Available via license: CC BY 4.0

Content may be subject to copyright.

QCD corrections to inclusive heavy hadron weak decays at Λ3

QCD=m3

Q

Thomas Mannel and Alexei A. Pivovarov

Theoretische Elementarteilchenphysik, Naturwiss.-techn. Fakultät, Universität Siegen,

57068 Siegen, Germany

(Received 16 August 2019; published 11 November 2019)

We present analytical calculations of the αscorrections for the coefficient of ρD=m3

Qterm in the heavy

quark expansion for the inclusive semileptonic decays of heavy hadrons such as B→Xcl¯

ν. The full

dependence of the coefficient on the final-state quark mass is taken into account. Our result leads to further

improvement of theoretical predictions for the precision determination of CKM matrix element jVcbj.

DOI: 10.1103/PhysRevD.100.093001

I. INTRODUCTION

The discovery of a Higgs boson a few years ago

completed the Standard Model (SM) of particle physics

which thereby became a highly predictive framework, i.e.,

it allows for performing precise calculations. On the

experimental side there is currently no hint at any particle

or interaction which is not described by the SM, even at the

highest possible energies. This implies that particle physics

is about to enter an era of precision measurements of the

SM parameters. In particular, accurate measurements

accompanied by precise theoretical calculations in the

flavor sector of the SM have already proven to have an

enormous reach at scales that are much larger than the

center-of-mass energy of any existing or projected colliders

[1]. Aside from large-scale experimental efforts, this

strategy also requires accurate theoretical computations.

The need in obtaining a high precision of theoretical

predictions in the flavor sector is urgent since the structure

of the quark mixing is expected to be rather sensitive

to possible effects from physics beyond the SM (BSM).

While the SM has successfully passed a variety of tests

within current precision (as a review, see e.g., [2–4]), any

further insights will require the use of even more accurate

theoretical calculations.

The weak decays of quarks mediated by charged currents

occur at a tree level and are believed to not have sizable

contributions from the BSM physics. However, the study of

such decays is important for the precise determination of

the numerical values of the SM parameters, in particular,

the CKM matrix elements. For heavy quarks (i.e., for heavy

hadrons) a reliable theoretical treatment of weak decays is

possible because the mass mQof a decaying heavy quark

constitutes a perturbative scale that is much larger than the

QCD infrared scale ΛQCD,mQ≫ΛQCD .

Heavy quark expansion (HQE) techniques provide a

systematic expansion of physical observables in powers

of the small parameter ΛQCD=mQ. Quantitatively, the

techniques work well for bottom quarks since a typical

hadronic scale associated with binding effects in QCD is

ΛQCD ∼400–800 MeV. With some reservations, the HQE

has been used for charmed quarks as well, though the

analysis is expected to be more of a qualitative nature, since

the charm-quark mass is not sufficiently large. Thus, the

HQE and the corresponding effective theory of heavy

quarks and soft gluons—Heavy Quark Effective Theory

(HQET)—have become the major tools of modern pre-

cision analysis in heavy quark flavor physics [5–8].

For determination of Vcb from the inclusive b→c

semileptonic transitions the HQE has brought an enormous

progress. The HQE expansion for the total rate and for

spectral moments have been driven to such a high accuracy

that the theoretical uncertainty in the determination of jVcbj

is now believed to be at the order of about one percent.

However, this assumes that higher order terms in ΛQCD=mQ

and αsðmQÞare of the expected size. In fact, the leading

order terms (i.e., the partonic rate) has been fully computed

to order α2

sðmQÞ, the first subleading terms of order

ðΛQCD=mQÞ2are known to OðαsðmQÞÞ while all higher-

order terms in the ðΛQCD=mQÞn(n¼3, 4, 5) are known

only at tree level.

In the present paper we analytically compute parts of

the QCD corrections to the contributions of order

ðΛQCD=mQÞ3, which is one of the not-yet-known pieces.

We point out that the size of these terms is expected to

be of the same order of magnitude as the partonic α3

sðmQÞ

contributions, likewise the terms of order α2

sðmQÞ

ðΛQCD=mQÞ2. However, at the level of current precision

these terms turn out to be small and hence their calculation

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to

the author(s) and the published article’s title, journal citation,

and DOI. Funded by SCOAP3.

PHYSICAL REVIEW D 100, 093001 (2019)

2470-0010=2019=100(9)=093001(9) 093001-1 Published by the American Physical Society

is to validate the assumption that they are of the

expected size.

Specifically we compute the coefficient of the power

suppressed dimension six Darwin term ρDat next-to-

leading order (NLO) of the strong coupling perturbation

theory with the full dependence on the final state quark

mass. Compared to the calculations at lower orders in

the HQE the present calculation has some new features

since the mixing of operators of different dimensionality

in HQET has to be taken into account for the proper

renormalization of the Darwin term coefficient.

