Vertex functions for d-wave mesons in the light-front approach
aa r X i v : . [ h e p - ph ] O c t Vertex functions for d-wave mesons in the light-front approach
Hong-Wei Ke ∗ and Xue-Qian Li † School of Science, Tianjin University, Tianjin 300072, China School of Physics, Nankai University, Tianjin 300071, China
Abstract
While the light-front quark model (LFQM) is employed to calculate hadronic transition matrixelements, the vertex functions must be pre-determined. In this work we derive the vertex functionsfor all d-wave states in this model. Especially, since both of D and S are 1 −− mesons, theLorentz structures of their vertex functions are the same. Thus when one needs to study theprocesses where D is involved, all the corresponding formulas for S states can be directlyapplied, only the coefficient of the vertex function should be replaced by that for D . The resultswould be useful for studying the newly observed resonances which are supposed to be d-wave mesonsand furthermore the possible 2S-1D mixing in ψ ′ with the LFQM. PACS numbers: 12.39.Ki, 14.40.-n ∗ [email protected] † [email protected] . INTRODUCTION In the hadronic physics the most tough job is to calculate the hadronic transition ma-trix elements which are fully governed by non-perturbative QCD. Because of lack of solidknowledge on non-perturbative QCD so far, one needs to invoke phenomenological models.Applications of such models to various processes have achieved relative successes so far.Among these models the light-front quark model(LFQM) is a relativistic model and has ob-vious advantages for dealing with the hadronic transitions where light hadrons are involved[1, 2]. The light-front wave function is manifestly Lorentz invariant and expressed in terms offractions of internal momenta of the constituents which are independent of the total hadronmomentum. This approach has been applied to many processes and thoroughly discussed inliteratures [3–13]. Generally the results qualitatively coincide with experimental observationand while taking the error ranges into account (both experimental and theoretical), they canbe considered to quantitatively agree with data.However earlier researches with the LFQM only concern the s-wave and p-wave mesonswhereas the higher orbital excited states have not been discussed yet. With the improve-ments of experimental facilities and rapid growth of database many new resonances havebeen observed and some of them are regarded as higher orbital excited states, for exam-ple ψ (3770) and ψ (4153) are suggested to be d-wave charmonia whose principal quantumnumbers are respectively n = 1 and n = 2[14]. New analysis on X (3872) indicates thatit has a possible charmonium assignment D [14]. Moreover, a d-wave state Υ(1 D ) wasobserved by CLEO[15]. The situation persuades us to extend our scope to involve d-wave.It is generally believed that only the lattice theory indeed deals with the non-perturbativeQCD effects from the first principle. So far, the lattice study is constrained by not only thecomputing abilities, but also the theory itself. Even so, remarkable progresses have beenmade on the two aspects. It is hoped that the lattice calculation will eventually solve all theproblems on hadrons, such as the hadron spectra, wavefunctions, even the hadronic tran-sition matrix elements. At present, the lattice calculation indeed shed some light on thewavefunctions[16–18]. For examples, the authors of Ref.