The recent theoretical progress in the description of semileptonic decays in
the framework of the heavy mass expansion is summarized. Both inclusive and
exclusive decays are considered.Comment: Contribution to the XXIXth Rencontres de Moriond (Electroweak
  Interactions and Unified Theories}, Meribel, France, March 12-19, 1994,
  LaTeX, 10 pages, two figues appended after \end{document} as uu-encoded and
  compressed .eps files, uses epsf., CERN-TH.7264/9

Mannel, Thomas

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The recent theoretical progress in the description of semileptonic decays in the framework of the heavy mass expansion is summarized. Both inclusive and exclusive decays are considered.The recent theoretical progress in the description of semileptonic decays in the framework of the heavy mass expansion is summarized. Both inclusive and exclusive decays are considered

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arXiv:hep-ph/9405353v1  23 May 1994CERN-TH.7264/94hep-ph/9405353Weak Decays of Heavy–Quark SystemsThomas MannelTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandAbstractThe recent theoretical progress in the description of semileptonic de-cays in the framework of the heavy mass expansion is summarized.Both inclusive and exclusive decays are considered.Contribution to the XXIXth Rencontres de Moriond,Electroweak Interactions and Unified Theories,Me´ribel, France, March 12–19, 1994.CERN-TH.7264/94May 19941 IntroductionHadrons containing heavy quarks have attracted a lot of theoretical attentionrecently, since there has been an improvment in the understanding of heavyhadrons. The Standard Model describes the decays of the heavy quarks, butdue to the confinement property of the strong interactions only the decays ofheavy hadrons may be observed. In order to make contact between the quarkand the hadron level, one has to deal with the QCD bound-state problem,which has not yet been solved. For this reason one has to use models todescribe the weak decays of hadrons.However, it has been observed recently by various authors [1, 2] that themodel dependence may be reduced substantially for b and c quarks. Themass of these quarks is much larger than the energy scale determined by thelight degrees of freedom, and it is convenient to switch to an effective theorydescription in which the dynamical degrees of freedom of the heavy quarkare “integrated out”; in other words, the heavy quark becomes an infinitelyheavy, static source of a colour field, moving with a fixed velocity.This “Heavy Quark Effective Theory” (HQET) corresponds to a sys-tematic expansion of the full QCD Green-functions in inverse powers of theheavy-quark mass(es). The leading terms of this heavy mass expansion maybe constructed from a renormalizable Lagrangian, which may be obtainedfrom QCD by following the usual steps to construct an effective theory [3].In the heavy mass limit, two new symmetries appear, which are notpresent in QCD. The first symmetry is the heavy flavour symmetry, which isdue to the fact that the interaction of the quarks with the gluons is flavourblind and in the heavy mass limit all heavy quarks act as a static sourceof colour. Formally this corresponds to an SU(2) symmetry relating b intoc quarks moving with the same velocity. The second symmetry is the spinsymmetry of the heavy quark. The interaction of the heavy quark spin withthe “chromomagnetic” field is inversely proportional to the heavy mass andhence vanishes in the infinite mass limit. As a consequence, the rotations forthe heavy quark spin become an SU(2) symmetry, which holds for a fixedvelocity of the heavy quark.Corrections to the limit 1/mQ = 0 may be studied systematically inthe framework of HQET. The corrections are given as power series expan-sions in two small parameters. The first small parameter is the strong cou-pling constant taken at the scale of the heavy quark αs(mQ). This type1of correction may be calculated systematically using perturbation theory inHQET. The second type of correction is characterized by the small param-eter Λ¯/mQ, where Λ¯ is a scale of the light QCD degrees of freedom, e.g.