II. HEAVY QUARK EXPANSION FOR HEAVY

FLAVOR DECAYS

In this section we set the stage by giving the basics for

the theoretical analysis of semileptonic decays, in particu-

lar, for the decay B→Xclν. A more detailed description

can be found in, e.g., [9].

The low-energy effective Lagrangian Leff for the semi-

leptonic b→cl¯

νltransitions reads

Leff ¼2ﬃﬃﬃ

2

pGFVcbð¯

bLγμcLÞð¯

νLγμlLÞþH:c:; ð1Þ

where GFis a Fermi constant, a subscript Ldenotes the left-

handed projection of fermion fields and Vcb is the relevant

CKM matrix element.

Using optical theorem one obtains the inclusive decay

rate B→Xcl¯

νlfrom taking an absorptive part of the

forward matrix element of the leading order transition

operator T(see e.g., [9])

T¼iZdxTfLeff ðxÞLeff ð0Þg;

ΓðB→Xcl¯

νlÞ∼ImhBjTjBi:ð2Þ

The transition operator Tis a nonlocal functional of the

quantum fields participating in the decay process. Since the

quark mass mQis a large scale compared to the infrared

scale of QCD, mQ≫ΛQCD; the relevant forward matrix

element still contains perturbatively calculable contribu-

tions. These contributions can be separated from the

nonperturbative pieces by employing effective field theory

tools which allows us for an efficient separation of the

kinematical mQand dynamical ΛQCD scales involved in the

decay process.

For a heavy hadron with the momentum pHand the mass

MH, a large part of the heavy-quark momentum pQis due

to a pure kinematical contribution due to its large mass

pQ¼mQvþΔwith v¼pH=MHbeing the velocity of the

heavy hadron. The momentum Δdescribes the soft-scale

fluctuations of the heavy quark field near its mass shell

originating from the interaction with light quarks and

gluons in the hadron. This is implemented by redefining

the heavy quark field QðxÞby separating a “hard”

oscillating phase and a “soft”field bvðxÞwith a typical

momentum Δof order ΛQCD,Δ∼ΛQCD,

QðxÞ¼e−imQðvxÞbvðxÞ:ð3Þ

Inserting this expression into (2) we get

T¼iZdxeimQv·xTf

˜

Leff ðxÞ

˜

Leff ð0Þg;ð4Þ

where

˜

Lis the same expression as Lwith the replacement

QðxÞ→bvðxÞ. This makes the dependence of the decay

rate on the heavy quark mass mQexplicit and allows us

for building up an expansion in ΛQCD=mQby matching the

transition operator Tin QCD onto an expansion in terms of

HQET operators [10,11].

Generally, the HQE for semileptonic weak decays are

written as (e.g., [12])

ΓðB→Xcl¯

νlÞ¼Γ0jVcbj2a0ðμÞ1þμ2

πðμÞ

2m2

b

þa2ðμÞμ2

GðμÞ

2m2

b

þaDðμÞρDðμÞþaLSðμÞρLSðμÞ

2m3

bþ…;

where Γ0¼G2

Fm5

b=ð192π3Þand mbis the b-quark mass.

The coefficients aiðμÞ,i¼0;2;D;LS depend on the

mass ratio m2

c=m2

band on the renormalization scale μ,

while μ2

πðμÞ,μ2

GðμÞ,ρDðμÞand ρLSðμÞare forward matrix

elements of local operators, usually called HQE parame-

ters. Note that the corresponding local operators in HQET

depend on the renormalization scale μas well, while the

entire expression for the rate is μindependent as such a

dependence mutually cancels between coefficient functions

and matrix elements.

The precise definition of the appropriate mass parameter

for the heavy quark field QðxÞis of utmost importance for

the precision of the predictions of the HQE and is thus

extensively discussed in the literature, see e.g., [13]. The

HQE parameter μ2

πðμÞis the kinetic energy parameter for

the B-meson in HQE; μ2

GðμÞis the chromo-magnetic

parameter. The term ρLSðμÞcontains the spin-orbital

interaction and ρDðμÞis the Darwin term which is of

our main interest in the present paper.

The power suppressed terms are becoming important

phenomenologically as the precision of experimental data

continues to improve. The coefficients aiðμÞhave a

perturbative expansion in the strong coupling constant

αsðμÞwith μ∼mQ. The leading coefficient a0is known

analytically to Oðα2

sÞaccuracy in the massless limit for the

final state quark [14]. At this order the mass corrections

have been analytically accounted for in the total width as an

THOMAS MANNEL and ALEXEI A. PIVOVAROV PHYS. REV. D 100, 093001 (2019)

093001-2

expansion in the mass of the final fermion in [15] and for

the differential distribution in [16].

