Subleadingpower corrections to the radiative leptonic B → γℓν decay in QCD
Received: March
Subleadingpower corrections to the radiative leptonic
YuMing Wang 0 1 2 3 5
YueLong Shen 0 1 2 4
0 Songling Road 238 , Qingdao, 266100 Shandong , P.R. China
1 Boltzmanngasse 5 , 1090 Vienna , Austria
2 Weijin Road 94 , 300071 Tianjin , China
3 Fakultat fur Physik, Universitat Wien
4 College of Information Science and Engineering, Ocean University of China
5 School of Physics, Nankai University
Applying the method of lightcone sum rules with photon distribution amplitudes, we compute the subleadingpower correction to the radiative leptonic B ! decay from the twisttwo hadronic photon contribution at nexttoleading order in QCD; and further evaluate the highertwist \resolved photon" corrections at leading order in s up to twistfour accuracy. QCD factorization for the vacuumtophoton correlation function with an interpolating current for the Bmeson is established explicitly at leading power b =mb employing the evanescent operator approach. Resummation of the parametrically large logarithms of m2= 2 entering the hard function of the leadingtwist factorization formula is achieved by solving the QCD evolution equation for the lightray tensor operator at two loops. The leadingtwist hadronic photon e ect turns out to preserve the symmetry relation between the two B ! form factors due to the helicity conservation, however, the highertwist hadronic photon corrections can yield symmetrybreaking e ect already at tree level in QCD. Using the conformal expansion of photon distribution amplitudes with the nonperturbative parameters estimated from QCD sum rules, the twisttwo hadronic photon contribution can give rise to approximately 30% correction to the leadingpower \direct photon" e ect computed from the perturbative QCD factorization approach. In contrast, the subleadingpower corrections from the highertwist twoparticle and threeparticle photon distribution amplitudes are estimated to be of O(3 approach. We further predict the partial branching fractions of B !
Heavy Quark Physics; Perturbative QCD

B
energy cut E
Ecut, which are of interest for determining the inverse moment of the
leadingtwist Bmeson distribution amplitude thanks to the forthcoming highluminosity
Belle II experiment at KEK. Keywords: Heavy Quark Physics, Perturbative QCD
1 Introduction 2 3 4
5
6
A Spectral representations B
Highertwist photon DAs
1
Introduction
Theoretical overview of B !
`
decay
Leadingtwist hadronic photon correction in QCD
The twisttwo hadronic photon correction at tree level
The twisttwo hadronic photon correction at one loop
Highertwist hadronic photon corrections in QCD
3.1
3.2
4.1
4.2
5.1
5.2
Numerical analysis
Conclusion
Highertwist twoparticle corrections
Highertwist threeparticle corrections
Theory inputs
Predictions for the B !
` form factors
the LHC phenomenologically. In these respects, the radiative leptonic decay B !
with an energetic photon in the
nal state is widely believed to provide a clean probe
of the strong interaction dynamics of a heavy quark system and to put stringent
constraints on the inverse moment of the leadingtwist Bmeson distribution amplitude (DA).
Factorization properties of B !
` have been investigated extensively at leading power
in
=mb with distinct QCD techniques [1, 2] and with the softcollinear e ective theory
(SCET) [3{5] which established the corresponding QCD factorization formula to all orders
in perturbation theory.
Subleadingpower corrections to the B !
` transition form factors were discussed
in QCD factorization at tree level [6], where the symmetrypreserving form factor (E )
was introduced to parameterize the nonlocal SCET matrix element without integrating
{ 1 {
out the hardcollinear scale. Systematic studies on the higherpower terms of the radiative
leptonic Bmeson decay amplitude in the heavy quark expansion are, however, still absent
in the framework of SCET beyond the leadingorder in
s
. Applying the dispersion
relations and the partonhadron duality, an alternative approach without identifying manifest
structures of the subleadingpower e ective operators was proposed [7] to estimate the
power suppressed soft contributions at tree level and was further extended [8] to compute
the softoverlap contribution at nexttoleadingorder (NLO) in QCD. Consequently, there
will be a price to pay for the dispersion approach when taking into account the hadronic
photon corrections and the endpoint contributions (the socalled Feynman mechanism)
by implementing the nonperturbative modi cations of the QCD spectral densities, as two
additional nonperturbative parameters (vector meson mass m and e ective threshold
parameter s0) are introduced when compared to the direct QCD calculation. It is then evident
that evaluating the higherpower terms in the expansion of
=mb individually with direct
QCD approaches is of particular interest to deepen our understanding of perturbative QCD
factorization for hard exclusive reactions.
The major objective of this paper is to perform QCD calculations of the
subleadingpower corrections induced by the hadronic component of the energetic photon at NLO
in the strong coupling constant. QCD factorization formula for the twoparticle hadronic
!
photon correction to the B
`
amplitude was demonstrated to be invalidated by
the rapidity divergence in the convolution integral of the hard scattering kernel with the
lightcone DAs of the Bmeson and of the photon [5]. Employing the technique of
lightcone sum rules (LCSR) with the twoparticle photon DAs, the power suppressed \resolved
photon" contribution was computed at twistfour accuracy and at leadingorder (LO) in
s [9{11], and was further updated [12] by including the NLO correction to the
leadingtwist hadronic photon DA contribution and by calculating the highertwist correction from
the threeparticle photon DAs at tree level. However, QCD factorization for the
vacuumtophoton correlation function with an interpolating current for the Bmeson is not explicitly
demonstrated with the operatorproductexpansion (OPE) technique at one loop in [12],
where the renormalization scheme dependence of 5 for the QCD amplitude in dimensional
regularization was not addressed in any detail. It is therefore necessary to perform an
independent calculation of the twisttwo hadronic photon correction to the B !
` form
factors at NLO in s by compensating the abovementioned gaps. To this end, we will apply
the standard perturbative matching procedure including the evanescent SCET operators to
establish QCD factorization formulae for the vacuumtoBmeson correlation function with
the Dirac matrix 5 de ned in naive dimensional regularization (NDR) (see [13, 14] for an
overview, and [15] for a discussion in the context of the pionphoton transition form factor).
The presentation is organized as follows. We rst summarize the theoretical status on
QCD calculations of the B !
` form factors with di erent techniques based upon the
heavy quark expansion and discuss the origin of subleadingpower corrections in section 2.
To construct the sum rules for the leadingtwist hadronic photon correction, we then
establish QCD factorization for the correlation function de ned with an interpolating current for
the Bmeson and with the weak transition current [u
(1
5) b] in section 3, where the
master formula of the hard matching coe cient entering the factorization formula at one
{ 2 {
loop will be derived with the implementation of the infrared (IR) subtraction including the
evanescent SCET operator. With the aid of the evolution equation of the twisttwo photon
DA at two loops, summation of the parametrically large logarithms of m2= 2 in the hard
function will be further preformed at nexttoleadinglogarithmic (NLL) accuracy
applying the momentumspace renormalization group (RG) approach. The NLL resummation
improved LCSR for the twisttwo hadronic correction to the B !
form factors will be
also presented here, taking advantage of the dispersion relation technique and the
partonhadron duality ansatz. The subleadingpower corrections to the B !
` decay amplitude
from both the twoparticle and threeparticle highertwist photon DAs displayed in [16] will
b
be computed with the LCSR approach at tree level in section 4, where a comparison of our
results with that obtained in [11, 12] will be also presented. Phenomenological impacts of
the various subleadingpower corrections with the nonperturbative parameters of the
photon DAs determined from QCD sum rules [17] will be explored in section 5, including the
dependence of the partial branching fractions of B !
` , with the phasespace cut of the
photon energy, on the inverse moment B. A summary of our main observations and future
perspectives will be presented in section 6. We further collect spectral representations of
the convolution integrals entering the leadingtwist factorization formulae for the
vacuumtophoton correlation function at oneloop accuracy and the operatorlevel de nitions of
the highertwist photon DAs up to the twistfour in appendices A and B, respectively.
