Isgur-Wise function in $B_c$ decays to charmonium with the Bethe-Salpeter method

The Isgur-Wise function vastly reduces the weak-decay form factors of hadrons containing one heavy quark. In this paper, we extract the Isgur-Wise functions from the instantaneous Bethe-Salpeter method, and give the numerical results for the $B_c$ decays to charmonium where the final states include $1S$, $1P$, $2S$ and $2P$. The overlapping integral of the wave functions for the initial and final states is the Isgur-Wise function, as the heavy quark effective theory does. In the case of accurate calculation, describing form factors need to introduce more relativistic corrections which are the overlapping integrals with the relative momentum between the quark and antiquark to Isgur-Wise function. The relativistic corrections to Isgur-Wise function provide greater contributions especially involving the excited state, and therefore are necessary to be adopted.


I. INTRODUCTION
Under the heavy quark effective theory (HQET), a semileptonic decay process can be related to a rotation of the heavy quark flavor or spin [1,2]. In the limit m Q → ∞ (Q denotes the heavy quark or anti-quark), this rotation is a symmetry transformation. The form factors depend only on the Lorentz boost γ = v · v which connects the rest frames of the initial state and final state. The transition can be described by a dimensionless function ξ(v · v ). Heavy-quark symmetry reduces the weak-decay form factors of heavy hadrons to this universal function. These relations were derived by Isgur and Wise firstly [3,4], so called Isgur-Wise function (IWF).
HQET vastly simplifies the calculations, and plays a crucial role in extracting the values of |V cb | and |V ub |. For example, the differential semileptonic decay rate for B → D in the heavy-quark limit can be model-independent described by [2] dΓ The decay rate depends on only two quantities, |V cb | and ξ(ω). If the differential semileptonic combined a improved variational approach [9], the baryon hyperspherical approach [10], a combination of linear confining and Hulthén potentials [11], the three-boy Schrödinger equation [12], a combination of Deng-Fan-type and harmonic potentials [13], a quantum isotonic nonlinear oscillator potential model [14] and so on. Choudhury et al. had studied the renormalization scale dependence of IW function by using a wave function with linear part of the Cornell potential as perturbation [15]. Yazarloo  and branching ratio of the B → D * ν process [16]. In a potential model, the Isgur-Wise function usually corresponds to the overlapping integral of the wave functions obtained by this model.
However the lowest order result is not accurate enough due to the heavy quark approximation. The symmetry-breaking corrections are needed when the study becomes more precise, since the masses of the heavy quarks or anti-quarks are not infinite actually. In addition the radiation correction can not be ignored. The HQET provides a systematic framework to analyze these corrections. For example, Luke analyzed the 1/m Q corrections for the more complicated case of weak decay form factors [17]. Falk  in slope, and all of 1/m Q corrections and QCD radiative corrections are distilled [22]. Other efforts of complements are too many to be list here.
The flavor-spin symmetry can be used not only in weak decays of ground hardons containing one heavy quark, but also in the processes involving orbitally and radially excited states containing one heavy quark. Isgur and Wise exploited the flavor-spin symmetry to obtain model-independent predictions in weak decays from pseudoscalar meson of a heavy quark Q i to P wave excited states of another heavy quark Q j in term of two Isgur-Wise functions [23]. Gan [26,27].
However, the validity of HQET is suspectable, when the systems contain two or more heavy degrees of freedom. Some work have explored the application of the heavy quark symmetry to describe the weak decays of hadrons containing two heavy quarks. Sanchis-Lozano estimated that the flavor symmetry losts, while the spin symmetry holds for the double-heavy meson [28]. As for the baryon containing two heavy quarks, two heavy quarks constitute a bosonic diquark whose mass could be regarded as infinite, and therefore HQET  [36,37]. Nowadays the instantaneous BS method has developed to be quite covariant, and the full Salpeter equations are solved for different J P (C) states [38][39][40][41]. So the relativistic correction which equate to symmetry-breaking correction has been taken into account. In this paper we shall not derive these corrections from HQET, but attempt to extract the IWF from the solutions of the instantaneous BS method. Note that we do not use the heavy quark limit. The results show that only the lowest order correction to IWF is not enough, and higher-order relativistic corrections need to be introduced for more accurate results.