The coefficient of the kinetic energy parameter is linked

to the coefficient a0by reparametrization invariance (e.g.,

[17]). The NLO correction to the coefficient of the chromo-

magnetic parameter a2ðμÞhas been investigated in [18]

where the hadronic tensor has been computed analytically

and the total decay rate has been then obtained by direct

numerical integration over the phase space. This calculation

allows for the application of different energy/momentum

cuts in the phase space necessary for the accurate com-

parison with experimental data.

The NLO strong interaction αsðμÞcorrection to the

chromo-magnetic coefficient a2ðμÞin the total decay rate

has been analytically computed in [19,20]. The techniques

of Refs. [19,20] allow also for an analytical computation

of various moments in the hadronic invariant mass or/and

that of the lepton pair. In the present paper we give the

NLO result for the coefficient aDðmQÞof ρDðmQÞin an

analytical form retaining the full dependence on the charm

quark mass.

III. NLO FOR DARWIN TERM ρD:

CALCULATIONS AND RESULTS

In this section we describe the actual computation of the

coefficient aDðμÞof the Darwin term. The present calcu-

lation follows the techniques used earlier for the determi-

nation of NLO corrections to the chromo-magnetic operator

coefficient in the total width [19–21]. Here we give a brief

outline of the calculational setup; for details of the

techniques, see [20].

We consider a normalized transition operator

˜

Tdefined

by

ImT¼Γ0jVcbj2˜

T:ð5Þ

The heavy quark expansion for the rate is constructed by

using a direct matching from QCD to HQET

˜

T¼C0O0þCv

Ov

mbþCπ

Oπ

2m2

bþCG

OG

2m2

bþCD

OD

2m3

b

;ð6Þ

where we retain only the Darwin term in the 1=m3

Qorder.

The local operators Oiin the expansion (6) are ordered by

their dimensionality:

O0¼¯

hvhvðdimension three in mass unitsÞ;ð7Þ

Ov¼¯

hvvπhvðdimension four in mass unitsÞ;ð8Þ

Oπ¼¯

hvπ2

⊥hvðdimension five in mass unitsÞ;ð9Þ

OG¼¯

hvσμνGμν hvðdimension five in mass unitsÞ;

ð10Þ

OD¼¯

hv½πμ

⊥;½πμ

⊥;πvhvðdimension six in mass unitsÞ:

ð11Þ

Here the field hvis the heavy quark field, the dynamics of

which are given by the QCD Lagrangian expanded to order

1=m3

Q. Furthermore, πμ¼iDμis the covariant derivative of

QCD and πμ¼vμðvπÞþπμ

⊥. The coefficients C0,Cv,CG,

CDof the operators are obtained by matching the appro-

priate matrix elements between QCD and HQET.

After taking the forward matrix element with the B-

meson state one can use the HQET equations of motion

for the field hvin order to eliminate the operator Ov.We

note that, in general, there is an additional operator

O5¼¯

hvðvπÞ2hvin the complete basis at dimension five;

however it will be of higher order in the HQE after using

equations of motion of HQET.

Note that one can use the full QCD fields for the heavy

quark expansion afterwards [22]. It is convenient to choose

the local operator ¯

b

=

vb defined in full QCD as a leading

term of the heavy quark expansion [23] as it is absolutely

normalized and provides a direct correspondence to the

quark parton model as the leading order of the HQE.

We no te t h at (6) is an operator relation and hence the

coefficient functions Ciare independent of any external

states. Thus these states can be freely chosen as long as they

comply with the requirements of HQE. Thus, for the

matching of QCD to HQET we can chose external states

built from gluons and heavy quarks, and the matching

procedure consists in computing matrix elements of the

relation (6) with partonic states built from quarks and gluons.

The coefficient function C0determines the total width of

the heavy quark and, at the same time, the leading power

contribution to the total width of a bottom hadron within

the HQE. At NLO the contributions to the transition

operator

˜

Tin (2) are represented by three-loop Feynman

diagrams shown in (1).

The leading order result is given by two-loop Feynman

integrals of a simple topology—the so called sunset-type

diagrams [24,25], while at the NLO level one has to

evaluate three-loop integrals with massive lines due to

massive c- and b-quarks. In Fig. 1we show a typical three-

loop diagram for the power corrections in the heavy quark

expansion.

We use dimensional regularization for both ultraviolet

and infrared singularities. We used the systems of symbolic

manipulations REDUCE [26] and Mathematica [27] with

special codes written for the calculation. For reduction of

integrals to master integrals the program LiteRed [28] was

used. The master integrals have been then computed

directly. Some of them have been checked with the program

HypExp [29].

For the Darwin term one takes an amplitude of quark

to quark-gluon scattering and projects it to an HQET

operator. We choose a momentum kgluon and take the

QCD CORRECTIONS TO INCLUSIVE HEAVY HADRON WEAK …PHYS. REV. D 100, 093001 (2019)

093001-3

structure ðϵvÞk2

⊥. There are several operators in HQET that

can have such a structure, for instance, ¯

hvðπvÞπ2

⊥hv. This

operator is irrelevant because it is of higher power on shell.

