Radiative Leptonic $B_c\to \gamma \ell\bar\nu$ Decay in Effective Field Theory beyond Leading Order

0.78 MB·English

by  Wei Wang

Radiative Leptonic B γ(cid:96)ν¯ Decay in Effective Field Theory
c
→
beyond Leading Order
Wei Wang1,2 and Rui-Lin Zhu ∗1,2
1 INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology,
Department of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai, 200240, China
2 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China†
Abstract
5
1 We study the radiative leptonic B γ(cid:96)ν¯ decays in the nonrelativistic QCD effective field
c
0 →
theory, and consider a fast-moving photon. As a result the interactions with the heavy quarks
2
can be integrated out, and thus we arrive at a factorization formula for the decay amplitude. We
r
p calculatenotonlytherelevantshort-distancecoefficientsatleadingorderandnext-to-leadingorder
A
in α , but also the nonrelativistic corrections at the order v 2 in our analysis. We find out that
s
| |
8 the QCD corrections can sizably decrease the branching ratio and thus is of great importance in

extractingthelong-distanceoperatormatrixelementsofB . Forthephenomenologicalapplication,
] c
h
we present our results for the photon energy, lepton energy and lepton-neutrino invariant mass
p
- distribution.
p
e
h
[

2
v
3
9
4
4
0
.
1
0
5
1
:
v
i
X
r
a
∗ Corresponding author
†Electronic address: [email protected]; Electronic address: [email protected]
1
I. INTRODUCTION
The search for new degrees of freedom can proceed under two distinctive directions. At
the high energy frontier, new particles have different signatures with the standard model
(SM) particles, and measurements of their production may provide definitive evidence on
their existence. On the other hand, it is likely that low energy processes will be influenced
through loop effects. Rare decays of heavy mesons, with tiny decay rates in the SM, are
sensitive to the new degrees of freedom and thus can be exploited as indirect searches of
these unknown effects, for a recent review see Ref. [1].
The B meson is the unique pseudo-scalar meson that is long lived and composed of
c
two different heavy flavors. Since this hadron is stable against strong interactions, its weak
decays provide a rich phenomena for the study of CKM matrix elements, and also a platform
to study the effects of weak interactions in a heavy quarkonium system [2, 3]. In the
past decades it has received growing attentions since the first observation by the CDF
collaboration [4]. This can be particularly witnessed by the recent LHCb measurements of
the B lifetime [5, 6], the decay widths of B J/ψπ and B J/ψ(cid:96)ν¯ [7, 8], and various
c c c
→ →
other decay modes [9–12]. One may expect that more decay channels of B can be measured
c
by the LHCb, ATLAS and CMS experiments [13–15].
On theoretical side, various approaches have been applied to calculate the decay width
of B decays [16–52], but most of them are phenomenological. Since both constituents of
c
the B are heavy and can only be treated nonrelativistically, an effective field theory can be
c
established [53]. Taking the B J/ψ(cid:96)ν¯ as the example, one may derive the conjectured
c
→
non-relativistic QCD (NRQCD) factorization formula for its decay amplitude:
(B J/ψ) C 0 c¯b B J/ψ c¯c 0 , (1)
A c → ∝ ij(cid:104) |Oi | c(cid:105)×(cid:104) |Oj | (cid:105)
where the ff(cid:48) are constructed by low energy operators. The short-distance, or hard, con-
Oi,j
tributions at the length scale 1/m are encapsulated into the coefficients C that can be
b,c ij
computed in perturbation theory.
