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Connected formulas for amplitudes in standard model

Song He, Yong Zhang

TL;DR

This work extends the twistor-string/connected-formula framework to realistic Standard Model processes by deriving four-dimensional connected formulas for tree-level QCD amplitudes with massless quarks and for Higgs plus multi-parton amplitudes. The QCD amplitudes are expressed as Parke-Taylor-type integrals decorated by a flavor- and ordering-dependent Jacobian, obtained from SYM decompositions, while the Higgs amplitudes are encoded using two auxiliary massless legs that capture the Higgs momentum and decouple from the Parke-Taylor factor. The authors verify the constructions through MHV and all-minus limits, factorization, and NMHV tests, and show that the Higgs case yields a natural form-factor representation for Tr F^2, opening avenues for form-factor studies within CHY/connected-formula language. The results suggest rich connections to Grassmannian, BCJ relations, and potential loop-level extensions, offering a compact, conceptually transparent framework for SM amplitudes and future explorations beyond four dimensions.

Abstract

Witten's twistor string theory has led to new representations of S-matrix in massless QFT as a single object, including Cachazo-He-Yuan formulas in general and connected formulas in four dimensions. As a first step towards more realistic processes of the standard model, we extend the construction to QCD tree amplitudes with massless quarks and those with a Higgs boson. For both cases, we find connected formulas in four dimensions for all multiplicities which are very similar to the one for Yang-Mills amplitudes. The formula for quark-gluon color-ordered amplitudes differs from the pure-gluon case only by a Jacobian factor that depends on flavors and orderings of the quarks. In the formula for Higgs plus multi-parton amplitudes, the massive Higgs boson is effectively described by two additional massless legs which do not appear in the Parke-Taylor factor. The latter also represents the first twistor-string/connected formula for form factors.

Connected formulas for amplitudes in standard model

TL;DR

This work extends the twistor-string/connected-formula framework to realistic Standard Model processes by deriving four-dimensional connected formulas for tree-level QCD amplitudes with massless quarks and for Higgs plus multi-parton amplitudes. The QCD amplitudes are expressed as Parke-Taylor-type integrals decorated by a flavor- and ordering-dependent Jacobian, obtained from SYM decompositions, while the Higgs amplitudes are encoded using two auxiliary massless legs that capture the Higgs momentum and decouple from the Parke-Taylor factor. The authors verify the constructions through MHV and all-minus limits, factorization, and NMHV tests, and show that the Higgs case yields a natural form-factor representation for Tr F^2, opening avenues for form-factor studies within CHY/connected-formula language. The results suggest rich connections to Grassmannian, BCJ relations, and potential loop-level extensions, offering a compact, conceptually transparent framework for SM amplitudes and future explorations beyond four dimensions.

Abstract

Witten's twistor string theory has led to new representations of S-matrix in massless QFT as a single object, including Cachazo-He-Yuan formulas in general and connected formulas in four dimensions. As a first step towards more realistic processes of the standard model, we extend the construction to QCD tree amplitudes with massless quarks and those with a Higgs boson. For both cases, we find connected formulas in four dimensions for all multiplicities which are very similar to the one for Yang-Mills amplitudes. The formula for quark-gluon color-ordered amplitudes differs from the pure-gluon case only by a Jacobian factor that depends on flavors and orderings of the quarks. In the formula for Higgs plus multi-parton amplitudes, the massive Higgs boson is effectively described by two additional massless legs which do not appear in the Parke-Taylor factor. The latter also represents the first twistor-string/connected formula for form factors.

Paper Structure

This paper contains 5 sections, 34 equations, 3 figures.

Table of Contents

  1. Introduction and summary of results
  2. Connected formula for massless QCD amplitudes
  3. Higgs plus multi-parton amplitudes
  4. Discussions
  5. $\mathcal{J}_{\rm ferm}$ for four quark lines

Figures (3)

  • Figure 1: Two cases for amplitudes with 2 fermion pairs: QCD vs. SYM .
  • Figure 2: Five cases for amplitudes with 3 fermion pairs: QCD vs. SYM .
  • Figure 3: Equivalent forms with gluon flavors reduced, which implies a vanishing identity.