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Systematics of One-Loop Scattering Amplitudes in N=4 Super Yang-Mills Theories

Mingxing Luo, Congkao Wen

TL;DR

This work analyzes one-loop scattering amplitudes in $N=4$ SYM through the CSW MHV-diagram paradigm, revealing a highly constrained set of generically independent loop integrals, $[n/2]-1$. It classifies one-loop MHV diagrams into 1PI and 1PR topologies and shows that all subleading color amplitudes $A_{n;c}$ can be constructed from the leading $A_{n;1}$ in the same way as in conventional field theory, with explicit transformation rules. The authors provide detailed proofs for both 1PI and 1PR cases, demonstrating parity-consistent reductions and matching known one-loop results in standard QFT, thereby supporting the MHV/CSW paradigm and its twistor-inspired underpinnings. While the results are most robust for lower negative-helicity counts, they argue for broader applicability and call for efficient computational methods to exploit the reduced integral basis.

Abstract

One-loop scattering amplitudes in N=4 super Yang-Mills (SYM) theories are analyzed in the paradigm of maximal helicity violating Feynman diagrams. There are very limited number of loop integrals to be evaluated. For a process with n external particles, there are only [n/2]-1 generically independent integrals. Furthermore, the relations between leading N_c amplitudes A_{n;1} and sub-leading amplitudes A_{n;c} are found to be identical to those obtained from conventional field theory calculations, which can be interpreted as an indirect support for the paradigm.

Systematics of One-Loop Scattering Amplitudes in N=4 Super Yang-Mills Theories

TL;DR

This work analyzes one-loop scattering amplitudes in SYM through the CSW MHV-diagram paradigm, revealing a highly constrained set of generically independent loop integrals, . It classifies one-loop MHV diagrams into 1PI and 1PR topologies and shows that all subleading color amplitudes can be constructed from the leading in the same way as in conventional field theory, with explicit transformation rules. The authors provide detailed proofs for both 1PI and 1PR cases, demonstrating parity-consistent reductions and matching known one-loop results in standard QFT, thereby supporting the MHV/CSW paradigm and its twistor-inspired underpinnings. While the results are most robust for lower negative-helicity counts, they argue for broader applicability and call for efficient computational methods to exploit the reduced integral basis.

Abstract

One-loop scattering amplitudes in N=4 super Yang-Mills (SYM) theories are analyzed in the paradigm of maximal helicity violating Feynman diagrams. There are very limited number of loop integrals to be evaluated. For a process with n external particles, there are only [n/2]-1 generically independent integrals. Furthermore, the relations between leading N_c amplitudes A_{n;1} and sub-leading amplitudes A_{n;c} are found to be identical to those obtained from conventional field theory calculations, which can be interpreted as an indirect support for the paradigm.

Paper Structure

This paper contains 7 sections, 15 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Review of existing results and the paradigm of MHV Feynman diagrams
  3. Classification of MHV Feynman diagrams
  4. The relation between $A_{n;1}$ and $A_{n;c}$
  5. The case of 1PI MHV Feynman diagrams
  6. The case of 1PR MHV Feynman diagrams
  7. Conclusion

Figures (5)

  • Figure 1: Graphic representation of Eq. (3.1), where $d$ is the degree of the algebraic curve, $q$ the number of external particles of negative helicity, and $l$ the number of loops. Only the lowest line $l=0$ and the point $(q,l)=(2,1)$ are relatively well understood.
  • Figure 2: (a) One loop quiver diagrams: (a) 1PI; (b) 1PR.
  • Figure 3: Graphic representation of Eq (4.5), a one-loop sub-leading MHV diagram is expressed in terms of a set of one-loop leading MHV diagrams. All the external lines inside the circle are reflected to outside, in the manner prescribed in Eq (4.6) and Figure 7 of [19].
  • Figure 4: Reflecting a tree inside the circle to the outside. Note the order of the external lines after the reflection and the overall sign.
  • Figure 5: Graphic representation of Eq (4.7), indicating terms from $A_{n;c}$ constructed out of $A_{n;1}$ with the lines originally inside the circle mingle with those originally outside.