Physics Letters B 666 (2008) 336–339

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Physics Letters B

www.elsevier.com/locate/physletb

O(α2s ) corrections to semileptonic decay b → clν̄l

Kirill Melnikov

Department of Physics and Astronomy, University of Hawaii, Honolulu, HI D96822, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 June 2008Received in revised form 15 July 2008Accepted 28 July 2008Available online 31 July 2008Editor: B. Grinstein

We compute the next-to-next-to-leading order (NNLO) QCD corrections to b → clν̄l decay rate at afully differential level. Arbitrary cuts on kinematic variables of the decay products can be imposed.Our computation can be used to study the NNLO QCD corrections to the total decay rate as well asto the lepton energy, hadronic invariant mass and hadronic energy moments and to incorporate thosecorrections into global fits of inclusive semileptonic B-decays.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Inclusive semileptonic decays of B-mesons into charmed finalstates are benchmark processes at B-factories. Because of rela-tively large rates and clean experimental signatures, these decayscan be studied with great precision. On the other hand, theo-retical description of semileptonic B decays is robust thanks tothe Operator Product Expansion (OPE) in inverse powers of theb-quark mass mb . The application of the OPE to semileptonic de-cays of B-mesons leads to the conclusion that both the total de-cay rate and various kinematic distributions can be described bypower series in ΛQCD/mb [1]. For infinitely heavy b-quark, thedecay rate B → Xclν̄l coincides with the rate computed at thequark level. For realistic values of bottom and charm masses, a fewnon-perturbative matrix elements that enter at order (ΛQCD/mb)

n ,n = 2,3, are accounted for in existing theoretical predictions.

In recent years, many measurements of moments of chargedlepton energy and hadronic invariant mass in B → Xclν̄l decayshave been performed by BaBar, Belle, CLEO, CDF and DELPHI [2–6].Comparison of these experimental results with theoretical predic-tions for corresponding observables leads to the determination ofthe Cabibbo–Kobayashi–Maskawa (CKM) matrix element |V cb|, bot-tom and charm quark masses and a number of non-perturbativeparameters such as μ2

π and μ2G [7,8]. Typical precision claimed in

these analyses is about one percent for |V cb| and mb and a fewpercent for mc and non-perturbative matrix elements [7,8].

To achieve such precision, advances in theoretical understand-ing of semileptonic B-decays were necessary, including subtle in-terplay between perturbative and non-perturbative physics andsignificant developments in technology of multi-loop computa-tions. While one-loop corrections to both b → clν̄l total decayrate [9] and a number of important differential distributions areknown since long [10], it is interesting to remark that phenomeno-

E-mail address: kirill@phys.hawaii.edu.

0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2008.07.089

logically relevant triple differential distribution in charged leptonenergy, leptonic invariant mass and hadronic invariant mass wascomputed through O(αs) only a few years ago [11,12]. This fact il-lustrates the complexity of perturbative calculations, when massiveparticles are involved, at a fully differential level.

Given the precision of available experimental measurements,good understanding of non-perturbative effects and a fairly largevalue of the strong coupling constant αs(mb) ≈ 0.24, it is expectedthat O(α2

s ) corrections to b → Xclν̄l decays are required for a con-sistent theoretical description. However, as was realized long ago,technical complexity of such an endeavor is daunting. To simplifythe problem, the O(α2

s ) corrections to b → Xclν̄l were computedin three specific kinematic points [13–15]. These results were usedin Ref. [13] to estimate the NNLO QCD corrections to Γ (b → Xclν̄l).Unfortunately, such a description is necessarily limited in its scopeeven for the total rate and a generalization of such an approachto more differential quantities, such as lepton energy and hadronicinvariant mass moments, is clearly out of question.

On the other hand, a subset of the NNLO QCD corrections, theBLM corrections [16], received significant attention recently. TheBLM corrections are associated with the running of the strongcoupling constant; they are potentially important since the QCDβ-function is large. For B decays, however, the BLM correctionsare known to be modest if proper definition of quark masses isadopted and judicious choice of the renormalization scale in thestrong coupling constant is made. The BLM effects are the easiestNNLO effects to calculate since they can be obtained from a one-loop computation if the latter is performed with a non-vanishinggluon mass [17]. For this reason, in the past, the BLM correctionsto b → clν̄l were calculated for the total rate and various kinematicmoments [18,19]. However, the NNLO QCD corrections beyond theBLM approximation, for which genuine two-loop computations arerequired, remained missing.

Calculation of these two-loop corrections became possible re-cently thanks to developments in numerical approaches to multi-loop computations [20]. These numerical methods benefit from the

K. Melnikov / Physics Letters B 666 (2008) 336–339 337

absence of mass hierarchy in the problem which is the case forb → c decays, since masses of bottom and charm quarks are close.The possibility to use the approach of Ref. [20] to describe de-cays of charged particles was recently pointed out in [21] whereelectron energy spectrum in muon decay was computed throughsecond order in the perturbative expansion in QED.

