The electromagnetic form factors of heavy-light pseudo-scalar and vector mesons

We work within DSEs/BSEs framework in Euclidean space. Under the rainbow-ladder (RL) approximation, the dressed-quark propagator can be obtained from the following gap equation,

and the general form of $S^{-1}(p)$ can be written as

Where $A(p^{2})$ and $B(p^{2})$ are scalar functions, $m$ is current quark mass, $Z_{1,2,4}$ are the renormalization constants, in this work we employ a mass-independent momentum-subtraction renormalisation scheme [40] and choose renormalization scale $\xi=19$ GeV [41]. $\int_{q}^{\Lambda}$ represents a translationally-invariant regularization of the four-dimensional integral with the regularization scale $\Lambda$. For the dressed-gluon propagator $D_{\mu\nu}(k)$, we employ the Qin-Chang model [26, 28]

where $\mathcal{P}_{\mu\nu}^{T}(k)=\delta_{\mu\nu}-{k_{\mu}k_{\nu}}/k^{2}$ is transverse projection operator and the effective interaction is

with $\mathcal{F}(k^{2})=\{1-\exp[(-k^{2}/(4m_{t}^{2})]\}/k^{2}$, $m_{t}=0.5$ GeV, $\tau=e^{2}-1$, $\Lambda_{\text{QCD}}=0.234$ GeV, $\gamma_{m}=12/25$ [41]. For model parameters $D$ and $\omega$, a typical choice is $\omega=0.5$ GeV with $D\omega=(0.82\ \text{GeV})^{3}$ for $u/d$, $s$ quark, $\omega=0.8$ GeV with $D\omega=(0.6\ \text{GeV})^{3}$ for $c$, $b$ quark [41, 37]. In this work we follow these values except $(D\omega)_{s}=(0.68\ \text{GeV})^{3}$, $(D\omega)_{c}=(0.66\ \text{GeV})^{3}$ ,$(D\omega)_{b}=(0.48\ \text{GeV})^{3}$, which are tweaked slightly to consider flavor dependence of the interaction [33], thus producing results closer to the experimental value (see Table. 1). More details of Eq. (1-4) are presented in Refs. [22, 24, 25, 26, 23].

Correspondingly, the meson’s Bethe-Salpeter amplitudes (BSAs), $\Gamma^{f\bar{g}}_{H}\left(k_{+},k_{-}\right)$, and the dressed quark-photon vertex, $\Gamma^{\gamma,f\bar{f}}_{\mu}\left(k_{+},k_{-}\right)$, can be obtained from (in)homogeneous BSEs, respectively,

and

where $f$ and $g$ denote the flavor of (anti-)quark, $k_{+}=k+\alpha P$; $k_{-}=k-(1-\alpha)P$ with the momentum partitioning parameter $\alpha\in[0,1]$. Although the physical observables do not depend on $\alpha$, in the actual calculation¹¹1If we define the vertex of parabola contour in $k^{2}_{\pm}$ complex plane as $(-\alpha^{2}\Delta_{f}^{2},0)$ and $(-(1-\alpha)^{2}\Delta_{\bar{g}}^{2},0)$ , we will have $1-\Delta_{\bar{g}}/M<\alpha<\Delta_{f}/M$, and the best $\alpha=\Delta_{f}/(\Delta_{f}+\Delta_{\bar{g}})$, where $M$ is meson’s mass., the selection of $\alpha$ should be careful to avoid parabola include the pole for the accuracy of the contour integral [44, 45]. The general form of $\Gamma(k_{+},k_{-})$ can be written as

where $\tau^{i}(k,P)$ is basis and $\mathcal{F}_{i}(k,P)$ is scalar function. For the pseudo-scalar/vector meson, we choose [28]

with $V_{\mu}^{T}=V_{\mu}-P_{\mu}(V\cdot P)/P^{2}$. Then the decay constant can be obtained easily after normalization of the meson’ BSAs [28].

As for the quark-photon vertex $\Gamma^{\gamma}_{\mu}(k_{+},k_{-})$, the general basis is

and $\Gamma^{\gamma}_{\mu}(k_{+},k_{-})$ should satisfy the vector Ward-Green-Takahashi identity (WGTI) [34, 39]

The only thing left is the BSE interaction kernel $K^{f\bar{g}}(k,q;P)$. In the case of flavor symmetry, the standard RL approximation kernel can be written as [41]

and generally, it works well for the flavor symmetric/slightly asymmetric ground state pseudo-scalar/vector mesons. However, for the highly flavor asymmetric system, such as $u\bar{c}$, $u\bar{b}$, $\dots$, Eq. (11) is difficult to be applied because of the lack of flavor asymmetry [46]. To consider this effect, we effectively average the kernel as [30]

Here a weight factor $\eta$ is introduced, and Eq. (12) can be regarded as an extension of RL approximation, that is, weight-RL. For the flavor symmetric case, $\eta=1-\eta=0.5$ and it will degenerate to the RL kernel, therefore the solution of Eq. (6) still satisfy the vector WGTI, which will be used in the calculation of electromagnetic form factor [34].

