The B c Decays to P -wave Charmonium by Improved Bethe-Salpeter Approach
aa r X i v : . [ h e p - ph ] F e b The B c Decays to P -wave Charmonium by ImprovedBethe-Salpeter Approach Zhi-hui Wang [1] , Guo-Li Wang [1] ∗ , Chao-Hsi Chang [2 , † Department of Physics,Harbin Institute of Technology,Harbin, 150001, China. CCAST (World Laboratory),P.O.Box 8730, Beijing 100190, China. Institute of Theoretical Physics,Chinese Academy of Sciences,P.O.Box 2735, Beijing 100190, China.
Abstract
We re-calculate the exclusive semileptonic and nonleptonic decays of B c meson to a P -wavecharmonium in terms of the improved Bethe-Salpeter (B-S) approach, which is developed recently.Here the widths for the exclusive semileptonic and nonleptonic decays, the form factors, and thecharged lepton spectrums for the semileptonic decays are precisely calculated. To test the concernedapproach by comparing with experimental measurements when the experimental data are available,and to have comparisons with the other approaches the results obtained by the approach and thoseby some approaches else as well as the original B-S approach, which appeared in literature, arecomparatively presented and discussed. PACS numbers:
Keywords: B c decays, P -wave charmonium, Bethe-Salpeter equation, New approach. ∗ gl [email protected] † [email protected] . INTRODUCTION The meson B c is the ground state of the double heavy (both of the components are heavy)quark-antiquark binding system (¯ bc ). In Stand Model (SM) it is an unique meson whichcarries two different heavy flavors explicitly, thus it decays weakly, that is very differentfrom the ground states of flavor-hidden double heavy mesons, such as η c and η b . Namely B c decays via weak interaction (via virtual W emitting or annihilating) only, while theground states of flavor-hidden double heavy mesons decay dominantly by annihilating togluons (strong interaction) or/and photons (electronic interaction). The meson B c has veryrich and experimentally accessible decay channels, so to study the decays of B c meson isspecially interesting. By comparing the experimental and theoretical results of the decaysof B c meson, we can also reach some insight into the binding effects of the heavy quark-antiquark system, which are of QCD nature, besides the knowledge of the weak interactionsuch as the CKM matrix elements etc.The meson B c was first experimentally discovered by the CDF collaboration at Fermilabthrough the semileptonic decay B c → J/ψ + l + ν [1], and soon it is confirmed not only byCDF itself via another decay channel B c → J/ψ + π [2], but also by the other collaborationD0 at Fermilab [3]. The latest experimental report for its lifetime and mass in PDG [4] is M B c = 6 . ± .
006 GeV and τ B c = (0 . ± . × − s. Because the cross sectionof B c production is comparatively small, so to discover it is quite difficult in experiment.Whereas according to the estimates [5–7], that LHC will produce about 5 × B c eventsper year, it is expected that more measurements of decays and production of the meson B c are available soon at LHC (LHCb, CMS, ATLAS), and it must push more studies of thedecays of B c meson forward. So both experimental and theoretical studies on B c meson nowbecome more interesting.In fact, the decays of B c meson can be divided into three categories: i). The anti-bottomquark ¯ b decays into ¯ c (or ¯ u ) with c -quark being as a spectator; ii). The charm quark c decaysinto s (or d ) with ¯ b -quark being as a spectator; iii). The two components, ¯ b and c , annihilateweakly. According to the decay products we may realize which one or two even three of thecategories play roles in a concerned decay, thus one can measure the CKM elements such as V bc , V ub , V cs , V cd through the decays. In the present paper, we are highlighting the decaysof B c meson to a P -wave charmonium, and one may easily to realize that the decays beingconsidered here belong to the category i). Since the lepton spectrum and the weak formfactors, which relate to the binding effects (wave functions) precisely, may be measurable in2emi-lepton decays as long as the experimental sample of decay events is great enough, sowe will share quite a lot of lights on them.In fact, one may find a lot of theoretical methods to treat the semi-leptonic and non-leptonic decays of B c meson, such as the varieties of relativistic constituent quark models[8–15] and QCD sum rules [16, 17] etc in the literature, and moreover one may realizethat among the relativistic constituent quark models, the method presented in Ref. [18] andadopted in Refs. [9, 10] is based on the instantaneous version [19] of the Bethe-Salpeter (B-S)equation [20], and the ‘instantaneous treatment’ is also extended to the weak-current matrixelements using the Mandelstam formulation [21], while the adopted approach in Ref. [8] isdifferent from the one presented in Ref. [18] only in the kernel of the B-S equation and the‘instantaneous treatment’ etc. Recently in [22] an improvement to that of [18] is proposed,and the relativistic effects in the binding systems and decays between the systems may beconsidered by the new development more properly, especially, considering the fact that, ofthe new development, the part (factor) for dealing with the binding effects has been appliedto study (test) the spectra of positronium (a QED binding system) [23] and double heavyflavor binding systems (QCD binding systems) [24] and quite satisfied results are obtained(see Refs. [23, 24]), so to test the new development [22] when experimental data are availablein foreseeable future, in this paper we try to apply the development to the decays of B c mesonto a P -wave charmonium and to compare the obtained results with those obtained by oldmethod in Ref. [18] and obtained by other theoretical approaches. Since we suspect that thedecays of B c meson to a P -wave charmonium might be more sensitive in testing the effectscaused by the improvement than the decays of B c meson to an S -wave charmonium, so herewe focus our attention on the decays of B c meson to a P -wave charmonium.The new development [22] contains two factors: one is about relativistic wave functionswhich describe bound states with definite quantum numbers, i.e. a relativistic form of wavefunctions (see Appendix C) which are solutions of the full Salpeter equation (see AppendixB). Note that here we solve the full equations Eqs. (B9, B10, B11), not only the first oneEq. (B9) as other authors did. The other factor of the improvement is about computing theweak-current matrix elements for the decays with the obtained relativistic wave functionsas input. It is more ’complete’ than that as done in Refs. [9, 10, 18], i.e. the ’complete’formula in Eq. (15).The paper is organized as follows: the formulations of the exclusive semi-leptonic andnon-leptonic decays are outlined in Sec. II. The newly developed formulations, mainly for the3atrix elements of the hadron weak decays, are presented in Sec. III. In Sec. IV, numericalcalculations for the exclusive semi-leptonic decays and non-leptonic decays are described,the results and comparisons among the various approaches are presented. Finally the Sec. Vis attributed to discussions. In Appendices, the formulations as necessary pieces for thecalculations of the decays are given. II. THE FORMULATIONS FOR EXCLUSIVE SEMI-LEPTONIC DECAYS ANDNON-LEPTONIC DECAYS
Let us now derive the formulations for the exclusive semi-leptonic and non-leptonic decaysprecisely (mainly quoted from Ref. [22]) for numerical calculations later on.In the following subsections we will focus light on the matrix elements of weak currents,and show how to present the amplitudes of the semileptonic or nonleptonic decays via thematrix elements of weak currents precisely. In fact, one may see that the newly developedmethod mainly is about the matrix elements of weak currents.
