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Bounded Collection of Feynman Integral Calabi-Yau Geometries.

Jacob L Bourjaily1, Andrew J McLeod1, Matt von Hippel1

  • 1Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark.

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Summary

We introduce Feynman integral rigidity, the smallest dimension for nonpolylogarithmic behavior. Massless Feynman integrals in four dimensions exhibit a rigidity bound of 2(L-1) at L loops for marginal cases.

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Area of Science:

  • Theoretical Physics
  • Quantum Field Theory
  • String Theory

Background:

  • Feynman integrals are fundamental in quantum field theory, calculating scattering amplitudes.
  • Understanding their analytic properties, like polylogarithmic behavior, is crucial for theoretical predictions.
  • The dimension dependence of Feynman integrals is a key aspect of their complexity.

Purpose of the Study:

  • To define and investigate the concept of 'rigidity' for Feynman integrals.
  • To establish bounds on the rigidity of specific classes of Feynman integrals.
  • To explore the connection between Feynman integrals and advanced geometric structures.

Main Methods:

  • Defining rigidity as the minimum dimension for nonpolylogarithmic behavior.
  • Analyzing marginal Feynman integrals with (L+1)D/2 propagators in D dimensions.
  • Investigating the geometric structures (e.g., Calabi-Yau geometries) associated with these integrals.

Main Results:

  • Proving a rigidity bound of 2(L-1) for massless Feynman integrals in four dimensions at L loops within the marginal class.
  • Demonstrating that marginal Feynman integrals in D dimensions generically involve Calabi-Yau geometries.
  • Providing examples of four-dimensional massless φ⁴ theory Feynman integrals that achieve the predicted rigidity bound across all loop orders.

Conclusions:

  • The concept of rigidity offers a new perspective on the complexity of Feynman integrals.
  • Marginal Feynman integrals exhibit deep connections to Calabi-Yau geometries.
  • The established rigidity bound provides a significant theoretical constraint for calculations in quantum field theory.