Beyond the Standard Model in B Decays: Three Topics

12 Pages·0.11 MB·English

by  A. L. Kagan

Beyond the Standard Model in B Decays: Three
Topics1
Alexander L. Kagan 2
2 PhysicsDepartment,UniversityofCincinnati,CincinnatiOH45221,USA
0
0
Abstract. Three new results are discussed: (a) A non-vanishing amplitude for the ‘wrong sign’
2
kaon decay B J/Y K or its CP conjugate is shown to be a necessary condition for obtaining
n different CP a→symmetries in B J/Y K . A significant effect would require a scale of new
a → S,L
physicsfarbelowtheweakscale,allbutrulingoutthispossibility.(b)Theleadingisospinbreaking
J
1 contributionstotheB→K∗g decayamplitudescanbecalculatedinQCDfactorization,providing
a sensitive probe of the penguin sector of the effective weak Hamiltonian. New physics models
3
whichreversethepredicted10 20%StandardModelamplitudehierarchycouldberuledoutwith

more precise data. (c) A slowly−falling g gh form factor can be ruled out using the h spectrum
1 ∗ ′ ′
v obtainedbyARGUSatthe¡ (1S).Thedecayb sgh ′isthereforehighlysuppressedandtheorigin
3 of the anomalously large B h ′Xs rate remai→ns unknown, perhaps requiring the intervention of
→
1 NewPhysics.
3
1
0
2 INTRODUCTION
0
/
h ThreerecentdevelopmentswithpotentialimplicationsfornewphysicsandBdecaysare
p
discussed. Recently, the BaBaR and Belle collaborations presented new measurements
-
p of the time-dependent CP asymmetries in B J/Y K and B J/Y K [1, 2]. In the
S L
e → →
Standard Model the two CP asymmetries are predicted to have the same magnitudebut
h
: oppositesign. Althoughthe measured values are equal within errors the CP asymmetry
v
i in the B J/Y KL channel is somewhat larger, which raises the question of whether
X a (B →y K ) = a (B y K ) is possible. The conditions which must be fulfilled
CP L CP S
r | → |6 | → |
a forthistohappenareexplainedbelow[3].Amodel-independentanalyisimpliesthatthe
associatedscaleofnewphysicsinteractonswouldhavetolieataprohibitivelylowscale
ofa fewGeV in ordertoobtainasignificanteffect.
Measurements of the exclusive B K g branching ratios have been reported by the
∗
→
CLEO, Belle and BaBar Collaborations, with the results (averaged over CP-conjugate
modes):
4.55+0.72 0.34 [4]
105Br(B¯0 K¯∗0g ) = 4.96−00.6.86±7 0.45 [5]
→  ± ±
4.23 0.40 0.22 [6]

± ±

1 Invitedtalkatthe9thInternationalSymposiumonHeavyFlavorPhysics,Caltech,Pasadena,Sept.10-
13,2001
2 WorksupportedbytheDepartmentofEnergy,GrantNo.DOEDEFG02-84ER-40153
3.76+0.89 0.28 [4]
105Br(B− K¯∗−g ) = 3.89−00.8.39±3 0.41 [5]
→  ± ±
3.83 0.62 0.22 [6]

± ±
The average branching ratios for the two modes are (4.44 0.35) 10 5 and (3.82
−
0.47) 10 5. When corrected for the difference in the B-m±eson lif·etimes, t /t =±
− B B¯0
· −
1.068 0.016[7], theseresultsimply
±
G (B¯0 K¯ 0g ) G (B K¯ g )
D → ∗ − −→ ∗− =0.11 0.07. (1)
0− ≡ G (B¯0 K¯∗0g )+G (B− K¯∗−g ) ±
→ →
Although there is no significant deviation of this quantity from zero, the fact that all
three experiments see a tendency for a larger neutral decay rate raises the question of
whether the Standard Model can account for isospin-breaking effects of order 10% in
thedecay amplitudes.
