Anatomy of relativistic pion loop corrections to the electromagnetic nucleon coupling

We present a relativistic formulation of pion loop corrections to the coupling of photons with nucleons on the light front. Vertex and wave function renormalization constants are computed to lowest order in the pion field, including their nonanalytic behavior in the chiral limit, and studied numerically as a function of the ultraviolet cutoff. Particular care is taken to explicitly verify gauge invariance and Ward-Takahashi identity constraints to all orders in the $m_{\pi }$ expansion. The results are used to compute the chiral corrections to matrix elements of local operators, related to moments of deep-inelastic structure functions. Finally, comparison of results for pseudovector and pseudoscalar coupling allows the resolution of a longstanding puzzle in the computation of pion cloud corrections to structure function moments.

Figure 1

Pion loop corrections to the photon–nucleon coupling in the PV pion-nucleon theory: (a) photon coupling to the bare nucleon, (b) wave function renormalization, (c) rainbow diagram with coupling to the nucleon, (d) rainbow diagram with coupling to the pion, (e) Kroll-Ruderman diagrams, (f) pion tadpole diagram with coupling to the pion-nucleon vertex, (g) pion bubble diagram with coupling to the pion.

Figure 2

Contributions to the vertex renormalization $Z_1^N$ from terms in Eq. (10) proportional to $1/D_{\pi }{D}_N^2$ (dashed line), $1/D_{\pi }{D}_N$ (dot-dashed line), $1/D_{\pi }$ (dotted line), and the sum (solid line), as a function of the $k_{\perp }$ momentum cutoff $\Lambda$.

Figure 3

Contributions to the vertex renormalization $Z_1^{\pi }$ from terms in Eq. (18) proportional to $1/D_{\pi }{D}_N$ (dot-dashed line), $1/D_{\pi }$ (dotted line), and the sum (solid line), as a function of the $k_{\perp }$ momentum cutoff $\Lambda$.

Figure 4

Contributions to the vertex renormalization $Z_1^{\mathrm{KR}}$ from terms in Eq. (23) proportional to $1/D_{\pi }{D}_N$ (dot-dashed line), $1/D_{\pi }$ (dotted line), and the sum (solid line), as a function of the $k_{\perp }$ momentum cutoff $\Lambda$.

Figure 5

Contributions to the vertex renormalization $Z_1$ from the photon–nucleon coupling $Z_1^N$ (solid line), photon–pion coupling $Z_1^{\pi }$ (dashed line), and Kroll-Ruderman terms $Z_1^{\mathrm{KR}}$ (dot-dashed line), as a function of the $k_{\perp }$ momentum cutoff $\Lambda$. Note that the sum of the pion tadpole and bubble contributions to $Z_1$ vanishes.

Figure 6

Nucleon tadpole (solid line) and pion bubble (dashed line) contributions to the vertex renormalization as a function of the $k_{\perp }$ momentum cutoff $\Lambda$.

Figure 7

Coupling of an electromagnetic current to a proton (with momentum $p$) dressed by a pion (with momentum $k$): (a), (b) wave function renormalization diagrams, (c) rainbow diagram with coupling to the proton, (d) rainbow diagram with coupling to the ${\pi }^{+}$, (e), (f) Kroll-Ruderman diagrams, (g) tadpole diagram with coupling to the $\pi \pi pp$ vertex, (h) bubble diagram with coupling to the pion. The current brings in a momentum $q$.