Strong Z_{c}^{+}(3900)\rightarrow J/\psi\pi^{+};\eta_{c}\rho^{+} decays in QCD

# Strong Z+c(3900)→J/ψπ+;ηcρ+ decays in QCD

S. S. Agaev Department of Physics, Kocaeli University, 41380 Izmit, Turkey Institute for Physical Problems, Baku State University, Az–1148 Baku, Azerbaijan    K. Azizi Department of Physics, Doǧuş University, Acibadem-Kadiköy, 34722 Istanbul, Turkey    H. Sundu Department of Physics, Kocaeli University, 41380 Izmit, Turkey
July 13, 2019
###### Abstract

The widths of the strong decays and are calculated. To this end, the mass and decay constant of the exotic state are computed by means of a two-point sum rule. The obtained results are then used to calculate the strong couplings and employing QCD sum rules on the light-cone supplied by a technique of the soft-meson approximation. We compare our predictions on the mass and decay widths with available experimental data and other theoretical results.

###### pacs:
12.39.Mk, 14.40.Rt, 14.40.Pq

## I Introduction

The exotic hadronic states, i.e. ones that can not be included into the quark-antiquark and three-quark bound schemes of the standard spectroscopy already attracted interests of physicists GellMann:1964nj (); Witten:1979kh (). The quantitative investigations of such states are connected with the invention of the QCD sum rule method Shifman:1979 (), which was employed for analysis of glueballs, hybrid resonances Shifman:1979 (); Balitsky:1982ps (); Reinders:1985 (), exotic four-quark mesons Braun:1985ah (); Braun:1988kv () and six-quark systems Larin:1986 (). But because of problems of old experiments, stemmed mainly from difficulties in detecting heavy resonances, the existence of the exotic states was not then certainly established.

The situation changed dramatically during the last decade when the Belle, BaBar, LHCb and BES collaborations began copiously to yield experimental data providing an information on the masses, decay width and quantum numbers of new exotic states. Starting from the discovery of the charmonium-like resonance by Belle Belle:2003 (), confirmed later by other experiments D0:2004 (), studying of new family of mesons became one of the interesting and rapidly growing branches of the high energy physics (see, the reviews Swanson:2006st (); Klempt:2007cp (); Godfrey:2008nc (); Voloshin:2007dx (); Nielsen:2010 (); Faccini:2012pj (); Esposito:2014rxa () and references therein).

There were attempts to describe the new charmonium-like resonances as excitations of the ordinary charmonium: In order to compute the charmonium spectrum, various quark-antiquark potentials were used and their mass and radiative transitions to other charmonium states were studied Barnes:2005pb (). It should be noted that some of new resonances allow interpretation as the excited states. But the bulk of the collected experimental data can not be included into this scheme, and hence for their explanation new-unconventional quark configurations are required. To this end various quark-gluon models were suggested. They differ from each other in elements of the substructure, and in mechanisms of the strong interactions between these elements that form bound states.

One of the most employed in this context models is the four-quark or the tetraquark model of the new resonances, already used to analyze the light-quark exotic mesons Braun:1985ah (); Braun:1988kv (). In the renewed tetraquark model, the hadronic bound state is formed by two heavy and two light quarks . These quarks may group into compact tetraquark state, where all quarks have overlapping wave functions Braaten:2013oba (). Their strong interaction can be studied by the quark potential models that include not only the 2-body potentials of pairwise quark interactions, but also the 3-body and 4-body potentials. In other model four quarks cluster into the colored diquark and antidiquark , which emerge as the elements of the substructure. Diquark-antidiquarks are organized in such a way that reproduce quantum numbers of the corresponding exotic states Maiani:2004vq (). In this picture the bound state forms due to not only the quark-antiquark potentials, but also owing to the diquark-antidiquark interactions. Alternatively, in the meson molecule picture, the quarks appear as color-singlet and mesons. Finally, in the hadro-quarkonium model suggested in Ref. Dubynskiy:2008mq (), the four-quarks create the bound system consisting of colorless and pairs of the heavy and light quarks, respectively. In the molecule and hadro-quarkonium models the strong interactions are mediated by the meson exchange.

