Constituent quark-light vector mesons effective couplings in a weak background magnetic field

Effective couplings between light SU(2) vector and axial mesons and constituent quarks are calculated in the presence of a background electromagnetic field by considering a one dressed gluon exchange quark-quark interaction. The effective coupling constants, obtained from a large quark mass expansion, are expressed in terms of the Lagrangian parameters of the initial model and of components of the quark and nonperturbative gluon propagators. In spite of many possible couplings, only a few coupling constants emerge. As a second step, constituent quark-vector and axial mesons effective coupling constants are redefined to show explicit dependence on a weak background magnetic field. Ratios between the effective coupling constants are found in the limit of large quark effective mass and numerical estimates are presented.

Figure 1

In these diagrams, the wavy line with a full dot is a (dressed) nonperturbative gluon propagator, a solid line represents a constituent quark, the single wavy line stands for a photon, the wavy line with a triangle in the vertex corresponds to the photon strength tensor $F^{\mu \nu }$, the dashed-dotted line represents a vector or axial meson $V_{\mu }^{\alpha }$, and the dashed-dotted line with a triangle in the vertex stands for the Abelian strength tensor ${\mathcal{F}}_{\mu \nu }^{\alpha }$ of a vector or axial meson $\alpha$.

Figure 2

The form factors $g_{qA}(Q)$ and $g_3^{B_1}(Q)$ from expressions (39) and (40) are shown for the gluon propagator $D_I(k)$ of Refs. [37, 38]. The thin solid line stands for $g_{qA}(Q)$ with $M^{*}=330\mathrm{MeV}$ (for $B_0=0$) and the dashed line for $M^{*}=380\mathrm{MeV}$, i.e., it corresponds to an effective mass at $e{B}_0=0.2{M^{*}}^2$. The thick solid line represents the anisotropic correction $g_3^{B_1}(Q)$ for $M^{*}=330\mathrm{MeV}$ whereas the dotted line stands for $M^{*}=380\mathrm{MeV}$ for $e{B}_0=0.2{M^{*}}^2$. To add the anisotropic correction to the form factor $g_{qA}$ it still must be multiplied by $(e{B}_0)/{M^{*}}^2$. The quadrupole fit of the rho meson-nucleon electric form factor from Ref. [51] is shown in the dotted-dashed line for $F_1^V(0)=2.8$.

Figure 3

The form factors $g_{qA}(Q)$ and $g_3^{B_1}(Q)$ from expressions (39) and (40) are shown for the gluon propagator $D_{II}(k)$ of Ref. [40]. The thin solid line $g_{qA}(Q)$ stands for $M^{*}=330\mathrm{MeV}$ (for $B_0=0$) and the dashed line for $M^{*}=380\mathrm{MeV}$ for $e{B}_0=0.2{M^{*}}^2$. The thick solid line represents the anisotropic correction $g_3^{B_1}(Q)$ for $M^{*}=330\mathrm{MeV}$ whereas the dotted line stands for $M^{*}=380\mathrm{MeV}$, i.e., it corresponds to an effective mass at $e{B}_0=0.2{M^{*}}^2$. To add the anisotropic correction to the form factor $g_{qA}$ it must be multiplied by $(e{B}_0)/{M^{*}}^2$. The quadrupole fit of the rho meson-nucleon electric form factor from Ref. [51] is shown in the dotted-dashed line for $F_1^V(0)=0.21$.

Figure 4

The rho squared electromagnetic radius (rqm) with the gluon propagator $D_I(k)$ from Refs. [37, 38] for $M^{*}(B_0=0)=330\mathrm{MeV}$ is shown as a function of the magnetic field in terms of the effective mass $(e{B}_0)/{M^{*}}^2$. The symbol $\times$ indicates the leading correction $⟨r_{\rho }^2(M^{*})⟩=-6\frac{d}{d{Q}^2}{g}_{qA}(Q)$ in expression (41) and the symbols $*$ represent the total values of $⟨r_{\rho }^2⟩(B_0)$ with the anisotropic correction $(e{B}_0)/{M^{*}}^2$.

Figure 5

The rho squared electromagnetic radius (rqm) with the gluon propagator $D_{II}(k)$ from Ref. [40] for $M^{*}(B_0=0)=330\mathrm{MeV}$ is shown as a function of the magnetic field in terms of the effective mass $(e{B}_0)/{M^{*}}^2$. The symbol $\times$ indicates the leading correction $⟨r_{\rho }^2(M^{*})⟩=-6\frac{d}{d{Q}^2}{g}_{qA}(Q)$ in expression (41) and the symbols $*$ represent the total values of $⟨r_{\rho }^2⟩(B_0)$ with the anisotropic correction $(e{B}_0)/{M^{*}}^2$.