Unified description of hadron-photon and hadron-meson scattering in the Dyson-Schwinger approach

We derive the expression for the nonperturbative coupling of a hadron to two external currents from its underlying structure in QCD. Microscopically, the action of each current is resolved to a coupling with dressed quarks. The Lorentz structure of the currents is arbitrary and thereby allows to describe the hadron’s interaction with photons as well as mesons in the same framework. We analyze the ingredients of the resulting four-point functions and explore their potential to describe a variety of processes relevant in experiments. Possible applications include Compton scattering, the study of two-photon effects in hadron form factors, pion photo- and electroproduction on a nucleon, nucleon-pion or pion-pion scattering.

Figure 1

Various contributions to the four-point function, here in the case of pion photoproduction on the nucleon, that must be recovered in a microscopic description: handbag (a) and cat’s-ears diagrams (b); $s$-channel and, via crossing, $u$-channel nucleon resonances (c); $t$-channel meson exchange (d).

Figure 2

Left panel: Dyson equation (6) for the scattering matrix in the three-quark system. Right panel: bound-state equation (9) for a baryon.

Figure 3

Quark-antiquark vertex (16) and its behavior on the meson bound-state pole, Eq. (18).

Figure 4

Upper panel: Diagrammatic representation of the scattering amplitude $J^{\mu \nu }$, Eq. (39), in the case of a baryon. A symmetrization of the two currents with indices $\mu$ and $\nu$ leads to an additional crossed diagram. Lower panel: Decomposition of the quantity ${\mathbf{\Lambda }}^{\mu }$, cf. Eqs. (37, 41, 44), that enters the scattering amplitude. For each of the first three graphs on the r.h.s. there are two further permutations with respect to the quark lines. Only the first two diagrams contribute in rainbow-ladder truncation.

Figure 5

Contributions to the nucleon-photon Compton amplitude $J^{\mu \nu }$ in rainbow-ladder truncation, cf. Eq. (63). All quark propagators are dressed, and each box includes further diagrams with permuted quark lines. In addition, the photon indices $\mu$ and $\nu$ must be symmetrized. $T$ represents the quark-antiquark and three-quark scattering matrices and $K$ the two-quark irreducible kernel. The hatched nucleon amplitudes represent the quantities defined in (60). Diagram (c) contains $t$-channel meson exchange whereas $s$- and $u$-channel nucleon resonances appear in diagram (d). To obtain the pion photoproduction amplitude, replace one of the quark-photon vertices by an on-shell pion amplitude; for nucleon-pion scattering, replace both.

Figure 6

$\pi \pi$ scattering amplitude $J^{\mu \nu }$ in rainbow-ladder truncation, cf. Eq. (69). A further symmetrization in the indices $\mu$ and $\nu$ duplicates the number of diagrams. To obtain the pion Compton scattering amplitude, replace the amplitudes attached to $\mu$ and $\nu$ by quark-photon vertices.