The long-distance, or soft part of, matrix elements have to be extracted in a nonpertur-
bative approach, for instance the Lattice QCD simulation, or constrained by much simpler
processes for instance the annihilation modes B (cid:96)ν¯ and B γ(cid:96)ν¯. However, the useful-
c c
→ →
ness of the B (cid:96)ν¯ is challenged by two aspects. Firstly its decay rate is given by
c
→
G2 m2 (cid:18) m2 (cid:19)2
Γ(B (cid:96)ν¯ ) = F V 2f2 m3 (cid:96) 1 (cid:96) , (2)
c → (cid:96) 8π | cb| Bc Bcm2 − m2
Bc Bc
in which the suppression factor m2/m2 arises from the helicity flip. As a result, the B
(cid:96) Bc c →
µν¯ andB eν¯ havetinybranchingfractionsthatmaybeoutofthedetectorcapabilityat
µ c e
→
the current experimental facilities. Secondly, there is only one physical observable, namely
the decay rate, and thus the B (cid:96)ν¯ is not capable to uniquely determine all, typically
c
→
more than one when relativistic corrections are taken into account, long-distance matrix
elements (LDMEs).
On the contrary, the B γ(cid:96)ν¯ can provide a wealth of information [54–58], in terms of
c
→
a number of observables ranging from the decay probabilities, polarizations to an angular
analysis. It is interesting to notice that the counterpart in B sector, B γ(cid:96)ν¯, has been
→
widely discussed towards the understanding of the B meson light-cone distribution ampli-
tudes [59–63]. The small branching fraction of B γ(cid:96)ν¯ can be compensated by the high
c
→
2
b γ b γ b γ b γ
W W
µ− b c
µ− µ− µ−
c¯ νµ c¯ W νµ c¯ νµ c¯ νµ
W W
(a) (b) (c) (d)
FIG. 1: Leading order Feynman diagrams for the radiative leptonic B γµν¯ decay in the SM.
c µ
→
The lepton µ can also be e or τ. The photon emission from a virtual W-boson shown in the second
panel is suppressed by 1/m2 compared to the other contributions.
W
luminosity at the ongoing hadron colliders and the under-design experimental facilities. The
main purpose of this paper is to explore the B γ(cid:96)ν¯ at next-to-leading order (NLO) in
c
→
α and in v 2, which shall catch up the progress in the B (cid:96)ν¯ [55, 64]. For the leptonic
s c
| | →
decay constant, the two-loop calculation is also available in Ref. [65].
The rest of this paper is organized as follows. In Sec. I, we will derive the formulas
for various partial decay widths of B γ(cid:96)ν¯. Sec. III is extensively devoted to the next-
c
→
to-leading order calculation. We will discuss the phenomenological results in Sec. IV. We
summarize our findings and conclude in Sec. V. We relegate the calculation details to the
Appendix.
In the SM, leading order (LO) Feynman diagrams for the B γ(cid:96)ν¯ decay are shown in
c
→
Fig 1. The photon emission from a virtual W-boson is suppressed by 1/m2 compared to
W
other contributions, and thus the second diagram in Fig. 1 can be neglected. Integrating
out the off-shell W-boson, we arrive at the effective electro-weak Hamiltonian
G
H = FV c¯γ (1 γ )b¯lγµ(1 γ )ν +h.c., (3)
eff cb µ 5 5
√2 − −
where V is the CKM matrix element. The decay amplitude, matrix element of the abo
cb
II. B γ(cid:96)ν¯
c
→
ve Hamiltonian between the B and γ(cid:96)ν¯ state,
c
= γl−ν¯ H B (4)
eff c
A (cid:104) | | (cid:105)
is responsible for the process B γ(cid:96)ν¯.
