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Tiziano Peraro - One of the best experts on this subject based on the ideXlab platform.
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analytic helicity amplitudes for two loop five gluon scattering the single minus case
arXiv: High Energy Physics - Phenomenology, 2018Co-Authors: Simon Badger, Christian Bronnumhansen, H B Hartanto, Tiziano PeraroAbstract:We present a compact analytic expression for the leading colour two-loop five-gluon amplitude in Yang-Mills theory with a single negative helicity and four positive helicities. The analytic result is reconstructed from numerical evaluations over finite fields. The numerical method combines integrand reduction, integration-by-parts identities and Laurent Expansion into a basis of pentagon functions to compute the coefficients directly from six-dimensional generalised unitarity cuts.
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tensor integrand reduction via Laurent Expansion
Journal of High Energy Physics, 2016Co-Authors: Valentin Hirschi, Tiziano PeraroAbstract:We introduce a new method for the application of one-loop integrand reduction via the Laurent Expansion algorithm, as implemented in the public C++ library Ninja. We show how the coefficients of the Laurent Expansion can be computed by suitable contractions of the loop numerator tensor with cut-dependent projectors, making it possible to interface Ninja to any one-loop matrix element generator that can provide the components of this tensor. We implemented this technique in the Ninja library and interfaced it to MadLoop, which is part of the public MadGraph5_aMC@NLO framework. We performed a detailed performance study, comparing against other public reduction tools, namely CutTools, Samurai, IREGI, PJFry++ and Golem95. We find that Ninja out-performs traditional integrand reduction in both speed and numerical stability, the latter being on par with that of the tensor integral reduction tool Golem95 which is however more limited and slower than Ninja. We considered many benchmark multi-scale processes of increasing complexity, involving QCD and electro-weak corrections as well as effective non-renormalizable couplings, showing that Ninja’s performance scales well with both the rank and multiplicity of the considered process.
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ninja automated integrand reduction via Laurent Expansion for one loop amplitudes
Computer Physics Communications, 2014Co-Authors: Tiziano PeraroAbstract:Abstract We present the public C++ library Ninja , which implements the Integrand Reduction via Laurent Expansion method for the computation of one-loop integrals. The algorithm is suited for applications to complex one-loop processes. Program summary Program title: Ninja Catalogue identifier: AETO_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AETO_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 3 No. of lines in distributed program, including test data, etc.: 74303 No. of bytes in distributed program, including test data, etc.: 530944 Distribution format: tar.gz Programming language: C++. Computer: Any computer with a compliant C++ compiler. Operating system: Unix-like (tested on Linux and Mac OS). RAM: Several thousands of bytes (it can vary depending on the complexity of the computation) Classification: 4.4, 11.1. External routines: A library of one-loop Master Integrals: OneLOop, LoopTools, or any other library implementing a suitable interface Nature of problem: Computation of one-loop integrals contributing to scattering amplitudes. Solution method: Semi-numerical implementation of the integrand reduction via Laurent Expansion, using a simplified polynomial division algorithm. Running time: Depending on the number of integrals and their complexity, between less than a millisecond up to several seconds per phase-space point, for the computation of a full amplitude.
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multi leg one loop massive amplitudes from integrand reduction via Laurent Expansion
arXiv: High Energy Physics - Phenomenology, 2013Co-Authors: Hans Van Deurzen, Pierpaolo Mastrolia, Edoardo Mirabella, Gionata Luisoni, Giovanni Ossola, Tiziano PeraroAbstract:We present the application of a novel reduction technique for one-loop scattering amplitudes based on the combination of the integrand reduction and Laurent Expansion. We describe the general features of its implementation in the computer code NINJA, and its interface to GoSam. We apply the new reduction to a series of selected processes involving massive particles, from six to eight legs.