The goal of this Letter is to present the computation of O(α2s )

corrections to b → Xclν̄l decay rate at a fully differential level.Our results can be used to calculate arbitrary observables relatedto inclusive b → c transition through NNLO in QCD. For example,second order QCD corrections to such popular observables as lep-ton energy, hadronic invariant mass and hadronic energy momentscan be studied in dependence of the cut on charged lepton en-ergy. Inclusion of the results of our computation into global fits,should lead to a reduction of the theoretical uncertainty in the de-termination of |V cb|, the bottom and charm quark masses and thenon-perturbative parameters that contribute to the decay rate.

2. Computation

In this section, we set up our notation and briefly describetechnical aspects of the computation. A detailed description of themethod can be found in [20,21].

Consider the decay b → Xclν̄l where the final state lepton ismassless. The differential decay rate can be written as

dΓ = G F |V cb|2m5b

192π3

(dF0 + a dF1 + a2 dF2

), (1)

where G F is the Fermi constant, mb is the b-quark pole mass,a = αs/π and αs is the MS strong coupling constant definedin the theory with five active flavors and renormalized at thescale mb . For numerical computations, we use mb = 4.6 GeV andmc = 1.15 GeV. While these numerical values for the quark massescannot be justified in the pole scheme, our choice is motivatedby an eventual necessity to transform the pole scheme computa-tion to a more suitable scheme. The values of the quark massesthat we employ in this Letter correspond to the central values ofmb,c in the “kinetic scheme” [22], derived in recent fits to inclusivesemileptonic B-decays [7,8].

To calculate the functions dF0−2, we have to account for dif-ferent processes. At leading order, dF0 is computed by squaringthe matrix element of the process b → clν̄l and summing or av-eraging over spins and colors, as appropriate. At next-to-leadingorder, dF1 receives contributions from virtual O(αs) correctionsto b → clν̄l and from the real-emission process b → clν̄l + g . Tocompute dF2, we require two-loop O(α2

s ) corrections to b → clν̄l ,one-loop O(αs) corrections to b → clν̄l + g and the double real-emission corrections b → clνl + X , where X refers to two gluonsor a quark–antiquark pair or a ghost–antighost pair. We will referto these corrections as double-virtual, real-virtual and double-real,respectively. In addition, we have to account for a variety of renor-malization constants, when computing higher order corrections.We do not include the process b → clν̄l + cc̄ in our calculationsince the energy release in this process is so small that it cannotbe treated perturbatively.

To calculate the NNLO QCD corrections, the method for multi-loop computations developed in [20,21] is employed; in those ref-erences a detailed discussion of many technical issues relevant forthe current computation can be found. One technical aspect thatwe improve upon relative to Refs. [20,21] is how virtual correctionsto single gluon emission process b → clν̄l + g are treated. In Refs.[20,21] these corrections were dealt with by an analytic reductionto master integrals followed by a numerical evaluation of those.This method, however, becomes impractical quite rapidly, once thenumber of external particles or the number of massive particlesin the problem increases. In principle, the real-virtual corrections

can be computed numerically, but for heavy-to-light decays this iscomplicated because some Feynman diagrams develop imaginaryparts. To handle these imaginary parts, we proceed as follows. Forall Feynman diagrams that contribute to real-virtual corrections, itturns out possible to identify a Feynman parameter that enters thedenominator of the integrand linearly. Let us call this Feynman pa-rameter x1. Then, a typical integral that has to be computed reads

I(0,1) =1∫

0

dx1 x−ε+n1

(−a + bx1 + i0)1+ε. (2)

Here n � −1, b > a > 0 and both a and b depend on other Feyn-man parameters and the kinematic variables. The two argumentsof the function I refer to lower and upper limits for x1 integration.To calculate I(0,1), we note that by extending upper integrationboundary to infinity, a solvable integral for arbitrary a,b and n isobtained. On the other hand, since

I(0,1) = I(0,∞) − I(1,∞), (3)

and because the denominator of the integrand in I(1,∞) is sign-definite, I(1,∞) can be computed numerically in a straightforwardway. It turns out that, up to minor modifications, this trick can beused to avoid dealing with the imaginary parts for all Feynmandiagrams that contribute to one-loop corrections to single gluonemission process in b → c decays.

Because couplings of quarks and leptons to the charged currentare chiral, proper treatment of the Dirac matrix γ5 in d = 4 − 2εdimensions is important. While this problem can be avoided in thecomputation of the total decay rate, for more differential quantitiesit becomes an issue. We use the approach of Ref. [23] where aconsistent framework for extending the axial vector current to d-dimensions is given.

