- Email: [email protected]
Factorization of radiative leptonic decays of B− and D− mesons
Available online at www.sciencedirect.com
ScienceDirect Nuclear Physics B 889 (2014) 778–800 www.elsevier.com/locate/nuclphysb
Factorization of radiative leptonic decays of B − and D − mesons Ji-Chong Yang, Mao-Zhi Yang ∗ School of Physics, Nankai University, Tianjin 300071, PR China Received 1 September 2014; accepted 30 October 2014 Available online 11 November 2014 Editor: Hong-Jian He
Abstract In this work, we study the factorization of the radiative leptonic decays of B − and D − mesons, the contributions of the order O(ΛQCD /mQ ) are taken into account. The factorization is proved to be valid explicitly at the order O(αs ΛQCD /mQ ). The hard kernel is obtained. The numerical results are calculated using the wave-function obtained in relativistic potential model. The O(ΛQCD /mQ ) contribution is found to be very important, the correction to the decay amplitudes of B − → γ eν¯ is about 20%–30%. For D mesons, the O(ΛQCD /mQ ) contributions are more important. © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3 .
1. Introduction The study of the heavy meson decays is an important field in high energy physics. In recent years, both experimental and theoretical studies have been improved greatly [1–3]. However, the limitation in understanding and controlling the non-perturbative effects in strong interaction is so far still a problem. Varies theoretical methods on how to deal with the non-perturbative effects have been developed. An important approach is to separate the hard and soft physics which is known as factorization [4,5]. This method has been greatly developed in recent years [6]. * Corresponding author.
E-mail address: [email protected] (M.-Z. Yang). http://dx.doi.org/10.1016/j.nuclphysb.2014.10.027 0550-3213/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3 .
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
779
The idea of factorization is to absorb the infrared (IR) behaviour into the wave-function, the matrix element can be written as the convolution of wave-function and hard kernel F = dkΦ(k)Thard (k) (1) The wave-function should be determined by non-perturbative methods. The radiative leptonic decay of heavy mesons provides a good opportunity to study the factorization approach, where strong interaction is involved only in one hadronic external state. Except for that, with a photon emitted out, more details about the wave-function of the hadronic bound state can be exploited. Many works has been done on the factorization of this decay mode. In Ref. [7], the 1-loop QCD correction is calculated in the large energy effective theory, and they found the factorization will depend on the transverse momentum. In Refs. [8] and [9], factorization is proved in leading order of 1/mQ expansion in the frame of QCD factorization [4,5], where the heavy quark is treated in the heavy quark effect theory (HQET) [3,10]. In Refs. [11,12], the factorization is constructed using the soft-collinear effective theory (SCET) [13,14]. In this work, factorization in the radiative leptonic decays of heavy mesons is revisited. The work of Refs. [8,9] is extended by taking into account the contributions of the order of O(ΛQCD /mQ ). The factorization is proved to be still valid explicitly. We also find the factorization is valid at any order of O(ΛQCD /mQ ). The numerical results shows that, the O(ΛQCD /mQ ) correction is very important for the B and D mesons, the correction can be as large as 20%–30%. The remainder of the paper is organized as follows. In Section 2, we discuss the kinematic of the radiative decay and the wave-function. In Section 3, we present the factorization at tree level. In Section 4, the 1-loop corrections of the wave-function are discussed. The factorization at 1-loop order is presented in Section 5. In Section 6, we briefly discuss the resummation of the large logarithms. The numerical results are presented in Section 7. And Section 8 is a summary. 2. The kinematic The B or D meson is constituted with a quark and an anti-quark, where one of the quarks is a heavy quark, and the other is a light quark. The Feynman diagrams at tree level of the radiative leptonic decay can be shown as Fig. 1. The contribution of Fig. 1d is suppressed by a factor of 1/Mw 2 , and can be neglected. The amplitudes of Fig. 1a, b and c can be written as p /γ −p /q μ −ieq GF VQq PL Q(pQ )(lPLμ ν¯ ) q(p ¯ q¯ )/ε∗γ √ 2p · p 2 γ q¯ /Q −p / γ + mQ ∗ −ieQ GF VQq μp (0) ε/γ Q(pQ )(lPLμ ν) ¯ Ab = q(p ¯ q¯ )PL √ 2pQ · pγ 2 pγ + p / l + ml ) −eGF VQq μ ∗ i(/ = )P Q(p ) l/ ε ν ¯ P A(0) q(p ¯ √ q¯ L Q Lμ c γ 2(pγ ·pl ) 2 A(0) a =
(2)
where pq¯ and pQ are the momenta of the anti-quark q¯ and quark Q, respectively, pγ , pl and pν are the momenta of photon, lepton and neutrino, εγ denotes the polarization vector of photon, and PLν is defined as γ μ (1 − γ5 ). We work in the rest-frame of the meson, and we choose the frame such that the direction of the photon momentum is on the opposite z axis, so the momentum of the photon can be written as pγ = (Eγ , 0, 0, −Eγ ), with 0 ≤ Eγ ≤ mQ /2.
780
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
Fig. 1. Tree level amplitudes, the double line represents the heavy quark propagator but not the HQET propagator.
To study the factorization, we consider the state of two free quark and anti-quark at first. The wave-function of the two quark and anti-quark state is defined as ¯ (3) Φ(kq , kQ ) = d 4 xd 4 y exp(ikq · x) exp(ikQ · y)0|v¯q¯ (x)[x, y]uQ (y)|qQ where [x, y] denotes the Wilson line [15]. And the matrix element is defined as μ
F = γ |v¯q¯ (x)PL uQ (y)|qQ ¯
(4)
The prove of factorization is to prove that, up to 1-loop order, the matrix element can be written as the convolution of the wave-function Φ and a hard-scattering kernel T , where T is IR-finite and independent of the external state. 3. Tree level factorization We start with the matrix elements at tree level. Using the definition of the wave-function in coordinate space Φαβ (x, y) = 0|q¯α (x)[x, y]Qβ (y)q¯ S (pq¯ ), Qs (pQ ) (5) where S and s are spin labels of q¯ and Q, respectively. We find (0) Φαβ (kq¯ , kQ ) = (2π)4 δ 4 (kq¯ − pq¯ )(2π)4 δ 4 (kQ − pQ )v¯α (pq¯ )uβ (pQ )
And then the matrix element can be written as d 4 kq¯ d 4 kQ (0) (0) F = Φ (kq¯ , kQ )T (0) (kq¯ , kQ ) = Φ (0) ⊗ T (0) (2π)4 (2π)4
(6)
(7)
With Eqs. (2) and (7), we obtain the hard scattering kernel at tree level as Ta(0) = −eq
ε/∗γ p / γ − 2εγ∗ · kq 2pγ · kq¯
μ
PL ,
(0)
Tb
μ
= −eQ PL
−/ p γ ε/∗γ 2kQ · pγ
(8)
In the expressions above, we have already assumed to consider the kinematical region Eγ ∼ mQ . The polarization vector of the photon does not have 0-component, as a result
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
781
(/kQ + mQ )/εuQ /2pγ · pQ is an order of O(Λ2QCD /m2Q ) contribution and is neglected. The reμ
maining terms of Tb(0) , and the transverse part of Ta(0) , which is 2eq ε · kq¯ PL /2pγ · kq¯ are order of O(ΛQCD /mQ ) contributions. Exchange the Lorentz index in Ac , we obtain A(0) c =
p γ ε/∗γ + 2εγ∗ · (pQ + pq¯ − pν )) −ieGF VQq μ (/ Q(pQ )(lPLμ ν¯ ) q(p ¯ q¯ )PL √ 2(pγ ·(pQ + pq¯ − pν )) 2
(9)
We find / γ ε/∗γ μp
+ 2ε · (pQ + pq¯ − pν )
Fc(0) = −ev¯q¯ PL
2pγ · (pQ + pq¯ − pν ) ∗ / γ ε/γ + 2ε · (kQ + kq¯ − pν ) μp
Tc(0) = −ePL
uQ (10)
2pγ · (kQ + kq¯ − pν )
This term is also an order of O(ΛQCD /mQ ) contribution. 4. 1-loop correction of wave-function The expansion of the decay amplitude can be written as [8] F = F (0) + F (1) + . . . = Φ (0) ⊗ T (0) + Φ (1) ⊗ T (0) + Φ (0) ⊗ T (1) + . . .
(11)
At the 1-loop level, the amplitude can be written as F (1) = Φ (1) ⊗ T (0) + Φ (0) ⊗ T (1)
(12)
The 1-loop corrections of Φ ⊗ T come from the QCD interaction and the Wilson line. The later can be written as [8,15] x n x (igs )n
4 μ [x, y] = exp igs d zzμ A (z) = d 4 zi ziμ Aμ (zi ) (13) n! n i y
y
The corrections are shown in Fig. 2. (1) We use Φq to represent the correction with the gluon from the Wilson line connected to the light quark external leg. So the correction in Fig. 2a can be written as Φq(1) (kq¯ , kQ ) =
4
d x × igs
4
d ye
ikq¯ ·x ikQ ·y
e
x 0|q¯q¯ (x)igs
dzzμ Aμ (z)Q(y) y
d 4 x2 q¯q¯ (x2 )/ A(x2 )qq¯ (x2 )q¯ S (pq¯ ), Qs (pQ )
(14)
After the integration, the result is Φq(1) ⊗ T (0) = igs2 CF
(/l + p / q¯ − mq¯ ) dd l 1 v¯q¯ γ ρ (2π)d l 2 (l + pq¯ )2 − m2q¯
k = pq¯ + αl,
1 0
∂T (0) ∂T (0) dα uQ ρ − ρ ∂kq ∂kQ kq =k ,kQ =K
K = pQ − αl
The procedure of the integration can be found in Appendix A.
(15)
782
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
(0)
Fig. 2. The 1-loop correction of wave-functions. Φ (1) ⊗ Ta
(0)
and Φ (1) ⊗ Tb
are established in this figure.
Similar to Φq , the correction in Fig. 2b can be written as (1)
ΦQ ⊗ T (0) 1 = −iCF gs2
dα
0
k = pq¯ + αl,
(0) dd l 1 ∂T (0) (/ ∂T p Q − /l + mQ ) ρ v¯q¯ γ uQ ρ − ρ (2π)d l 2 ∂kq ∂kQ (pQ − l)2 − m2Q kq =k ,kQ =K
K = pQ − αl
(16)
We use ΦWfc to denote the correction shown in Fig. 2c. We find (1) ΦWfc
⊗T
(0)
g 2 CF =− s 2
dd l (2π)d
1
1 dα
0
dβ 0
1 ∂ 2 (0) ∂ v ¯ − T uQ q¯ ∂kq ∂kQ l2 kq =k ,kQ =K
(17)
The corrections shown in Fig. 2d, 2e and 2f can be denoted as Φbox , ΦextQ and Φextq . We find that, they have the same forms as the free particle 1-loop QCD corrections. 5. 1-loop factorization For simplicity, we denote x = m2Q ,
y = 2pQ · pγ ,
z = 2pγ · pq¯ ,
w = 2pQ · pq¯
(18)
The definitions of x, y and z are the same as Ref. [8] at order O(ΛQCD /mQ )0 , while w is a new scalar that appears at the order of O(ΛQCD /mQ ) contributions which will be shown later, it represents the effect of the transverse momentum. To calculate the hard-scattering amplitude, we need to calculate all 1-loop corrections of Fa , Fb and Fc . We take a small mass mq for the light quark to regulate the collinear IR divergences. The soft IR divergences will not appear explicitly in this factorization procedure. We use MS [19]
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
783
Fig. 3. 1-loop QCD correction of Fa .