c
→
A. Differential decay widths
Since there is no strong interaction connection between the leptonic and hadronic part,
the decay amplitude can be decomposed into two individual sectors:
(cid:26)
G
= FV 0 c¯γ (1 γ )b B γl−ν¯ ¯lγµ(1 γ )ν 0
cb µ 5 c 5
A √2 (cid:104) | − | (cid:105)×(cid:104) | − | (cid:105)
(cid:27)
+ γ c¯γ (1 γ )b B u¯ γµ(1 γ )v , (5)
µ 5 c l 5 ν
(cid:104) | − | (cid:105)× −
3
with the matrix elements encoding the hadronic effects:
0 c¯γ (1 γ )b B , γ c¯γ (1 γ )b B . (6)
µ 5 c µ 5 c
(cid:104) | − | (cid:105) (cid:104) | − | (cid:105)
The first one defines the B decay constant
c
0 c¯γ γ b B (p ) = if p , (7)
(cid:104) | µ 5 | c Bc (cid:105) Bc Bc,µ
while the B γ transition is parametrized by two form factors:
c
→
V(L2)
γ((cid:15),k) c¯γ b B (p ) = e (cid:15) (cid:15)∗νpρ kσ, (8)
(cid:104) | µ | c Bc (cid:105) − p k µνρσ Bc
Bc · (cid:18) p (cid:15)∗(cid:19) ie
γ((cid:15),k) c¯γ γ b B (p ) = ieA(L2) (cid:15)∗ k Bc · f p p (cid:15)∗, (9)
(cid:104) | µ 5 | c Bc (cid:105) µ − µ p k − p k Bc Bcµ Bc ·
Bc · Bc ·
with the momentum transfer L = p k. Here and throughout this work we adopt the
Bc −
convention (cid:15)0123 = +1. The above equations are similar with the parameterization of the
B γ form factors as given in Ref. [66]. The last term in Eq. (9) that is proportional to
→
the B decay constant has been added in order to maintain the gauge invariance of the full
c
amplitude [67, 68], and see appendix A for a derivation.
Substituting Eqs. (7), (8), (9) into Eq. (5), we obtain
G (cid:26) (cid:18) p (cid:15)∗(cid:19) iv(s ) (cid:27)
= i FV ef u¯ γµ(1 γ )v [1+a(s )] (cid:15)∗ k Bc · l (cid:15) (cid:15)∗νpρ kσ ,
A − √2 cb Bc (cid:96) − 5 ν l µ − µ p k − p k µνρσ Bc
Bc · Bc ·
(10)
where s = L2 and terms due to lepton mass corrections have been neglected. Apparently,
l
this expression is gauge invariant. For the sake of simplicity, we have defined two abbrevia-
tions in the above 1
A(s ) V(s )
a(L2) l , v(s ) l . (11)
l
≡ f ≡ f
Bc Bc
In terms of the decay constant and form factors, the differential decay width for the
B γ(cid:96)−ν¯ is given as
c (cid:96)
→
d2Γ 1
= 2
dE dE 64m π3|A|
k l Bc
α f2 V 2G2m (cid:20)
= em Bc| cb| F Bc(1 x ) a2(cid:0)x2 +2x (x 1)+2(x 1)2(cid:1)
4π2x2 − k × k k l − l −
k
(cid:0) (cid:1)
+2a (v +1)x2 +2(v +1)x (x 1)+2(x 1)2 +2vx (x +2x 2)
k k l − l − k k l −
(cid:21)
(cid:0) (cid:1)
+v2 x2 +2x (x 1)+2(x 1)2 +x2 +2x x 2x +2x2 4x +2 ,
k k l − l − k k l − k l − l
(12)
1 One shall distinguish the form factor v from the relative velocity v to be defined in the following.
4
where x = 2E /m and y = 2E /m , and E and E is the energy of the photon and
k k Bc l Bc k l
charged lepton in the B rest frame, respectively. One can integrate out the E and obtain
c l
dΓ α f2 V 2G2m2 x (1 x )((1+a)2 +v2)
= em Bc| cb| F Bc k − k . (13)
dE 12π2
k
The differential distributions can also be converted to
d2Γ = m2Bc −sl V 2α f2 G2(1 x ) 1 (cid:20)a2(cid:0)x2 +2x (x 1)+2(x 1)2(cid:1)
ds dcosθ 32m π2| cb| em Bc F − k x2 k k l − l −
l l Bc k
(cid:0) (cid:1)
+2a (v +1)x2 +2(v +1)x (x 1)+2(x 1)2 +2vx (x +2x 2)
k k l − l − k k l −
(cid:21)
(cid:0) (cid:1)
+v2 x2 +2x (x 1)+2(x 1)2 +x2 +2x x 2x +2x2 4x +2 ,
k k l − l − k k l − k l − l
(14)
using the relation:
m2 s
E = Bc − l, (15)
k
2m
Bc
1
(cid:2) (cid:3)
E = (m2 +s ) (m2 s )cosθ . (16)
l 4m Bc l − Bc − l l
Bc
The θ is the polar angle between the lepton (cid:96) flight direction and the opposite direction of
l
the B meson in the rest frame of the (cid:96)ν¯ pair. Likewise one can integrate out the θ
c (cid:96) l
dΓ α f2 V 2G2(m2 s )s ((1+a)2 +v2)
= em Bc| cb| F Bc − l l . (17)
ds 24π2m3
l Bc
B. NRQCD factorization
The factorization properties for the B γ(cid:96)ν¯ depend on the kinematics of the photon.
