Five-Point Off-Shell Conformal Integrals

• Five-point off-shell conformal integrals are multi-loop Feynman-type integrals with five arbitrary external momenta that yield five independent conformal cross ratios.
• They play a central role in maximally supersymmetric Yang–Mills theory, with evaluations performed via hypergeometric series, Feynman parameterizations, and differential equations.
• Advanced regularization methods preserving dual conformal invariance ensure pure, uniform transcendental weight, leading to compact representations with multiple polylogarithms and symbol alphabets.

A five-point off-shell conformal integral is a multi-loop Feynman-type integral with five generic, non-null external momenta or coordinates. Such integrals are central to computations in maximally supersymmetric Yang–Mills theory, the conformal bootstrap, and the analytic structure of amplitudes and correlation functions, particularly in dimensions $d=4$ and $d=6$. The "off-shell" qualifier denotes that all invariants (e.g., $p_i^2$ or $x_{ij}^2$) are arbitrary and not restricted to null or massless kinematics, maximizing the number of independent conformal cross ratios. These integrals exhibit rich analytic behavior, powerful symmetry constraints (notably conformal or dual conformal invariance), and frequently admit representations in terms of hypergeometric series, differential equation systems, or canonical symbols.

1. General Definition and Conformal Structure

The canonical position-space form for a five-point off-shell conformal integral in four dimensions is

$$I(1,2,3,4,5) = \int \frac{d^4 x_a\, d^4 x_b}{\pi^4} \frac{N(x_i, x_a, x_b)}{D(x_i, x_a, x_b)}$$

where $x_{1,\ldots,5}$ are fixed external points, $x_{a,b}$ are loop integration variables, $N$ encodes possible numerators (e.g., Gram determinants for purity and uniform transcendental weight) and $D$ is a product of quadratic propagators. The off-shell condition means that none of the $x_{ij}^2$ vanish, so the function ultimately depends only on five algebraically independent cross ratios. A generic basis of cross ratios for five points is, for instance,

$$u_1 = \frac{x_{13}^2 x_{24}^2}{x_{12}^2 x_{34}^2}, \quad u_2 = \frac{x_{14}^2 x_{25}^2}{x_{15}^2 x_{24}^2}, \quad u_3 = \frac{x_{15}^2 x_{23}^2}{x_{12}^2 x_{35}^2}, \quad u_4 = \frac{x_{13}^2 x_{25}^2}{x_{15}^2 x_{23}^2}, \quad u_5 = \frac{x_{14}^2 x_{35}^2}{x_{13}^2 x_{45}^2}$$

Any such choice is equivalent up to reparametrization in $\mathbb{C}^5$ (Pal et al., 2021).

In dual coordinates (momentum space), five-point off-shell conformal integrals also admit representations in terms of cross ratios built from invariants $p_{ij}^2$ and may involve more elaborate numerator structures to ensure conformal covariance under inversion.

2. Classification of Five-Point Off-Shell Conformal Topologies

At two loops, all five-point off-shell integrals arising in $\mathcal{N}=4$ SYM correlation functions or amplitudes fall into several projective topologies, with six master families after imposing Gram determinant constraints and removing spurious singularities (Kuo et al., 26 Dec 2025, Bercini et al., 11 Jan 2024):

• Scalar double-box: Seven propagators; two-loop ladder/box configuration.
• Kissing boxes: Products of one-loop box integrals sewn along a single propagator.
• Penta-box: Eight propagators; one loop contains a pentagon subgraph.
• Double-pentagon: Nine propagators; each loop encompasses five lines.

These integrals may appear with numerator factors constructed from Gram determinants or square roots—such as

$$\Delta_5 = \sqrt{ (x_{13}^2 x_{24}^2 - x_{12}^2 x_{34}^2 - x_{14}^2 x_{23}^2)^2 - 4 x_{12}^2 x_{23}^2 x_{34}^2 x_{14}^2 }$$

—to ensure uniform transcendental weight and pure leading singularities (Kuo et al., 26 Dec 2025).