[16] suggested a method to computethe spectra and wavefunctions of hadron excited states and applied their method to the U (1) lattice gauge theory. Abada et. al. [17] study the pion light-cone wave function onthe lattice by considering the three-point Green functions. On other aspects, for example,the readers who are interested in the unquenched lattice calculation on the charmonia whosetotal quantum numbers J P C are determined simultaneously, are suggested to refer to thoseenlightening papers [19]. Some works about the radiative decays of charmonia are included inconcerned references[20, 21]. More similar works can be found in most recent literatures[22–24]. In fact, due to the rapid progress of lattice calculations, people are more tempted totrust those results, but it by no means implies that we should abandon phenomenologicalmodels because those models are directly invented to manifest the physical mechanisms and2oreover, they are simpler and applicable in practice.To evaluate the transition rate in the LFQM one needs to know the wave functions ofparent and daughter hadrons. For any S +1 L J state, its wavefunction is constructed as thecorresponding spinors multiplying the so-called vertex function which should be theoreticallyderived. It is also noted the wavefunctions for s-wave and p-wave have been derived and theirexplicit forms are given in Ref.[7]. Following the same strategy we obtain the wavefunctionfor all d-wave states.The traditional LFQM was employed to study the decay constants and form factors ofweak decays [4–6], but to maintain the Lorentz covariance other contributions such as Z-diagram[5] or zero-mode[25–28] contributions must be included.Thus a covariant LFQM[6] has been suggested which systematically includes the zero-mode contributions. In the traditional LFQM the constituent quarks in the bound stateare required to be on their mass shells, nevertheless in the covariant LFQM approach theconstituent quarks in the meson are off-shell while only the meson is on its mass shell. In thisapproach, one writes down the transition amplitudes where all quantities and the integrationsmaintain their four-dimensional forms, then integrates out the light-front momentum p − ina proper way. While carrying out the contour integration the antiquark is enforced to beon its mass shell. The integrand in the remaining three-dimension integration reduces intoa form where all quantities can be expressed in terms of the conventional wavefunctions.During this procedure, some extra contributions emerge comparing with the original scheme(see the text for details).In this work after this introduction we derive the phenomenological vertex functions ford-wave in the conventional light-front approach in section II. Then in section III we presenttheir forms in the covariant light-front approach. In section IV we discuss some formula for D states and the section V is devoted to a brief summary. II. VERTEX FUNCTIONS IN THE CONVENTIONAL LIGHT-FRONT AP-PROACH
Let us first derive the vertex functions in the conventional light-front approach.In the conventional light-front approach a meson with the total momentum P and spin J can be written as [7] | M ( P, S +1 L J , J Z ) i = Z { d ˜ p }{ d ˜ p } π ) δ ( ˜ P − ˜ p − ˜ p ) X λ λ Ψ JJ z LS ( ˜ p , ˜ p , λ , λ ) | q ( p , λ ) ¯ q ( p , λ ) i , (1)where the flavor and color indices are omitted; q and ¯ q correspond to the quark andantiquark in the meson and p and p are the on-shell light-front momenta of quark and3ntiquark, p is the three-momentum ( p − p ) / p i = ( p + i , p i ⊥ ) , p i ⊥ = ( p i , p i ) , p − i = m + p i ⊥ p + i , { d ˜ p i } ≡ dp + i d p i ⊥ π ) , | q ( p , λ ) ¯ q ( p , λ ) i = b † λ ( p ) d † λ ( p ) | i . The light-front momenta p and p are expressed via the variables p and x i ( i = 1 ,
2) as p +1 = x P + , p +2 = x P + , x + x = 1 ,p ⊥ = x P ⊥ + p ⊥ , p ⊥ = x P ⊥ − p ⊥ . (2) P p − p FIG. 1: The vertex for meson-quark-antiquark.
