Quadratic relations between Feynman integrals

Broadhurst, David J.

doi:10.22323/1.303.0053

Cited by 23 publications

(30 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These quadratic relations resemble those discovered by Broadhurst and Roberts in Ref. [23]. The difference is that our relations explicitly depend on one parameter q 2 /m 2 while the relations discovered in Ref.…”

Section: Equal Mass Sunrise Integral In D =supporting

confidence: 80%

“…Using the quadratic relations for the maximally cut sunrise integral with equal masses in D = 2 we obtain for any given L the quadratic relations for the moments of the product of the Bessel and Macdonald functions. However, we were not able to derive the closed formula for these relations for general L. The found relations, in a sense, generalize the quadratic relations obtained by Broadhurst and Roberts [23] to the case of arbitrary incoming momentum. However, it is not easy to obtain the latter from the former.…”

Section: Resultsmentioning

confidence: 59%

“…Similar functions appear in Refs. [21][22][23]. For any specified L it is easy to construct the matrix M for transition to moments basis using the matrix R which shifts the power of x in the integrand.…”

Section: Equal Mass Sunrise Integral In D =mentioning

confidence: 99%

See 2 more Smart Citations

Differential equations, recurrence relations, and quadratic constraints for L-loop two-point massive tadpoles and propagators.

Lee

Pomeransky

2019

J. High Energ. Phys.

View full text Add to dashboard Cite

We consider L-loop two-point tadpole (watermelon) integral with arbitrary masses, regularized both dimensionally and analytically. We derive differential equation system and recurrence relations (shifts of dimension and denominator powers). Since the L-loop sunrise integral corresponds to the (L + 1)-loop watermelon integral with one cut line, our results are equally applicable to the former. The obtained differential system has a Pfaffian form and is linear in dimension and analytic regularization parameters. In general case, the solutions of this system can be expressed in terms of the Lauricella functions F (L) C with generic parameters. Therefore, as a by-product, we obtain, to our knowledge for the first time, the Pfaffian system for F At m = 1/3 we have one relationWe note that we could have easily proceeded to more loops.

show abstract

Section: Equal Mass Sunrise Integral In D =supporting

confidence: 80%

Section: Resultsmentioning

confidence: 59%

See 1 more Smart Citation

Differential equations, recurrence relations, and quadratic constraints for L-loop two-point massive tadpoles and propagators.

Lee

Pomeransky

2019

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…Recent mathematical literature on intersection numbers of twisted cycles and co-cycles include application to Gel'fand-Kapranov-Zelevinski systems [13,66,67] and to quadratic relations [25][26][27][28][29][30][31].…”

Section: Linear and Quadratic Relationsmentioning

confidence: 99%

“…We show how intersection numbers can be used to establish linear and quadratic relations for Feynman integrals, and, more generally, for Aomoto-Gel'fand generalized hypergeometric functions. The former set of relations yields results that are equivalent to the known integration-by-parts identities (IBPs) [24], while the latter allow for a systematic classification of relations which, for certain type of integrals were originally detected within the application of number-theoretic methods to Feynman integrals, giving rise to interesting conjectures [25][26][27][28][29], proven to be true quite recently [30,31]. A special set of quadratic relations have been presented in [32], and it would be interesting to investigate if they can be classified as Twisted Riemann Period Relations [4].…”

Section: Introductionmentioning

confidence: 99%

Decomposition of Feynman integrals by multivariate intersection numbers

Frellesvig

Gasparotto

Laporta

et al. 2021

J. High Energ. Phys.

View full text Add to dashboard Cite

We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers, and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

show abstract

Holonomic Anti-Differentiation and Feynman Amplitudes

Koutschan¹

2021

Texts &Amp; Monographs in Symbolic Computation

View full text Add to dashboard Cite

Quadratic relations between Feynman integrals

Cited by 23 publications

References 14 publications

Differential equations, recurrence relations, and quadratic constraints for L-loop two-point massive tadpoles and propagators.

Differential equations, recurrence relations, and quadratic constraints for L-loop two-point massive tadpoles and propagators.

Decomposition of Feynman integrals by multivariate intersection numbers

Holonomic Anti-Differentiation and Feynman Amplitudes

Contact Info

Product

Resources

About