scheme to regulate the ultraviolet (UV) divergences, in D = 4 − demission, we define NUV as 2 (19) − γE + log(4π)
We take the factorization scale the same as the renormalization scale, so we use the same μ in F (1) and Φ (1) ⊗ T (0) . NUV =
5.1. 1-loop correction of Ta(0) The Feynman diagrams of the 1-loop corrections of Ta are shown in Fig. 3. We denote the correction of the electric–magnetic (EM) vertex, the one shown in Fig. 3a, as F (1)EM . To show the effect of the transverse momentum explicitly, we establish the longitudinal part and transverse part separately. The matrix element at tree level can be written as (0)
(0)
Fa(0) = Fa + Fa⊥ , (0)
Fa⊥ = eq v¯q¯
(0)
Fa = −eq v¯q¯
2ε · pq¯ μ P uQ 2pγ · pq¯ L
ε/p /γ μ P uQ 2pγ · pq¯ L (20)
The transverse part is at order O(ΛQCD /mQ ). The corrections to each part is represented separately as (1)EM
Fa(1)EM = Fa
(1)EM
+ Fa⊥ / γ + mq ) i(−/ p q¯ − /l + p i(−/ p q¯ − /l + mq ) dd l (1)EM Fa = CF gs2 v¯q¯ iγρ (−ieq )/ε d 2 2 (2π) (pq¯ + l) − mq (pq¯ + l − pγ )2 − m2q i/ pγ μ −i × iγ ρ P uQ 2pq¯ · pγ L l 2 / γ − mq ) p q¯ + /l − p (/ p q¯ + /l − mq ) (/ dd l (1)EM Fa⊥ = ieq CF gs2 v¯q¯ γρ ε/ (2π)d (pq¯ + l)2 − m2q (pq¯ + l − pγ )2 − m2q p / q¯ μ1 ×γρ PL 2 uQ 2pq¯ · pγ l
(21)
784
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
After performing the momentum-integration, the result is 2pγ · pq¯ 2pγ · pq¯ (1)EM (0) αs CF Fa = Fa NUV − log + 2 log 4π μ2 m2q 2pq¯ · pγ (1)EM (0) αs CF Fa⊥ = Fa⊥ NUV − log +1 4π μ2
(22) (1)
where CF is defined as CF = (N 2 − 1)/2N = 4/3 for QCD. The transverse part F⊥ is an order O(αs ΛQCD /mQ ) contribution. Accordingly the 1-loop correction of the wave-function is (1) (0) (1) (0) (0) Φq ⊗ Ta , which can also be written as Φq ⊗ (Ta + Ta⊥ ): 2/ pγ ε/∗γ (/l + p / q¯ − mq ) dd l 1 μ (0) (1) 2 Φq ⊗ Ta = ieq CF gs v ¯ p / PL uQ q ¯ γ 2 2 (2π)d l 2 4p · (p + l)p · p (l + pq¯ ) − mq¯ γ γ 1 dd l 1 (0) Φq(1) ⊗ Ta⊥ = −ieq gs2 CF dα (2π)d l 2 0
4/εγ (pγ · (pq¯ + αl) − 4/ pγ εγ · (pq¯ + αl) (/l + p / q¯ − mq¯ ) × v¯q¯ PLμ uQ (23) (2pγ · (pq¯ + αl))2 (l + pq¯ )2 − m2q¯ Φq(1) ⊗ T⊥(0) is calculated in the light-cone coordinate (see also Appendix A in Ref. [7]), the result is m2q CF αs (0) (0) (0) (1) Φq ⊗ Ta = 2 Φq(1) ⊗ Ta⊥ = 0 (24) T NUV − log 2 + 2 , 4π a μ Using Eq. (12), we can obtain the hard-scattering kernel 2kq¯ · pγ 2kq¯ · pγ (0) CF αs (0) CF αs Ta(1)EM = Ta − 4 + T + 1 log − log a⊥ 4π 4π μ2 μ2
(25)
We find that, the hard-scattering kernel of the longitudinal direction is the same as the result in Ref. [8]. The correction of the weak vertex is shown in Fig. 3b. The result can be written as [20] Fa(1)weak
/ q¯ p /γ −p i y xz x x v¯q¯ ε/ NUV − log 2 + y1 − 2 log 2 + log −z y 16π2 y x −y μ wy x z xz 2 + x −4 log 2 − 2 + xy 5 log + 8 log + 5 y y (x − y)2 y 2 y 3z xz y x y 2 + −3y1 + 8 log 2 − 4 + y 2 log + 4 log − 3 x z (x − y)2 y 2 y y y x 2 yz 7y1 + 13 log + 19 log + 14 + x z (x − y)2 y 2 x z xy 2 z −5y1 + 4 log + 14 log − 17 + y y (x − y)2 y 2 3 z x y z μ − 3 log + 2 log + 7 PL y + 1 y y (x − y)2 y 2 log xy / q¯ μ p /γ −p xz μ1 + / Q (1 − γ5 ) (26) 4pq¯ p − 2mQ PR log 2 uQ −z 2(x − y) y y
= −ieq CF gs2
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
with
xz x x y1 = −π + 2 Li2 1 − − log2 2 + log2 y y y 2
785
(27)
In the expressions above, we have already neglected the terms which are irrelevant for the discussion, because they will not contribute to the matrix element up to order O(ΛQCD /mQ ) when convoluted with the distribution amplitudes, due to their Dirac structures [8]. In the rest of this section, we will always use this simplification if possible. (1) (0) The correction of wave-function is ΦQ ⊗ Ta , which can be written as (1) (0) ΦQ ⊗ Ta
/γ p /γ mQp z z μ i PL − 1 1 − + NUV log 2pq¯ · pγ y y y 16π 2 π2 1 z 2 x 2 yμ − 2− + log 2 − log 1+ 3 4 zmQ y μ /γ mQp z x x + uQ − 1 log 2 − log 2 − 2 y y μ μ
= −2ieq CF gs2 v¯q¯ ε/
and
i z mQ μ (1) (0) ΦQ ⊗ Ta⊥ = −2ieq CF gs2 v ¯ P log + 1 NUV − q¯ L y y 16π 2 mQ y x x − log 2 − 2 log − log 2 + 4 ε/uQ y z μ μ
(28)
(29)
With Eq. (6) and Eq. (12), the hard-scattering kernel can be obtained through the following equation (30) Φ (0) ⊗ Ta(1)weak = Fa(1)weak − Φ (1) ⊗ Ta(0) The correction in Fig. 3c is 2pγ · pq¯ CF αs (0) Fa(1)wfc = − Fa NUV − log + 1 4π μ2
(31)
We find that the correction of the wave-function also vanishes as Ref. [8], i.e. (1)
ΦWfc ⊗ Ta(0) = 0
(32)
Then we obtain T
wfc(1)
2pγ · kq¯ αs CF (0) (0) = −1 Ta + Ta⊥ log 4π μ2
(33)
The corrections of the external legs and the box diagram are equal to the relevant corrections to the wave-functions because we use the renormalization scale equal to the factorization scale, so (1)extq
Ta
= Ta(1)extQ = Ta(1)box = 0
(34)
786
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
Fig. 4. 1-loop QCD correction of Fb . (0)
5.2. 1-loop correction of Tb
The corrections of Tb are also order O(ΛQCD /mQ ) contributions as the tree level, the Feynman diagrams of the 1-loop corrections of Tb are shown in Fig. 4. The correction of the weak vertex is y x x x2 (1)weak 2 i Fb = −ieQ CF gs v ¯ − log − 2 + log + N q ¯ UV y − x (x − y)2 y 16π 2 μ2
y x x /γ 2wyy2 wy log y (3 log x−y + log x−y ) μ p + + PL 2 w−z y (w − z) y y x 2yy2 (w − z) + y(log x (3 log x−y + log x−y )) μ + 2PL p /q (35) ε/uQ 2(w − z)2 with y2 =
3 y −w+z 1 x x log log 2(w − z) w+x −y −z x −y +w−z y −w+z
and the correction of the wave-function is 1 (0) (1) Φq ⊗ Tb ∼ O m2Q
(36)
(37)
the hard kernel can be obtained using Φ (0) ⊗ Tbweak(1) = Fbweak(1) The results of the EM vertex is (1)EM Fb
/ γ ε/ i / −2y x x p μ mQ ε P log + 2NUV − log 2 y x −y y y 16π 2 L μ x π2 6x − y x y − −2 Li2 1 − + − log uQ y x 3 x −y y
= −ieQ CF gs2 v¯q¯
(38)
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
787
/ γ ε/ i x μp v ¯ P − log + 2 u N q ¯ Q UV L y 16π 2 μ2 2 m2Q m2Q 2pγ · kQ π2 (1)EM (0) CF gs = Tb log 2 − 4 − −2 Li2 1 − + Tb 4π 2pγ · kQ 3 μ m2Q 2 2 6mQ − 2pγ · kQ mQ − 2 log 2pγ · kQ mQ − 2pγ · kQ (1)
(0)
ΦQ ⊗ Tb
= −2ieQ CF gs2
+ ieQ CF gs2
μ m2Q PL ε/mQ 4pγ · kQ i v ¯ u × log q ¯ Q 2pγ · kQ 16π 2 2pγ · kQ m2Q − 2pγ · kQ
(39)
The results in Fig. 4c is (1)wfc
Fb
i x x x y(2x − y) μ = −ieQ Cf gs2 v ¯ P log + −NUV − 1 + log 2 − q¯ L (x − y) y 16π 2 μ (x − y)2 p /γ 2x x x x (2x − 3y)y + log −3NUV − 5 + 3 log 2 + + ε/uQ y x −y y y μ (x − y)2 x x x mQ ε/ (2x − 3y)y + −3NUV − 5 + 3 log 2 + log + u Q x −y y y μ (x − y)2
(1)
(0)
ΦWfc ⊗ Tb
=0 C α x x s F (0) (1)wfc = Tb Tb −1 + log 2 − 4π x−y μ 2x x x + −5 + 3 log 2 + + y x −y μ i μ − ieQ CF gs2 v¯q¯ PL ε/uQ 16π 2 mQ x x × −5 + 3 log 2 + y x −y μ
x y(2x − y) log 2 y (x − y) x (2x − 3y)y log y (x − y)2 +
x y(2x − 3y) log + y (x − y)2
(40) pQ →kQ
And the correction of the external legs and the box correction are also equal to each other, so we also have (1)extq
Tb
(1)extQ
= Tb
(1)box
= Tb
=0
(41)
(0)
5.3. 1-loop correction of Tc
The correction of the distribution function with gluons in Wilson line does not have correspondent 1-loop QCD corrections. And because the momentum of light quark and heavy quark (0) show up together as kQ + kq¯ in Tc , we find ∂ ∂ − (42) T (0) = 0 ∂kQ ∂kq¯ c and we obtain (1)q
Tc
= Tc(1)Q = Tc(1)wfc = 0
(43)
788
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
All the other corrections are similar as Eq. (34) and Eq. (41), we find (1)extq
Tc
(1)trangle
= Tc(1)extQ = Tc
=0
(44)
5.4. 1-loop result summery With Eqs. (25), (30), (33), (34), (38)–(41), (43) and (44), we can establish the order αs hardscattering kernel. We find that, T (1) is IR-finite up to order O(αs ΛQCD /mQ ), so the factorization is proved up to the order of O(1/mQ ) corrections. It is well known that, the factorization will fail with the light mesons. However, we find that the ΛQCD /mQ expansion is irrelevant by briefly investigating the factorization at higher orders of O(ΛQCD /mQ ). In the calculations above, we find that the IR divergences in the corrections of the external legs and the box diagrams are cancelled exactly, while the remaining IR divergences are all collinear divergences which show up in the amplitudes with one of the vertexes of the gluon propagator connecting to the external light quark, which are denoted in Fig. 3a and Fig. 4a. In Ta(1)EM , there are no neglected O(ΛQCD /mQ )2 or higher order contributions. As a result, (1)weak . We also find that, there are both the IR divergences at higher orders only survive in Tb (1)weak (1) (0) higher order IR divergences in Fb and Φq ⊗ Tb . We shall investigate whether those IR divergences can be cancelled. We concentrate on the IR region of the loop integration, using l → 0, we find weak(1)
Fb IR
l
= lim −ieQ CF gs l →0
×
ρ /Q 2pQp
−l ρ + 2pγ p /γ
−/ pq¯ dd l 1 μ v¯q γρ P (2π)d l 2 (l + pq¯ )2 − m2q L ρ
ρ
− 2pQp / γ − 2pγ p / Q + 2γ ρ pQ · pγ + 2(pQ − pγ )ρ mQ (2pQ · pγ )2
ε/uQ (45)
(0)
We also bring back the neglected higher order terms in the numerator of Tb , and the correction of the wave-function is k/Q − p / γ + mQ ε/ (kQ − pγ )2 − m2Q l (1) p / q¯ dd l 1 μ (0) 2 Φq ⊗ T = lim −ieQ CF gs v¯q γ ρ P IR l →0 (2π)d l 2 (l + pq¯ )2 − m2q L (0)
Tb
μ
= eQ PL
−l
pQ − p / γ + mQ ) −γρ 2pQ · pγ − 2(pQ − pγ )ρ (/ × ε/uQ (2pQ · pγ )2
(46)
we find (1) Fb IR − Φ (1) ⊗ T (0) IR = 0
(47)
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
789
This result indicates that, the factorization is valid at any order of O(ΛQCD /mQ ), as a result, the valid region of factorization in Ref. [8] is extended. The failure of the factorization approach in the light meson decays is due to the bad convergence behaviour in the O(αs ) expansion. We then concentrate on the contribution of the hard scattering kernel to the amplitude. The amplitude can be obtained by replacing the wave-function with the one obtained in Ref. [16] 1 1 (0) d 3 kΨ (k) √ M|0 Φ (kq , kQ ) = √ 3 2 3 3 2 + m2 δ k 2 + m2
(k Q − k)δ
kq0 − k − k (48) × δ (kq¯ + k)δ ¯ Q0 q Q with M=
i+
↓) − bi+ (k,
↓)dqi+ (−k,
↑), bQ (k, ↑)dqi+ (−k, Q
i
= 4π mP λ3 e−λP |k| Ψ (k) P
(49)
The matrix element can be written as F μ (μ) =
d 4 kq¯ Ψ (k) Tr M · T (0)μ (kq¯ , kQ ) + T (1)μ (kq¯ , kQ ) 2 2 2 2
3 (k Q − k)δ
kq0 (50) × δ 3 (k q¯ + k)δ ¯ − k + mq δ kQ0 − k + mQ
3 1 √ 3 (2π) 6
d 3k
d 4 kQ
After convolution with the wave-functions of the heavy mesons, some of the terms will have identical contributions to the matrix element up to order O(ΛQCD/mQ ) due to their Dirac structures. As a result, we find that the F μ can be simplified as a function of four different types of Dirac structures, and can be written as 1 3 μ d 3 kΨ (k) Tr Cn (pQ , pq¯ , μ)M · Kn (pQ , pq¯ ) F (μ) = (51) √ 3 (2π) 6 n
and pQ = ( m2 + k 2 , k)
denote the on-shell momenta of the light with pq = ( m2q + k 2 , −k) Q anti-quark and the heavy quark in the bound state. And the Kn are defined as (0)
K2 (kQ , kq¯ ) = Ta⊥
(0)
K4 (kQ , kq¯ ) = eQ
K1 (kQ , kq¯ ) = Ta ,
(0) μ
K3 (kQ , kq¯ ) = Tb ,
PL ε/mQ 2pγ · kQ
(52)
Except for C1 K1 , all the other products are contribution of order O(ΛQCD /mQ ), for clarity, we define C1 = C10 + C11 , with C1m represents order O(ΛQCD /mQ )m contribution, the coefficients are αs CF xz x y x 2π 2 C10 (pq , pQ , μ) = 1 + − log 2 + y1 − 2 log 2 + log − 4 + 4π x −y y 3 μ y y x z y x (53) + 2 log2 − 2 log log 2 + 2 log 2 + 2 log 2 − 5 z z μ μ μ
790
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
C11 (pq , pQ , μ) =
w x z xz 2 − 2 + xy 5 log + 8 log + 5 −4 log x y y (x − y)2 y y2 3 xz y x z y 2 + y 2 log + 4 log − 3 + −3y1 + 8 log 2 − 4 x z (x − y)2 y 2 y 2 y y x z + 7y1 + 13 log + 19 log + 14 x z (x − y)2 y xz x z + −5y1 + 4 log + 14 log − 17 y y (x − y)2 yz z x + − 3 log + 2 log + 7 y 1 y y (x − y)2 w x 2w 4xz xz − log + − 2 log 2 2(x − y) y y y y 2w 4xz y y x x + − 2 log 2 log − 2 log − log 2 + 4 y z z y μ μ 2 y x 2z y x π − log2 + log log 2 + log 2 (54) − 2− 3 z z y μ μ αs CF xz x y xz xz − log 2 + y1 − 2 log 2 + log + log 2 4π y yw μ y y 4zx y x y x − log 2 log − 2 log − log 2 + 4 yw z z μ μ
C2 (pq , pQ , μ) = 1 +
C3 (pq , pQ , μ)
2x x π2 6x − y x y log 2 − 2 − − Li2 1 − + − log y x 3 x −y y μ x x x y(2x − y) − 1 + log 2 − log + x −y y μ (x − y)2 2x x x x x (2x − 3y)y y + log −5 + 3 log 2 + + − log 2 − 2 + y x−y y y −x μ (x − y)2 μ y x x x 2wyy2 wy log y (3 log x−y + log x−y ) x2 log + + + y w−z (x − y)2 (w − z)2
αs CF =1+ 4π
(55)
C4 (pq , pQ , μ) 2y y(w − z) + log y (3y log x + y log y ) 2 αs C F 2xz2 x x−y x−y = − 2z 4π yw (w − z)2 2y x x x x y(2x − 3y) − log log + −5 + 3 log 2 + + x −y y x−y y μ (x − y)2 with x, y, z and w defined in Eq. (18), and y1 , y2 defined in Eq. (27) and Eq. (36).
