Recursive Construction of Higgs-Plus-Multiparton Loop Amplitudes: The Last of the Phi-nite Loop Amplitudes

TL;DR

This work extends on-shell recursion techniques to finite one-loop amplitudes involving a Higgs-like scalar field phi coupled to QCD via the large-m_t effective operator. By treating H as the real part of phi and decomposing into self-dual/anti-self-dual sectors, the authors derive compact, all-n expressions for three infinite families of finite phi amplitudes (including phi with a quark pair and arbitrary gluons). They solve recursion relations for all-plus and quark-gluon configurations, verify factorization and soft limits, and present complete results for two- and three-parton final states with partial four-parton results, offering analytic inputs for Higgs-plus-jet backgrounds. The results provide essential building blocks for NLO Higgs cross sections and establish a scalable framework for constructing divergent phi amplitudes via unitarity and bootstrap methods. Overall, the paper advances analytic, recursion-based methods for Higgs-related multi-jet processes in the large-m_t limit, with direct implications for precision Higgs phenomenology at the LHC.

Abstract

We consider a scalar field, such as the Higgs boson H, coupled to gluons via the effective operator H tr G_{mu nu} G^{mu nu} induced by a heavy-quark loop. We treat H as the real part of a complex field phi which couples to the self-dual part of the gluon field-strength, via the operator phi tr G_{SD mu nu} G_{SD}^{mu nu}, whereas the conjugate field phi^dagger couples to the anti-self-dual part. There are three infinite sequences of amplitudes coupling phi to quarks and gluons that vanish at tree level, and hence are finite at one loop, in the QCD coupling. Using on-shell recursion relations, we find compact expressions for these three sequences of amplitudes and discuss their analytic properties.

Figures (7)

• Figure 1: (a) An $L$ type primitive amplitude, in which the fermion line turns left on entering the loop (following the arrow). (b) In an $R$ type primitive amplitude, the fermion line turns right.
• Figure 2: (a) Graphs in which the external fermion line passes to the left of the loop are assigned to $L$ type. (b) Graphs in which it passes to the right are called $R$ type. Gluons, fermions or scalars run in the loop. The same decomposition is used when gluons are emitted from the external fermion line.
• Figure 3: Diagram corresponding to the recursion relation (\ref{['mhvrec']}) for $A_n^{(0)}(\phi,1^-,2^+,\dots,j^-,\dots,n^+)$.
• Figure 4: Two cuts of a one-loop $\phi$ amplitude which become the same cut in the soft limit $k_\phi \to 0$: (a) the cut in the $s_{i\cdots j}$ channel, and (b) the cut in the $s_{\phi,i\cdots j}$ channel.
• Figure 5: Diagram corresponding to the recursion relation (\ref{['recallplusphi']}) for $A_n^{(1)}(\phi,1^+,2^+,\ldots,n^+)$.