Conformal four-point ladder integrals in diverse dimensions and polylogarithms

Technical Summary: Conformal Four-Point Ladder Integrals in Diverse Dimensions and Polylogarithms

Problem Statement
The paper addresses the computation and structural analysis of conformal four-point ladder integrals in arbitrary space-time dimensions $D$. These integrals, associated with massless $\phi^3$ theory and fishnet conformal field theories (CFT), are central to understanding scattering amplitudes and correlation functions in theories with enhanced symmetries, such as $\mathcal{N}=4$ Super Yang-Mills (SYM). While previous work established all-loop results for these integrals in arbitrary dimensions using the graph-building operator technique and conformal quantum mechanics [70], the resulting expressions were formulated in terms of Gegenbauer polynomials. Although analytically general, this form was noted to be of limited practical utility for explicit evaluation. The specific challenge addressed here is to systematically rewrite these representations in terms of classical polylogarithms and rational functions, particularly for even dimensions and specific choices of propagator powers, and to explore the underlying shift identities that relate integrals across different dimensions and loop orders.

Methodology
The authors employ a combination of operator techniques and conformal quantum mechanics to analyze the integrals. The core methodology involves:

1. Graph-Building Operator and Green's Function Representation: The $L$-loop ladder integrals are represented as the Green's function of a Hamiltonian in conformal quantum mechanics. This allows the generating function of the integrals to be expressed via an eigenfunction expansion involving Gegenbauer polynomials and an integral over a spectral parameter $\nu$.
2. Dimensional Shift Operators: The authors utilize a specific differential operator, $R_d = \frac{1}{z-\bar{z}}(z\partial_z - \bar{z}\partial_{\bar{z}})$, which acts as a shift operator translating the dimension of the space-time from $D$ to $D+2$. This operator is applied to the integral representations to relate results in higher even dimensions to the two-dimensional case.
3. Loop Shift Operators: A complementary operator, constructed from the Hamiltonian $H_\beta$ and the dimensional shift operator, is used to establish recursion relations that increase the number of loops from $L$ to $L+1$.
4. Analytic Continuation and Residue Calculations: For integer values of the propagator power parameter $\beta$, the integral over the spectral parameter $\nu$ is evaluated by closing the contour and calculating residues. This process reveals the pole structure and allows the reduction of the integral representation to sums involving polylogarithms.
5. Factorization in Two Dimensions: For the specific case of $D=2$ (where $\lambda = D/2 - 1 = 0$), the authors demonstrate a remarkable factorization of the integral into a product of functions depending separately on the cross-ratios $z$ and $\bar{z}$. This factorized form serves as the base case for generating results in higher even dimensions via the dimensional shift operator.

Key Contributions and Results

• Explicit Polylogarithmic Representations: The primary result is the derivation of explicit analytical expressions for conformal ladder integrals in arbitrary even dimensions $D$ and for the specific case of $\beta=1$ (corresponding to physical propagator powers in $D=4$). The authors show that the previously derived Gegenbauer polynomial representation can be systematically rewritten as linear combinations of classical polylogarithms ($\text{Li}_s(z)$) with coefficients that are rational functions of the cross-ratios $z$ and $\bar{z}$.
• Dimensional and Loop Shift Identities: The paper verifies that the derived representations satisfy specific shift identities.
  • Dimensional Shift: The operator $R_d$ successfully maps the solution for dimension $D$ to $D+2$, allowing the reduction of the problem in any even dimension to the two-dimensional case.
  • Loop Shift: An operator is constructed that increases the loop number $L \to L+1$. The paper also discusses an alternative loop recursion operator derived in [76] and demonstrates its consistency with the constructed operator for $\beta=1$.
• Two-Dimensional Factorization: For generic $\beta$ in $D=2$, the authors derive a factorized representation of the integrals. This result, which expresses the integral as a derivative of a product of two functions (one depending on $z$, the other on $\bar{z}$), is extended to all even dimensions using the dimensional shift operator.
• Regularization of Infrared Singularities: The study of integer $\beta \ge 2$ reveals a mechanism for regularizing infrared singularities. The authors show that for specific combinations of $D$ and $\beta$ where singularities would typically arise (e.g., $\beta = D/2 + k$), the application of the dimensional shift operator $R_d$ annihilates the singular terms, rendering the integrals regular in the physical region.
• Explicit Examples: The paper provides explicit formulas for the integrals in dimensions $D=4, 6, 8, 10$ for $\beta=1$, detailing the specific rational coefficients and polylogarithmic weights.

Significance and Claims
The authors claim that these results bridge the gap between the abstract operator-based construction of conformal integrals and practical, explicit analytical forms. By expressing the integrals in terms of classical polylogarithms, the work makes the results more accessible for practical applications in quantum field theory calculations. Furthermore, the demonstration of factorization in two dimensions and its translation to higher even dimensions highlights a deep structural simplicity in these conformal integrals.

The paper posits that these findings may aid in revealing further underlying symmetries, such as antipodal self-duality in fishnet theories, and provides a robust framework for studying more general families of conformal integrals. The work is presented as a continuation of previous studies [70], refining the general representation into a form that is both analytically explicit and computationally tractable for even dimensions. The authors remain modest regarding future implications, focusing on the utility of the derived identities and the structural insights provided by the factorization and shift operators.