(56)
(57)
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
791
6. Large logarithm resummation In the expression of F μ , large logarithms show up. Those large logarithms need to be resummed so that the result is phenomenologically reliable.We concentrate on the large logarithms at order O(ΛQCD /mQ )0 , because terms like (ΛQCD log
mQ ΛQCD )/mQ mQ
are suppressed and not
large. In this section, we use y = 2pγ · pQ = 2Eγ mQ for simplicity. We concentrate on C10 , which can also be written as C10 (pq , pQ , μ) x y y y y αs (μ)CF −2 Li2 1 − − 2 log2 − log + 2 log =1+ 4π x y x −y x x
x μ2 x π2 x π2 x log 2 + 3 log 2 − log2 2 − 6 − + log2 −3− (58) y 12 z 4 μ μ μ When μ ∼ mQ , the large logarithms are the same as Ref. [8]. However, when μ is set to μ ∼ mQ ΛQCD , the large logarithms are different from Ref. [8] by + 2 log
mQ 1 1 mQ − log2 2 + log 2 (59) 2 2 μ μ This is because the corrections of wave-functions in this work are not calculated in HQET framework as Ref. [8], in which mQ will not appear in the result. 6.1. Renormalization group equation (RGE) evolution We start with the RGE at order O(ΛQCD /mQ )0 , and use the method introduced in Ref. [13]. The RGE of C10 is μ
∂ 0 C (μ) = γ (μ)C10 (μ) ∂μ 1
(60)
with γ (μ) defined as anomalous dimension of C10 , which can be obtained by using the counter terms of C10 K1 . The counter terms at the order O(ΛQCD /mQ )0 can be written as αs (μ)CF 2 z Z= 2 log − 3 (61) 4π
y We can use the counter terms and the β function in QCD to calculate γ (μ) 3
∂ ∂ −1 γ (μ) = Z β = −g + O g , +β Z μ 2 ∂μ ∂g
(62)
The result is
αs (μ)CF μ2 z γ (μ) = − 2 log + 2 log 2 − 3 2π y μ
(63)
With this result, we can solve the RGE of the coefficient. Assume the coefficient can be written as a hard function multiplied by a jet function [9,21], so C(μ) = H (μ)J (μ), γ (μ) = γH (μ) + γJ (μ) ∂ ∂ μ H (μ) J (μ) + H (μ) μ J (μ) = γH (μ)H (μ)J (μ) + γJ (μ)H (μ)J (μ) (64) ∂μ ∂μ
792
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
The RGE of the hard function is ∂ μ H (μ) = γH (μ)H (μ) ∂μ
(65)
(0)
By splitting the coefficient C1 into H and J , we assume that the natural scale of H is mQ , while the natural scale of J is mQ ΛQCD , which is also the case for γ (μ) with a log(μ2 /y) term and a log(μ2 /z) term. So we split the γ (μ) such that γH (μ) is the sum of the log(μ2 /y) term and an undetermined constant n , so αs (μ)CF μ2
γH (μ) = − 2 log +n (66) 2π y Similar as Refs. [8,14], we find αs (μ)CF μ γHLO = − 2 log π mQ 2Eγ αs (μ)CF
α 2 (μ) μ γHNLO = − (67) n − 2 log − 2CF B s 2 log 2π mQ mQ (2π) The γHNLO is the same as Refs. [8,14], so the solution is f0 H (μ) = exp + f1 H (mQ ) αs (mQ ) 4πCF 1 f0 = αs (mQ ) −2 2 − 1 + log r β0 αs (mQ ) r CF β1 CF
y 1 2 f1 = − 3 1 − r + r log r − log r + log r n − 2 log 2 β0 x β0 −
2CF B (r − 1 − log r) β02 2N
(68) 34C 2
10C N
(μ) f A f A A with r = ααs s(m , β0 = 11C − 2CF Nf , where CA = 3 for QCD, 3 − 3 and β1 = 3 − 3 Q) Nf is the number of the flavour of quarks taken into account, and B can only been derived from 2-loop calculations. In Ref. [14], by comparing the result with B → Xs γ and B → Xu l v¯ in 5Nf π2 Ref. [22], B is found to be B = CA ( 67 18 − 6 ) − 9 . So the result of H is αs (mQ )Cf μ y μ μ H (μ) = H (mQ ) exp + 4 log log − 2n log −4 log2 4π mQ x mQ mQ 2 (69) + O αs
6.2. The resummation The hard function can be derived by using the method introduced in Ref. [9]. With xγ defined as xγ = 2Eγ /mQ , the result is 2Eγ ˆ H μ 2Eγ αs (μ)CF 1 2 2Eγ =1+ log xγ + 2 log −2 Li2 (1 − xγ ) − 2 log − (70) 4π μ 1 − xγ μ
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
and H (mQ ) = Hˆ
2Eγ mQ
793
(71)
we also find SCET H (mQ ) = C3,6 (mQ )
(72)
n
With = 3, the evaluation of the RGE will correctly resum the large logarithms, which can be shown explicitly by expanding the solution of RGE of hard function μ αs (mQ )CF y μ μ H (μ) = H (mQ ) 1 + + 4 log log − 6 log −4 log2 4π mQ x mQ mQ 2 + O αs x y y y y αs (mQ )CF −2 Li2 1 − − 2 log2 − log + 2 log =1+ 4π x y x−y x x x π2 x x 2 x (73) + O αs2 + 2 log log 2 + 3 log 2 − log 2 − 6 − y 12 μ μ μ Comparing Eqs. (60) and (64) with Eq. (73), we find that there are no more large logarithms in the remaining terms, which gives the jet function 2 αs (μ)CF π2 2 μ J (μ) = 1 + log −3− (74) 4π z 4 The γJ (μ) can been obtained by subtracting γH (μ) from γ (μ) γJ (μ) =
αs (μ)CF μ 2 log √ π z
The solution of RGE of jet function is √ 2 2 √ αs ( z)CF 2 μ J (μ) = J ( z) × exp log 2 + O αs 4π z
(75)
(76)
We find that, Eq. (76) can also correctly resum the large logarithms which will show up in jet function when μ is evaluated to lower than ΛQCD mQ . When μ > ΛQCD mQ , the resummed result at order O(αs (ΛQCD /mQ )0 ) is y αs (mQ )CF y 0 C1 = 1 + −2 Li2 1 − − 2 log2 4π x x 2 y y π y log + 2 log − 6 − − x −y x x 12 αs (mQ )Cf μ y μ μ 2 × exp + 4 log log − 6 log −4 log 4π mQ x mQ mQ 2 2 αs (μ)Cf μ π × 1+ log2 −3− (77) 4π z 4 √ At the leading order of O(ΛQCD /mQ ), z can be written as z = 2pγ · kq¯ = 2k+ Eγ . Using m ΛQCD = 200 MeV, mb = 4.98 GeV, and k+ = ΛQCD , Eγ = 4Q , we can show the evaluation of coefficient C10 in Fig. 5.
794
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
m
Fig. 5. C10 as function of μ, with Eγ = 4Q , k+ = ΛQCD . The solid line is the resummed result, and the dotted line is the un-resummed one.
7. Numerical applications The amplitude is derived in Eq. (50), which can be decomposed as [6,23] ρ γ |qΓ ¯ μ Q|P = μνρσ ε ν pP pγσ FV + i εμ pP · pγ − pγμ ε · pP FA
(78)
The contribution in Fig. 1c depends on not only Eγ but also on pν and pl . For simplicity, we treat this term separately, the form factors is written as μ ρ μ γ |qΓ ¯ μ Q|P + Fc pP = μνρσ ε ν pP pγσ FV + i εμ pP · pγ − pγμ ε · pP FA + Fc pP 1 3 1 1 d 3 kΨ (k) FV = √ 3 (2π) 2 pq0 pQ0 (pq0 + mq )(pQ0 + mQ ) mP Eγ 6 0 × 2eq C1 mQ − C1 pq0 eq + eQ 2pq0 C3 1 3 1 1 d 3 kΨ (k) FA = √ 3 (2π) 2 pq0 pQ0 (pq0 + mq )(pQ0 + mQ ) mP Eγ 6 2pq0 mQ 2pq0 mQ × 2eq C1 mQ − C10 pq0 eq − eq C2 − eQ 2pq0 C3 + eQ C4 (79) Eγ Eγ The relation FA = FV at leading order is explicitly broken at order O(ΛQCD /mQ ) as expected [24]. We evaluate this integral using [16] mD = 1.9 GeV,
mB = 5.1 GeV,
mb = 4.98 GeV,
mc = 1.54 GeV
ΛQCD = 200 MeV,
mu = md = 0.08 GeV
λB = 2.8 GeV−1 ,
λD = 3.4 GeV−1
(80)
The result of FA,V of B → γ eνe is shown in Fig. 6, the O(ΛQCD /Eγ ) contribution is more important at the region Eγ → 0 which is clearly shown in the figure. The numerical results of FA,V are inconvenient to use when calculate the decay widths. For simplicity, we use some simple forms to fit the numerical results. For the form factors at the order O(αs (ΛQCD /mQ )0 ), we use the single-pole form FAαs (Eγ ) = FVαs (Eγ ) =
f (0) f (0) = q 2 − m∗2 m2P − m∗2 − 2mP Eγ
(81)
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
795
Fig. 6. Form factors of B → γ eνe as functions of Eγ . The results are presented as the ratios of the form factors at 1-loop order to the one at the tree level and the leading order of the ΛQCD /mQ expansion, which is denoted as F0 . The solid line is FA /F0 = FV /F0 , where FA,V are at the order of O(αs (ΛQCD /mQ )0 ). The dotted line is FV /F0 and the dashed line is FA /F0 , both at the order of O(αs ΛQCD /mQ ). Table 1 The results of the parameters in the form factors in Eqs. (81) and (82). B → γ eνe D → γ eνe
m∗ (GeV)
f (0) (GeV)
AV
BV
AA
BA
5.37 1.98
−0.63 −0.15
0.27 −0.00095
0.45 −0.54
0.32 −0.27
−0.67 −0.059
where q = pP − pγ . While up to the order O(αs ΛQCD /mQ ), inspired by the form factors in Ref. [17] the form factors are fitted as αs ΛQCD ΛQCD 2 mQ FA,V (Eγ ) = AA,V + BA,V (82) Eγ Eγ The predicted results for the form factors are more reliable at the region Eγ ΛQCD because we have neglected the higher order terms of ΛQCD /Eγ in the numerical calculation, so we choose the region Eγ > 2ΛQCD to fit the parameters in Eqs. (81) and (82). The fitting is given in Table 1 and shown in Figs. 7 and 8. On the other hand, Fc can be related to the decay constant [16,17] by μ
μ
Fc pP = ie
pl · εPL μ pl · ε 0|qγ ¯ μ (1 − γ5 )Q|P = −iefP pP pl · p γ pl · p γ
(83)
Using the fitted result of FA , FV , the result of Fc , and using the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements [25,26] Vcd = 0.226,
Vub = 0.0047
(84)
we obtain the result for the branching ratios. There are IR divergences in the radiative leptonic decays in the case that the photon is soft or the photon is collinear with the emitted lepton. The-
796
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
Fig. 7. Fit of the form factors of B → γ eνe . The solid line is for the result at the leading order of ΛQCD /mQ . The ‘×’ points and the dotted line are for FV while the dashed line and the ‘+’ points are for FA .