A similar classification exists in momentum space for planar five-point integrals with up to two off-shell legs, relevant, e.g., for QCD computations with two massive vector bosons (Abreu et al., 9 Aug 2024).

3. Analytic Representations: Series, Differential Equations, and Symbols

Five-point off-shell conformal integrals can be analytically constructed by several complementary techniques:

• A-hypergeometric (GKZ) Series: The general $N$-point conformal "star" integral in $d=4$,

$$I(p; Q) = \int_{\mathbb{R}^4} d^4 Q \prod_{i=1}^N |Q - Q_i|^{-2p_i}$$

(with $\sum p_i = 4$ for $N=5$), factors into leg-dependent prefactors times a convergent five-variable hypergeometric (GKZ) sum,

$$P_5(\mu; u) = \sum_{n_i \ge 0} \frac{u_1^{n_1} \dots u_5^{n_5}}{n_1!\dots n_5!} \prod_{k=1}^5 \Gamma\left(\ell_k(\vec{n}; \mu)\right)$$

with $\mu_i = p_i$, and $\ell_k$ linear in $n_i$ and $\mu_j$ (Pal et al., 2021).

• Momentum-Space Feynman Parameterizations: The five-point off-shell momentum-space correlator is expressed as a $(n-1)(n-2)/2 = 6$-fold integral over the edge-momenta $q_{ij}$ of a 4-simplex, with arbitrary function $\widehat{F}$ of five momentum-space cross ratios (Bzowski et al., 2019). This form trivially solves the conformal Ward identities for any choice of $\widehat{F}$.
• Differential Equation (DE) Method and Symbol Calculus: By IBP reduction (e.g., with LiteRed or Kira) (Bercini et al., 11 Jan 2024, Kuo et al., 26 Dec 2025), the two-loop five-point system closes on a finite basis of master integrals (typically 6–8 topologies), which satisfy first-order systems of DEs in the cross ratios. In a pure basis, the DEs take canonical form,

$d\vec{I}(u) = \epsilon\, dA(u)\, \vec{I}(u)$

where $A(u)$ is a sum of constant matrices times logarithmic differentials $d\log W_k(u)$, and the "letters" $W_k$ include both rational and square-root functions of the cross ratios. The solutions at symbol level are combinations of nested iterated integrals (Chen integrals) and multiple polylogarithms up to weight 4 at two loops (Kuo et al., 26 Dec 2025, Abreu et al., 9 Aug 2024). The symbol alphabet is highly non-trivial, containing up to several hundred letters for two-mass five-point planar kinematics (Abreu et al., 9 Aug 2024).

4. Regularization Methods and Manifest Dual Conformal Invariance

A major computational advance is the use of regularization schemes that preserve dual conformal invariance (DCI) at every stage. The method of regions with mixed (dimensional plus analytic) regulators enables a decomposition where every region's integrand and result are DCI (Bork et al., 15 Sep 2025):

• Introduce regulators $\epsilon$ (for dimension) and $\alpha_i$ (analytic, one per propagator).
• Impose DCI constraints per inversion: the sum $\sum_i \alpha_i\, \theta_{li}$ must vanish at external points and $-2$ at internal points (per propagator touch-structure $\theta_{li}$).
• After region expansion, each non-vanishing contribution (e.g., 32 out of 43 for the two-loop pentabox) is a product of $\Gamma$-functions, resulting upon regulator removal in a polynomial in the cross-ratio logarithms $L_i = \ln u_i$ plus zeta values, with all poles canceling.

This approach both simplifies computations and guarantees that the final answers are manifestly in terms of physical cross ratios only, bypassing non-DCI artifacts present in conventional regularizations (e.g., standard dimensional regularization).