Λ¯ ∼ mhadron −mQ. In the effective theory approach this type of correctionenter through operators of higher dimension, the matrix elements of whichhave to be parametrized in general by additional form factors.Inclusive decays may be treated in the heavy mass expansion by per-forming an operator product expansion much as is done in deep inelasticscattering [4]. The leading term in this expansion turns out to be the de-cay of a free quark, and corrections may be parametrized by forward matrixelements of higher dimensional operators.In this contribution I shall focus on semileptonic decays of heavy mesonsand consider in the next section exclusive heavy-to-heavy transitions, withsome emphasis on the model-independent determination of Vcb. In section 3we summarize the recent developments in the application of the heavy massexpansion to inclusive semileptonic rates and decay spectra.2 Exclusive Semileptonic DecaysThe heavy-quark spin flavour symmetry strongly reduces the number of formfactors in exclusive heavy-to-heavy transitions [1]; in fact, for mesonic heavy-to-heavy transitions there is only one form factor, called the Isgur–Wisefunction. For a transition between heavy ground state mesons H (eitherpseudoscalar or vector) with heavy flavour f (f ′) moving with velocities v(v′), one obtains in the heavy-quark limit〈H(f ′)(v′)|h¯(f ′)v′ Γh(f)v |H(f)(v)〉 = ξ(vv′)CΓ(v, v′). (1)Here h(f)v is the field operator annihilating a heavy quark of flavour f , movingwith velocity v; Γ is some arbitrary Dirac matrix. The Isgur–Wise functionξ(vv′) contains all the non-perturbative information for the heavy-to-heavytransition, while the coefficient CΓ(v, v′) may be calculated from the symme-tries. Furthermore, heavy-quark symmetry fixes the value of ξ at the pointv = v′ to beξ(vv′ = 1) = 1, (2)since the current h¯(f′)v Γh(f)v is one of the generators of heavy-flavour symme-try.2In the following we shall treat the b and the c quarks as heavy. Then(1) implies that the decays B → Dℓν¯ℓ and B → D∗ℓν¯ℓ are described by asingle form factor, the Isgur–Wise function. In general, the relevant matrixelements are given in terms of six form factors〈D(v′)|c¯γµb|B(v)〉 = √mBmD[ξ+(y)(vµ + v′µ) + ξ−(y)(vµ − v′µ)](3)〈D∗(v′, ǫ)|c¯γµb|B(v)〉 = i√mBmD∗ξV (y)εµαβρǫ∗αv′βvρ〈D∗(v′, ǫ)|c¯γµγ5b|B(v)〉 = i√mBmD∗[ξA1(y)(vv′ + 1)ǫ∗µ − ξA2(y)(ǫ∗v)vµ−ξA2(y)(ǫ∗v)v′µ],where we have defined y = vv′. In the heavy-quark limit, these form factorsare related to the Isgur–Wise function ξ byξi(y) = ξ(y) for i = +, V, A1, A3, ξi(y) = 0 for i = −, A2. (4)The normalization statement (2) may be used to perform a model-indepen-dent determination of Vcb from semileptonic heavy-to-heavy decays by ex-trapolating the lepton spectrum to the kinematic endpoint v = v′. Using themode B → D(∗)ℓν one obtains the relationlimv→v′1√(vv′)2 − 1dΓd(vv′)=G2F4π3|Vcb|2(mB −mD∗)2m3D∗ |ξA1(1)|2. (5)In the heavy-quark limit the form factor ξA1 reduces to the Isgur–Wise func-tion and is unity at the non-recoil point; aside from |Vcb| everything in theright-hand side is known.However, there are corrections to the normalization of the form factor ξA1at zero recoil, which may be addressed using HQET. A complete discussion ofthe corrections may be found in the review article by Neubert [2], includingreference to the original papers. Here we only state the final resultξA1(1) = x6/25[1 + 1.561αs(mc)− αs(mb)π− 8αs(mc)3π(6)+z{2554− 1427x−9/25 +118x−12/25 +825ln x}− αs(m¯)πz21− z ln z](7)+ δm2 (8)3where we use the abbreviationsx =αs(mc)αs(mb), z =mcmb, mc < m¯ < mb.Contributions (6) and (7) originate from QCD radiative corrections. Sincefor scales abovemb and belowmc the currents are conserved, logarithmic cor-rections from running will only be induced from scales between mb and mc.The contribution (6) is the correction in leading and next-to-leading logarith-mic approximation; including the non-logarithmic one-loop contributions.Corrections of order z = mc/mb may only be induced by QCD radiativecorrections. The first term in (7) is obtained by keeping the contributions oforder 1/mb in the matching at the scale mb, performing the renormalizationgroup running for these operators and matching at the scale mc. In thisway a resummation of logarithmic terms of the form αsz ln z is achieved.However, since z ∼ 0.3 is not particularly small, one