Some Remarks on Non-planar Feynman Diagrams

Bielas, Krzysztof; Dubovyk, Ievgen; Gluza, J.; Riemann, T.

doi:10.5506/aphyspolb.44.2249

Cited by 22 publications

(19 citation statements)

References 5 publications

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“…In the following, the MB-suite will be described in some detail. It comprises several tools for dimensionally regulated Feynman integrals in the momentum space: (i) Transform them into Feynman integrals expressed by Feynman parameters (textbook knowledge); (ii) Use the proper version of the AMBRE package [7][8][9][10] controlled for automation procedures by the PlanarityTest.m package [11,12] -transform them into the Mellin-Barns integrals, valid at initial parameters which include a finite shift of dimension, d = 4−2 , and with original integration paths parallel to the imaginary axis; (iii) Use MB.m or MBresolve.m [13,14] -perform an analytical continuation in , approaching small and (iv) -expand the Mellin-Barnes integrals as series in small ; (v) Use barnesroutines.m from the MBtools web page [15] perform simplifications using Barnes lemmas. (vi) At this stage, the original representation of the Feynman integral in terms of several finite MB integrals has been formulated.…”

Section: Introductionmentioning

confidence: 99%

New Prospects for the Numerical Calculation of Mellin--Barnes Integrals in Minkowskian Kinematics

Dubovyk¹,

Gluza²,

Jeliński³

et al. 2017

Acta Phys. Pol. B

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During the last several years, remarkable progress has been made in numerical calculations of dimensionally regulated multi-loop Feynman diagrams using Mellin-Barnes (MB) representations. The bottlenecks were non-planar diagrams and Minkowskian kinematics. The method has been proved to work in a highly non-trivial physical application (two-loop electroweak bosonic corrections to the Z → bb decay), and cross-checked with the sector decomposition (SD) approach. In fact, both approaches have their pros and cons. For multidimensional integrals, depending on masses and scales involved, they are complementary. A powerful top-bottom approach to the numerical integration of multidimensional MB integrals is automatized in the MB-suite AMBRE/PlanarityTest/MBtools/MBnumerics/ CUBA. New key elements are: a dedicated use of the Cheng-Wu theorem for non-planar topologies and of shifts and deformations of the integration contours. An alternative bottom-up approach starting with complex 1-dimensional MB integrals, based on the exploration of steepest descent integration contours in Minkowskian kinematics, is also discussed. Shortand long-term prospects of the MB method for multi-loop applications to LHC-and LC-physics are discussed.

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Section: Introductionmentioning

confidence: 99%

New Prospects for the Numerical Calculation of Mellin--Barnes Integrals in Minkowskian Kinematics

Dubovyk¹,

Gluza²,

Jeliński³

et al. 2017

Acta Phys. Pol. B

Self Cite

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“…The MB method has been well developed in recent years and there are useful software packages available at the MBtools webpage in the hepforge archive [32]: MB [33], MBresolve [34], AMBRE 1 [35] and barnesroutines (D. Kosower). Further, one may use PlanarityTest [36], AMBRE 2 [37] and AMBRE 3 [38], as well as MBsums [39], which are available from the AMBRE webpage [40]. For our purposes, we have derived MB representations with AMBRE and used the package MB, aided by MBresolve and barnesroutines, for a derivation of an expansion in terms of = (4 − d)/2.…”

Section: Strictly Numerical Two-loop Integration Techniquesmentioning

confidence: 99%

The two-loop electroweak bosonic corrections to sin2⁡θeffb

Dubovyk¹,

Freitas²,

Gluza³

et al. 2016

Physics Letters B

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The prediction of the effective electroweak mixing angle sin 2 θ b eff in the Standard Model at two-loop accuracy has now been completed by the first calculation of the bosonic two-loop corrections to the Zbb vertex. Numerical predictions are presented in the form of a fitting formula as function of M Z , M W , M H , m t and ∆α, α s . For central input values, we obtain a relative correction of ∆κ (α 2 ,bos) b = −0.9855 × 10 −4 , amounting to about a quarter of the fermionic corrections, and corresponding to sin 2 θ b eff = 0.232704. The integration of the corresponding two-loop vertex Feynman integrals with up to three dimensionless parameters in Minkowskian kinematics has been performed with two approaches: (i) Sector decomposition, implemented in the packages FIESTA 3 and SecDec 3, and (ii) Mellin-Barnes representations, implemented in AMBRE 3/MB and the new package MBnumerics. arXiv:1607.08375v2 [hep-ph] 30 Sep 2016 * In fitting programs like Gfitter [ 14] it is assumed that the validity of (3) was established by data preparation. In this respect we would like to mention that a natural language for the unfolding of measured cross-sections into pseudoobservables (not discussed here) and the relation of pseudo-observables to theory predictions has been worked out in the S-matrix approach [15-17]. We will discuss this topic elsewhere.

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“…We had to develop two new tools. For the treatment of non-planar Feynman integrals, we developed AMBRE 3 [72,73,74]. The package MBnumerics [75] delivers a stable 8-digit numerical treatment of Feynman integrals with presently up to four dimensionless scales in the Minkowskian region.…”

Section: The Bosonic Zbb Topologiesmentioning

confidence: 99%

“…The MB-representation is derived with calls to the packages PlanarityTest [72,79] and AM-BRE 3 [73,74]. The U-and F-polynomials are:…”

Section: Pos(ll2016)075mentioning

confidence: 99%

30 years, 750 integrals, and 1 desert

Dubovyk¹,

Freitas²,

Gluza³

et al. 2016

Proceedings of Loops and Legs in Quantum Field Theory — PoS(LL2016)

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The one-loop corrections to the weak mixing angle sin 2 θ b eff , derived from the Zbb vertex, are known since 1985. It took another 30 years to calculate the complete electroweak two-loop corrections to sin 2 θ b eff . The main obstacle was the calculation of the O(700) bosonic two-loop vertex integrals with up to three mass scales, at s = M 2 Z . We did not perform the usual integral reduction and master evaluation, but chose a completely numerical approach, using two different calculational chains. One method relies on publicly available sector decomposition implementations. Further, we derived Mellin-Barnes (MB) representations, exploring the publicly available MB suite. We had to supplement the MB suite by two new packages: AMBRE 3, a Mathematica program, for the efficient treatment of non-planar integrals and MBnumerics for advanced numerics in the Minkowskian space-time. Our preliminary result for LL2016, the "dessert", for the electroweak bosonic two-loop contributions to sin 2 θ b eff is:This contribution is about a quarter of the corresponding fermionic corrections and of about the same magnitude as several of the known higher-order QCD corrections. The sin 2 θ b eff is now predicited in the Standard Model with a relative error of 10 −4 [1].

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Some Remarks on Non-planar Feynman Diagrams

Cited by 22 publications

References 5 publications

New Prospects for the Numerical Calculation of Mellin--Barnes Integrals in Minkowskian Kinematics

New Prospects for the Numerical Calculation of Mellin--Barnes Integrals in Minkowskian Kinematics

The two-loop electroweak bosonic corrections to sin2⁡θeffb

30 years, 750 integrals, and 1 desert

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