One disentangles the mixing of such operators with the

Darwin term by using two quark momenta k1and k2and

picks up the structure ðk1k2Þthat emerges in the coefficient

of the Darwin term. The other operators can have k2

1or k2

2

structure. The coefficient aDis defined in front of the

meson matrix element. After taking the matrix element one

can use the equation of motion of HQET to reduce the

number of the operators in the basis. One more conven-

tional step is to trade the leading order operator ¯

hvhvfor the

QCD operator ¯

b

=

vb that provides correspondence to the

parton model.

The final expression for the coefficient aDis then

aDðμÞ¼2CDðμÞþ3

4ðCvðμÞCHQET

DðμÞ−C0ðμÞCbvb

DðμÞÞ;

ð12Þ

where CHQET

DðμÞis the NLO coefficient of the operator

ODðμÞin the HQET Lagrangian, and Cbvb

DðμÞis the NLO

coefficient of the operator ODðμÞin the expansion of ¯

b

=

vb.

At the leading order in αsðμÞwe find

aLO

DðμÞ¼−5r4−8r3þ24r2þ36r2logðrÞ−88r

þ48 logðrÞþ77;ð13Þ

where r¼m2

c=m2

bthat agrees with [30]. At LO the

coefficient aLO

DðμÞcan depend on μthrough quark masses.

The coefficient contains a logarithmic singularity logðrÞ

at small r. This singularity reflects the mixing to hidden/

intrinsic charm contribution [31]. At higher powers even

more singular terms (like 1=r) can appear [32]. The

matching is performed by integrating out the charm quark

simultaneously with the hard modes of the b-quark. This

means that we treat m2

c=m2

bas a number fixed in the limit

mb→∞, and therefore our results cannot be used to

extrapolate to the limit mc→0.

An important check of a loop computation consists in

verifying the cancellation of poles after performing the

appropriate renormalization of the physical quantity in

question. Since in the case at hand this is quite delicate, we

briefly discuss the renormalization of the ρDcoefficient

CDðμÞin Eq. (6) at NLO within our computation.

We write the coefficient CDðμÞat NLO as

CDðμÞ¼CLO

DðμÞþαsðμÞ

4π

CNLO

DðμÞ;

and then single out the pole contribution to the NLO

coefficient CNLO

DðμÞin the form

CNLO

DðmbÞ¼αsðmbÞ

4π1

ϵ

CNLO−pol

DþCNLO−fin

D:ð14Þ

The contribution to the coefficient CNLO−pol

Dfrom one-

particle irreducible diagrams reads

CNLO−pol

D¼CA−

17r4

3þ16r3

3−28r2þ36r2logðrÞ

þ32r

3þ16 logðrÞþ53

3

þCF−

1181r4

8þ207r3þ87r2

þ285

2r2logðrÞ−419rþ72 logðrÞþ2181

8:

Here CA¼Ncand CF¼ðN2

c−1Þ=ð2NcÞare Casimir

invariants of the gauge group SUðNcÞwith Nc¼3for

QCD. The pole part of the coefficient CNLO−pol

Dcontains

different functional dependencies on r, and it is instructive

to see how the cancellation works in the present case.

The proper cancellation of these poles requires one to

consider the mixing between HQE operators of different

dimensionality which is known to be possible in HQET

[33–38]. The anomalous dimensions of the operators are

numbers independent of rwhile the functions of r

FIG. 1. Diagrams for the contribution at NLO level, (left)—partonic type, right—power correction type with an insertion of an

external gluon

THOMAS MANNEL and ALEXEI A. PIVOVAROV PHYS. REV. D 100, 093001 (2019)

093001-4

appearing in CDshould cancel in the renormalization of

the coefficient. This is a rather strong restriction because

only the coefficient functions of the lower power operators

[which are basically C0ðrÞand CvðrÞ] can be used for

the pole cancellation in CD. The leading order CGcoef-

ficient is proportional to CLO

0ðrÞ. Indeed, the explicit

expression reads

CLO

G¼2−16r−24r2lnðrÞþ16r3−2r4¼2CLO

0:ð15Þ

These properties of HQE are important for the implemen-

tation of the renormalization procedure of the Darwin-term

coefficient.