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Integrand reduction of one-loop scattering amplitudes through Laurent series Expansion
Journal of High Energy Physics, 2012Co-Authors: Pierpaolo Mastrolia, Edoardo Mirabella, Tiziano PeraroAbstract:We present a semi-analytic method for the integrand reduction of one-loop amplitudes, based on the systematic application of the Laurent Expansions to the integrand-decomposition. In the asymptotic limit, the coefficients of the master integrals are the solutions of a diagonal system of equations, properly corrected by counterterms whose parametric form is known a priori. The Laurent Expansion of the integrand is implemented through polynomial division. The extension of the integrand-reduction to the case of numerators with rank larger than the number of propagators is discussed as well.
Mark W Coffey - One of the best experts on this subject based on the ideXlab platform.
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bernoulli identities zeta relations determinant expressions mellin transforms and representation of the hurwitz numbers
Journal of Number Theory, 2018Co-Authors: Mark W CoffeyAbstract:Abstract The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, is generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet L series, to Eisenstein series, and to general Mellin transforms. The Hurwitz numbers H ˜ n occur in the Laurent Expansion about the origin of a certain Weierstrass ℘ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general ℘ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence. Other results concern the Matter numbers of the equianharmonic case of the ℘ function.
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bernoulli identities zeta relations determinant expressions mellin transforms and representation of the hurwitz numbers
arXiv: Number Theory, 2016Co-Authors: Mark W CoffeyAbstract:The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet $L$ series, to Eisenstein series, and to general Mellin transforms. The Hurwitz numbers $\tilde{H}_n$ occur in the Laurent Expansion about the origin of a certain Weierstrass $\wp$ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general $\wp$ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence.
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an asymptotic form for the stieltjes constants gamma k a and for a sum s gamma n appearing under the li criterion
Mathematics of Computation, 2011Co-Authors: Charles Knessl, Mark W CoffeyAbstract:We present several asymptotic analyses for quantities associated with the Riemann and Hurwitz zeta functions. We first determine the leading asymptotic behavior of the Stieltjes constants γk(a). These constants appear in the regular part of the Laurent Expansion of the Hurwitz zeta function. We then use asymptotic results for the Laguerre polynomials Lαn to investigate a certain sum Sγ(n) involving the constants γk(1) that appears in application of the Li criterion for the Riemann hypothesis. We confirm the sublinear growth of Sγ(n)+n, which is consistent with the validity of the Riemann hypothesis.
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addison type series representation for the stieltjes constants
Journal of Number Theory, 2010Co-Authors: Mark W CoffeyAbstract:The Stieltjes constants γk(a) appear in the coefficients in the regular part of the Laurent Expansion of the Hurwitz zeta function ζ(s,a) about its only pole at s=1. We generalize a technique of Addison for the Euler constant γ=γ0(1) to show its application to finding series representations for these constants. Other generalizations of representations of γ are given.
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an effective asymptotic formula for the stieltjes constants
Mathematics of Computation, 2010Co-Authors: Charles Knessl, Mark W CoffeyAbstract:The Stieltjes constants γ k appear in the coefficients in the regular part of the Laurent Expansion of the Riemann zeta function ζ(s) about its only pole at s = 1. We present an asymptotic expression for γ k for k >> 1. This form encapsulates both the leading rate of growth and the oscillations with k. Furthermore, our result is effective for computation, consistently in close agreement (for both magnitude and sign) for even moderate values of k. Comparison to some earlier work is made.
Pierpaolo Mastrolia - One of the best experts on this subject based on the ideXlab platform.
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multi leg one loop massive amplitudes from integrand reduction via Laurent Expansion
arXiv: High Energy Physics - Phenomenology, 2013Co-Authors: Hans Van Deurzen, Pierpaolo Mastrolia, Edoardo Mirabella, Gionata Luisoni, Giovanni Ossola, Tiziano PeraroAbstract:We present the application of a novel reduction technique for one-loop scattering amplitudes based on the combination of the integrand reduction and Laurent Expansion. We describe the general features of its implementation in the computer code NINJA, and its interface to GoSam. We apply the new reduction to a series of selected processes involving massive particles, from six to eight legs.
