The semileptonic decays B/B_{s}\to(\pi,K)(l^{+}l^{-},l\nu,\nu\bar{\nu}) in the perturbative QCD approach beyond the leading-order

# The semileptonic decays B/Bs→(π,K)(l+l−,lν,ν¯ν) in the perturbative QCD approach beyond the leading-order

Wen-Fei Wang, and Zhen-Jun Xiao111Email Address: xiaozhenjun@njnu.edu.cn Department of Physics and Institute of Theoretical Physics,
Nanjing Normal University, Nanjing, Jiangsu 210046, People’s Republic of China
July 16, 2019
###### Abstract

In this paper we first calculate the form factors of and transitions by employing the perturbative QCD (pQCD) factorization approach with the inclusion of the next-to-leading-order(NLO) corrections, and then we calculate the branching ratios of the corresponding semileptonic decays (here denotes and ). Based on the numerical calculations and phenomenological analysis, we found the following results: (a) For and transition form factors , the NLO pQCD predictions for the values of and their -dependence agree well with those obtained from other methods; (b) For and decay modes, the NLO pQCD predictions for their branching ratios agree very well with the measured values; (c) By comparing the pQCD predictions for with the measured decay rate we extract out the magnitude of : ; (d) We also defined several ratios of the branching ratios, and , and presented the corresponding pQCD predictions, which will be tested by LHCb and the forthcoming Super-B experiments.

###### pacs:
13.20.He, 12.38.Bx, 14.40.Nd

## I Introduction

The semileptonic decays and with are very interesting decays modes and playing an important role in testing the standard model (SM) and in searching for the new physics (NP) beyond the SM, such as the determination of and the extractions of the transition form factors of meson to pion and/or kaon. For the charged current decays, the ”Tree” diagrams provide the leading order contribution. For the neutral current and decays, however, the leading order SM contributions come from the photon penguin, the Z penguin and the box diagrams, as shown in Fig. 1, where the symbol denotes the corresponding one-loop SM contributions.

On the experiment side, some decay modes among the all considered decays have been measured by CLEO, BaBar and Belle experiments babar-83-032007 (); cleo-99-041802 (); belle-648-139 (); babar-1204-3933 (); babar-82-112002 (). The decays are now under studying and will be measured by the LHCb and the forthcoming Super-B experiments D-2012 (); Buras-2012 ().

On the theory side, the considered semileptonic decays strongly depend on the values and the shape of the form factors. At present, there are various approaches to calculate the transition form factors, such as the lattice QCD technique HPQCD-2006 (), the light cone QCD sum rules (LCSRs) pball-98 (); pball-05 (); jhep04-014 (); prd83-094031 (), as well as the perturbative QCD (pQCD) factorization approach li-65-014007 (); yang-npb642 (); yang-epjc23-28 (); lu-79-014013 (); huang-71 (). The direct perturbative calculations of the one-gluon exchange diagram for the meson transition form factors suffer from the end-point singularities. Because of these end-point singularities, it was claimed that the transition form factors is not calculable perturbatively in QCD huang-ref-17 ().

In the pQCD factorization approach li-papers (), however, a form factor is generally written as the convolution of a hard amplitude with initial-state and final-state hadron distribution amplitudes. In fact, in the endpoint region the parton transverse momenta is not negligible. If the large double logarithmic term and large logarithms are resummed to all orders, the relevant Sudakov form factors from both the resummation and the threshold resummationpap-resum (); li-resum () can cure the endpoint singularity which makes the perturbative calculation of the hard amplitudes infrared safe, and then the main contribution comes from the perturbative regions.

In Refs. li-65-014007 (); lu-79-014013 (), for example, the authors calculated the li-65-014007 () and form factors lu-79-014013 () at the leading order by employing the pQCD factorization approach and found that the values of the corresponding form factors coming from the pQCD factorization approach agree well with those obtained by using other methods. In a recent paperli-85074004 (), Li, Shen and Wang calculated the next-to-leading order (NLO) corrections to the transition form factors at leading twist in the factorization theorem. They found that the NLO corrections amount only up to of the form factors at the large recoil region of the pion.

In this paper, based on the assumption of the flavor symmetry, we first extend the NLO results about the form factors as presented in Ref. li-85074004 () to the cases of and directly, and then calculate the -dependence of the differential decay rates and the branching ratios of the considered semileptonic decay modes, and furthermore extract based on our calculations.

