QCD corrections to inclusive heavy hadron weak decays at Λ QCD 3 / m Q 3

Abstract

We present analytical calculations of the αs corrections for the coefficient of ρD/mQ3 term in the heavy quark expansion for the inclusive semileptonic decays of heavy hadrons such as B→Xcℓν¯. 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 |Vcb|.

Full text

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
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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¯
νlÞ¼Γ0jVcbj2a0ðμÞ1þμ2
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
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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
⊥;πvhvðdimension 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
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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
aDðμÞ¼2CDðμÞþ3
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
4π1
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
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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
8CFCvðrÞ−5
4CAþ31
8CFC0ðrÞð19Þ
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
¼−57.1591−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
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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)
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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
aDðμÞ¼2CDðμÞþ3
4CvðμÞ1þαsðμÞ
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
3Li2ð−ffiffiffi
r
pÞ
þð3408r5=2þ5312r3=2−144r4þ120r2þ880 ffiffiffi
r
pÞLi2ðffiffiffi
r
pÞ
þð168r4þ256r3−1304r2þ928rþ424ÞLi2r−1
r
þ−408r5=2−
2368r3=2
3−192r4−32r3þ144r2−
440 ffiffiffi
r
p
3þ80Li2ðrÞ
−648π2r5=2−
2800
3π2r3=2þ44π2r4−
1615r4
48 þ16π2r3
3−
123184r3
45
−34π2r2−
363827r2
180 þ164r4þ880r3
3−
443r2
3þ584rþ140log2ðrÞ
þ18677r4
15 −
54296r3
45 þ1648r2
9þ392
15r2þ6424r
3−
4496
45r−
20603
9logð1−rÞ
þ1296r5=2þ5600r3=2
3þ880 ffiffiffi
r
p
3log ð1−ffiffiffi
r
pÞþ−1296r5=2−
5600r3=2
3−
880 ffiffiffi
r
p
3log ðffiffiffi
r
pþ1Þ
−
2093r4
60 þ34466r3
45 þ41815r2
18 þ−296r4−
352r3
3−
6904r2
3−576r−
1912
3logð1−rÞ
þ3752r
3−
2096
3logðrÞþ220204r
45 −
440π2ffiffiffi
r
p
3þ392
15r−
40π2
3−
91603
720 :ðA6Þ
Here Li2ðzÞis a dilogarithm.
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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
3Li2ð−ffiffiffi
r
pÞ
þð456r5=2þ400r3=2−336r2−280 ffiffiffi
r
pÞLi2ðffiffiffi
r
pÞþ−208r3þ712r2
3−360rþ216Li2r−1
r
þ−76r5=2−
200r3=2
3−24r3þ72r2−72rþ140 ffiffiffi
r
p
3þ24Li2ðrÞ
−76π2r5=2−
200
3π2r3=2þ329r4
36 þ4π2r3þ749r3
9þ16π2r2þ12941r2
45
þ−
172r3
3þ268r2
3−216rþ40log2ðrÞ
þ48r4−
10114r3
45 þ134r2þ238
15r2þ280r
3−
62
r−
40
9logð1−rÞ
þ152r5=2þ400r3=2
3−
280 ffiffiffi
r
p
3log ð1−ffiffiffi
r
pÞþ−152r5=2−
400r3=2
3þ280 ffiffiffi
r
p
3log ðffiffiffi
r
pþ1Þ
−26r4þ11794r3
45 −673r2þ136r3
3−144r2þ128r−
320
3logð1−rÞþ508r
3−
1060
3logðrÞ
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
9þOðr3=2Þ:ðA9Þ
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