A. The semileptonic decays of B c meson The Fig. 1 is a typical Feymann diagram responsible for a semileptonic decay of B c mesonto a charmonium. The corresponding amplitude for the decay can be written as: B c , P p m c ¯ c, P f p m p ′ m ′ p ′ m ′ l + ν l FIG. 1: The Feynman diagram of a semileptonic decay of B c meson to a charmonium. T = G F √ V bc ¯ u ν l ( p ν ) γ µ (1 − γ ) υ l ( p l ) h χ c ( h c )( P f ) | J µ | B c ( P ) i , (1)where V bc is the CKM matrix element, h χ c ( h c )( P f ) | J µ | B c ( P ) i is the hadronic weak-currentmatrix element responsible for the decay, and P , P f , p ν and p l are the momenta of initial4tate B c , the finial P -wave state of ( c ¯ c ) (i.e. h c , χ c , χ c , χ c and their excited states), theneutrino and the charged lepton respectively.Generally, the form factors are defined in terms of the matrix elements of weak currentresponsible for the decays appearing in Eq. (1). Namely for the decay of B c meson to scalarcharmonium χ c , the form factors s + and s − are defined as follows: h χ c ( P f ) | V µ | B c ( P ) i = 0 , h χ c ( P f ) | A µ | B c ( P ) i = s + ( P + P f ) µ + s − ( P − P f ) µ . (2)For the decay of B c meson to vector charmonium χ c , the relevant form factors f , u , u and g are defined as follows: h χ c ( P f ) | V µ | B c ( P ) i = f ( M + M f ) ε µ + [ u P µ + u P µf ] ε · PM , h χ c ( P f ) | A µ | B c ( P ) i = 2 gM + M f iǫ µνρσ ε ν P ρ P f σ . (3)For the decay of B c meson to vector charmonium h c , the relevant form factors V , V , V and V are defined as follows: h h c ( P f ) | V µ | B c ( P ) i = V ( M + M f ) ε µ + [ V P µ + V P µf ] ε · PM , h h c ( P f ) | A µ | B c ( P ) i = 2 V M + M f iǫ µνρσ ε ν P ρ P f σ . (4)For the decay of B c meson to tenser charmonium χ c , the relevant form factors k , c , c and h are defined as follows: h χ c ( P f ) | A µ | B c ( P ) i = k ( M + M f ) ε µα P α M + ε αβ P α P β M ( c P µ + c P µf ) , h χ c ( P f ) | V µ | B c ( P ) i = 2 hM + M f iε αβ P α M ǫ µβρσ P ρ P f σ . (5)In the case without considering polarization, we have the squared decay-amplitude withthe polarizations in final states being summed:Σ s ν ,s l ,S χc ( hc ) | T | = G F | V bc | l µν h µν , (6)where l µν is the leptonic tensor: l µν = Σ s ν ,s l ¯ υ l ( p l ) γ µ (1 − γ ) u ν l ( p ν )¯ u ν l ( p ν ) γ ν (1 − γ ) υ l ( p l ) , h µν ≡ Σ S χc ( hc ) h B c ( P ) | J µ + | χ c ( h c )( P f ) ih χ c ( h c )( P f ) | J ν | B c ( P ) i = − αg µν + β ++ ( P + P f ) µ ( P + P f ) ν + β + − ( P + P f ) µ ( P − P f ) ν + β − + ( P − P f ) µ ( P + P f ) ν + β −− ( P − P f ) µ ( P − P f ) ν + iγǫ µνρσ ( P + P f ) ρ ( P − P f ) σ , (7)where the functions α , β ++ , β + − , β − + , β −− , γ are related to the form factors and we putthe relations in Appendix A precisely.The total decay width Γ can be written as:Γ = 12 M (2 π ) Z d ~P f E f d ~p l E l d ~p ν E ν (2 π ) δ ( P − P f − p l − p ν )Σ s ν ,s l ,S χc ( hc ) | T | , (8)where E f , E l and E ν are the energies of the charmonium, the charged lepton and the neutrinorespectively. If we define x ≡ E l /M, y ≡ ( P − P f ) /M , the differential width of the decaycan be reduced to: d Γ dxdy = | V bc | G F M π ( αM ( y − m l M )+ β ++ " x (1 − M f M + y ) − x − y ! + m l M x + 4 M f M − y − m l M ! +( β + − + β − + ) m l M − x + y − M f M + m l M ! + β −− m l M y − m l M ! − " γy − M f M − x + y + M l M ! + 2 γ M l M − M f M ! , (9)here M is the mass of the meson B c , M f is the mass of the charmonium in final state, and thetotal width of the decay is just an integration of the differential width i.e. Γ = R dx R dy d Γ dxdy .Thus the key problem for calculating the semileptonic decays is turned to calculating thehadronic weak-current matrix elements. B. The nonleptonic decays of B c meson In this subsection we mainly consider the nonleptonic two-body decays to a P -wavecharmonium, i.e. decays B c → M M where M is a P -wave charmonium and M is acommon meson. Fig. 2 is the Feynman diagram for the decays via the relevant effectiveHamiltonian H eff [25, 26]: H eff = G F √ ( V cb h c ( µ ) O cb + c ( µ ) O cb i − V tb V ∗ tq X i =3 C i ( µ ) O i !) + h.c., (10)6 c M M FIG. 2: The Feynman diagram of a nonleptonic decay of B c to two mesons M (a charmonium)and M (a common meson). where G F is the Fermi constant, q = d, s , V ij are the CKM matrix elements and c i ( µ ) are thescale-dependent Wilson coefficients. O i are the operators constructed by four quark fieldsand have J µ J µ structure as follows: O cb = [ V ud ( ¯ d α u α ) V − A + V us (¯ s α u α ) V − A + V cd ( ¯ d α c α ) V − A + V cs (¯ s α c α ) V − A ](¯ c β b β ) V − A ,O cb = [ V ud ( ¯ d α u β ) V − A + V us (¯ s α u β ) V − A + V cd ( ¯ d α c β ) V − A + V cs (¯ s α c β ) V − A ](¯ c β b α ) V − A ,O = (¯ q α b α ) V − A X q ′ (¯ q ′ β q ′ β ) V − A , O = (¯ q β b α ) V − A X q ′ (¯ q ′ α q ′ β ) V − A ,O = (¯ q α b α ) V − A X q ′ (¯ q ′ β q ′ β ) V + A , O = (¯ q β b α ) V − A X q ′ (¯ q ′ α q ′ β ) V + A ,O = 32 (¯ q α b α ) V − A X q ′ e q ′ (¯ q ′ β q ′ β ) V + A , O = 32 (¯ q β b α ) V − A X q ′ e q ′ (¯ q ′ α q ′ β ) V + A ,O = 32 (¯ q α b α ) V − A X q ′ e q ′ (¯ q ′ β q ′ β ) V − A , O = 32 (¯ q β b α ) V − A X q ′ e q ′ (¯ q ′ α q ′ β ) V − A , (11)where (¯ q q ) V − A = ¯ q γ µ (1 − γ ) q . The operators O and O are the current-current (tree)operators, O , ..., O are the QCD-penguin operators and O , ..., O are the electroweakpenguin operators. Since we calculate the decay up-to leading order, we just consider thecontribution of O and O .Here we apply the so-called naive factorization to H eff i.e. the operators O i [27], so thenonleptonic two-body decay amplitude T can be reduced to a product of a transition matrixelement of a weak current h M | J µ | B c i and an annihilation matrix element of another weakcurrent h M | J µ | i : T = h M M | H eff | B c i ≈ h M | J µ | B c ih M | J µ | i , (12)while the annihilation matrix element is relating to a decay constant directly. The reasonwhy we adopt the naive factorization here is that it works well enough due to the fact that7ll the decays concerned in this paper are ‘constrained’ to those in them the quark c as a‘spectator’ goes from initial B c meson into the final meson M always, thus as pointed bythe authors of [28, 29], in the concerned cases the corrections to the naive factorization aresuppressed.Since M = χ c ( h c ), the matrix element h M | J µ | B c i is just the hadronic weak-currentmatrix element appearing in the previous subsection, but different from it by momentumtransfer being fixed (owing to the decays are of one to two-body). The annihilation matrixelement h M | J µ | i with J µ = (¯ q q ) V − A is related to the decay constant of a ‘common meson’ M and can be measured via proper processes generally.Precisely, let us now ‘restrict ourselves’ to analyze the B c nonleptonic decays to the P -wave charmonium and the π + , ρ + , etc, which are governed by the weak decay ¯ b → ¯ cu ¯ d , or tothe P -wave charmonium and K + , K ∗ , etc, which are governed by the weak decay ¯ b → ¯ cu ¯ s .As an example, under naive factorization, we have the decay amplitude of B c → χ c ρ + asfollows: T ( B c → χ c ρ + ) = G F √ V cb V ∗ ud a ( µ ) h χ c | J µ | B c ih ρ + | J µ | i , (13)here a = c + N c c and N c = 3 is the number of colors.Since h M | J µ | i is relating to the decay constant of the meson M directly, so to calculatethe widths of the non-leptonic decays is straightforward when the weak-current transitionmatrix elements h M | J µ | B c ( P ) i are well calculated. Thus one may see that the problemto calculate the non-leptonic decays is essentially attributed to calculating the hadronicweak-current matrix elements h M | V µ | B c ( P ) i and h M | A µ | B c ( P ) i appearing in the abovesubsection for semileptonic decays. III. COMPUTATION OF THE TRANSITION-MATRIX ELEMENTS FORWEAK-CURRENTS