Another possibility to describe the four-quark state is the Born-Oppenheimer tetraquark structure proposed recently in Ref. Braaten:2013boa (). In this approach the heavy quarks and are considered as being embedded in the configuration of gluon and light-quark fields, which are not a flavor singlet, but have isospin 1. The exotic mesons can also be considered in the framework of the traditional hybrid models, as particles consisting of the heavy quarks and a gluon . The gluonic excitation in this approach is treated as a constituent-particle with definite quantum numbers. It should be noted, however that none of these models firmly succeeded in analysis of the variety of the available experimental data: in order to describe features of the observed exotic states one should involve different models.

The meson masses and decay widths were calculated, and their quantum numbers analyzed using all of the aforementioned models and various theoretical methods. Theoretical approaches within QCD include the lattice simulations to explore the exotic and excited charmonium spectroscopy Dudek:2007wv (); Liu:2012ze (); Moir:2013ub (), the calculations based on the different quark potential models Swanson:2006st (); Voloshin:2007dx (), and QCD sum rule method (see, for review Ref. Voloshin:2007dx (); Nielsen:2010 ()). The evaluation of the masses, decay constants and widths of numerous exotic states and comparison of the obtained results with the accumulated experimental data yield valuable information on the quark-gluon structure of new states and mechanisms of the strong interactions between their building elements. Despite remaining problems, one can state that now important parts of the whole picture of exotic multi-quark states are clearer than in the beginning of the decade.

The states discovered by BESIII in the process Ablikim:2013mio (), were observed by the Belle collaboration Liu:2013dau (), as well. Their existence were also confirmed in Ref. Xiao:2013iha () on the basis of the CLEO-c data analysis. The decays demonstrate that are tetraquark states with constituents and . Observation of the neutral partner of in the process was reported in Ref. Ablikim:2015tbp (). Theoretical investigations of the states encompass different models and approaches (see, Refs. Wang:2010rt (); Dias:2013xfa (); Wang:2013vex (); Deng:2014gqa (); Esposito () and references therein).

In this work we evaluate the widths of the strong decays and of (in what follows denoted as ) considering it as the diquark-antidiquark state. For these purposes, we compute the mass and decay constant of , as well as the couplings of the strong and vertices allowing us to find the required decay widths. For calculation of the mass and decay constant we employ QCD two-point sum rule, whereas in the case of the strong couplings apply methods of QCD light-cone sum rule (LCSR) supplemented by the soft-meson approximation Braun:1989 (); Ioffe:1983ju (); Braun:1995 (). The latter is necessary because state contains the four valence quarks, as a result, the light-cone expansion of the correlation functions inevitably reduces to the short-distance expansion in terms of local matrix elements. In the context of LCSR approach this correspondences to the vanishing meson momentum. In the present work we adopt the zero-momentum limit for the mesons referring to the approach itself as the soft-meson approximation. This approximation is rather simple, and, as we shall see, leads to nice agreement with the experimental data Ablikim:2013mio (); Liu:2013dau (). Within the sum rule method state was studied previously in Refs. Wang:2010rt (); Dias:2013xfa (); Wang:2013vex (). Thus, in order to calculate strong couplings and decay widths of state in Ref. Dias:2013xfa () QCD three-point sum rule method was employed.

This article is organized in the following way. In section II, we calculate the mass and decay constant of the state within two-point QCD sum rule approach. Section III is devoted to calculation of the strong and vertices, where the sum rules for the couplings and are derived. Here we also calculate the widths of the decay channels under consideration. Numerical computations of the mass, decay constant, strong couplings, and decay widths are performed in Section IV. The obtained results are compared with the available experimental data, as well as with existing theoretical calculations. This Section contains also our conclusions. The explicit expression of the spectral density necessary for computation of the mass and decay constant of state is moved to Appendix A A.

## Ii The mass and decay constant of the Zc state

In order to calculate the mass and decay constant of the state in the framework of QCD sum rules, we start from the two-point correlation function

 Πμν(q)=i∫d4xeiq⋅x⟨0|T{JZcμ(x)JZc†ν(0)}|0⟩, (1)

where the interpolating current with required quantum numbers is given by the following expression

 JZcν(x) = iϵ~ϵ√2{[uTa(x)Cγ5cb(x)][¯¯¯dd(x)γνC¯¯cTe(x)] (2) −[uTa(x)Cγνcb(x)][¯¯¯dd(x)γ5C¯¯cTe(x)]}.

Here we have introduced the short-hand notations and . In Eq. (2) are color indexes and is the charge conjugation matrix.