c
→
In this work, we will not study the soft-photon contribution as discussed in B decays [69],
and leave it for future work. In the region where the photon is a collinear (fast-moving)
object, its interaction with heavy quarks is highly virtual and thus should be encoded in the
short distance coefficients. In the NRQCD scheme, we only need retain those color-singlet
operator matrix elements that connect the B state to the vacuum. To the desired order,
c
one expects the following factorization formula:
(cid:115) (cid:34) (cid:35)
2 cf (cid:18) i (cid:19)2
f = cf 0 χ†ψ B (p) + 2 0 χ† ←→D ψ B (p) + (v4) , (18)
Bc m 0(cid:104) | c b| c (cid:105) m2 (cid:104) | c −2 b| c (cid:105) O
Bc Bc
(cid:115) (cid:34) (cid:35)
2 cV cV (cid:18) i (cid:19)2
V = 0 0 χ†ψ B (p) + 2 0 χ† ←→D ψ B (p) + (v4) , (19)
m m (cid:104) | c b| c (cid:105) m3 (cid:104) | c −2 b| c (cid:105) O
Bc Bc Bc
(cid:115) (cid:34) (cid:35)
2 cA cA (cid:18) i (cid:19)2
A = 0 0 χ†ψ B (p) + 2 0 χ† ←→D ψ B (p) + (v4) , (20)
m m (cid:104) | c b| c (cid:105) m3 (cid:104) | c −2 b| c (cid:105) O
Bc Bc Bc
5
where v denotes half relative velocity between the charm and bottom quarks in the meson,
cf,V,A and cf,V,A are the dimensionless short-distance coefficients that can be expanded in
0 2
terms of the strong coupling constant 2. We shall calculate the one-loop corrections to
the cf,V,A, but give only the LO results for cf,V,A since the latter ones are already power-
0 2
suppressed. ψ and χ† represent Pauli spinor fields that annihilate the heavy quark Q and
Q Q
¯
anti-quark Q, respectively. Besides, one need note that the state H(p) in QCD has the
| (cid:105)
standard normalization: H(p(cid:48)) H(p) = 2E (2π)3δ3(p p(cid:48)), while an additional factor 2E
p p
(cid:104) | (cid:105) −
is abandoned in the nonrelativistic normalization where H(p(cid:48)) H(p) = (2π)3δ3(p p(cid:48)).
(cid:104) | (cid:105) −
III. NEXT-TO-LEADING ORDER CALCULATION
A. Kinematics
Let p and p represent the momenta for the heavy quark Q and anti-quark Q¯(cid:48). Without
1 2
loss of generality, one may adopt the decomposition:
p = αP q, (21)
1 Bc −
p = βP +q, (22)
2 Bc
where P is the total momentum of the quark pair. q is a half of the relative momentum
Bc
between the quark pair with P q = 0. α and β are the energy fraction for Q and Q¯(cid:48) in
Bc ·
the meson, respectively. The explicit expressions for all the momentum in the rest frame of
the B meson are given by
c
Pµ = (E +E ,0), (23)
Bc 1 2
qµ = (0,q), (24)
pµ = (E , q), (25)
1 1
−
pµ = (E ,q). (26)
2 2
In the rest frame, the meson momentum becomes purely timelike while the relative momen-
(cid:112) (cid:112) (cid:112)
tum is spacelike. One can obtain the relations α = m2 q2/( m2 q2 + m2 q2) and
(cid:112) b − (cid:112) b − c −
β = 1 α with the on-shell conditions E = m2 q2, E = m2 q2, and q2 = q2.