From the section above, we can see that to calculate the weak currents matrix elements h M | J µ | B c ( P ) i is the key problem for the concerned semileptonic and nonleptonic decays,so let us now explain the reason why and show how to apply the newly developed method[22] to calculate the matrix elements. In fact it is also to prepare necessary formulae forfinal numerical calculations.Here the weak-current matrix elements are for ‘transitions’ from a state of a double heavymeson to another double heavy meson. Due to the mass difference of the two states, the8elativistic effects for the transitions are great, that a proper formulation to deal with therelativistic effects is desired. It is known that the approach of relativistic B-S equation for thebound states and Mandelstam formulation for the transition matrix elements may be takeninto account quite well, and furthermore the B-S equation and Mandelstam formulationeven under ‘instantaneous approximation’ still works, because here the involved mesons aredouble heavy. While the newly developed method [22], which applies the ‘instantaneousapproximation’ to the current matrix elements and B-S equation completely, should bebetter than the original one in Ref. [18], where the ‘instantaneous approximation’ is appliedincompletely. The ’completeness’ here means to apply it to the B-S equation, the solutions(B-S wave functions) and the transition matrix element (under Mandelstam formulation)properly, and let us outline it below.According to the Mandelstam formulation [21], the corresponding hadronic matrix ele-ments of weak current between the double heavy meson B c in initial state and the doubleheavy meson χ c ( h c ) in final state, appearing in Eq. (1), Eq. (12) and Eq. (13), can be writtenas: h χ c ( h c )( P f ) | J µ | B c ( P ) i = i Z d qd q ′ (2 π ) T r h χ χ c ( h c ) ( P ′ , q ′ )( p − m ) χ Bc ( P, q ) V cb γ µ (1 − γ ) δ ( p − p ′ ) i = i Z d q (2 π ) T r h χ χ c ( h c ) ( P ′ , q ′ )( α P + q − m ) χ Bc ( P, q ) V cb γ µ (1 − γ ) i , (14)where p = α P + q ( α ≡ m m + m ), p = α P − q ( α ≡ m m + m ) are the momenta of c -quarkand ¯ b -quark respectively inside B c meson; p ′ = α ′ P f + q ′ ( α ′ ≡ m ′ m ′ + m ′ ), p ′ = α ′ P f − q ′ ( α ′ ≡ m ′ m ′ + m ′ ) are the momenta of c -quark and ¯ c -quark respectively inside the P -wave charmonium χ c ( h c ); moreover, for the final result (the last line of Eq. (14)) we have P = P f + p l + p ν and q ′ = α P + q − α ′ P f .The newly developed method [22] essentially is to apply the ‘instantaneous approxima-tion’ to the current matrix elements and the B-S equation completely, to outline it and for‘applying the instantaneous approximation’ in a covariant way, we need to decompose therelative momentum q into two components: the time-like one q µ k and the space-like one q µ ⊥ as follows: q µ = q µ k + q µ ⊥ , q µ k ≡ P · qM P µ , q µ ⊥ ≡ q µ − q µ k ,P ′ µ = P ′ µ k + P ′ µ ⊥ , P ′ µ k ≡ ( P · P ′ ) M P µ , P ′ µ ⊥ ≡ P ′ µ − P ′ µ k ;9nd q ′ µ = q ′ µ k + q ′ µ ⊥ , q ′ µ k ≡ ( P · q ′ /M ) P µ , q ′ µ ⊥ ≡ q ′ µ − q ′ µ k , where M is the mass of the meson B c , and we may further have two Lorentz invariantvariables q P ≡ P · qM and q T ≡ q − q ⊥ .The ‘instantaneous approximation’ applying to the matrix element is just to carry outthe integration of dq µ k by a contour one on Eq. (14) precisely and to obtain the result below: h χ c ( h c )( P f ) | J µ | B c ( P ) i = i Z d q (2 π ) T r h χ χ c ( h c ) ( P ′ , q ′ )( α P + q − m ) χ Bc ( P, q ) V cb γ µ (1 − γ ) i = Z d q ⊥ (2 π ) T r (h ¯ ϕ ′ ++ ( q ′⊥ ) PM ϕ ++ ( q ⊥ ) + ¯ ϕ ′ ++ ( q ′⊥ ) PM ψ + − ( q ⊥ ) − ¯ ψ ′− + ( q ′⊥ ) PM ϕ ++ ( q ⊥ ) − ¯ ψ ′ + − ( q ′⊥ ) PM ϕ −− ( q ⊥ )+ ¯ ϕ ′−− ( q ′⊥ ) PM ψ − + ( q ⊥ ) − ¯ ϕ ′−− ( q ′⊥ ) PM ϕ −− ( q ⊥ ) i γ µ (1 − γ ) ) , (15)where: ϕ ++ ( q ⊥ ) = Λ +1 ( q ⊥ ) η ( q ⊥ )Λ +2 ( q ⊥ ) M − ω − ω , ¯ ϕ ′ ++ ( q ′ P ⊥ ) = Λ ′ +2 ( q ′ P ⊥ )¯ η ( q ′ P ⊥ )Λ ′ +1 ( q ′ P ⊥ ) E f − ω ′ − ω ′ ,ϕ −− ( q ⊥ ) = − Λ − ( q ⊥ ) η ( q ⊥ )Λ − ( q ⊥ ) M + ω + ω , ¯ ϕ ′−− ( q ′ P ⊥ ) = − Λ ′− ( q ′ P ⊥ )¯ η ( q ′ P ⊥ )Λ ′− ( q ′ P ⊥ ) E f + ω ′ + ω ′ ,ψ − + ( q ⊥ ) = Λ − ( q P ⊥ ) η ( q ⊥ )Λ +2 ( q P ⊥ ) M − ω − ω ′ − E f , ¯ ψ ′− + ( q ′ P ⊥ ) = Λ ′− ( q ′ P ⊥ )¯ η ′ ( q ′ P ⊥ )Λ ′ +1 ( q ′ P ⊥ ) M − ω − ω ′ − E f ,ψ + − ( q ⊥ ) = Λ +1 ( q P ⊥ ) η ( q ⊥ )Λ − ( q P ⊥ ) M + ω + ω ′ − E f , ¯ ψ ′ + − ( q ′ P ⊥ ) = Λ ′ +2 ( q ′ P ⊥ ) ¯ η ′ ( q ′ P ⊥ )Λ ′− ( q ′ P ⊥ ) M + ω + ω ′ − E f , (16) ϕ ij ( q ⊥ ) , ψ ij ( q ⊥ ) and ¯ ϕ ′ ij ( q ′ P ⊥ ) , ¯ ψ ′ ij ( q ′ P ⊥ ) are B-S wave functions as the B-S equation solu-tions under ‘complete instantaneous approximation’ [23, 24] and with ‘energy projection’Λ ± of the mesons in initial and finial states properly. The precise definitions of the ‘energyprojection’ and the B-S ‘vertex’ η P , ¯ η P ( η ′ P , ¯ η ′ P ) are presented in Appendix B. One mayalso see that the four equations, Eqs. (B9, B10, B11), are B-S equations under the completeinstantaneous approximation, instead of the incomplete instantaneous approximation whichonly considering the Eq. (B9).Namely the ‘improvements’ from the ‘newly development method’ are attributed to: i).with the complete instantaneous approximation to current matrix element, as a result, thereare six terms in the squared bracket of Eq. (15) instead of the first term h χ c ( h c )( P f ) | J µ | B c ( P ) i = Z d q ⊥ (2 π ) T r n ¯ ϕ ′ ++ ( q ′⊥ ) PM ϕ ++ ( q ⊥ ) γ µ (1 − γ ) o (17)10s only kept; ii). the B-S wave functions hidden in ϕ ij ( q ⊥ ) , ψ ij ( q ⊥ ) and ¯ ϕ ′ ij ( q ′ P ⊥ ) , ¯ ψ ′ ij ( q ′ P ⊥ )are solved under complete instantaneous approximation to the B-S equation. For the pointi), since the considered double heavy meson, B c or χ c ( h c ), is weak binding system i.e. thebinding energy ε ≡ M − ω − ω (or ε ≡ E f − ω ′ − ω ′ ) is small ( εM ≪ O (1)), thusfrom Eq. (16) we are sure that ϕ ++ ( q ⊥ ) and ¯ ϕ ′ ++ ( q ′ P ⊥ ) are much greater than the others ϕ ij ( q ⊥ ) , ψ ij ( q ⊥ ) and ¯ ϕ ′ ij ( q ′ P ⊥ ) , ¯ ψ ′ ij ( q ′ P ⊥ ), so that using the Eq. (17) instead of Eq. (15) isa very good approximation, which we have precisely examined by considering the decay B c → χ c lν l as an example: in fact, the contributions of the second term and third term ofEq. (15) to the form factor are less than the one of first term of Eq. (15) roughly by a factor10 − ∼ − times. If the first three terms are considered, the decay width is 1 . × − GeV, while if only the first term is considered, the decay width is 1 . × − GeV, i.e. thetwo results are very similar. So the approximation is very good and we may use Eq. (17)instead of Eq. (15) to compute the weak-current matrix elements safely.
IV. NUMERICAL CALCULATIONS AND RESULTS WITH PROPER COMPAR-ISONS
In this section, based on the formulations obtained in the paper, we evaluate the decaywidths for semileptonic and nonleptonic decays and some interesting quantities else forsemileptonic decays, such as form factors and charged lepton spectrum etc and then discussthem briefly.First of all, we need to fix the parameters appearing in the framework. We adjustedthe parameters a = e = 2 . λ = 0 .
