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N=4 SYM NMHV Loop Amplitude in Superspace

Yu-tin Huang

TL;DR

This work constructs the six-point NMHV one-loop amplitude in N=4 SYM in a full superspace form using SU(4)_R fermionic variables, enabling all external particle configurations to be obtained by expanding in the Grassmann parameters. The authors analyze unitarity cuts in three channels, derive the B1 and B2 box-coefficient structures, and demonstrate explicit all-gluino and all-scalar amplitudes by explicit eta-expansion, confirming consistency with known gluonic results. They also discuss how MHV-vertex (CSW) methods can reproduce NMHV loop amplitudes by reducing to two-propagator diagrams and outline prospects for extending the approach to higher-point or higher-loop cases. Overall, the paper provides a compact, superspace-based framework for NMHV loop amplitudes and reinforces the relevance of CSW-type reconstruction in N=4 SYM.

Abstract

Here we construct N=4 SuperYang-Mills 6 point NMHV loop amplitude (amplitudes with three minus helicities) as a full superspace form, using the $SU(4)_{R}$ anti-commuting spinor variables. Amplitudes with different external particle and cyclic helicity ordering are then just a particular expansion of this fermionic variable. We've verified this by explicit expansion obtaining amplitudes with two gluino calculated before. We give results for all gluino $A(Λ^{-}Λ^{-}Λ^{-}Λ^{+}Λ^{+}Λ^{+})$and all scalar $A(φφφφφφ)$scattering amplitude. A discussion of using MHV vertex approach to obtain these amplitudes are given, which implies a simplification for general loop amplitudes.

N=4 SYM NMHV Loop Amplitude in Superspace

TL;DR

This work constructs the six-point NMHV one-loop amplitude in N=4 SYM in a full superspace form using SU(4)_R fermionic variables, enabling all external particle configurations to be obtained by expanding in the Grassmann parameters. The authors analyze unitarity cuts in three channels, derive the B1 and B2 box-coefficient structures, and demonstrate explicit all-gluino and all-scalar amplitudes by explicit eta-expansion, confirming consistency with known gluonic results. They also discuss how MHV-vertex (CSW) methods can reproduce NMHV loop amplitudes by reducing to two-propagator diagrams and outline prospects for extending the approach to higher-point or higher-loop cases. Overall, the paper provides a compact, superspace-based framework for NMHV loop amplitudes and reinforces the relevance of CSW-type reconstruction in N=4 SYM.

Abstract

Here we construct N=4 SuperYang-Mills 6 point NMHV loop amplitude (amplitudes with three minus helicities) as a full superspace form, using the anti-commuting spinor variables. Amplitudes with different external particle and cyclic helicity ordering are then just a particular expansion of this fermionic variable. We've verified this by explicit expansion obtaining amplitudes with two gluino calculated before. We give results for all gluino and all scalar scattering amplitude. A discussion of using MHV vertex approach to obtain these amplitudes are given, which implies a simplification for general loop amplitudes.

Paper Structure

This paper contains 10 sections, 43 equations, 2 figures.

Table of Contents

  1. 1.Introduction
  2. 2. The Construction
  3. 2.1 $B_{1}$ Coefficient $\Rightarrow$ $t_{123}$ cut
  4. 2.2 $B_{2}$ Coefficient $\Rightarrow$$t_{234}$ cut
  5. 3. Amplitudes with all gluinos and all scalars
  6. 3.1 $A(\Lambda_{1}^{\emph{1}+}\Lambda_{2}^{\emph{2}+}\Lambda_{3}^{\emph{3}+}\Lambda_{4}^{\emph{1}-}\Lambda_{5}^{\emph{2}-}\Lambda_{6}^{\emph{3}-})$
  7. 3.2 $A(\phi_{1}\phi_{2}\phi_{3}\phi_{4}\phi_{5}\phi_{6})$
  8. 4. A brief discussion on the MHV vertex approach
  9. 5. Conclusion
  10. 6. Acknowledgements

Figures (2)

  • Figure 1: Here we show for a particular case of the gluonic NMHV loop amplitude, different assignment of helicity for the propagators will change the MHV or $\overline{MHV}$ nature of each vertex. In the upper graph the propagators has the same helicity on each vertex while in the lower they are opposite. This result in a MHV vertex on the left for the upper graph and an $\overline{MHV}$ vertex on the left for the lower graph. In practice one has to sum these two possibilities which is the reason we have two terms in eq.(5)-(7)
  • Figure 2: MHV diagrams for NMHV loop amplitude, includes the one-particle-irreducible and one-particle-reducible graph. The $m_{i}$s label external momenta that are adjacent to the propagator. For the first graph we need to integrate over all three propagators, while only $L_{2}$$L_{3}$ are integrated in the second. One also needs to sum over all possible ways of assigning external momenta to the vertices