LÈ ,� � S

� to�

find whichregionsare favorablefor increasingRb

§ . Figures8� a~ � –8� d� �show£ the shift in g¹ L

b§

as~ a function of pairs of theseparam-eters,� using the Yukawa coupling � 109� corresponding� to atop�

quarkmassof 174GeV andthetheoreticalpreferencefortan� ���

1 � weÄ implementthe latter by settingg¹ Lb§ �

0�

for pa-rametersª thatwould give tan ��� 1� . As shownin TableI, oneneeds� � g¹ L

Èb§ �0� 0.0067�

in orderto explainthe observedvalue

FIG. 8. The dependenceof g Lb1

on the varioussupersymmetric� parameters.Since g� L

?b1 dependsonly� on massratios in our approximation,theunits� of massarearbitrary,with themassesof allthe�

charginosand squarkswhich are not beingvaried� set to unity.

4296 54BAMERT,D

BURGESS,CLINE, LONDON, AND NARDI

ofl Rb§ . The values of the masses are taken to be

M ��� M ����� mÖ f� � mÖ f

� ��  1 ¡ in arbitraryunits¢ ,� exceptfor thosethat�

areexplicitly variedin eachfigure. In Fig. 8£ a~ ¤ weÄ lookat~ the situationin which tan ¥ L ¦ tan

� §R ¨ 1, in contradiction

to�

condition © 108ª ,� andvary the scalarmixing angleandthemassof mostly tñ R� scalar£ in the limit of zerosquarkmixing.Theú

signof g¹ Lb§

has¥

thewrongvalue,aspredictedby Eq. « 98y ¬

.Figure­

8® b� ¯ shows£ the same situation except that nowtan� °

LÈ ±0² tan

� ³R� ´ 1, in accordancewith Eq. µ 108¶ . Thenthe

sign£ of g¹ LÈb§ is negative,asdesired,andhasthe right size for

substantial£ rangesof · S¸ and~ M

� ¹. In Fig. 8º c� » weÄ keepall the

masses� nearly degenerateand set ¼ S¸ ½ 0�

to show the depen-dence�

on tan ¾ LÈ and~ tan ¿ R

� . It is easyto seethat g¹ LÈb§ hasthe

correct� sign and largestmagnitudeÀ whichÄ is also almostaslarge�

asneededÁ whenÄ condition  108à is�

satisfied.Finally inFig.­

8Ä d� Å weÄ show the dependenceon the massesof themostly W-ino fermion and on Æ R

� whenÄ ÇS¸ È 0�

andtan� É

LÈ Ê0Ë 1, showingagainthe preferencefor mixing angles

obeyingl Eq. Ì 108Í ,� as well assomeenhancementwhenthereis�

a hierarchybetweenthe two charginomasses.One�

might thereforeget the impressionthat it is easytoexplain� Rb

§ using° supersymmetriccontributionsto the Zbbvertex.� Theproblemis thatto geta largeenoughcontributiononel is driven to a ratherspecialregion of parameterspace,whichÄ comesclose to satisfying condition Î 108Ï . As men-tioned�

above,the consequentcondition Ð 112Ñ prevents¢ onefrom¤

making the chargino massesarbitrarily heavy. This,coupled� with the suppressionin R

¦b§ whenÄ the squarksare

heavierthanthecharginos,meansthatall therelevantsuper-symmetric£ particles must be relatively light, except thecharged� Higgs bosonwhich hasto be heavyto suppressthewrong-signÄ contributionfrom H

» Ò$Ótñ loops.�

Thus in the ex-ample~ of Fig. 8Ô c� Õ ,� the preferredvaluesof c5 R

� Ö 1, s� LÈ ×0Ø 1,

s� R� Ù c5 L

È Ú 0�

imply that mÖ f� Û(Ü sin£ Ý and~ mÖ f

� Þ�ß(à cos� á ,� whileâäã M�

2 å 0,�

which arepreciselythe circumstancesof the su-persymmetric¢ modelsconsideredin Refs. æ 27