21 GeV , Λ QCD = 0 .
27 GeV, m b = 4 .
96 GeV, m c = 1 .
62 GeV and V for the B-S kernel as those in Refs. [24, 30, 31], which as the bestinput for spectroscopy, then the spectra of the mesons and the masses M B c = 6 .
276 GeV, M χ c = 3 .
414 GeV, M χ c = 3 .
510 GeV, M χ c = 3 .
555 GeV, M h c = 3 .
526 GeV etc [24], whichare used in this paper, are obtained, moreover the decay constants, average energies as wellas annihilations of quarkonia are fitted [30–32].With the obtained B-S wave functions (under the formulation defined in Appendix B)and as a next step, we substitute the functions into ϕ ++ ( q ⊥ ) and ¯ ϕ ′ ++ ( q ′ P ⊥ ), so that they arerelated to the components of the B-S wave functions precisely as depicted in Appendix C.With the formula Eq. (17), finally we represent the hadronic transition weak-current matrixelements as proper integrations of the components of the B.-S. wave functions. As final re-11 ABLE I: The semileptonic decay widths (in the unit 10 − GeV)Mode This work [12] [13] [15] [10] [16] [17] B + c → χ c eν . ± .
46 1.27 2.52 1.55 1.69 2.60 ± B + c → χ c τ ν . ± .
12 0.11 0.26 0.19 0.25 0.7 ± B + c → χ c eν . ± .
45 1.18 1.40 0.94 2.21 2.09 ± B + c → χ c τ ν . ± .
10 0.13 0.17 0.10 0.35 0.21 ± B + c → χ c eν . ± .
39 2.27 2.92 1.89 2.73 B + c → χ c τ ν . ± .
07 0.13 0.20 0.13 0.42 B + c → h c eν . ± .
10 1.38 4.42 2.4 2.51 2.03 ± ± B + c → h c τ ν . ± .
20 0.11 0.38 0.21 0.36 0.20 ± ± sults of this paper, the decay widths for the semileptonic and nonleptonic decays and someinteresting quantities else for the semileptonic decays, such as form factors and charged lep-ton spectrum etc, are straightforwardly calculated numerically. In the following subsectionswe present the results for the semileptonic decays and nonleptonic decays separately. A. The semi-leptonic decays
When the weak current transition matrix element for a definite semi-leptonic decay iscalculated precisely and the values of the CKM matrix elements | V ud | = 0 . | V us | = 0 . | V bc | = 0 . α , β ++ , β + − , β − + , β −− , γ appearing in the spectrum of the chargedlepton (see Eq. (9)) are related to the form factors directly as shown in Appendix A precisely.Therefore when we calculate and present the results for semi-leptonic decays, not only thoseof the decay widths but also the spectrums of the charged lepton in the decays are considered.Since τ lepton is quite massive and m µ ≃ m e is quite a good approximation for the B c mesondecays, so when we calculate and present the widths and the spectrums of the charged leptonfor the decays, only the cases that the lepton being electron or τ are considered.Note that since the input B-S wave functions by solving the B-S equation for the double12 - -s + B c →c c0 t m -t (GeV ) u -u g-fB c →c c1 t m -t(GeV ) -k-c c hB c →c c2 t m -t (GeV ) V V V -V B c → h c t m -t (GeV ) FIG. 3: The form factors of the B c decays to a P -wave charmonium defined as in Eq.(A1), Eq.(A2),Eq.(A3) and Eq.(A4) and t = q = ( P − P f ) = M + M f − M E f ( t m is the maximum of t ). heavy mesons which are involved in the transition matrix elements of weak current haveuncertainties, due to the parameters fitting to fix the B-S kernel and quark masses, the wayto solve the B-S equation numerically, and the approximation from Eq. (15) to Eq. (17)for the transition matrix elements of the weak currents is taken etc, so in the numericalresults obtained finally there are certain errors. To consider the uncertainties caused bythe input parameters, we changed all the input parameters simultaneously within 5% ofthe center values, then we get the uncertainties of numerical results for the semi-leptonicdecays and the non-leptonic decays shown in Table. I. We find that the uncertainties of thedecays B c → h c ( χ c ) + e + ν e vary up to 30% of center values, while the uncertainties of13 p fi e |GeV d G / ( G d | p fi e | ) G e V - |p fi t | GeV d G / ( G d | p fi t | ) G e V - FIG. 4: The energy spectrums of the charged lepton in the B c semileptonic decays to P -wavecharmoniums. The left figure is for B c → χ c , , ( h c ) eν and the right figure is for B c → χ c , , ( h c ) τ ν .Where the solid lines are the results for χ c , the dash lines are for χ c , the dot lines are for χ c andthe dot-dash lines are for h c . B c → h c ( χ c ) + τ + ν τ are up to 60% in Table. I, the reason is that the phase spaces for B c → h c ( χ c ) + τ + ν τ are smaller than the ones for B c → h c ( χ c ) + e + ν e because of theheavy τ lepton, and the the uncertainties for the former are more sensitive to the changesof the phase space than the latter.To compare with the results obtained by the other approaches, we present the decaywidths calculated out this work with error bar and the results obtained by the other ap-proaches by putting them together in a table i.e. Table I.In addition we also present the obtained form factors and the spectrums of the chargedlepton in the decays in Fig. 3 and Fig. 4 respectively. To compare with the results of theprevious work Ref. [10], we draw the curves of the spectrums of charged lepton obtained bythis work and the work Ref. [10] in Fig. 5. Whereas in order to see the tendency of the formfactors and the lepton spectrum clearly and we suspect that at present stage it is enough, soin the figures we draw the curves with the center values but not involve the errors precisely.14 p fi e |GeVB c →c c0 d G / ( G d | p fi e | ) G e V - |p fi e |GeVB c →c c1 d G / ( G d | p fi e | ) G e V - |p fi e |GeVB c →c c2 d G / ( G d | p fi e | ) G e V - |p fi e |GeVB c → h c d G / ( G d | p fi e | ) G e V - FIG. 5: The energy spectrums of the charged lepton in the B c semileptonic decays to P -wavecharmoniums respectively. The solid lines are the results of this work, the dash lines are the resultsof [10]. B. The non-leptonic decays
The exclusive non-leptonic decays are of two-body in final states, thus the hadronictransition matrix elements of weak-currents appearing in Eq. (13) have a fixed momentumtransfer t = m M (the mass squared of the other meson M in the decay B c → M M and M = χ c or h c ). In fact the transition matrix elements have been already calculated in theabove subsection of semi-leptonic decays. To calculate the decay widths, from Eq. (13), nowwe need to calculate the annihilation matrix element of the weak current such as h M | J µ | i additionally. It is known that the annihilation matrix element is related to the ‘decay15 ABLE II: The nonleptonic decay widths (in the unit 10 − GeV)Mode This work [12] [13] [15] [10] B + c → χ c π + (0 . ± . a a a a a B + c → χ c π + (0 . ± . a a a a a B + c → χ c π + (0 . ± . a a a a a B + c → h c π + (1 . ± . a a a a a B + c → χ c ρ + (0 . ± . a a a a a B + c → χ c ρ + (0 . ± . a a a a a B + c → χ c ρ + (0 . ± . a a a a a B + c → h c ρ + (2 . ± . a a a a a B + c → χ c K + (0 . ± . a a a a a B + c → χ c K + (0 . ± . a a a a a B + c → χ c K + (0 . ± . a a a a a B + c → h c K + (0 . ± . a a a a a B + c → χ c K ∗ + (0 . ± . a a a a a B + c → χ c K ∗ + (0 . ± . a a a a a B + c → χ c K ∗ + (0 . ± . a a a a a B + c → h c K ∗ + (0 . ± . a a a a a constant’ f M directly, and the decay constant f P , f V or f A of a pseudoscalar meson, avector meson or an axial vector meson may be extracted from experimental data for thepure leptonic decays of the relevant mesons, but they may also be calculated by models,such as the one in Ref. [30] although there are some debates. In this work we adopt the valuesof the decay constants: f π = 0 .
130 GeV, f ρ = 0 .
205 GeV, f K = 0 .
156 GeV, f K ∗ = 0 . a = 1 .
14 for non-leptonic decays as done in most references, and16
ABLE III: Branching ratios (in %) of B c decays calculated for the B c lifetime τ B c = 0 .
453 ps and a = 1 .