The paper is organized as follows. In section II, we give the useful formulas for the B c decays to charmonium. In section III, we give the relativistic wave function for 0 − state in the instantaneous BS method. In section IV, we extract the IWF and give the analytical results.
In section V, we give the numerical results and discussions. We summarize and conclude in section VI, and put the Salpeter equation and some wave functions in the appendix A.

II. FORM FACTORS AND SEMILEPTONIC DECAY WIDTH
For the B + c → (cc) + ν processes shown in figure 1, the transition amplitude element reads where (cc) denotes charmonium; V cb is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element; J µ ≡ V µ − A µ is the charged current responsible for the decays; P and P f are the momenta of the initial B + c and the final charmonium, respectively. The hadronic transition element can be written as the overlapping integral over the initial and final relativistic BS wave functions within Mandelstam formalism. We would not solve the full BS equation, but the instantaneous one, namely, the full Salpeter equation. We perform the instantaneous approximation to the transition element [42] and write it as where ϕ ++ P denotes the positive energy component of the instantaneous BS wave function of the initial state; ϕ ++ P f ≡ γ 0 (ϕ ++ P f ) † γ 0 is the Dirac conjugate of the positive energy component of the final state; m 1 and m 2 are the masses of quark and antiquark in the final state, respectively, and q = q − m 1 m 1 +m 2 P f is the relative momentum between them. In this paper, we keep only the positive energy component ϕ ++ of the relativistic wave functions, because the contributions from other components are much smaller than 1% in transition of B c → (cc) [43]. This matrix element can also be written in the framework in which the momentum q is the integral argument by means of a suitable Jacobi transformation, where q = q + α P f , α = m 1 m 1 +m 2 and θ is the angle between q and P f . The Eq. (4) is more convenient, because some matrix elements we calculate in this paper involve a P -wave final state [37].
For B + c → P + ν (here P denotes η c or χ c0 ), the hadronic matrix element can be written as where S + and S − are the form factors. For B + c → V + ν (here V denotes J/ψ, h c or χ c1 ), the hadronic matrix element can be written as where µ is the polarization vector of the final vector meson; t 1 , t 2 , t 3 and t 4 are the form factors. For B + c → T + ν (here T denotes χ c2 ), the hadronic matrix element can be written as where αβ is the polarization tensor of the final tensor meson; t 1 , t 2 , t 3 and t 4 are the form factors.
The summation formulas for polarization of the final vector meson are The summation formulas for polarization of the final tensor meson are Where S µν = −g µν + . Finally, the semileptonic decay width can be expressed as where P is the three-dimensional momentum of the final lepton, and P f is the threedimensional momentum of the final meson. In this paper, we only calculate the form factors but no longer calculate the decay widths.

III. RELATIVISTIC WAVE FUNCTION
Usually, the nonrelativistic wave function for a pseudoscalar is written as [36] Ψ where M and P are the mass and momentum of the meson, respectively; q is the relative momentum between the quark and antiquark in the meson, and the radial wave function f ( q) can be obtained numerically by solving the Schrodinger equation.
But in our method, we solve the full Salpeter equation. The form of wave function is relativistic and depends on the J P (C) quantum number of the corresponding meson. For a pseudoscalar, the relativistic wave function can be written as the four items constructed by P , q ⊥ and γ-matrices [44] where q = p 1 − α 1 P = α 2 P − p 2 is the relative momentum between quark (with momentum p 1 and mass m 1 ) and antiquark (momentum p 2 and mass m 2 ), Taking into account the last two equations in Eq. (A9), we obtain the relations where the quark energy . The wave function corresponding to the positive energy projection has the form where The normalization condition reads be calculated numerically. In this paper, besides the wave function for 0 − state, we also need the wave functions for the states of 1 −− (J/ψ), 1 +− (h c ), 0 ++ (χ c0 ), 1 ++ (χ c1 ) and 2 ++ (χ c2 ). We put the 2 ++ state wave function in the appendix A and the others can be referred to [45].

