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Conformal four-point ladder integrals in diverse dimensions and polylogarithms

S. E. Derkachov, A. P. Isaev, L. A. Shumilov

TL;DR

This work analyzes conformal four-point ladder integrals in arbitrary dimensions via an operator-based representation tied to conformal quantum mechanics. The authors establish dimensional and loop-shift identities and demonstrate a two-dimensional factorization that maps higher-D results to the $D=2$ case, enabling extension to even dimensions using a dimension-raising operator $R_d$. For β=1 in any even dimension, the ladder integrals reduce to expressions in terms of classical polylogarithms with rational coefficients, while a two-dimensional factorization framework yields systematic constructions for general integer β (β=2,3,...) and provides infrared regularization. These results illuminate the analytic structure and hidden symmetries of conformal ladder integrals and offer tools for fishnet/CFT computations.

Abstract

In the paper, the family of conformal four-point ladder diagrams in arbitrary space-time dimensions is considered. We use the representation obtained via explicit calculation using the operator approach and conformal quantum mechanics to study their properties, such as symmetries, loop and dimensional shift identities. In even integer dimensions, latter allows one to reduce the problem to two-dimensional case, where the notable factorization holds. Additionally, for a specific choice of propagator powers, we show that the representation can be written in the form of linear combinations of classical polylogarithms (with coefficients that are rational functions) and explore the structure of the resulting expressions.

Conformal four-point ladder integrals in diverse dimensions and polylogarithms

TL;DR

This work analyzes conformal four-point ladder integrals in arbitrary dimensions via an operator-based representation tied to conformal quantum mechanics. The authors establish dimensional and loop-shift identities and demonstrate a two-dimensional factorization that maps higher-D results to the case, enabling extension to even dimensions using a dimension-raising operator . For β=1 in any even dimension, the ladder integrals reduce to expressions in terms of classical polylogarithms with rational coefficients, while a two-dimensional factorization framework yields systematic constructions for general integer β (β=2,3,...) and provides infrared regularization. These results illuminate the analytic structure and hidden symmetries of conformal ladder integrals and offer tools for fishnet/CFT computations.

Abstract

In the paper, the family of conformal four-point ladder diagrams in arbitrary space-time dimensions is considered. We use the representation obtained via explicit calculation using the operator approach and conformal quantum mechanics to study their properties, such as symmetries, loop and dimensional shift identities. In even integer dimensions, latter allows one to reduce the problem to two-dimensional case, where the notable factorization holds. Additionally, for a specific choice of propagator powers, we show that the representation can be written in the form of linear combinations of classical polylogarithms (with coefficients that are rational functions) and explore the structure of the resulting expressions.

Paper Structure

This paper contains 15 sections, 8 theorems, 163 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conformal four-point ladder integrals in diverse dimensions
  3. Formulation of the problem and introductory remarks
  4. Dimensional shift for generic propagator index $\beta$
  5. Loop shift for generic propagator index $\beta$
  6. Ladder integrals for even dimensions and integer $\beta$
  7. Ladder integrals and polylogarithms for $\beta=1$
  8. Two-dimensional factorization for generic $\beta$
  9. Consideration of conformal ladder diagrams for $\beta=2,3,\ldots$
  10. Conclusions
  11. The limit $\lambda \to 0$ for the function $\widetilde{\Phi}^{(\beta)}_L(z ,\bar{z};\lambda)$
  12. Explicit check of the loop shift identity for $\lambda = 1, \beta = 1$
  13. Symmetrical polynomials in $z, \bar{z}$ in the case $\beta = 1$
  14. Derivation of factorization in two-dimensional case
  15. Derivation of the loop recursion for the operator $\widetilde{R}_\ell$ in the case $\beta=1$

Key Result

Proposition 1

The function $\widetilde{\Phi}^{(\beta)}_L(z,\bar{z};\lambda)$, introduced in (beta0), obeys equation I.e., since $\lambda = D/2-1$, the operator $R_d$ translates the expression for the conformal L-loop 4-point ladder integral for dimension $D$ to the expression for the conformal L-loop 4-point ladder integral for dimension $D+2$.

Figures (2)

  • Figure 1: The $L$-loop ladder diagram in momentum space for massless $\varphi^3$ theory.
  • Figure 2: Dual graphical representation of the integral (\ref{['br1']}) written in the form (\ref{['genf07']}). The integrations are performed over boldface vertices and each line with index $\alpha$: $x_i \stackrel{\alpha}{-\!\!\!-\!\!\!-\!\!\!-} x_j$, corresponds to the propagator $x_{ij}^{-2\alpha}$

Theorems & Definitions (8)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8