14. Decay Br Decay Br B + c → χ c eν . ± . B + c → χ c τ ν . ± . B + c → χ c eν . ± . B + c → χ c τ ν . ± . B + c → χ c eν . ± . B + c → χ c τ ν . ± . B + c → h c eν . ± . B + c → h c τ ν . ± . B + c → χ c π + . ± . B + c → χ c ρ + . ± . B + c → χ c π + . ± . B + c → χ c ρ + . ± . B + c → χ c π + . ± . B + c → χ c ρ + . ± . B + c → h c π + . ± . B + c → h c ρ + . ± . B + c → χ c K + . ± . B + c → χ c K ∗ + . ± . B + c → χ c K + . ± . B + c → χ c K ∗ + . ± . B + c → χ c K + . ± . B + c → χ c K ∗ + . ± . B + c → h c K + . ± . B + c → h c K ∗ + . ± . the experimental value of B c lifetime τ B c = 0 .
453 ps as well, we calculate branching ratiosof the decays and put them in Table III.
V. DISCUSSIONS AND CONCLUSIONS
In Sec. IV, the form factors (Fig. 3), energy spectrums of the charge leptons (Fig. 4 andFig. 5), decay widths (Table I) for the semileptonic decays, and the decay widths for non-leptonic decays (Table II) are presented. Specially in tables some comparisons with otherapproaches else are also given. Thus one may read off a lot of interesting matters already.Since the form factors for the semi-leptonic decays, which are directly related to overlap-ping integrations of the components of the B-S wave functions of the initial and final statesas shown in Appendix C, are comparatively difficult to be measured, so in Fig. 3 we show thebehaviors of the form factors briefly (without errors). Whereas the energy spectrums of thecharged lepton in the decays may be measured not so difficult, as long as the event exampleis great enough and the abilities of the detector are strong enough, and to see the differences17etween the spectrums of electron and τ lepton clearly in Fig. 4 we plot the curves withcenter values without theoretical uncertainties. Moreover to see the differences between thiswork and the ones [10], in Fig. 5 we plot the spectrums of electron obtained by this work vsthe ones [10] obtained by previous approach and for both of them only center value withouttheoretical uncertainties are taken. Since the spectrums of muon ( µ ) is very similar to thatof electron in exclusive semi-leptonic decays, thus we do not present the spectrums of muonat all. From Fig. 4 we can see the difference in the energy spectrums among the B c decays todifferent P -wave charmonia clearly, although the results of electron is greater than the oneof τ lepton. From Fig. 5 we can see that the difference in the energy spectrums of electrondue to different approaches: the difference caused by newly improved approach and by theprevious approach can be quite sizable and can be tested experimentally in future. Forthe widths of the decays, from Table I and Table II, both the semi-leptonic decays and thenon-leptonic decays, one may see that in general the results of this work fall into the regionof the predictions by various models, but the distribution of the predictions is quite wide,so future experimental data will be critical and may conclude which one of the predictionsis more reliable.Considering the fact that the substantial tests of the B c -meson decays have not beenstarted yet, although the meson B c has been observed at Tevatron for years and LHCis running now, according to the estimates of the production at LHC, one may believereasonably that the tests of the predictions on the B c decays will be started with LHC moremeasurements available. From theoretical point of view, we think that the newly improvedapproach works better than the previous one, this trust need to be tested by experiments.We would also like to note here that according to the estimates [33–36] of the productionat an e − e + collider running at CM energy √ S ≃ m Z ( m Z is Z-boson mass) with very highluminosity ( L = 10 ∼ cm − s − ) i.e. a “Super- Z -Factory” and considering the advantages,may be more suitable to test the approaches by measuring the decays precisely than that todo them at hadronic collider such as Tevatron or LHC, because at such a Super- Z -Factorynumerous B c mesons may be produced and the energy-momentum of the produced B c meson,as the e − e + one of the collider, is precisely known in an e + − e − collider environment.18 cknowledgments This work was supported in part by Natural Science Foundation of China (NSFC) underGrant No.10875032, No.10805082, No.10875155, No.10847001. This research was also sup-ported in part by the Project of Knowledge Innovation Program (PKIP) of Chinese Academyof Sciences, Grant No. KJCX2.YW.W10.
Appendix A: The functions α , β ++ , β + − , β − + , β −− , γ Here according to the P -wave charmonium appearing in the final state we present theuseful functions α , β ++ , β + − , β − + , β −− , γ how precisely to relate to the form factors in turn. a). When B c decays to χ c :Since the matrix elements of weak currents are described in terms of two form factors( s + , s − ): h χ c ( P f ) | V µ | B c ( P ) i = 0 , h χ c ( P f ) | A µ | B c ( P ) i = s + ( P + P f ) µ + s − ( P − P f ) µ , then the functions are read as β ++ = s , β + − = β − + = s + s − , β −− = s − . (A1) b). When B c decays to χ c :Since the matrix elements of weak currents can be described in terms of four form factors( f, u , u , g ): h χ c ( P f ) | V µ | B c ( P ) i = f ( M + M f ) ε µ + [ u P µ + u P µf ] ε · PM , h χ c ( P f ) | A µ | B c ( P ) i = 2 gM + M f iǫ µνρσ ε ν P ρ P f σ , then the functions are read as α = f + 4 M g ~p f ,β ++ = f M f − M g y + 12 " M M f (1 − y ) − f u + + M ~p f M f u ,β + − = β − + = g ( M − M f ) − f M f − f ( u + + u − ) − M E f M f f ( u + − u − ) + u + u − M ~p f M f ,β −− = − g ( M + 2 M E f + M f ) + f M f − M E f M f + 1 ! f u − + u − M ~p f M f ,γ = − f g (A2)19hen setting f = f ( M + M f ) , u + = ( u + u )2 M , u − = ( u − u )2 M , g = gM + M f . c). When B c decays to h c :Since the matrix elements of weak currents can be described in terms of four invariantform factors ( V , V , V , V ): h h c ( P f ) | V µ | B c ( P ) i = V ( M + M f ) ε µ + [ V P µ + V P µf ] ε · PM , h h c ( P f ) | A µ | B c ( P ) i = 2 V M + M f iǫ µνρσ ε ν P ρ P f σ , then the functions are read as α = f + 4 M g ~p f ,β ++ = f M f − M g y + 12 " M M f (1 − y ) − f a + + M ~p f M f a ,β + − = β − + = g ( M − M f ) − f M f − f ( a + + a − ) − M E f M f f ( a + − a − ) + a + a − M ~p f M f ,β −− = − g ( M + 2 M E f + M f ) + f M f − M E f M f + 1 ! f a − + a − M ~p f M f ,γ = − f g , (A3)when setting f = V ( M + M f ) , a + = ( V + V )2 M , a − = ( V − V )2 M , g = V M + M f . d). When B c decays to χ c :Since the matrix elements of weak currents can be described in terms of four form factors( k, c , c , h ): h χ c ( P f ) | A µ | B c ( P ) i = k ( M + M f ) ε αµ P α M + ε αβ P α P β M ( c P µ + c P µf ) , h χ c ( P f ) | V µ | B c ( P ) i = 2 hM + M f iε αβ P α M ǫ µβρσ P ρ P f σ , where ε αβ ( ε αµ ) is the polarization tensor of tensor meson, then the functions are read as α = c k + 4 M h ~p f ) ,β ++ = ck M f − ch M y + 23 c c + 43 ck c + M (1 − y ) + M f M f − ! + k M (1 − y ) + M f M f − ! ,β + − = β − + = − ck M f + ch M − M f ) + k M (1 − y ) + M f M f ! − + 23 c c + c − ck c + M (1 − y ) + M f M f + 12 ! + 23 ck c − M (1 − y ) + M f M f − ! ,β −− = ck M f − ch M + M f ) − M y ) + 23 c c − + 43 ck c − − M (1 − y ) + M f M f − ! + k M (1 − y ) + M f M f + 12 ! ,γ = − ch k , (A4)when setting c = M ~p f M f , k = k (1 + M f M ) , c + = c + c M , c − = c − c M , h = hM ( M + M f ) . Appendix B: The B-S equation under ‘complete instantaneous approximation’
In this appendix we outline the ‘complete instantaneous approximation’ onto the Bethe-Salpeter equation when it has an instantaneous kernel, which describes a double heavy mesonquite well.The Bethe-Salpeter equation [20] is read as( p − m ) χ p ( q )( p + m ) = i Z d k (2 π ) V ( P, k, q ) χ p ( k ) , (B1)where χ p ( q ) is B-S wave function of the relevant bound state, P is the four momentum ofthe meson state and p , p , m , m are the momenta and constituent masses of the quarkand anti-quark respectively. From the definition, they relate to the total momentum P andrelative momentum q as follows: p = α P + q, α ≡ m m + m ,p = α P − q, α ≡ m m + m . The interaction kernel V ( P, k, q ) for a double heavy system, being instantaneous approx-imately, can be treated as a potential after doing instantaneous approximation, i.e. thekernel take the simple form (in the rest frame) [19] V ( P, k, q ) ⇒ V ( | ~k − ~q | ) . For various usages, we divide the relative momentum q into two parts, q µ = q µ k + q µ ⊥ , q µ k ≡ P · qM P µ , q µ ⊥ ≡ q µ − q µ k , M is the mass of the meson, and we may have two Lorentz invariant variables: q P ≡ P · qM , q T ≡ q − q ⊥ . For the convenience below, let us introduce the definitions: ϕ p ( q µ ⊥ ) ≡ i Z d q p π χ p ( q µ k , q µ ⊥ ) , η ( q µ ⊥ ) ≡ Z d k ⊥ (2 π ) V ( k ⊥ , q ⊥ ) ϕ p ( k µ ⊥ ) , (B2)then the B-S equation can be rewritten as χ ( q k , q ⊥ ) = S ( p ) η ( q ⊥ ) S ( p ) . (B3)Owing to Eqs. (B2, B3), it is reasonable and for convenience we may call η ( q ⊥ ) as ‘instan-taneous B-S vertex’. The propagator of quark or anti-quark may be decomposed: S i ( p i ) = Λ + i ( q ⊥ ) J ( i ) q P + α i M − w i + iǫ + Λ − i ( q ⊥ ) J ( i ) q P + α i M − w i + iǫ , where i =1, 2 for quark and anti-quark respectively, and J ( i ) = ( − i +1 , ω = q m + q T , ω = q m + q T , and Λ ± , Λ ± are the generalized energy projection operators,Λ ± ( q ⊥ ) ≡ ω [ PM ω ± ( m + q ⊥ )] , Λ ± ( q ⊥ ) ≡ ω [ PM ω ∓ ( m + q ⊥ )] , (B4)and have the properties:Λ + iP ( q µP ⊥ ) + Λ − iP ( q µP ⊥ ) = PM , Λ ± iP ( q µP ⊥ ) PM Λ ∓ iP ( q µP ⊥ ) = 0 , Λ ± iP ( q µP ⊥ ) PM Λ ± iP ( q µP ⊥ ) = Λ ± iP ( q µP ⊥ ) , (B5)The instantaneous approximation to the B-S equation is to do contour integration over q P on both sides of Eq. (B3), and obtains: ϕ p ( q ⊥ ) = Λ +1 ( q ⊥ ) η ( q ⊥ )Λ +2 ( q ⊥ ) M − ω − ω − Λ − ( q ⊥ ) η ( q ⊥ )Λ − ( q ⊥ ) M + ω + ω , (B6)If we introduce the notations: ϕ ±± p ( q ⊥ ) ≡ Λ ± ( q ⊥ ) PM ϕ p ( q ⊥ ) PM Λ ± ( q ⊥ ) , (B7)we have ϕ p ( q ⊥ ) = ϕ ++ p ( q ⊥ ) + ϕ + − p ( q ⊥ ) + ϕ − + p ( q ⊥ ) + ϕ −− p ( q ⊥ ) , (B8)22ith the properties Eq. (B5) and notations Eq. (B7), the full Salpeter equation Eq. (B6)can be written as ( M − ω − ω ) ϕ ++ p ( q ⊥ ) = Λ +1 ( q ⊥ ) η ( q ⊥ )Λ +2 ( q ⊥ ) , (B9)( M + ω + ω ) ϕ −− p ( q ⊥ ) = − Λ − ( q ⊥ ) η ( q ⊥ )Λ − ( q ⊥ ) , (B10) ϕ + − p ( q ⊥ ) = ϕ − + p ( q ⊥ ) = 0 . (B11)The normalization condition for the B-S equations now is read as: Z q T dq T π T r [ ¯ ϕ ++ PM ϕ ++ PM − ¯ ϕ −− PM ϕ −− PM ] = 2 P . (B12)The couple equations Eq. (B9), Eq. (B10) and Eq. (B11) with the normalization conditionEq. (B12) are the final B-S (Salpeter) equation under ‘complete instantaneous approxima-tion’ vs the previous one i.e. Salpeter equation [19] where only Eq. (B9) is considered.In addition, note that in the model used here for the double heavy quark-antiquarksystems, the QCD-inspired interaction kernel V , being instantaneous approximately anddictating the Cornell potential which is composed by a linear scalar interaction plus a vectorinteraction, is read as: V ( ~q ) = V s ( ~q ) + V v ( ~q ) γ ⊗ γ ,V s ( ~q ) = − ( λα + V ) δ ( ~q ) + λπ ~q + α ) ,V v ( ~q ) = − π α s ( ~q )( ~q + α ) , (B13)where the QCD running coupling constant α s ( ~q ) = π − N f a + ~q / Λ QCD ) ; the constants λ, α, a, V and Λ QCD are the parameters characterizing the potential.
Appendix C: The reduced wave functions ϕ ++ ( ~q ) and the form factors In the appendix we present the reduced wave functions ϕ ++ ( ~q ) (and ψ + − ( ~q )) whichdirectly relate to the solutions by newly solving the obtained coupled equations Eq. (B9),Eq. (B10) and Eq. (B11) under a new approach. The key point of the new approach is tosolve the B-S equation according to the quantum numbers of the concerned bound statesrespectively [24, 30, 31], i.e. to solve the equation under the new approach we need to give23he most general formulation for the wave function first. Therefore for the present usage,in this appendix, we precisely quote the solutions for the low-laying bound states B c mesonwith quantum numbers J P = 0 − , χ c with quantum numbers J P C = 0 ++ , χ c with quantumnumbers J P C = 1 ++ , χ c with quantum numbers J P C = 2 ++ and h c with quantum numbers J P C = 1 + − from [24, 30, 31], and then we write down the reduced wave functions ϕ ++ ( ~q )and the form factors accordingly.When the weak-current matrix elements are computed precisely, as an intermediate step,the form factors can be