2
Theoretical overview of B
`
decay
!
The radiative leptonic B !
` decay amplitude is de ned by the following matrix element
(p) `(p`) (p ) `
(1
5)
[u
(1
5) b] B (pB) :
A(B
into account by the rede nition of the axial form factor FA(n p).
At leading power in
=mb the QCD factorization formula for the B !
form factors
can be readily derived with the SCET technique [3, 4]
FV; LP(n p) = FA; LP(n p) =
n p
Qu mB f~B( ) C?(n p; )
Z 1
0
d!
B+(!; )
!
J?(n p; !; ) :
(2.4)
{ 3 {
The hard function C
? arises from matching the QCD weak current u
the corresponding SCET current and the oneloop expression is given by [18, 19]
5) b onto
C
? = 1
4
s CF 2 ln2
+
3r
1
2
r
n p
2
12
ln r +
? entering the SCET factorization
formula (2.4) reads [3, 4, 8]
HJEP05(218)4
J
? = 1 +
s CF
4
ln2
2
n p (!
n p)
2
6
1 + O( s2) :
Setting
as a hardcollinear scale of order p
of the parametrically large logarithms in the hard function yields
mb and performing the NLL resummation
FV; LP(n p) = FA; LP(n p)
=
Qu mB
n p B( )
h
U2(n p; h2; ) f~B( h2)i [U1(n p; h1; ) C?(n p; h1)]
1 +
de ned in [6] and the manifest expressions of the evolution functions U1 and U2 can be
The subleadingpower corrections from photon radiation o the heavy quark and from
highertwist Bmeson DAs were addressed [6] by computing the two diagrams for the
W
amplitude in QCD. Since the factorization property of the nonlocal
subleadingpower correction from photon radiation o the light quark has not been explored
yet, we will only focus on the local subleadingpower contribution to the B !
As discussed in [5] the subleadingpower contribution can be further generated by the
e ective matrix element h (p)jOjB (pB)i with the SCET operator O
[qs hv]s [ ]c
containing no photon eld, due to the unsuppressed interactions of photons with any numbers
of collinear quark and gluon
elds. The collinear matrix element h (p)j[
]cj0i de nes
the photon DAs on the light cone, making the photon behave in analogy to an energetic
vector meson. Consequently, these terms are also referred to as the \hadronic (resolved)
photon" contributions in di erent contexts. QCD calculations of such power suppressed
corrections to the B !
DAs accuracy, with the LCSR approach in the following.
` decay form factors will be carried out, to the twistfour photon
{ 4 {
(2.5)
(2.6)
1
;
(2.7)
To obtain the sum rules for the form factors FV (n p) and FA(n p), we construct the
vacuumtophoton correlation function with an interpolating current for the Bmeson
Z
5) b(x); b(0) 5 u(0)gj0i ;
(3.1)
where q = p` + p refers to the fourmomentum of the leptonneutrino pair. QCD
factorization for the correlation function (3.1) can be demonstrated with the technique of the
lightcone OPE at (p + q)2
following power counting scheme
mb2 and q
2
b
m2. For de niteness, we will employ the
The twisttwo hadronic photon correction at tree level
QCD factorization for the twisttwo contribution to the correlation function (3.1) can be
justi ed by investigating the QCD amplitude
d4x ei q x hq(z p) q(z p)jT fu(x)
? (1
5) b(x); b(0) 5 u(0)gj0i ;
(3.3)
where z indicates the momentum fraction carried by the collinear quark and z
1
Evaluating the tree diagram displayed in gure 1 leads to
F (0)(p; q) =
=
i
i
2 z(p + q)2 + z q2
n q
n q
where the convolution integral of z0 is represented by an asterisk. hOA; (z; z0)i(0) indicates
the partonic matrix element of the SCET operator OA; at tree level
hOA; (z; z0)i = hq(z p) q(z p)jOA; (z0)j0i = (z p)
? 6 n (1 + 5) (z p) (z
z0) + O( s);
where the general de nition of the collinear operator in moment space reads
Z
0
{ 5 {
Oj; (z0) =
n p Z
2
j =
? 6 n (1 + 5) :
d e i z0 n p ( n) Wc( n; 0) j (0) ;
The collinear Wilson line with the convention of the covariant derivative D
Wc( n; 0) = P
Exp i gs
d n Ac( n)
:
(3.4)
(3.5)
(3.6)
(3.7)
To establish the hardcollinear factorization for the QCD amplitude (3.3), we further
decompose the SCET operator OA; into the lightray operators de ning the photon DAs
displayed in [16]
with
OA; = O1; + O2; + OE; ;
1 =
? 6 n ;
2 =
n
2
;
E =
Expanding the operator matching equation including the evanescent operator
we can readily derive the treelevel factorization formula for the correlation function (3.1)
p + q
i
p
u
u¯
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
at the LO in the strong coupling constant, gives rise to
X
i
(p; q) =
i
(z ; )
Employing the de nition of the Bmeson decay constant in QCD
hB (pB)jb 5 uj0i =
i
fB m2B ;
we can derive the hadronic dispersion relation of (3.1) as follows
where s0 is the e ective threshold of the Bmeson channel. The treelevel LCSR for the B !
` form factors can be obtained by matching the factorization formula (3.13) and (3.15)
with the aid of the partonhadron duality approximation and the Borel transformation
HJEP05(218)4
fB mB
q2). With the power counting scheme for the threshold parameter
and the Borel mass entering the sum rules (3.16)
s0
mb2
M 2
O(mb ) ;
z0
=mb ;
the heavyquark scaling of the hadronic photon correction at leading twist can be
established
i
(3.15)
(z; )
(3.16)
(3.17)
(3.18)
(3.19)
FV2;PpLhToton
FA2P;pLhToton
O
mb
which is indeed suppressed by a factor of =mb compared with the direct photon
contribution (see [5] for more details)
*
(p) qs A6?( )
1
i n D s
6
n
2
(1
5) hv B (pB)
mb
1=2
:
3.2
The twisttwo hadronic photon correction at one loop
In this subsection we will proceed to derive the NLO sum rules for the twisttwo hadronic
photon correction to the B !
form factors and to perform resummation of the large
b
logarithms of m2= 2 in the hard function at NLL accuracy. To this end, we will need to
demonstrate QCD factorization for the vacuumtophoton correlation function (3.1) at one
loop, applying the technique of the lightcone OPE. For the sake of determining the NLO
matching coe cients entering the factorization formulae of
(p; q), we will rst evaluate
the oneloop diagrams for the QCD matrix element F (p; q) displayed in gure 2.
The oneloop QCD correction to the weak vertex diagram shown in gure 2(a) can be
readily computed as
F (1;w)eak =
gs2CF
z(p+q)2 +zq2
u(zp) (z 6p+ 6 l)
dDl
1
(2 )D [(zp+l)2 +i0][(zp+q +l)2
5)(z 6p+ 6q+ 6 l+mb) (z 6p+ 6 l+mb) 5v(zp) ;
(3.20)
{ 7 {
+
3=2
;
O
(a)
(b)
(c)
(d)
where the external partons are already taken to be on the massshell due to the insensitivity
of the hard matching coe cients on the IR physics.
With the power counting scheme speci ed in (3.2), one can identify the leadingpower contributions of the scalar integral
HJEP05(218)4
I1 =
Z
dDl
1
(2 )D [(z p + l)2 + i0][(z p + q + l)2
mb2 + i0][l2 + i0]
;
(3.21)
from the hard and collinear regions as expected. Applying the method of regions [20],
the collinear contribution of I1 vanishes in dimensional regularization due to the resulting
scaleless integral. The collinear contribution of I1 may not vanish with a di erent
regularization scheme, however, it will be always cancelled by the corresponding IR subtraction
term. Reducing the Dirac algebra of F (1;w)eak with the NDR scheme of the Dirac matrix 5
and preforming the loopmomentum integration leads to
F (1;w);ehak =
+
+
where r1 = (z p + q)2=mb2 and r2 = q2=mb2.
displayed in gure 2(b) can be written as
F (1;B) =
dDl
(z 6p+ 6q+ 6 l+mb) 5 (z 6p 6 l)
1
(2 )D [(z p l)2 +i0][(z p+q +l)2
which again depends on the precise prescription of 5 in the complex Ddimensional space.