Fig. 8. Fit of the form factors of D → γ eνe . The solid line is for the result at the leading order of ΛQCD /mQ . The ‘×’ points and the dotted line are for FV while the dashed line and the ‘+’ points are for FA .
oretically this IR divergences can be canceled by adding the decay rate of the radiative leptonic decay with the pure leptonic decay rate, in which one-loop correction is included [27]. The radiative leptonic decay cannot be distinguished from the pure leptonic decay in experiment when the photon energy is smaller than the experimental resolution to the photon energy. So the decay rate of the radiative leptonic decay depend on the experimental resolution to the photon energy Eγ which is denoted by Eγ . The dependence of the branching ratios on the resolution are listed in Table 2. Using Eγ = 10 MeV [28], the branching ratios are given in Table 3. We find that, in general, the 1-loop results are smaller than the tree level results. The 1-loop correction is found to be
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
797
Table 2 The branching ratios with different photon resolution Eγ . Eγ
BR(B → eνe γ )
BR(D → eνe γ )
Eγ
BR(B → eνe γ )
BR(D → eνe γ )
5 MeV 10 MeV 15 MeV
1.77 × 10−6 1.66 × 10−6 1.60 × 10−6
3.10 × 10−5 2.81 × 10−5 2.64 × 10−5
20 MeV 25 MeV 30 MeV
1.56 × 10−6 1.53 × 10−6 1.48 × 10−6
2.53 × 10−5 2.45 × 10−5 2.38 × 10−5
Table 3 The branching ratios of the decay modes. BR
O((
B → eνe γ D → eνe γ
αs ΛQCD 0 mQ ) )
2.38 × 10−6 9.62 × 10−6
BR
ΛQCD O(αs ( m )0 ) Q
1.01 × 10−6 2.71 × 10−6
BR
O(
αs ΛQCD mQ )
1.66 × 10−6 2.81 × 10−5
important. For B meson, the correction to the decay amplitude at the order O(ΛQCD/mQ )0 is about 10% to 30% due to the large logarithms. The contribution of the order O(αs ΛQCD /mQ ) are generally not negligible. For B meson, the correction of the order O(ΛQCD /mQ ) contribution to the decay amplitude can be as large as 30%. For D mesons, the mass of c quark is not large enough, the order O(ΛQCD /mQ ) contributions is much more important, it is necessary to include higher order corrections in ΛQCD/mQ expansion. 8. Conclusion In this paper, we study the factorization of the radiative leptonic decays of B − and D − mesons. Compared with the work in Ref. [8], the factorization is extended to include the O(ΛQCD /mQ ) contributions, and the transverse momentum is also considered. The factorization is proved explicitly at 1-loop order, the valid region of the factorization is extended. The hard kernel at order O(αs ΛQCD /mQ ) is obtained. We use the wave function obtained in Ref. [16] to derive the numerical results. The branching ratios of B − → γ eν¯ is found to be at the order of 10−6 , which is close to the previous works [7,17,29–31]. In the previous works, the results of D mesons are different from each other from 10−3 to 10−6 , our results agree with 10−5 . We also find that the O(ΛQCD /mQ ) contribution is very important even for B meson, the correction to the decay amplitude is about 20%–30%, which can affect the branching ratios about 50%. This is because of the importance of O(ΛQCD /Eγ ) contributions. In previous works, O(ΛQCD /Eγ ) contributions is neglected. For a typical region, Eγ ∼ mQ /4, which is also the leading region of the phase space of the tree level, the neglected contributions can be up to 20%. As a result, the correction to the branching ratios can be up to 40% at tree level. Acknowledgements This work is supported in part by the National Natural Science Foundation of China under contracts Nos. 11375088, 10975077, 10735080, 11125525.
798
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
Appendix A. IBP reductant relation of wave function The integral is
Φq(1) (kq¯ , kQ ) =
4
4
d x
d ye
× igs
ikq¯ ·x ikQ ·y
e
x 0|q¯q¯ (x)igs
dzzμ Aμ (z)Q(y) y
d 4 x2 q¯q¯ (x2 )/ A(x2 )qq¯ (x2 )q¯ S (pq¯ ), Qs (pQ )
(A.1)
After variable substitution using z = x + αy, we find x
1 dzzμ A (z) = μ
y
d y + α(x − y) μ Aμ y + α(x − y)
0
1 =
dα(x − y)μ Aμ y + α(x − y)
(A.2)
0
After the contraction and then integral over x2 , p, the result is Φq(1) ⊗ T (0) = −CF gs2
1
4
d x
4
d y
dα
dd l (2π)d
0 ˙
˙
× eiy (kQ +αl−pQ ) eix (kq¯ −αl−pq¯ ) v¯q¯ γ · (x − y)
/ q¯ + mq¯ ) (0) 1 (−/l − p T uQ 2 l (l + pq¯ )2 − m2q¯
(A.3)
The integral over x and kq¯ can be worked out term by term. However, we find IBP reduction relation [18] an elegant way to do so. Consider this integral d 4 kq Γ (kq ) d 4 xe−i(pq¯ +αl−kq¯ )·x (A.4) (2π)4 It is unchanged when kq is shifted, so under the infinitesimal transformation kq → kq + βK The integral transforms as Γ (kq ) d 4 xe−i(pq¯ +αl−kq¯ )·x ∂ 4 −i(pq¯ +αl−kq¯ )·x → βK · Γ (kq ) d xe ∂kq The Lie algebra leads to d 4 kq ∂ 4 −i(pq¯ +αl−kq¯ )·x K· Γ (kq ) d xe =0 ∂kq (2π)4
(A.5)
(A.6)
(A.7)
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
so that
d 4 kq Γ (kq )K · x (2π)4
d 4 xe−i(pq¯ +αl−kq¯ )·x = iK ·
∂Γ (kq ) ∂kq kq =pq¯ +αl
799
(A.8)
So after integrate over x, kq¯ , y and kQ , the result is Φq(1) ⊗ T (0) = igs2 CF
(/l + p / q¯ − mq¯ ) dd l 1 v¯q¯ γ ρ d 2 (2π) l (l + pq¯ )2 − m2q¯
k = pq¯ + αl,
K = pQ − αl
1 dα 0
∂T (0) ∂T (0) − uQ ρ ρ ∂kq ∂kQ kq =k ,kQ =K
(A.9)
References [1] P.F. Harrison, H.R. Quinn, BaBar Collaboration, The BaBar physics book: physics at an asymmetric B factory, SLAC-R-0504; J. Nash, Int. J. Mod. Phys. A 17 (2002) 2982–2998, arXiv:hep-ex/0201002; H. Tajima, Int. J. Mod. Phys. A 17 (2002) 2967–2981, arXiv:hep-ex/0111037. [2] M. Artuso, E. Barberio, S. Stone, PMC Phys. A 3 (2009) 3, arXiv:0902.3743. [3] M. Neubert, Int. J. Mod. Phys. A 11 (1996) 4173–4240, arXiv:hep-ph/9604412. [4] R.P. Feynman, Photon–Hadron Interactions, Benjamin, Reading, MA, 1972. [5] J.C. Collins, D.E. Soper, G. Sterman, Adv. Ser. Dir. High Energy Phys. 5 (1988) 1–91, arXiv:hep-ph/0409313. [6] M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Phys. Rev. Lett. 83 (1999) 1914–1917, arXiv:hep-ph/9905312; M. Beneke, G. Buchalla, M. Neuber, C.T. Sachrajda, Nucl. Phys. B 591 (2000) 313–418, arXiv:hep-ph/0006124; M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Nucl. Phys. B 606 (2001) 245–321, arXiv:hep-ph/0104110v1. [7] G.P. Korchemsky, D. Pirjol, T.-M. Yan, Phys. Rev. D 61 (2000) 114510, arXiv:hep-ph/9911427v3. [8] S. Descotes-Genon, C.T. Sachrajda, Nucl. Phys. B 650 (2003) 356–390, arXiv:hep-ph/0209216v2. [9] S.W. Bosch, R.J. Hill, B.O. Lange, M. Neubert, Phys. Rev. D 67 (2003) 094014. [10] N. Isgur, M.B. Wise, Phys. Lett. B 232 (1989) 113; H. Georgi, Phys. Lett. B 240 (1990) 447. [11] R.J. Hill, M. Neubert, Nucl. Phys. B 657 (2003) 229–256, arXiv:hep-ph/0211018. [12] E. Lunghi, D. Pirjol, D. Wyler, Nucl. Phys. B 649 (2003) 349–364. [13] C.W. Bauer, S. Fleming, M. Luke, Phys. Rev. D 63 (2000) 014006. [14] C.W. Bauer, S. Fleming, D. Pirjol, I.W. Stewart, Phys. Rev. D 63 (2001) 114020. [15] K.G. Wilson, Phys. Rev. D 10 (1974) 2445. [16] M.-Z. Yang, Eur. Phys. J. C 72 (2012) 1880. [17] J.-C. Yang, M.-Z. Yang, Mod. Phys. Lett. A 27 (2012) 1250120, arXiv:1204.2383. [18] A.G. Grozin, Int. J. Mod. Phys. A 26 (2011) 2807–2854, arXiv:1104.3993v4. [19] W.A. Bardeen, A.J. Buras, D.W. Duke, T. Muta, Phys. Rev. D 18 (1978) 3998. [20] G. ’t Hooft, M. Veltman, Nucl. Phys. B 153 (1979) 365–401; J.C. Romão, Modern techniques for one-loop calculations (unpublished), which can be found at http://www. researchgate.net/publication/228530611_Modern_Techniques_for_One_Loop_Caculations. [21] T.T. Jouttenus, Phys. Rev. D 81 (2010) 094017. [22] G.P. Korchemsky, G. Sterman, Phys. Lett. B 340 (1994) 96; R. Akhoury, I.Z. Rothstein, Phys. Rev. D 54 (1996) 2349. [23] M. Wirbel, B. Stech, M. Bauer, Z. Phys. C, Part. Fields 29 (1985) 637–642; D. Fakirov, B. Stech, Nucl. Phys. B 133 (1978) 315. [24] F. Hussain, G. Thompson, arXiv:hep-ph/9502241. [25] K. Nakamura, et al., Particle Data Group, J. Phys. G 37 (2010) 075021, and 2011 partial update for the 2012 edition. [26] A.G. Akeroyd, F. Mahmoudi, J. High Energy Phys. 1010 (2010) 038; T. Konstandin, T. Ohlsson, Phys. Lett. B 634 (2006) 267–271.
800
[27] [28] [29] [30] [31]
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
C.-H. Chang, et al., Phys. Rev. D 60 (1999) 114013. K. Aamodt, et al., ALICE Collaboration, J. Instrum. 3 (2008) S08002. D. Atwood, G. Eilam, A. Soni, Mod. Phys. Lett. A 11 (1996) 1061. C.-Q. Geng, C.-C. Lih, W.-M. Zhang, Mod. Phys. Lett. A 25 (2000) 2087. C.-D. Lu, G.-L. Song, Phys. Lett. B 562 (2003) 75.
