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D-Dimensional Modular Assembly of Higher-Derivative Four-Point Contact Amplitudes Involving Fermions

John Joseph M. Carrasco, Sai Sasank Chava, Alex Edison, Aslan Seifi

TL;DR

This work introduces a D-dimensional, modular framework for constructing four-point higher-derivative contact amplitudes built from gauge-invariant kinematic blocks, color factors, and permutation-invariant Mandelstam polynomials. By decomposing amplitudes into Leg Godt scalar blocks, Leg Godt color blocks, and Leg Godt spin blocks, the approach achieves a scalable, non-redundant operator basis that naturally incorporates evanescent operators and is amenable to the double-copy program. The method yields explicit constructions across two- and four-fermion sectors and various bosonic combinations, enabling systematic mapping to SMEFT/LEFT operators and facilitating connections to maximal SYM, NLSM, and Z-theory within a unified color-kinematics framework. The framework’s modularity and compatibility with double-copy constructions provide a versatile platform for exploring higher-derivative EFTs in gauge theories and gravity, with practical applications to phenomenology and string-inspired theories.

Abstract

We present a novel robust framework for systematically constructing D-dimensional four-point higher-derivative contact amplitudes. Our modular block ("LEGO"-like) approach builds amplitudes directly from manifestly gauge-invariant kinematic blocks, color-weight factors, and scalar Mandelstam polynomials. Symmetries (Bose/Fermi) are imposed algebraically, acting as filters on combinations of compatible pieces. This framework operates entirely in D dimensions, naturally incorporating evanescent operators crucial for loop-level consistency. Scaling to arbitrary mass dimension is achieved in a highly controlled manner using permutation-invariant scalar polynomials, avoiding combinatorial explosion. A key feature is its manifest compatibility with the double-copy program, allowing the systematic generation of operator towers not only for gauge theories but also for gravity and other theories within the double-copy web.

D-Dimensional Modular Assembly of Higher-Derivative Four-Point Contact Amplitudes Involving Fermions

TL;DR

This work introduces a D-dimensional, modular framework for constructing four-point higher-derivative contact amplitudes built from gauge-invariant kinematic blocks, color factors, and permutation-invariant Mandelstam polynomials. By decomposing amplitudes into Leg Godt scalar blocks, Leg Godt color blocks, and Leg Godt spin blocks, the approach achieves a scalable, non-redundant operator basis that naturally incorporates evanescent operators and is amenable to the double-copy program. The method yields explicit constructions across two- and four-fermion sectors and various bosonic combinations, enabling systematic mapping to SMEFT/LEFT operators and facilitating connections to maximal SYM, NLSM, and Z-theory within a unified color-kinematics framework. The framework’s modularity and compatibility with double-copy constructions provide a versatile platform for exploring higher-derivative EFTs in gauge theories and gravity, with practical applications to phenomenology and string-inspired theories.

Abstract

We present a novel robust framework for systematically constructing D-dimensional four-point higher-derivative contact amplitudes. Our modular block ("LEGO"-like) approach builds amplitudes directly from manifestly gauge-invariant kinematic blocks, color-weight factors, and scalar Mandelstam polynomials. Symmetries (Bose/Fermi) are imposed algebraically, acting as filters on combinations of compatible pieces. This framework operates entirely in D dimensions, naturally incorporating evanescent operators crucial for loop-level consistency. Scaling to arbitrary mass dimension is achieved in a highly controlled manner using permutation-invariant scalar polynomials, avoiding combinatorial explosion. A key feature is its manifest compatibility with the double-copy program, allowing the systematic generation of operator towers not only for gauge theories but also for gravity and other theories within the double-copy web.

Paper Structure

This paper contains 51 sections, 78 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Relation to prior work
  3. Roadmap
  4. Scalar building blocks and Kinematics
  5. Leg Godt: Scalar kinematics definite-parity blocks
  6. Special cases
  7. Two mass families:
  8. Entirely agnostic to mass:
  9. Useful Decompositions
  10. Color Blocks
  11. Leg Godt: Color factor definite-parity blocks
  12. Mixing color and kinematics
  13. Spin building blocks (Spacetime Parity Conserving)
  14. Leg Godt: Spinor multilinear definite-parity blocks
  15. Two fermions + two scalars (2F+2S)
  16. ...and 36 more sections

Figures (1)

  • Figure 1: Schematic workflow for the "LEGO-like" modular bootstrap of four-point contact amplitudes. Spin-dependent blocks (from $F, \Gamma$), scalar kinematic polynomials (in $s,t,u$), and color factors are combined according to well-defined compatibility. Symmetry constraints (Bose/Fermi) are applied during the merge step to yield the full D-dimensional, gauge-invariant amplitude. An optional subsequent step (dashed) can relate these to predictions in a wide web of EFT theories via double copy. It is straightforward to encode these contact structures at the operator level having identified the distinct predictive building blocks.