It is straightforward to verify that the leadingpower contributions to the Bmeson vertex
diagram also arise from the hard and collinear regions. Evaluating the hard contribution
{ 8 {
to F (1;B) with the method of regions in the NDR scheme of 5 yields
F (1;B);h =
1 r1
1 r3
(1 r3) Li2 1
1 r1
1 r3
1
1
+ln
2
m2
b
+(3 r1 r3 1)
1 r3
1 3 r1 +3 F (0) ;
with r3 = (p + q)2=mb2.
gure 2(c) can be computed as
gure 2(d)
F (1;b)ox =
dDl
The selfenergy correction to the intermediate bottomquark propagator displayed in
F (1;w)fc =
Furthermore, the wave function renormalization of the external quarks will be cancelled
precisely by the corresponding collinear subtraction term and hence will not contribute to
the perturbative matching coe cients.
Now we turn to compute the oneloop correction to the box diagram displayed in
(2 )D [(z p+l)2 +i0][(z p+q +l)2
mb2 +i0][(u p l)2 +i0][l2 +i0]
dDl
(2 )D [(z p+l)2 +i0][(z p+q +l)2
mb2 +i0][(u p l)2 +i0][l2 +i0]
(D
4)
D
D
4 l2 +
2 ?
n l
n q
n l n (u p+q)+l2
;
(3.26)
where the reduction of the Dirac algebra is achieved with the NDR scheme of 5 in the
second step and l2
?
g? l l . Performing the loopmomentum integration we nd that
the oneloop box diagram only contributes at O( ), vanishing in four dimensional space.
Such observation is in analogy to the hardcollinear factorization for the hadronic photon
correction to the pionphoton form factor at leadingtwist accuracy [15].
Adding up di erent pieces together, we obtain the oneloop QCD correction to the
fourpoint QCD matrix element as follows
F (
1
)(p; q) = T A(1;)hard(z0; (p + q)2; q2) hOA; (z; z0)i(0) + : : :
=
X
i=1;2;E
Ti(;1h)ard(z0; (p + q)2; q2)
hOi; (z; z0)i(0) + : : : ;
(3.27)
{ 9 {
where the explicit expression of the NLO hard amplitude is given by
Ti(;1h)ard NDR =
1 r1 +
1 r3
3
1 r1
1 r1
1 r3
1
+ln
2
m2
b
+
+
+
+
+
1 r1
+
4
r2
+2 ln(1 r2)+
3
1 r1
15
2
Ci(;0h)ard ;
ln2(1 r1)
6
r3
where the parameter z in the de nition of r1 should be apparently understood as z0.
We are now in a position to derive the master formulae for the hard functions
C1;2(z0; (p + q)2; q2) by implementing the ultraviolet (UV) renormalization and the IR
subtraction. Expanding the operator matching condition (3.10) at O( s) gives rise to
X Ti(
1
)(z0; (p+q)2; q2) hOi; (z; z0)i(0)
= X hCi(
1
)(z0; (p+q)2; q2) hOi; (z; z0)i(0) +Ci(0)(z0; (p+q)2; q2) hOi; (z; z0)i(
1
)i : (3.29)
The UV renormalized oneloop SCET matrix elements hOi; i
(
1
) can be further written
hOi; i
(
1
) =
X hMi(j1;)b;aRre + Zi(j1)i
hOj; i
(0) ;
j
i
i
as [21]
where Mi(j1;)b;aRre are the bare matrix elements dependent on the IR regularization scheme and
Zi(j1) are the UV renormalization constants at one loop. When both UV and IR divergences
are coped with dimensional regularization, the bare SCET matrix elements vanish due to
the resulting scaleless integrals from the corresponding oneloop diagrams. Comparing the
coe cients of hOi; i
(0) (i = 1; 2) on both sides of (3.29) with the aid of (3.30) yields
The SCET operators O1; and O2; do not mix into each other, which can be veri ed
explicitly by computing the oneloop correction to the two SCET matrix elements
hOi; i
(
1
) = Zi(i1) hOi; i
(0) ; with i = 1; 2 :
The collinear subtraction term Zi(i1) hOi; i
pseudoscalar current b 5 u will remove the divergent terms of the NLO QCD amplitude Ti(
1
)
(0) and the UV renormalization of the QCD
(a)
(b)
(c)
to guarantee that the perturbative matching coe cients entering the factorization formulae
of the correlation function (3.1) are free of singularities and are entirely from the
hardscale dynamics of
. We further turn to determine the IR subtraction term ZE(1i) (i = 1; 2)
originated from the renormalization mixing of the evanescenet operators OE; into the
physical SCET operators O1;
and O2; . As discussed in [21{23], the renormalization
constants ZE(1i) (i = 1; 2) will be determined by implementing the prescription that the IR
nite matrix element of the evanescent operator OE; vanishes, when applying dimensional
regularization only to the UV divergences and regularizing the IR singularities with any
other scheme di erent from the dimensions of spacetime. In accordance with (3.30) this
amounts to
HJEP05(218)4
ZE(1i) =
M E(1i);;boare :
Inserting (3.33) into (3.31) leads to the following master formula
where Ti(;1h)a;rrdeg is the regularized terms of the NLO hard contribution to the QCD matrix
element F
as presented in (3.28) and i = 1; 2.
hOE; i
We proceed to compute the oneloop matrix element of the evanescent SCET operator
(
1
) by evaluating the e ective diagrams shown in
gure 3. Employing the SCET
Feynman rules, we nd that only the diagram (a) with a collineargluon exchange between
two collinear quarks could give rise to a nontrivial contribution to M E(1i);;boare. Evaluating
this oneloop SCET diagram explicitly yields
hOE; (z; z0)i(
1
)
i gs2 CF
Z
dDl
1
(2 )D [(z p + l)2 + i0][(l
z p)2 + i0][l2 + i0]
which only generates a nonvanishing contribution proportional to the SCET matrix
element hOE; i
(0) at O( ) with the NDR scheme of 5. Explicitly, we obtain
C(0)
E
M E(1i);;boare = 0 ; with i = 1; 2 ;
from which the oneloop hard matching coe cients can be written as
C(
1
) = Ti(;1h)a;rrdeg :
i
(3.33)
(3.34)
(3.35)
(3.36)
(3.37)
Now we are ready to demonstrate the factorizationscale independence of the
factorization formula for the vacuumtophoton correlation function (3.1)
To this end, we need to make use of the evolution equation for the leadingtwist photon DA
The explicit expression of the oneloop evolution kernel Ve0 is given by [26, 27]
Ve0(z; z0) = 2 CF
(z
z0) +
(z0
z)
CF (z
z0) ;
(3.42)
z
z0 z0
1
z
+
where the plus function is de ned as
It is then straightforward to write down
f (z; z0) + = f (z; z0)
(z
z0)
dt f (t; z0) :
d mb( )
d ln
z
z0 z
=
1
z0
0
X
n=0
with the renormalization kernel Ve (z; z0) expanded perturbatively in QCD
and the RG equation for the bottomquark mass [24, 25]
Ve (z; z0) =
4
s n+1
Ven(z; z0) ;
X
n=0
s( ) n+1
4
m(n) ;
m(0) = 6 CF :
0
i
(3.38)
(3.40)
(3.41)
(3.43)
(3.44)
d
dependence at one loop arises from the UV renormalization of the
pseudoscalar QCD current de ning the correlation function (3.1). Distinguishing the
renormalization scale , due to the nonconservation of the pseudoscalar current in QCD, from
the factorization scale
governing the RG evolution in SCET, we are led to conclude
that the factorization formula (3.38) of
(p; q) is indeed independent of the scale
at
oneloop accuracy.