­ çand~ è 28

­ é. Fig-

ure° 8ê d� ë ,� on the otherhand,hasits maximumvalueof Rb§ at~

c5 R� ì s� R

� í c5 LÈ îï s� L

È ð 1, implying tan ñ�ò 1 andthus from Eq.ó112ô that

� õmÖ f� öU÷`øúù mÖ f

� û`ü 2­

M�

W . Becausethe lightestcharginomass� is constrainedby experimentallower limits, there islittle parameterspacefor getting a large hierarchybetweenthe�

two charginomasses,as one would want in the presentexample� in orderto getthefull shift14 ofl ý 0.0067

�in g¹ L

b§. Our

analysis~ allows one to pinpoint just wherethe favorablere-gions¨ arefor solving the Rb

§ problem.¢We�

thusseethat it is possibleto understandmanyof theconclusions� in the literature þ 2­ 7–31ÿ onl supersymmetryandR¦

b§ using° somerather simple analytic formulas. Thesein-

clude� thepreferencefor smallvaluesof tan � as~ well aslightHiggsinosandsquarks.

VI. FUTURE TESTS

If we excludethe possibility that the experimentalvalueofl Rb

§ is simply a 3.7� statistical£ fluctuation,we canexpect

that,�

once the LEP Collaborationshave completed theiranalyses~ of all the data collectedduring the five yearsofrunningª at the Z

¬pole,¢ the ‘‘ R

¦b§ crisis’’� will becomean even

more� seriousproblemfor the standardmodel. � Of�

course,itis wise to keepin mind that theremay be a simpleexplana-tion,�

namelythatsomesystematicuncertaintiesin theanaly-sis£ of the experimentaldataarestill not well understoodorhave¥

been underestimated.� Inè

Secs. III –V we have dis-cussed� a variety of modelsof new physicswhich could ac-count� for the experimentalmeasurementof Rb

§ . The nextobviousl stepis to considerwhich other measurementsmaybe�

usedto revealthe presenceof this new physics.The most direct method of finding the new physics is

clearly� the discoveryof new particleswith the correctcou-plings¢ to the Z

¬and~ the b

ëquark.« However,failing that, there

are~ some indirect tests. For example,many of the new-physics¢ mechanismswhich havebeenanalyzedin this paperwillÄ affect the rate for somerare B

�decays�

in a predictableway.Ä Theratesfor theraredecaysB

� �X�

s� l� l�

and~ B� �

X�

s� � �are~ essentiallycontrolled by the Zbs� effective� vertex � bs

§ � ,�since£ additionalcontributions� such£ asbox diagramsandZ- �interference� �

are~ largely subleading.15 Inè

the SM, in the ap-proximation¢ made throughoutthis paperof neglectingthebë

-quark massand momentum,a simple relation holds be-tween�

thedominantmÖ tË vertex� effectsin Rb§ and~ in theeffec-

tive�

Zb¬ ¯s� vertex� �

bs§ � :

�bs§ � ,SM � V tbË* V tsË�

V tbË � 2 ����� ,SM,� � 113 whereÄ !#"%$ ,SM is definedas in Eq. & 26' withÄ the SM formfactor as given in Eqs. ( 27) and~ * 38

+ ,. The meaningof Eq.-

113. is�

that, within the SM, the Zb¬ ¯s� effective� vertex mea-

surable£ in Z¬

-mediatedB�

decays�

representsa direct/

measure-�mentof the mÖ tË -dependentvertexcorrectionscontributingtoRb§ ,� moduloa ratio of the relevantCKM matrix elements.In

particular,¢ both correctionsvanish in the mÖ tË 0 0�

limit. Thequestion« is now the following: how is this relationaffectedby�

the new physicsinvokedin Secs.III –V to explainRb§ ?

Consider1

first the tree-levelbë

-bë 2

mixing effectsanalyzedin�

Sec.III. It is straightforwardto relate the correctionsofthe�

LH and RH Zb¬ ¯bë

couplings� to new tree-levelmixing-inducedFCNC couplingsg¹ L

È,R�bs

§. In this caseEq. 3 54 5 reads

g¹ L,Rbs§ 687 9

w: ; g¹ w: <>= LÈ

,R� ?

