Tensor form factors of B\to K_{1} transition from QCD light cone sum rules

The tensor form factors of into p–wave axial vector meson transition are calculated within light cone QCD sum rules method. The parametrizations of the tensor form factors based on the series expansion are presented.

PACS number(s): 11.55.Hx, 13.75.Gx, 13.75.Jz

## 1 Introduction

Rare decays due to the flavor–changing neutral current transitions constitute one of the most important classes of decays in carefully checking the predictions of the Standard Model (SM) at tree level, since they are forbidden in SM at loop level. In SM the flavor changing neutral current (FCNC) processes proceed through the electroweak penguin and box diagrams. These decays are also very suitable in looking for new physics (NP) beyond the SM, via contributions of the new particles to the penguin and box diagrams, that are absent in the SM.

The decay has been observed in [1, 2]. Moreover, the forward–backward asymmetry has been measured in [3, 4]. The longitudinal polarization and forward–backward asymmetry of and the isospin asymmetry of and are also measured by BaBar Collaboration in [5] and [6], respectively. The experimental results are more or less in agreement with the predictions of SM. However, the precision of experiments is currently too low to make the final conclusion. The situation should considerably be improved at LHCb.

The radiative decays of meson, involving , where is the orbitally excited state, is observed by BELLE. The other radiative and semileptonic decay modes involving and are hopefully expected to be measured soon.

Similar to the decay the decay is also a very good object for probing the new physics effects beyond the SM. Here the problem becomes more sophisticated due to the mixing of and state. The physical states and are determined by

 (|K1(1400)⟩|K1(1270)⟩)=(cosθ−sinθsinθcosθ)(|K1A⟩|K1B⟩) . (1)

In the present work we calculate the tensor form factors for the transition in the framework of the light cone QCD sum rules method (LCSR) (for more about LCSR see [7, 8]).

The paper is organized in the following way. In section 2 we derive the LCSR for the tensor form factors describing the transition. Section 3 is devoted to the numerical analysis of the sum rules for the form factors. We also summarize our results in this section.

## 2 Light cone QCD sum rules for the tensor form factors of the B→K1A(B) transition

The decay is described by transition at quark level. The effective Hamiltonian responsible for the transition is given by,

 H=−4GF√2VtbV∗ts10∑i=1Ci(μ)O(μ) , (2)

where the form of the local Wilson operators is given in [9]. This effective Hamiltonian leads to the following result for the decay amplitude

 M = GF2√2αemπVtbV∗ts{Ceff9¯sγμ(1−γ5)b¯ℓγμℓ+C10¯sγμ(1−γ5)b¯ℓγμγ5ℓ (3) − 2mbq2C7¯siσμνqν(1+γ5)¯ℓγμℓ} ,

where the Wilson coefficient , with , contains both the perturbative and the long distance contribution parts. The explicit expression of , , and are given in [9]. The long distance effects generated by the four–quark operators with the –quark have recently been calculated for the and decays in [10] and it is obtained that below the charmonium region of this effect can change the value of around 5% and 20% for and transitions, respectively. Similar calculations for transition has not yet been calculated. For simplicity, in the following discussions we denote and as .

It follows from Eq.(3) that for the calculation of the transition, the matrix elements and are needed. For the transition, these matrix elements are defined in terms of the form factors as follows:

 ⟨K1(p,λ)∣∣¯sγμ(1−γ5)b∣∣B(pB)⟩ = −i2mB+mK1ϵμναβε(λ)∗νpαBpβAK1(q2) − [(mB+mK1)ε(λ)∗μVK11(q2)−(pB+p)μ(ε(λ)∗pB)VK12(q2)mB+mK1] + 2mK1(ε(λ)∗pB)q2qμ[VK13(q2)−VK10(q2)] , ⟨K1(p,λ)∣∣¯sσμνqν(1+γ5)b∣∣B(pB)⟩ = 2TK11(q2)ϵμναβε(λ)∗νpαBpβ (5) − iTK12(q2)[(m2B−m2K1)ε(λ)∗μ−(ε(λ)∗q)(pB+p)μ] − iTK13(q2)(ε(λ)∗q)[qμ−q2m2B−m2K1(pB+p)μ],

where . There are the following relations between the form factors:

 VK13(q2) = mB+mK12mK1VK11(q2)−mB−mK12mK1VK12(q2) , VK13(0) = VK10(0),  \rm and, TK11(0) = TK12(0) . (6)

To be able to calculate the form factors responsible for the transition we consider the following two correlation functions:

 Πμ = i∫d4xeiqx⟨K1(p,λ)∣∣T{¯s(x)γμ(1−γ5)b(x)¯b(0)iγ5d(0)}∣∣0⟩ , (7) Πμν = i∫d4xeiqx⟨K1(p,λ)∣∣T{¯s(x)σμν¯b(x)b(0)iγ5d(0)}∣∣0⟩ . (8)

In order to construct the sum rules for the form factors responsible for the transition these correlation functions should be calculated in two different languages, in terms of hadrons and quark and gluon degrees of freedom. The calculation of the correlation function in terms of quark and gluon degrees of freedom is carried out at virtualities and . Using the operator product expansion, the sum rules are obtained by equating these two representations through the dispersion relations.

