Nucleon Compton scattering in the Dyson-Schwinger approach

We analyze the nucleon’s Compton scattering amplitude in the Dyson-Schwinger/Faddeev approach. We calculate a subset of diagrams that implements the nonperturbative handbag contribution as well as all $t$-channel resonances. At the quark level, these ingredients are represented by the quark Compton vertex whose analytic properties we study in detail. We derive a general form for a fermion two-photon vertex that is consistent with its Ward-Takahashi identities and free of kinematic singularities, and we relate its transverse part to the on-shell nucleon Compton amplitude. We solve an inhomogeneous Bethe-Salpeter equation for the quark Compton vertex in rainbow-ladder truncation and implement it in the nucleon Compton scattering amplitude. The remaining ingredients are the dressed quark propagator and the nucleon’s bound-state amplitude which are consistently solved from Dyson-Schwinger and covariant Faddeev equations. We verify numerically that the resulting quark Compton vertex and nucleon Compton amplitude both reproduce the $\pi \gamma \gamma$ transition form factor when the pion pole in the $t$ channel is approached.

Figure 1

Graphical representation of the nucleon Compton scattering amplitude in rainbow-ladder truncation, Eq. (2) with Eq. (6). All propagators are dressed, and permutations in the quark lines and symmetrizations of the photon legs are not displayed. Diagram (a) corresponds to the term with ${\mathbf{\Lambda }}^{\{\mu }G{\mathbf{\Lambda }}^{\nu \}}$ and diagrams (b)–(c) to ${\mathbf{\Lambda }}^{\mu \nu }$. In terms of Eq. (8), diagram (a) illustrates the sum of the first and third lines, diagram (b) the second and diagram (c) the fourth line. The kernels that connect the spectator quarks, i.e., the factors ($1-K_a$) in Eq. (8), have been absorbed in the hatched amplitudes. Diagram (b) contains the 1PI quark two-photon vertex and diagram (a) provides the Born parts, i.e., the pure handbag diagrams, to the full quark Compton vertex. Note that the electromagnetic current of Eq. (1) in rainbow-ladder truncation has the same form as diagram (b) if the two-photon vertex is replaced by the quark-photon vertex.

Figure 2

Upper panel: Separation of the fermion Compton vertex into Born terms and a 1PI part. Lower panel: Inhomogeneous BSE for the quark Compton vertex in rainbow-ladder truncation, where the $q\overline{q}$ kernel $K$ additionally simplifies to a dressed gluon exchange.

Figure 3

Kinematics in the fermion Compton vertex.

Figure 4

Sketch of the Compton scattering phase space in the variables $\tau$ and ${\tau }^{\prime }$ at fixed $t$ (upper panel) and in the variables $X$ and $Z$ for arbitrary $t\ne 0$ (lower panel). The abbreviations RCS, VCS, VVCS, and TCS denote real, virtual, doubly virtual and timelike Compton scattering. The colored area describes the spacelike region.

Figure 5

Separation of the fermion Compton vertex into Born terms and a 1PI part.

Figure 6

$t$-channel meson poles in the quark Compton vertex (left panel) and the nucleon Compton scattering amplitude (right panel). At the pion pole, the four-point functions separate into the $\pi \gamma \gamma$ transition current and the pion’s bound-state amplitude (left panel) or the nucleon-pion current as the pole residue of the nucleon’s pseudoscalar current (right panel).

Figure 7

Dressing functions ${\tilde{f}}_{1,1}(t)$ and ${\tilde{f}}_{4,1}(t)$ of the quark Compton vertex in the basis (62) at $p^2=0$, $Z=0$ and $\sigma =1$. We plot the zeroth Chebyshev moments in $z$ and $y$. The scalar and pion pole locations are shown by dotted vertical lines.

Figure 8

Reconstruction of $F_{\pi \gamma \gamma }$ from the quark Compton vertex (left panel) and the nucleon Compton amplitude (right panel) via Eq. (186). The thick (red) line in each panel is the result of the direct calculation for the $\pi \gamma \gamma$ transition current. The thin lines are the extrapolated values for various values of $t$, decreasing from bottom to top (left panel) or from right to left (right panel), as indicated by the legend.

Figure 9

Inhomogeneous BSE for the quark Compton vertex, Eqs. (18, b1).

Figure 10

Notation and kinematics for the handbag diagrams of Eq. (12) in the nucleon’s rainbow-ladder truncated Compton scattering amplitude.