Numerical integration of one-loop Feynman diagrams for N-photon amplitudes

TL;DR

This paper investigates a fully numerical approach to one-loop, IR-safe amplitudes by combining a Feynman-parameter representation with contour deformation and Monte Carlo integration. It demonstrates the method on massless-electron-loop N-photon scattering, presenting results for N=4, 5, and 6 (with left-handed couplings) and validating against known analytic results where available. Key contributions include a practical contour-deformation strategy, a UV subtraction scheme for the divergent case, and a tailored Monte Carlo sampling approach that targets singular regions. The findings show that the method is viable up to N=6, with discussion on potential extensions to larger N and to more general theories such as QCD.

Abstract

In the calculation of cross sections for infrared-safe observables in high energy collisions at next-to-leading order, one approach is to perform all of the integrations, including the virtual loop integration numerically. One would use a subtraction scheme that removes infrared and collinear divergences from the integrand in a style similar to that used for real emission graphs. Then one would perform the loop integration by Monte Carlo integration along with the integrations over final state momenta. In this paper, we have explored how one can perform the numerical integration. We have studied the N-photon scattering amplitude with a massless electron loop in order to have a case with a singular integrand that is not, however, so singular as to require the subtractions. We report results for N = 4, N = 5 with left-handed couplings, and N=6.

Figures (5)

• Figure 1: Feynman diagram for the $N$-photon amplitude.
• Figure 2: Illustration of the double parton scattering singularity.
• Figure 3: Four photon amplitude. We plot $|{\cal M}|/\alpha^{2}$ for helicities $+$$+$$-$$-$ versus $\theta - \pi/2$, where $\theta$ is the scattering angle. The curve is the analytical result from Ref. lightbylight. The points are the result of numerical integration.
• Figure 4: Five vector boson amplitude. We plot $\sqrt s\,|{\cal M}|/\alpha^{5/2}$ with helicities $+$$-$$-$$+$$+$. The vector boson is massless and couples to the electron with the left-handed part of the photon coupling. An arbitrarily chosen final state was rotated about the $y$-axis through angle $\theta$.
• Figure 5: Six photon amplitude. We plot $s\,|{\cal M}|/\alpha^{3}$. An arbitrarily chosen final state was rotated about the $y$-axis through angle $\theta$. The points are the result of numerical integration. At the top, the points in the range 10000 to 25000 are for helicities $+$$+$$-$$-$$-$$-$ and are compared with the analytical results of Ref. MahlonTahoe. In the middle, the points in the range from 2000 to 8000 are for helicities $+$$-$$-$$+$$+$$-$. There is no analytical result for this helicity combination. At the bottom, we show numerical results for helicities $+$$+$$+$$+$$+$$+$ and $+$$+$$+$$+$$+$$-$. According to Ref. Mahlon, the amplitude should vanish for these helicity choices. The results for $+$$+$$+$$+$$+$$+$ are computed at $\theta = 0, 0.2, 0.4, \dots$ while the results for $+$$+$$+$$+$$+$$-$ are computed at $\theta = 0.1, 0.3, 0.5, \dots$.