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To be published in the Proceedings of the Workshop on B Factories: The State of the Art in Accelerators,

arXiv:hep-ph/9207270v1 29 Jul 1992To be published in the Proceedings of the Workshop on B Factories: The State of the Art in Accelerators,Detectors, and Physics, Stanford, California, April 6–10, 1992.

Recent Developments in the Theory of Heavy-Quark Decays∗Matthias NeubertStanford Linear Accelerator Center, Stanford, CA 94309 USAABSTRACTI report on recent developments in the heavy-quark effective theory and its application to B meson decays.The parameters of the effective theory, the spin-flavor symmetry limit, and the leading symmetry-breakingcorrections to it are discussed. The results of a QCD sum rule analysis of the universal Isgur-Wise func-tions that appear at leading and subleading order in the 1/mQ expansion are presented.

I illustrate thephenomenological applications of this formalism by focusing on two specific examples: the determination ofVcb from the endpoint spectrum in semileptonic decays, and the study of spin-symmetry violating effects inratios of form factors. I also briefly comment on nonleptonic decays.1IntroductionThe theoretical description of hadronic processes in-volving the decay of a heavy quark has recentlyexperienced great simplification due to the discov-ery of new symmetries of QCD in the limit wheremQ →∞[1, 2].

The properties of a hadron contain-ing the heavy quark then become independent of itsmass and spin, and the complexity of the hadronicdynamics results from the strong interactions amongthe light degrees of freedom only.The so-calledheavy-quark effective theory (HQET) provides anelegant framework to analyze such processes [3]. Itallows a systematic expansion of decay amplitudesin powers of 1/mQ.In the formal limit of infinite heavy-quark masses,the spin-flavor symmetries impose restrictive con-straints on weak decay amplitudes.

In the case ofsemileptonic transitions between two heavy pseu-doscalar or vector mesons, for instance, the largeset of hadronic form factors reduces to a single uni-versal function, the so-called Isgur-Wise form factorξ(v·v′). This function only depends on the change ofvelocities that the heavy mesons undergo during thetransition.

It is normalized at zero recoil (v = v′).This observation offers the exciting possibility of be-ing able to extract in a model-independent way theweak mixing parameter Vcb from the measurementof semileptonic decays of beauty mesons or baryons,∗Financial support from the BASF Aktienge-sellschaft and the German National Scholarship Foun-dation is gratefully acknowledged. This work was alsosupported by the Department of Energy, contract DE-AC03-76SF00515.without limitations arising from the ignorance oflong-distance dynamics.The heavy-quark symmetries greatly simplify thephenomenology of semileptonic weak decays in thelimit where the heavy-quark masses can be consid-ered very large compared to other hadronic scalesin the process.But clearly, a careful analysis ofsymmetry-breaking corrections is essential for anyphenomenological application.

Already at leadingorder in the heavy-quark expansion the spin-flavorsymmetries are violated by hard-gluon exchange.The corresponding corrections are of perturbativenature and are known very accurately to next-to-leading order in renormalization-group improvedperturbation theory [4, 5, 6]. At order 1/mQ, on theother hand, one is forced to introduce a larger setof universal form factors, which are nonperturbativehadronic quantities such as the Isgur-Wise functionitself [7, 8].

These functions characterize the prop-erties of the light degrees of freedom in the back-ground of the static color source provided by theheavy quark. Their understanding is at the heart ofnonperturbative QCD.

In this talk I review recentprogress in this direction. I discuss the parametersof HQET, the leading QCD and 1/mQ correctionsto the infinite quark-mass limit, and some specificapplications of the effective theory to semileptonicand nonleptonic B decays.2Parameters of HQETThe construction of HQET is based on the observa-tion that, in the limit mQ ≫ΛQCD, the velocity vµof a heavy quark is conserved with respect to soft

processes. It is then possible to remove the mass-dependent piece of the momentum operator by afield redefinition.

To this end, one introduces a fieldhQ(v, x), which annihilates a heavy quark Q withvelocity v (v2 = 1, v0 ≥1), by [3]hQ(v, x) = (1 + /v)2exp(imQv·x) Q(x). (1)If Pµ is the total momentum of the heavy quark, thenew field carries only the residual momentum kµ =Pµ −mQvµ, which is of order ΛQCD.