According to the QCD factorization formula (3.38) for the correlation function
(p; q),
we cannot avoid the parametrically large logarithms of O(ln (mb2= 2)) by adopting a
universal scale
in the hard matching coe cient and in the photon DA. We will perform
resummmation of the abovementioned large logarithms at NLL accuracy by applying the
twoloop RG equation of the twisttwo photon DA and by setting the factorization scale
as
mb. The NLO evolution kernel Ve1 in QCD can be decomposed as follows [28{30]
Ve1(z; z0) =
2
Nf CF VeN (z; z0) + CF CA VeG(z; z0) + CF2 VeF (z; z0) ;
(3.45)
where the explicit expressions of the evolution functions can be found in [29]. Symmetry
properties of the RG evolution equation (3.39) imply the series expansion of the
leadingtwist photon DA in terms of the Gegenbauer polynomials
1
n=0
(z; ) = 6 z z X an( ) Cn3=2(2z
(0)(s; q2) =
s
b
s
q
mb2) :
The twoloop evolution of the Gegenbauer moment an( ) can then be obtained as follows
( ) hqqi( ) an( ) = ETN;LnO( ; 0) ( 0) hqqi( 0) an( 0)
+
s( ) X
where k; n = 0; 2; 4; : : : and the explicit expressions of the RG functions ET(N;n)LO and the
o diagonal mixing coe cients can be found in appendix A of [15]. In contrast to the LO
evolution in QCD, the Gegenbauer coe cients an( ) do not renormalize multiplicatively
at NLO accuracy. Inserting (3.46) and (3.47) into the NLO factorization formula (3.38)
gives rise to the NLL resummation improved expression
(p; q) = gem Qu n p ( ) hqqi( )
where the perturbative function Kn((p + q)2; q2) is determined by
Kn =
dz hCi(0)(z; (p + q)2; q2) + Ci(
1
)(z; (p + q)2; q2)i h6 z z Cn3=2(2 z
i
1) :
To construct the sum rules for the twisttwo hadronic photon correction to the B !
`
form factors, we need to derive the dispersion representation for the NLL factorization
formula (3.48). Applying the spectral representations of the convolution integrals collected
in appendix A, we can readily obtain
i
2 gem Qu n p n q ( ) hqqi( )
(p) hg?
i
n v i
(p + q)2
i0
where the LO spectral function (0)(s; q2) is given by
(3.48)
(3.49)
(3.50)
(3.51)
The resulting NLO spectral function (
1
)(s; q2) is rather involved and can be written as
(
1
)(s; q2) = ( 2) ln
2
m2
b
1
0
dz ln
+ ln
+
+
(z r3 +z r2 1) 0 (z; )+
z z
(z; )
(r3 1)
z r2
1
z r3 +z r2
1
4 z
1
1 z r2 P z r3 +z r2 1
;
;
+2 ln(r3 1) 0 (z = 1; )
+
r2
d
1 r2 dz
r2(r2 2)(1+z)+2 z
r2 (1 r2) zz
2 (1 3 z r2) 1
r3 r2 z r2 (1 z r2) r3
0
dz
(r3 1) ;
(z; )
z
(z; )
1
r2 r3
(3.52)
HJEP05(218)4
where P indicates the principlevalue prescription. Finally, the NLL sum rules for the
hadronic photon correction to the B !
form factors at leading twist can be written as
fB mB
It is evident that the twisttwo hadronic photon correction preserves the symmetry relation
of the two form factors FV and FA at leading power in
=mb.
Highertwist hadronic photon corrections in QCD
In this section we will aim at computing the highertwist hadronic photon corrections to
s, up to the twistfour accuracy, from the LCSR
approach. Following the discussion on a general classi cation of the photon DAs [16], we
will need to take into account the subleadingpower contributions arising from the lightcone
matrix elements of both the twobody and threebody collinear operators. To achieve this
goal, we rst demonstrate QCD factorization for the twoparticle and threeparticle
highertwist contributions to the vacuumtophoton correlation function (3.1) and then construct
the treelevel sum rules for the form factors FV and FA following the standard strategy.
4.1
Highertwist twoparticle corrections
Employing the lightcone expansion of the bottomquark propagator and keeping the
subleadingpower contributions to the correlation function (3.1) leads to
Z
d4k Z
(2 )4
i
Z
d4k Z
(2 )4
d4x ei (q k) x
k2
k
m2 h (p)ju(x)
(1 + 5) u(0)j0i
d4x ei (q k) x
k2
b
mb
b
m2 h (p)ju(x)
(1
5) u(0)j0i :
(4.1)
Making use of the de nitions of the highertwist photon DAs displayed in appendix B, it
is straightforward to write down
i
4 gem Qq (p q)
(
" 2AP;2HT((p+q)2; q2; z)
[(z p+q)2
+
2AP;3HT((p+q)2; q2; z) #
+
2VP;3HT((p+q)2; q2; z) # )
where the explicit expressions of the invariant functions 2VP(HAT); i (i = 2 ; 3) are given by
2PHT((p + q)2; q2; z) = 2 mb f3 ( ) (a)(z; )
V;2
2VP;3HT((p + q)2; q2; z) =
2 mb2 hqqi( ) A(z; ) ;
2AP;2HT((p + q)2; q2; z) = 4 mb f3 ( ) (v)(z; ) + A(z; )
2AP;3HT((p + q)2; q2; z) =
2 mb2 hqqi( ) A(z; )
2 h (z; ) :
hqqi( ) A(z; ) ;
2 h (z; ) hqqi( ) ;
(4.3)
The two new functions (v)(z; ) and h (z; ) introduced in (4.3) are de ned by
Z z
0
Z z
0
(v)(z; ) = 2
d
(v)( ; ) ;
h (z; ) =
4
d (z
) h ( ; ) :
(4.4)
The resulting LCSR for the twoparticle highertwist hadronic photon corrections to the
B !
` form factors can be further derived as follows
FV2 P(AH)T; p;LhoLton(n p)
m2B
q2 exp
=
fB mB
exp
1
q2 M 2 exp
q2)2 exp
q2)2 ds0
d h
q
2
ds
s
2 (mb2
d
exp
V (A);2 s0; q2; z = f1(s0; q2)
2PHT
d
dz
1
2 z 2VP(HAT);3(s0; q2; z)
V (A);2 s; q2; z = f1(s; q2)
2PHT
2VP(HAT);3(s0; q2; z)
i
s
z=f1(s0;q2)
z=f1(s0;q2)
z=f1(s;q2)
)
;
(4.5)
where we have de ned f1(s; q2) = (mb2
q2) to compactify the above expressions.
Several comments on the treelevel sum rules for the highertwist corrections to the
form factors FV and FA presented in (4.5) are in order.
It is evident from (4.3) that the highertwist twoparticle hadronic photon corrections
can lead to the symmetrybreaking contributions to the B !
` form factors already
at tree level, in agreement with the observation made in [12]. However, it needs to
be pointed out that the subleadingtwist e ects do not always violate the symmetry
relation of the two B !
form factors at leading power in
=mb [8].
Applying the powercounting scheme for the threshold parameter (3.17) and the
endpoint behaviours of the twoparticle photon DAs
(v)(z; ),
(a)(z; ), A(z; ) and
h (z; ), we can readily identify the heavyquark scaling for the twoparticle
highertwist corrections
FV2;PpHhTot;oLnL(n p)
FA2P;pHhTot;oLnL(n p)
mb
3=2
;
(4.6)
B !
of the pionphoton form factor).
which is of the same power as the twisttwo hadronic photon contribution obtained
in (3.18) and is suppressed by only one factor of
=mb compared with the \direct"
photon contribution (3.19). We are then led to conclude that there is generally no
correspondence between the heavyquark expansion and the twist expansion for the
` form factors in the LCSR approach (see [31] for a discussion in the context
F (p; q) de ned in (3.3) at tree level.