ScienceDirect Nuclear Physics B 889 (2014) 778–800 www.elsevier.com/locate/nuclphysb
Factorization of radiative leptonic decays of B − and D − mesons Ji-Chong Yang, Mao-Zhi Yang ∗ School of Physics, Nankai University, Tianjin 300071, PR China Received 1 September 2014; accepted 30 October 2014 Available online 11 November 2014 Editor: Hong-Jian He
Abstract In this work, we study the factorization of the radiative leptonic decays of B − and D − mesons, the contributions of the order O(ΛQCD /mQ ) are taken into account. The factorization is proved to be valid explicitly at the order O(αs ΛQCD /mQ ). The hard kernel is obtained. The numerical results are calculated using the wave-function obtained in relativistic potential model. The O(ΛQCD /mQ ) contribution is found to be very important, the correction to the decay amplitudes of B − → γ eν¯ is about 20%–30%. For D mesons, the O(ΛQCD /mQ ) contributions are more important. © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3 .
1. Introduction The study of the heavy meson decays is an important field in high energy physics. In recent years, both experimental and theoretical studies have been improved greatly [1–3]. However, the limitation in understanding and controlling the non-perturbative effects in strong interaction is so far still a problem. Varies theoretical methods on how to deal with the non-perturbative effects have been developed. An important approach is to separate the hard and soft physics which is known as factorization [4,5]. This method has been greatly developed in recent years [6]. * Corresponding author.
E-mail address: [email protected] (M.-Z. Yang). http://dx.doi.org/10.1016/j.nuclphysb.2014.10.027 0550-3213/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3 .
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
779
The idea of factorization is to absorb the infrared (IR) behaviour into the wave-function, the matrix element can be written as the convolution of wave-function and hard kernel F = dkΦ(k)Thard (k) (1) The wave-function should be determined by non-perturbative methods. The radiative leptonic decay of heavy mesons provides a good opportunity to study the factorization approach, where strong interaction is involved only in one hadronic external state. Except for that, with a photon emitted out, more details about the wave-function of the hadronic bound state can be exploited. Many works has been done on the factorization of this decay mode. In Ref. [7], the 1-loop QCD correction is calculated in the large energy effective theory, and they found the factorization will depend on the transverse momentum. In Refs. [8] and [9], factorization is proved in leading order of 1/mQ expansion in the frame of QCD factorization [4,5], where the heavy quark is treated in the heavy quark effect theory (HQET) [3,10]. In Refs. [11,12], the factorization is constructed using the soft-collinear effective theory (SCET) [13,14]. In this work, factorization in the radiative leptonic decays of heavy mesons is revisited. The work of Refs. [8,9] is extended by taking into account the contributions of the order of O(ΛQCD /mQ ). The factorization is proved to be still valid explicitly. We also find the factorization is valid at any order of O(ΛQCD /mQ ). The numerical results shows that, the O(ΛQCD /mQ ) correction is very important for the B and D mesons, the correction can be as large as 20%–30%. The remainder of the paper is organized as follows. In Section 2, we discuss the kinematic of the radiative decay and the wave-function. In Section 3, we present the factorization at tree level. In Section 4, the 1-loop corrections of the wave-function are discussed. The factorization at 1-loop order is presented in Section 5. In Section 6, we briefly discuss the resummation of the large logarithms. The numerical results are presented in Section 7. And Section 8 is a summary. 2. The kinematic The B or D meson is constituted with a quark and an anti-quark, where one of the quarks is a heavy quark, and the other is a light quark. The Feynman diagrams at tree level of the radiative leptonic decay can be shown as Fig. 1. The contribution of Fig. 1d is suppressed by a factor of 1/Mw 2 , and can be neglected. The amplitudes of Fig. 1a, b and c can be written as p /γ −p /q μ −ieq GF VQq PL Q(pQ )(lPLμ ν¯ ) q(p ¯ q¯ )/ε∗γ √ 2p · p 2 γ q¯ /Q −p / γ + mQ ∗ −ieQ GF VQq μp (0) ε/γ Q(pQ )(lPLμ ν) ¯ Ab = q(p ¯ q¯ )PL √ 2pQ · pγ 2 pγ + p / l + ml ) −eGF VQq μ ∗ i(/ = )P Q(p ) l/ ε ν ¯ P A(0) q(p ¯ √ q¯ L Q Lμ c γ 2(pγ ·pl ) 2 A(0) a =
(2)
where pq¯ and pQ are the momenta of the anti-quark q¯ and quark Q, respectively, pγ , pl and pν are the momenta of photon, lepton and neutrino, εγ denotes the polarization vector of photon, and PLν is defined as γ μ (1 − γ5 ). We work in the rest-frame of the meson, and we choose the frame such that the direction of the photon momentum is on the opposite z axis, so the momentum of the photon can be written as pγ = (Eγ , 0, 0, −Eγ ), with 0 ≤ Eγ ≤ mQ /2.
780
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
Fig. 1. Tree level amplitudes, the double line represents the heavy quark propagator but not the HQET propagator.
To study the factorization, we consider the state of two free quark and anti-quark at first. The wave-function of the two quark and anti-quark state is defined as ¯ (3) Φ(kq , kQ ) = d 4 xd 4 y exp(ikq · x) exp(ikQ · y)0|v¯q¯ (x)[x, y]uQ (y)|qQ where [x, y] denotes the Wilson line [15]. And the matrix element is defined as μ
F = γ |v¯q¯ (x)PL uQ (y)|qQ ¯
(4)
The prove of factorization is to prove that, up to 1-loop order, the matrix element can be written as the convolution of the wave-function Φ and a hard-scattering kernel T , where T is IR-finite and independent of the external state. 3. Tree level factorization We start with the matrix elements at tree level. Using the definition of the wave-function in coordinate space Φαβ (x, y) = 0|q¯α (x)[x, y]Qβ (y)q¯ S (pq¯ ), Qs (pQ ) (5) where S and s are spin labels of q¯ and Q, respectively. We find (0) Φαβ (kq¯ , kQ ) = (2π)4 δ 4 (kq¯ − pq¯ )(2π)4 δ 4 (kQ − pQ )v¯α (pq¯ )uβ (pQ )
And then the matrix element can be written as d 4 kq¯ d 4 kQ (0) (0) F = Φ (kq¯ , kQ )T (0) (kq¯ , kQ ) = Φ (0) ⊗ T (0) (2π)4 (2π)4
(6)
(7)
With Eqs. (2) and (7), we obtain the hard scattering kernel at tree level as Ta(0) = −eq
ε/∗γ p / γ − 2εγ∗ · kq 2pγ · kq¯
μ
PL ,
(0)
Tb
μ
= −eQ PL
−/ p γ ε/∗γ 2kQ · pγ
(8)
In the expressions above, we have already assumed to consider the kinematical region Eγ ∼ mQ . The polarization vector of the photon does not have 0-component, as a result
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
781
(/kQ + mQ )/εuQ /2pγ · pQ is an order of O(Λ2QCD /m2Q ) contribution and is neglected. The reμ
maining terms of Tb(0) , and the transverse part of Ta(0) , which is 2eq ε · kq¯ PL /2pγ · kq¯ are order of O(ΛQCD /mQ ) contributions. Exchange the Lorentz index in Ac , we obtain A(0) c =
p γ ε/∗γ + 2εγ∗ · (pQ + pq¯ − pν )) −ieGF VQq μ (/ Q(pQ )(lPLμ ν¯ ) q(p ¯ q¯ )PL √ 2(pγ ·(pQ + pq¯ − pν )) 2
(9)
We find / γ ε/∗γ μp
+ 2ε · (pQ + pq¯ − pν )
Fc(0) = −ev¯q¯ PL
2pγ · (pQ + pq¯ − pν ) ∗ / γ ε/γ + 2ε · (kQ + kq¯ − pν ) μp
Tc(0) = −ePL
uQ (10)
2pγ · (kQ + kq¯ − pν )
This term is also an order of O(ΛQCD /mQ ) contribution. 4. 1-loop correction of wave-function The expansion of the decay amplitude can be written as [8] F = F (0) + F (1) + . . . = Φ (0) ⊗ T (0) + Φ (1) ⊗ T (0) + Φ (0) ⊗ T (1) + . . .
(11)
At the 1-loop level, the amplitude can be written as F (1) = Φ (1) ⊗ T (0) + Φ (0) ⊗ T (1)
(12)
The 1-loop corrections of Φ ⊗ T come from the QCD interaction and the Wilson line. The later can be written as [8,15] x n x (igs )n
4 μ [x, y] = exp igs d zzμ A (z) = d 4 zi ziμ Aμ (zi ) (13) n! n i y
y
The corrections are shown in Fig. 2. (1) We use Φq to represent the correction with the gluon from the Wilson line connected to the light quark external leg. So the correction in Fig. 2a can be written as Φq(1) (kq¯ , kQ ) =
4
d x × igs
4
d ye
ikq¯ ·x ikQ ·y
e
x 0|q¯q¯ (x)igs
dzzμ Aμ (z)Q(y) y
d 4 x2 q¯q¯ (x2 )/ A(x2 )qq¯ (x2 )q¯ S (pq¯ ), Qs (pQ )
(14)
After the integration, the result is Φq(1) ⊗ T (0) = igs2 CF
(/l + p / q¯ − mq¯ ) dd l 1 v¯q¯ γ ρ (2π)d l 2 (l + pq¯ )2 − m2q¯
k = pq¯ + αl,
1 0
∂T (0) ∂T (0) dα uQ ρ − ρ ∂kq ∂kQ kq =k ,kQ =K
K = pQ − αl
The procedure of the integration can be found in Appendix A.
(15)
782
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
(0)
Fig. 2. The 1-loop correction of wave-functions. Φ (1) ⊗ Ta
(0)
and Φ (1) ⊗ Tb
are established in this figure.
Similar to Φq , the correction in Fig. 2b can be written as (1)
ΦQ ⊗ T (0) 1 = −iCF gs2
dα
0
k = pq¯ + αl,
(0) dd l 1 ∂T (0) (/ ∂T p Q − /l + mQ ) ρ v¯q¯ γ uQ ρ − ρ (2π)d l 2 ∂kq ∂kQ (pQ − l)2 − m2Q kq =k ,kQ =K
K = pQ − αl
(16)
We use ΦWfc to denote the correction shown in Fig. 2c. We find (1) ΦWfc
⊗T
(0)
g 2 CF =− s 2
dd l (2π)d
1
1 dα
0
dβ 0
1 ∂ 2 (0) ∂ v ¯ − T uQ q¯ ∂kq ∂kQ l2 kq =k ,kQ =K
(17)
The corrections shown in Fig. 2d, 2e and 2f can be denoted as Φbox , ΦextQ and Φextq . We find that, they have the same forms as the free particle 1-loop QCD corrections. 5. 1-loop factorization For simplicity, we denote x = m2Q ,
y = 2pQ · pγ ,
z = 2pγ · pq¯ ,
w = 2pQ · pq¯
(18)
The definitions of x, y and z are the same as Ref. [8] at order O(ΛQCD /mQ )0 , while w is a new scalar that appears at the order of O(ΛQCD /mQ ) contributions which will be shown later, it represents the effect of the transverse momentum. To calculate the hard-scattering amplitude, we need to calculate all 1-loop corrections of Fa , Fb and Fc . We take a small mass mq for the light quark to regulate the collinear IR divergences. The soft IR divergences will not appear explicitly in this factorization procedure. We use MS [19]
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
783
Fig. 3. 1-loop QCD correction of Fa .
