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All Planar Two-Loop Amplitudes in Maximally Supersymmetric Yang-Mills Theory.

Anne Spiering1,2, Matthias Wilhelm2, Chi Zhang2,3

  • 1Humboldt Universität zu Berlin, Institut für Physik, 10117 Berlin, Germany.

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|March 7, 2025
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Summary

We calculated key integrals for scattering amplitudes in maximally supersymmetric Yang-Mills theory. This breakthrough enables computations for any particle number and helicity configuration.

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Area of Science:

  • High-energy theoretical physics
  • Quantum field theory
  • Supersymmetry

Background:

  • Scattering amplitudes in quantum field theories are crucial for understanding particle interactions.
  • Two-loop calculations in maximally supersymmetric Yang-Mills theory are complex and require advanced techniques.
  • Planar integrals, such as double-pentagon and pentabox integrals, are essential components of these calculations.

Purpose of the Study:

  • To calculate general planar dual-conformally invariant double-pentagon and pentabox integrals in four dimensions.
  • To derive onefold integral representations for these elliptic integrals.
  • To enable the calculation of all two-loop planar scattering amplitudes in maximally supersymmetric Yang-Mills theory.

Main Methods:

  • Derivation of onefold integral representations for elliptic integrals.
  • Utilizing polylogarithms of weight three.
  • Applying consistency conditions to determine integral symbols.
  • Combining new results with previously calculated double-box integrals.

Main Results:

  • General planar dual-conformally invariant double-pentagon and pentabox integrals in four dimensions have been calculated.
  • Onefold integral representations for these integrals were successfully derived.
  • The symbols for these integrals were determined using consistency conditions.
  • These calculations provide the necessary components for all two-loop planar scattering amplitudes.

Conclusions:

  • The calculation of double-pentagon and pentabox integrals is complete.
  • These results, along with the double-box integral, are sufficient to determine any two-loop planar scattering amplitude in maximally supersymmetric Yang-Mills theory.
  • The methodology allows for the calculation of amplitudes for arbitrary numbers of particles and helicity configurations.