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Title: Bootstrapping two-loop Feynman integrals for planar $$ \mathcal{N}=4 $$ sYM

Abstract

We derive analytic results for the symbol of certain two-loop Feynman integrals relevant for seven- and eight-point two-loop scattering amplitudes in planar $$ \mathcal{N}=4 $$ super-Yang-Mills theory. We use a bootstrap inspired strategy, combined with a set of second-order partial differential equations that provide powerful constraints on the symbol ansatz. When the complete symbol alphabet is not available, we adopt a hybrid approach. Instead of the full function, we bootstrap a certain discontinuity for which the alphabet is known. Then we write a one-fold dispersion integral to recover the complete result. At six and seven points, we find that the individual Feynman integrals live in the same space of functions as the amplitude, which is described by the 9- and 42-letter cluster alphabets respectively. Starting at eight points however, the symbol alphabet of the MHV amplitude is insufficient for individual integrals. In particular, some of the integrals require algebraic letters involving four-mass box square-root singularities. We point out that these algebraic letters are relevant at the amplitude level directly starting with N2MHV amplitudes even at one loop.

Authors:
 [1]; ORCiD logo [2]; ORCiD logo [3]
  1. Johannes Gutenberg University, Mainz (Germany); Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Munchen (Germany)
  2. SLAC National Accelerator Lab., Menlo Park, CA (United States); Stanford Univ., CA (United States)
  3. Univ. of California, Los Angeles, CA (United States)
Publication Date:
Research Org.:
SLAC National Accelerator Lab., Menlo Park, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1490625
Grant/Contract Number:  
AC02-76SF00515
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Volume: 2018; Journal Issue: 10; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 1/N Expansion; Scattering Amplitudes; Supersymmetric Gauge Theory

Citation Formats

Henn, Johannes, Herrmann, Enrico, and Parra-Martinez, Julio. Bootstrapping two-loop Feynman integrals for planar $ \mathcal{N}=4 $ sYM. United States: N. p., 2018. Web. doi:10.1007/jhep10(2018)059.
Henn, Johannes, Herrmann, Enrico, & Parra-Martinez, Julio. Bootstrapping two-loop Feynman integrals for planar $ \mathcal{N}=4 $ sYM. United States. doi:10.1007/jhep10(2018)059.
Henn, Johannes, Herrmann, Enrico, and Parra-Martinez, Julio. Tue . "Bootstrapping two-loop Feynman integrals for planar $ \mathcal{N}=4 $ sYM". United States. doi:10.1007/jhep10(2018)059. https://www.osti.gov/servlets/purl/1490625.
@article{osti_1490625,
title = {Bootstrapping two-loop Feynman integrals for planar $ \mathcal{N}=4 $ sYM},
author = {Henn, Johannes and Herrmann, Enrico and Parra-Martinez, Julio},
abstractNote = {We derive analytic results for the symbol of certain two-loop Feynman integrals relevant for seven- and eight-point two-loop scattering amplitudes in planar $ \mathcal{N}=4 $ super-Yang-Mills theory. We use a bootstrap inspired strategy, combined with a set of second-order partial differential equations that provide powerful constraints on the symbol ansatz. When the complete symbol alphabet is not available, we adopt a hybrid approach. Instead of the full function, we bootstrap a certain discontinuity for which the alphabet is known. Then we write a one-fold dispersion integral to recover the complete result. At six and seven points, we find that the individual Feynman integrals live in the same space of functions as the amplitude, which is described by the 9- and 42-letter cluster alphabets respectively. Starting at eight points however, the symbol alphabet of the MHV amplitude is insufficient for individual integrals. In particular, some of the integrals require algebraic letters involving four-mass box square-root singularities. We point out that these algebraic letters are relevant at the amplitude level directly starting with N2MHV amplitudes even at one loop.},
doi = {10.1007/jhep10(2018)059},
journal = {Journal of High Energy Physics (Online)},
issn = {1029-8479},
number = 10,
volume = 2018,
place = {United States},
year = {2018},
month = {10}
}

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Figures / Tables:

Table 1 Table 1: Number of unconstraint coefficients in the heptagon Steinmann ansatz after applying various constraints for both two-loop heptagon integrals.

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    Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.