Our computation can be checked in a number of ways. First,the double-virtual, real-virtual and double-real corrections are di-vergent when taken separately but these divergences must cancelin physical observables. We have checked these cancellations for avariety of observables, from the inclusive rate to various momentswith cuts on both charged lepton energy and the hadronic invari-ant mass. Second, in the limit mc → mb , the NNLO QCD correctionsto the decay rate b → clν̄l are described by the so-called zero-recoil form factors computed through O(α2

s ) long time ago [15].We have checked that in the limit mc → mb our computationreproduces the zero-recoil form factors. Third, we can use pub-lished results for the BLM corrections to the total rate and chargedlepton energy, hadronic invariant mass and hadronic energy mo-ments [19] to check parts of our computation related to masslessquark contributions to gluon vacuum polarization. Finally, consid-ering the limit mc � mb , we reproduce the NNLO QCD correctionsto b → ulν̄l decay rate reported in Ref. [24].

3. Results

We are now in position to discuss the results of our computa-tion. We consider a number of observables, mostly for illustrationpurposes. We present the results in the pole mass scheme anduse the strong coupling constant renormalized at the scale mb .While the pole mass scheme is known to be an unfortunate choiceinasmuch as the convergence of the perturbative expansion is con-cerned, we decided to present our results in this way for clarity.However, we emphasize that the impact of the NNLO QCD cor-rections, computed in this Letter, on the determination of |V cb|,heavy quark masses and the non-perturbative parameters, includ-ing kinetic and chromomagnetic heavy quark operators, can onlybe assessed once the pole mass scheme is abandoned in favor ofa more suitable quark mass definition and the NNLO QCD correc-tions are included into the fit.

338 K. Melnikov / Physics Letters B 666 (2008) 336–339

Table 1Lepton energy moments

n Ecut, GeV L(0)n L(1)

n L(2,BLM)n L(2)

n

0 0 1 −1.77759 −1.9170 3.401 0 0.307202 −0.55126 −0.6179 1.112 0 0.10299 −0.1877 −0.2175 0.3940 1 0.81483 −1.4394 −1.5999 2.631 1 0.27763 −0.49755 −0.5667 1.002 1 0.09793 −0.17846 −0.20875 0.382

Table 2Hadronic energy moments

n Ecut, GeV H (0)n H (1)

n H (2,BLM)n H (2)

n

1 1 0.334 −0.57728 −0.6118 1.022 1 0.14111 −0.23456 −0.2343 0.362

To present the results, we follow Ref. [19] and define

Ln(Ecut) = 〈(El/mb)nθ(El − Ecut)dΓ 〉

〈dΓ0〉 , (4)

Hn(Ecut) = 〈(Eh/mb)nθ(El − Ecut)dΓ 〉〈dΓ0〉 , (5)

where 〈· · ·〉 denotes average over the phase-space of all final stateparticles, El,h is the energy of the charged lepton or hadronic sys-tem in the b-quark rest frame and

dΓ0 = G F |V cb|2m5b

192π3dF0. (6)

The lepton energy moments introduced in Eq. (4) can be writ-ten as

Ln = L(0)n + aL(1)

n + a2(β0L(2,BLM)n + L(2)

n) + · · · , (7)

where ellipses stands for higher order terms in the perturbativeexpansion in QCD. Similar decomposition can be performed for thehadronic energy moments Hn . In addition, we use β0 = 11−2/3N f

and define the non-BLM corrections L(2)n , H(2)

n as the difference ofthe complete O(α2

s ) correction and the BLM correction computedwith N f = 3.

In Tables 1, 2 the results for lepton energy and hadronicenergy moments with and without a cut on the lepton en-ergy are displayed. The numerical accuracy of L(0,1)

n , H(0,1)n and

L(2,BLM)n , H(2,BLM)

n is about 0.1–0.2% whereas the numerical accu-racy of L(2)

n , H(2)n is about 1–3%. It is possible to improve on the

accuracy but this requires somewhat large CPU time. Neverthe-less, for all practical applications the achieved numerical accuracyis sufficient.

There are a few interesting observations that follow from Ta-bles 1, 2. Quite generally, the non-BLM corrections and the BLMcorrections have opposite signs; given their relative magnitude andthe value of β0, it is easy to see that the O(α2

s ) corrections areabout twenty percent smaller than what the BLM-based estimatessuggest. The relative magnitude of the non-BLM and BLM correc-tions is largely independent of n and of whether the lepton energycut is applied.