Recently it has been shown that in the heavy-quark limit the decay amplitudes for
these processes can be calculated in a model-independent way using a QCD factoriza-
tion approach [8, 9], which is similar to the scheme developed for the analysis of non-
leptonictwo-bodydecaysofBmesons[10].ToleadingorderinL /m onefindsthatthe
b
amplitudesforthedecays B¯0 K¯ 0g and B K g coincide.Here wereporton sub-
∗ − ∗−
→ →
sequent work [11], in which the QCD factorization approach was used to estimate the
leading isospin-breaking contributions for the B K g decay amplitudes in the Stan-
∗
dardModel.Theseareduetoannihilationgraphsw→hichenteratorderL /m .Becauseof
b
theirrelationtomatrixelementsofpenguinoperators, wewillseethatisospin-breaking
effectsin B K g decays aresensitiveprobes ofphysicsbeyondtheStandard Model.
∗
→
Finally,theCLEO collaborationandmorerecently,asweheard atthisworkshop,the
BaBaR collaboration havemeasured very large rates for fast h production in B h X
′ ′ s
→
decays:
6.2 1.6 1.3+0 10 4; CLEO [12],
BR (B→h ′Xs)ph ′>2 GeV =(6.8+±.17.0±±1+0.5×−11.50×−4; − BaBar [13]. (2)
− −
The experimental cut on ph is beyond the kinematic limit for most b c decays. A
′ →
majority of the events lie at large recoil mass, consistent with a three-body or higher
multiplicitydecay.Theh yieldfromtheb ccomponentisdominatedbyintermediate
′
charmonia decays, which contribute only→ 1.1 10 4 [14] to the branching ratio.
−
Charmless h production proceeding via the≈quark×content of the h has been estimated
′ ′
using factorization [15, 16], giving a contribution to the branching ratio of 1 10 4
−
≈ ×
withaquasitwo-bodyrecoilspectrumthatispeakedatlowenergiesinconflictwiththe
observedspectrum.
The surprisinglylarge h yield led Atwood and Soni [14] to propose that it is associ-
′
atedwiththegluoniccontentoftheh viathesubprocess b s(g h g).Makingthe
′ ∗ ′
key assumptionthat the form factor remains constant up to→q2 m2→, where q is the vir-
tual gluon’smomentum, they showed that the large h yield co∼uld ebasily be reproduced
′
intheStandardModel.Moreover,theyobservedthatthethree-bodydecayleadstoanh
′
recoil spectrum that is consistent with observation. Hou and Tseng [17] argued that the
factorofa implicitinH shouldbeevaluatedatthescaleofmomentumtransferthrough
s
theg gh vertex. Howeverthis would only introduce a mild logarithmicsuppressionof
∗ ′
theform factorversusq2 and thereforestilllead toalarge h yield.
′
Assymptotically, perturbative QCD (pQCD) predicts that the leading form factor
contributions should fall like 1/q2. The question is in what region of q2 does this
behaviour set in? We will see [20] that the h spectrum measured in ¡ (1S) h X
′ ′
→
decays by the ARGUS Collaboration [18] rules out form factors which fall slowly in
the range q2 m2. However, a rapidly falling form factor [16] representative of pQCD
predictions [2≤1, 2b2] is consistent with the data. The corresponding b sgh branching
′
→
ratio in the Standard Model is about a factor of 20 smaller than the measured values in
Eq.(2).
Y
CAN THE CP ASYMMETRIES IN B J/ K BE DIFFERENT?
S,L
→
Therelevantquantitiesare[23]
q A¯
l B S,L, (3)
S,L
≡ p A
B S,L
where
A¯ A(B¯ y K ), A A(B y K ). (4)
S,L S,L S,L S,L
≡ → ≡ →
The neutral B and K meson mass eigenstates are defined in the usual way in terms of
flavoreigenstates,
B = p B q B¯ , K = p K q K¯ . (5)
L,H B B S,L K K
| i | i± | i | i | i± | i
Thetime-dependentCP asymmetriesare givenby
Iml 1 l 2
a (B y K )= 2 S,L sinD m t+ −| S,L| cosD m t. (6)
CP → S,L − 1+ l 2 B 1+ l 2 B
S,L S,L
| | | |
In the limit of no direct CP violation ( l =1) the asymmetries reduce to the simple
S,L
forma (B y K )= Iml sinD |m t,|as in theStandard Model.