Phenomenological parts of the correlation functions (7) and (8) can be obtained by inserting complete set of hadrons with the same quantum numbers as the interpolating current, and separating the ground state one can easily obtain

 Πμ = −⟨K1(p,λ)∣∣¯sγμ(1−γ5)b∣∣B(pB)⟩⟨B(pB)∣∣¯biγ5d∣∣0⟩p2B−m2B+⋯ , (9) Πμν = (10)

where “” describes the contributions coming from higher states and continuum, and the matrix element is given in Eq. (2). The second matrix element in Eq. (9) is expressed in the standard way

 ⟨B(pB)∣∣¯biγ5d∣∣0⟩=fBm2Bmb , (11)

where is the –decay constant and is the –quark mass. The matrix element is defined as

 ⟨K1(p,λ)∣∣¯sσμνb∣∣B(pB)⟩ = −iA(q2)[ε(λ)∗μ(p+pB)ν−ε(λ)∗ν(p+pB)μ] (12) + iB(q2)(ε(λ)∗μqν−ε(λ)∗νqμ)+i2C(q2)m2B−mK1(pμqν−pνqμ) .

Contracting Eq. (12) with the momentum and using the relation

 σμνγ5=−i2ϵμναβσαβ ,

the following relations among and can easily be obtained:

 TK11(q2) = A(q2) , TK12(q2) = A(q2)−q2m2B−m2K1B(q2) , TK13(q2) = B(q2)+C(q2) . (13)

Using Eqs. (11) and (12), for the phenomenological parts of the correlation functions we get

 Πμ = −fBm2Bmb1p2B−m2B{−2im2B−m2K1ϵμναβε(λ)∗νpαBpβAK1(q2) − [(mB+mK1)ε(λ)∗μVK11(q2)−Pμ(ε(λ)∗q)VK12(q2)mB+mK1] + 2mK1ε(λ)∗qq2qμ[VK13(q2)−V0(q2)]} , Πμν = −fBm2Bmb1p2B−m2B{−iA(q2)(ε(λ)∗μPν−ε(λ)∗νPμ)+iB(q2)(ε(λ)∗μqν−ε(λ)∗νqμ) (15) + 2iC(q2)ε(λ)∗qm2B−m2K1(pμqν−pνqμ)} ,

where .

We now proceed to calculate the theoretical part of the correlation functions. The calculation is performed by using the background field approach [11]. In the large virtuality region, where and , the operator product expansion is applicable to the correlation functions. In light cone sum rules the method is based on the expansion of the non–local quark–antiquark operators in powers of the deviation from the light cone. In obtaining the expression of the correlation functions the propagator of heavy quark and the matrix elements of the non–local operators and between the vacuum and axial–vector meson are needed, where are the Dirac matrices (in our case or ), and is the gluon field strength tensor.

The expression of the heavy quark operator is given in [12]

 SQ=SfreeQ(x)+igs∫d4k(2π)4e−ikx∫du[to0.0pt/k+mb2(m2b−k2)2Gμν(ux)σμν+um2b−k2xμGμν(ux)γν] , (16)

where is the free quark operator and we adopt the convention for covariant derivative .

Two particle distribution amplitudes for the axial vector mesons are presented in [13, 14]

 ⟨K1(p,λ)|¯sα(x)qδ(0)|0⟩ = −i4∫dueiupx (17) × {fK1mK1[to0.0pt/pγ5ε(λ)∗xpxϕ∥(u)+(to0.0pt/ε(λ)∗−ε(λ)∗xpxto0.0pt/p)γ5g(a)⊥(u) − to0.0pt/xγ5ε(λ)∗x2(px)2m2K1¯g3(u)+ϵμνρσε(λ)∗νpρxσγμg(v)⊥4] + fA⊥[12(to0.0pt/pto0.0pt/ε(λ)∗−to0.0pt/ε(λ)∗to0.0pt/p)γ5ϕ⊥(u)−12(to0.0pt/pto0.0pt/x−to0.0pt/xto0.0pt/p)γ5ε(λ)∗x(px)2m2K1¯h(t)∥(u) − 14(to0.0pt/ε(λ)∗to0.0pt/x−to0.0pt/xto0.0pt/ε(λ)∗)γ5m2K1px¯h3(u)+i(ε(λ)∗x)m2K1γ5h(p)∥(u)2] + O(x2)}δα ,

where

 ¯g3(u) = g3(u)+ϕ∥−2g(a)⊥(u) , ¯h(t)∥(u) = h(t)∥(u)−12ϕ⊥(u)−12h3(u) , ¯h3(u) = h3(u)−ϕ⊥(u) , (18)

and , are the twist–2, , , and are twist–3, and and are twist–4 functions. The three particle distribution amplitudes are define as

 ⟨K1(p,λ)∣∣¯s(x)γαγ5gsGμν(ux)q(0)∣∣0⟩ = pα(pνε(λ)∗μ−pμε(λ)∗ν)fA3K1A+⋯ , ⟨K1(p,λ)∣∣¯s(x)γαgs˜Gμν(ux)q(0)∣∣0⟩ = ipα(pμε(λ)∗ν−pνε(λ)∗μ)fV3K1V+⋯ , (19)

where

 A=∫DαeiPx(α1+uα3)A(α1,α2,α3) ,

and

 ∫Dα=∫10dα1∫10dα2∫10dα3 δ(1−α1−α2−α3) .