may as well choose toperform the matching from full QCD to the effective theory in only one step,neglecting the running between mb and mc. In this way one may retain thefull dependence on z, at the price, however,of a scale ambiguity (mc < m¯ <mb) in the choice of αs. The second correction in (7) is obtained by thisprocedure, subtracting the linear term in z, which is already contained inthe first term.Finally there are recoil corrections to the normalization of ξA1. It hasbeen shown [5] that the terms linear in Λ/mc and Λ/mc have to vanish dueto heavy-quark symmetry. Thus the first non-vanishing recoil corrections areof order (Λ/mc)2, (Λ/mb)2 and Λ2/(mbmc). These contributions may onlybe estimated, since they need an input beyond heavy-quark effective theory.There are various estimates for these corrections, which are compatible withone anotherδm2 = −2% . . .− 3% [6], δm2 = 0 . . .− 5% [7], δm2 = 0% . . .− 8% [8]Adding all the corrections to the normalization, one obtains for the formfactor ξA1ξA1(1) = 0.96± 0.03, (9)where the error quoted is due to the uncertainty of the 1/m2 corrections.This result has been used to extract Vcb from data. In Fig. 1 the latestdata [9] are shown. Three different forms of the Isgur–Wise function have41 1.1 1.2 1.3 1.4 1.5w00.020.040.06| Vcb|  ξ(w)Figure 1: Latest data [9] for the product |Vcb|ξ(vv′) as afunction of ω = vv′. Figure taken from [11].been used for the extrapolation; however, the difference between the curvesis almost invisible. From this fit one obtains|Vcb|(τB1.5ps)1/2= 0.039± 0.006. (10)This value of |Vcb| is compatible with the value obtained in other ways, e.g.from inclusive decays [10].3 Inclusive Decays of Heavy MesonsIn order to obtain a 1/mQ expansion for inclusive decay rates one uses anoperator product expansion [4] as in deep inelastic scattering. Starting from5the effective HamiltonianHeff =GF√2Vbq(b¯γµ(1− γ5)q)(ℓ¯γµ(1− γ5)ν), (11)one writes the total inclusive rate asΓ =12mB∑X(2π)4δ4(PB − q − PX)|〈Xs|Heff |B(v)〉|2. (12)The matrix element appearing in (12) contains a large scale, the mass of theb quark. To make the dependence on this scale explicit, one redefines the bquark field by removing a phase factor corresponding to an on-shell b quarkmoving with the velocity of the mesonb(x) = exp(−imbvx)Qv(x). (13)Inserting this in (12),Γ =12mB∫d4x exp(imbvx)〈B(v)|H˜eff(x)H˜†eff(0)|B(v)〉, (14)where the tilde denotes the effective Hamiltonian with b replaced by Qv. Oncethe phase factor is extracted from the matrix element, it no longer dependson the large mass, and a short-distance expansion may be performed for theoperator product appearing in the matrix element. The relevant momentumis mbv and the short-distance expansion has the form∫d4xeimbvxT[H˜eff(x)H˜†eff(0)]=∞∑n=0(12mb)nCn+3(µ)On+3(µ),where On are operators of dimension n, renormalized at scale µ, and Cn arethe corresponding Wilson coefficients.The lowest-order term of the operator product expansion is the dimension-three operator O3 = Q¯vQv, and its forward matrix element is normalizedbecause of heavy-quark symmetries. Evaluating this contribution yields thefree-quark decay rate.All dimension-four operators are proportional to the equations of mo-tion O4 ∝ Q¯v(ivD)Qv, and the first non-trivial contribution comes from6dimension-five operators and are of order of 1/m2b . For mesonic decays thereare two matrix elements of dimension-five operators:〈B(v)|h¯(b)v (iD)2h(b)v |B(v)〉 = 2mbλ1 and 〈B(v)|h¯(b)v igσµνGµνh(b)v |B(v)〉 = 12mbλ2,where 4λ2 = m2B∗ −m2B. The parameter λ1 is not as easily accessible, butthe expectations from QCD sum rules are λ1 = −0.52± 0.12 GeV2 [12].In terms of these two parameters the non-perturbative corrections to theinclusive decay B → Xuℓν is given by the expressionΓ(B → Xuℓν) = Γb[1 +λ1 − 9λ22m2b], Γb =G2Fm5b192π3|Vub|2; (15)a similar expression may be obtained for the rate for B → Xcℓν.The spectrum of the charged lepton in inclusive decays may be calculatedby similar means. The rate is written as a product of the hadronic andleptonic tensordΓ =G2F4mB|VQq|2WµνΛµνd(PS), (16)where d(PS) is the phase-space differential. The operator product expansionalong the lines described above is then performed for the two currents ap-pearing in