− 1 b − 2 c − −
B. Convariant projection method
In the following calculation, we will adopt the covariant spin-projector method, which
can be applied to all orders in v.
The Dirac spinors for the B system may be written as
c
(cid:114) (cid:18) (cid:19)
E +m ξ
1 b λ
u (p ,λ) = −→ , (27)
b 1 2E (cid:126)σ·p1 ξ
1 E1+mb λ
2 Throughoutthispaper,weshallusethesuperscripts(0)and(1)toindicatetheLOandNLOcontributions
in α and the subscripts 0 and 2 to denote the LO and NLO contributions in the velocity.
s
6
(cid:114) (cid:18) −→ (cid:19)
vc(p2,λ) = E2 +mc E(cid:126)σ2+·pm2cξλ , (28)
2E ξ
2 λ
where ξ is the two-component Pauli spinors and λ is the polarization parameters. It is
λ
straightforward to derive the covariant form of the spin-singlet combinations of spinor bilin-
ears:
(cid:88) 1 1 1
c
Π (q) = i u (p ,λ )v¯ (p ,λ ) λ λ 00
0 b 1 1 c 2 2 1 2
− (cid:104)2 2 | (cid:105)⊗ √N
c
λ1,λ2
i /p +E +E 1
= (α/p /q+m ) Bc 1 2γ (β/p +/q m ) c ,
4√2E E ω Bc − b E +E 5 Bc − c ⊗ √N
1 2 1 2 c
(29)
with the auxiliary parameter ω = √E +m √E +m . Here 1 is the unit matrix in the
1 b 2 c c
fundamental representation of the color SU(3) group.
C. Perturbative matching
Due to the simplicity of the final state, one can directly match the QCD currents onto the
NRQCD ones. To determine the values of c and c , we follow the spirit that those short-
0 2
distance coefficients are insensitive to the long-distance hadronic dynamics. As a convenient
choice, one can replace the physical B− meson by a free c¯b pair of the quantum number
c
1S[1], so that both the full amplitude, [c¯b(1S[1]) γ(cid:96)ν¯], and the NRQCD operator matrix
0 0
A →
elements can be directly accessed in perturbation theory. The short-distance coefficients
c can then be solved by equating the QCD amplitude and the corresponding NRQCD
i
A
amplitude, order by order in α . For this purpose, we introduce a decay constant and two
s
form factors at the free quark level:
0 c¯γ γ b c¯b(1S[1]) = i(cid:102)g , (30)
µ 5 0 µ0
(cid:104) | | (cid:105)
1
γ((cid:15),k) c¯γ b c¯b(1S[1]) = e V(cid:15) (cid:15)∗νpρ kσ, (31)
(cid:104) | µ | 0 (cid:105) − k p µνρσ Bc
(cid:18)· Bc p (cid:15)∗(cid:19) 1
γ((cid:15),k) c¯γ γ b c¯b(1S[1]) = ieA (cid:15)∗ k Bc · ie (cid:102)p p (cid:15)∗. (32)
(cid:104) | µ 5 | 0 (cid:105) µ − µ p k − p k Bcµ Bc ·
Bc · Bc ·
Analogous to (18,19,20), one can write down the matching formula:
f (cid:18) (cid:19)2
c i
(cid:102) = cf 0 χ†ψ c¯b(1S[1]) + 2 0 χ† ←→D ψ c¯b(1S[1]) , (33)
0(cid:104) | c b| 0 (cid:105) (m +m )2(cid:104) | c −2 b| 0 (cid:105)
b c
(cid:34) (cid:35)
1 cV (cid:18) i (cid:19)2
V = cV 0 χ†ψ c¯b(1S[1]) + 2 0 χ† ←→D ψ c¯b(1S[1]) , (34)
m +m 0 (cid:104) | c b| 0 (cid:105) (m +m )2(cid:104) | c −2 b| 0 (cid:105)
b c b c
(cid:34) (cid:35)
1 cA (cid:18) i (cid:19)2
A = cA 0 χ†ψ c¯b(1S[1]) + 2 0 χ† ←→D ψ c¯b(1S[1]) , (35)
m +m 0(cid:104) | c b| 0 (cid:105) (m +m )2(cid:104) | c −2 b| 0 (cid:105)
b c b c
where we have adopted the nonrelativistic normalization.