The operator Oπfrom HQE after the insertion of one

more Oπfrom the Lagrangian can mix with ODthat

produces the pole structure proportional to C0ðrÞ

OR

πðμÞ¼OB

πþγπD

αsðμÞ

4πϵ

1

mb

ODðμÞ:ð16Þ

The relation (16) means that the gh ¯

hvertex computed in

perturbation theory within HQET with one insertion of Oπ

gets a contribution from higher powers of the HQET

Lagrangian (see, e.g., [33]). By the same token the operator

OGðμÞfrom HQE after the insertion of one more OGðμÞ

from the HQET Lagrangian can mix with ODðμÞ

OR

GðμÞ¼OB

GþγGD

αsðμÞ

4πϵ

1

mb

ODðμÞ;ð17Þ

that produces the pole structure proportional to C0ðrÞagain

because of Eq. (15). The cross-insertions (OGfrom HQE to

Oπfrom the Lagrangian and vice versa) renormalize the

spin-orbit operator at the order 1=m2

b. This type of mixing

is known for a long time from the computation of the

renormalization group running of the coefficients of the

HQET Lagrangian at the order of 1=m2

b.

In the literature the renormalization is considered often

for the static heavy fields when the contributions of

reiterated terms in the HQET Lagrangian are accounted

for through the bi-local operators [36,37]. We consider the

standard approach and treat higher order terms as pertur-

bations (see [35,39]). In our case the inclusion of these

mixings does not suffice to cancel all the poles in CDas

there are other structures than C0ðrÞnecessary. The

operator Ovcan mix with double insertions of higher

dimensional terms, i.e.,

OR

vðμÞ¼OB

vþγvD

αsðμÞ

4πϵ

1

m2

b

ODðμÞ;ð18Þ

and the counterterm proportional to ODemerges from two

insertions of the operators Oπ. The mixing matrix γvD is

unknown. But the effect of such a mixing leads to the

appearance of the coefficient CvðrÞin the expression for

the poles. One can now fit the pole function with two

entries C0ðrÞand CvðrÞ.

Thus we infer the corresponding mixing anomalous

dimensions and find that the combination

−CAþ23

8CFCvðrÞ−5

4CAþ31

8CFC0ðrÞð19Þ

cancels the poles in both color structures CFand CAfor the

entire mcdependence. The solution in Eq. (19) is unique.

The presence of the coefficient Cvmeans an admixture to

the operator Ov. At this level it is impossible to confirm the

two mixings as the mixing matrices are still not uniquely

given in the literature and γvD is completely new. An

independent computation of mixing matrices could be a

useful check of our computation. Because the term r2is

present only in the mixing with Ovone can extract γvD.But

it is impossible to separate γmD and γkD as only their sum is

extracted with our current method.

Thus we arrive at a finite coefficient for the ρDðmbÞ, the

full analytical expression for the NLO correction to aDðmbÞ

is given in the Appendix. Here we discuss a numerical

impact of our result. With αsðμÞnormalized at μ¼mband

for a typical value of r¼m2

c=m2

b¼0.07 one finds

aDðmbÞ¼−57.159 þαsðmbÞ

4πð−56.594CAþ408.746CFÞ

¼−57.159 þαsðmbÞ

4πð375.213Þ

¼−57.1591−αsðmbÞ

4π

6.564:ð20Þ

For αsðmbÞ¼0.2,

aDðmbÞ¼−57.159ð1−0.10Þ;ð21Þ

the NLO contribution shifts the ρD-coefficient by 10%.

We give numerical estimates of the coefficient aDðmbÞ

for the value of the mass ratio r¼0.07 since this value is

often used in phenomenological analysis (e.g., [12,40]).

Because the quark masses are not direct observables the

numerical values for them depend on the precise definition.

For the bottom quark mass the pole mass definition is the

most convenient one and mpol

b¼4.8GeV is a commonly

accepted numerical value. The charm quark is much lighter

than the bottom quark and the pole mass definition for

its mass is not adequate. The MS-scheme mass definition is

usually used. The Particle Data Group provides a numerical

value for the charm quark running mass as mcðmcÞ¼

1.275 0.025 GeV. In phenomenological analyses of

the semileptonic weak decays the kinetic mass definition

for the bottom quark is also used. Thus, with a broad variety

of mass definitions we decide to provide numerical

QCD CORRECTIONS TO INCLUSIVE HEAVY HADRON WEAK …PHYS. REV. D 100, 093001 (2019)

093001-5

estimates for the ρDcoefficient for some typical value of the

mass ratio only.

IV. DISCUSSION

The technical details of the calculation will be discussed

in a more detailed paper, where we also plan to calculate

moments of various distributions. However, the results

presented here already have a few interesting consequences.

The first remark concerns the dependence on the mass of

the charm quark which appears in the ratio r¼m2

c=m2

b.