In the limitmQ →∞the effective Lagrangian for the stronginteractions of the heavy quark becomesLeff= ¯hQ iv·D hQ −δmQ ¯hQhQ,(2)where Dµ is the covariant derivative, and δmQ de-notes the residual mass of the heavy quark in theeffective theory [9].Note that there is some ambiguity associated withthe construction of HQET, since the heavy-quarkmass used in the definition of the field hQ is notuniquely defined. In fact, for HQET to be consis-tent it is only necessary that δmQ and kµ be of or-der ΛQCD, i.e., stay finite in the limit mQ →∞.A redefinition of mQ by a small amount ∆in-duces changes in these quantities.

In particular, ifmQ →mQ+∆, then δmQ →δmQ−∆. Hence thereis a unique choice m∗Q for the heavy-quark mass inthe construction of the effective theory such that theresidual mass vanishes, δmQ = 0.

This prescriptionprovides a nonperturbative definition of the heavy-quark mass, which has been implicitly adopted inmost previous analyses based on HQET. Yet it isimportant to notice that the mass m∗Q is a non-trivial parameter of the theory.

For instance, onecan associate the difference ¯Λ between this massand the mass of a meson M (or baryon) contain-ing the heavy quark with the energy carried by thelight constituents. That ¯Λ is in fact a parametercharacterizing the properties of the light degrees offreedom becomes explicit in the relation¯Λ = mM −m∗Q = ⟨0 | ¯q (iv·←−D) Γ hQ |M(v)⟩⟨0 | ¯q Γ hQ |M(v)⟩,(3)which can be derived from the equations of motionof HQET [9].

Here Γ is an appropriate Dirac matrixsuch that the currents interpolate the heavy mesonM.The two parameters m∗Q and ¯Λ characterize thestatic properties of the heavy quark and of the lightdegrees of freedom. Their ratio determines the sizeof power corrections to the infinite quark-mass limit.Assuming ¯Λ ≃0.50 GeV one expects ¯Λ/2m∗b ≃5%and ¯Λ/2m∗c ≃20% for the leading power correctionsrelevant to processes involving B or D mesons, re-spectively.

This estimate is confirmed by detailedcomputations (see below).Because of the spin-flavor symmetry the non-trivial dynamical properties of a hadron containingthe heavy quark are entirely related to its light con-stituents.Consider, for instance, a transition be-tween two heavy mesons (pseudoscalar or vector),M →M ′, induced by a weak current.At lead-ing order in the heavy-quark expansion the associ-ated hadronic matrix element factorizes into a triv-ial kinematical part, which depends on the mass andspin-parity quantum numbers of the mesons, and areduced matrix element, which describes the elastictransition of the light degrees of freedom. Introduc-ing spin wave-functions byM(v) = √mM(1 + /v)2−γ5; JP = 0−,/ǫ; JP = 1−,(4)one finds⟨M ′| ¯hQ′Γ hQ |M⟩= −ξ(w) TrM′(v′) Γ M(v),(5)where w = v · v′, and ξ(w) is the universal Isgur-Wise form factor [2, 4].

It measures the overlap ofthe wave functions of the light degrees of freedom inthe two mesons moving at velocities v and v′. Theconservation of the vector current implies that thereis complete overlap if v = v′, such that at zero recoilξ(1) = 1.Let us now focus on semileptonic decays of Bmesons.

It is convenient to define a set of heavy-meson form factors hi(w) by⟨D(v′)| Vµ | ¯B(v)⟩= √mB mDhh+(w) (v + v′)µ + h−(w) (v −v′)µi,⟨D∗(v′)| Vµ | ¯B(v)⟩= i√mB mD∗hV (w) ǫµναβ ǫ∗ν v′α vβ,(6)⟨D∗(v′)| Aµ | ¯B(v)⟩= √mB mD∗hhA1(w) (w + 1) ǫ∗µ−hA2(w) ǫ∗·v vµ −hA3(w) ǫ∗·v v′µi,where Vµ = ¯c γµ b and Aµ = ¯c γµγ5 b, and ǫµ is thepolarization vector of the D∗meson. In the infinite

quark-mass limit one finds from (5)h+(w) = hV (w) = hA1(w) = hA3(w) = ξ(w),h−(w) = hA2(w) = 0. (7)These relations summarize the symmetry con-straints imposed on the weak matrix elements.The mass parameter ¯Λ and the Isgur-Wise func-tions are fundamental hadronic quantities that ap-pear at leading order of the heavy-quark expansion.They can only be computed using nonperturbativetechniques such as lattice gauge theory or QCD sumrules.

