Poincaré covariant pseudoscalar and scalar meson spectroscopy in Wigner-Weyl phase

The coupled quark Dyson-Schwinger and meson Bethe-Salpeter equations in rainbow-ladder truncation for spin-0 mesons are solved in the Wigner-Weyl phase in the chiral limit and beyond, retaining only the ultraviolet finite terms of the phenomenologically most successful Maris-Tandy interaction. This allows one to reveal and discuss the scalar and pseudoscalar meson masses in a chirally symmetric setting without additional medium effects. Independent of the current-quark mass, the found solutions are spacelike, i.e., have negative squared masses. The current-quark mass dependence of meson masses, leptonic decay constants and chiral condensate are illustrated in the Wigner-Weyl phase.

Figure 1

Propagator functions $A(p^2)$ (left panel) and $M(p^2)$ (right panel) along the real axis obtained from Newton-Krylov optimization for different initial functions $B_0(p^2)$ at $\omega =0.5\mathrm{GeV}$ and $D=16.0{\mathrm{GeV}}^2$ in the WW phase for $m_q=0$ (solid red curve), $m_q=10\mathrm{MeV}$ (dashed blue curve), $m_q=20\mathrm{MeV}$ (dash-dotted green curve), and $m_q=30\mathrm{MeV}$ (dotted magenta curve). Solutions in the NG phase are depicted in black with line styles that correspond to the WW phase.

Figure 2

Nonzero parts of the pseudoscalar BS partial amplitudes in the WW phase for $m_q=5\mathrm{MeV}$, $\omega =0.5\mathrm{GeV}$, and $D=16.0{\mathrm{GeV}}^2$. All other parts are $⪅{10}^{-14}$.

Figure 3

Nonzero parts of the scalar BS partial amplitudes in the WW phase for $m_q=5\mathrm{MeV}$, $\omega =0.5\mathrm{GeV}$, and $D=16.0{\mathrm{GeV}}^2$. All other parts are $⪅{10}^{-15}$.

Figure 4

Nonzero parts of the pseudoscalar BS partial amplitudes in the NG phase (pion) for $m_q=5\mathrm{MeV}$, $\omega =0.5\mathrm{GeV}$, and $D=16.0{\mathrm{GeV}}^2$. All other parts are $⪅{10}^{-14}$.

Figure 5

Nonzero parts of the scalar BS partial amplitudes in the NG phase (sigma) for $m_q=5\mathrm{MeV}$, $\omega =0.5\mathrm{GeV}$, and $D=16.0{\mathrm{GeV}}^2$. All other parts are $⪅10^{-14}$.

Figure 6

Bound state masses in the WW phase for equal quarks in the pseudoscalar (solid curve) and scalar (dashed curve) channel up to the critical current-quark mass $m_q^{\mathrm{cr}}=31\mathrm{MeV}$, $\omega =0.5\mathrm{GeV}$, and $D=16.0{\mathrm{GeV}}^2$.

Figure 7

Leptonic decay constant in the WW phase for equal quarks in the pseudoscalar channel up to the critical current-quark mass $m_q^{\mathrm{cr}}=31\mathrm{MeV}$, $\omega =0.5\mathrm{GeV}$, and $D=16.0{\mathrm{GeV}}^2$. The scalar leptonic decay constant is numerically zero, $f_{\sigma }⪅{10}^{-14}\mathrm{GeV}$, for all current-quark masses as in the NG phase [71].

Figure 8

Chiral condensate in WW phase up to the critical current-quark mass $m_q^{\mathrm{cr}}=31\mathrm{MeV}$, $\omega =0.5\mathrm{GeV}$ and $D=16.0{\mathrm{GeV}}^2$.