This ratio is kept at a fixed value as mb,mc→∞and the

behavior of the coefficients close to r¼0is given by

aLO

DðmbÞ¼−20ð1−rÞ4þ…ð22Þ

aNLO

DðmbÞ¼−CF8ð1−rÞ3ð29 þ312lnð1−rÞÞþ…;ð23Þ

where the standard representation

aDðμÞ¼aLO

DðμÞþαsðμÞ

4π

aNLO

DðμÞ

has been used. Note that the behavior of the coefficient at

the border of the available phase space depends on the

definition of the c-quark mass. Here we use the MS-scheme

definition for the charm quark mass. Note that theoretically

the end point region r→1can be described by using

HQET for both charm and bottom quarks. In the literature

there are calculations within such a setup. However, our

ultimate goal in the paper is to provide a phenomenological

estimate relevant for the description of data. The mass ratio

r¼0.07 is rather small and the approximation r∼1is a

bad starting point for phenomenology. The HQET setup for

the charm quark does not work quantitatively or is almost

marginal. So, although theoretically the HQET approach in

the end region of the phase space is justified, it does not

work numerically with any reasonable accuracy for the

actual values of quark masses.

In Fig. 2(left panel) we plot the dependence on rin the

full kinematically allowed region range 0≤r≤1.We

show the ratio

aNLO

DðmbÞð1−rÞ

aLO

DðmbÞ;ð24Þ

while the right panel of Fig. 2focuses on the physical

region around r¼0.07.

The plots show that the mass dependence in the physical

region is weak, while it is sizable over the full range. As we

discussed above, the massless limit cannot be taken, since in

the case of a b→utransition additional operators have to be

taken into account. Indeed, the operator product expansion

(OPE) for massless charm quark should be modified by

adding a new four quark operator ¯

hc¯

ch in the HQE [31].This

operator has dimension six and mixes with the ρDterm at

renormalization.The sequential renormalization of the pairof

the operators ð¯

hc¯

ch; ρDÞallows then to find the finite

expressions for the corresponding coefficient functions.

The IR divergence of the form lnðmb=mcÞat mc→0is

now hidden in the definition of the renormalized operators

¯

hc¯

ch and ρD. Nevertheless, numerically at small rthe NLO

corrections become first even smaller and have a zero at r∼

0.005 before starting to grow. A similarly strong dependence

on rhas been observed also for the QCD corrections in the

coefficient of μ2

G[20].

Although the NLO corrections are not untypically large,

they will have a visible impact on the determination of Vcb .

This is mainly due to the fact that the coefficient in front

of ρDin the total rate is quite large, see (21). While a

detailed analysis will require to repeat the combined fit as,

e.g., in [40], we may obtain a tendency from an approxi-

mate formula given in Eq. (12) in this paper. According

to (21) the NLO correction corresponds to a reduction of

the contribution of ρDby 10%; thus Eq. (12) of [40] implies

a shift in the central value of

ΔVcb

Vcb ¼−0.3%;ð25Þ

which is about a third of the current theoretical uncertainty.

FIG. 2. Mass dependence of the NLO Coefficient of ρD. Left panel: The ratio (24) over the full range of r; right panel: The ratio of the

NLO coefficient to the LO one in the physically interesting region.

THOMAS MANNEL and ALEXEI A. PIVOVAROV PHYS. REV. D 100, 093001 (2019)

093001-6

However, parametrically this correction is of the same

size as the yet unknown corrections of order α2

sΛ2

QCD=m2

b

and α3

s, which would need to be included in a full analysis

up to order αsΛ3

QCD=m3

b. We note in passing that the

corrections αsρ2

LS are not needed, since these are included

in the known αsμ2

Gcontributions [41]. Nevertheless, the

contribution of ρDis significant due to the large coefficient

in front of ρDand hence we expect that the impact of this

correction is the largest one.

ACKNOWLEDGMENTS

This research was supported by the Deutsche

Forschungsgemeinschaft (DFG, German Research

Foundation) under grant 396021762—TRR 257 “Particle

Physics Phenomenology after the Higgs Discovery.”

APPENDIX

The matching coefficient of the “operator”ρDin the

HQET Lagrangian in NLO at μ¼mbgets a correction [11]:

1þαsðmbÞ

4π

2CA:ðA1Þ

After using equations of motion the final ρD-coefficient is

expressed through the coefficient of the relevant operator

in HQE (direct contribution) and the contributions due to

HQET Lagrangian and the choice of the full QCD operator

at the leading power through the relation

aDðμÞ¼2CDðμÞþ3

4CvðμÞ1þαsðμÞ

4π

2CA−

3

4C0:

ðA2Þ

At LO one obtains

aLO

DðμÞ¼−5r4−8r3þ24r2þ36r2logðrÞ

−88rþ48 logðrÞþ77 ðA3Þ

that agrees with [30].

