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Sunset integrals with up to three mass scales in chiral perturbation theory: a comparative study of the Mellin-Barnes representation technique

Balasubramanian Ananthanarayan, Sumit Banik, Véronique Bernard, Samuel Friot, Shayan Ghosh, Ulf-G. Meißner

Abstract

Sunset integrals are among the simplest of two-loop integrals that appear in perturbative quantum field theories and possess up to four distinct mass scales. By means of integration by parts identities, they can be written in terms of four distinct master integrals. In this article, we discuss the independent configurations of on-shell and off-shell sunset master integrals with one, two and three mass scales that arise in chiral perturbation theory. We derive Mellin-Barnes integral representations of these integrals and analytically solve them using various methods to obtain exact results in the form of single and double convergent series of the hypergeometric type, for the values of the mass parameters that allow us to do so. We then discuss how to analytically continue the results to other regions of the parameters and conclude by discussing a few applications in chiral perturbation theory.

Sunset integrals with up to three mass scales in chiral perturbation theory: a comparative study of the Mellin-Barnes representation technique

Abstract

Sunset integrals are among the simplest of two-loop integrals that appear in perturbative quantum field theories and possess up to four distinct mass scales. By means of integration by parts identities, they can be written in terms of four distinct master integrals. In this article, we discuss the independent configurations of on-shell and off-shell sunset master integrals with one, two and three mass scales that arise in chiral perturbation theory. We derive Mellin-Barnes integral representations of these integrals and analytically solve them using various methods to obtain exact results in the form of single and double convergent series of the hypergeometric type, for the values of the mass parameters that allow us to do so. We then discuss how to analytically continue the results to other regions of the parameters and conclude by discussing a few applications in chiral perturbation theory.

Paper Structure

This paper contains 34 sections, 227 equations, 13 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The Mellin-Barnes method
  3. Mellin-Barnes representations of sunsets using $\texttt{AMBRE}$
  4. Hypergeometric representations using $\texttt{MBConicHulls}$
  5. $H_{\{2,1,1\}} \left(m, M, M; m^2 \right)$
  6. $H_{\{1,1,1\}} \left(m_1, m_2, m_3; m_1^2 \right)$
  7. $H_{\{2,1,1\}} \left(m_1, m_2, m_3; m_3^2 \right)$
  8. $H_{\{1,1,1\}} \left(m_1, m_2, m_2; m_3^2 \right)$
  9. $H_{\{1,1,1\}} \left(m_1, m_2, m_3; p^2 \right)$
  10. Discussion
  11. Two mass scale sunsets
  12. $H_{\{1,1,1\}}(m,M,M;m^2)$ in the straight contours approach
  13. Solving the MB representation
  14. Three mass scale sunsets
  15. $H_{\{1,1,1\}}(m_1,m_2,m_3;m_1^2)$ in the straight contours approach
  16. ...and 19 more sections

Figures (13)

  • Figure 1: The sunset diagram
  • Figure 2: Singularity structure, in the complex $z$-plane, of the MB integrand of the second term of the MB representation of $H_{\{1,1,1\}}(m,M,M;m^2)$ as given in Eq. \ref{['Eq:H111SCMB']}, for (small) positive $\text{Re}(\epsilon)$. The dashed vertical line segment represents the MB straight contour.
  • Figure 3: Singularity structure, in the complex $z$-plane, of the MB integrand of the MB representation of $H_{\{1,1,1\}}(m,M,M;m^2)$ as given in Eq. \ref{['MBrepH111']}, for positive real $\epsilon$. The blue dashed curve represents the MB non-straight contour
  • Figure 4: Singularity structure, in the complex $z$-plane, of the MB integrand of the MB representation of $H_{\{2,1,1\}}(m,M,M;m^2)$ as given by Eq. \ref{['MBrepH211mM']} (upper figure) and by Eq. \ref{['MBrepH211mM']} where Eq. \ref{['Gamma_relation']} is used (lower figure), for small positive $\text{Re}(\epsilon)$. The blue dashed curves represents the MB non-straight contours.
  • Figure 5: Singularity structure of integral $I_2$
  • ...and 8 more figures