For flavor asymmetric meson, an automatic average of this weight has been presented in Ref. [30]. In this work, we directly determine the $\eta$ by the pseudo-scalar meson’s mass to obtain a relatively realistic interaction. Once the weight factor is fixed, the decay constant of the pseudo-scalar meson, the mass and decay constant of the vector meson can all be well predicted (see Table. 2 and Figure. 1). It is worth noting that Eq. (12) should be considered as an effective kernel, the theoretical explore for flavor dependence and strict beyond-RL kernel is still ongoing [47, 48, 49, 33, 46, 50]. .

The generalized impulse approximation allows electromagnetic processes to be described in terms of dressed quark propagators, bound state BSAs, and the dressed quark-photon vertex. These couplings are given by [34, 35, 37]

Where $P-Q/2$, $P+Q/2$ and $Q$ are incoming meson, outgoing meson and incoming photon momenta, which are constrained by on-shell condition

with $M$ is meson’s mass. The other elements in Eq. (13) are the dressed-quark propagators $S(q)$, the meson’s BSAs $\Gamma_{H}(k_{+},k_{-})$ and dressed quark-photon vertex $\Gamma^{\gamma}_{\mu}(k_{+},k_{-})$. In this work we determine them from homogeneous/inhomogeneous BSEs in the moving frame, correspondingly, $k_{+}=k+(1-\alpha)P+Q/2$, $k_{-}=k+(1-\alpha)P-Q/2$, $k_{p}=k-\alpha P$, with $\alpha$ is the momentum partitioning parameter of the outgoing meson and physical observables are independent of it. Then the need for interpolation or extrapolation of the meson’s BSAs/quark-photon vertex can be avoided [35].

Consider the coupling of a photon to the quark and antiquark, this interaction should be written as the sum of two terms (see Figure. 2)

where $\hat{Q}$ is the quark or antiquark electric charge. For pseudo-scalar meson, the only form factor is defined by [34]

and for vector meson, the three form factors are defined by [35]

with

and $\mathcal{P}_{\mu\nu}^{T}(k)=\delta_{\mu\nu}-{k_{\mu}k_{\nu}}/k^{2}$. Where $G_{E}(0)=F(0)=e$, $e=1,0$ defines the meson’s charge. In the impulse approximation, as long as the relation between the dressed quark propagator and the quark-photon vertex satisfies WGTI, and the BSE kernel is independent of the meson momentum, the conservation of electromagnetic current will be preserved after the meson’s BSAs are canonical normalized [34]. Besides, $G_{M}$(0) and $G_{\mathcal{Q}}(0)$ can be identified with the magnetic moment, $\mu$, and the quadrupole moment, $\mathcal{Q}$, of a vector meson [35, 37].

With all the above at hand, we first consider the electromagnetic form factor for ground state heavy-light pseudo-scalar mesons. These are the simplest quark-antiquark bound states embodying confinement, and the lightest pseudo-scalar meson, $\pi$, is also the Goldstone mode of DCSB. In Figure. 3, we present a comparison of the electromagnetic form factors of the pion and kaon with current experimental results. Our predictions demonstrate a good agreement with the experimental data.

In the case of pseudo-scalar channel, the only electromagnetic form factor, $F(Q^{2})$, corresponds to the charge distribution of the system²²2Although the exact form of this relation is still up for debate [67, 68].. It is well known that the charge radius can be defined as

which denotes the distribution range of charge. The numerical results of Eq. (24) can be found in Table. 3. As a comparison, we also collected some predictions from other approaches.

For the full charge radius of these mesons, each approaches reports results of the same order of magnitude, that is, $\sqrt{\left\langle r^{2}\right\rangle}<1$ fm. It is no surprise because fm is the order of magnitude of a nucleon’s size. Nevertheless, the charge radii obtained from contact interaction model (CI) are lower than others, since CI is a relatively simple model [12]. However, it still provides useful qualitative results. CI predicts that the charge radius of $\pi$ is the largest of these ground state pseudo-scalar meson, and our results confirm this conclusion. This is interesting because $\pi$ is the one with the most significant DCSB effect, which seems to suggest that DCSB, while generating mass, also tends to increase the size of the system.