First row in Table 1 provides the NNLO QCD corrections to thetotal decay rate b → clν̄l in the pole mass scheme. Such correctionswere estimated earlier in Ref. [13]. Note that in Ref. [13] the nu-merical results are given for the ratio of quark masses mc/mb = 0.3and also the BLM corrections are defined with N f = 4, rather thanN f = 3. Calculating the non-BLM corrections for the set of parame-

ters employed in [13], we find L(2)0 ≈ 1.73 which is to be compared

with the estimate L(2)0 ≈ 0.9(3), reported in [13].1

The results of Ref. [13] were used in Ref. [25] to estimate theimpact of the QCD corrections on Γ (B → Xclνl). In Ref. [25] theperturbative corrections to b → clν̄l decay rate are described by afactor Apert, defined as

Γ (b → Xclν̄l) = Apert(r)〈dΓ0〉, (8)

where r = mc/mb . Apert depends on the adopted scheme for thequark masses. In the kinetic mass scheme, Apert(0.25) = 0.908 isquoted. To arrive at this result, Ref. [25] uses L(2)

0 = 1.4 which isabout a factor 2.5 smaller than the corresponding entry in Table 1.Correcting for this discrepancy, we derive

Apert(0.25) = 0.919. (9)

We believe that this value for the perturbative renormalization fac-tor in the kinetic scheme for mc/mb = 0.25 should be employed inglobal fits of semileptonic B-decays.

Further analysis of entries in Table 1 suggests that the QCDcorrections in general and the non-BLM corrections in particu-lar mostly affect the overall normalization rather than shapes ofkinematic distributions. This follows from the approximate inde-pendence of L(1,2)

n /L(0)n of n and also of whether or not the cut on

the lepton energy is imposed. It is therefore possible to speculatethat the non-BLM corrections computed in this Letter will mostlyaffect the extraction of |V cb| whereas their influence on, e.g., theb-quark mass determination will be minor. Of course, this is onlya speculation and it needs to be verified explicitely.

Concerning the |V cb|, we note that the increase of the perturba-tive renormalization factor Apert by 10 × 10−3 implies the changein the value of |V cb| by about −0.25 × 10−3, comparable with thecurrent uncertainty [7,8]. However, since our results are derived inthe pole mass scheme, a translation to a more suitable scheme isrequired to assess the importance of non-BLM corrections. Such atranslation introduces additional non-BLM corrections to the finalexpression for the width, related to non-BLM corrections in therelationship between the pole and short-distance quark masses.Those corrections are included in [7] which uses Apert = 0.908from Ref. [25] but they are absent, alongside with all other non-BLM corrections, in [8]. It is possible that the non-BLM correctionsto the widths and masses somewhat compensate each other,2 re-ducing the importance of the second order non-BLM effects on the|V cb| determination in [8]. To settle this issue, the O(α2

s ) QCD cor-rections computed in this Letter should be included into global fitsof semileptonic b-decays.

4. Conclusions

In this Letter, the computation of the NNLO QCD correctionsto the fully differential b → clν̄l decay rate is reported. The dif-ferential nature of the computation makes it possible to applyarbitrary cuts on the kinematic variables of final state particles.This result allows to extend the existing determinations of theCKM matrix element |V cb|, the bottom and charm quark massesand the non-perturbative parameters μ2

π and μ2G from global fits

to semileptonic decays of B-mesons, by including the NNLO QCDcorrections exactly. We note that for a consistent high-precisionanalysis of semileptonic B-decays, also O(αs) corrections to Wil-son coefficients of non-perturbative kinetic and chromomagnetic

1 The value of L(2)0 quoted in Eq. (10) of Ref. [13] is 1.4 but this value is in error

due to an unfortunate mistake in the interpolating procedure. Correcting for thismistake, one obtains L(2)

0 = 0.9(3).2 I am grateful to Z. Ligeti for emphasizing this point to me.

K. Melnikov / Physics Letters B 666 (2008) 336–339 339

operators are required. Such a correction is available for the ki-netic operator [26] but is still missing for the chromomagnetic.

We presented a few results for charged lepton energy momentsand hadronic energy moments with and without a cut on the lep-ton energy in the pole mass scheme. These results suggest that themagnitude of the non-BLM corrections does not depend stronglyon the kinematics; the non-BLM corrections are approximately 2%for all the moments considered. We therefore expect that the non-BLM NNLO QCD corrections will mostly affect the determination of|V cb| decreasing its central value by a fraction of a percent whereastheir impact on the quark masses and the non-perturbative param-eters will probably be quite mild.

As a final remark, we note that it would be interesting to ex-tend this calculation in two ways. First, one may consider semilep-tonic decays of B-mesons into massive leptons. Such an extension,relevant for the description of B → Xc + τ + ν̄τ decay is straight-forward. Second, it is interesting to extend the current calculationsto allow for a massless quark in the final state. This is a difficultproblem but it is highly relevant for the determination of the CKMmatrix element |V ub| from semileptonic b → u transitions.

Acknowledgements

Discussions with F. Petriello and useful correspondence withT. Becher are gratefully acknowledged. I would like to thankA. Czarnecki and A. Pak for informing me about their results priorto publication. This research is partially supported by the DOE un-der grant number DE-FG03-94ER-40833.

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