represented as overlapping integrations of the components appearingin the B-S solutions, thus in this appendix we also give the formulas of the form factors interms of the ‘overlapping integrations’. a). For B c meson with quantum numbers J P = 0 − The B-S wave function (solution of Eq. (B9), Eq. (B10) and Eq. (B11) of B c meson with J P = 0 − is read as: ϕ Bc ( ~q ) = M " PM f ( ~q ) n − 6 q ⊥ ( w + w ) m w + m w o + f ( ~q ) n q ⊥ ( w − w ) m w + m w o γ , (C1)where M, P are the mass and the total momentum of the meson B c , q ⊥ = (0 , ~q ), ~q is therelative momentum of quark and anti-quark in the meson, so q ⊥ = − ~q .Then we can rewrite the reduced wave function: ϕ ++ Bc ( ~q ) = b " b + PM + b q ⊥ + b q ⊥ PM γ , (C2)where b = M (cid:18) f ( ~q ) + f ( ~q ) m + m w + w (cid:19) , b = w + w m + m ,b = − ( m − m ) m w + m w , b = ( w + w )( m w + m w ) . In Appendix. B in Eq. (B2), we have η ( q ⊥ ) = Z d kV ( ~k ) M " PM f ( ~k ) n − 6 k ⊥ ( w + w ) m w + m w o + f ( ~k ) n k ⊥ ( w − w ) m w + m w o γ , (C3)where w = q m − k ⊥ , w = q m − k ⊥ , V ( ~k ) = V s ( ~k ) + V v ( ~k ) γ ⊗ γ .According to Eq. (C1), η ( q ⊥ ) = Z d k ( V s ( ~k ) + V v ( ~k ) γ ⊗ γ )24 " PM f ( ~k ) n − 6 k ⊥ ( w + w ) m w + m w o + f ( ~k ) n k ⊥ ( w − w ) m w + m w o γ = M " g PM + g + g q ⊥ M + g P q ⊥ M γ . (C4) g = Z d k [ V s − V v ] f ( ~k ) , g = Z d k [ V s − V v ] f ( ~k ) ,g = Z d k [ V s + V v ] ~k · ~q | ~q | f ( ~k ) ( w − w ) m w + m w , g = Z d k [ V s + V v ] f ( ~k ) ( w + w ) m w + m w . So we can also write down the wave function of ψ + − ( q ⊥ ), ψ + − ( q ⊥ ) = Λ +1 ( q P ⊥ ) η ( q ⊥ )Λ − ( q P ⊥ ) M + ω + ω ′ − E f = " n PM + n + n q ⊥ + n q ⊥ PM γ . (C5)set tt = w w ( M + ω + ω ′ − E f ) , where the symbol ′ , denotes of the final state, and n = tt [ g M ( − q + m m − w w ) + g M ( m w − m w ) + g ( w + w ) q + g ( m + m ) q ] ,n = tt [ g M ( m w − m w ) + g M ( q + m m − w w ) + g ( m − m ) q + g ( w − w ) q ] ,n = tt [ − g M ( w + w ) − g M ( m − m ) + g ( q + m m + w w ) + g ( m w + m w )] ,n = tt [ g M ( m + m ) + g M ( w − w ) − g ( m w + m w ) − g ( − q + m m + w w )] . b). For the charmonium χ c ( J P C = 0 ++ ) and the form factors s + and s − The B-S wave function (solution of Eq. (B9), Eq. (B10) and Eq. (B11) under new methodto solve the coupled equations) of χ c is read as: ϕ χ c ( ~q ′ ) = f ′ ( ~q ′ ) q ′⊥ + f ′ ( ~q ′ ) P f q ′⊥ M f + f ′ ( ~q ′ ) M f + f ′ ( ~q ′ ) P f , (C6)with constraints on the components of wave function, for the charmonium, m ′ = m ′ , w ′ = w ′ , we get: f ′ ( ~q ′ ) = f ′ ( ~q ′ ) q ′ ⊥ M f m ′ , f ′ ( ~q ′ ) = 0 , where M f , P f are the mass and the total momentum of final meson χ c , q ′⊥ = (0 , ~q ′ ), ~q ′ is therelative momentum of quark and anti-quark in the meson, so q ′ ⊥ = − ~q ′ . Then the reducedwave function ϕ ++ P ( ~q ′ ) as: ϕ ++ χ c ( ~q ′ ) = a " q ′⊥ + a P f q ′⊥ M f + a + a P f M f , (C7)25ith a = 12 f ′ ( ~q ′ ) + f ′ ( ~q ′ ) m ′ w ′ ! , a = w ′ m ′ , a = q ′ ⊥ m ′ , a = 0 . The wave function of ¯ ψ ′− + ( q ′ P ⊥ ) is¯ ψ ′− + ( q ′ P ⊥ ) = Λ ′− ( q ′ P ⊥ )¯ η ′ ( q ′ P ⊥ )Λ ′ +1 ( q ′ P ⊥ ) M − ω − ω ′ − E f = n ′ q ′⊥ + n ′ q ′⊥ P f M f + n ′ + n ′ P f M f . (C8)Set tt ′ = w ′ ( M − ω − ω ′ − E f ) , where n ′ = tt ′ [ − g ′ q ′ + 2 g ′ M f m ′ ] , n ′ = 0 ,n ′ = tt ′ [ − g ′ m ′ q ′ + 2 g ′ M f m ′ ] , n ′ = tt ′ [ − g ′ w ′ q ′ + 2 g ′ M f m ′ w ′ ] , and g ′ = Z d k ′ [ V s − V v ] ~k ′ · ~q ′ | ~q ′ | f ′ ( ~k ′ ) , g ′ = Z d k ′ [ V s − V v ] ~k ′ · ~q ′ | ~q ′ | f ′ ( ~k ′ ) ,g ′ = Z d k ′ [ V s + V v ] f ′ ( ~k ′ ) k ′ ⊥ M f m ′ , g ′ = 0 . With Eq. (17), the form factors may be presented by overlapping integrations: s + = 12 Z d q (2 π ) a b M M f h a b M f + α E f ( a b E f + M f + a b ~q · ~P f )+ b ( M f q + α M f ~q · ~P f ) + M ( a b q − α a b E f − α M f )+ M q cos θ | ~P f | (1 − E f M )( a b E f − a b M f + M f + a b ~q · ~P f ) , (C9) s − = 12 Z d q (2 π ) a b M M f h a b M f + α E f ( a b E f + M f + a b ~q · ~P f )+ b ( M f q + α M f ~q · ~P f ) − M ( a b q − α a b E f − α M f ) − M q cos θ | ~P f | (1 + E f M )( a b E f − a b M f + M f + a b ~q · ~P f ) , (C10)where α = α ′ = m ′ m ′ + m ′ . c). For the charmonium χ c ( J P C = 1 ++ ) and form factors the f , u , u , g The B-S wave function (solution of Eq. (B9), Eq. (B10) and Eq. (B11) under new methodto solve the coupled equations) of χ c is read as: ϕ χ c ( ~q ′ ) = iǫ µναβ P νf q ′ α ⊥ ε β [ f ′ ( ~q ′ ) M f γ µ + f ′ ( ~q ′ ) P f γ µ + f ′ ( ~q ′ ) q ′⊥ γ µ + if ′ ( ~q ′ ) ǫ µρσδ P fσ q ′⊥ ρ γ δ γ /M f ] /M f , (C11)26here ε is the polarization vector of axial vector meson and with the constraint on thecomponents: f ′ ( ~q ′ ) = 0 , f ′ ( ~q ′ ) = f ′ ( ~q ′ ) M f m ′ , Then the reduced wave function ϕ ++ P ( ~q ′ ) as: ϕ ++ χ c ( ~q ′ ) = iǫ µναβ P νf q ′ α ⊥ ε β a [ M f γ µ + a γ µ P f + a γ µ q ′⊥ + a γ µ P f q ′⊥ ] /M f , (C12)with a = 12 f ′ ( ~q ′ ) + f ′ ( ~q ′ ) w ′ m ′ ! , a = − m ′ w ′ , a = 0 , a = − w ′ . With Eq. (17), the form factors may be presented by overlapping integrations: f = Z d q (2 π ) a b M f ( M + M f ) h(cid:16) a M f q − a b M f q − a b E f ~q · ~P f +( a − a b )( ~q · ~P f ) + α a E f ~P f + a b α ( E f − E f M f ) − α a E f ~q · ~P f + α a b E f ~q · ~P f + b E f M f ( q − α ~q · ~P f ) ) + q θ − (cid:16) M f ( a − a b )+ b E f ( M f + a ~q · ~P f − α a ~P f ) (cid:19)(cid:21) , (C13) u = Z d q (2 π ) a b MM f (cid:20) α E f M (cid:16) a b M f + M f b ~q · ~P f + a ( α E f M f + b ( M f q + ( ~q · ~P f ) − α ~q · ~P f E f )) (cid:17) − α E f q cos θM | ~P f | (cid:16) ( a − a b ) M f + b E f ( − α a E f + M f + a ~q · ~P f + α a M f ) (cid:17) − E f M q cos θ | ~P f | (cid:16) a b M f + M f b ~q · ~P f + a ( α E f M f + b ( M f q + ( ~q · ~P f ) − α ~q · ~P f E f )) (cid:17) + q M | ~P f | ( − M f + (2 E f + M f ) cos θ ) (cid:16) M f ( a − a b )+ b E f ( M f + a ~q · ~P f − α a ~P f ) (cid:19)(cid:21) , (C14) u = Z d q (2 π ) a b MM f (cid:20) − M (cid:16) b M f q + ( a b + α a E f )( α E f − ~q · ~P f ) (cid:17) + E f q cos θM | ~P f | (cid:16) α M f b E f + ( a b − a )( ~q · ~P f − α E f ) (cid:17) + 1 M q cos θ | ~P f | (cid:16) a b M f + M f ( b ~q · ~P f α b E f ~q · ~P f ) + a ( α E f M f + b ( M f q + ( ~q · ~P f ) − α ~q · ~P f E f )) (cid:17) − E f q M | ~P f | (3 cos θ − (cid:18) M f ( a − a b ) + b E f ( M f + a ~q · ~P f − α a ~P f ) (cid:19)(cid:21) , (C15) g = Z d q (2 π ) a b ( M + M f ) M f " E f M ( q cos θ | ~P f | − α ) (cid:16) a ( b E f + b ~q · ~P f ) + M f + a b E f ( q − α ~q · ~P f ) (cid:17) + q M (cos θ − (cid:16) b ( M f + a ( ~q · ~P f − α E f )) (cid:17) . (C16) d). For the charmonium h c ( J P C = 1 + − ) and form factors the V , V , V , V The B-S wave function (solution of Eq. (B9), Eq. (B10) and Eq. (B11) under new methodto solve the coupled equations) of h c is read as: ϕ h c ( ~q ′ ) = q ′⊥ · ε " f ′ ( ~q ′ ) + f ′ ( ~q ′ ) P f M f + f ′ ( ~q ′ ) q ′⊥ + f ′ ( ~q ′ ) P f q ′⊥ M f γ , (C17)with the constraint on the components of the wave function, f ′ ( ~q ′ ) = 0 , f ′ ( ~q ′ ) = − f ′ ( ~q ′ ) M f m ′ , Then we have the reduced wave function ϕ ++ h c ( ~q ′ ): ϕ ++ h c ( ~q ′ ) = q ′⊥ · εa " a P f M f + a q ′⊥ + a q ′⊥ P f M f γ , (C18) a = 12 f ′ ( ~q ′ ) + f ′ ( ~q ′ ) w ′ m ′ ) ! , a = m ′ w ′ , a = 0 , a = 1 w ′ . With Eq. (17), the form factors may be presented by overlapping integrations: V = Z d q (2 π ) a b M f ( M + M f ) q θ − h a b E f + a b E f − b M f + a b ~q · ~P f i , (C19) V = Z d q (2 π ) a b M f " E f M ( α − q cos θ | ~P f | ) (cid:16) α a b E f + α a b E f ~q · ~P f + b M f + a b ~q · ~P f (cid:17) q M | ~P f | ( − M f + (2 E f + M f ) cos θ ) − α E f q cos θM | ~P f | !(cid:16) a b E f + a b E f − b M f + a b ~q · ~P f (cid:17)i , (C20) V = Z d q (2 π ) M a b M f " E f M ( α − q cos θ | ~P f | ) (cid:16) − a b q + a − a b E f α (cid:17) +( α E f q cos θM | ~P f | + E f q M | ~P f | (3 cos θ − (cid:16) a b E f + a b E f − b M f + a b ~q · ~P f (cid:17)i , (C21)28 = − Z d q (2 π ) a b ( M + M f ) M M f q θ −
1) [ a ( b + b E f α ) + b a ] . (C22) e). For the charmonium χ c ( J P C = 2 ++ ) and form factors the k , c , c , h The B-S wave function (solution of Eq. (B9), Eq. (B10) and Eq. (B11) under new methodto solve the coupled equations) of χ c is read as: ϕ χ c ( ~q ′ ) = ε µν q ′ ν ⊥ { q ′ µ ⊥ [ f ′ ( ~q ′ ) + P f M f f ′ ( ~q ′ ) + q ′⊥ M f f ′ ( ~q ′ ) + P f q ′⊥ M f f ′ ( ~q ′ )]+ γ µ [ M f f ′ ( ~q ′ )+ P f f ′ ( ~q ′ )+ q ′⊥ f ′ ( ~q ′ )] + iM f f ′ ( ~q ′ ) ǫ µαβδ P fα q ′⊥ β γ δ γ } , (C23)with the constraint on the components of the wave function: f ′ ( ~q ′ ) = [ q ′ ⊥ f ′ ( ~q ′ ) + M f f ′ ( ~q ′ )] M f m ′ , f ′ ( ~q ′ ) = 0 , f ′ = 0 , f ′ = f ′ ( ~q ′ ) M f m ′ , where ε µν is a tensor for J = 2. Then we have the reduced wave function ϕ ++ χ c ( ~q ) as: ϕ ++ χ c ( ~q ′ ) = ε µν q ′ ν ⊥ { q ′ µ ⊥ [ a + a P f M f + a q ′⊥ M f + a q ′⊥ P f M f ] + γ µ [ a + a P f M f + a q ′⊥ M f + a P f q ′⊥ M f ] } , (C24)with a = q ′ ⊥ M f m ′ n + ( f ′ ( ~q ′ ) w ′ − f ′ ( ~q ′ ) m ′ ) M f m ′ w ′ , a = ( f ′ ( ~q ′ ) w ′ − f ′ ( ~q ′ ) m ′ ) M f m ′ w ′ ,a = 12 n + f ′ ( ~q ′ ) M f m ′ w ′ , a = 12 ( − w ′ m ′ ) n + f ′ ( ~q ′ ) M f m ′ w ′ ,a = M f n , a = M f m ′ w ′ n , a = 0 , a = M f w ′ n ,n = 12 ( f ′ ( ~q ′ ) + f ′ ( ~q ′ ) m ′ w ′ ) , n = 12 ( f ′ ( ~q ′ ) − f ′ ( ~q ′ ) w ′ m ′ ) . With Eq. (17), the form factors may be presented by overlapping integrations: k = Z d q (2 π ) b M f ( M + M f ) " q θ − (cid:16) − α E f ( M f ( − a − a b E f + a b M f ) + a ( b E f + b ~q · ~P f )) − ( α a b E f − a b M f + a b ~q · ~P f ) − α E f ( M f ( − a − a b E f + a b M f )+ a ( b E f + b ~q · ~P f ) + 2 a b E f ) (cid:17) − E f ( α − q cos θ | ~P f | ) (cid:16) − a M f + a b E f q + a b E f M f + a b M f ~q · ~P f − α a b E f ~q · ~P f (cid:17) − E f q cos θ | ~P f | (1 − cos θ ) (( M f ( − a − a b E f + a b M f )+ a ( b E f + b ~q · ~P f ) + 2 a b E f )) ) i , (C25)29 = Z d q (2 π ) b MM f (cid:20) α E f M (cid:18) α E f M ( − α a b E f − α a b E f ~q · ~P f + a b M f + a b M f ~q · ~P f + a M f ( b q + α E f − α b ~q · ~P f )) − α a b E f ~q · ~P f /M (cid:17) − E f q cos θM | ~P f | (cid:18) − M ( α E f M ) ( M f ( − a − a b E f + a b M f ) + a ( b E f + b ~q · ~P f )) − α E f M ( α a b E f − a b M f + a b ~q · ~P f )+2 α E f M ( − α a b E f − α a b E f ~q · ~P f + a b M f + a b M f ~q · ~P f + a M f ( b q + α E f − α b ~q · ~P f )) − α a b E f ~q · ~P f /M − α E f M ( α a b E f + a b M f + a b ~q · ~P f ) (cid:19) + q M | ~P f | ( − M f + (2 E f + M f ) cos θ ) (cid:16) M ( a b q + a M f + α a b E f − α a M f − a − a b M f + α a b E f ) − α E f ( M f ( − a − a b E f + a b M f ) + a ( b E f + b ~q · ~P f )) − ( α a b E f − a b M f + a b ~q · ~P f ) − α E f ( M f ( − a − a b E f + a b M f ) + a ( b E f + b ~q · ~P f )+2 a b E f ))) − q E f cos θ M | ~P f | (3 M f − (2 E f + 3 M f ) cos θ ) (cid:16) ( M f ( − a − a b E f + a b M f ) + a ( b E f + b ~q · ~P f ) + 2 a b E f ) (cid:17)i , (C26) c = Z d q (2 π ) b MM f (cid:20) α E f M (cid:16) α E f ( a b q + a M f + α a b E f − α a M f ) − ( − a b q − a b M f + α a E f ) (cid:17) + E f q cos θM | ~P f | (cid:16) ( − α E f )( M f ( − a − a b E f + a b M f ) + a ( b E f + b ~q · ~P f )) − α ( α a b E f − a b M f a b ~q · ~P f ) − α E f ( a b q + a M f + α a b E f − α a M f )+( − a b q − a b M f + α a E f ) − α E f ( a b q + a M f + α a b E f − α a M f − a − a b M f + α a b E f ))+ q M | ~P f | ( − M f + (2 E f + M f ) cos θ ) (cid:16) a b q + a M f + α a b E f − α a M f − a − a b M f + α a b E f ) − q E f M | ~P f | (3 cos θ − (cid:16) − α E f ( M f ( − a − a b E f + a b M f ) + a ( b E f + b ~q · ~P f )) − ( α a b E f − a b M f + a b ~q · ~P f ) − α E f ( M f ( − a − a b E f + a b M f )30 a ( b E f + b ~q · ~P f ) + 2 a b E f )) (cid:17) − q cos θ M | ~P f | [(4 E f + M f ) cos θ − (2 E f + M f )]( ( M f ( − a − a b E f + a b M f ) + a ( b E f + b ~q · ~P f ) + 2 a b E f )) ) ] , (C27) h = Z d q (2 π ) b ( M + M f ) M f " ( q cos θ | ~P f | − α ) E f M ( a b q + a b M f + α a E f )+ α E f q cos θM | ~P f | ( a − a b M f + α a b E f ) − α E f q cos θM | ~P f | b ( α a E f + a M f − a ~q · ~P f ) − q M ~P f ( − M f + (2 E f + M f ) cos θ )( a − a b M f + α a b E f )+ q E f M ~P f (3 cos θ − b ( α a E f + a M f − a ~q · ~P f )) q θ − (cid:18) − α E f M (2( b M f ( a α − a ) + a ( b + α b E f )) + 4 a b ) (cid:19) + E f q cos θ M | ~P f | (1 − cos θ )(2( b M f ( a α − a ) + a ( b + α b E f )) + 4 a b ) . 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