Direct extraction of one-loop integral coefficients

We present a general procedure for obtaining the coefficients of the scalar bubble and triangle integral functions of one-loop amplitudes. Coefficients are extracted by considering two-particle and triple unitarity cuts of the corresponding bubble and triangle integral functions. After choosing a specific parametrization of the cut loop momentum we can uniquely identify the coefficients of the desired integral functions simply by examining the behavior of the cut integrand as the unconstrained parameters of the cut loop momentum approach infinity. In this way we can produce compact forms for scalar integral coefficients. Applications of this method are presented for both QCD and electroweak processes, including an alternative form for the recently computed three-mass triangle coefficient in the six-photon amplitude $A_6(1^{-},2^{+},3^{-},4^{+},5^{-},6^{+})$. The direct nature of this extraction procedure allows for a very straightforward automation of the procedure.

Figure 1

A generic quadruple cut used to isolate the scalar box integral $D_0(K_i^2,K_j^2,K_k^2)$.

Figure 2

The triple-cut used to compute the scalar triangle coefficient of $C_0(K_1^2,K_2^2)$.

Figure 3

The two-particle cut for computing the scalar bubble coefficient of $B_0(K_1^2)$.

Figure 4

Triple-cut six-photon amplitude in the $12:34:56$ channel.

Figure 5

Triple cut in the $14:23:56$ channel.

Figure 6

The scalar bubble integral with a leg of mass $K_1^2$.

Figure 7

The scalar triangle with its three legs of mass $K_1^2$, $K_2^2$, and $K_3^2$.