Generic Mass Banana Integrals

• Generic mass banana integrals are l-loop two-point Feynman integrals with l+1 propagators and varying masses that define families of Calabi-Yau (l-1)-folds.
• They are formulated using Feynman parameters and are governed by inhomogeneous Picard-Fuchs differential equations alongside a compact D-module representation.
• Recent work computes their holonomic rank as 2^(l+1)-1, unifying momentum-space, position-space, and algebraic-geometric techniques in their analysis.

Generic mass banana integrals are $l$-loop two-point Feynman integrals with $l+1$ propagators carrying possibly different masses, and they can be formulated as functions of the external momentum squared together with the internal squared masses. In two dimensions, the general $l$-loop banana integral can be written in Feynman parameters as

$${\cal F}_{\sigma_l}(t, \xi_{i}) = \int_{\sigma_l} \frac{\mu_l}{P_l(t,\xi_i;x)},$$

with

$$P_l(t,\xi_i;x) = \left( t - \left( \sum_{i=1}^{l+1} \xi_i^2 x_i \right) \left( \sum_{i=1}^{l+1} x_i^{-1}\right) \right) \prod_{i=1}^{l+1} x_i,$$

where the zero locus $P_l=0$ defines a family of Calabi-Yau $(l-1)$-folds in projective space (Klemm et al., 2019). A recent development gives a compact $\mathcal{D}$-module description in dimensional regularization: a set of $\ell+3$ simple differential operators annihilates the $\ell$-loop generic mass banana integral, the singular locus is contained in the set of Landau singularities of the first and second type, and the holonomic rank has been computed up to $l+1$0 (Flieger, 6 Aug 2025).

1. Graph family, kinematics, and basic representations

The banana graph $l+1$1 is a graph with two vertices connected by $l+1$2 edges, each with possibly different mass $l+1$3, and the corresponding Feynman integral is denoted $l+1$4 (Kreimer, 2022). In the notation used for the generic $l+1$5-loop problem, one sets $l+1$6 and collects the external invariant and internal squared masses into variables $l+1$7, with $l+1$8 associated with the external momentum squared and $l+1$9 with the squared masses (Flieger, 6 Aug 2025).

Several equivalent descriptions coexist. In the parametric formulation, the integral is a relative period on a Calabi-Yau family, and the homogeneous cycles give the maximal cut while the full Feynman integral corresponds to a relative period and obeys inhomogeneous Picard-Fuchs differential equations (Klemm et al., 2019). In position space, an $l$0-banana diagram is just a product of $l$1 Green’s functions,

$l$2

and this viewpoint leads to high-order differential equations whose Fourier transforms factor through the Picard-Fuchs operator (Mishnyakov et al., 2024). For cut banana graphs, there is also a recursive iterated-integral description of the imaginary part $l$3, and the full result can be recovered through dispersion (Kreimer, 2022).

These formulations emphasize different structures. The parametric form makes the period interpretation explicit; the position-space form makes the product structure explicit; and the cut/dispersive form organizes thresholds, monodromy, and iterative integration. A plausible implication is that the generic-mass problem is naturally multiform: it is simultaneously a problem in Feynman integral technology, differential equations, and algebraic geometry.

2. Differential-operator formulation and the $l$4-ideal

A central recent result is the construction of a left ideal $l$5 in the Weyl algebra over the space of kinematic invariants and internal squared masses, generated by $l$6 differential operators that annihilate the generic $l$7-loop banana integral in dimensional regularization (Flieger, 6 Aug 2025). The generating set consists of one Euler operator, $l$8 second-order operators, and one symmetric differential operator: $l$9

$${\cal F}_{\sigma_l}(t, \xi_{i}) = \int_{\sigma_l} \frac{\mu_l}{P_l(t,\xi_i;x)},$$0

and

$${\cal F}_{\sigma_l}(t, \xi_{i}) = \int_{\sigma_l} \frac{\mu_l}{P_l(t,\xi_i;x)},$$1

The Euler operator reflects the scaling or homogeneity of the integral, each $${\cal F}_{\sigma_l}(t, \xi_{i}) = \int_{\sigma_l} \frac{\mu_l}{P_l(t,\xi_i;x)},$$2 relates variations in the external momentum and squared masses, and $${\cal F}_{\sigma_l}(t, \xi_{i}) = \int_{\sigma_l} \frac{\mu_l}{P_l(t,\xi_i;x)},$$3 reflects the symmetry of banana integrals under permutations of the mass arguments (Flieger, 6 Aug 2025). The operators were constructed by Griffiths-Dwork pole reduction, ensuring that all boundary terms vanish after integration and giving truly annihilating differential operators.