Highertwist threeparticle corrections
We will proceed to compute the highertwist threeparticle hadronic photon corrections to
` form factors at tree level with the sum rule technique. Following the standard
strategy, we rst compute the threeparticle contribution to the fourpoint QCD amplitude
F (p; q) (3.3) displayed in gure 4. Keeping the onegluon part for the lightcone expansion
of the bottomquark propagator in the background gluon/photon eld [32, 33]
and employing the de nitions of the threeparticle photon DAs in appendix B, we obtain
(p; q) i gem Qq (p q)
Z 1
0
dv
Z
[D i
(
3AP;2((p+q)2; q2; i; v)
[(( q +v g) p+q)2
where the integration measure is de ned as
"
#
+
i "
Z 1
0
[(( q +v g) p+q)2
d q
d q
q
q
g) :
(4.9)
Z q
0
Z q
0
Implementing the continuum subtraction with the aid of the partonhadron duality
relation and preforming the Borel transformation in the variable (p + q)2
the desired sum rules for the threeparticle hadronic photon corrections at tree level
! s gives rise to
mb +mu
fB mB exp
= Qq hqqi( )
0
m2B
M 2
( Z f1(s0;q2)
FV3 P(A;L);Lphoton(n p)
d q
3P
V (A);2 s0; q2; q; q; g; v =
(1
q f2( q; s; q2)) 1
b
s
M 2
+
Z s0
m2
b
mb2 q2
ds
0
Z f1(s;q2)
Z 1 q
d q
V (A);2 s; q2; q; q; g; v =
3P
exp
s0
M 2
d g
g
f2( q;s;q2)
f2( q; s; q2)
g
g; q; g; v = 1
The resulting expressions for the invariant functions 3VP(A);i (i = 2; 3) are given by
3VP;2 =
where we have introduced the following notations
(4.10)
(4.11)
(4.12)
+
+
+
0
0
0
0
d q
1
d q
f2( q;s0;q2)
f2( q;s0;q2)
Z f1(s0;q2)
Z 1 q
d h
ds0
ds
Z s0
m2
b
exp
0
d2 h
ds2 exp
s0
M 2
s
M 2
d
dv 2 ( q +v g) V (A);3 s0; q2; q; q; g; v
3P
d g
s0 q
2
g 2 (mb2 q2)
V (A);3 s0; q2; q; q; g; v
3P
v=f2( q;s0;q2)= g
(1
q f2( q; s0; q2))
Z f1(s;q2)
Z 1 q
d q
d g
s q
2
g 2 (mb2 q2)
f2( q;s;q2)
3P
V (A);3 s; q2; q; q; g; v
i
i
v=f2( q;s0;q2)= g
(1
q f2( q; s; q2))
v=f2( q;s;q2)= g
)
;
where for brevity we have introduced the auxiliary function f2( q; s; q2) de ned by
In accordance with the power counting scheme for the threshold parameter and the
endpoint behaviours of the threeparticle photon DAs entering the sum rules (4.12), we can
deduce the heavyquark scaling of the threeparticle hadronic photon corrections
FV3;Pp;hLoLton(n p)
FA3P;p; hLoLton(n p)
mb
5=2
;
which is suppressed by one factor of
=mb compared with the highertwist twoparticle
contributions to the B !
` form factors at tree level as presented in (4.5). It remains
interesting to verify whether the NLO QCD corrections to the threeparticle hadronic photon
contributions can give rise to a dynamically enhancement to remove the powersuppression
mechanism of the LO contributions (see [34, 35] for a discussion in the context of the NLO
sum rules for the B !
yet higherorder corrections for future work.
form factors) and we will leave explicit QCD calculations of the
Collecting the di erent pieces together, the resulting expressions for the B !
form factors including the subleadingpower contributions from the treelevel b u !
amplitude in QCD and from the hadronic photon corrections can be written as
`
W
FV (n p) = FV; LP(n p)+FVL;CNLP(n p)+FV2;PpLhToton(n p)+FV2;PpHhTot;oLnL(n p)+FV3;Pp;hLoLton(n p) ;
FA(n p) = FA; LP(n p)+FAL;CNLP(n p)+FA2P;pLhToton(n p)+FA2P;pHhTot;oLnL(n p)+FA3P;p;hLoLton(n p)
+
Q` fB
v p
;
where the last term proportional to the electric charge of the lepton comes from the
rede nition of the axialvector form factor as discussed in section 2. The detailed
expressions of the individual terms displayed on the righthand side of (4.15) are given
by (2.7), (2.8), (3.53), (4.5) and (4.12), respectively. We mention in passing that the LCSR
calculations of the hadronic photon corrections to the B !
` decay form factors
presented here su er from the systematic uncertainty due to the partonhadron duality ansatz
in the Bmeson channel. Future development of the subleadingpower contributions to the
radiative leptonic Bmeson decays in the framework of SCET including a proper treatment
of the rapidity divergences will be in demand for a modelindependent QCD analysis.
Several comments on the general structure of the B !
` form factors (4.15) are
in order.
Both the pointlike (shortdistance) and the hadronic (longdistance) photon
contributions to the B !
` amplitude were computed with same correlation
function (3.1) in [12] (see also [9, 10]), where the leadingpower pointlike photon
contribution was represented by the triangle quark diagrams. By contrast, we apply
the QCD factorization approach for the evaluation of the shortdistance photon
effect and employ the LCSR method with the photon DAs for the computation of the
(4.13)
(4.14)
subleading power hadronic photon corrections. Since both the shortdistance and
longdistance photon contributions can be de ned by hadronic matrix elements of
the corresponding e ective operators in SCET [5], we are allowed to compute the
different hadronic matrix elements contributing to the B !
` amplitude with the aid
of distinct QCD techniques, and such \hybrid" computation scheme as implemented
in this work is free of the doublecounting issue. However, it needs to be pointed
out that an additional source of the systematic uncertainty could be introduced in
the \hybrid" approach, due to the scheme dependence of separating the leading and
subleading power contributions.
The subleading power contributions to the B !
` transition form factors were
also computed from the dispersion approach [7, 8, 36] by investigating the B !
`
amplitude with the partonhadron duality ansatz. In particular, the highertwist
corrections to the form factors due to the higher Fock states of the Bmeson and to
the transverse momentum of the lightquark in the valence state were computed at
tree level [36], where the soft contributions from the twist ve and six Bmeson DAs
were also estimated in the factorization approximation.
5
Numerical analysis
We are now ready to explore the phenomenological implications of the hadronic photon
corrections to the B !
` amplitude computed from the LCSR approach. To this end,
we will proceed by specifying the nonperturbative models of the twoparticle and
threeparticle photon DAs, the rst inverse moment B( ) and the logarithmic moments 1( )
and
2( ) of the leadingtwist Bmeson DA, and by determining the Borel mass and the
hadronic threshold parameter entering the sum rules for the subleadingpower resolved
photon contributions. Having at our disposal the theory predictions for the form factors
FV and FA, we will further explore the opportunity of constraining the inverse moment
B( ) taking advantage of the improved measurements at the Belle II experiment in the
near future.