In order to derive QCD sum rule expression we first calculate the correlation function in terms of the physical degrees of freedom. Performing integral over in Eq. (1), we get

 ΠPhysμν(q)=⟨0|JZcμ|Zc(q)⟩⟨Zc(q)|JZc†ν|0⟩m2Zc−q2+...

where is the mass of the state, and dots stand for contributions of the higher resonances and continuum states. We define the decay constant through the matrix element

 ⟨0|JZcμ|Zc(q)⟩=fZcmZcεμ, (3)

with being the polarization vector of state. Then in terms of and , the correlation function can be written in the following form

 ΠPhysμν(q)=m2Zcf2Zcm2Zc−q2(−gμν+qμqνm2Zc)+… (4)

The Borel transformation applied to Eq. (4) yields

 Bq2ΠPhysμν(q)=m2Zcf2Zce−m2Zc/M2(−gμν+qμqνm2Zc)+… (5)

The same function in QCD side, , has to be determined employing of the quark-gluon degrees of freedom. To this end, we contract the heavy and light quark fields and find

 ΠQCDμν(q)=−i2∫d4xeiqxϵ~ϵϵ′~ϵ′{Tr[γ5˜Saa′u(x) ×γ5Sbb′c(x)]Tr[γμ˜Se′ec(−x)γνSd′dd(−x)] −Tr[γμ˜Se′ec(−x)γ5Sd′dd(−x)]Tr[γν˜Saa′u(x) ×γ5Sbb′c(x)]−Tr[γ5˜Sa′au(x)γμSb′bc(x)] ×Tr[γ5˜Se′ec(−x)γνSd′dd(−x)]+Tr[γν˜Saa′u(x) ×γμSbb′c(x)]Tr[γ5˜Se′ec(−x)γ5Sd′dd(−x)]}, (6)

where

 ˜Sijc(q)(x)=CSijTc(q)(x)C.

Here the heavy-quark propagator is given by the expression Reinders:1984sr ()

 Sijc(x)=i∫d4k(2π)4e−ikx[δij(⧸k+mc)k2−m2c −gGαβij4σαβ(⧸k+mc)+(⧸k+mc)σαβ(k2−m2c)2 +g212GAαβGAαβδijmck2+mc⧸k(k2−m2c)4+…]. (7)

In Eq. (7) the short-hand notation

 Gαβij≡GαβAtAij,A=1,2…8,

is used, where are color indexes, and with being the standard Gell-Mann matrices. The first term in Eq. (7) is the free (perturbative) massive quark propagator, next ones are nonperturbative gluon corrections. In the nonperturbative terms the gluon field strength tensor is fixed at

The light-quark propagator employed in our work reads

 Sijq(x)=i⧸x2π2x4δij−mq4π2x2δij−⟨q¯¯¯q⟩12 ×(1−imq4⧸x)δij−x2192m20⟨q¯¯¯q⟩(1−imq6⧸x)δij −igGαβij32π2x2[σαβ⧸x+⧸xσαβ]+…. (8)

The correlation function has also the following decomposition over the Lorentz structures

 ΠQCDμν(q)=ΠQCD(q2)gμν+˜ΠQCD(q2)qμqν. (9)

The QCD sum rule expression for the mass and decay constant can be derived after choosing the same structures in both and . We choose to work with the term and invariant function , which can be represented as the dispersion integral

 ˜ΠQCD(q2)=∫∞4m2cρQCD(s)s−q2+..., (10)

where is the corresponding spectral density.

The QCD sum rule calculations requires utilization of some consecutive operations: we recall only the main steps in the computational scheme used in the present work to derive the spectral density . Thus, having employed the transformation

 1(x2)n=∫dDt(2π)De−it⋅xi(−1)n+12D−2nπD/2 (11)

we first replace, where it is necessary, by , and calculate the integral. As a result, we get the delta function with a combination of the momenta in its argument. This Dirac delta is used to remove one of the momentum integrals. The remaining integrations over and over the momentum require invoking the Feynman parametrization and performing rearrangements of denominators in obtained expressions. Then we carry out integration over and perform the last integral over by means of the formulas

 ∫d4k1(k2+L)α=iπ2(−1)αΓ(α−2)Γ(α)[−L]α−2, (12)

and

 ∫d4kkμkν(k2−2Akq+Aq2−Bm2c)α = iπ2(−1)α+1Γ(α−3)Γ(α)[−L]α−3[gμν2+A2(α−3)Lqμqν].