5. Pure Bases, Leading Singularities, and Symbol Alphabets

Purity and uniform transcendental weight play a decisive role in constructing bases of five-point off-shell conformal integrals. By diagonalizing all leading singularities—i.e., normalizing maximal cuts to unity—one builds a canonical "pure" basis comprising six topologies (Kuo et al., 26 Dec 2025):

• Each basis element is defined with a numerator (e.g., square roots, Gram determinants) specifically tuned to eliminate spurious residues and ensure unit leading singularity.
• Mapping these integrals to four-mass two-loop Feynman families under suitable conformal frame-fixing yields a computationally tractable approach for integration.
• The full symbol-level solution involves an alphabet with dozens to hundreds of "letters," incorporating new five-point square roots such as $\lambda_{m,ij,kl}$ and one-loop four-mass Gram determinants $\Delta_i$ (Kuo et al., 26 Dec 2025, Abreu et al., 9 Aug 2024).

For planar two-loop five-point integrals with two off-shell legs, the canonical symbol alphabet reaches 570 letters, including both "even" (rational) and "odd" (involving square roots) sectors (Abreu et al., 9 Aug 2024).

6. Special Cases: Large-Spin Limit, Two-Mass Kinematics, and OPE Limits

In $\mathcal{N}=4$ SYM, two-loop five-point integrals appear prominently in the computation of half-BPS correlators for operators of varying spin and "weight" sectors (Bercini et al., 11 Jan 2024). Detailed analysis via the DE method in various kinematic regimes (such as plane kinematics and null limits) reveals:

• Linearly reducible Schwinger parameterizations permit full analytical expressions in terms of multiple polylogarithms in the plane.
• In certain limits (e.g., $u_1,u_3 \to 0$), the integrals exhibit universal logarithmic divergence structures with coefficients that are themselves MPLs in the remaining invariants.
• Special kinematic limits (OPE, Euclidean, eikonal) allow extraction of block decompositions or exponentiation patterns (e.g., the cusp anomalous dimension structure).

The large-spin and small polarization regimes also display conjectured all-loop behaviors for structure constants, with subtleties in analytic continuation (Bercini et al., 11 Jan 2024). For integrals with two off-shell legs—important in both amplitude computations and double Lagrangian insertions in Wilson loops—analytic and numerical studies establish not just uniform transcendentality, but positivity properties within specific "amplituhedron" regions (Abreu et al., 9 Aug 2024).

7. Generalizations and Higher Dimensional Analogues

The dual conformal symmetry admits meaningful generalizations to $d=6$ through the introduction of "dotted" propagators (powers $1/(p^2)^2$). The unique linear combination of one- and two-loop five-point integrals in $d=6$ can be constructed to maintain full DCI and satisfy iterative "BDS-like" exponentiation relations, closely paralleling the four-dimensional structure (Bork et al., 2020): $$M_5^{(L)} = X_L[M_5^{(1)}, \dots, M_5^{(L-1)}] + f^{(L)}(\epsilon) M_5^{(1)}(L\epsilon) + C^{(L)} + O(\epsilon)$$ where $X_L$ is a specific polynomial of lower-loop amplitudes and $f^{(L)}$, $C^{(L)}$ encode cusp and collinear anomalous dimensions. The all-loop conjecture asserts that the full exponentiation and cross-ratio dependence persists in $d=6$ (Bork et al., 2020).

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References:

• (Bork et al., 15 Sep 2025): Method of regions for dual conformal integrals
• (Kuo et al., 26 Dec 2025): Notes on off-shell conformal integrals and correlation functions at five points
• (Abreu et al., 9 Aug 2024): Two-Loop Five-Point Two-Mass Planar Integrals and Double Lagrangian Insertions in a Wilson Loop
• (Bzowski et al., 2019): Conformal $n$-point functions in momentum space
• (Pal et al., 2021): Conformal Integrals in four dimensions
• (Bercini et al., 11 Jan 2024): Two loop five point integrals: light, heavy and large spin correlators
• (Bork et al., 2020): Dual Conformal Symmetry and Iterative Integrals in Six Dimensions