CP S,L S,L B
Weneedt→orewritel −intermsofthedecay amplitudesintokaonflavoreigenstates,
S,L
A¯ A(B¯ y K), A¯ A(B¯ y K¯), (7)
K K¯
≡ → ≡ →
A A(B y K), A A(B y K¯). (8)
K K¯
≡ → ≡ →
Allowing for the possibility that the ‘wrong-sign’ kaon amplitudes A¯ and A receive
K K¯
newphysicscontributions(theyare negligibleintheStandard Model),oneobtains
1 a
l = l ± , (9)
S,L
± 1 b
(cid:18) ± (cid:19)
where l q q A¯ /p p A , and a and b are proportional to ratios of wrong-sign to
B K K¯ B K K
≡
right-signkaonamplitudes
p A¯ q A
a K K , b K K¯ . (10)
≡ q A¯ ≡ p A
K K¯ K K
In the Standard Model and more generally, in any model in which the wrong-sign
amplitudes are negligibly small (a=b=0), this reduces to l = l so that Im(l +
S,L S
l )=0. The sinD m t term therefore has equal magnitudebut oppos±itesign for thetwo
L b
CPasymmetries.
From the general relation l +l = l 2(a b)/(1 b2) we learn that a necessary
S L
and sufficient condition for l = l to be sa−tisfied is−the presence of non-vanishing
S L
6
wrong-signamplitudes with a=b. As an example, if each of theright-sign and wrong-
6
sign amplitudes is dominated by a single contribution, then a b . If in addition,
Rel Iml O(1), as in the Standard Model, and Arg[a] A|rg|[b≈] | |O(1), we obtain
Im(l ∼+l )∼ a b . Thus the difference in CP asymm∼etries is ∼of the order of the
S L
∼ | | ∼ | |
ratioofwrong-signtoright-signkaon amplitudes.
Constraints on New Physics Scenarios
Consider the wrong-sign decay B0 J/Y K. As the final state does not contain a d
→
quark, the decay must proceed via annihilation of the B meson. It must therefore be
mediatedbysix-quarkoperators, withan effectiveHamiltonianoftheform
g
bscc¯d¯d¯, (11)
M5
wheregisadimensionlesscoupling,Misthescaleofnewphysics(thecolorindicesand
Dirac structure of the operators have been suppressed). An estimate of the wrong-sign
amplitudeinthefactorizationapproximationyields
g
AK¯ ∼ M5 fBfKfY mBmY (e Y ·pK). (12)
Forpurposesofcomparison,wenotethatintheStandardModeltheright-signamplitude
inthefactorizationapproximationisgivenby
G
AK = FVcba2fY F1mY (e Y pK), (13)
√2 ·
where F is the B K form factor, and a is a function of the current-current operator
1 2
Wilson coefficient→s C and C arising from W exchange. The observed B J/Y K
1 2 S
→
amplitudeisreproduced ifa .25.
2
We can get an order of mag≈nitude upper bound on the quantity g/M2 by integrating
out the charm quarks at one-loop, yielding a contribution to the effective Hamiltonian
mediatingcharmlesshadronicB decays oforder
1 g
bsd¯d¯. (14)
16p 2M2
Upper bounds on the strenghths of such operators can be obtained by considering their
contributions to the rare decays B p K or, more specifically, to the wrong-sign
± ± S
kaon decays B p K¯0 [24] in th→e factorization approximation, yielding g/M2 <
± ±
10 5 GeV 2. Ins→erting this bound into our estimate for A in Eq. (12) and equatin∼g
− − K¯
the result with the expression for A in Eq. (13) (for a .25) implies that a scale of
K 2
≈
new physics M of a few GeV is required in order to obtain significant wrong-sign kaon
amplitudes.
We know of only one scenario with such a potentially low scale for new flavor-
changing interactions: supersymmetric models with a light bottom squark b˜ of mass
2-5.5 GeV and light gluinos of mass 12-16 GeV [25], which have been proposed to
enhance the b quark production cross section at hadron colliders. Among the new
operators which can arise at tree-level are several of the form d¯bb˜ b˜. Stringent upper
∗
bounds on their strenghths from rare B decays have been obtained in [26]. Interactions
of the desired form in Eq. (11) would be generated from these operators if the R-parity
violating Yukawa couplings mediating b˜ c¯d¯ and b˜ c¯s¯ decays were also present.
