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On-shell superamplitudes in N<4 SYM

Henriette Elvang, Yu-tin Huang, Cheng Peng

TL;DR

Elvang, Huang, and Peng develop an on-shell superspace approach for pure ${\cal N}<4$ SYM using two CPT-conjugate superfields to capture the full spectrum; tree-level MHV and NMHV amplitudes are obtained by truncation from ${\cal N}=4$ and the ${\Phi}$-${\Psi}$ formalisms. They analyze the large-$z$ behavior of super-BCFW shifts, identifying a 'bad' shift that contributes to 1-loop bubble terms in ${\cal N}=0,1,2$ and derive bubble coefficients in a manifestly supersymmetric way, showing their sum for 4-point amplitudes equals $-\beta_0\, {\cal A}_{4}^{\text{tree}}$ with the standard 1-loop beta-function coefficient. The paper further solves SUSY Ward identities in ${\cal N}<4$ YM, and establishes a precise 4d–6d correspondence that relates 4d ${\cal N}^K\text{MHV}$ sectors to 6d structures, including dimensional reductions to ${\cal N}=(1,0)$ leading to ${\cal N}=2$ in 4d. Extensions to ${\cal N}<8$ supergravity are outlined, highlighting broad applicability of on-shell methods to lower-supersymmetry theories and higher-dimensional relations.

Abstract

We present an on-shell formalism for superamplitudes of pure N<4 super Yang-Mills theory. Two superfields, Phi and Phi^+, are required to describe the two CPT conjugate supermultiplets. Simple truncation prescriptions allow us to derive explicit tree-level MHV and NMHV superamplitudes with N-fold SUSY. Any N=0,1,2 tree superamplitudes have large-z falloffs under super-BCFW shifts, except under [Phi,Phi^+>-shifts. We show that this `bad' shift is responsible for the bubble contributions to 1-loop amplitudes in N=0,1,2 SYM. We evaluate the MHV bubble coefficients in a manifestly supersymmetric form and demonstrate for the case of four external particles that the sum of bubble coefficients is equal to minus the tree superamplitude times the 1-loop beta-function coefficient. The connection to the beta-function is expected since only bubble integrals capture UV divergences; we discuss briefly how the minus sign arises from UV and IR divergences in dimensional regularization. Other applications of the on-shell formalism include a solution to the N^{K}MHV N=1 SUSY Ward identities and a clear description of the connection between 6d superamplitudes and the 4d ones for both N=4 and N=2 SYM. We outline extensions to N<8 supergravity.

On-shell superamplitudes in N<4 SYM

TL;DR

Elvang, Huang, and Peng develop an on-shell superspace approach for pure SYM using two CPT-conjugate superfields to capture the full spectrum; tree-level MHV and NMHV amplitudes are obtained by truncation from and the - formalisms. They analyze the large- behavior of super-BCFW shifts, identifying a 'bad' shift that contributes to 1-loop bubble terms in and derive bubble coefficients in a manifestly supersymmetric way, showing their sum for 4-point amplitudes equals with the standard 1-loop beta-function coefficient. The paper further solves SUSY Ward identities in YM, and establishes a precise 4d–6d correspondence that relates 4d sectors to 6d structures, including dimensional reductions to leading to in 4d. Extensions to supergravity are outlined, highlighting broad applicability of on-shell methods to lower-supersymmetry theories and higher-dimensional relations.

Abstract

We present an on-shell formalism for superamplitudes of pure N<4 super Yang-Mills theory. Two superfields, Phi and Phi^+, are required to describe the two CPT conjugate supermultiplets. Simple truncation prescriptions allow us to derive explicit tree-level MHV and NMHV superamplitudes with N-fold SUSY. Any N=0,1,2 tree superamplitudes have large-z falloffs under super-BCFW shifts, except under [Phi,Phi^+>-shifts. We show that this `bad' shift is responsible for the bubble contributions to 1-loop amplitudes in N=0,1,2 SYM. We evaluate the MHV bubble coefficients in a manifestly supersymmetric form and demonstrate for the case of four external particles that the sum of bubble coefficients is equal to minus the tree superamplitude times the 1-loop beta-function coefficient. The connection to the beta-function is expected since only bubble integrals capture UV divergences; we discuss briefly how the minus sign arises from UV and IR divergences in dimensional regularization. Other applications of the on-shell formalism include a solution to the N^{K}MHV N=1 SUSY Ward identities and a clear description of the connection between 6d superamplitudes and the 4d ones for both N=4 and N=2 SYM. We outline extensions to N<8 supergravity.

Paper Structure

This paper contains 25 sections, 126 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction and summary of results
  2. On-shell formalism for pure SYM: $\Phi$-$\Phi^\dagger$ formalism
  3. On-shell superfields and MHV superamplitudes in ${\cal N}=4$ SYM
  4. ${\cal N}=1$ SYM on-shell superfields $\Phi$ and $\Phi^\dagger$ and MHV superamplitude
  5. MHV vertex expansion
  6. Pure ${\cal N}=0,1,2,3,4$ SYM: $\Phi$-$\Psi$ formalism
  7. MHV superamplitudes for $0\le \mathcal{N} \le4$
  8. NMHV superamplitudes for $0\le \mathcal{N} \le4$
  9. On the range of ${\cal N}$ in SYM
  10. Super-BCFW
  11. Bubble contributions to 1-loop amplitudes
  12. Which bubble coefficients contribute?
  13. Intermediate state sums
  14. Evaluation of the bubble cuts in ${\cal N}=1,2$ SYM
  15. Bubbles and the 1-loop $\beta$-function coefficient
  16. ...and 10 more sections

Figures (4)

  • Figure 1: Example of 1-loop double cut.
  • Figure 2: Two different "bubble cuts" of a 1-loop MHV superamplitude.
  • Figure 3: The projection of the 6d $\mathcal{N}=(1,1)$ multiplet onto the 4d $\mathcal{N}=4$ multiplet. The two axes are the weights of the states with respect to the two $U(1)$'s of little group $SU(2) \times SU(2)$. The diagonal line represents the $U(1)$ subgroup of the 4d little group.
  • Figure 4: A proposed (not proven) scenario connecting minHV amplitudes to all N$^K$MHV amplitudes via the 6d reconstruction.