Massive one-loop conformal Feynman integrals and quadratic transformations of multiple hypergeometric series

Ananthanarayan, B.; Banik, Sumit; Friot, Samuel; Ghosh, Shayan

doi:10.1103/physrevd.103.096008

Cited by 18 publications

(21 citation statements)

References 10 publications

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“…Therefore, by the study of appropriate classes of multiple MB integrals, the method of [5] opens promising horizons in hypergeometric functions theory and, in particular, for the determination of the analytic continuations of many classes of hypergeometric series, whatever the number of their variables is, in terms of others multivariable hypergeometric series. Obviously, a large number of applications can follow in physics, as already shown in the recent works [10] and [11] in the context of the study of Feynman integrals in quantum field theory.…”

Section: Introductionmentioning

confidence: 94%

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On the evaluation of the Appell $F_2$ double hypergeometric function

Ananthanarayan,

Bera,

Friot

et al. 2021

Preprint

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The transformation theory of the Appell F2(a, b1, b2; c1, c2; x, y) double hypergeometric function is used to obtain a set of series representations of F2 which provide an efficient way to evaluate F2 for real values of its arguments x and y and generic complex values of its parameters a, b1, b2, c1 and c2 (i.e. in the nonlogarithmic case). This study rests on a classical approach where the usual double series representation of F2 and other double hypergeometric series that appear in the intermediate steps of the calculations are written as infinite sums of one variable hypergeometric series, such as the Gauss 2F1 or the 3F2, various linear transformations of the latter being then applied to derive known and new formulas. Using the three well-known Euler transformations of F2 on these results allows us to obtain a total of 44 series which form the basis of the Mathematica package AppellF2, dedicated to the evaluation of F2. A brief description of the package and of the numerical analysis that we have performed to test it are also presented.

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Section: Introductionmentioning

confidence: 94%

“…We refer the reader to Eqs. ( 10) and (11) for the notation of Kampé de Fériet series and their mirror partners used in the expressions presented in this appendix.…”

Section: Appendix: Series Representations Of the Appell F 2 Functionmentioning

confidence: 99%

On the evaluation of the Appell $F_2$ double hypergeometric function

Ananthanarayan,

Bera,

Friot

et al. 2021

Preprint

Self Cite

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“…Furthermore, it will show how an interesting cross-fertilization between the latter naturally appears, thanks to our method. The reader will find all calculational details in the companion paper [24].…”

Section: Cross-fertilization Between Qft and Hypergeometric Functions...mentioning

confidence: 99%

“…This is where QFT enters into the game, to provide an even better solution than just furnishing the proof of our conjectured expression of this convergence region. Indeed, by choosing another set of conformal variables for the massive one-loop conformal 3-point integral (see [32]), we show in [24] that it is possible to derive an alternative (degenerate, nonresonant) threefold MB representation of the 3-point integral from which we derive 14 alternative series representations. One of them gives birth to an interesting new quadratic transformation formula for H C in terms of H B 12 , valid in the red region R 1 of Fig.…”

Section: Cross-fertilization Between Qft and Hypergeometric Functions...mentioning

confidence: 99%

“…We have also considered in [24] the same exercise in a less trivial situation where, instead of generic propagator powers for the conformal 3-point integral, we choose them equal to unity. In this case, the corresponding threefold MB representations belong to the degenerate resonant class and the calculations are less straightforward.…”

Section: Cross-fertilization Between Qft and Hypergeometric Functions...mentioning

confidence: 99%

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Multiple Series Representations of $N$-fold Mellin-Barnes Integrals

Ananthanarayan,

Banik,

Friot

et al. 2020

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Mellin-Barnes (MB) integrals are well-known objects appearing in many branches of mathematics and physics, ranging from hypergeometric functions theory to quantum field theory, solid state physics, asymptotic theory, etc. Although MB integrals have been studied for more than one century, to our knowledge there is no systematic computational technique of the multiple series representations of N -fold MB integrals (for any given positive integer N ). We present here a breakthrough in this context, which is based on simple geometrical considerations related to the gamma functions present in the numerator of the MB integrand. The method rests on a study of N -dimensional conic hulls constructed out of normal vectors of the singular (hyper)planes associated with each of the gamma functions. Specific sets of these conic hulls are shown to be in one-to-one correspondence with series representations of the MB integral. This provides a way to express the series representations as linear combinations of multiple series, the derivation of which does not depend on the convergence properties of the latter. Our method can be applied to N -fold MB integrals with straight as well as nonstraight contours, in the resonant and nonresonant cases and, depending on the form of the MB integrand, it gives rise to convergent series representations or diverging asymptotic ones. When convergent series are obtained the method also allows, in general, the determination of a single "master series" for each linear combination, which considerably simplifies convergence studies and numerical checks.

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Yangian Ward identities for fishnet four-point integrals

Corcoran

Loebbert

Miczajka

2022

J. High Energ. Phys.

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We derive and study Yangian Ward identities for the infinite class of four-point ladder integrals and their Basso-Dixon generalisations. These symmetry equations follow from interpreting the respective Feynman integrals as correlation functions in the biscalar fishnet theory. Alternatively, the presented identities can be understood as anomaly equations for a momentum space conformal symmetry. The Ward identities take the form of inhomogeneous extensions of the partial differential equations defining the Appell hypergeometric functions. We employ a manifestly conformal tensor reduction in order to express these inhomogeneities in compact form, which are given by linear combinations of Basso-Dixon integrals with shifted dimensions and propagator powers. The Ward identities naturally generalise to a one-parameter family of D-dimensional integrals representing correlators in the generalised fishnet theory of Kazakov and Olivucci. When specified to two spacetime dimensions, the Yangian Ward identities decouple. Using separation of variables, we explicitly bootstrap the solution for the conformal 2D box integral. The result is a linear combination of Yangian invariant products of Legendre functions, which reduce to elliptic K integrals for an isotropic choice of propagator powers. We comment on differences in the transcendentality patterns in two and four dimensions and their relations to discontinuities.

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Massive one-loop conformal Feynman integrals and quadratic transformations of multiple hypergeometric series

Cited by 18 publications

References 10 publications

On the evaluation of the Appell $F_2$ double hypergeometric function

On the evaluation of the Appell $F_2$ double hypergeometric function

Multiple Series Representations of $N$-fold Mellin-Barnes Integrals

Yangian Ward identities for fishnet four-point integrals

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