While no lattice results are available so far,QCD sum rules [10] have often been used success-fully to compute hadron masses, decay constants,and form factors. This method has been recentlyapplied to the analysis of form factors in HQET[11, 12, 13].From the study of the correlator oftwo heavy-light currents in the effective theory onefinds that [11]¯Λ = 0.50 ± 0.07 GeV,(8)corresponding to heavy-quark masses m∗b ≃4.8GeV and m∗c ≃1.4 GeV.

The Isgur-Wise functionis obtained from the analysis of a three-current-correlator.The result can be parameterized interms of a pole-type functionξ(w) ≃2w + 1β(w);β(w) = 2 + 0.6w . (9)It explicitly obeys the normalization conditionξ(1) = 1 and exhibits dipole behavior at large re-coil.3Symmetry-Breaking CorrectionsFrom the fact that the mass of the charm quarkis not particularly large compared to a hadronicscale such as ¯Λ one expects substantial symmetry-breaking corrections to the relations (7).

These haveto be incorporated in any phenomenological anal-ysis based on HQET if the effective theory is tobe more reliable than a particular model for thehadronic form factors. The leading corrections comefrom hard-gluon exchange and from terms of order1/m∗Q in the heavy-quark expansion.

I will discussthese corrections separately below. Fortunately, itturns out that at least at zero recoil they can be cal-culated in an almost model-independent way, suchthat reliable predictions beyond the infinite quark-mass limit are still possible.Table 1: The universal form factors at leading andsubleading order in HQET.functionnormalizationbroken symmetriesξ(v · v′)ξ(1) = 1noξ3(v · v′)nospin, flavorχ1(v · v′)χ1(1) = 0flavorχ2(v · v′)nospin, flavorχ3(v · v′)χ3(1) = 0spin, flavorIn order to make the heavy-quark symmetry limitand the leading symmetry-breaking corrections to itexplicit, I writehi(w) =hαi + βi(w) + γi(w) + .

. .iξ(w),(10)where α+ = αV = αA1 = αA3 = 1 and α−= αA2 =0, the functions βi(w) are the short-distance per-turbative corrections, and γi(w) contain the 1/m∗cand 1/m∗b corrections.

The ellipses represent higher-order terms.3.1QCD CorrectionsThe form factors receive perturbative correctionsdue to the coupling of hard gluons to the heavyquarks. The corresponding coefficients βi(w) in (10)are complicated functions of w, αs(m∗c), αs(m∗b), andthe mass ratio m∗c/m∗b.

Their calculation is, how-ever, purely perturbative and can make use ofthe powerful methods of the renormalization group[4, 5, 6]. The coefficients βi(w) are known to next-to-leading logarithmic order and are tabulated inRefs.

[6].3.21/m∗Q CorrectionsAt subleading order in the heavy-quark expansionthe currents no longer have the simple structure asin (5). Instead, there appear higher-dimensional op-erators such as12m∗Q¯hQ′Γ i /D hQ,(11)whose hadronic matrix elements give rise to newuniversal form factors.In total, four additionalfunctions are required to describe all 1/m∗Q cor-rections to transitions between two heavy mesons

[7, 14]. Their properties are collected in Table 1.The conservation of the vector current implies thattwo of these functions vanish at zero recoil.

As aconsequence, the hadronic form factors h+(w) andhA1(w) are protected against 1/m∗Q corrections atw = 1. This is the content of Luke’s theorem [7].The subleading universal functions can again becalculated using QCD sum rules in the effective the-ory.

One finds [13]ξ3(w) ≃13hξ(w) −κ (w −1)i,χ1(w) ≃23w −1w + 1h4w + 72κ −ξ(w)i+ 18 χ3(w),χ2(w) ≃0,χ3(w) ≃κ8h1 −ξ(w)i. (12)Nonperturbative effects are contained in the Isgur-Wise function and the parameter κ ≃0.16, whichis proportional to the mixed quark-gluon conden-sate ⟨¯qσµνGµνq⟩.One does indeed find that thefunctions χ1(w) and χ3(w) vanish at w = 1.