The operator algebra also has an explicit commutator structure: $${\cal F}_{\sigma_l}(t, \xi_{i}) = \int_{\sigma_l} \frac{\mu_l}{P_l(t,\xi_i;x)},$$4 This identifies a compact differential system for the generic-mass family that does not require integration-by-parts identities or Gram determinant eliminations to state (Flieger, 6 Aug 2025).

3. Singular locus and Landau singularities

For the ideal generated by these operators, the singular locus is shown to be contained within the set of Landau singularities of the first and second type (Flieger, 6 Aug 2025). The explicit description is

$${\cal F}_{\sigma_l}(t, \xi_{i}) = \int_{\sigma_l} \frac{\mu_l}{P_l(t,\xi_i;x)},$$5

where the second product is over all projectively nonequivalent sign assignments $${\cal F}_{\sigma_l}(t, \xi_{i}) = \int_{\sigma_l} \frac{\mu_l}{P_l(t,\xi_i;x)},$$6.

The factor $${\cal F}_{\sigma_l}(t, \xi_{i}) = \int_{\sigma_l} \frac{\mu_l}{P_l(t,\xi_i;x)},$$7 corresponds to possible thresholds when internal or external momenta or masses vanish, while the second factor encodes branch points and leading Landau singularities (Flieger, 6 Aug 2025). This description places the differential system directly against the expected physical singular structure.

A closely related hierarchy appears in the study of maximal-cut banana amplitudes. In that setting, one compares coordinate-space equations, their Fourier transforms, and Picard-Fuchs equations from the parametric representation; for generic masses, the momentum-space operator factors through the Picard-Fuchs operator,

$${\cal F}_{\sigma_l}(t, \xi_{i}) = \int_{\sigma_l} \frac{\mu_l}{P_l(t,\xi_i;x)},$$8

so that all Picard-Fuchs solutions are solutions of the Fourier-transformed equation, but not conversely (Mishnyakov et al., 2023). This factorization hierarchy and the $${\cal F}_{\sigma_l}(t, \xi_{i}) = \int_{\sigma_l} \frac{\mu_l}{P_l(t,\xi_i;x)},$$9-ideal description are not identical constructions, but both organize the differential constraints around the same singular geometry. This suggests that the annihilating-ideal viewpoint is compatible with earlier observations that the physically relevant solutions are isolated by lower-order period equations.

4. Holonomic rank and master integrals

The Macaulay matrix method is used to compute the holonomic rank of the proposed ideal (Flieger, 6 Aug 2025). The procedure expands the system generated by the differential operators, increases the order stepwise, and expresses monomials in derivatives as linear combinations; the number of standard monomials after reduction gives the holonomic rank.

For $$P_l(t,\xi_i;x) = \left( t - \left( \sum_{i=1}^{l+1} \xi_i^2 x_i \right) \left( \sum_{i=1}^{l+1} x_i^{-1}\right) \right) \prod_{i=1}^{l+1} x_i,$$0, the computed rank is

$$P_l(t,\xi_i;x) = \left( t - \left( \sum_{i=1}^{l+1} \xi_i^2 x_i \right) \left( \sum_{i=1}^{l+1} x_i^{-1}\right) \right) \prod_{i=1}^{l+1} x_i,$$1

with explicit values

$$P_l(t,\xi_i;x) = \left( t - \left( \sum_{i=1}^{l+1} \xi_i^2 x_i \right) \left( \sum_{i=1}^{l+1} x_i^{-1}\right) \right) \prod_{i=1}^{l+1} x_i,$$2

This matches the number of master integrals for the generic mass banana family in dimensional regularization (Flieger, 6 Aug 2025).