5.1
Theory inputs
In analogy to the leadingtwist photon DA, we employ the conformal expansion for the
twistthree DAs de ned by the chiraleven lightcone matrix elements
5
2
V ( i; ) = 540 !V ( ) ( q
A( i; ) = 360 q q g2 1 +
"
3
64
(v)(z; ) = 5 3 2
1 +
15 !V ( ) 5 !A( )
3
30 2 + 35 4 ;
(a)(z; ) =
1
2 (5 2
1) 1 +
3) ;
3
16
(5.1)
HJEP05(218)4
with
= 2 z
1, and for the chiralodd twistfour DAs
A(z; ) = 40 z2 z2 3 ( )
2
2
g (1
g (1
q) [4 7 ( q + q)] ;
q) q q g ;
q) q q g ;
where the anomalous dimension matrix ! is given by [16, 37]
! =
Due to the FerraraGrilloParisiGatto theorem [38], the twistfour parameters
corresponding to the \P"wave conformal spin satisfy the following relations
( )
+( ) + 1( )
1+( ) (1 2 g)+ 2( ) (3 4 g) ;
( )+ +( ) + 1( )+ 1+( ) (1 2 g)+ 2( ) (3 4 g) ;
de ned by the chiralodd lightray matrix elements. Here, we have truncated the conformal
expansion of the photon lightcone DAs up to the nexttoleading conformal spin (i.e.,
``P"wave). The renormalizationscale dependence of the twistthree parameters can be
written as
f3 ( ) =
s( )
s( 0)
!A( )
!V ( ) + !A( )
!
f = 0
f3 ( 0) ;
f =
s( )
s( 0)
!= 0
3
CF + 3 CA ;
1( ) + 11 2( ) 2 2+( ) =
(5.2)
(5.3)
(5.4)
(5.5)
( 0)
2:1
1:0
0:2
0:2
+( 0)
0
1( 0)
0:4
0:4
0
0
renormalization scale 0 = 1:0 GeV.
The scale evolution of the nonperturbative parameters at twistfour accuracy is given by
1( ) =
2 ( ) =
s( )
s( 0)
s( )
s( 0)
s( )
s( 0)
( +
qq)= 0
Q(3) qq = 0
+ = 3 CA
rqq =
3 CF ;
Q(3) =
CF ;
13
3
1( 0) ;
+
2 ( 0) ;
5
3
CF ;
( ) =
+
1 ( ) =
s( )
s( 0)
s( )
s( 0)
(
qq)= 0
Q(5) qq = 0
= 4 CA
3 CF ;
11
2
Q(
1
) =
CA
3 CF ;
Q(5) = 5 CA
CF :
8
3
( 0) ;
+
where the anomalous dimensions of these twistfour parameters at one loop are given by [16]
we will take the interval
the semileptonic B
!
tion equation of B+(!; ) [41]
Numerical values of the input parameters entering the photon DAs up to twistfour are
collected in table 1, where we have assigned 100 % uncertainties for the estimates of the
twistfour parameters from QCD sum rules [17]. The second Gegenbauer moment of the
leadingtwist photon DA will be further taken as a2( 0) = 0:07
0:07 as obtained in [16].
The magnetic susceptibility of the quark condensate (1 GeV) = (3:15
0:3) GeV 2
computed from the QCD sum rule approach including the O( s) corrections [16] and the quark
condensate density hqqi(1 GeV) =
will be also employed for the numerical estimates in the following.
(246+2189 MeV)3 determined by the GMOR relation [39]
The key quantity entering the leadingpower factorization formula of the B !
`
form factors is the rst inverse moment of the Bmeson DA
B( ), whose determination
has been discussed extensively in the context of exclusive Bmeson decays with distinct
QCD approaches (see [34, 40] for more discussions). To illustrate the phenomenological
consequences of the subleadingpower corrections from the hadronic photon contributions
B(1 GeV) = 354+3380 MeV implied by the LCSR calculations of
form factors with Bmeson DAs on the lightcone [34]. The
renormalizationscale dependence of B( ) at one loop can be determined from the
evoluB( )
B( 0)
= 1 +
4
s( 0) CF ln
0
2
2 ln
0
4 1( 0) + O( s2) ;
(5.8)
where the inverselogarithmic moments n( 0) are de ned as [6]
n( 0) =
B( )
Z 1 d!
0
!
lnn
0
!
B+(!; 0) :
(5.9)
Numerically we will employ 1(1GeV) = 1:5
1 consistent with the NLO QCD sum rule
calculation [42] and
2(1GeV) = 3
2 from [6]. Furthermore, the static Bmeson decay
constant f~B( ) will be expressed in terms of the QCD decay constant fB
f~B( ) = fB
1 +
and the determination fB = (192:0
be taken in the numerical analysis.
Following the discussions presented in [6, 34], the hard scales h1 and h2 entering the
leadingpower factorization formula will be chosen as
h1 =
h2 2 [mb=2; 2 mb] around
the default value mb and the factorization scale in (2.7) will be varied in the interval
1 GeV
2 GeV with the central value
= 1:5 GeV. In contrast, the factorization scale
entering the LCSR for the hadronic photon corrections will be taken as
around the default choice mb. In addition, we adopt the numerical values of the bottom
quark mass mb(mb) = 4:193+00::002323 GeV [44] in the MS scheme from nonrelativistic sum
rules. Finally, we turn to determine the Borel mass M 2 and the threshold parameter
s0 in the LCSR for the hadronic photon contributions. Applying the standard strategies
presented in [34] (see also [45] for a review) gives rise to following intervals
4:3) MeV from the FLAG Working Group [43] will
s0 = (37:5
2:5) GeV2 ;
M 2 = (18:0
3:0) GeV2 ;
(5.11)
which is consistent with the determinations from the LCSR of the B !
form factors [46].
5.2
Predictions for the B !
` form factors
We are now in a position to explore the phenomenological signi cance of the hadronic
photon corrections to the B !
` form factors. To develop a better understanding of the
heavy quark expansion for the bottom sector, we plot the photonenergy dependence of
the leadingpower contribution, the subleadingpower local correction and the
subleadingpower twoparticle and threeparticle hadronic photon e ects in
gure 5. It is apparent
that the twisttwo hadronic photon contribution at NLL can generate sizeable destructive
interference with the leadingpower \direct photon" contribution: approximately O(30%)
for n p 2 [3 GeV ; mB] with
B( 0) = 354 MeV. However, both the twoparticle
highertwist and the threeparticle hadronic photon contributions turn out to be numerically
insigni cant at tree level and will only shift the leadingpower prediction by an amount of
O (3
FVL;CNLP at tree level displayed in (2.8) will give rise to O(3%) correction at n p = mB and
5)% for n p 2 [3 GeV ; mB]. Furthermore, the subleadingpower local contribution
O(10%) correction at n p = 3 GeV. On account of the observed pattern for the separate
terms contributing to the B !
form factors numerically, we are led to conclude that the
power suppressed contributions to the radiative leptonic Bmeson decay are dominated by
the leadingtwist hadronic photon correction with the default theory inputs.
form factor FV (2 E ) as displayed in (4.15) with the central values of theory inputs. The individual
contributions correspond to the leadingpower contribution at NLL computed from the QCD
factorization approach (FVN;LLLP, black), the subleadingpower local contribution at LO (FVL;CNLP, green), the
twoparticle leadingtwist hadronic photon correction at NLL (FV2;PpLhTo;tNonLL, blue), the twoparticle
highertwist hadronic photon correction at leadinglogarithmic (LL) accuracy (FV2;PpHhTot;oLnL, yellow),
the threeparticle leadingtwist hadronic photon correction at LL (FV3;Pp;hLoLton, red).
We further turn to investigate the numerical impact of the perturbative correction
at NLO and the QCD resummation of the parametrically large logarithms of m2= 2 for
b
the leadingtwist hadronic photon contribution computed from the LCSR technique. It
is evident from
gure 6 that the NLO QCD correction can decrease the treelevel
prediction of the twisttwo hadronic photon contribution by an amount of O (20
for the factorization scale varied in the interval [3:0; 5:0] GeV and the NLL
resumma40)%
tion e ect can yield O (10 %) enhancement to the NLO QCD results within the same
range of . Hence, the dominant radiative correction to the leadingtwist hadronic
photon contribution originates from the NLO QCD correction to the hard matching
coefcient entering the factorization formula (3.38) rather than from resummation of the
large logarithms m2= 2
b
. However, the renormalization scale dependence of the
resummation improved theory predictions in the allowed region indeed becomes weaker compared
with the NLO calculation.