In the momentum representation, the wavefunction Ψ JJ z LS for the state S +1 L J can bedecomposed into the formΨ JJ z LS ( ˜ p , ˜ p , λ , λ ) = 1 √ N c h LS ; L z S z | LS ; J J z i R SS z λ λ ( x, p ⊥ ) ϕ LL z ( x, p ⊥ ) , (3)where ϕ LL z ( x, p ⊥ ) describes the relative momentum distribution of the quark (antiquark)in the meson and L is the orbital angular momentum between the constituents. R SS z λ λ transforms a light-front helicity ( λ , λ ) eigenstate to a state with definite spin ( S, S z ) andit is expressed as R SS z λ λ ( x, p ⊥ ) = 1 √ M ( M + m + m ) ¯ u ( p , λ )(¯ /P + M )Γ S v ( p , λ ) , Γ = γ ( for S = 0) , Γ = − / ˆ ε ( S z ) ( for S = 1) , (4)with ¯ P = p + p , M = m + p ⊥ x + m + p ⊥ x and ˜ M ≡ q M − ( m − m ) . For more details,readers are suggested to refer Ref.[7]. 4n the LFQM, the harmonic oscillator wavefunctions are employed to describe the relative3-momentum distribution of quark and antiquark in a meson. For d-wave the harmonicoscillator wavefunction is[29] ϕ Lz ( x, p ⊥ ) = ˆ ε µν ( L z ) K µ K ν √ β ϕ, (5)where K = ( p − p ) / ϕ is the harmonic oscillator wavefunction for s wave and itsexplicit expression is with ϕ = 4( πβ ) / s dp z dx exp( − p z + p ⊥ β ) ,p z = x M − m + p ⊥ x M . (6)Substituting Eq.(3) into Eq.(2), we deduce the expressionΨ JJ z S ( ˜ p , ˜ p , λ , λ ) = 1 √ N c h S ; L z S z | S ; J J z i R SS z λ λ ( x, p ⊥ ) ϕ Lz ( x, p ⊥ )= 1 √ N c ϕ √ M ( M + m + m ) ¯ u ( p , λ )(¯ /P + M )Γ ( s +1 D J ) v ( p , λ ) , (7)with Γ ( D ) = s β [ − / ˆ ε ( J z )3 ( K · ¯ P M − K ) + / ¯ P K · ¯ P K · ˆ ε ( J z ) M − /KK · ˆ ε ( J z )] , Γ ( D ) = s β ˆ ε µν ( J z ) K µ K ν γ , Γ ( D ) = s β ˆ ε µν ( J z ) γ [ γ ν γ µ ( K · ¯ P ) − M K M − γ µ K ν ( K · ¯ P − M /K ) + K µ K ν ] , Γ ( D ) = s β ˆ ε µνα ( J z ) γ β ( K µ K ν g αβ + K µ K α g νβ + K α K ν g µβ ) , (8)where relations h L z | J z i = ˆ ε ∗ µν ( L z )ˆ ε µν ( J z ), h L z S z | J z i = − q ˆ ε ∗ µν ( L z )ˆ ε ∗ µ ( S z )ˆ ε ν ( J z ) , h L z S z | J z i = i q ǫ αβµν ˆ ε ∗ αω ( L z )ˆ ε ∗ β ( S z )ˆ ε µω ′ ( J z ) g ωω ′ ¯ P ν M and h L z S z | J z i = ˆ ε µνα ( J z )[ ˆ ε ∗ µν ( L z ) ˆ ε ∗ α ( S z ) + ˆ ε ∗ µα ( L z ) ˆ ε ∗ ν ( S z ) + ˆ ε ∗ να ( L z ) ˆ ε ∗ µ ( S z )]are used.One can further simplify these wavefunctions in terms of the Dirac equation /p u ( p ) = m u ( p ) and /p v ( p ) = − m v ( p ), so that all scalar products of vectors are replaced by only M , m and m via a simple algebra, thus the wave function isΨ JJ z S ( ˜ p , ˜ p , λ , λ ) = ¯ u ( p , λ ) h ′ ( s +1 D J ) Γ ′ ( s +1 D J ) v ( p , λ ) , (9)5here h ′ ( D ) = − s N c √ M √ √ M β [ M − ( m − m ) ][ M − ( m + m ) ] ϕ,h ′ ( D ) = s N c M β ϕ,h ′ ( D ) = s N c s
23 1˜ M β ϕ,h ′ ( D ) = s N c
13 1˜ M β ϕ, (10)and Γ ′ ( D ) = [ γ µ − w ( D ) ( p − p ) µ ]ˆ ε µ , Γ ′ ( D ) = γ K µ K ν ˆ ε µν , Γ ′ ( D ) = γ [ 1 w a ( D ) γ ν γ µ + 1 w b ( D ) γ µ K ν + 1 w c ( D ) γ ν K µ + 1 w d ( D ) K µ K ν ]ˆ ε µν , Γ ′ ( D ) = [ K µ K ν ( γ α + 2 K α w ( D ) ) + K µ K α ( γ ν + 2 K ν w ( D ) ) + K α K ν ( γ µ + 2 K µ w ( D ) )]ˆ ε µνα , (11)with w ( D ) = ( m + m ) − M M + m + m ,w a ( D ) = − M [ M − ( m + m ) ][ M − ( m − m ) ] ,w b ( D ) = − M (2 M + m + m )[ M − ( m − m ) ] ,w c ( D ) = 6 M ( M − m − m )[ M − ( m − m ) ] ,w b ( D ) = M m − m ,w ( D ) = M + m + m . (12)It is interesting to ask where the QCD which definitely governs the physical processes, getsinvolved or how can one implant the QCD information into our calculation in the LFQM.In Ref.[30] the authors derived an effective Hamiltonian for bound states in the light-frontframe based on the standard Lagrangian of QCD. The vertex function is the effective couplingbetween the bound state and its constituent quarks, thus as the wavefunction of the boundstate is obtained the effective vertex function is in hand. After a long discussion about the6 P p − p FIG. 2: The Feynman diagram for a meson annihilation.