L,R

@ b§* A L,R

B s� C w: 1. D 114EHenceg¹ L

È,R�bs

§involve the samegaugecouplingsand mixing

matrices� that determinethe deviation/

from¤

the SM of theflavor-diagonalF

bë

couplings.�It is also true that, for manymodelsof new physics,the

loop correctionsto the Zbbë

vertex� would changethe effec-tive�

Zb¬ ¯s� vertex� in much the sameway, thereforeinducing

computable� modificationsto the SM electroweakpenguindiagrams.G

In thesemodels,for eachloop diagraminvolving14An additionalconstraintis that the lightest Higgs bosonmass

mhH 0 vanishes� at treelevel whentan IKJ 1, anda very largesplitting

betweenthe top squarkmassesis neededfor the one-loopcorrec-tions to mh

H 0 to�

be large enough.This is why Ref. L 29M finds lessthanthedesiredshift in Rb

1 in theminimal supersymmetricstandardmodel.We thankJ. Lopezfor clarifying this point.

15Due to the absence of ZN

- O interferenceP

and of largerenormalization-group-inducedQCD corrections, the processB Q XsR S�S representstheoreticallythecleanestproof of theeffectiveZbN ¯sT vertex� U 32

5 V.

54 4297R0

b1 AND NEW PHYSICS:A COMPREHENSIVEANALYSIS

the�

new statesfW

,� fW X and~ their coupling to the bë

quark« g¹ f f� Y

b§ ,�

there�

will be a similar diagramcontributingto Z bs§ [ that�

canbe�

obtainedby the simplereplacementg¹ f f� \

b§ ] g¹ f f

� ^s� . For ex-

ample,_ the generalanalysisof tñ -quark mixing effects pre-sented` in Sec. IV can be straightforwardly applied toZ-mediatedB decays.

aDeviationsfrom the SM predictions

for the B b Xs� l� c l� d

and_ B e Xs� fgf decaya

ratescanbe easilyevaluatedh by means of a few simple replacementslikei j

tbË k 2 l V tbË* V tsË and_ m n

tË o bp q 2 r V tË s bp* V tË t s� inu

all our equations.16

To a largeextent,this is alsotruefor supersymmetryv SUSYw x

models.Indeed,the analysisof the SUSY contributionstothey

Zb¬ ¯s� form

zfactor { 34

+ |can} teachmuchaboutSUSYeffects

inu

R~

bp . And oncea particularregionof parameterspacesuit-

able_ to explainthe Rbp problem� is chosen,a definitenumeri-�

cal� prediction� for the B � Xs� l� � l� �

and_ B � Xs� �g� decaya

ratescan} be made.

Thisú

brief discussionshowsthat, for a largeclassof new-physics� models,the new contributionsto Rb

p and_ to the ef-fective � bs

p � vertex� arecomputablein termsof thesamesetofnew-physics� parameters.Therefore,for all thesemodels,theassumption_ that somenew physicsis responsiblefor the de-viations� of Rb

p from the SM predictionwill imply a quanti-tativey

prediction of the corresponding deviations forZ¬

-mediatedB�

decays.a

However,�

this statementcannot be applied to all new-physics� possibilities.For example,if a new Z � boson

�is re-

sponsible` for the measuredvalueof Rbp ,� thenno signal can

be�

expectedin B�

decays,a

sincein this casethe new physicsrespects� the GIM mechanism.This would also be true ifmÖ bp -dependenteffectsareresponsiblefor theobserveddevia-

tionsy

in Rbp as_ could happen,for example,in the very large

tany �

region� of multi-Higgs-doubletor SUSY models.Moregenerally,� the loop contributionsof the new statesf

W,� fW � can}

be�

different, since g¹ f f� �

s� is not necessarilyrelatedto g¹ f f� �

bp ,�

and_ in particular,wheneverthe new physicsinvolved in Rbp

couples} principally to thethird generation,it is quitepossiblethaty

no sizeableeffect will show up in B�

decays.a

Still, thestudy` of B � Xs� l� � l

� �and_ B � Xs� �g� could} help to distinguish

between�

modelsthat do or do not significantly affect thesedecays.a

Unfortunately,�

at presentonly upperlimits havebeenseton� the branching ratios for B � Xs� l� � l