We write the coefficient of ρDterm after taking matrix

elements as

aDðμÞ¼aLO

DþαsðμÞ

4π

aNLO

DðμÞ;ðA4Þ

and then

aNLO

DðμÞ¼aNLO;cf

DðμÞCFþaNLO;ca

DðμÞCA:ðA5Þ

The CFcolor part reads at NLO

aNLO;cf

DðmbÞ¼−1776r5=2−

6464r3=2

3−144r4þ120r2−

880 ﬃﬃﬃ

r

p

3Li2ð−ﬃﬃﬃ

r

pÞ

þð3408r5=2þ5312r3=2−144r4þ120r2þ880 ﬃﬃﬃ

r

pÞLi2ðﬃﬃﬃ

r

pÞ

þð168r4þ256r3−1304r2þ928rþ424ÞLi2r−1

r

þ−408r5=2−

2368r3=2

3−192r4−32r3þ144r2−

440 ﬃﬃﬃ

r

p

3þ80Li2ðrÞ

−648π2r5=2−

2800

3π2r3=2þ44π2r4−

1615r4

48 þ16π2r3

3−

123184r3

45

−34π2r2−

363827r2

180 þ164r4þ880r3

3−

443r2

3þ584rþ140log2ðrÞ

þ18677r4

15 −

54296r3

45 þ1648r2

9þ392

15r2þ6424r

3−

4496

45r−

20603

9logð1−rÞ

þ1296r5=2þ5600r3=2

3þ880 ﬃﬃﬃ

r

p

3log ð1−ﬃﬃﬃ

r

pÞþ−1296r5=2−

5600r3=2

3−

880 ﬃﬃﬃ

r

p

3log ðﬃﬃﬃ

r

pþ1Þ

−

2093r4

60 þ34466r3

45 þ41815r2

18 þ−296r4−

352r3

3−

6904r2

3−576r−

1912

3logð1−rÞ

þ3752r

3−

2096

3logðrÞþ220204r

45 −

440π2ﬃﬃﬃ

r

p

3þ392

15r−

40π2

3−

91603

720 :ðA6Þ

Here Li2ðzÞis a dilogarithm.

QCD CORRECTIONS TO INCLUSIVE HEAVY HADRON WEAK …PHYS. REV. D 100, 093001 (2019)

093001-7

Note that in the course of the computation of the ρD-coefficient a new master integral Nmhas appeared compared to

our previous results for the chromo-magnetic operator in [20]. A relevant combination of the basic master integrals Np,Nm

(see [20] for more details) turns out to be ðNpþNmÞ=2and both Npand Nmshould be evaluated at the leading order in

the ϵ-expansion.

A corresponding expression for the CAcolor part of the NLO coefficient looks rather similar and reads

aNLO;ca

DðmbÞ¼−152r5=2−

400r3=2

3−336r2þ280 ﬃﬃﬃ

r

p

3Li2ð−ﬃﬃﬃ

r

pÞ

þð456r5=2þ400r3=2−336r2−280 ﬃﬃﬃ

r

pÞLi2ðﬃﬃﬃ

r

pÞþ−208r3þ712r2

3−360rþ216Li2r−1

r

þ−76r5=2−

200r3=2

3−24r3þ72r2−72rþ140 ﬃﬃﬃ

r

p

3þ24Li2ðrÞ

−76π2r5=2−

200

3π2r3=2þ329r4

36 þ4π2r3þ749r3

9þ16π2r2þ12941r2

45

þ−

172r3

3þ268r2

3−216rþ40log2ðrÞ

þ48r4−

10114r3

45 þ134r2þ238

15r2þ280r

3−

62

r−

40

9logð1−rÞ

þ152r5=2þ400r3=2

3−

280 ﬃﬃﬃ

r

p

3log ð1−ﬃﬃﬃ

r

pÞþ−152r5=2−

400r3=2

3þ280 ﬃﬃﬃ

r

p

3log ðﬃﬃﬃ

r

pþ1Þ

−26r4þ11794r3

45 −673r2þ136r3

3−144r2þ128r−

320

3logð1−rÞþ508r

3−

1060

3logðrÞ

þ12π2r−

2009r

15 þ140π2ﬃﬃﬃ

r

p

3þ238

15r−4π2−

47137

180 :ðA7Þ

Note that there is no 1=r singularity at small r. The small rexpansion for the CFstructure is

aNLO;cf

DðmbÞ¼−72 log2ðrÞ−

2096 logðrÞ

3−84π2−

5815

144 −

440π2ﬃﬃﬃ

r

p

3

þ1

9rð1080 log2ðrÞþ7896 logðrÞ−1392π2þ80111ÞþOðr3=2Þ;ðA8Þ

and for the CApart is

aNLO;ca

DðmbÞ¼−68 log2ðrÞ−

1060 logðrÞ

3−40π2−

7481

36 þ140π2ﬃﬃﬃ

r

p

3

þr−36 log2ðrÞþ740 logðrÞ

3þ72π2−

2134

9þOðr3=2Þ:ðA9Þ

[1] J. Charles, O. Deschamps, S. Descotes-Genon, H. Lacker,

A. Menzel, S. Monteil, V. Niess, J. Ocariz et al.,Phys.

Rev. D 91, 073007 (2015).

[2] J. N. Butler et al. (Quark Flavor Physics Working Group

Collaboration), arXiv:1311.1076.