This paper is organized as follows. In Sec.II, we collect the distribution amplitudes of the mesons and the mesons being used in the calculation and give the -dependent NLO expressions of the corresponding transition form factors. In Sec.III, based on the factorization formulism, we calculate and present the expressions for the transition form factors in the large recoil regions. The numerical results and relevant discussions are given in Sec. IV. And Sec. V contains the conclusions and a short summary.

## Ii The theoretical framework and NLO corrections

For the sake of simplicity, we use denotes both the and meson and denotes final meson or from now on. As usual, we treat meson as a heavy-light system. In the meson rest frame, with the stands for the mass of meson, we define the meson momentum and the final meson (say or meson) momentum in the light-cone coordinates:

 p1=mB√2(1,1,0T),p2=mB√2η(0,1,0T), (1)

with the energy fraction carried by the final meson (here ). The light spectator momenta in the meson and in the final meson are parameterized as

 k1=(x1mB√2,0,k1T),k2=(0,x2ηmB√2,k2T). (2)

Because of the final pseudoscalar meson moving along the minus direction with large momentum, the plus component of its parton¡¯s momentum should be very small, so it’s dropped in the expression of . But the four components of should be of the same order, i.e. , with , being the quark mass. However, since is mainly in the minus direction with , the hard amplitudes will not depend on the minus component as explained below. This is the reason why we do not give in Eq.(2) explicitly.

In Ref. li-85074004 (), the authors derived the -dependent NLO hard kernel for the transition form factor. We here use their results directly for transition processes, and extend the expressions of form factors to the ones for both and transitions, under the assumption of flavor symmetry. As given in Eq.(56) of Ref. li-85074004 (), the NLO hard kernel can be written as

 H(1) = F(x1,x2,η,μf,μ,ζ1)H(0) (3) = αs(μf)CF4π[214lnμ2m2B−(lnm2Bζ21+132)lnμ2fm2B+716ln2(x1x2) +18ln2x1+14lnx1lnx2+(2lnm2Bζ21+78lnη−14)lnx1 +(78lnη−32)lnx2+(154−716lnη)lnη −12lnm2Bζ21(3lnm2Bζ21+2)+10148π2+21916]H(0).

where li-85074004 (), is the factorization scale and set to the hard scales or as defined in the Appendix, is the energy fraction carried by the final meson, and finally the renormalization scale is defined as li-85074004 ()

 ts(μf)={Exp[c1+(lnm2Bζ21+54)lnμ2fm2B]xc21xc32}2/21μf, (4)

with the coefficients

 c1 = −(154−716lnη)lnη+12lnm2Bζ21(3lnm2Bζ21+2)−10148π2−21916, c2 = −(2lnm2Bζ21+78lnη−14), c3 = −78lnη+32. (5)

In this paper, we use the same distribution amplitudes for meson and for the and meson as those used in Refs. li-85074004 (); xiao-85094003 (); pball-05 (); pball-0605004 ().

 ϕB(x,b) = NBx2(1−x)2exp[−12(xmBωb)2−ω2bb22], (6)

and

 ϕBs(x,b) = NBsx2(1−x)2exp[−12(xmBsωBs)2−ω2Bsb22], (7)

where the normalization factors are related to the decay constants through

 ∫10dxϕB(s)(x,b=0) = fB(s)2√6. (8)

Here the shape parameter has been fixed at  GeV by using the rich experimental data on the mesons with  GeV. Correspondingly, the normalization constant is . For meson, we adopt the shape parameter  GeV with  GeV, then the corresponding normalization constant is . In order to analyze the uncertainties of theoretical predictions induced by the inputs, we can vary the shape parameters and by 10%, i.e.,  GeV and  GeV, respectively.

For the and mesons, we adopt the same set of distribution amplitudes (the leading twist-2 ) and with as defined in Refs. pball-05 (); pball-0605004 (); pball-pi ()):

 ϕAi(x) = 3fi√6x(1−x)[1+a1C3/21(t)+a2C3/22(t)+a4C3/24(t)], (9) ϕPi(x) = fi2√6[1+(30η3−52ρ2i)C1/22(t)−3{η3ω3+920ρ2i(1+6a2)}C1/24(t)], (10) ϕσi(x) = fi2√6x(1−x)[1+(5η3−12η3ω3−720ρ2i−35ρ2ia2)C3/22(t)], (11)

where , are the mass ratios ( here GeV and GeV are the chiral mass of pion and kaon), are the Gegenbauer moments, while are the Gegenbauer polynomials

