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<p>We initiate a systematic study on how to extract planar integrands of (supersymmetric) scattering amplitudes with <i>L</i> loops and <i>n</i> legs in Aharony-Bergman-Jafferis-Maldacena (ABJM) theory from the recently proposed (bosonic) generating function for squared amplitudes with <i>N</i> := <i>n</i>+<i>L</i> dual points; the latter enjoys a hidden permutation symmetry <i>S</i><sub><i>N</i></sub> and is given by a linear combination of weight-3 planar <i>f</i> -graphs that can be recast as bipartite graphs, which manifest important properties of ABJM amplitudes. We provide evidence that it contains sufficient information to reconstruct individual amplitudes, despite the absence of squared amplitudes at odd loops. The extraction of the four-point amplitude is already non-trivial and closely parallels the extraction of five-point amplitudes in <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 4 super Yang-Mills (SYM) from weight-4 <i>f</i>-graphs: we comment on this similarity and provide new results for <i>n</i> = 4 ABJM loop integrand up to <i>L</i> = 6. For higher multiplicities, based on Yangian invariants (including BCFW building blocks for tree amplitudes) and an appropriate basis of planar dual conformal invariant(DCI) integrands, we disentangle six-point integrands up to two loops and eight-point tree amplitude from the squared amplitudes. Our results suggest that ABJM amplitudes of arbitrary multiplicity and loop order can be reconstructed from squared amplitudes, closely paralleling the role of <i>f</i>-graphs in <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 4 SYM.</p>

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Loops and legs: ABJM amplitudes from f-graphs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月19日

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  • Song He,
  • Yao-Qi Zhang

摘要

We initiate a systematic study on how to extract planar integrands of (supersymmetric) scattering amplitudes with L loops and n legs in Aharony-Bergman-Jafferis-Maldacena (ABJM) theory from the recently proposed (bosonic) generating function for squared amplitudes with N := n+L dual points; the latter enjoys a hidden permutation symmetry SN and is given by a linear combination of weight-3 planar f -graphs that can be recast as bipartite graphs, which manifest important properties of ABJM amplitudes. We provide evidence that it contains sufficient information to reconstruct individual amplitudes, despite the absence of squared amplitudes at odd loops. The extraction of the four-point amplitude is already non-trivial and closely parallels the extraction of five-point amplitudes in N \( \mathcal{N} \) = 4 super Yang-Mills (SYM) from weight-4 f-graphs: we comment on this similarity and provide new results for n = 4 ABJM loop integrand up to L = 6. For higher multiplicities, based on Yangian invariants (including BCFW building blocks for tree amplitudes) and an appropriate basis of planar dual conformal invariant(DCI) integrands, we disentangle six-point integrands up to two loops and eight-point tree amplitude from the squared amplitudes. Our results suggest that ABJM amplitudes of arbitrary multiplicity and loop order can be reconstructed from squared amplitudes, closely paralleling the role of f-graphs in N \( \mathcal{N} \) = 4 SYM.