We further plot the photonenergy dependence of the ratio
RF2PVL;Tphoton(n p)
FV2;PpLhTo;tNonLL(n p)=FV2;PpLhTo;tLonL(n p) characterizing the perturbative QCD
corrections at NLL in
gure 6, where the theory uncertainties due to the variations of the
renormalization scale
are also displayed.
form factor FV (mB) at LL (dashed), NLO (dotted), and NLL (solid) accuracy,
respectively. Right : the photonenergy dependence of the ratio RF2PVL;Tphoton(n p)
p)=FV2;PpLhTo;tLonL(n p) with the uncertainties from the variations of the renormalization scale .
FV2;PpLhTo;tNonLL(n
B !
Taking into account the fact that the QCD sum rule calculation of the second
Gegenbauer moment of the twisttwo photon DA a2( 0) su ers from the large theory uncertainties
due to the strong sensitivity to the input parameters [16], we plot the leadingtwist hadronic
photon correction to the vector form factor FV (n p) in a wide range of a2( 0) in gure 7.
One can readily observe that the variation of the Gegenbauer moment a2( 0) 2 [ 0:2; 0:2]
can only give rise to a minor impact on the theory prediction of the B !
form factor
FV (mB) at maximal recoil numerically. However, the \Pwave" conformal spin
contribution from the leadingtwist photon DA will become signi cant for the evaluation of the
form factor FV (n p) with the decrease of the photon energy: approximately O(35%) at
n p = 3 GeV. To further understand the systematic uncertainty due to the truncation
of the conformal expansion at \Pwave", we also display the theory predictions for the
` form factors including the \Dwave" e ect from the fourth Gegenbauer moment
a4( 0) in gure 7. It is apparent that the sensitivity of the leadingtwist hadronic photon
contribution on a4( 0) is rather weak numerically for n p 2 [3 GeV; mB] in the \reasonable"
interval
0:2
a4( 0)
0:2. In the light of such observation, the yet higher Gegenbauer
moments of the twisttwo photon DA are not expected to bring about notable impact on
the prediction of the subleadingpower contribution to the B !
` form factors induced
by the photon lightcone DAs.
We present our nal predictions for the B !
` form factors including the newly
computed twoparticle and threeparticle hadronic photon corrections with theory
uncertainties in gure 8. The dominant theory uncertainties originate from the rst inverse moment
B( 0), the factorization scale
entering the leadingpower \direct photon" contribution,
and the second Gegenbauer moment a2( 0) of the twisttwo photon DA. However, the
symmetry breaking e ect between the two B !
form factors due to the subleadingpower
and n p = mB (right panel) on the second Gegenbauer moment of the photon lightcone DA a2( 0)
with di erent values of the fourth Gegenbauer moment: a4( 0) = 0:2 (dashed), a4( 0) = 0 (solid)
and a4( 0) =
0:2 (dotted).
` form factors as well as their di erence
computed from (4.15) with the theory uncertainties from variations of di erent input parameters
added in quadrature.
local contribution and the highertwist hadronic photon corrections su ers from much less
uncertainty than the individual form factors at 3 GeV
n p
mB. Having in our hands
the theoretical predictions for the B !
constraints on the inverse moment
` form factors, we proceed to discuss the theory
B( 0) taking advantage of the future measurements
on the (partially) integrated branching fractions with a photonenergy cut to get rid of
the soft photon radiation. It is straightforward to derive the di erential decay width for
` ; E
inverse moment B( 0) for Ecut = 1:5 GeV (blue band) and Ecut = 2:0 GeV (green band).
Ecut) on the rst
B !
` in the rest frame of the Bmeson (see also [6, 8])
d (B !
` )
d E
6 2
em G2F jVubj2 mB E3
1
FV2 (n p) + FA2 (n p) ;
(5.12)
and the integrated branching fractions with the phasespace cut on the photon energy read
BR(B !
` ; E
Ecut) = B
Z mB=2
Ecut
dE
d (B !
d E
` )
(5.13)
where B indicates the lifetime of the Bmeson. Our predictions for the partial branching
fractions of the radiative leptonic decay B !
` including the hadronic photon corrections
to the form factors are displayed in gure 9 with the variation of the inverse moment B( 0)
in the interval [0:2; 0:6] GeV. It can be observed that the integrated branching fractions
BR(B !
` ; E
Ecut) grow rapidly with the decrease of the inverse moment due to
the dependence of the two form factors on 1= B( 0) at leadingpower in
=mb. Since
the photonenergy cut E
1 GeV implemented in the Belle measurements [47] is not
su ciently large to perform perturbative QCD calculations of the B !
we will not employ the experimental bound BR(B !
` ; E
Ecut) < 3:5
form factors,
10 6 with
the full Belle data sample reported in [47] for the determination of B( 0) at the moment.
Instead, we prefer to explore the solid theory constraints on the
rst inverse moment
by comparing our predictions of the (partially) integrated branching fractions with the
improved measurements at the Belle II experiment, with the tighter phasespace cut on
the photon energy, thanks to the much higher designed luminosity of the SuperKEKB
accelerator.
tion to the B !
We computed perturbative QCD corrections to the leadingtwist hadronic photon
contribu` form factors employing the LCSR method. QCD factorization for the
B !
vacuumtophoton correlation function (3.1) has been demonstrated explicitly at one loop
with the OPE technique and the NDR scheme of the Dirac matrix 5 including the
evanescent SCET operator. The perturbative matching coe cient entering the NLO factorization
formula (3.38) was obtained by applying the method of regions and the factorizationscale
independence of the correlation function (3.1) was further veri ed at O( s) with the
evolution equations of the twisttwo photon DA and of the bottomquark mass. Resummation
of the parametrically large logarithms of O(ln(mb2= 2)) was achieved at NLL accuracy with
the twoloop RG equation of the lightray tensor operator. Implementing the continuum
subtraction with the aid of the partonhadron duality and the Borel transformation, the
NLL resummation improved LCSR for the twisttwo hadronic photon correction to the
form factors was subsequently constructed with the spectral representations of the
factorization formula (3.38). The subleadingpower correction to the B !
from the leadingtwist photon DA was shown to preserve the symmetry relation between
the two form factors due to the helicity conservation, in agreement with the observation
made in [12].
Along the same vein, we proceed to compute the twoparticle and threeparticle
highertwist hadronic photon corrections to the B !
` form factors at tree level, up to the
twistfour accuracy. The symmetry relation between the two form factors FV (n p) and
FA(n p) was found to be violated by both the twoparticle and threeparticle highertwist
e ects of the photon lightcone DAs. In addition, our calculations explicitly indicate that
the correspondence between the heavyquark expansion and the twist expansion is generally
invalid for the soft contributions to the exclusive Bmeson decays, in analogy to the similar
pattern observed in the context of the pionphoton form factor [31].1
Adding up di erent pieces contributing to the B !
` amplitude, we further
investigated the phenomenological impacts of the subleadingpower hadronic photon
contributions, employing the conformal expansion of the photon DAs at the \Pwave" accuracy.
Numerically, the NLL twisttwo hadronic photon correction was estimated to give rise to
an approximately O(30%) reduction of the leadingpower contribution, computed from
QCD factorization, with the default values of theory inputs. By contrast, the highertwist
hadronic photon contributions at LO in O( s) was found to be of minor importance at
3 GeV
n p
mB, albeit with the rather conservative uncertainty ranges for the
nonperturbative parameters collected in table 1. Moreover, we observed that the dominant
radiative e ect of the leadingtwist hadronic photon contribution comes from the NLO
b
QCD correction instead of the QCD resummation of the parametrically large logarithms
m2= 2. To understand the systematic uncertainty from the truncation of the Gegenbauer
expansion at the second order, we explored the numerical impact of the fourth moment
of the leadingtwist photon DA in a wide interval 4( 0) 2 [ 0:2; 0:2] and observed that
1The large scale Q2 in the pionphoton transition form factor F
! (Q2) plays a similar role of the
heavyquark mass in Bmeson decays.