7
One can organize the full amplitudes defined in Eqs. (30,31,32) in powers of the relative
momentum between c¯ and b, denoted by q. To the desired accuracy, one can truncate the
series at (q2), with the first two Taylor coefficients. We will compute both amplitudes
O
at LO in α in subsection IIID, and the calculation at NLO in α will be conducted in
s s
subsection IIIE.
The NRQCD matrix elements encountered in the above equations are particularly simple
at LO in α :
s
(cid:112)
0 χ†ψ c¯b(1S[1]) (0) = 2N ,
0 c
(cid:104) | | (cid:105)
i (cid:112)
0 χ†( ←→D)2ψ c¯b(1S[1]) (0) = 2N q2, (36)
(cid:104) | −2 | 0 (cid:105) c
where the factor √2N is due to the spin and color factors of the normalized c¯b(1S[1]) state.
c 0
The computation of these matrix elements to (α ) will be addressed in subsection IIIF.
s
O
D. Tree-level amplitude
Adopting the above notation, one can easily obtain the tree-level amplitude for the decay
constant
0 c¯γ γ b c¯b(1S[1]) (0) = Tr[Π (q)γ γ ]
µ 5 0 0 µ 5
(cid:104) | | (cid:105)
(cid:112) (E +m )(E +m )+q2
= ipµ 2N 1 b 2 c
Bc c2(cid:112)E E (E +m )(E +m )(E +E )
1 2 1 b 2 c 1 2
(cid:112) (cid:18) q2 (cid:19)
= ig 2N 1 , (37)
µ0 c − 8m2
red
where the qµ terms have been omitted and
m m
b c
m = , (38)
red
m +m
b c
is defined as the reduced mass of the c¯b system.
The vector current is similarly evaluated as:
i(k/ /p +m ) i(/p k/+m )
γ c¯γ b c¯b(1S[1]) (0) = Tr[Π (q)iee/(cid:15)∗ − 2 c γ ]+Tr[Π (q)γ 1 − b iee/(cid:15)∗]
(cid:104) | µ | 0 (cid:105) 0 c (k p )2 m2 µ 0 µ(p k)2 m2 b
− 2 − c 1 − − b
e√2N e e
c c b
= ( + )
−4w√E E E k p +Ek q E k p Ek q
1 2 2 · Bc · 1 · Bc − ·
(cid:26) (cid:27)
E (cid:15) (cid:15)∗νkρpσ +E(E +E +m m )(cid:15) (cid:15)∗νkρqσ . (39)
× bc µνρσ Bc 1 2 b − c µνρσ
We have introduced the abbreviation E = E + E , and E = (E + m )(E + m ) + q2.
1 2 bc 1 b 2 c
Here e = 2/3 and e = 1/3 is the electric charge of the c and b quark, respectively.