scheme to regulate the ultraviolet (UV) divergences, in D = 4 − demission, we define NUV as 2 (19) − γE + log(4π)
We take the factorization scale the same as the renormalization scale, so we use the same μ in F (1) and Φ (1) ⊗ T (0) . NUV =
5.1. 1-loop correction of Ta(0) The Feynman diagrams of the 1-loop corrections of Ta are shown in Fig. 3. We denote the correction of the electric–magnetic (EM) vertex, the one shown in Fig. 3a, as F (1)EM . To show the effect of the transverse momentum explicitly, we establish the longitudinal part and transverse part separately. The matrix element at tree level can be written as (0)
(0)
Fa(0) = Fa + Fa⊥ , (0)
Fa⊥ = eq v¯q¯
(0)
Fa = −eq v¯q¯
2ε · pq¯ μ P uQ 2pγ · pq¯ L
ε/p /γ μ P uQ 2pγ · pq¯ L (20)
The transverse part is at order O(ΛQCD /mQ ). The corrections to each part is represented separately as (1)EM
Fa(1)EM = Fa
(1)EM
+ Fa⊥ / γ + mq ) i(−/ p q¯ − /l + p i(−/ p q¯ − /l + mq ) dd l (1)EM Fa = CF gs2 v¯q¯ iγρ (−ieq )/ε d 2 2 (2π) (pq¯ + l) − mq (pq¯ + l − pγ )2 − m2q i/ pγ μ −i × iγ ρ P uQ 2pq¯ · pγ L l 2 / γ − mq ) p q¯ + /l − p (/ p q¯ + /l − mq ) (/ dd l (1)EM Fa⊥ = ieq CF gs2 v¯q¯ γρ ε/ (2π)d (pq¯ + l)2 − m2q (pq¯ + l − pγ )2 − m2q p / q¯ μ1 ×γρ PL 2 uQ 2pq¯ · pγ l
(21)
784
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
After performing the momentum-integration, the result is 2pγ · pq¯ 2pγ · pq¯ (1)EM (0) αs CF Fa = Fa NUV − log + 2 log 4π μ2 m2q 2pq¯ · pγ (1)EM (0) αs CF Fa⊥ = Fa⊥ NUV − log +1 4π μ2
(22) (1)
where CF is defined as CF = (N 2 − 1)/2N = 4/3 for QCD. The transverse part F⊥ is an order O(αs ΛQCD /mQ ) contribution. Accordingly the 1-loop correction of the wave-function is (1) (0) (1) (0) (0) Φq ⊗ Ta , which can also be written as Φq ⊗ (Ta + Ta⊥ ): 2/ pγ ε/∗γ (/l + p / q¯ − mq ) dd l 1 μ (0) (1) 2 Φq ⊗ Ta = ieq CF gs v ¯ p / PL uQ q ¯ γ 2 2 (2π)d l 2 4p · (p + l)p · p (l + pq¯ ) − mq¯ γ γ 1 dd l 1 (0) Φq(1) ⊗ Ta⊥ = −ieq gs2 CF dα (2π)d l 2 0
4/εγ (pγ · (pq¯ + αl) − 4/ pγ εγ · (pq¯ + αl) (/l + p / q¯ − mq¯ ) × v¯q¯ PLμ uQ (23) (2pγ · (pq¯ + αl))2 (l + pq¯ )2 − m2q¯ Φq(1) ⊗ T⊥(0) is calculated in the light-cone coordinate (see also Appendix A in Ref. [7]), the result is m2q CF αs (0) (0) (0) (1) Φq ⊗ Ta = 2 Φq(1) ⊗ Ta⊥ = 0 (24) T NUV − log 2 + 2 , 4π a μ Using Eq. (12), we can obtain the hard-scattering kernel 2kq¯ · pγ 2kq¯ · pγ (0) CF αs (0) CF αs Ta(1)EM = Ta − 4 + T + 1 log − log a⊥ 4π 4π μ2 μ2
(25)
We find that, the hard-scattering kernel of the longitudinal direction is the same as the result in Ref. [8]. The correction of the weak vertex is shown in Fig. 3b. The result can be written as [20] Fa(1)weak
/ q¯ p /γ −p i y xz x x v¯q¯ ε/ NUV − log 2 + y1 − 2 log 2 + log −z y 16π2 y x −y μ wy x z xz 2 + x −4 log 2 − 2 + xy 5 log + 8 log + 5 y y (x − y)2 y 2 y 3z xz y x y 2 + −3y1 + 8 log 2 − 4 + y 2 log + 4 log − 3 x z (x − y)2 y 2 y y y x 2 yz 7y1 + 13 log + 19 log + 14 + x z (x − y)2 y 2 x z xy 2 z −5y1 + 4 log + 14 log − 17 + y y (x − y)2 y 2 3 z x y z μ − 3 log + 2 log + 7 PL y + 1 y y (x − y)2 y 2 log xy / q¯ μ p /γ −p xz μ1 + / Q (1 − γ5 ) (26) 4pq¯ p − 2mQ PR log 2 uQ −z 2(x − y) y y
= −ieq CF gs2
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
with
xz x x y1 = −π + 2 Li2 1 − − log2 2 + log2 y y y 2
785
(27)
In the expressions above, we have already neglected the terms which are irrelevant for the discussion, because they will not contribute to the matrix element up to order O(ΛQCD /mQ ) when convoluted with the distribution amplitudes, due to their Dirac structures [8]. In the rest of this section, we will always use this simplification if possible. (1) (0) The correction of wave-function is ΦQ ⊗ Ta , which can be written as (1) (0) ΦQ ⊗ Ta
/γ p /γ mQp z z μ i PL − 1 1 − + NUV log 2pq¯ · pγ y y y 16π 2 π2 1 z 2 x 2 yμ − 2− + log 2 − log 1+ 3 4 zmQ y μ /γ mQp z x x + uQ − 1 log 2 − log 2 − 2 y y μ μ
= −2ieq CF gs2 v¯q¯ ε/
and
i z mQ μ (1) (0) ΦQ ⊗ Ta⊥ = −2ieq CF gs2 v ¯ P log + 1 NUV − q¯ L y y 16π 2 mQ y x x − log 2 − 2 log − log 2 + 4 ε/uQ y z μ μ
(28)
(29)
With Eq. (6) and Eq. (12), the hard-scattering kernel can be obtained through the following equation (30) Φ (0) ⊗ Ta(1)weak = Fa(1)weak − Φ (1) ⊗ Ta(0) The correction in Fig. 3c is 2pγ · pq¯ CF αs (0) Fa(1)wfc = − Fa NUV − log + 1 4π μ2
(31)
We find that the correction of the wave-function also vanishes as Ref. [8], i.e. (1)
ΦWfc ⊗ Ta(0) = 0
(32)
Then we obtain T
wfc(1)
2pγ · kq¯ αs CF (0) (0) = −1 Ta + Ta⊥ log 4π μ2
(33)
The corrections of the external legs and the box diagram are equal to the relevant corrections to the wave-functions because we use the renormalization scale equal to the factorization scale, so (1)extq
Ta
= Ta(1)extQ = Ta(1)box = 0
(34)
786
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
Fig. 4. 1-loop QCD correction of Fb . (0)
5.2. 1-loop correction of Tb
The corrections of Tb are also order O(ΛQCD /mQ ) contributions as the tree level, the Feynman diagrams of the 1-loop corrections of Tb are shown in Fig. 4. The correction of the weak vertex is y x x x2 (1)weak 2 i Fb = −ieQ CF gs v ¯ − log − 2 + log + N q ¯ UV y − x (x − y)2 y 16π 2 μ2
y x x /γ 2wyy2 wy log y (3 log x−y + log x−y ) μ p + + PL 2 w−z y (w − z) y y x 2yy2 (w − z) + y(log x (3 log x−y + log x−y )) μ + 2PL p /q (35) ε/uQ 2(w − z)2 with y2 =
3 y −w+z 1 x x log log 2(w − z) w+x −y −z x −y +w−z y −w+z
and the correction of the wave-function is 1 (0) (1) Φq ⊗ Tb ∼ O m2Q
(36)
(37)
the hard kernel can be obtained using Φ (0) ⊗ Tbweak(1) = Fbweak(1) The results of the EM vertex is (1)EM Fb
/ γ ε/ i / −2y x x p μ mQ ε P log + 2NUV − log 2 y x −y y y 16π 2 L μ x π2 6x − y x y − −2 Li2 1 − + − log uQ y x 3 x −y y
= −ieQ CF gs2 v¯q¯
(38)
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
787
/ γ ε/ i x μp v ¯ P − log + 2 u N q ¯ Q UV L y 16π 2 μ2 2 m2Q m2Q 2pγ · kQ π2 (1)EM (0) CF gs = Tb log 2 − 4 − −2 Li2 1 − + Tb 4π 2pγ · kQ 3 μ m2Q 2 2 6mQ − 2pγ · kQ mQ − 2 log 2pγ · kQ mQ − 2pγ · kQ (1)
(0)
ΦQ ⊗ Tb
= −2ieQ CF gs2
+ ieQ CF gs2
μ m2Q PL ε/mQ 4pγ · kQ i v ¯ u × log q ¯ Q 2pγ · kQ 16π 2 2pγ · kQ m2Q − 2pγ · kQ
(39)
The results in Fig. 4c is (1)wfc
Fb
i x x x y(2x − y) μ = −ieQ Cf gs2 v ¯ P log + −NUV − 1 + log 2 − q¯ L (x − y) y 16π 2 μ (x − y)2 p /γ 2x x x x (2x − 3y)y + log −3NUV − 5 + 3 log 2 + + ε/uQ y x −y y y μ (x − y)2 x x x mQ ε/ (2x − 3y)y + −3NUV − 5 + 3 log 2 + log + u Q x −y y y μ (x − y)2
(1)
(0)
ΦWfc ⊗ Tb
=0 C α x x s F (0) (1)wfc = Tb Tb −1 + log 2 − 4π x−y μ 2x x x + −5 + 3 log 2 + + y x −y μ i μ − ieQ CF gs2 v¯q¯ PL ε/uQ 16π 2 mQ x x × −5 + 3 log 2 + y x −y μ
x y(2x − y) log 2 y (x − y) x (2x − 3y)y log y (x − y)2 +
x y(2x − 3y) log + y (x − y)2
(40) pQ →kQ
And the correction of the external legs and the box correction are also equal to each other, so we also have (1)extq
Tb
(1)extQ
= Tb
(1)box
= Tb
=0
(41)
(0)
5.3. 1-loop correction of Tc
The correction of the distribution function with gluons in Wilson line does not have correspondent 1-loop QCD corrections. And because the momentum of light quark and heavy quark (0) show up together as kQ + kq¯ in Tc , we find ∂ ∂ − (42) T (0) = 0 ∂kQ ∂kq¯ c and we obtain (1)q
Tc
= Tc(1)Q = Tc(1)wfc = 0
(43)
788
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
All the other corrections are similar as Eq. (34) and Eq. (41), we find (1)extq
Tc
(1)trangle
= Tc(1)extQ = Tc
=0
(44)
5.4. 1-loop result summery With Eqs. (25), (30), (33), (34), (38)–(41), (43) and (44), we can establish the order αs hardscattering kernel. We find that, T (1) is IR-finite up to order O(αs ΛQCD /mQ ), so the factorization is proved up to the order of O(1/mQ ) corrections. It is well known that, the factorization will fail with the light mesons. However, we find that the ΛQCD /mQ expansion is irrelevant by briefly investigating the factorization at higher orders of O(ΛQCD /mQ ). In the calculations above, we find that the IR divergences in the corrections of the external legs and the box diagrams are cancelled exactly, while the remaining IR divergences are all collinear divergences which show up in the amplitudes with one of the vertexes of the gluon propagator connecting to the external light quark, which are denoted in Fig. 3a and Fig. 4a. In Ta(1)EM , there are no neglected O(ΛQCD /mQ )2 or higher order contributions. As a result, (1)weak . We also find that, there are both the IR divergences at higher orders only survive in Tb (1)weak (1) (0) higher order IR divergences in Fb and Φq ⊗ Tb . We shall investigate whether those IR divergences can be cancelled. We concentrate on the IR region of the loop integration, using l → 0, we find weak(1)
Fb IR
l
= lim −ieQ CF gs l →0
×
ρ /Q 2pQp
−l ρ + 2pγ p /γ
−/ pq¯ dd l 1 μ v¯q γρ P (2π)d l 2 (l + pq¯ )2 − m2q L ρ
ρ
− 2pQp / γ − 2pγ p / Q + 2γ ρ pQ · pγ + 2(pQ − pγ )ρ mQ (2pQ · pγ )2
ε/uQ (45)
(0)
We also bring back the neglected higher order terms in the numerator of Tb , and the correction of the wave-function is k/Q − p / γ + mQ ε/ (kQ − pγ )2 − m2Q l (1) p / q¯ dd l 1 μ (0) 2 Φq ⊗ T = lim −ieQ CF gs v¯q γ ρ P IR l →0 (2π)d l 2 (l + pq¯ )2 − m2q L (0)
Tb
μ
= eQ PL
−l
pQ − p / γ + mQ ) −γρ 2pQ · pγ − 2(pQ − pγ )ρ (/ × ε/uQ (2pQ · pγ )2
(46)
we find (1) Fb IR − Φ (1) ⊗ T (0) IR = 0
(47)
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
789
This result indicates that, the factorization is valid at any order of O(ΛQCD /mQ ), as a result, the valid region of factorization in Ref. [8] is extended. The failure of the factorization approach in the light meson decays is due to the bad convergence behaviour in the O(αs ) expansion. We then concentrate on the contribution of the hard scattering kernel to the amplitude. The amplitude can be obtained by replacing the wave-function with the one obtained in Ref. [16] 1 1 (0) d 3 kΨ (k) √ M|0 Φ (kq , kQ ) = √ 3 2 3 3 2 + m2 δ k 2 + m2
(k Q − k)δ
kq0 − k − k (48) × δ (kq¯ + k)δ ¯ Q0 q Q with M=
i+
↓) − bi+ (k,
↓)dqi+ (−k,
↑), bQ (k, ↑)dqi+ (−k, Q
i
= 4π mP λ3 e−λP |k| Ψ (k) P
(49)
The matrix element can be written as F μ (μ) =
d 4 kq¯ Ψ (k) Tr M · T (0)μ (kq¯ , kQ ) + T (1)μ (kq¯ , kQ ) 2 2 2 2
3 (k Q − k)δ
kq0 (50) × δ 3 (k q¯ + k)δ ¯ − k + mq δ kQ0 − k + mQ
3 1 √ 3 (2π) 6
d 3k
d 4 kQ
After convolution with the wave-functions of the heavy mesons, some of the terms will have identical contributions to the matrix element up to order O(ΛQCD/mQ ) due to their Dirac structures. As a result, we find that the F μ can be simplified as a function of four different types of Dirac structures, and can be written as 1 3 μ d 3 kΨ (k) Tr Cn (pQ , pq¯ , μ)M · Kn (pQ , pq¯ ) F (μ) = (51) √ 3 (2π) 6 n
and pQ = ( m2 + k 2 , k)
denote the on-shell momenta of the light with pq = ( m2q + k 2 , −k) Q anti-quark and the heavy quark in the bound state. And the Kn are defined as (0)
K2 (kQ , kq¯ ) = Ta⊥
(0)
K4 (kQ , kq¯ ) = eQ
K1 (kQ , kq¯ ) = Ta ,
(0) μ
K3 (kQ , kq¯ ) = Tb ,
PL ε/mQ 2pγ · kQ
(52)
Except for C1 K1 , all the other products are contribution of order O(ΛQCD /mQ ), for clarity, we define C1 = C10 + C11 , with C1m represents order O(ΛQCD /mQ )m contribution, the coefficients are αs CF xz x y x 2π 2 C10 (pq , pQ , μ) = 1 + − log 2 + y1 − 2 log 2 + log − 4 + 4π x −y y 3 μ y y x z y x (53) + 2 log2 − 2 log log 2 + 2 log 2 + 2 log 2 − 5 z z μ μ μ
790
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
C11 (pq , pQ , μ) =
w x z xz 2 − 2 + xy 5 log + 8 log + 5 −4 log x y y (x − y)2 y y2 3 xz y x z y 2 + y 2 log + 4 log − 3 + −3y1 + 8 log 2 − 4 x z (x − y)2 y 2 y 2 y y x z + 7y1 + 13 log + 19 log + 14 x z (x − y)2 y xz x z + −5y1 + 4 log + 14 log − 17 y y (x − y)2 yz z x + − 3 log + 2 log + 7 y 1 y y (x − y)2 w x 2w 4xz xz − log + − 2 log 2 2(x − y) y y y y 2w 4xz y y x x + − 2 log 2 log − 2 log − log 2 + 4 y z z y μ μ 2 y x 2z y x π − log2 + log log 2 + log 2 (54) − 2− 3 z z y μ μ αs CF xz x y xz xz − log 2 + y1 − 2 log 2 + log + log 2 4π y yw μ y y 4zx y x y x − log 2 log − 2 log − log 2 + 4 yw z z μ μ
C2 (pq , pQ , μ) = 1 +
C3 (pq , pQ , μ)
2x x π2 6x − y x y log 2 − 2 − − Li2 1 − + − log y x 3 x −y y μ x x x y(2x − y) − 1 + log 2 − log + x −y y μ (x − y)2 2x x x x x (2x − 3y)y y + log −5 + 3 log 2 + + − log 2 − 2 + y x−y y y −x μ (x − y)2 μ y x x x 2wyy2 wy log y (3 log x−y + log x−y ) x2 log + + + y w−z (x − y)2 (w − z)2
αs CF =1+ 4π
(55)
C4 (pq , pQ , μ) 2y y(w − z) + log y (3y log x + y log y ) 2 αs C F 2xz2 x x−y x−y = − 2z 4π yw (w − z)2 2y x x x x y(2x − 3y) − log log + −5 + 3 log 2 + + x −y y x−y y μ (x − y)2 with x, y, z and w defined in Eq. (18), and y1 , y2 defined in Eq. (27) and Eq. (36).