HJEP05(218)4
the dependence of the twisttwo hadronic photon correction to the B !
on
4( 0) was rather moderate at n p
3 GeV, at least in the framework of the LCSR
method. Our main theory predictions for the B !
` form factors with the
uncertainties from variations of di erent input parameters added in quadrature were displayed in
gure 8 and the poor constraint on the rst inverse moment of the Bmeson DA
brought about one of the major uncertainties for the theory calculations. In this respect,
the improved measurements of the partial branching fractions BR(B !
` ; E
with the tighter phasespace cut on the photon energy to validate the perturbative QCD
calculations from the Belle II experiment will be of value to provide solid constraints on the
B( 0)
Ecut)
inverse moment
B( 0), when combined with the theory predictions including the power
suppressed contributions of di erent origins.
B !
Further improvements of the theory descriptions of the B !
` form factors in QCD
can be pursued in distinct directions. First, it would be of interest to perform the NLO
QCD corrections to the twistthree hadronic photon corrections with the LCSR approach
for a systematic understanding of the highertwist contributions. The technical challenge
of accomplishing this task lies in the demonstration of QCD factorization for the
vacuumtoBmeson correlation function (3.1) in the presence of the nontrivial mixing of the
twoparticle and threeparticle lightray operators under the QCD renormalization. Second,
exploring the subleadingpower contributions to the radiative leptonic Bmeson decay in
the framework of SCET directly will be indispensable for deepening our understanding of
factorization properties for more complicated exclusive Bmeson decays, where the rapidity
divergences of the convolution integrals entering the corresponding factorization formulae
already emerge at leading power in the heavy quark expansion. Earlier attempts to
address this ambitious question have been undertaken in di erent contexts (for an incomplete
list, see for instance [5, 48{50]). Third, computing the subleadingpower corrections to the
form factors from the highertwist Bmeson DAs will be of both conceptual and
phenomenological value to investigate general properties of the twist expansion in heavyquark
e ective theory (see [8] for a preliminary discussion with an incomplete decomposition of
the threeparticle vacuumtoBmeson matrix element on the lightcone). To this end, we
will need to employ the RG equations for these highertwist Bmeson DAs at one loop
following the discussions presented in [51], where the evolution equations of the twistfour
Bmeson DAs at one loop were demonstrated to be completely integrable and therefore
can be solved exactly. We are therefore anticipating dramatic progress toward better
understanding of the strong interaction dynamics of the radiative leptonic decay B !
Acknowledgments
Y.M.W acknowledges support from the National Youth Thousand Talents Program, the
Youth Hundred Academic Leaders Program of Nankai University, the Fundamental
Research Grant of National Universities, and the NSFC with Grant No. 11675082 and
11735010. The work of Y.L.S is supported by Natural Science Foundation of Shandong
Province, China under Grant No. ZR2015AQ006. { 29 {
Here we collect the dispersion representations of the various convolution integrals entering
the NLL factorization formula of the vacuumtophoton correlation function (3.50) for the
sake of constructing the LCSR for the B ! ` form factors.
:
1
1
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
1 r2
1 r2
0
+
0
1 r3
0
(r3 1) 2
r2 r3 6
1
1
1
1
1
1
0
0
Ims
Ims
Ims
Ims
Ims
sgn(r2 r3) ln 1
dz
1
z
(z; )
1 z r3 z r2
1 r2
(z r3 +z r2 1) :
z r3 +z r2 1+i 0 r1 r3
+ (r3 r2) ln2(1 r2) :
z r3 +z r2 1
1 r2
(z; )
:
1 r2
r1 r2
1 r1
1 r3
(z; )
:
1 r1
1 r2
1
1 r1
1 r3
1
:
1
1
1
1
1
1
z r3 +z r2 1+i 0 r1 r2
ln j1 z r3 z r2j (z r3 +z r2 1)
(z r3 +z r2 1) 0 (z; )
(z; )
+ (r3 r2) ln2(1 r2) :
dz (z; )
0
0
0
0
0
Ims
Ims
0
1
1
1
1
1
+ (r3 r2)
0
(z; )
ln2(1 r2) :
z r3 +z r2 1+i 0 r1 r3
z r2 (z r3 +z r2)
z r3 +z r2 1+i 0 r1 r2 r2
1 r2 ;
r3 r2
2 r2
4 r2 r2
4
(z r3 +z r2 1) :
2 r2
4
+2
+2
(r3 r2)
z r3 +z r2 1+i 0 r3(r1 r3)
z r2
(r3 r2)+
4 z (r3 1)
(1 z r2)(r3 r2)
2 (1 3 z r2)
z r3 +z r2 1+i 0 2 (r1 r2) 1 r1
1 r2
r3 r2
(z; ) 3 (r3 1) 0
1
1
1
1
1
1
1
2
r2
r2(1 r2)zz
(r2 r3) ln(1 r2)
r2 r3
1 r2
1 r2
(r3 1)
dz ln(z r3 +z r2 1) (z r3 +z r2 1) 00(z; )
ln(r3 1) 0 (z = 1; )
dz ln(z r3 +z r2 1) (z r3 +z r2 1) 0 (z; )
6
1
dz (z; )
Ims
Ims
Ims
0
Z 1
0
0
r2 r3
dz (z; )
16 r2 15 (r3 1)
2 (1 r2) r3 r2
z r2
dz
0
Z 1
0
(z; )
z
:
1 r2
r3 r2
1
r1 r3
4 z
3
15
2
1 r2
r3 r2
;
(A.7)
(A.8)
(A.9)
(A.10)
ln j1 r3j
:
(A.11)
(A.12)
In this appendix we will collect the operatorlevel de nitions of the twoparticle and
threeparticle photon DAs on the lightcone up to the twistfour accuracy as presented in [16].
( ) (z; )+
A(z; )
x
2
= i gem Qq hqqi( ) (p
+ 2 gem Qq hqqi( )
i
q x
dz eiz p x
0
5 q(0)j0i
p x
h (p)jq(x) Wc(x; 0) gs G (v x) q(0)j0i
dz eiz p x h (z; ) :
0
Z 1
dz eiz p x (a)(z; ) :
= i gem Qq hqqi( ) (p
) [D i] ei( q+v g)p x S( i; ) :
h (p)jq(x) Wc(x; 0) gs Ge (v x) i 5 q(0)j0i
= i gem Qq hqqi( ) (p
) [D i] ei( q+v g)p x Se( i; ) :
h (p)jq(x) Wc(x; 0) gs Ge (v x)
= gem Qq f3 ( ) p (p
h (p)jq(x) Wc(x; 0) gs G (v x) i
= gem Qq f3 ( ) p (p
) [D i] ei( q+v g)p x V ( i; ) :
h (p)jq(x) Wc(x; 0) gem Qq F (v x) q(0)j0i
gs G (v x) q(0)j0i
= i gem Qq hqqi( ) (p
) [D i] ei( q+v g)p x S ( i; ) :
q(0)j0i = gem Qq f3 ( )
dz eiz p x (v)(z; ) :
p
) [D i] ei( q+v g)p x A( i; ) :
p
p
p
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
(B.8)
Z
Z
Z
5 q(0)j0i
Z
q(0)j0i
Z
= gem Qq hqqi( ) p
gem Qq hqqi( ) p
gem Qq hqqi( )
gem Qq hqqi( )
h
h
(p x
(p x
p
p x )(p
p x )(p
p x
p x
g? ( $ )
i Z
i Z
p
p
) Z
) Z
[D i] ei( q+v g)p x T1( i; )
[D i] ei( q+v g)p x T2( i; )
[D i] ei( q+v g)p x T3( i; )
[D i] ei( q+v g)p x T4( i; ) : (B.9)
(p x
) Z
[D i] ei( q+v g)p x T4 ( i; )+: : :
(B.10)
p x
0
integration measure
G
e
Z
d q
d q
q
q
g) :
(B.11)
Open Access.
HJEP05(218)4
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