c b
−
One can perform the Taylor expansion of the amplitudes in powers of qµ:
∂ (0) 1 ∂2 (0)
(q) = (0)+ A qµ + A qµqν +.... (40)
A A ∂qµ |q=0 2!∂qµ∂qν |q=0
8
Those terms linear in q should be dropped since this auxiliary momentum introduced at the
quark level has no correspondence at the hadron level. In this paper, the ( q 2) contribu-
O | |
tions will be retained. In order to simplify the calculation in the covariant derivation, one
shall use the following replacement:
q 2 Pµ Pν
qµqν | | ( gµν + Bc Bc). (41)
→ D 1 − P2
− Bc
The result for the axial-vector current is a bit lengthy:
(cid:112) 1
γ c¯γ γ b c¯b(1S[1]) (0) = ie 2N
µ 5 0 c (cid:112)
(cid:104) | | (cid:105) − 4 E E (E +m )(E +m )
1 2 1 b 2 c
(cid:26)
k p E +k qE(E E +m m )
(cid:15)∗e · Bc bc · 1 − 2 b − c
× µ c E k p +Ek q
2 · Bc ·
k p E +k qE(E E +m m )
(cid:15)∗e · Bc bc · 1 − 2 b − c
− µ b E k p Ek q
1 · Bc − ·
2(E E +m m )(E (cid:15)∗ p +E(cid:15)∗ q)
+q e 1 − 2 b − c 2 · Bc ·
µ c
E k p +Ek q
2 · Bc ·
2(E E +m m )(E (cid:15)∗ p E(cid:15)∗ q)
q e 1 − 2 b − c 1 · Bc − ·
µ b
− E k p Ek q
1 · Bc − ·
2E (E (cid:15)∗ p +E(cid:15)∗ q)
+p e bc 2 · Bc ·
Bcµ c E(E k p +Ek q)
2 · Bc ·
2(E E (cid:15)∗ p +E(cid:15)∗ q(E +q2)
p e 1 bc · Bc · bc
− Bcµ b E(E k p Ek q)
1 · Bc − ·
E (cid:15)∗ p +E(cid:15)∗ q(E E +m m )
k e bc · Bc · 1 − 2 b − c
µ c
− E k p +Ek q
E (cid:15)∗ p +E2 (cid:15)∗· qB(cE E· +m m )(cid:27)
+k e bc · Bc · 1 − 2 b − c . (42)
µ b
E k p Ek q
1 · Bc − ·
In order to extract the form factor, we only need to keep the (cid:15) term which corresponds
µ
A
to Feynman gauge (cid:15) p = 0, but we have explicitly checked the gauge invariance up to v2
· Bc
order.
The tree-level NRQCD matrix elements for the c¯b have been given in Eq. (36), and thus
the above results in Eqs. (37,39,42) lead to the tree-level Wilson coefficients
cf,0 = 1, (43)
0
z˜4
cf,0 = , (44)
2 −8z2
e e
cV,0 = c b, (45)
0 −2z − 2
(cid:18)e (3z2 +2z +11) e (11z2 +2z +3)(cid:19)
cV,0 = z˜2 c + b , (46)
2 − 48z3 48z2
e e
cA,0 = b c, (47)
0 2 − 2z
9
b μ−
c¯ ν
μ
(a) (b) (c)
(d) (e)
FIG. 2: Typical NLO Feynman diagrams for the radiative leptonic B γµν¯ decay in the
c µ
→
SM. The other four diagrams can be easily obtained by interchanging the bottom and anti-charm
quarks lines.
(cid:18)e [(3z2 +2z +11)+8z(1 z)m /E ]
cA,0 = z˜2 c − b k
2 − 48z3
e [(11z2 +2z +3) 8z(1 z)m /E ](cid:19)
b b k
− − . (48)
− 48z2
In the above results, we have defined z = m /m and z˜ = 1 + z. cf,0 means the LO of
c b i
Wilson coefficient cf. It is interesting to notice that the Wilson coefficients cA,0 depends
i 2
on the energy of the emitted photon, which will induce nontrivial behaviors as will be
demonstrated later.
E. NLO amplitudes in QCD
Typical one-loop diagrams for the QCD corrections to the B γ(cid:96)ν¯ decay are shown
c (cid:96)
→
in Fig. 2. In calculating the one-loop amplitudes, we use the dimensional regularization to
regulate the ultraviolet (UV) and infrared (IR) divergence.
The diagram (a) in Fig. 2 contributes to the NLO decay constant:
(cid:112) C α (cid:20) 1 2 µ2 6lnz(cid:21)
(cid:102)1 = 2N F s + +3ln 2+2t , (49)
0,a c 4π (cid:15)ˆ (cid:15)ˆ m2 − 1 − z +1
UV IR b
with
1 (cid:18) (cid:20) 1 16m2 v 2(cid:21)(cid:19)
t = π2 iπ ln red| | ,
1 2 v − (cid:15)ˆ − µ2
IR
| |
q
v = . (50)
2m
red
We have introduced the abbreviation
1 1
= γ +ln4π. (51)
E
(cid:15)ˆ (cid:15) −
UV,IR UV,IR
10