� � �3+5–37¡ and_

B ¢ Xs� £g£ ¤ 32+ ¥

. Sincetheselimits area few timeslargerthanthey

SM predictions,theycannothelp to pin downthecorrectsolution` to the R

~bp problem.� However,future measurements

of� theserare decaysat B factoriescould well confirm thatnew physicsis affectingthe rateof b

ë-quarkproductionin Z

decays,a

as well as give somehints as to its identity. If nosignificant` deviationsfrom theSM expectationsaredetected,thisy

would alsohelp to restrict the remainingpossibilities.

VII.¦

CONCLUSIONS

Until�

recently,the SM hasenjoyedenormoussuccessinexplainingh all electroweakphenomena.However,a number

of� chinks have startedto appearin its armour. There arecurrently} severaldisagreementsbetweentheory and experi-ment§ at the2 level

©or greater.TheyareR

~bp ª¬«

bp /­ ® had ¯ 3.7

+ °g±,�

R~

ct ²¬³ ct /­ ´ hadµ ¶ 2.5

· ¸g¹,� the inconsistencybetweenA

ºe»0¼ as_ mea-

sured` at LEP with thatdeterminedat SLC ½ 2.4¾g¿ ,� andA FBÀ0¼ ÁÃÂÅÄÁ

2.0ÆgÇ . Takentogether,the datanow excludethe SM at the98.8%y

confidencelevel.OfÈ

theabovediscrepancies,it is essentiallyonly R~

bp whichÉ

causes} problems.If Rbp by�

itself is assumedto be accountedforz

by new physics,thenthe fit to the datadespitethe otherdiscrepanciesa

is reasonableÊÌË min2 /­NÍ

DF Î 15.5/11Ï —the othermeasurementscould thus be regardedsimply as statisticalfluctuations.

InÐ

this paperwe haveperformeda systematicsurveyofnew-physics� models in order to determinewhich featuresgive� correctionsto Rb

p of� the right sign andmagnitude.Themodelsconsideredcan be separatedinto two broad class-es:h thosein which new Zb

Ñ ¯bÒ

couplings} appearat treelevel,by�

ZÑ

or� bÒ

-quarkmixing with newparticles,andthosewhichgive� loop correctionsto the Zbb

Òvertex.� The latter type in-

cludes} tÓ -quarkmixing andmodelswith new scalarsandfer-mions.§ We did not considertechnicolormodelsor newgaugebosons�

appearingin loopssincethesecasesaremuchmoremodeldependent.

The new physics can modify either the left-handedorright-handed� Zb

Ñ ¯bÒ

couplings,} gÔ Lbp

or� gÔ Rbp. To increaseR

~bp toy

itsexperimentalh value, Õ gÔ L

bp

must§ be negativeandhavea mag-nitude typical of a loop correctionwith large Yukawa cou-plings.� Thus Ö gÔ L

×bp could} eitherbea small tree-leveleffect,ora_ largeone-loopeffect. On the otherhand,the SM valueofgÔ R

bp

isu

oppositein sign to its LH counterpartandis aboutfivetimesy

smaller.Thereforeone would needa large tree-levelmodificationto gÔ R

Øbp toy

explainfor Rbp .

Here�

areour results.(1)Ù

Tree-level effects. ItÐ

is straightforwardto explainR~

bp ifu

they

Z or� bÒ

mix with newparticles.With Z Ú Z Û mixing thereare_ constraintsfrom neutral-currentmeasurements,but thesedoa

not excludeall models.UsingbÒ

-bÒ Ü

mixing§ is easiersincethey

experimentalvalue of R~

bp can} be accommodatedby

bÒ

L× -bÒ

LÝ or� bÒ

RØ -bÒ

RÞ mixing.§ If the mixing is in the LB bÒ

sector,`theny

solutionsare possibleso long as I3ß

L×à¬áãâ 1/2. An addi-

tionaly

possibility with I3ß

L×ä¬å 0

æand very large LH mixing,

thoughy

perhapsunappealing,is still viable.For RH bÒ

mixing,§ifu

Iç

3ß

R

èêé0æ

then small mixing is permitted,while if Iç

3ß

R

ëíì0,æ

large©

mixing is necessary.Interestingly,the requiredlargebÒ

-mixing anglesare still not ruled out phenomenologically.A numberof papersin the literaturehaveappealedto b