In ad-dition, restricting to the diagrams usually includedin a sum rule analysis one finds no contribution tothe spin-symmetry violating form factor χ2(w), andobtains the parameter-free predictionξ3(1) = 13. (13)Corrections to this relation are expected to be small.In Table 2, I show the theoretical prediction forthe sum of the symmetry-breaking corrections tothe various heavy-meson form factors, based on themost recent calculation of QCD corrections [6] andthe above sum rule results.

The relations betweenthe corrections γi(w) and the subleading universalfunctions are given in Ref. [14].4Phenomenological ApplicationsThe theoretical results summarized in Table 2 forma solid basis for a comprehensive analysis of semilep-tonic B decays to subleading order in HQET.

Somespecific applications, as well as the extension to non-leptonic decays, are discussed below. I do not ad-dress here the important issue of decay constants ofheavy mesons.

The reader interested in this subjectis referred to Refs. [11, 12].4.1Measurement of VcbAs a first application let me focus on the extractionof the quark-mixing parameter Vcb from the extrap-Table 2:Total symmetry-breaking correctionsδi(w) = βi(w) + γi(w) in %.wδ+δ−δVδA1δA2δA31.02.6−9.531.0−1.5−34.1−1.91.12.4−9.529.6−0.9−31.7−0.91.23.1−9.429.20.6−29.60.91.34.9−9.529.82.8−27.63.41.47.3−9.631.15.7−25.86.41.510.4−9.733.29.0−24.210.0olation of semileptonic B decay rates to zero recoil.This subject has been discussed in detail in Ref.

[15].In general, one finds thatlimw→11[w2 −1]1/2dΓ( ¯B →D∗ℓ¯ν)dw= G2F4π3 |Vcb|2 (mB −mD∗)2 m3D∗η∗2,limw→11[w2 −1]3/2dΓ( ¯B →D ℓ¯ν)dw= G2F48π3 |Vcb|2 (mB + mD)2 m3D η2,(14)with η∗= η = 1 in the infinite quark-mass limit.Because of Luke’s theorem the decay rate for ¯B →D∗ℓ¯ν is protected against 1/m∗Q corrections at zerorecoil. Thus to subleading order in HQET the co-efficient η∗deviates from unity only due to radia-tive corrections.

Ignoring terms of order 1/m∗2Q , onefinds that η∗= 1 + δ∗QCD with δ∗QCD ≃−0.01 [6].On the other hand, Luke’s theorem does not ap-ply for ¯B →D ℓ¯ν decays because the decay rateis helicity-suppressed at zero recoil [14, 15]. In thiscase η = 1 + δQCD + δ1/m∗Q with δQCD ≃0.05 andδ1/m∗Q =¯Λ2 1m∗c+ 1m∗bmB −mDmB + mD21−2 ξ3(1),(15)which gives δ1/m∗Q ≃0.02.Note that the 1/m∗Qcorrections are suppressed by the Voloshin-Shifmanfactor [(mB−mD)/(mB+mD)]2 ≃0.23 [1], and thatthe corrections to the sum rule prediction ξ3(1) =1/3 are expected to be small.

Since the canonicalsize of 1/m∗2Q corrections is 1 −5%, I thus concludethat the theoretical uncertainty in η is comparableto that in η∗. Hence one should extract Vcb from

both decay modes, using the theoretical numbersη∗≃0.99,η ≃1.07,(16)which are expected to have an accuracy of betterthan 5%.Until now such an analysis has only been per-formed for ¯B →D∗ℓ¯ν [15]. Using the updated valuefor the total branching ratio as measured by CLEO,B( ¯B →D∗ℓ¯ν) = 4.4 ± 0.5% [16], I findVcb = 0.040 ± 0.005(17)for τB = 1.3 ps.4.2Ratios of Form FactorsIt has been emphasized in Ref.

[13] that a measure-ment of spin-symmetry-breaking effects in ratios ofthe various form factors that describe ¯B →D∗ℓ¯νtransitions would not only offer the possibility of anontrivial test of HQET beyond the leading order,but also provide valuable information about non-perturbative QCD. In the limit where the leptonmass is neglected, two axial form factors, A1(q2)and A2(q2), and one vector form factor, V (q2), areobservable in these decays.

The ratiosR1 =1 −q2(mB + mD∗)2 V (q2)A1(q2),(18)R2 =1 −q2(mB + mD∗)2 A2(q2)A1(q2)become equal to unity in the infinite quark-masslimit and are thus sensitive measures of symmetry-breaking effects.To subleading order in HQET, I writeRi = 1 + εQCDi+ ε1/m∗Qi;i = 1, 2. (19)The theoretical prediction for εi as a function of q2is shown in Table 3.