 C3/21(t) = 3t,C1/22(t)=12(3t2−1),C3/22(t)=32(5t2−1), C1/24(t) = 18(3−30t2+35t4),C3/24(t)=158(1−14t2+21t4). (12)

The Gegenbauer moments appeared in Eqs. (9-11) are the following pball-05 (); pball-0605004 ()

 aπ1 = 0,aK1=0.06±0.03,aπ,K2=0.25±0.15, aπ4 = −0.015,ηπ,K3=0.015,ωπ,K3=−3. (13)

## Iii Form factors and semileptonic decays

### iii.1 B(s)→π,K form factors

The form factors for transition are defined by npb592-3 ()

 ⟨P(p2)|¯b(0)γμq(0)|B(p1)⟩ = [(p1+p2)μ−m2B−m2Pq2qμ]F+(q2) (14) +m2B−m2Pq2qμF0(q2),

where is the momentum transfer to the lepton pairs. In order to cancel the poles at , should be equal to . For the sake of the calculation, it is convenient to define the auxiliary form factors and

 ⟨P(p2)|¯b(0)γμq(0)|B(p1)⟩=f1(q2)p1μ+f2(q2)p2μ (15)

in terms of and , the form factors and are defined as

 F+(q2) = 12[f1(q2)+f2(q2)], F0(q2) = 12f1(q2)[1+q2m2B−m2P]+12f2(q2)[1−q2m2B−m2P]. (16)

As for the tensor operator, there’s only one independent form factor, which is important for the semi-leptonic decay

 ⟨P(p2)|¯b(0)σμνq(0)|B(p1)⟩ = i[p2μqν−qμp2ν]2FT(q2)mB+mP, ⟨P(p2)|¯b(0)σμνγ5q(0)|B(p1)⟩ = ϵμναβpα2qβ2FT(q2)mB+mP. (17)

The above form factors are dominated by a single gluon exchange in the lowest order and in the large recoil regions. The factorization formula for the form factors is written as li-65-014007 (); yang-epjc23-28 ()

 ⟨P(p2)|¯b(0)γμq(0)|B(p1)⟩ = g2sCFNc∫dx1dx2d2k1Td2k2Tdz+d2zT(2π)3dy−d2yT(2π)3 (18) ×e−ik2⋅y⟨P(p2)|¯q′γ(y)qβ(0)|0⟩eik1⋅z⟨0|¯bα(0)q′δ(z)|B(p1)⟩Tγβ;αδHμ.

In the hard-scattering kernel, the transverse momentum in the denominators are retained to regulate the endpoint singularity. The masses of the light quarks and the mass difference between the quark and the meson are neglected. The terms proportional to in the numerator are dropped, because they are power suppressed compared to other terms. In the transverse configuration b-space and by including the Sudakov form factors and the threshold resummation effects, we obtain the form factors as following,

 f1(q2) = 16πCFm2B∫dx1dx2∫b1db1b2db2ψB(x1,b1) (19) ×{[r0(ϕp(x2)−ϕt(x2))⋅h1(x1,x2,b1,b2)−r0x1ηm2Bϕσ(x2)⋅h2(x1,x2,b1,b2)] ⋅αs(t1)exp[−Sab(t1)] ⋅αs(t2)exp[−Sab(t2)]},
 f2(q2) = 16πCFm2B∫dx1dx2∫b1db1b2db2ψB(x1,b1) (20) ×{[[(x2η+1)ϕa(x2)+2r0((1η−x2)ϕt(x2)−x2ϕp(x2)+3ϕσ(x2))] ⋅h1(x1,x2,b1,b2)−r0x1m2B(1+x2η)ϕσ(x2)⋅h2(x1,x2,b1,b2)]⋅αs(t1)exp[−Sab(t1)]
 FT(q2) = 8πCFm2B∫dx1dx2∫b1db1b2db2(1+rP)ψB(x1,b1) (21) ×{[  r0x1m2Bϕσ(x2)⋅h2(x1,x2,b1,b2) ⋅αs(t1)exp[−Sab(t1)]

where is a color factor, , and is the mass of the final pseudoscalar meson, and is the mass of the quarks involved in the final meson. The functions and , the scales , and the Sudakov factors are given in the Appendix A of this paper. One should note that and as given in Eqs. (19-21) do not including the NLO correction. In order to include the NLO corrections, the in Eqs. (19-21) should be changed into , where the NLO factor has been defined in Eq. (3).