Here , , and are the respective momentum fractions carried by , quarks and gluon in the meson. Using these definitions, and after lengthy calculations for the theoretical parts of the correlation functions, we obtain

 Correlation function= 14∫du{fKimKi[ε(λ)∗αΦ(i)a∂∂QαTr(Γto0.0pt/PSQ)−ga⊥Tr(ΓSQto0.0pt/ε(λ)∗) −12m2Ki¯g(ii)3ε(λ)∗α∂∂Qα∂∂QβTr(ΓSQγβ)+iϵαβρσgv⊥4ε(λ)∗βpρ∂∂QσTr(ΓSQγαγ5)] +f⊥Ki[12ϕ⊥(u)Tr[ΓSQ(to0.0pt/pto0.0pt/ε(λ)∗−to0.0pt/ε(λ)∗to0.0pt/p)]−12m2Ki¯h(t)(ii)∥ε(λ)∗α∂∂Qα∂∂QβTr[ΓSQ(to0.0pt/pγβ−γβto0.0pt/p)] +h(i)34m2Ki∂∂QαTr[ΓSQγ5(to0.0pt/ε(λ)∗γα−γαto0.0pt/ε(λ)∗)]+12h(p)∥m2Kiε(λ)∗α∂∂QαTr[ΓSQ]} +14∫dv∫Dαi1{m2b−[q+(α1+vα3)p]2}2{2vpq[fA3iA(αi)+fV3iV(αi)]Tr(Γto0.0pt/ε(λ)∗to0.0pt/p)} ,

where

 SQ = mb+to0.0pt/Qm2b−Q2 ,  \rmwith ,   Q=q+pu , Φ(i)a = ∫u0[ϕ(v)∥−ga⊥(v)]dv , f(i) = ∫u0f(v)dv , f(ii) = ∫u0dv∫v0dv′f(v′) ,

and correspond to , respectively, is equal to or . After taking derivatives and traces, equating expressions of correlation functions (2), (15) and (2), and performing Borel transformation with respect to the variable in order to suppress the higher states and continuum contributions, one can obtain the sum rules for the transition form factors. Here we present the sum rules only for the tensor form factors, since , , and are calculated within the same framework in [15]:

 Ai(q2) = −m2bf⊥i2m2BfBem2B/M2∫10du{1ue−s(u)/M2θ[s0−s(u)][ϕ⊥(u) − mifimbf⊥i(ug(a)⊥(u)+ϕ(i)a+gv⊥(u)4)]−14e−s(u)/M2mifiumbf⊥i(m2b+q2)gv⊥(u) × (θ[s0−s(u)]uM2+δ[u−u0]s0−q2)} − mb2m2BfBem2B/M2∫10vdv∫e−s(k)/M2dα1dα3fA3iA(αi)+fV3iV(αi)(α1+vα3)2 × {θ[s0−s(k)]−(m2b−q2)(θ[s0−s(k)](α1+vα3)M2+δ[k−u0]s0−q2)} , Bi(q2) = −m2bf⊥i2m2BfBem2B/M2∫10du{1ue−s(u)/M2θ[s0−s(u)][ϕ⊥(u) − mifimbf⊥i(−(2−u)g(a)⊥(u)+ϕ(i)a+(1−2u)gv⊥(u)4)] − 1ue−s(u)/M214mifimbf⊥i[2m2b−(m2b−q2)(1−2u)]g(v)⊥(u) × (θ[s0−s(u)]uM2+δ[u−u0]s0−q2)} − mb2m2BfBem2B/M2∫10vdv∫e−s(k)/M2dα1dα3fA3iA(αi)+fV3iV(αi)(α1+vα3)2 × {θ[s0−s(k)]−(m2b−q2)(θ[s0−s(k)](α1+vα3)M2+δ[k−u0]s0−q2)} , Ci(q2) = mbmifi2fBem2B/M2∫10du{1ue−s(u)/M2[(2ϕ(i)a(u)−g(v)⊥(u)2) (21) × (θ[s0−s(u)]uM2+δ[u−u0]s0−q2)} ,

where or , and

 s(n)=m2b−(1−n)q2n,k≡α1+vα3,\rm~{}and, u0=m2b−q2s0−q2 .

In these expressions, we neglect terms . Using Eq. (2) one can easily obtain the corresponding sum rules for