Unfortunately,anupperboundoforder10→5 ontheprod→uctofthesetwocouplingsfrom
−
box-graph contributions to K K¯ mixing implies that the wrong-sign kaon amplitudes
wouldbenegligiblysmall,i.e.−, a b 10 5. Althoughthisresult does notconstitutea
−
∼ ∼
no-go theorem, it appears that the possibility of significantly different CP asymmetries
inB J/Y K decaysisextremelyunlikelyduetotherequirementofsuchalowscale
S,L
→
fornewinteractions.
g
ISOSPIN VIOLATION IN B K
∗
→
IntheStandard Model,theeffectiveweak Hamiltonianforb sg transitionsis
→
H = GF (cid:229) l (s) C Qp+C Qp+ (cid:229) C Q , (15)
eff √2 p 1 1 2 2 i i
p=u,c i=3,...,8
(cid:16) (cid:17)
where l (s) =V V , Qp are the current–current operators, Q are the local four-
p p∗s pb 1,2 3,...,6
quark QCD penguin operators, and Q and Q are the electro-magnetic and chromo-
7 8
magnetic dipole operators. (We adopt the conventions of [10]; in particular, C 1
1
≈
is the largest coefficient.) The Wilson coefficients C and the matrix elements of the
i
renormalizedoperators Q depend on therenormalizationscaleµ.
i
At leadingpowerin L /m theB K g decay amplitudeisgivenby
b ∗
→
G
iA = F l (s)ac K¯ (k,h )g (q,e ) Q B¯ , (16)
lead c 7 ∗ 7
√2 h | | i
where the next-to-leading order (NLO) result for the coefficient ac =C +... can be
7 7
foundin[8]. TheellipsesdenoteO(a )hard spectatorinteractioncontributions.
s
The leading isospin-breaking effects arise from the diagrams shown in Fig. 1. Their
contributions to the decay amplitudes can be parametrized as A =b A , where q is
q q lead
the flavor of the spectator antiquark in the B¯ meson. We neglect NLO terms of order
a C while retaining terms of order a C . This is justified, because the penguin
s 3,...,6 s 1,8
b s b s b s
c
q q
q q q q
FIGURE 1. Spectator-dependent contributions from local 4-quark operators (left), the chromo-
magnetic dipole operator (center), and the charm penguin (right). Crosses denote alternative photon
attachments.
coefficientsC arenumericallyverysmall.Also,itisasafeapproximationtoneglect
3,...,6
terms of order a l (s)/l (s). It then suffices to evaluate the contributions of the 4-quark
s u c
operatorsshowninthefirst diagramat treelevel.
The QCD factorization approach gives an expression for the coefficients b in terms
q
ofconvolutionsofhard-scatteringkernelswithlight-conedistributionamplitudesforthe
K and Bmesons.Theresultcan bewrittenas
∗
12p 2f Q f f m
bq = mbT1B→BK∗aqc7 (cid:18)mK⊥b∗ K1+ 6Kl ∗BmKB∗ K2(cid:19), (17)
where TB K∗ is a form factor in the decomposition of the B K matrix element of
thetenso1r→current,andm denotestherunningb-quarkmass.l →isa∗hadronicparameter
b B
which enters the first inverse moment of the relevant B meson distribution amplitude.
Heavy-quark scaling laws imply that b scales like L /m . However, because of the
q b
large numerical factor 12p 2 in the numerator the values of b turn out to be larger than
q
anticipatedin [8, 9].
ThedimensionlessquantitiesK andK arelinearfunctionsoftheWilsoncoefficients.
1 2
The latterare multipliedby convolutionintegrals of hard scattering kernels with meson
distribution amplitudes. The integrals associated with the four-quark QCD penguin
operatorsandcurrent-currentoperatorsexistforanyreasonablechoiceofthedistribution
amplitudes,whichshowsthattheQCDfactorizationapproachholdsatsubleadingpower
fortheirmatrixelements,totheorderweareworking.However,theconvolutionintegral
associated with the chromomagnetic dipole operator suffers from a logarithmic end
point singularity, indicating that at subleading power factorization breaks down for this
matrixelement. We regulatethe singularityby introducinga cutoff. A large uncertainty
isassignedtothisestimatesincethiscontributionmustbedominatedbysoft physics.
