MHV Rules for Higgs Plus Multi-Gluon Amplitudes

TL;DR

The paper develops a novel MHV-rule framework for Higgs boson amplitudes with an arbitrary number of gluons by splitting the heavy-top induced HGG interaction into selfdual and anti-selfdual parts, yielding holomorphic and anti-holomorphic MHV towers. Higgs amplitudes are obtained as the sum of a φ-generated (MHV) and a φ†-generated (anti-MHV) tower, enabling compact tree-level expressions for Higgs+ n gluons up to n≤5 and all-plus/all-minus configurations. It also analyzes the soft-Higgs limit and verifies consistency with pure gauge theory results, while extending the approach to a second effective model based on a G^3 operator and exploring SUSY embeddings. The work provides a structured path toward applying twistor/MHV techniques to non-supersymmetric theories and to higher-dimension operators, with potential implications for loop-level calculations and phenomenology such as Higgs production with multiple jets. Overall, it demonstrates that holomorphic/anti-holomorphic decompositions can render complex Higgs-plus-parton amplitudes tractable within an MHV-like perturbation theory.

Abstract

We use tree-level perturbation theory to show how non-supersymmetric one-loop scattering amplitudes for a Higgs boson plus an arbitrary number of partons can be constructed, in the limit of a heavy top quark, from a generalization of the scalar graph approach of Cachazo, Svrcek and Witten. The Higgs boson couples to gluons through a top quark loop which generates, for large top mass, a dimension-5 operator H tr G^2. This effective interaction leads to amplitudes which cannot be described by the standard MHV rules; for example, amplitudes where all of the gluons have positive helicity. We split the effective interaction into the sum of two terms, one holomorphic (selfdual) and one anti-holomorphic (anti-selfdual). The holomorphic interactions give a new set of MHV vertices -- identical in form to those of pure gauge theory, except for momentum conservation -- that can be combined with pure gauge theory MHV vertices to produce a tower of amplitudes with more than two negative helicities. Similarly, the anti-holomorphic interactions give anti-MHV vertices that can be combined with pure gauge theory anti-MHV vertices to produce a tower of amplitudes with more than two positive helicities. A Higgs boson amplitude is the sum of one MHV-tower amplitude and one anti-MHV-tower amplitude. We present all MHV-tower amplitudes with up to four negative-helicity gluons and any number of positive-helicity gluons (NNMHV). These rules reproduce all of the available analytic formulae for Higgs + n-gluon scattering (n<=5) at tree level, in some cases yielding considerably shorter expressions.

Figures (4)

• Figure 1: Tree diagrams with MHV vertices which contribute to the NMHV amplitude $A_n(\phi,\ldots,m_1^-,\ldots,m_2^-,\ldots,m_3^-,\ldots)$. The scalar $\phi$ is represented by a dashed line and negative-helicity gluons, $g^-$, by solid lines. Positive-helicity gluons $g^+$ emitted from each vertex are indicated by dotted semicircles, with labels showing the bounding $g$ lines in each MHV vertex.
• Figure 2: Skeleton diagram showing the labelling of $n$ gluons for amplitudes $A_n$ with four negative-helicity gluons. Positive-helicity gluons $g^+$ emitted from each vertex are indicated by dotted lines with labels showing the bounding $g^+$ lines in each MHV vertex.
• Figure 3: NNMHV tree diagrams contributing to the amplitude $A_n(\phi,g_{m_1}^-,g_{m_2}^-,g_{m_3}^-,g_{m_4}^-)$.
• Figure 4: Two representations for $A_n(\phi,1^-,2^-,3^-,\ldots,n^-)$ in the MHV-rules approach. In (a), we illustrate how attaching only negative-helicity gluons requires just three-point MHV vertices. In (b), the shaded circle represents the coupling of an off-shell gluon to many on-shell negative helicity gluons, which is obtained by summing MHV graphs of the type shown in (a).