Lorentz structure and the features of the dynamics of the vertex functions, they derive thevertex function which has exactly the form of Eq.(9) in our work. In Ref.[30], the radial wavefunction was obtained by solving the eigenvalue equation numerically. Obviously all QCDinformation (both short-distance and long-distance effects) is involved in the Hamiltonianand as well as in the solution. As argued in literature[5, 29], the solution can be wellapproximated by a Gaussian function with model parameters to be fixed by fitting data.Thus following Ref.[7] we choose a Gaussian wave function where the QCD information isincluded in the model parameter β .We can apply these wavefunctions to deal with concrete physical processes. For example,when we calculate the rate of D state annihilation through a vector current (in Fig.2), thetransition amplitude is written as A convµ = N c Z d ˜ p Ψ J z ¯ v ( p , λ ) q p +2 γ µ u ( p , λ ) q p +1 = N c Z dx d p ⊥ π h ′ D √ x x Tr[Γ ′ D ( /p − m ) γ µ ( /p + m )]= N c Z dx d p ⊥ π h ′ D √ x x Tr { [ γ ν − ( p − p ) ν w D ]( /p − m ) γ µ ( /p + m ) } ˆ ε ν . (13) III. VERTEX FUNCTIONS IN THE COVARIANT LIGHT-FRONT APPROACH
Comparing with the conventional LFQM where both p and p are on their mass shells,in the covariant light-front approach the quark and antiquark are off-shell, but the totalmomentum P = p + p is the on-shell momentum of the meson, i.e. P = M where M isthe mass of the meson. Obviously, the covariant LFQM is closer to the physical reality.If one tries to obtain the covariant vertex functions based on an underlying principle, i.e.QCD, he should invoke a reasonable theoretical framework. To directly obtain the covariant7ertex functions in the 4-dimensional momentum space, the authors of Ref.[6] suggested tosolve the Bethe-Salpeter (B-S) equations for the bound states [31, 32]. The kernel of theB-S equation includes the Coulomb piece which is induced by the one-gluon exchange aswell as its higher-order corrections, and the confinement piece which incorporates the non-perturbative QCD but is not derivable so far. Genrally for solving the B-S equation, theinstantaneous approximation is usually taken.In the concrete calculations of the observable physical quantity in terms of the LFQM thefinal result is eventually reduced into an integration over the four-momentum. Fortunately,by doing so, we may not really need the explicit covariant wavefunctions defined in thefour-momentum space. Namely, we try to reduce the integration into a simple form whereonly three-momentum wavefunctions remain by a mathematical manipulation, then we areable to relate the corresponding integrand to the conventional vertex function which is welldefined in the three-momentum space.Since the Lorentz structures of the covariant vertices are the same as that of the conven-tional vertex functions we rewrite these covariant vertex functions in Eq.(9) as iH ( D ) [ γ µ − W ( D ) ( p − p ) µ ] ε µ ,iH ( D ) γ K µ K ν ε µν ,iH ( D ) γ [ 1 W a ( D ) γ ω γ µ + 1 W b ( D ) γ µ K ω + 1 W c ( D ) γ ω K µ + 1 W d ( D ) K µ K ω ] ε µν ,iH ( D ) [ K µ K ν ( γ α + 2 K α W ( D ) ) + K µ K α ( γ ν + 2 K ν W ( D ) ) + K α K ν ( γ µ + 2 K µ W ( D ) )] ε µνα , (14)where H ( S +1 D J ) and W ( S +1 D J ) are functions in the 4-dimensional space. Practically, thevertex function(s) is(are) included in a transition matrix element, for example, the amplitudeof D state annihilation via a vector current is written as A covµ = − i N c π Z d p H ( D ) N N Tr { [ γ ν − ( p − p ) ν W D ]( − /p + m ) γ µ ( /p + m ) } ε ν = − i N c π Z d p H ( D ) N N s ε ν , (15)where s = Tr { [ γ ν − ( p − p ) ν W D ]( − /p + m ) γ µ ( /p + m ) } , N = p − m + iǫ and N = p − m + iǫ .One first needs to integrate over p − as discussed in Ref.[6, 7]. Integrating over p − is completedby a contour integration where the antiquark is set on shell. Then the integration turns into N c π Z dx d p ⊥ h ( D ) x x ( M − M ) ˆ s ˆ ε ν , (16)where w ( D ) and ˆ ε ν replace W ( D ) and ε ν in Eq.(15) respectively.Following Ref.[7] we have the relation h ( D ) = ( M − M ) √ x x h ′ ( D ) . (17)8n additional factor ( M − M ) √ x x was introduced when comparing the decay constant f P of pseudoscalar meson obtained in the two approaches as depicted in the appendix Aof Ref.[7]. The legitimacy is guaranteed because the decay constant is free of zero modecontribution. Then the authors have applied the relation into the vertex functions for S andP waves. To show the reasonability of such replacement, we substitute Eq.(17) into Eq.(16)to obtain a new expression whose form is similar to the right side of Eq.