IV. ISGUR-WISE FUNCTION
Since B c and charmonium are the weak-binding states, the approximation is taken in this paper. This approximation requires the three-dimensional relative momentum | q| between quarks much less than the masses of quarks, but its contribution will be suppressed by the wave function f i ( q) in the large | q| interval. After performing this approximation and the trace on the matrix element Eq. (4), the dependence of all the form factors on the overlapping integrals of the wave functions for the initial state and the final state becomes transparent. For instance, one type of overlapping integrals are where f i denotes the wave function of initial state, and f i denotes the wave function of final state. Two wave functions from the same meson is very close numerically, i.e., f 1 ≈ f 2 and f 1 ≈ f 2 . So the four overlapping integrals in Eq. (18) are approximately equal, and for convenience they are replaced by their average which is denoted as where C is the normalized coefficient; v, v are the four dimensional velocities of the initial state and final state respectively, and f f = . There are other overlapping integrals with the relative momentum q being inserted. They may be the relativistic corrections to the function ξ 00 . We denote them as ξ qx , where subscript q denotes the power of the relative momentum q , subscript x denotes the power of cos θ, and θ is the angle between q and P f , i.e., and so on. When the final state is S-wave meson, we keep the first six functions and abandon the higher order O(q 4 ); When the final state is P-wave meson whose wave function includes a q, ξ 00 disappears, thus we reserve the first eight functions and abandon the higher order O(q 5 ). The normalized coefficients based on the normalized formulas are shown in Table. I for each process. Taking the process B c → η c as an example, the initial and final states are So the normalized wave function of 0 − state is 2 √ M f , and the normalized coefficient is The form factors of semileptonic decay B c → η c ν can be written as where where The function ξ 00 may be directly related to the Isgur-Wise function appearing in HQET for 0 − → 0 − decays. Because the form factors in this process will degenerate into those in the nonrelativistic limit if only the function ξ 00 is considered [2], The other functions are the relativistic corrections (1/m i corrections) to the leading order IWF ξ 00 , where i denote a quark or anti-quark in the initial and final mesons. The number of q contained in the function ξ qx (subscript q) corresponds to the order of the correction.
Note that there should have been another type of overlapping integrals with the relative momentum q in the initial state. For example, where β is the angle between q and P f . Due to the relation q = q + α P f , α = m 1 m 1 +m 2 , this overlapping integral Eq. (25) is decomposed into ξ 11 + α| P f |ξ 00 . So the item involving αξ 00 should be considered as relativistic correction of the same order as ξ 11 . Generally, the item involving α n ξ qx is the q + n order relativistic correction (1/m q+n i correction) which can be confirmed in Eq. (22) and (23). The process 0 − → 1 −− is the same as above case. The leading order result is agree with HQET [2], i.e., It is very natural that the leading order results in this paper are entirely consistent with HQET for 0 − → 0 − or 1 −− processes. Because for the leading order results, the terms involving / q disappear and ω i = m i , so that the BS wave functions degenerate into the nonrelativistic case, i.e., A pseudoscalar meson and its corresponding vector have the same radial wave function Ψ in the nonrelativistic limit. But in this paper, the radial wave functions are obtained by solving BS equation, and the Ψ in Eq. (27) corresponds to the normalized wave function 2 √ M f i . They are not exactly the same numerically. And then, the numerical results of IWF is not close to HQET and contains part of the relativistic correction.
For P-wave meson as the final state, the nonrelativistic wave functions are usually written as 0 ++ : and these states have the same radial wave function Φ. Similarly in this paper, the radial wave functions are obtained by solving BS equation, and the Φ in Eq. (28) corresponds to the normalized wave function. In these cases, ξ 00 disappears and ξ 11 is IWF. Due to the change of orbital angular momentum, this weak decay process can not correspond to a scattering process simply. We give the leading order results in the case that only the function ξ 11 is considered, These results are not agree with Ref. [32]. The latter analyzes the reduction of form factors in the heavy quark limit, and there are two IWFs ξ E , ξ F v α for B c to P-wave charmonium.