[3] A. J. Bevan et al. (BABAR and Belle Collaborations), Eur.

Phys. J. C 74, 3026 (2014).

[4] S. Forte, A. Nisati, G. Passarino, R. Tenchini, C. M. C.

Calame, M. Chiesa, M. Cobal, G. Corcella et al.,Eur. Phys.

J. C 75, 554 (2015).

THOMAS MANNEL and ALEXEI A. PIVOVAROV PHYS. REV. D 100, 093001 (2019)

093001-8

[5] M. A. Shifman and M. B. Voloshin, Sov. J. Nucl. Phys. 41,

120 (1985).

[6] H. Georgi, Phys. Lett. B 240, 447 (1990).

[7] M. Neubert, Phys. Rep. 245, 259 (1994).

[8] A. V. Manohar and M. B. Wise, Cambridge Monogr. Part.

Phys., Nucl. Phys., Cosmol. 10, 1 (2000).

[9] I. I. Y. Bigi, M. A. Shifman, N. G. Uraltsev, and A. I.

Vainshtein, Phys. Rev. Lett. 71, 496 (1993).

[10] T. Mannel, W. Roberts, and Z. Ryzak, Nucl. Phys. B368,

204 (1992).

[11] A. V. Manohar, Phys. Rev. D 56, 230 (1997).

[12] D. Benson, I. I. Bigi, T. Mannel, and N. Uraltsev, Nucl.

Phys. B665, 367 (2003).

[13] A. A. Penin and A. A. Pivovarov, Phys. Lett. B 443, 264

(1998).

[14] T. van Ritbergen, Phys. Lett. B 454, 353 (1999).

[15] A. Pak and A. Czarnecki, Phys. Rev. Lett. 100, 241807

(2008).

[16] K. Melnikov, Phys. Lett. B 666, 336 (2008).

[17] T. Becher, H. Boos, and E. Lunghi, J. High Energy Phys. 12

(2007) 062.

[18] A. Alberti, P. Gambino, and S. Nandi, J. High Energy Phys.

01 (2014) 147.

[19] T. Mannel, A. A. Pivovarov, and D. Rosenthal, Phys. Lett. B

741, 290 (2015).

[20] T. Mannel, A. A. Pivovarov, and D. Rosenthal, Phys. Rev. D

92, 054025 (2015).

[21] T. Mannel, A. A. Pivovarov, and D. Rosenthal, Nucl. Part.

Phys. Proc. 263–264, 44 (2015).

[22] T. Mannel and K. K. Vos, J. High Energy Phys. 06 (2018)

115.

[23] A. V. Manohar and M. B. Wise, Phys. Rev. D 49, 1310

(1994).

[24] S. Groote, J. G. Korner, and A. A. Pivovarov, Nucl. Phys.

B542, 515 (1999).

[25] S. Groote, J. G. Korner, and A. A. Pivovarov, Ann. Phys.

(Amsterdam) 322, 2374 (2007);Phys. Lett. B 443, 269

(1998).

[26] A. C. Hearn, REDUCE, User’s manual Version 3.8. Santa

Monica, CA, USA (2004).

[27] Wolfram Research, Inc., Mathematica, Version 9.0,

Champaign, IL (2012).

[28] R. N. Lee, J. Phys. 523, 012059 (2014).

[29] T. Huber and D. Maitre, Comput. Phys. Commun. 178, 755

(2008).

[30] M. Gremm and A. Kapustin, Phys. Rev. D 55, 6924 (1997).

[31] I. Bigi, T. Mannel, S. Turczyk, and N. Uraltsev, J. High

Energy Phys. 04 (2010) 073.

[32] T. Mannel, S. Turczyk, and N. Uraltsev, J. High Energy

Phys. 11 (2010) 109.

[33] A. F. Falk, B. Grinstein, and M. E. Luke, Nucl. Phys. B357,

185 (1991).

[34] C. W. Bauer and A. V. Manohar, Phys. Rev. D 57, 337

(1998).

[35] M. Finkemeier and M. McIrvin, Phys. Rev. D 55, 377

(1997).

[36] C. Balzereit and T. Ohl, Phys. Lett. B 386, 335 (1996).

[37] B. Blok, J. G. Korner, D. Pirjol, and J. C. Rojas, Nucl. Phys.

B496, 358 (1997).

[38] C. L. Y. Lee, Report No. CALT-68-1663.

[39] C. W. Bauer, A. F. Falk, and M. E. Luke, Phys. Rev. D 54,

2097 (1996).

[40] P. Gambino and C. Schwanda, Phys. Rev. D 89, 014022

(2014).

[41] M. Fael, T. Mannel, and K. Keri Vos, J. High Energy Phys.

02 (2019) 177.

QCD CORRECTIONS TO INCLUSIVE HEAVY HADRON WEAK …PHYS. REV. D 100, 093001 (2019)

093001-9