### iii.2 Semileptonic B and Bs meson decays

For the charged current and decays, as illustrated in Fig.1(a) and 1(d), the quark level transitions are the transitions with , the effective Hamiltonian for such transitions is operators ()

 Heff(b→ul¯νl)=GF√2Vub¯uγμ(1−γ5)b⋅¯lγμ(1−γ5)νl, (22)

where is the Fermi-coupling constant, is one of the Cabbibo-Kobayashi-Maskawa (CKM) matrix elements. The corresponding differential decay widths can be written as lu-79-014013 (); p-73-115006 ()

 dΓ(b→ul¯νl)dq2 = G2F|Vub|2192π3m3Bq2−m2l(q2)2 ⎷(q2−m2l)2q2 ⎷(m2B−m2P−q2)24q2−m2P (23) ×{(m2l+2q2)[q2−(mB−mP)2][q2−(mB+mP)2]F2+(q2) +3m2l(m2B−m2P)2F20(q2)},

where is the mass of the lepton. If the produced lepton is or , the corresponding mass terms could be neglected, the above expression then becomes

 dΓ(b→ul¯νl)dq2=G2F|Vub|2192π3m3Bλ3/2(q2)|F+(q2)|2, (24)

where is the phase-space factor.

For those flavor changing neutral current one-loop decay modes, such as with and decays, as illustrated in Fig.1, the quark level transitions are the transitions, the corresponding effective Hamiltonian for such transitions is

 Heff=−GF√2VtbV∗tq10∑i=1Ci(μ)Oi(μ), (25)

where , are the Wilson coefficients and the local operators are given by operators ()

 O1 = (¯qαcα)V−A(¯cβbβ)V−A,O2=(¯qαcβ)V−A(¯cβbα)V−A, O3 = (¯qαbα)V−A∑q′(¯q′βq′β)V−A,O4=(¯qαbβ)V−A∑q′(¯q′βq′α)V−A, O5 = (¯qαbα)V−A∑q′(¯q′βq′β)V+A,O6=(¯qαbβ)V−A∑q′(¯q′βq′α)V+A, O7 = emb8π2¯qσμν(1+γ5)bFμν, O9 = αem8π(¯lγμl)[¯qγμ(1−γ5)b],O10=αem8π(¯lγμγ5l)[¯qγμ(1−γ5)b], (26)

where , .

For the decays with transition, for example, the decay amplitude can be written as operators ()

 A(b→sl+l−) = GF2√2αemπV∗tsVtb{C10[¯sγμ(1−γ5)b][¯lγμγ5l] (27) +Ceff9(μ)[¯sγμ(1−γ5)b][¯lγμl] −2mbCeff7(μ)[¯siσμνqνq2(1+γ5)b][¯lγμl]},

where and are the effective Wilson coefficients, defined as

 Ceff7(μ) = C7(μ)+C′b→sγ(μ), (28) Ceff9(μ) = C9(μ)+Ypert(^s)+YLD(^s). (29)

Here the term represents the short distance perturbative contributions and has been given in Ref. pap-ypert ()

 Ypert(^s) = h(^mc,^s)C0−12h(1,^s)(4C3+4C4+3C5+C6) (30) −12h(0,^s)(C3+3C4)+29(3C3+C4+3C5+C6),

with , , and , while the functions and in above equation are of the form

 h(z,^s) = −89lnmbμ−89lnz+827+49x (31) −29(2+x)√|1−x|⎧⎪ ⎪⎨⎪ ⎪⎩(ln|√1−x+1√1−x−1|−iπ),(x≡4z2^s<1),2arctan1√x−1,(x≡4z2^s>1), h(0,^s) = 827−89lnmbμ−49ln^s+49iπ. (32)

The term in Eq. (29) refers to the long-distance contributions from the resonant states and will be neglected because they could be excluded by experimental analysisprl-106161801 (); bbns2009 (). The term in Eq. (28) is the absorptive part of and is given byc7eff-pr ()

 C′b→sγ(μ)=iαs{29η14/23[GI(xt)−0.1687]−0.03C2(μ)}, (33)

with

 GI(xt)=xt(x2t−5xt−2)8(xt−1)3+3x2tlnxt4(xt−1)4, (34)

where