Threshold and pseudothreshold values of the sunset diagram

Berends, F.A.; Davydychev, Andrei I.; Ussyukina, N.I.

doi:10.1016/s0370-2693(98)00166-x

Cited by 58 publications

(93 citation statements)

References 26 publications

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“…In the two-loop calculation of the pseudoscalar meson masses and decay constants, these are the only two-mass configurations that arise. Results for the pseudothresholds, calculated directly using an integral representation of the sunsets, are given in [4]. We rederived the three pseudothreshold results…”

Section: Evaluation Using Tarcermentioning

confidence: 99%

“…Therefore, an alternative semi-analytic result is presented here for this mass configuration. The method used to derive the one-dimensional integral representation given in this section has been taken from the work of [4].…”

Section: Conementioning

confidence: 99%

“…(3.15) of [8] for k = 0 and Eq. (27) of [4] for k = 1. The case of k = 9, which we have checked numerically, agrees with Eq.…”

Section: Conementioning

confidence: 99%

“…The relations from [32] have been implemented independently using FORM [43]. The expansions around s = 0 were also derived using the methods of [3,20] and numerical results have been compared with the results from analytical expressions of [4,24,27].…”

Section: Introductionmentioning

confidence: 99%

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An analytic approach to sunset diagrams in chiral perturbation theory: Theory and practice

Ananthanarayan¹,

Bijnens²,

Ghosh³

et al. 2016

Eur. Phys. J. A

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We demonstrate the use of several code implementations of the Mellin-Barnes method available in the public domain to derive analytic expressions for the sunset diagrams that arise in the two-loop contribution to the pion mass and decay constant in three-flavoured chiral perturbation theory. We also provide results for all possible two-mass configurations of the sunset integral, and derive a new one-dimensional integral representation for the one mass sunset integral with arbitrary external momentum. Thoroughly annotated Mathematica notebooks are provided as ancillary files, which may serve as pedagogical supplements to the methods described in this paper.

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Section: Evaluation Using Tarcermentioning

confidence: 99%

Section: Conementioning

confidence: 99%

“…(3.15) of [8] for k = 0 and Eq. (27) of [4] for k = 1. The case of k = 9, which we have checked numerically, agrees with Eq.…”

Section: Conementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An analytic approach to sunset diagrams in chiral perturbation theory: Theory and practice

Ananthanarayan¹,

Bijnens²,

Ghosh³

et al. 2016

Eur. Phys. J. A

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show abstract

“…Their analytical expansions at p 2 = 0 and p 2 = ∞ were given in [2] and [3]. Their values at threshold p 2 = −(m 1 + m 2 + m 3 ) 2 and pseudothresholds (p 2 = −(m 1 − m 2 + m 3 ) 2 , p 2 = −(m 1 + m 2 − m 3 ) 2 , p 2 = −(−m 1 + m 2 + m 3 ) 2 ) were presented in [4]. The analytical expansions at pseudothresholds were found in [5].…”

Section: Introductionmentioning

confidence: 99%

The analytical values of the sunrise master integrals for one of the masses equal to zero

2002

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Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points

Lee

Smirnov

2018

J. High Energ. Phys.

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This is a sequel of our previous paper where we described an algorithm to find a solution of differential equations for master integrals in the form of an ǫ-expansion series with numerical coefficients. The algorithm is based on using generalized power series expansions near singular points of the differential system, solving difference equations for the corresponding coefficients in these expansions and using matching to connect series expansions at two neighboring points. Here we use our algorithm and the corresponding code for our example of four-loop generalized sunset diagrams with three massive and two massless propagators, in order to obtain new analytical results. We analytically evaluate the master integrals at threshold, p 2 = 9m 2 , in an expansion in ǫ up to ǫ 1 . With the help of our code, we obtain numerical results for the threshold master integrals in an ǫ-expansion with the accuracy of 6000 digits and then use the PSLQ algorithm to arrive at analytical values. Our basis of constants is build from bases of multiple polylogarithm values at sixth roots of unity.

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Threshold and pseudothreshold values of the sunset diagram

Abstract: Analytic results for the threshold and pseudothreshold values of the sunset diagram with arbitrary masses are obtained in terms of dilogarithms of ratios of the masses.Comment: 11 pages, plain LaTe

Cited by 58 publications

References 26 publications

An analytic approach to sunset diagrams in chiral perturbation theory: Theory and practice

An analytic approach to sunset diagrams in chiral perturbation theory: Theory and practice

The analytical values of the sunrise master integrals for one of the masses equal to zero

Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points

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