The same number appears in a different framework: for the banana graph with distinct edge masses, the number of master integrals is $$P_l(t,\xi_i;x) = \left( t - \left( \sum_{i=1}^{l+1} \xi_i^2 x_i \right) \left( \sum_{i=1}^{l+1} x_i^{-1}\right) \right) \prod_{i=1}^{l+1} x_i,$$3, with $$P_l(t,\xi_i;x) = \left( t - \left( \sum_{i=1}^{l+1} \xi_i^2 x_i \right) \left( \sum_{i=1}^{l+1} x_i^{-1}\right) \right) \prod_{i=1}^{l+1} x_i,$$4 the number of edges (Kreimer, 2022). Since $$P_l(t,\xi_i;x) = \left( t - \left( \sum_{i=1}^{l+1} \xi_i^2 x_i \right) \left( \sum_{i=1}^{l+1} x_i^{-1}\right) \right) \prod_{i=1}^{l+1} x_i,$$5, the two statements agree numerically: $$P_l(t,\xi_i;x) = \left( t - \left( \sum_{i=1}^{l+1} \xi_i^2 x_i \right) \left( \sum_{i=1}^{l+1} x_i^{-1}\right) \right) \prod_{i=1}^{l+1} x_i,$$6 This agreement is one of the main structural checks supporting the conjecture that the $$P_l(t,\xi_i;x) = \left( t - \left( \sum_{i=1}^{l+1} \xi_i^2 x_i \right) \left( \sum_{i=1}^{l+1} x_i^{-1}\right) \right) \prod_{i=1}^{l+1} x_i,$$7 operators generate the full holonomic annihilating $$P_l(t,\xi_i;x) = \left( t - \left( \sum_{i=1}^{l+1} \xi_i^2 x_i \right) \left( \sum_{i=1}^{l+1} x_i^{-1}\right) \right) \prod_{i=1}^{l+1} x_i,$$8-ideal (Flieger, 6 Aug 2025).

The paper formulates this as a conjecture: the operators $$P_l(t,\xi_i;x) = \left( t - \left( \sum_{i=1}^{l+1} \xi_i^2 x_i \right) \left( \sum_{i=1}^{l+1} x_i^{-1}\right) \right) \prod_{i=1}^{l+1} x_i,$$9 generate the full holonomic annihilating ideal for the generic mass $P_l=0$0-loop banana integral, and the rank is always $P_l=0$1 (Flieger, 6 Aug 2025). Since this is stated as a conjecture, the claim is not that completeness has been proved for arbitrary $P_l=0$2, but that all available evidence up to $P_l=0$3 is consistent with it.

5. Geometric and differential-equation frameworks

Generic mass banana integrals admit a broad geometric formulation in terms of GKZ systems and relative periods on families of Barth-Nieto Calabi-Yau $P_l=0$4-folds (Klemm et al., 2019). In this framework, the exponents of monomials in $P_l=0$5 define the Newton polytope, the periods satisfy a GKZ system of linear PDEs in the moduli, and the full banana amplitude is obtained by solving inhomogeneous Picard-Fuchs equations for a relative period. Explicit results were developed up to three loops, including the full mass dependence, and the general construction recovers known special-mass cases (Klemm et al., 2019).

A complementary position-space framework treats banana diagrams as products of propagators. For generic unequal masses, the product $P_l=0$6 satisfies a linear differential equation of order $P_l=0$7, while the Fourier-transformed operator in momentum space contains the Picard-Fuchs operator as a rightmost factor (Mishnyakov et al., 2024). In a related review of maximal-cut amplitudes, the order of the generic-mass Picard-Fuchs operator in $P_l=0$8 is stated as

$P_l=0$9

which is much less than the order of the Fourier-transformed position-space operator for the same $(l-1)$0 (Mishnyakov et al., 2023). The differential-ideal picture of generic mass banana integrals therefore sits among several non-equivalent, but strongly connected, operator formalisms.