While we only need IWF ξ 11 for these processes in the leading order. Ref. [32] does not further describe the used IWFs. The difference needs further examination. Note that the above results are not confined to the processes of B c to charmonium, but hold true for any possible process where the initial and final mesons are corresponding J P C states. In the next section, we will give the numerical results and discussions on the specific processes.

V. RESULTS AND DISCUSSIONS
The parameters used in this paper: Γ Bc = 1.298 × 10 −12 GeV, G F = 1.166 × wave functions for ηc wave functions for χc1 wave functions for χc2 slightly differ. Because these decay processes are just related by a rotation of the heavy-quark spin or the meson spin, and this rotation is a symmetry transformation in the infinite-mass limit. Note that the infinite-mass limit is not used in this paper, but this spin-symmetry reflected in the results automatically, as Fig. 4-5 shows. This indicates that spin-symmetry still mantains though the initial and final states are both the double-heavy mesons. When wave functions for ψ(2S)  wave functions for χ The solid line is the Isgur-Wise function, the dash and dot-dash one are the first order correction functions, the dot one is the second order correction function, and so on, in every subfigure. the configuration of initial or final state changes, for example, the final η c turns into η c (2S), the behaviors of IWFs become significantly different from before. Next we will discuss these four modes one by one.  The mode 1S → 1S has been extensively studied in HQET. B c and η c are related by the replacement b → c, while η c and J/ψ are related by the transformation c ⇑ → c ⇓ here. These two rotations (flavour and spin rotations) are symmetry transformations in the infinite-mass limit. So the radial wave functions of these mesons will be identical in this limit, and the corresponding IWF whose general form is the overlapping integral of the initial and final wave functions will be the same as the normalization formula at zero recoil. It is very natural that ξ(1) = 1 in HQET. In this paper we solve the full Salpeter equations without the infinite-mass limit, and the normalized radial wave function is approximated to 2 √ M f for 1S state. The normalized wave functions have little difference for η c and J/ψ (two dominate wave functions), which is consistent with Eq. (27). But the discrepancy between B c and the former two is in the order of 30% (peak value), as Fig. 2(a)-2(c) shows. This indicates that in the double-heavy system the spin-symmetry keeps, while the flavour-symmetry breaks.
The masses of quark and antiquark are in the same order of magnitude, and therefore the change of flavour will lead to a great impact. Although the behaviors of IWF ξ 00 are the same as ξ in HQET, they are not strict unity at zero recoil in this paper, as Fig. 4(a)-4(b) shows. The relativistic correction reflected in IWF ξ 00 is around 10% at zero recoil. In the mode 1S → 1S, it is convenient to fit the IWF as where ρ 2 is the slope parameter and c is the curvature parameter which characterizes the shape of the IWF. The slope and curvature by fitting are 2.25 and 1.74 respectively in B c → η c , and they are 2.38 and 1.98 respectively in B c → J/ψ. The result is agree with the rule that the slope is bigger as the (reduced) mass is heavier [11,31]. The other functions ξ qx are the relativistic corrections to IWF ξ 00 . The more q the correction function contains, the less contribution it makes. We may call the correction function with one relative momentum q as the first order correction, the correction function with two q as the second order correction, and so on. The values of ξ 11 is about 1/20 of ξ 00 . Because the decay width is proportional to modular square of amplitude, the first order correction may reach 1/10 of the leading order result. It is important for accurate calculation. Our previous study shows that the higher order relativistic corrections also have considerable contributions, and the total relativistic correction can reach around 20% at the level of decay width [45].