I propose a measurement ofthese quantities as an ideal test of the heavy-quarkexpansion for b →c transitions. In particular, notethat the large values of R1 result from both largeQCD and 1/m∗Q corrections.The latter ones areto a large extent model-independent since the sub-leading universal functions only appear in the 1/m∗bterms [13].

Thus the sizeable deviation of R1 fromthe symmetry limit R1 = 1 is an unambiguous pre-diction of HQET. A measurement of this ratio withan accuracy of 10% could provide valuable informa-tion about the size of higher-order corrections.

TheTable 3: Theoretical predictions for the symmetry-breaking corrections εi in %.q2 [GeV2]εQCD1ε1/m∗Q1εQCD2ε1/m∗Q210.6912.019.10.5−11.08.5711.718.20.5−10.36.4511.317.50.5−9.64.3311.016.80.5−8.92.2110.716.20.5−8.30.0910.415.60.5−7.7ratio R2, on the other hand, receives only very smallQCD corrections and is sensitive to the subleadingform factors ξ3(w) and χ2(w).It can be used totest the sum rule predictions (12). For the practi-cal feasibility of such tests it seems welcome that thetheoretical predictions for both ratios are almost in-dependent of q2 (R1 ≃1.3 and R2 ≃0.9), such thatit suffices to measure the integrated ratios.4.3Nonleptonic DecaysAs a final application, let me briefly comment onnonleptonic two-body decays of B mesons.

In thiscase, the heavy-quark symmetries do not yield re-lations as restrictive as those for semileptonic tran-sitions.One still has to rely on the factorizationhypothesis, under which the complicated hadronicmatrix elements of the weak Hamiltonian reduce toproducts of decay constants and matrix elementsof current operators, which are of the same type asthose encountered in semileptonic processes. It is atthis stage that the heavy-quark symmetries can beadvantageously incorporated, leading to essentiallymodel-independent predictions for the factorized de-cay amplitudes.This provides for the first timea clean framework in which to test factorization.The procedure would be as follows: One extractsthe Isgur-Wise function from data on semileptonicB decays and incorporates the leading symmetry-breaking corrections as discussed above.

This deter-mines the functions hi(w), which suffice to computeall matrix elements that appear in the factorized de-cay amplitudes for nonleptonic processes. Besidesdecay constants, these amplitudes contain two pa-rameters, a1 and a2, which are related to the Wil-son coefficients of the effective Hamiltonian.

Theywould be universal numbers if factorization were ex-

act. In cases where the relevant decay constants areknown, a case-by-case determination of a1 or a2 pro-vides a test of factorization.

In other cases, one mayrely on factorization to obtain estimates for yet un-known decay constants such as fDS. Both strategieshave been pursued by various authors, and we referthe interested reader to the literature [17].5ConclusionsI have presented a short overview of recent devel-opments in the theory of heavy-quark decays.

Thespin-flavor symmetries that QCD reveals for heavyquarks lead to relations among the hadronic formfactors which describe semileptonic decays of heavymesons or baryons. The heavy-quark effective the-ory provides a convenient framework for the analysisof such processes.

It allows a separation of short-and long-distance phenomena in such a way thatthe nontrivial dynamical information is parameter-ized in terms of universal functions, which describethe properties of the light degrees of freedom in thebackground of the static color source provided bythe heavy quark.These functions are fundamen-tal, nonperturbative quantities of QCD. I have pre-sented explicit expressions for them obtained fromQCD sum rules.

In the near future, similar resultsshould be obtainable from lattice gauge theory.If the leading symmetry-breaking corrections aretaken into account, the heavy-quark effective theoryforms a solid, almost model-independent basis for ananalysis of many weak decay processes. I have dis-cussed the determination of Vcb from the endpointspectrum in semileptonic B decays, and the studyof symmetry-breaking effects in ratios of form fac-tors, which offers nontrivial tests of the heavy-quarkexpansion beyond leading order.

I have also empha-sized that the use of the spin-flavor symmetry mayprovide a cleaner basis for tests of factorization innonleptonic two-body decays of B mesons.Acknowledgement: Part of the work reported herehas been done in a most enjoyable collaborationwith A. Falk and M. Luke.REFERENCES[1] M.B. Voloshin and M.A.

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