Effective field theory for large logarithms in radiative corrections to electron proton scattering

Radiative corrections to elastic electron proton scattering are analyzed in effective field theory. A new factorization formula identifies all sources of large logarithms in the limit of large momentum transfer, $Q^2\gg m_e^2$. Explicit matching calculations are performed through two-loop order. A renormalization analysis in soft-collinear effective theory is performed to systematically compute and resum large logarithms. Implications for the extraction of charge radii and other observables from scattering data are discussed. The formalism may be applied to other lepton-nucleon scattering and $e^{+}{e}^{-}$ annihilation processes.

Figure 1

Scattering of proton from electromagnetic source.

Figure 2

First order radiative corrections to electron scattering from static source.

Figure 3

Second order radiative corrections to electron scattering from static source. Diagrams involving photon emission from the initial state electron are not shown.

Figure 4

Radiative correction $\delta$ from Eq. (26), in static source limit for $E=1\mathrm{GeV}$, $\mathrm{\Delta }E=5\mathrm{MeV}$, computed at first (bottom, blue curve) and second (top, red curve) in $\alpha$.

Figure 5

Radiative correction factor $\delta$ from Eq. (26) in resummed perturbation theory for the static source limit of electron proton scattering, with $E=1\mathrm{GeV}$, $\mathrm{\Delta }E=5\mathrm{MeV}$. The bands represent the impact of varying $\min (Q^2,E^2)/2<{\mu }_H^2<2\max (Q^2,E^2)$ and $\min (m^2,\mathrm{\Delta }{E}^2)/2<{\mu }_L^2<2\max (m^2,\mathrm{\Delta }{E}^2)$, using leading log resummation (blue horizontal stripes), next-to-leading log resummation (red vertical stripes), and complete next-to-leading order resummation (black solid band).

Figure 6

Same as Fig. 5, but including recoil and nuclear charge corrections (i.e., two-photon exchange and proton vertex corrections). The total radiative correction $\delta$ is given by Eq. (66).

Figure 7

Comparison of complete next-to-leading order resummed correction (solid black band) to naive exponentiations using different factorization scales for the two-photon exchange correction: ${\mu }^2=M^2$ (dotted red line) and ${\mu }^2=Q^2$ (dashed blue line). See text for details.

Figure 8

Expansion in momentum regions of amplitudes for electron scattering in the static source limit. Diagram on the left hand side is in the full theory (QED), and diagrams on the right hand side are in the effective theory. Soft and collinear photons are represented by curly lines and curly lines superimposed on solid lines, respectively.

Figure 9

Same as Fig. 8, but for electron scattering with real photon emission.