In the mode 1S → 1P , the configuration of initial state is 1S, while the configuration of final state is 1P . Their orbital angular momenta are different, so the symmetry transformations exist only between the final states, i.e., spin rotations. χ c0 , χ c1 and χ c2 are spin triplet states that are related by the rotation of total spin component (the component of total spin in the direction of orbital angular momentum), while h c and the former three are related by the transformation c ⇑ → c ⇓ . These two spin rotations are symmetry transformations in the infinite-mass limit, so the normalized radial wave functions of these mesons will be identical. In this paper we study these mesons without the infinite-mass limit, and their normalized radial wave functions are approximated to 2| q |h respectively. Their numerical results are almost the same, as Figs. 2(d)-2(f) shows, which is consistent with Eq. (28). This indicates that the spin-symmetry keeps in the P-wave charmonium though the quark and anti-quark have the same masses. Because P-wave function contains a q , ξ 00 disappears, and IWF is ξ 11 which behavior is obvious different from χ c0 , χ c1 and χ c2 , the total relativistic corrections are 50%, 64%, 34% and 14% at the level of decay width respectively [45]. The total relativistic correction of B c → χ c2 is unusually small, because the different orders corrections cancel each other out. This can be seen in the following analysis of form factors.
In the mode 1S → 2S, the configuration of initial state is different from the final state.
Similarly, the only symmetry transformation is the spin rotation c ⇑ → c ⇓ which relates η c (2S) with ψ(2S). Their normalized wave functions are almost the same, as Fig. 3(a)-3(b) shows, which is consistent with Eq. shows, and the leading order form factors have the same behaviors due to Eq. (24) and (26).
The behaviors of the other correction functions in this mode are similar to 1S → 1S, but they make more contributions here. For example, ξ 11 and ξ 20 are about one fifth and one eighth of ξ 00 at the maximum recoil, respectively. So the relativistic corrections become greater, and our previous study shows they are about 19%-28% larger than those in the mode 1S → 1S [45].
Comparing to the mode 1S → 1P , the analysis about the symmetry and normalized wave functions is the same in 1S → 2P , as Figs. 5(c)-5(e) shows. The IWF ξ 11 is also zero at zero recoil, but increases more slowly. The ξ 20 and ξ 22 are comparable to IWF ξ 11 , and are no more decreasing but increasing as the momentum recoil is increasing. So the relativistic corrections may be more significant in this mode. They are about 10%-16% larger than those in the mode 1S → 1P [45].
The above analysis of relativistic corrections is qualitative, because the kinematic factors multiplied by IWFs are different and complex. In order to discuss these relativistic  BS equation for a quark-antiquark bound state generally is written as [46] ( / p 1 − m 1 )χ P (q)( / p 2 + m 2 ) = i d 4 k (2π) 4 V (P, k, q)χ P (k), where p 1 , p 2 ; m 1 , m 2 are the momenta and masses of the quark and antiquark, respectively; χ P (q) is the BS wave function with the total momentum P and relative momentum q; V (P, k, q) is the kernel between the quark-antiquark in the bound state. P and q are defined as p 1 = α 1 P + q, α 1 = m 1 m 1 + m 2 , p 2 = α 2 P − q, α 2 = m 2 m 1 + m 2 . (A2) We divide the relative momentum q into two parts, q P || and q P ⊥ , a parallel part and an orthogonal one to P , respectively q µ = q µ P || + q µ P ⊥ , where q µ P || ≡ (P · q/M 2 )P µ , q µ P ⊥ ≡ q µ − q µ P || , and M is the mass of the relevant meson. Correspondingly, we have two Lorentz-invariant variables If we introduce two notations as below η(q µ P ⊥ ) ≡ k 2 P T dk P T ds (2π) 2 V (k P ⊥ , s, q P ⊥ )ϕ(k µ p ⊥ ), ϕ(q µ p ⊥ ) ≡ i dq P 2π χ P (q µ P || , q µ P ⊥ ). (A5) Then the BS equation can take the form as follow χ P (q µ P || , q µ P ⊥ ) = S 1 (p µ 1 )η(q µ P ⊥ )S 2 (p µ 2 ).
The propagators of the relevant particles with masses m 1 and m 2 can be decomposed as with ω i P = m 2 i + q 2 P T , where i = 1, 2 for quark and antiquark, respectively, and J(i) = (−1) i+1 .