In Eqs. (12) and (LABEL:eq:F2) we use the notations

 A=2r(w+z−1),B=r(w+z)(w−1),

and

 L=r2(w−1){q2wz(w+z−1)−m2cw+zr}, r=1w2+(w+z)(z−1). (14)

By applying the replacement

 Γ(D2−n)(−1L)D2−n→(−1)n−1(n−2)!(−L)n−2ln(−L), (15)

in the obtained expression, we get the imaginary part of the correlation function. The remaining integrals over the Feynman parameters and in some simple cases can be carried out explicitly, or kept in their original form supplemented as a factor by the Heaviside function . The results of our calculations of the spectral density performed within this scheme are collected in Appendix A.

Applying the Borel transformation on the variable to the invariant amplitude , equating the obtained expression with the relevant part of , and subtracting the continuum contribution, we finally obtain the required sum rule. Thus, the mass of the state can be evaluated from the sum rule

 m2Zc=∫s04m2cdssρQCD(s)e−s/M2∫s04m2cdsρ(s)e−s/M2, (16)

whereas to extract the numerical value of the decay constant we employ the formula

 (17)

The last two expressions are required sum rules to evaluate the state’s mass and decay constant, respectively.

## Iii The Strong Vertices ZcJ/ψπ and Zcηcρ

This section is devoted to the calculation of the widths of the and decays. To this end we calculate the strong couplings and using methods of the QCD sum rules on the light-cone in conjunction with the soft-meson approximation.

### iii.1 The ZcJ/ψπ Vertex

We start our analysis from the vertex aiming to calculate : we consider the correlation function

 Πμν(p,q)=i∫d4xeipx⟨π(q)|T{JJ/ψμ(x)JZc†ν(0)}|0⟩, (18)

where

 JJ/ψμ(x)=¯¯ci(x)γμci(x), (19)

and is defined by Eq. (2). Here , and are the momenta of , and , respectively. A sample diagram describing the process is depicted in Fig. 1.

To derive the sum rules for the coupling, we calculate in terms of the physical degrees of freedom. Then it is not difficult to obtain

 ΠPhysμν(p,q) = ⟨0|JJ/ψμ|J/ψ(p)⟩p2−m2J/ψ⟨J/ψ(p)π(q)|Zc(p′)⟩ (20) ×⟨Zc(p′)|JZc†ν|0⟩p′2−m2Zc+….

where the dots denote contribution of the higher resonances and continuum states.

We introduce the matrix elements

 ⟨0|JJ/ψμ|J/ψ(p)⟩=fJ/ψmJ/ψεμ, ⟨Zc(p′)|JZc†ν|0⟩=fZcmZcε′∗ν −(p⋅ε′)(p′⋅ε∗)]gZcJ/ψπ, (21)

where are the decay constant, mass and polarization vector of the meson, and is the polarization vector of the state. It is worth noting that the matrix element in the last row of Eq. (21) is defined in the gauge-invariant form.

Having used these matrix elements we can rewrite the correlation function as

 ΠPhysμν(p,q)=fJ/ψfZcmZcmJ/ψgZcJ/ψπ(p′2−m2Zc)(p2−m2J/ψ) ×⎛⎝m2Zc+m2J/ψ2gμν−p′μpν⎞⎠+… =ΠPhysπ(p2,(p+q)2)gμν+˜ΠPhysπ(p2,(p+q)2)p′μpν. (22)

For calculation of the strong coupling under consideration we choose to work with the structure . Then, for the corresponding invariant function, we get

 ΠPhysπ(p2,(p+q)2)=fJ/ψfZcmZcmJ/ψgZcJ/ψπ(p′2−m2Zc)(p2−m2J/ψ) ×m2Zc+m2J/ψ2+Π(RS:C)(p2,(p+q)2), (23)

where is the contribution arising from the higher resonances and continuum states, that can be written down as the double dispersion integral:

 Π(RS:C)(p2,(p+q)2)=∫∫ρh(s1,s2)ds1ds2(s1−p2)(s2−p′2) +∫ρh1(s1)ds1(s1−p2)+∫ρh2(s2)ds2(s2−p′2). (24)

This formula contains also single dispersion integrals that are necessary to make the whole expression finite: As we shall see below, they play an important role in the soft-meson approximation adopted in the present work.