(56)
(57)
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
791
6. Large logarithm resummation In the expression of F μ , large logarithms show up. Those large logarithms need to be resummed so that the result is phenomenologically reliable.We concentrate on the large logarithms at order O(ΛQCD /mQ )0 , because terms like (ΛQCD log
mQ ΛQCD )/mQ mQ
are suppressed and not
large. In this section, we use y = 2pγ · pQ = 2Eγ mQ for simplicity. We concentrate on C10 , which can also be written as C10 (pq , pQ , μ) x y y y y αs (μ)CF −2 Li2 1 − − 2 log2 − log + 2 log =1+ 4π x y x −y x x
x μ2 x π2 x π2 x log 2 + 3 log 2 − log2 2 − 6 − + log2 −3− (58) y 12 z 4 μ μ μ When μ ∼ mQ , the large logarithms are the same as Ref. [8]. However, when μ is set to μ ∼ mQ ΛQCD , the large logarithms are different from Ref. [8] by + 2 log
mQ 1 1 mQ − log2 2 + log 2 (59) 2 2 μ μ This is because the corrections of wave-functions in this work are not calculated in HQET framework as Ref. [8], in which mQ will not appear in the result. 6.1. Renormalization group equation (RGE) evolution We start with the RGE at order O(ΛQCD /mQ )0 , and use the method introduced in Ref. [13]. The RGE of C10 is μ
∂ 0 C (μ) = γ (μ)C10 (μ) ∂μ 1
(60)
with γ (μ) defined as anomalous dimension of C10 , which can be obtained by using the counter terms of C10 K1 . The counter terms at the order O(ΛQCD /mQ )0 can be written as αs (μ)CF 2 z Z= 2 log − 3 (61) 4π
y We can use the counter terms and the β function in QCD to calculate γ (μ) 3
∂ ∂ −1 γ (μ) = Z β = −g + O g , +β Z μ 2 ∂μ ∂g
(62)
The result is
αs (μ)CF μ2 z γ (μ) = − 2 log + 2 log 2 − 3 2π y μ
(63)
With this result, we can solve the RGE of the coefficient. Assume the coefficient can be written as a hard function multiplied by a jet function [9,21], so C(μ) = H (μ)J (μ), γ (μ) = γH (μ) + γJ (μ) ∂ ∂ μ H (μ) J (μ) + H (μ) μ J (μ) = γH (μ)H (μ)J (μ) + γJ (μ)H (μ)J (μ) (64) ∂μ ∂μ
792
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
The RGE of the hard function is ∂ μ H (μ) = γH (μ)H (μ) ∂μ
(65)
(0)
By splitting the coefficient C1 into H and J , we assume that the natural scale of H is mQ , while the natural scale of J is mQ ΛQCD , which is also the case for γ (μ) with a log(μ2 /y) term and a log(μ2 /z) term. So we split the γ (μ) such that γH (μ) is the sum of the log(μ2 /y) term and an undetermined constant n , so αs (μ)CF μ2
γH (μ) = − 2 log +n (66) 2π y Similar as Refs. [8,14], we find αs (μ)CF μ γHLO = − 2 log π mQ 2Eγ αs (μ)CF
α 2 (μ) μ γHNLO = − (67) n − 2 log − 2CF B s 2 log 2π mQ mQ (2π) The γHNLO is the same as Refs. [8,14], so the solution is f0 H (μ) = exp + f1 H (mQ ) αs (mQ ) 4πCF 1 f0 = αs (mQ ) −2 2 − 1 + log r β0 αs (mQ ) r CF β1 CF
y 1 2 f1 = − 3 1 − r + r log r − log r + log r n − 2 log 2 β0 x β0 −
2CF B (r − 1 − log r) β02 2N
(68) 34C 2
10C N
(μ) f A f A A with r = ααs s(m , β0 = 11C − 2CF Nf , where CA = 3 for QCD, 3 − 3 and β1 = 3 − 3 Q) Nf is the number of the flavour of quarks taken into account, and B can only been derived from 2-loop calculations. In Ref. [14], by comparing the result with B → Xs γ and B → Xu l v¯ in 5Nf π2 Ref. [22], B is found to be B = CA ( 67 18 − 6 ) − 9 . So the result of H is αs (mQ )Cf μ y μ μ H (μ) = H (mQ ) exp + 4 log log − 2n log −4 log2 4π mQ x mQ mQ 2 (69) + O αs
6.2. The resummation The hard function can be derived by using the method introduced in Ref. [9]. With xγ defined as xγ = 2Eγ /mQ , the result is 2Eγ ˆ H μ 2Eγ αs (μ)CF 1 2 2Eγ =1+ log xγ + 2 log −2 Li2 (1 − xγ ) − 2 log − (70) 4π μ 1 − xγ μ
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
and H (mQ ) = Hˆ
2Eγ mQ
793
(71)
we also find SCET H (mQ ) = C3,6 (mQ )
(72)
n
With = 3, the evaluation of the RGE will correctly resum the large logarithms, which can be shown explicitly by expanding the solution of RGE of hard function μ αs (mQ )CF y μ μ H (μ) = H (mQ ) 1 + + 4 log log − 6 log −4 log2 4π mQ x mQ mQ 2 + O αs x y y y y αs (mQ )CF −2 Li2 1 − − 2 log2 − log + 2 log =1+ 4π x y x−y x x x π2 x x 2 x (73) + O αs2 + 2 log log 2 + 3 log 2 − log 2 − 6 − y 12 μ μ μ Comparing Eqs. (60) and (64) with Eq. (73), we find that there are no more large logarithms in the remaining terms, which gives the jet function 2 αs (μ)CF π2 2 μ J (μ) = 1 + log −3− (74) 4π z 4 The γJ (μ) can been obtained by subtracting γH (μ) from γ (μ) γJ (μ) =
αs (μ)CF μ 2 log √ π z
The solution of RGE of jet function is √ 2 2 √ αs ( z)CF 2 μ J (μ) = J ( z) × exp log 2 + O αs 4π z
(75)
(76)
We find that, Eq. (76) can also correctly resum the large logarithms which will show up in jet function when μ is evaluated to lower than ΛQCD mQ . When μ > ΛQCD mQ , the resummed result at order O(αs (ΛQCD /mQ )0 ) is y αs (mQ )CF y 0 C1 = 1 + −2 Li2 1 − − 2 log2 4π x x 2 y y π y log + 2 log − 6 − − x −y x x 12 αs (mQ )Cf μ y μ μ 2 × exp + 4 log log − 6 log −4 log 4π mQ x mQ mQ 2 2 αs (μ)Cf μ π × 1+ log2 −3− (77) 4π z 4 √ At the leading order of O(ΛQCD /mQ ), z can be written as z = 2pγ · kq¯ = 2k+ Eγ . Using m ΛQCD = 200 MeV, mb = 4.98 GeV, and k+ = ΛQCD , Eγ = 4Q , we can show the evaluation of coefficient C10 in Fig. 5.
794
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
m
Fig. 5. C10 as function of μ, with Eγ = 4Q , k+ = ΛQCD . The solid line is the resummed result, and the dotted line is the un-resummed one.
7. Numerical applications The amplitude is derived in Eq. (50), which can be decomposed as [6,23] ρ γ |qΓ ¯ μ Q|P = μνρσ ε ν pP pγσ FV + i εμ pP · pγ − pγμ ε · pP FA
(78)
The contribution in Fig. 1c depends on not only Eγ but also on pν and pl . For simplicity, we treat this term separately, the form factors is written as μ ρ μ γ |qΓ ¯ μ Q|P + Fc pP = μνρσ ε ν pP pγσ FV + i εμ pP · pγ − pγμ ε · pP FA + Fc pP 1 3 1 1 d 3 kΨ (k) FV = √ 3 (2π) 2 pq0 pQ0 (pq0 + mq )(pQ0 + mQ ) mP Eγ 6 0 × 2eq C1 mQ − C1 pq0 eq + eQ 2pq0 C3 1 3 1 1 d 3 kΨ (k) FA = √ 3 (2π) 2 pq0 pQ0 (pq0 + mq )(pQ0 + mQ ) mP Eγ 6 2pq0 mQ 2pq0 mQ × 2eq C1 mQ − C10 pq0 eq − eq C2 − eQ 2pq0 C3 + eQ C4 (79) Eγ Eγ The relation FA = FV at leading order is explicitly broken at order O(ΛQCD /mQ ) as expected [24]. We evaluate this integral using [16] mD = 1.9 GeV,
mB = 5.1 GeV,
mb = 4.98 GeV,
mc = 1.54 GeV
ΛQCD = 200 MeV,
mu = md = 0.08 GeV
λB = 2.8 GeV−1 ,
λD = 3.4 GeV−1
(80)
The result of FA,V of B → γ eνe is shown in Fig. 6, the O(ΛQCD /Eγ ) contribution is more important at the region Eγ → 0 which is clearly shown in the figure. The numerical results of FA,V are inconvenient to use when calculate the decay widths. For simplicity, we use some simple forms to fit the numerical results. For the form factors at the order O(αs (ΛQCD /mQ )0 ), we use the single-pole form FAαs (Eγ ) = FVαs (Eγ ) =
f (0) f (0) = q 2 − m∗2 m2P − m∗2 − 2mP Eγ
(81)
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
795
Fig. 6. Form factors of B → γ eνe as functions of Eγ . The results are presented as the ratios of the form factors at 1-loop order to the one at the tree level and the leading order of the ΛQCD /mQ expansion, which is denoted as F0 . The solid line is FA /F0 = FV /F0 , where FA,V are at the order of O(αs (ΛQCD /mQ )0 ). The dotted line is FV /F0 and the dashed line is FA /F0 , both at the order of O(αs ΛQCD /mQ ). Table 1 The results of the parameters in the form factors in Eqs. (81) and (82). B → γ eνe D → γ eνe
m∗ (GeV)
f (0) (GeV)
AV
BV
AA
BA
5.37 1.98
−0.63 −0.15
0.27 −0.00095
0.45 −0.54
0.32 −0.27
−0.67 −0.059
where q = pP − pγ . While up to the order O(αs ΛQCD /mQ ), inspired by the form factors in Ref. [17] the form factors are fitted as αs ΛQCD ΛQCD 2 mQ FA,V (Eγ ) = AA,V + BA,V (82) Eγ Eγ The predicted results for the form factors are more reliable at the region Eγ ΛQCD because we have neglected the higher order terms of ΛQCD /Eγ in the numerical calculation, so we choose the region Eγ > 2ΛQCD to fit the parameters in Eqs. (81) and (82). The fitting is given in Table 1 and shown in Figs. 7 and 8. On the other hand, Fc can be related to the decay constant [16,17] by μ
μ
Fc pP = ie
pl · εPL μ pl · ε 0|qγ ¯ μ (1 − γ5 )Q|P = −iefP pP pl · p γ pl · p γ
(83)
Using the fitted result of FA , FV , the result of Fc , and using the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements [25,26] Vcd = 0.226,
Vub = 0.0047
(84)
we obtain the result for the branching ratios. There are IR divergences in the radiative leptonic decays in the case that the photon is soft or the photon is collinear with the emitted lepton. The-
796
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
Fig. 7. Fit of the form factors of B → γ eνe . The solid line is for the result at the leading order of ΛQCD /mQ . The ‘×’ points and the dotted line are for FV while the dashed line and the ‘+’ points are for FA .