Toleading-orderinsmallquantitiesthetheoreticalexpressionfortheisospin-breaking
parameter (see Eq. (1)) is D =Re(b b ). A dominantuncertainty in theprediction
0 d u
forD 0 comesfromthetenso−rformfact−orT1B→K∗,recentestimatesofwhichrangefrom
0.32+−0.04 [27] to 0.38 0.06 [28]. On the other hand, a fit to the B K g branching
0.02 ± → ∗
fracti−ons yields the lower value 0.27 0.04 [9]. To good approximation the result for
±
D 0 is inversely proportional to the value of the form factor. We take T1B→K∗ =0.3 (at
µ =−m ) as a reference value. Values for the remaining input parameters together with
b
theiruncertaintiescan befound inRef. [11].
Combiningall sources ofuncertaintyweobtaintheStandard Model result
0.3
D =(8.0+2.1)% . (18)
0− −3.2 ×T1B→K∗
The three largest contributions to the error from input parameter variations are due to
l (+1.0%), the divergent integral in the matrix element of Q ( 1.2%), and the decay
B 2.5 8 ±
cons−tant f ( 0.8%). The perturbativeuncertainty is about 1%. Our result is in good
B
agreementwi±ththecurrentcentralexperimentalvalueofD ±includingitssign,whichis
0
predictedunambiguously.Byfarthemostimportantsource−ofisospin-breakingisdueto
thefour-quarkpenguinoperatorQ ,whosecontributiontoD isabout9%(atµ m ).
6 0 b
Theothertermsaremuchsmaller.Inparticular,thecontributio−nofthechromo-ma≈gnetic
dipoleoperator,forwhich factorizationdoes nothold,isless than1% inmagnitudeand
thereforenumericallyinsignificant.Hence,theimportantisospin-breakingcontributions
can be reliably calculated usingQCD factorization. It followsfrom our result that these
effectsmainlytestthemagnitudeandsignoftheratioRe(C /C )ofpenguincoefficients.
6 7
Because of their relation to matrix elements of penguin operators, isospin-breaking
effects in B K g decays are sensitive probes of physics beyond the Standard Model.
∗
In particular→, scenarios in which the sign of D is flipped could be ruled out in the
0
near future with more precise data. For simplicit−y lets restrict ourselves to new physics
models which do not enlarge the Standard Model operator basis, where this possibility
corresponds to flipping the sign of Re(C /C ). As a specific example, consider the
6 7
minimal supersymmetric standard model (MSSM) with minimal flavor violation, and
withtanb enhancedcontributionstoB X g decaystakenintoaccountbeyondleading-
s
→
order [29]. In this scenario new contributions to Q , and Q are too small to have
3,...,6 8
a significant effect. For low tanb , ReC (m ) is negative as in the Standard Model.
7 b
However, for tanb > 27 ReC (m ) can take on both positive or negative values with
7 b
positive values bec∼oming more probable as tanb increases [30]. Therefore, isospin
breakingcouldruleoutsignificantregionsofMSSMparameterspaceatlargetanb .The
sign of ReC can also be flipped in supersymmetric models with non-minimal flavor
7
violation (independently of tanb ) via gluino/down-type squark loop graphs, without
affecting the sign of C . A more detailed treatment of new physics effects, including
6
thepossibilityofan enlarged operatorbasiswillbepresented elsewhere.
h ¡
CONSTRAINTS ON THE G G FORM FACTOR FROM (1S)
∗ ′
DECAYS
Theeffectiveh g g couplingcan bewrittenas
′ ∗
H(q2)e ba µn qa kb e µ1e n2, (19)
where q= p p is thevirtual gluon’smomentum,k is the‘on-shell’gluon’smomen-
b s
tum,and H(q2−) is theg gh transitionform factor. TheQCD axial anomalydetermines
∗ ′
2
)
2 1.5
q
(
H
1
PSfrag repla
ements (a)
0.5
(b) (
)
2 4 6 8 10
2 2
q [GeV ℄
FIGURE2. ThreechoicesfortheformfactorH(q2),asdescribedinthetext:(a)theslowlyfallingform
factor,(b)arapidlyfallingformfactorrepresentativeofperturbativeQCDcalculations,(c)anintermediate
example.
theformfactoratsmallmomentumtransfer,i.e.,intheq2 0limit.Anestimateofthis
limitfromthedecay ratefor J/y h g gives[14]H(0) →1.8GeV 1.
