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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 factorization 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 chromomagnetic 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 contribution 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 supersymmetric 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 ′