Tracing masses of ground-state light-quark mesons

We describe a symmetry-preserving calculation of the light-quark meson spectrum, which combines a description of pion properties with reasonable estimates of the masses of heavier mesons, including axial-vector states. The kernels used in formulating the problem are essentially nonperturbative, incorporating effects of dynamical chiral symmetry breaking (DCSB) that were not previously possible to express. Our analysis clarifies a causal connection between DCSB and the splitting between vector and axial-vector mesons and exposes a key role played by the anomalous chromomagnetic moment of dressed quarks in forming the spectrum.

Figure 1

Dashed curve: Dressed-quark anomalous chromomagnetic moment distribution computed using $\eta =0.65$. Solid curve: Electromagnetic moment distribution computed from the inhomogeneous Bethe-Salpeter equation for the dressed-quark-photon vertex. In the chiral limit and the absence of DCSB, both curves are zero. Moreover, absent DCSB-induced terms in the dressed-quark-gluon vertex, Eq. (6), both curves would be an order-of-magnitude smaller [13]. (The Euclidean constituent-quark mass $M^E:=\left\{p\right|p>0,p^2=M^2\left(p^2\right)\}=0.35$GeV.)

Figure 2

Illustration of the procedure used to determine meson masses. Solid curve: $a_1$ meson, nonperturbative kernel. Dot-dash-dash curve: $a_1$, kernel derived from Eq. (7) only (Ball-Chiu, BC). Dashed curve: $a_1$, kernel derived from just the first term in Eq. (7) (1BC, a minimal renormalization improvement [18] of the leading order—RL, rainbow-ladder—kernel [10]). Dot-dashed curve: $\rho$ meson, nonperturbative kernel. Dot-dash-dot curve: $\rho$, BC kernel. Dotted curve: $\rho$, 1BC kernel. Points: Values of $1/\Gamma (k=0;P^2)$ in the given channel computed with the kernel described. Padé approximants are constructed in each case, and the location of the zero is identified with $(-m_{\mathrm{meson}}^2)$.