′ −
→ ≈
As we havediscussed,thecrucial issuethat needs to be addressed in determiningthe
b sgh decay rate is the dependence of H on q2. A simple model for the ggh vertex
′ ′
→
[16] in which a pseudoscalar current is coupled perturbatively to two gluons through
quarkloopsyieldsaform factorwhichcan beparametrized as
H m2
0 h
H(q2)= ′ . (20)
q2 m2
h
− ′
The dependence of H on q2 is subleading, but it insures the absence of a pole at
0
q2 = m2 . To first approximation it can be modeled by a constant to be identified with
h
′
the low energy coupling extracted from J/y h g , e.g., H 1.8 GeV 1. The above
′ 0 −
→ ≈
parametrization agrees well with recent pQCD calculations of the form factor [21, 22]
in which hard amplitudesinvolvingquark and gluonexchanges are convolutedwith the
h quark and gluonwavefunctions,particularlyifH 1.7 GeV 1.
′ 0 −
Below wewillconsiderthreerepresentativechoices≈forH(q2):
a) The slowly falling form factor of Ref. [17], H(q2) = Nfa s(q2)cosq /(p fh )
′ ≈
2.1a (q2)/a (m2 ) GeV 1 (q is the pseudoscalar mixing angle). It gives BR(b
s s h −
′ p →
sgh ′) 6.8 10−4 for ph >2 GeV.
≈ × ′
b) The rapidly falling form factor of Eq. (20), with H 1.7 GeV 1 (at low q2 it is
0 −
matched onto the value of the previous form factor a≈t q2 =m2 ). It gives BR(b
h
′ →
sgh ′) 3 10−5 for ph >2 GeV.
≈ × ′
c) An intermediate purely phenomenological form factor H(q2) (cid:181) 1/(q2+M2) with
M =2.2 GeV, whichgivesBR(b sgh ′) 4.4 10−4 for ph >2 GeV.
→ ≈ × ′
Thethreeform factors are plottedin Fig.2.
Theg gh formfactorinducesthedecay3S[1] gg(g h g)ofthedominantcolor-
∗ ′ 1 → ∗→ ′
singlet¡ (1S)Fockstate.Moreover,theregionofq2relevantforfasth productioninthis
′
decayhas largeoverlapwiththeregionofq2 relevantforfast h productioninb sgh
′ ′
decays.Therefore,theh spectrumin¡ (1S)decayscouldpotentiallyconstrainthe→g gh
′ ∗ ′
form factor, and at the same time tell us if the subprocess b sgh can account for the
′
h yieldin B decays [20]. →
′
The ARGUS collaboration mesured inclusive h production at the ¡ (1S) [18]. Un-
′
fortunately, due to limited it was not possible to subtract the continuum contribu-
tion from the h spectrum, so that modeling of the direct ¡ (1S) decay and contin-
′
uum components was required. The term ‘direct’ refers to hadronic ¡ (1S) decays with
e+e ¡ (1S) qq¯ events subtracted (the latter are included in the continuum data
−
sample→),essentia→llyleaving¡ (1S) gggevents.TheARGUSfitfordirectdecaysgives
dn/dz 16.8b e−10.6z, where n is→the h ′ multiplicity, and z 2Eh /m¡ . However, it is
possibl≈e to obtain a model-independent upper bound on th≡e h m′ultiplicity in each z
′
bin by assuming that the raw h yield is entirely due to direct ¡ (1S) decays (note that
′
the total number of direct ¡ (1S) decays in the data sample is known).3 Of particular
interest is the bound for the highest energy bin (.7 < z < 1.0) for which we obtain
n <(6.5 1.3) 10 4, where the error is statistical only. The model-dependent fit
z>.7 −
givesa slight±lylarge×r result, 8.8 10 4. We note here that in the near future theCLEO
−
andCLEO-Ccollaborationswill×beabletomeasuretheh spectrumin¡ (1S)decaysfar
′
moreprecisely.
We have calculated the h spectrum, dG (3S[1] gggh )/dz for each of the three
g gh form factors considered′ aboveat leading-1ord→er in QC′D, using a four-body phase
∗ ′
space Monte Carlo [20]. Relativistic corrections have not been taken into account. The
spectrahavebeennormalizedwithrespecttotheleading-orderthreegluondecaywidth,
G (3S[1] ggg),toobtaintheleading-orderh multiplicities.Theresultsarecomparedto
1 → ′
theARGUSfitinFig.3.Furthermore,comparisonwiththemodel-independentARGUS
upperboundforz>.7gives
41 (a) slowlyfalling,
ntheory
> 1 (b)pQCD, (21)
nargus ∼ 
(cid:18) (cid:19)z>.7 13 (c) intermediate.