Ò-bÒ î

mixing to explainRbp . Our ‘‘master formula’’ ï 8¡ ð and_ Table

IIÐ

includeall of thesemodels,aswell asmanyothers.(2)Ù

Loops: t-tÓ ñ mixing.ò InÐ

the presenceof tÓ -tÓ ó mixing,§they

SM radiative correctioncan be reduced,dependingonthey

weakisospinquantumnumbersof the tÓ ô as_ well ason theLHõ

andRH mixing angles.However,we found that it is notpossible� to completelyexplainR

~bp via� this method.The best

weÉ cando is to decreasethediscrepancybetweentheoryandexperimenth to about2ö . Sucha scenariopredictsthe exist-enceh of a light ÷ùø 100 GeVú charge} 2/3 quark,decayingpri-marily§ to Wb.

(3)Ù

Loops: Diagonal couplings to the Z. We consideredmodelswith exotic fermionsandscalarcouplingto both theZÑ

and_ bÒ

quark.û We assumedthat the couplingsto the ZÑ

are_16Forü

example,theparticularcaseof mixing of thetop quarkwitha new isosingletT ý , and the correspondingeffectsinducedon theZbs vertex,þ wasstudiedin Ref. ÿ 33

� �through�

ananalysisvery simi-lar to that of Sec.IV.

4298 54BAMERT,�

BURGESS,CLINE, LONDON, AND NARDI

diagonal,a

i.e., thereare no flavor-changingneutral currents�FCNC’s� . The correction � gÔ L

bp

can} then be written in asimple` form, Eq. � 82

� . The key point is that gÔ L

bp

isu

propor-tionaly

to Iç

3ß

L×f� � I

ç3ß

RØf� ,� where I

ç3ß

L×

,RØf

�isu

the third componentofweakÉ isospinof the fermion field f

WL×

,RØ in the loop. This ex-

plains� at a glancewhy many models,suchas multi-Higgs-doubleta

models and Zee-type models, have difficulty ex-plaining� R

~bp . Since the dominant contributions in these

models§ typically havetop-typequarks Iç 3ß

L

��� 12� ,� Iç

3ß

R

���0æ �

cir-}culating} in the loop, they give correctionsof the wrong signtoy

R~

bp . However,theseconsiderationsdid permit us to con-

struct` viable modelsof this type which do explainRbp . Two

such` examplesaregiven in Sec.V D, andmany otherscanbe�

invented.(4)Ù

Loops: Nondiagonal couplings to the Z. We alsoex-amined_ modelswith exotic fermionsandscalarswhich wereallowed_ to havenondiagonalcouplingsto the Z. SuchFC-NC’s�

canoccurwhenparticlesof differentweakisospinmix.The�

correction � gÔ Lbp

isu

much more complicated � Eq.� �

86� ���

thany

in the previouscase;evenits sign is not obvious.However, there are several interesting limiting cases

whereÉ it againbecomestransparent.The contributionsto R~

bp

of� supersymmetryfall into this category,which we discussedin somedetail.

NoteÍ

added. After completingthis work we becameawareof� Ref. � 38

+ ,� which discussesa different regionof parameter

space` in SUSYmodelsthantheonewe focusedon. Becauseof� our criterionof explainingtheentireRb

p discrepancya

ratherthany

only reducingits statisticalsignificance,we excludetheregion� in question.

ACKNOWLEDGMENTS

This�

researchwas financially supportedby NSERC ofCanada!

andFCAR of Quebec.�

E.N. wishesto acknowledgethey

pleasanthospitalityof thePhysicsDepartmentat McGillUniversity,�

during the final stageof this work. D.L. wouldlike©

to thankKen Raganfor helpful conversations.

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BURGESS,CLINE, LONDON, AND NARDI