The literature also includes direct representations of cuts as iterated integrals. For arbitrary $(l-1)$1 and arbitrary masses, the imaginary part of the banana graph obeys a recursive formula built from the two-edge kernel $(l-1)$2 and nested one-dimensional integrals (Kreimer, 2022). The hierarchy of function classes then grows with $(l-1)$3: $(l-1)$4 gives elementary functions, $(l-1)$5 gives elliptic integrals, and $(l-1)$6 gives iterated multiple integrals over algebraic functions of increasing complexity (Kreimer, 2022). This suggests a natural compatibility between the increasing holonomic rank and the increasing analytic complexity.

6. Canonical forms, K3 geometry, and computational developments

At three loops with four distinct non-zero masses in $(l-1)$7, a system of canonical differential equations has been constructed for the full set of master integrals (Duhr et al., 30 Jul 2025). The system takes the $(l-1)$8-factorized form

$(l-1)$9

with $\mathcal{D}$0, and the maximal cut defines a family of K3 surfaces whose periods and derivatives control the transcendental content (Duhr et al., 30 Jul 2025). The construction introduces 23 auxiliary functions defined as integrals over rational functions and K3 periods; twisted cohomology relations reduce 10 of these to expressions in terms of rational functions and K3 periods and derivatives, and after exploiting permutation symmetries only 2 functionally independent $\mathcal{D}$1-functions need to be defined (Duhr et al., 30 Jul 2025).

A related algorithmic account describes a two-step procedure toward $\mathcal{D}$2-factorization: filtered basis construction via maximal cuts and Baikov representation, followed by a recursive $\mathcal{D}$3-factorizing rotation (epsilon-collaboration et al., 4 Mar 2026). For the three-loop unequal-mass banana example, the number of master integrals including super-sectors is 15, the same number also stated in the canonical-system construction (Duhr et al., 30 Jul 2025). Both treatments emphasize that the geometric structure is encoded in the periods and their differential constraints, while the final solution is expressed in terms of Chen iterated integrals to arbitrary order in $\mathcal{D}$4 (Duhr et al., 30 Jul 2025).

Numerical methods have also been developed from general positivity properties. Scalar Feynman integrals in the Euclidean region are completely monotonic, and within a certain range of parameters many are in fact Stieltjes functions (Ditsch et al., 25 Jun 2026). The resulting CM bootstrap and Padé-approximant constructions apply to banana integrals with generic masses; the method was applied up to four-loop banana diagrams, and the paper discusses the 20-loop banana integral as an illustration of large-loop scalability (Ditsch et al., 25 Jun 2026). In some kinematic regions, the CM bootstrap outperforms AMFlow, while the Stieltjes/Padé method gives rigorous error control and fast analytic continuation in the cut complex plane (Ditsch et al., 25 Jun 2026).

7. Special loci, equal-mass limits, and related geometries

Although generic masses lead to the full multiscale problem, special mass loci often reveal simplified geometry. For the three-loop equal-mass banana graph, all master integrals can be written as linear combinations of iterated integrals of modular forms for $\mathcal{D}$5, and also in terms of elliptic polylogarithms evaluated at rational points (Broedel et al., 2019). The same congruence subgroup appears as in the two-loop equal-mass sunrise, and the third-order Picard-Fuchs operator is the symmetric square of the sunrise operator (Broedel et al., 2019).

Other specializations expose K3 factorizations. For the three-loop banana integral with three equal masses, the associated two-parameter family of K3 surfaces admits a modular parametrization, and the maximal cut can be written as a product of two copies of the maximal cuts of the two-loop equal-mass sunrise integral (Duhr et al., 21 Feb 2025). In dimensional regularization, the corresponding canonical differential equations can be expressed in terms of meromorphic modular forms, with a rigorous proof that the differential forms have only simple poles and define independent cohomology classes (Duhr et al., 24 Nov 2025).

These special cases do not reduce the generic-mass problem to equal-mass modularity, but they supply comparison points. The data indicate that equal-mass banana integrals are often controlled by modular or elliptic structures, whereas generic masses typically require K3 periods, twisted cohomology, GKZ systems, or higher-dimensional Calabi-Yau techniques (Klemm et al., 2019). A plausible implication is that the compact $\mathcal{D}$6-ideal for the generic-mass family provides a unifying operator-level description even when explicit function spaces vary strongly across special loci and loop orders.