Fig. 8. Fit of the form factors of D → γ eνe . The solid line is for the result at the leading order of ΛQCD /mQ . The ‘×’ points and the dotted line are for FV while the dashed line and the ‘+’ points are for FA .
oretically this IR divergences can be canceled by adding the decay rate of the radiative leptonic decay with the pure leptonic decay rate, in which one-loop correction is included [27]. The radiative leptonic decay cannot be distinguished from the pure leptonic decay in experiment when the photon energy is smaller than the experimental resolution to the photon energy. So the decay rate of the radiative leptonic decay depend on the experimental resolution to the photon energy Eγ which is denoted by Eγ . The dependence of the branching ratios on the resolution are listed in Table 2. Using Eγ = 10 MeV [28], the branching ratios are given in Table 3. We find that, in general, the 1-loop results are smaller than the tree level results. The 1-loop correction is found to be
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
797
Table 2 The branching ratios with different photon resolution Eγ . Eγ
BR(B → eνe γ )
BR(D → eνe γ )
Eγ
BR(B → eνe γ )
BR(D → eνe γ )
5 MeV 10 MeV 15 MeV
1.77 × 10−6 1.66 × 10−6 1.60 × 10−6
3.10 × 10−5 2.81 × 10−5 2.64 × 10−5
20 MeV 25 MeV 30 MeV
1.56 × 10−6 1.53 × 10−6 1.48 × 10−6
2.53 × 10−5 2.45 × 10−5 2.38 × 10−5
Table 3 The branching ratios of the decay modes. BR
O((
B → eνe γ D → eνe γ
αs ΛQCD 0 mQ ) )
2.38 × 10−6 9.62 × 10−6
BR
ΛQCD O(αs ( m )0 ) Q
1.01 × 10−6 2.71 × 10−6
BR
O(
αs ΛQCD mQ )
1.66 × 10−6 2.81 × 10−5
important. For B meson, the correction to the decay amplitude at the order O(ΛQCD/mQ )0 is about 10% to 30% due to the large logarithms. The contribution of the order O(αs ΛQCD /mQ ) are generally not negligible. For B meson, the correction of the order O(ΛQCD /mQ ) contribution to the decay amplitude can be as large as 30%. For D mesons, the mass of c quark is not large enough, the order O(ΛQCD /mQ ) contributions is much more important, it is necessary to include higher order corrections in ΛQCD/mQ expansion. 8. Conclusion In this paper, we study the factorization of the radiative leptonic decays of B − and D − mesons. Compared with the work in Ref. [8], the factorization is extended to include the O(ΛQCD /mQ ) contributions, and the transverse momentum is also considered. The factorization is proved explicitly at 1-loop order, the valid region of the factorization is extended. The hard kernel at order O(αs ΛQCD /mQ ) is obtained. We use the wave function obtained in Ref. [16] to derive the numerical results. The branching ratios of B − → γ eν¯ is found to be at the order of 10−6 , which is close to the previous works [7,17,29–31]. In the previous works, the results of D mesons are different from each other from 10−3 to 10−6 , our results agree with 10−5 . We also find that the O(ΛQCD /mQ ) contribution is very important even for B meson, the correction to the decay amplitude is about 20%–30%, which can affect the branching ratios about 50%. This is because of the importance of O(ΛQCD /Eγ ) contributions. In previous works, O(ΛQCD /Eγ ) contributions is neglected. For a typical region, Eγ ∼ mQ /4, which is also the leading region of the phase space of the tree level, the neglected contributions can be up to 20%. As a result, the correction to the branching ratios can be up to 40% at tree level. Acknowledgements This work is supported in part by the National Natural Science Foundation of China under contracts Nos. 11375088, 10975077, 10735080, 11125525.
798
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
Appendix A. IBP reductant relation of wave function The integral is
Φq(1) (kq¯ , kQ ) =
4
4
d x
d ye
× igs
ikq¯ ·x ikQ ·y
e
x 0|q¯q¯ (x)igs
dzzμ Aμ (z)Q(y) y
d 4 x2 q¯q¯ (x2 )/ A(x2 )qq¯ (x2 )q¯ S (pq¯ ), Qs (pQ )
(A.1)
After variable substitution using z = x + αy, we find x
1 dzzμ A (z) = μ
y
d y + α(x − y) μ Aμ y + α(x − y)
0
1 =
dα(x − y)μ Aμ y + α(x − y)
(A.2)
0
After the contraction and then integral over x2 , p, the result is Φq(1) ⊗ T (0) = −CF gs2
1
4
d x
4
d y
dα
dd l (2π)d
0 ˙
˙
× eiy (kQ +αl−pQ ) eix (kq¯ −αl−pq¯ ) v¯q¯ γ · (x − y)
/ q¯ + mq¯ ) (0) 1 (−/l − p T uQ 2 l (l + pq¯ )2 − m2q¯
(A.3)
The integral over x and kq¯ can be worked out term by term. However, we find IBP reduction relation [18] an elegant way to do so. Consider this integral d 4 kq Γ (kq ) d 4 xe−i(pq¯ +αl−kq¯ )·x (A.4) (2π)4 It is unchanged when kq is shifted, so under the infinitesimal transformation kq → kq + βK The integral transforms as Γ (kq ) d 4 xe−i(pq¯ +αl−kq¯ )·x ∂ 4 −i(pq¯ +αl−kq¯ )·x → βK · Γ (kq ) d xe ∂kq The Lie algebra leads to d 4 kq ∂ 4 −i(pq¯ +αl−kq¯ )·x K· Γ (kq ) d xe =0 ∂kq (2π)4
(A.5)
(A.6)
(A.7)
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
so that
d 4 kq Γ (kq )K · x (2π)4
d 4 xe−i(pq¯ +αl−kq¯ )·x = iK ·
∂Γ (kq ) ∂kq kq =pq¯ +αl
799
(A.8)
So after integrate over x, kq¯ , y and kQ , the result is Φq(1) ⊗ T (0) = igs2 CF
(/l + p / q¯ − mq¯ ) dd l 1 v¯q¯ γ ρ d 2 (2π) l (l + pq¯ )2 − m2q¯
k = pq¯ + αl,
K = pQ − αl
1 dα 0
∂T (0) ∂T (0) − uQ ρ ρ ∂kq ∂kQ kq =k ,kQ =K
(A.9)
References [1] P.F. Harrison, H.R. Quinn, BaBar Collaboration, The BaBar physics book: physics at an asymmetric B factory, SLAC-R-0504; J. Nash, Int. J. Mod. Phys. A 17 (2002) 2982–2998, arXiv:hep-ex/0201002; H. Tajima, Int. J. Mod. Phys. A 17 (2002) 2967–2981, arXiv:hep-ex/0111037. [2] M. Artuso, E. Barberio, S. Stone, PMC Phys. A 3 (2009) 3, arXiv:0902.3743. [3] M. Neubert, Int. J. Mod. Phys. A 11 (1996) 4173–4240, arXiv:hep-ph/9604412. [4] R.P. Feynman, Photon–Hadron Interactions, Benjamin, Reading, MA, 1972. [5] J.C. Collins, D.E. Soper, G. Sterman, Adv. Ser. Dir. High Energy Phys. 5 (1988) 1–91, arXiv:hep-ph/0409313. [6] M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Phys. Rev. Lett. 83 (1999) 1914–1917, arXiv:hep-ph/9905312; M. Beneke, G. Buchalla, M. Neuber, C.T. Sachrajda, Nucl. Phys. B 591 (2000) 313–418, arXiv:hep-ph/0006124; M. Beneke, G. Buchalla, M. Neubert, C.T. Sachrajda, Nucl. Phys. B 606 (2001) 245–321, arXiv:hep-ph/0104110v1. [7] G.P. Korchemsky, D. Pirjol, T.-M. Yan, Phys. Rev. D 61 (2000) 114510, arXiv:hep-ph/9911427v3. [8] S. Descotes-Genon, C.T. Sachrajda, Nucl. Phys. B 650 (2003) 356–390, arXiv:hep-ph/0209216v2. [9] S.W. Bosch, R.J. Hill, B.O. Lange, M. Neubert, Phys. Rev. D 67 (2003) 094014. [10] N. Isgur, M.B. Wise, Phys. Lett. B 232 (1989) 113; H. Georgi, Phys. Lett. B 240 (1990) 447. [11] R.J. Hill, M. Neubert, Nucl. Phys. B 657 (2003) 229–256, arXiv:hep-ph/0211018. [12] E. Lunghi, D. Pirjol, D. Wyler, Nucl. Phys. B 649 (2003) 349–364. [13] C.W. Bauer, S. Fleming, M. Luke, Phys. Rev. D 63 (2000) 014006. [14] C.W. Bauer, S. Fleming, D. Pirjol, I.W. Stewart, Phys. Rev. D 63 (2001) 114020. [15] K.G. Wilson, Phys. Rev. D 10 (1974) 2445. [16] M.-Z. Yang, Eur. Phys. J. C 72 (2012) 1880. [17] J.-C. Yang, M.-Z. Yang, Mod. Phys. Lett. A 27 (2012) 1250120, arXiv:1204.2383. [18] A.G. Grozin, Int. J. Mod. Phys. A 26 (2011) 2807–2854, arXiv:1104.3993v4. [19] W.A. Bardeen, A.J. Buras, D.W. Duke, T. Muta, Phys. Rev. D 18 (1978) 3998. [20] G. ’t Hooft, M. Veltman, Nucl. Phys. B 153 (1979) 365–401; J.C. Romão, Modern techniques for one-loop calculations (unpublished), which can be found at http://www. researchgate.net/publication/228530611_Modern_Techniques_for_One_Loop_Caculations. [21] T.T. Jouttenus, Phys. Rev. D 81 (2010) 094017. [22] G.P. Korchemsky, G. Sterman, Phys. Lett. B 340 (1994) 96; R. Akhoury, I.Z. Rothstein, Phys. Rev. D 54 (1996) 2349. [23] M. Wirbel, B. Stech, M. Bauer, Z. Phys. C, Part. Fields 29 (1985) 637–642; D. Fakirov, B. Stech, Nucl. Phys. B 133 (1978) 315. [24] F. Hussain, G. Thompson, arXiv:hep-ph/9502241. [25] K. Nakamura, et al., Particle Data Group, J. Phys. G 37 (2010) 075021, and 2011 partial update for the 2012 edition. [26] A.G. Akeroyd, F. Mahmoudi, J. High Energy Phys. 1010 (2010) 038; T. Konstandin, T. Ohlsson, Phys. Lett. B 634 (2006) 267–271.
800
[27] [28] [29] [30] [31]
J.-C. Yang, M.-Z. Yang / Nuclear Physics B 889 (2014) 778–800
C.-H. Chang, et al., Phys. Rev. D 60 (1999) 114013. K. Aamodt, et al., ALICE Collaboration, J. Instrum. 3 (2008) S08002. D. Atwood, G. Eilam, A. Soni, Mod. Phys. Lett. A 11 (1996) 1061. C.-Q. Geng, C.-C. Lih, W.-M. Zhang, Mod. Phys. Lett. A 25 (2000) 2087. C.-D. Lu, G.-L. Song, Phys. Lett. B 562 (2003) 75.