(13). However,the trace in Eq. (16) involves the zero mode contribution which makes its form differentfrom that in Eq.(13). Generally, after the contour integration over p − , h ( S +1 D J ) , w ( S +1 D J ) and ˆ ε replace H ( S +1 D J ) , W ( S +1 D J ) and ε respectively with the following relation to thecorresponding quantities of the conventional LFQM h ( S +1 D J ) = ( M − M ) √ x x h ′ ( S +1 D J ) . (18)Here we only concern the form of the covariant vertex function for the D-wave, includ-ing its Lorentz structure and coefficient, as well as its relations to the conventional vertexfunction. The details about the S- and P wave vertex functions were discussed in earlierliterature[7]. When one needs to calculate a transition matrix in the covariant light-frontquark model, he must know those vertex functions. Jaus has analyzed the case of the covari-ance of the transition matrix [6], and in his work, a general form of vertex function is usedand the three-momentum conservation is automatically guaranteed. He [6] indicates thatthe general form of the vertex function h must be functions of ˆ N i = x i ( M − M ) ( i = 1 , h adopted in this work coincides with this requirement. IV. THE FORMULA FOR D STATE
For a J P C = 1 −− state, the the orbital momentum between the two constituents may be L = 0 (s-wave) or L = 2 (d-wave) and their total spin is 1 ( S = 1). In Ref.[7] the authorsgave the meson-quark-antiquark vertex for S state as iH V [ γ µ − W V ( p − p ) µ ] . (19)Carrying out the contour integration over p − , H V and W V turn into h V and w V h V = ( M − M ) s x x N c √ M ϕ,w V = M + m + m , where the subscript V only refers to S state.The Lorentz structure of the vertex functions for D and S states are the same becausethey have the same quantum number J P C . The difference between the s-wave and d-waveis included in the coefficient functions h M and w M .9he decay amplitude of an S state via a vector current is proportional to [7] A µ = − i N c (2 π ) Z d p iH V N N T r { γ µ ( /p + m )[ γ ν − ( p − p ) ν W V ]( − /p + m ) } ˆ ε ν , (20)which is the same as Eq.(15), except H D and W D are replaced by H V and W V respectively.Integrating over p − H D , W D , H V and W V reduce into h D , w D , h V and w V . The decayconstant for S state reads f V = N c π M Z dx d p ⊥ h V x x ( M − M ) [ x M − m ( m − m ) − p ⊥ + m + m w V p ⊥ ] , (21)so that one would obtain the decay constant of the D state by replacing h V and w V by h D and w D , thus it is f D = N c π M Z dx d p ⊥ h D x x ( M − M ) [ x M − m ( m − m ) − p ⊥ + m + m w D p ⊥ ] . (22)In fact since the Lorentz structure of the vertex functions for D and S are the sameall the formula for D can be deduced from those for S . For example, the form factors f, g, a + and a − of P → D ( V ) decay can be obtained by simply replacing h V and w V of f, g, a + and a − given in Ref.[6, 7].With these formula we will be able to explore some new resonances of angular excitedstates, or furthermore to study the mixing of 2 S − D which was proposed to explain thefamous ρ − π puzzle for ψ ′ [33–35] in this model. V. A BRIEF SUMMARY
In this paper we deduce the vertex functions (or wave functions) for the d-wave in theconventional and covariant light-front quark model.For the D state the J P C is 1 −− and the Lorentz structure of its wave function is thesame as that for the S state so we obtain some useful formula for D from the formulafor S given in Ref.[6, 7]. It is noted we just discuss the vertex functions ( i Γ M ) for theincoming meson whereas for the outgoing meson the corresponding vertex functions shouldbe i ( γ Γ † M γ )[7].Since we adopt the Gaussian-type function for the radial part of the whole wavefunctioninstead of a solution obtained by solving the Schr¨odinger equation or the B-S equation, thesimplification definitely brings up certain theoretical uncertainties, but as more data will becollected in the future, the more precise model parameter(s) will be determined and eventhe form of the wavefunction can be improved, thus we may do a better job along the line.These vertex functions can be employed when one calculates the transition rates in thismodel. In the future we will study some concrete physical transitions where d-wave mesons10re involved in terms of these vertex functions. The results will be compared with dataand the consistency would tell us the accuracy degree of the model and the derived vertexfunctions. Once the validity of the model is verified via some processes, we can further discusssome long-standing puzzles and help to identify new resonances which are continuouslyobserved at BES and BELLE and elsewhere. Acknowledgments
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