Meson electro-magnetic form factors in an extended Nambu–Jona-Lasinio model including heavy quark flavors

Based on an extended NJL model including heavy quark flavors, we calculate the form factors of pseudo-scalar and vector mesons. After taking into account the vector-meson-dominance effect, which introduces a form factor correction to the quark vector coupling vertices, the form factors and electric radii of π+ and K+ pseudo-scalar mesons in the light flavor sector fit the experimental data well. The magnetic moments of the light vector mesons ρ+ and K*+ are comparable with other theoretical calculations. The form factors in the light-heavy flavor sector are presented to compare with future experiments or other theoretical calculations.


Introduction
The Nambu-Jona-Lasinio (NJL) model [1,2] has been widely used in hadron physics as an effective model to study chiral symmetry in the degree of quark freedom. Usually this model deals with light hadrons composed of only light quark flavors u, d, s with SU f (3) symmetry [3][4][5][6].
In hadron systems including heavy flavors, such as light-heavy mesons, although the chiral symmetry is broken due to the mass of the heavy quark, a complementary heavy flavor symmetry emerges and the so-called heavy quark effective theory (HQET) was formulated for this using the technique of 1/m Q expansion [7][8][9][10][11][12][13][14]. In Ref. [15], the NJL model was extended to include heavy quark flavors to investigate such light-heavy mesons like D ( * ) and B ( * ) mesons.
In our previous work [16], we also tried to extend the NJL model to include heavy flavors by expanding the NJL interaction strengths in the inverse power of constituent quark masses according to HQET. Based on this extension, we obtained the meson masses and meson-quark coupling constants of all light and lightheavy mesons in a unified way. Furthermore, the decay widths of the mesons were calculated from those effective meson quark couplings [17].
In this work, we further calculate the electromagnetic form factors of mesons within this extended model. Electromagnetic form factors play an important role in our understanding of hadronic structure. The form factors of the pseudo-scalar mesons π and K were measured in several experiments [18][19][20] and in some previous theoretical works, the form factors of π and K mesons were studied in the NJL model [21,22]. After considering the effect of vector-meson-dominance of the vector mesons, such as the ρ meson, in the calculation, typically the form factors of π fit the experimental data well. Furthermore, the form factor of π was also studied in case of finite temperature with the NJL model [23]. Certainly the form factor of π was studied in many other theoretical approaches, such as the Dyson-Schwinger equation using a confining quark propagator [24], light-cone or covariant quark wave functions [25,26], and the lattice QCD method [27,28]. Also, with the QCD factorization approach [29][30][31][32], the form factor can be extrapolated to higher energy regions by taking into account the perturbative QCD contribution.
The form factors of vector mesons have a rather more complicated structure. Consequently they can provide us with more information about vector mesons, such as magnetic moments and quadrupole moments. Presently there are only theoretical results about the form factors of vector mesons. Some works have used the constituent quark model and the light front dynamics [33][34][35] or Dyson-Schwinger equations [36]. Lattice QCD calculation have been performed with the three-point functions method [37], and the background field method using only two-point functions [38,39]. The magnetic moments of vector mesons were also calculated by dynamics with the external magnetic field [40], and with QCD sum rules [41].
There are a few papers studying the form factor of light-heavy mesons [42]. These focus on the electroweak form factors. From the heavy flavor symmetry, those form factors should be unifying described by the Isgur-Wise function when the heavy flavor mass goes to infinity.
Here, we perform a systematic calculation of the meson form factors, including pseudo-scalar mesons and vector mesons, of both the light flavor sector and the light-heavy flavor sector, within the extended NJL model. In the next section, we will introduce our model and formalism. The numerical results and discussion will be presented in Section 3.

Extended NJL model
To deal with both light and heavy mesons in the Nambu-Jona-Lasinio (NJL) model, in Ref. [16] the fourfermion point interactions are modified to where λ a are the generator of SU(3) in color space and q,q =u, d, s, c, b including both the light and the heavy flavors. Here the second part of the interaction is required to improve the spectra of light vector mesons and the factor of 1/(m q m q ) guarantees that the symmetry of heavy flavors will still be held in the heavy quark limit according to HQET. By solving the Bethe-Salpeter equation (BSE), we obtain the meson masses and their coupling constants with quarks. We will use the effective Lagrangian to describe the quark interaction in mesons. In the case of π and ρ, the effective Lagrangian reads Here the couplings g πq ,g πq and g ρq are treated as constants since the energy of immediate quarks is truncated to the low energy region in the NJL model. In Ref. [17], we have calculated the strong and radiative decays of vector mesons. In this work, we will use the above effective meson Lagrangian to further calculate the form factors of mesons.

Form factor of pseudo-scalar mesons
The definition of the form factor of a pseudo-meson is given by where q = p 1 −p 2 is the transfer momentum. Its Feynman diagrams are shown in Fig. 1 where m 1 and m 2 are the masses of the constituent quarks in the pseudo-scalar meson. Using the Feynman rules, the amplitude reads where F (1) and F (2) are the form factors of the quark and anti-quark respectively, Q i is the electron charge of the i-th quark, is the propagator of the i-th quark, and g andg are the coupling constants of the pseudo-scalar meson obtained in our previous work [16]. In the Breit frame, p 0 2 −p 0 1 = 0 and p 1 = −p 2 . We introduce where p≡ 1 2 (p 1 +p 2 )=(p 0 1 ,0), q =(0,q). Taking the direction of the z-axis along momentum p 1 , we find where and Note that the denominator D of the integrand is invariant under the transformation k→−k.
The electromagnetic radius will be further obtained from the derivative of the form factor via r= 6 dF dq 2 1/2 We have where is the radius of the i-th quark.

