Feynman Integral Calculus

Smirnov, V. A.

doi:10.1007/3-540-30611-0

Cited by 38 publications

(23 citation statements)

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“…And because loop integration remains considerably harder than integrand construction, it is extremely convenient to reuse integrals for many different computations or reduce them to a smaller set of master integrals, see e.g. [120][121][122][123] and references therein. While we are not using integral reduction in this work, we are simplifying the work required to find integrand-level representations, after which integration-by-parts identities could be exploited.…”

Section: From Generalized To Prescriptive Unitaritymentioning

confidence: 99%

Prescriptive unitarity

Bourjaily

Herrmann

Trnka

2017

J. High Energ. Phys.

116

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We introduce a prescriptive approach to generalized unitarity, resulting in a strictly-diagonal basis of loop integrands with coefficients given by specifically-tailored residues in field theory. We illustrate the power of this strategy in the case of planar, maximally supersymmetric Yang-Mills theory (SYM), where we construct closed-form representations of all (n-point N k MHV) scattering amplitudes through three loops. The prescriptive approach contrasts with the ordinary description of unitarity-based methods by avoiding any need for linear algebra to determine integrand coefficients. We describe this approach in general terms as it should have applications to many quantum field theories, including those without planarity, supersymmetry, or massless spectra defined in any number of dimensions.

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Section: From Generalized To Prescriptive Unitaritymentioning

confidence: 99%

Prescriptive unitarity

Bourjaily

Herrmann

Trnka

2017

J. High Energ. Phys.

116

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“…, α L }. Note that the β integration is not projective here (but could be projectivized using the Cheng-Wu theorem [28,29]). Rescaling these parameters,…”

Section: Feynman Parameterizationmentioning

confidence: 99%

Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms

et al. 2018

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We describe a family of finite, four-dimensional, L-loop Feynman integrals that involve weight-(L+1) hyperlogarithms integrated over (L-1)-dimensional elliptically fibered varieties we conjecture to be Calabi-Yau manifolds. At three loops, we identify the relevant K3 explicitly and we provide strong evidence that the four-loop integral involves a Calabi-Yau threefold. These integrals are necessary for the representation of amplitudes in many theories-from massless φ^{4} theory to integrable theories including maximally supersymmetric Yang-Mills theory in the planar limit-a fact we demonstrate.

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“…Before getting started, however, we should perhaps clarify our terminology. What we will call 'Feynman parameters' are sometimes more specifically referred to as 'Schwinger parameters' or even 'α parameters' [38,39]-distinguished by how these integrals are de-projectivized. For the most part, de-projectivization is a trivial distinction; but to be clear, when we have reason to de-projectivize any integral explicitly, we will always use δ(α i − 1) instead of Feynman's choice of δ(( i α i )−1).…”

Section: Feynman Parameter Integrals In Dual Coordinatesmentioning

confidence: 99%

“…which can for example be computed directly with HyperInt 8 or by using standard algorithms for the evaluation of Mellin-Barnes integrals [38,39,45]. In this case, the fundamental strip of M{I 2mh }(z) is (µ, ν) = (0, 1) with a third order pole at z = 0.…”

Section: Analytic Extraction Of Divergent and Finite Coefficientsmentioning

confidence: 99%

Manifestly dual-conformal loop integration

2019

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Local, manifestly dual-conformally invariant loop integrands are now known for all finite quantities associated with observables in planar, maximally supersymmetric Yang-Mills theory through three loops. These representations, however, are not infrared-finite term by term and therefore require regularization; and even using a regulator consistent with dual-conformal invariance, ordinary methods of loop integration would naïvely obscure this symmetry. In this work, we show how any planar loop integral through at least two loops can be systematically regulated and evaluated directly in terms of strictly finite, manifestly dual-conformal Feynmanparameter integrals. We apply these methods to the case of the two-loop ratio and remainder functions for six particles, reproducing the known results in terms of individually regulated local loop integrals, and we comment on some of the novelties that arise for this regularization scheme not previously seen at one loop.

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Feynman Integral Calculus

Cited by 38 publications

References 0 publications

Prescriptive unitarity

Prescriptive unitarity

Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms

Manifestly dual-conformal loop integration

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