In the standard LCSR approach in order to get sum rules for the strong couplings Braun:1995 (); Aliev:1996 (); Khodjamirian:1999 () one applies to Eq. (23) double Borel transformation in variables and that vanishes the single dispersion integrals leaving in the physical side of the LCSR only contributions of the ground state and the double spectral integral. In other words, effects of higher resonances and continuum states on the sum rule are under control and modeled by Borel transformed double integral. Then, using the quark-hadron duality assumption one replaces the spectral density by its theoretical counterpart , and subtracts the contribution of the resonance and continuum states from the theoretical side of the sum rules.

But in the case under consideration, the situation differs from the standard one. In fact, calculation of the function in the context of the perturbative QCD reveals its interesting features. As is seen from Eqs. (2) and(18), the tetraquark state contains four quarks at the same space-time location, therefore contractions of the and quark fields given at with the relevant fields at from the meson yield expressions where the remaining light quarks are sandwiched between the pion and vacuum states forming local matrix elements. In other words, we encounter with the situation when dependence of the correlation function on the meson distribution amplitudes disappears and integrals over the meson DAs reduce to overall normalization factors. Within framework of LCSR method such situation is possible in the kinematical limit , when the light-cone expansion reduces to the short-distant one. As a result, instead of the expansion in terms of DAs one gets expansion over the local matrix elements Braun:1995 (). In this limit and relevant invariant amplitudes in the correlation function depend only on one variable . Here we adopt this approach, and following Ref.  Braun:1995 () refer to the limit as the soft-meson approximation bearing in mind that it actually implies calculation of the correlation function with the equal initial and final momenta , and dealing with the obtained double pole terms.

The soft-meson approximation considerably simplifies the QCD side of the sum rules, but leads to more complicated expression for its hadronic representation. In the soft   limit, as it has been just emphasized above, the ground state contribution depends only on the variable . With some accuracy, it can be written in the form

 ΠPhysπ(p2)=fJ/ψfZcmZcmJ/ψgZcJ/ψπ(p2−m2)2m2, (25)

where . The Borel transformation in the variable applied to this correlation function yields

 ΠPhysπ(M2)=fJ/ψfZcmZcmJ/ψgZcJ/ψπ ×1M2e−m2/M2m2. (26)

But because within the soft-meson approximation we employ the one-variable Borel transformation, now the single dispersion integrals also contribute to the hadronic part of the sum rules. Non-vanishing contributions correspond to transitions from the exited states in the channel with to the ground state (the similar arguments are valid for the channel, as well). They are not suppressed relative to the ground state contribution even after the Borel transformation Braun:1995 (); Ioffe:1983ju (). Hence, taking into account all unsuppressed contributions to , denoted below as , the hadronic part of the sum rules can be schematically written in the form Braun:1995 ()

 ΠPhysπ(M2)≃1M2{fJ/ψfZcmZcmJ/ψm2gZcJ/ψπ +AM2}e−m2/M2. (27)

It is evident, that the terms emerge here as an undesired contamination and make extraction of the strong coupling problematic. In order to remove them from the hadronic part it is convenient to follow a prescription suggested in Ref. Ioffe:1983ju () and act by the operator

 (1−M2ddM2)M2em2/M2 (28)

to both sides of the sum rules.

Now we need to calculate the correlation function in terms of the quark-gluon degrees of freedom and find the QCD side of the sum rules. Having contracted heavy quarks fields we get

 ΠQCDμν(p,q)=∫d4xeipxϵ˜ϵ√2[γ5˜Sibc(x)γμ ×˜Seic(−x)γν+γν˜Sibc(x)γμ˜Seic(−x)γ5]αβ ×⟨π(q)|¯¯¯uaα(0)ddβ(0)|0⟩, (29)

where and are the spinor indexes.

To proceed we use the expansion

 ¯¯¯uaαddβ→14Γjβα(¯¯¯uaΓjdd), (30)

where is the full set of Dirac matrixes

 Γj=1, γ5, γλ, iγ5γλ, σλρ/√2,

Within the LCSR method we have also to use the light-cone expansion for the -quark propagator

 ⟨0|T{c(x)¯¯c(0)}|0⟩=i∫d4k(2π)4e−ikx⧸k+mck2−m2c −igs∫d4k(2π)4e−ikx∫10dv⎡⎣12⧸k+mc(k2−m2c)2Gμν(vx)σμν −⧸k+mck2−m2cvxμGμν(vx)γν]+… (31)

But because the light quark fields in the local matrix elements are fixed at the point , in the second piece of the propagator [Eq. (31)] we expand the gluon field strength tensor at keeping only the leading order term that is equivalent to usage the first two terms from (see, Eq. (7)) in the calculations .

