Dressed quark mass dependence of pion and kaon form factors

The structure of hadrons is described well by the Nambu–Jona-Lasinio (NJL) model, which is a chiral effective quark theory of QCD. In this work we explore the electromagnetic structure of the pion and kaon using the three-flavor NJL model in the proper-time regularization scheme, including effects of the pion cloud at the quark level. In the calculation there is only one free parameter, which we take as the dressed light quark ($u$ and $d$) mass. In the regime where the dressed light quark mass is approximately $0.25$ GeV we find that the calculated values of the kaon decay constant, current quark masses, and quark condensates are consistent with experiment- and QCD-based analyses. We also investigate the dressed light quark mass dependence of the pion and kaon electromagnetic form factors, where comparison with empirical data and QCD predictions also favors a dressed light quark mass near $0.25\mathrm{GeV}$.

Figure 1

Quark self-energy in the mean-field approximation. The solid line represents a dressed quark propagator.

Figure 2

Random phase approximation for the quark-antiquark $T$ matrix.

Figure 3

The pseudoscalar meson bubble diagram, ${\Pi }_k\left(p^2\right)$, where $k=\pi ,K$. The dashed line represents a pion $(\alpha ,\beta =1,2,3)$ or a kaon $(\alpha ,\beta =4,5,6,7)$.

Figure 4

Diagram representing the pseudoscalar meson decay constant. The dashed line represents a pseudoscalar meson and the wavy line an external axial-vector field.

Figure 5

Feynman diagrams for the meson electromagnetic current.

Figure 6

Feynman diagrams for the quark electromagnetic current with a pion cloud.

Figure 7

Pion-cloud self-energy diagram for the light quarks.

Figure 8

Dressing of the quark electromagnetic current from vector mesons.

Figure 9

Pion form factor with $M=0.25\mathrm{GeV}$ (see Table 1). The data show the experimental values from Amendolia et al. [6] and Huber et al. [7] and projected values from Huber et al. [8].

Figure 10

Pion form factor with $M=0.4\mathrm{GeV}$ (see Table 1). The data show the experimental values from Amendolia et al. [6] and Huber et al. [7] and projected values from Huber et al. [8].

Figure 11

Kaon form factor with $M=0.25\mathrm{GeV}$ (see Table 1). The experimental values are taken from Amendolia et al. [9] and Dally et al. [55].

Figure 12

Kaon form factor with $M=0.4\mathrm{GeV}$ (see Table 1). The experimental values are taken from Amendolia et al. [9] and Dally et al. [55].

Figure 13

Results of the ratio of the kaon form factor to the pion form factor $F_K\left(Q^2\right)/F_{\pi }\left(Q^2\right)$ for the case $M=0.25$ GeV.

Figure 14

Graphical representation of the general quark electromagnetic vertex ${\Gamma }^{\mu }$.