Two-loop five-point all plus helicity Yang-Mills amplitude

TL;DR

The paper addresses the challenge of computing the two-loop five-point all-plus Yang-Mills amplitude by combining four-dimensional unitarity with augmented recursion. The cut-constructible part $F_5^{cc}$ is obtained from 4D cuts and standard one-loop integral bases, with IR structure inferred from the all-$ o$-$ ext{epsilon}$ promotion of the one-loop amplitude, yielding a result consistent with known expressions and interpretable as a $D=8$ box in disguise. The rational piece $R_5^{(2)}$ is derived via augmented recursion using a Risager shift and axial-gauge currents to handle double poles, producing a compact, cyclically symmetric form $R^{(2)}_5 = rac{i}{6 ig angle12ig angle ig angle23ig angle ig angle34ig angle ig angle45ig angle ig angle51ig angle} (R^a_5 + R^b_5)$. Collectively, the approach reproduces the full amplitude with only one-loop integral structures, hinting at broader applicability to pseudo-one-loop amplitudes without explicit two-loop integration.

Abstract

We re-compute the recently derived two-loop five-point all plus Yang-Mills amplitude using Unitarity and Recursion. Recursion requires augmented recursion to determine the sub-leading pole. Using these methods the simplicity of this amplitude is understood.

Figures (6)

• Figure 1: Contributions to the two-loop amplitudes involving an all-plus loop (indicated by $\bullet$)
• Figure 2: The labelling and internal helicities of the quadruple cut.
• Figure 3: The origin of the double poles in $s_{de}$. The diagram has an explicit pole and an additional pole can arise from the triangle integral.
• Figure 4: The non-factorising contribution to the pole. We must also include the case with the helicities on $\alpha$ and $\beta$ reversed.
• Figure 5: Sources of $s_{\alpha\beta}$ poles in $\tau^1$