In order to determine the required local matrix elements we consider first the perturbative components of the heavy-quark propagators. To this end, it is convenient to take sums over the color indexes. Using the overall color factor , color factors of the propagators, as well as the projector onto a color-singlet state , it is easy to demonstrate that for the perturbative contribution, the replacement

 (32)

is legitimate. For nonperturbative contributions, forming as a product of the perturbative part of one of the propagators with the term from the other one, we find, for example,

 ϵabcϵdecδeiGρδib14Γjβα(¯¯¯uaΓjdd)→−14Γjβα(¯¯¯uΓjGρδd).

This rule allows us to insert into quark matrix elements the gluon field strength tensor that effectively leads to three-particle components and corresponding matrix elements of the pion. We neglect the terms appearing from the product of one-gluon components of the heavy propagators. Having finished a color summation one can calculate the traces over spinor indexes.

The spectral density has been found employing the approach outlined in the Section II. Calculations demonstrate that the pion local matrix element that, in the soft-meson limit, contributes to the both structures of is

 ⟨0|¯¯¯d(0)iγ5u(0)|π(q)⟩=fπμπ, (33)

where

 μπ=m2πmu+md=−2⟨¯¯¯qq⟩f2π. (34)

The second equality in Eq. (34) is the relation between , , the quark masses and the quark condensate arising from the partial conservation of axial vector current (PCAC).

Choosing the structure for our analysis it is straightforward to derive the corresponding spectral density

 ρQCDπ(s)=fπμπ(s+2m2c)√s(s−4m2c)12√2π2s. (35)

The continuum contribution can be subtracted in a standard manner after replacement. The final sum rule to evaluate the strong coupling reads

 gZcJ/ψπ=2fJ/ψfZcmZcmJ/ψ(m2Zc+m2J/ψ) ×(1−M2ddM2)M2 ×∫s04m2cdse(m2Zc+m2J/ψ−2s)/2M2ρQCDπ(s). (36)

The width of the decay can be found applying the standard methods and definitions for the strong coupling alongside with other matrix elements [Eq. (21)] and parameters of the state. Our calculations give

 Γ(Zc→J/ψπ)=g2ZcJ/ψπm2J/ψ24πλ(mZc, mJ/ψ,mπ) ×⎡⎣3+2λ2(mZc, mJ/ψ,mπ)m2J/ψ⎤⎦, (37)

where

 λ(a, b, c)=√a4+b4+c4−2(a2b2+a2c2+b2c2)2a.

The final expressions (36) and (37) will be used for numerical analysis of the decay channel .

### iii.2 The Zcηcρ Vertex

The coupling can be calculated utilizing the correlation function

 Πν(p,q)=i∫d4xeipx⟨ρ(q)|T{Jηc(x)JZc†ν(0)}|0⟩, (38)

where the current is defined as

 Jηc(x)=¯¯ci(x)iγ5ci(x).

In order to find the hadronic representation of the correlation function we define the matrix element:

 ⟨0|Jηc|ηc(p)⟩=fηcm2ηc2mc, (39)

with and being the meson’s mass and decay constant. The vertex is defined as in Eq. (21)

 ⟨ηc(p)ρ(q)|Zc(p′)⟩=[(q⋅p′)(ε∗⋅ε′) −(q⋅ε′)(p′⋅ε∗)]gZcηcρ, (40)

but with and being now the momentum and polarization vector of the -meson, respectively.

Then the calculation of the hadronic representation is straightforward and yields

 ΠPhysν(p,q) = ⟨0|Jηc|ηc(p)⟩p2−m2ηc⟨ηc(p)ρ(q)|Zc(p′)⟩ (41) ×⟨Zc(p′)|JZc†ν|0⟩p′2−m2Zc+….

Employing the corresponding matrix elements we find for the ground state contribution

 ΠPhysν(p,q)=fηcfZcmZcm2ηcgZcηcρ2mc(p′2−m2Zc)(p2−m2ηc) ×(m2ηc−m2Zc2ϵ∗ν+p′⋅ϵ∗qν)+… (42)

In the soft-meson limit only the structure survives. The relevant invariant amplitude is given by the formula

 ΠPhysρ(p2)=fηcfZcmZcm2ηcgZcηcρ4mc(p2−˜m2)2(m2ηc−m2Zc)+… (43)

where . The Borel transformation of yields

 ΠPhysρ(M2)=fηcfZcmZcm2η