Evidently, only the rapidly falling (pQCD) form factor is consistent with the ARGUS
dataatlargeenergies.Theintermediateandslowlyfallingformfactorsareingrosscon-
flict, givingorder of magnitudeor greater excesses. Note that the bulk of the h yield is
′
expectedto originatefrom long-distancefragmentationofthethreegluonconfiguration
andshouldthereforebequitesoft, inagreement withtheARGUSdata.
We have checked that significant contributions from decays of higher Fock states of
the¡ (1S)inducedbytheg gh coupling,e.g.,1S[8],3P[8] ggh ,wouldonlyhardenthe
h spectra.Furthermore,re∗lativ′isticcorrections,n0ext-tJo-l→eading′orderQCDcorrections,
′
andothertheoreticalrefinementswouldbeinsufficienttoeliminatethelargeexcessesfor
z>.7 in Eq. (21). Finally, we find that even in the extreme case that the gluon is given
a ‘phenomenological’ 1 GeV mass, as advocated in Ref. [31] (to improve agreement
3 I am grateful to Axel Lindner and Dietrich Wegener for supplying the raw h yields and efficiencies
′
fromRef.[19].
0.12
ARGUS
0.1
dz (a)
= 0.08
n
PSfrag repla
ements d 0.06
0.04
(
)
0.02
(b)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
z
FIGURE3. Theh multiplicityspectrumdn/dzindirect¡ (1S)decaysforthethreeg gh formfactors
′ ∗ ′
(a)Slowlyfalling,(b)pQCD(c)intermediate,andthemodel-dependentARGUSfit,asdescribedinthe
text.
betweenthepredictedandobservedphotonspectraneartheend-pointfor¡ (1S)decays),
theslowlyfallingform factorstillgivesan orderofmagnitudeexcess.
CONCLUSION
We have seen that a necessary condition for obtaining significantly different time-
dependentCPasymmetriesinB J/Y K isanon-vanishingamplitudeforthe‘wrong
S,L
→
sign’ kaon decay B J/Y K or its CP conjugate. We note that it may be possi-
→
ble to test directly for the presence of these amplitudes by searching for wrong-sign
B0 J/Y (K¯ K+p )decays. Amodel-independentanalysisshowsthatthescaleof
∗ −
→ →
new physics needs to be prohibitively low, of order a few GeV, all but ruling out this
possibility. The only potential example we have found, in the framework of supersym-
metric models with an ultra-light bottom squark and light gluinos, would lead to gross
violationofK K¯ mixingconstraints.
We have se−en that isospin breaking effects in B K g decays can be calculated
∗
→
model-indepndently in the QCD factorization framework. In the Standard Model the
decay rate for B¯0 K¯ 0g is predicted to be about 10–20% larger than that for B
∗ −
K g , in agreemen→t with the measured central values. The direction of the inequali→ty
∗−
between these amplitudes is particularly sensitive to the sign of the Wilson coefficient
ratioRe(C /C ).Asanapplication,moreprecisedatawhichwillbeavailableinthenot
6 7
too-distant future could rule out models in which the sign of ReC is flipped relative to
7
theStandard Model.
Finally, we have seen that slowly falling g gh form factors which can explain the
∗ ′
large B h X rate via the subprocess b s(g gh ) are ruled out by ARGUS data
′ s ∗ ′
onfasth→productionin¡ (1S)decays.How→ever,th→erapidlyfallingformfactorpredicted
′
byperturbativeQCD iscompatible.Unfortunately,thecorrespondingB h X ratefor
′ s
→
ph > 2 GeV is a factor of 20 smaller than observed. An enhanced b sg chromo-
′ →
magneticdipoleoperator, motivatedby thelowsemileptonicbranchingratio andcharm
multiplicityinBdecays,couldimproveagreementwithexperiment[16].TheCLEOand
CLEO-Ccollaborationswillbeabletomakemuchmoreprecisemeasurementsoftheh
′