One-Loop Gluonic Amplitudes from Single Unitarity Cuts

TL;DR

The paper introduces a single-cut unitarity framework to determine massless one-loop gluon amplitudes directly from $(n+2)$-point tree amplitudes. By performing cuts in either $D=4$ or $D=4-2\epsilon$ dimensions, it recovers the cut-constructible parts or the full amplitude, respectively, and demonstrates the procedure on all-plus, mostly-plus, and MHV configurations, including scalar-loop and $\mathcal{N}=4$ SYM cases. The method leverages CSW rules with massive scalars to generate the required tree inputs and shows how spurious poles cancel across diagrams, reproducing known results and highlighting the approach’s versatility and potential for extension to massive theories. Overall, it offers a complementary, gauge-invariant, and potentially efficient means of obtaining one-loop QCD and SYM amplitudes from on-shell data.

Abstract

We show that one-loop amplitudes in massless gauge theories can be determined from single cuts. By cutting a single propagator and putting it on-shell, the integrand of an n-point one-loop integral is transformed into an (n+2)-particle tree level amplitude. The single-cut approach described here is complementary to the double or multiple unitarity cut approaches commonly used in the literature. In common with these approaches, if the cut is taken in four dimensions, one finds only the cut-constructible parts of the amplitude, while if the cut is in D=4-2 epsilon dimensions, both rational and cut-constructible parts are obtained. We test our method by reproducing the known results for the fully rational all-plus and mostly-plus QCD amplitudes A^{(1)}_4(1^+,2^+,3^+,4^+) and A^{(1)}_5(1^+,2^+,3^+,4^+,5^+). We also rederive expressions for the scalar loop contribution to the four-gluon MHV amplitude, A_4^{(1,N=0)}(-,-,+,+) which has both cut-constructible and rational contributions, and the fully cut-constructible n-gluon MHV amplitude in N=4 Supersymetric Yang-Mills, A_4^{(1,N=4)}(-,-,+,...,+).

Figures (10)

• Figure 1: Schematic of a the $C_{i,i-1}$ single cut. The propagator between momenta $p_i$ and $p_{i-1}$ is cut, leading to two additional (dashed) outgoing external scalar particles. This cut is not sensitive to scalar integrals that have $p_i$ and $p_{i-1}$ emitted from the same vertex.
• Figure 2: Examples of master integrals which $(a)$ can and $(b)$ cannot be found by a $C_{i,i-1}$ cut.
• Figure 3: The MHV topologies associated with a $C_{1,4}$ cut of the four-gluon all-plus amplitude. Other cuts are obtained from this one by cyclic relabeling of the external particles.
• Figure 4: The MHV topologies associated with a $C_{1,5}$ cut of the all-plus five-gluon amplitude. Here we have set $\eta=p_5$ to eliminate any three-point vertex with containing two scalars and $p_5$. Other cuts are obtained from this one by cyclic relabeling of the external particles.
• Figure 5: The schematic structure of the three-point vertex with one massive scalar leg on-shell and one off-shell. A dotted leg represents the pinching of a propagator (scalar) or the absence of a spurious term associated with that particular gluon. When applied to a generic diagram, this identity leads to two contributions, one term which pinches a propagator and a second term which has no spurious pole.