the hadronic tensor. Redefining the heavy-quark fields as in (13)one finds that the momentum transfer variable relevant for the short-distanceexpansion is mbv − q, where q is the momentum transfer to the leptons.Again the contribution of the dimension-three operators yields the free-quark decay spectrum, and there are no contributions from dimension-fouroperators. The 1/m2b corrections are parametrized in terms of λ1 and λ2 andthe result for the lepton spectrum in B → Xuℓν is given by1ΓbdΓdy= 2y2(3− 2y) + 10y23λ1m2b+ 2y(6 + 5y)λ2m2b−λ1 + 33λ23m2bδ(1− y)− λ13m2bδ′(1− y), (17)where y = 2Eℓ/mb is the rescaled energy of the charged lepton.Aside from regular terms, the result also exhibits δ-function singulari-ties at the endpoint, indicating that higher terms in the operator productexpansion become important; here (mbv − q)2 becomes small.70 0.5 1 1.5 2 2.5El  [GeV]00.20.40.60.811/Γ b  dΓ/dE l  [GeV -1 ]Figure 2: Charged-lepton spectrum in B → Xuℓν¯ decays.The solid line is (19) with the ansatz (20), the dashed lineshows the prediction of the free-quark decay model.It has been shown [13] that the most singular terms of the short-distanceexpansion may be resummed into a structure function, analogous to a par-ton distribution function known from deep inelastic scattering. It is definedformally asf(k+) =12mB〈B(v)| h¯v δ(k+ − iD+) hv |B(v)〉 with k+ = k0 + k3. (18)Using this function, the result for the spectrum becomesdΓdEℓ=G2F |Vub|212π3E2ℓ (3mb − 4Eℓ)Λ¯∫2Eℓ−mbdk+ f(k+), (19)where Λ¯ = mB −mb. The function f is genuinely non-perturbative; in order8to illustrate its effect, we choose a simple one-parameter modelf(k+) =32π2Λ¯(1− x)2 exp{− 4π(1− x)2}Θ(1− x) ; x = k+Λ¯, (20)and Fig. 2 shows the resulting spectrum for Λ¯ = 570 GeV. Due to non-perturbative effects, the spectrum now extends beyond the parton modelendpoint mb/2 up to the “physical” endpoint mB/2. However, before theseresults may be confronted with data, the QCD radiative corrections need tobe taken into account. First results have been published [14], but the issueis still in a state of flux.References[1] N. Isgur and M. Wise, Phys. Lett. B232 (1989) 113 and B237 (1990)527; M. Voloshin and M. Shifman, Sov. J. Nucl. Phys. 45 (1987) 292and 47 (1988) 511; a more complete set of references can be found inone of the reviews [2].[2] For reviews see: H. Georgi: contribution to the Proceedings of TASI–91,by R. K. Ellis et al. (eds.) (World Scientific, 1991); B. Grinstein: con-tribution to High Energy Phenomenology, R. Huerta and M. A. Peres(eds.) (World Scientific, 1991); N. Isgur and M. Wise: contribution toHeavy Flavors, A. Buras and M. Lindner (eds.) (World Scientific, 1992);M. Neubert, SLAC–PUB 6263 (1993) (to appear in Phys. Rep.); T. Man-nel, contribution to QCD–20 years later, P. Zerwas and H. Kastrup(eds.) (World Scientific, 1993).[3] T. Mannel, W. Roberts and Z. Ryzak, Nucl. Phys. B 368 (1992) 204.[4] J. Chay, H. Georgi and B. Grinstein, Phys. Lett. B247 (1990) 399;I.I. Bigi, N. G. Uraltsev and A. I. Vainshtein, Phys. Lett. B293 (1992)430; I. I. Bigi et al., TPI-MINN-92/67-T (1992) and Phys. Rev. Lett. 71(1993) 496; B. Blok et al., Phys. Rev. D49 (1994) 3356; A. V. Manoharand M. B. Wise, Phys. Rev. D49 (1994) 1310; T. Mannel, Nucl. Phys.B423 (1994) 396.[5] M. Luke, Phys. Lett. B252 (1990) 447.9[6] A. Falk and M. Neubert, Phys. Rev. D49 (1994) 3356.[7] T. Mannel, preprint CERN-TH.7162/94 (1994).[8] M. Shifman, N. Uraltsev and A. Vainshtein, preprint TPI-MINN-94/13-T (1994).[9] G. Crawford et al. (CLEO Collaboration), to appear in the Proceedingsof the 16th International Symposium on Lepton and Photon Interac-tions, Ithaca, New York, 1993.[10] A review on the experimental situation for inclusive semileptonic decaysis given by S. Stone, in B Decays, S. Stone (ed.), World Scientific, 1992.[11] M. Neubert, preprint CERN-TH.7225/94.[12] P. Ball and V. Braun, Phys. Rev. D49 (1994) 2472.[13] M. Neubert, Phys. Rev. D49 (1994) 2472; I. Bigi et al., preprintCERN-TH.7129/94 (1994); T. Mannel and M. Neubert, preprint CERN-TH.7156/94 (1994).[14] A. Falk et al., preprint UCSD/PTH-93-38 (1993); see also the secondpaper of ref. [13].10

Weak Decays of Heavy-Quark Systems

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