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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
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