Form factor of vector mesons
The definition of the form factor of the vector meson reads [43,44] ρ + (p 2 ,λ 2 )|ψγ μ ψ|ρ + (p 1 ,λ 1 ) where (p 1 ) and (p 2 ) are the polarization vectors of the initial and the final vector meson respectively. Based on the Feynman diagrams, the LHS of Eq. (21) can be written as where Still working in the Breit frame and taking the zaxis along the momentum p 1 , the polarization vectors are chosen to be To retrieve F 1 , we take the time component in Eq. (21) and find that Then F 1 can be obtained via the transverse polarization To retrieve F 2 , we take the spatial components in Eq. (21) and find that Still each form factor F j is a charge weight average of form factors of quark and anti-quark in the vector meson, and

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Explicitly we obtain where and where We will not consider the form factor F 3 in this work.

Vector meson dominance and quark loop correction
According to the vector-meson-dominance picture, the π and K form factor are dominated by the ρ, ω and φ intermediate vector meson states [21]. In the NJL model, the vector-meson-dominance is represented by the correction to quark-photon vertex as shown in the Feynman diagram in Fig. 2. The correction will introduce a form factor to the constituent quark [5]. For the i-th quark where K V is the NJL vector coupling constant and J (T)

V V
represents the transverse vector loop integral [4,16]. The meson form factor will be modified to

Numerical results
The parameters of the extended NJL model were fixed by fitting the meson mass spectra and decay con-stants in a previous work [16]. The input parameters were the current masses of light quarks and the constituent masses of heavy quarks, two coupling constants and the 3-dimensional cutoff: Due to charge conservation, the form factor should be normalized to F (q 2 = 0) = 1 for any hadron carrying +1 charge. We will make a self-consistent calculation by using the theoretical values of meson masses and the quark coupling constants together. This will guarantee the strict normalization of the form factor at q 2 = 0 [23]. The theoretical values of pseudo-scalar and vector mesons are listed in Table 1 and Table 2 respectively.

Pseudo-scalar mesons
The form factors of π + and K + are compared with experimental data in Fig. 3 and Fig. 4 respectively. The theoretical results fit the experimental data well. In the theoretical calculation, the quark loop correction is included to account for the important effect of vector meson dominance.  Fig. 3. The form factor of π + compared to the experimental data from Ref. [20].
The heavy-light pseudo-scalar mesons like D + , D + s and B + have no experimental data for form factor yet. In Fig. 5, we present the form factors of all positive pseudoscalar mesons.
The form factor at low momentum q 2 can be well illustrated by the electromagnetic radius. The radii of all positive pseudo-scalar mesons are listed in Table 3.
As had been observed in Ref. [21], it is the quark loop correction of vector-meson-dominance that makes the π Fig. 4. The form factor of K + compared to the experimental data from Ref. [19]. where are the "intrinsic" charge radius and the quark loop correction respectively. The quark loop correction decreases as the quark mass increases, so the lighter quark has a larger radius than its heavier partner in any meson. We show the individual form factors of quark and anti-quark in π and K mesons in Fig. 6 and also list the individual quark radii in Table 3. Just like the π + and K + , the radii of the light-heavy mesons r 2 D + 1/2 , r 2 D + s 1/2 decrease as the meson mass increases. However the radius of the B + meson increases by roughly a factor 2. This is mainly because, in a lightheavy meson, the heavy quark's contribution is much smaller than that of the light one. If we ignore the contribution of the heavy quark, in the B + meson, the u-quark has a 2/3 charge weight of contribution. The d-quark, on the other hand, has only a 1/3 charge weight in the D + and D + s mesons. The form factors of individual constituent quarks in D and B are shown in Fig. 7.

Vector mesons
The electric form factors F 1 of vector mesons are shown in Fig. 8. The electric radii are listed in Table 4. Because all vector meson masses are close to their thresholds, their bound energies are small and their radii are larger than their pseudo-scalar partners. The magnetic form factors F 2 are presented in Fig. 9. They are connected to the magnetic momentum through [38] The magnetic moments are also listed in Table 4. The magnetic moments are given in the unit of nuclear magneton μ n . Generally, the magnetic momentum decreases as the meson mass increases. In our results, the magnetic moments of D * and D * s are smaller than those of the light mesons ρ and K. However, the magnetic moment of B * is larger than that of D * and D * s . The reason is still that the main contribution comes from the light quark but the charge of the u is larger than that of the d and s by a factor of 2. Up to now, no experimental data is available. We compare our results for the ρ + and K * + mesons with other theoretical work [33,37].

Summary
With the extended NJL model including heavy flavors, we have made a systematic calculation of the form factors of mesons, including the pseudo-scalar mesons and their vector partners, of both the light flavor sector and the light-heavy flavor sector. The form factors of the π and K mesons fit the experimental data. Other form factors of mesons, especially of the light-heavy mesons, are presented here to compare with future experiments and other theoretical calculations such as lattice calculation.
We would like to thank Professor Shi-Lin Zhu for useful discussions.