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Algorithms for minimal Picard–Fuchs operators of Feynman integrals

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Abstract

In even space-time dimensions, the multi-loop Feynman integrals are integrals of rational function in projective space. By using an algorithm that extends the Griffiths–Dwork reduction for the case of projective hypersurfaces with singularities, we derive Fuchsian linear differential equations and the Picard–Fuchs equations, with respect to kinematic parameters for a large class of massive multi-loop Feynman integrals. With this approach, we obtain the differential operator for Feynman integrals to high multiplicities and high loop orders. Using recent factorisation algorithms, we give the minimal-order differential operator in most of the cases studied in this paper. Amongst our results are that the order of Picard–Fuchs operator for the generic massive two-point \(n-1\)-loop sunset integral in two dimensions is \(2^{n}-\left( {\begin{array}{c}n+1\\ \left\lfloor \frac{n+1}{2}\right\rfloor \end{array}}\right) \) supporting the conjecture that the sunset Feynman integrals are relative periods of Calabi–Yau of dimensions \(n-2\). We have checked this explicitly till six loops. As well, we obtain a particular Picard–Fuchs operator of order 11 for the massive five-point tardigrade non-planar two-loop integral in four dimensions for generic mass and kinematic configurations, suggesting that it arises from K3 surface with Picard number 11. We determine as well Picard–Fuchs operators of two-loop graphs with various multiplicities in four dimensions, finding Fuchsian differential operators with either Liouvillian or elliptic solutions.

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Notes

  1. A Magma implementation is available at https://github.com/lairez/periods.

  2. The superscript denotes the number of independent mass parameters. When the mass parameters are identified with s different values, we use the notation \([r_1^{a_1} r_2^{a_2}\cdots r_s^{a_s}]\) such that \(\sum _{i=1}^r a_i r_i=n\). When all the mass parameters are different \(m_1\ne \cdots \ne m_n\) we use \([1^n]\).

  3. For all the Picard–Fuchs operator considered in this work, the degree refers to the degree in t of the polynomial multiplying the higher-order derivative term. By homogeneity, the degree of the polynomial coefficient decreases with the derivative order.

  4. These values correspond to the position of the singular fibres of the pencil of K3 surfaces associated with the three-loop sunset (see the discussion in [64] and [59]).

  5. They correspond to the positions of singular fibres of the pencil of Calabi–Yau threefold (see the discussion in [59] and [64] ).

  6. This is sometimes called the Dunce’s cap graph, e.g. [68]. But since we are generalising to the multi-loop case, we will call these graphs ice-cream cone with multi-scoops.

  7. We thank Charles Doran for this reference.

  8. This is a Frobenius basis of solutions that can be obtained from the indicial equation near \(t=0\) [71]. The indicial equation near the point \(t=\alpha \) is the equation on the exponents of a solution to the differential equation behaving as \((t-\alpha )^\rho \). In the following, we will consider \(\alpha =0\) or \(\alpha =\infty \).

  9. The solutions of a product operator AB are solutions of the inhomogeneous equation \(B(y) = u\), where u is a solution of \(A(u) = 0\). Using variation of parameters, the solutions of this inhomogeneous equations can be expressed in terms of the solutions of B and that of A, using only the operations defining Liouvillian functions.

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Acknowledgements

We thank David Broadhurst, Francis Brown, Charles Doran, Andrew Harder, and Andrey Novoseltsev for discussions and comments. We specially thank Alexandre Goyer and Marc Mezzarobba for help in factoring differential operators. We are grateful to IHES for making their computer resources available. This work has been supported by the ANR grant “Amplitude” ANR-17- CE31-0001-01, the ANR grant “SMAGP” ANR-20-CE40-0026-01, the ANR grant “De Rerum Natura” ANR-19-CE40-0018, and by the European Research Council under the European Union’s Horizon Europe research and innovation programme, grant agreement 101040794 (10000 DIGITS).

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  1. Inria, Uni. Paris-Saclay, Palaiseau, France

    Pierre Lairez

  2. Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, 91191, Gif-sur-Yvette Cedex, France

    Pierre Vanhove

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Lairez, P., Vanhove, P. Algorithms for minimal Picard–Fuchs operators of Feynman integrals. Lett Math Phys 113, 37 (2023). https://doi.org/10.1007/s11005-023-01661-3

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