For slightly/moderately flavor asymmetric meson, such as $u\bar{s}$, $u\bar{c}$, $s\bar{c}$, $s\bar{b}$, $c\bar{b}$, …, the charge radii predicted by these methods are generally consistent. However, In the case of extremely flavor symmetry breaking, such as $u\bar{b}$, these results begin to deviate from each other. CI reports 0.34 fm for $B^{+}$, but the result of AM is 0.926 fm, even larger than the proton radius, 0.841(1) fm [70]. In this work, the predicted charge radius of $B^{+}$ is rather close to light-front framework (LFF) $\sim 0.62$ fm. More comparison can be found in Table. 3.

According to Eq. (15), the contributions of different quarks to the system can be extracted, this gives us a glimpse into the internal structure of heavy-light meson. In Table. 3 and Figure. 4, both the separated contributions and full results are presented, we will discuss this in conjunction with vector mesons in the next subsection.

Compared with the pseudo-scalar case, the heavy-light vector meson’s electromagnetic form factor have received much less attention. On the one hand, it is difficult to measure in experiment, on the other hand, its theoretical calculation is more complicated. However, noteworthy distinctions or unexpected similarities between pseudo-scalar and vector mesons, such as their charge radii, can help us to better understand the internal structure of these hadrons. Therefore, in this work, we calculate the EFFs of pseudo-scalar/vector mesons with momentum transfer $Q^{2}<2\ \text{GeV}^{2}$ uniformly, except $B^{*}$ because the pole of the quark propagator in the complex plane limits the computable region [36].

The electric form factor of vector meson, $G_{E}(Q^{2})$, can be compared with the pseudo-scalar meson’s $F(Q^{2})$, because both of them correspond to the charge distribution. The results of this comparison are presented in Figure. 4, and a clear pattern can be noticed immediately.

As mentioned earlier, Eq. (15) allows us to separate the contributions of the lighter and heavier dressed-quark. For $u\bar{q}$, $q=u/d,s,c,b$ systems, with the increase of current quark mass of $q$, the form factor of the lighter quark becomes steeper while the form factor of the heavier quark flattens out (see Figure. 4, upper panel). The extracted charge radii are presented in the Table. 3 (PS) and Table. 4 (VC). Obviously, the distribution range of $u$ and $\bar{q}$ (anti-)quarks gradually expands and contracts. Especially, for $u$ quark in $u\bar{b}$ system, $\sqrt{2/3\langle r^{2}\rangle_{u}}=$ 0.618 fm (PS), 0.655 fm (VC), it is very close to the charge radius of $B^{+}$ and $B^{*+}$, that is, 0.619 fm and 0.657 fm. This reveals that, in $u\bar{b}$ systems, the $\bar{b}$ quark is basically stationary, with the charge distribution almost entirely contributed by the $u$ quark.

For the $s\bar{q}$, $c\bar{q}$ systems, in the middle panel of Figure. 4 we constructed four fictitious states, $\pi_{s},\pi_{c},\rho_{s},\rho_{c}$, which are constituted from $q=u/d$-like quarks with current masses equal to $s$ and $c$ quarks. Again, flavor symmetry breaking leads to the splitting of the form factor, the distribution range of lighter and heavier quarks gradually expands and contracts, respectively. We note that similar conclusions have also been reported by the ENJL model [14].

The full results of the electric form factors of heavy-light mesons are presented in the lower panel of Figure. 4. Qualitatively, the results of pseudo-scalar and vector channel are not much different, however, the charge radius of vector meson is larger than that of its pseudo-scalar counterpart (see Table. 3 and Table. 4). This suggests that spin-dependent interactions will expand the size of the meson, which is consistent with the conclusion in the case of flavor symmetry [37].

Differing from pseudo-scalar mesons, vector mesons have two more form factors, $G_{M}(Q^{2})$ and $G_{\mathcal{Q}}(Q^{2})$, corresponding to magnetic moment and quadrupole moment, respectively. The full results are plotted in Figure. 5 (charged mesons) and Figure. 6 (neutral mesons). For heavy-light mesons, the form factor is qualitatively consistent as the flavor asymmetry increases. However, in the case of charge neutrality, the significant deviation from zero has been exhibited, which reveals a non-trivial internal structure.

The extracted magnetic moments and quadrupole moments are presented together in Table. 4. It is easy to see that when the charge radius of dressed-quark rises, the corresponding magnetic moment and quadrupole moment also increases. In Table. 5, the magnetic moments are listed in the unit of nuclear magneton $\mu_{n}$ to compare with the results of other approaches. Once again, the results given by each model/framework are basically the same for slightly/moderately flavor asymmetric meson, but in the extremely flavor asymmetric case, there are significant differences between them. Our predictions of the magnetic moment are basically consistent with the ENJL model’s results. The possible reason is that NJL is a similar framework to DSEs/BSEs, and the magnetic moment is not sensitive to the form of interaction. However, due to the absence of experimental data, these predictions still need to be verified by more approaches.