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Integrand reduction of one-loop scattering amplitudes through Laurent series Expansion
Journal of High Energy Physics, 2012Co-Authors: Pierpaolo Mastrolia, Edoardo Mirabella, Tiziano PeraroAbstract:We present a semi-analytic method for the integrand reduction of one-loop amplitudes, based on the systematic application of the Laurent Expansions to the integrand-decomposition. In the asymptotic limit, the coefficients of the master integrals are the solutions of a diagonal system of equations, properly corrected by counterterms whose parametric form is known a priori. The Laurent Expansion of the integrand is implemented through polynomial division. The extension of the integrand-reduction to the case of numerators with rank larger than the number of propagators is discussed as well.
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feynman diagrams and differential equations
International Journal of Modern Physics A, 2007Co-Authors: Mario Argeri, Pierpaolo MastroliaAbstract:We review in a pedagogical way the method of differential equations for the evaluation of D-dimensionally regulated Feynman integrals. After dealing with the general features of the technique, we discuss its application in the context of one- and two-loop corrections to the photon propagator in QED, by computing the Vacuum Polarization tensor exactly in D. Finally, we treat two cases of less trivial differential equations, respectively associated to a two-loop three-point, and a four-loop two-point integral. These two examples are the playgrounds for showing more technical aspects about: Laurent Expansion of the differential equations in D (around D = 4); the choice of the boundary conditions; and the link among differential and difference equations for Feynman integrals.
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vertex diagrams for the qed form factors at the 2 loop level
Nuclear Physics, 2003Co-Authors: R Bonciani, Pierpaolo Mastrolia, E RemiddiAbstract:Abstract We carry out a systematic investigation of all the 2-loop integrals occurring in the electron vertex in QED in the continuous D -dimensional regularization scheme, for on-shell electrons, momentum transfer t =− Q 2 and finite squared electron mass m e 2 = a . We identify all the master integrals (MIs) of the problem and write the differential equations in Q 2 which they satisfy. The equations are expanded in powers of ϵ =(4− D )/2 and solved by the Euler's method of the variation of the constants. As a result, we obtain the coefficients of the Laurent Expansion in ϵ of the MIs up to zeroth order expressed in close analytic form in terms of harmonic polylogarithms.
Zhuohui Zhang - One of the best experts on this subject based on the ideXlab platform.
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Intertwining Operator for $Sp(4,\mathbb{R})$ and Orthogonal Polynomials
arXiv: Representation Theory, 2019Co-Authors: Zhuohui ZhangAbstract:We calculate the $(\mathfrak{g},K)$ module structure for the principal series representation of $Sp(4,\mathbb{R})$. Furthermore, we introduced a hypergeometric generating function together with an inverse Mellin transform technique as an improvement to the method to calculate the intertwining operators. We have shown that the matrix entries of the simple intertwining operators for $Sp(4,\mathbb{R})$-principal series are Hahn polynomials, and the matrix entries of the long intertwining operator can be expressed as the constant term of the Laurent Expansion of some hypergeometric function.
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intertwining operator for sp 4 mathbb r and orthogonal polynomials
arXiv: Representation Theory, 2019Co-Authors: Zhuohui ZhangAbstract:We calculate the $(\mathfrak{g},K)$ module structure for the principal series representation of $Sp(4,\mathbb{R})$. Furthermore, we introduced a hypergeometric generating function together with an inverse Mellin transform technique as an improvement to the method to calculate the intertwining operators. We have shown that the matrix entries of the simple intertwining operators for $Sp(4,\mathbb{R})$-principal series are Hahn polynomials, and the matrix entries of the long intertwining operator can be expressed as the constant term of the Laurent Expansion of some hypergeometric function.
Stefan Weinzierl - One of the best experts on this subject based on the ideXlab platform.
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planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularization parameter
Physical Review Letters, 2018Co-Authors: Luise Adams, Ekta Chaubey, Stefan WeinzierlAbstract:We compute systematically for the planar double box Feynman integral relevant to top pair production with a closed top loop the Laurent Expansion in the dimensional regularization parameter $ϵ$. This is done by transforming the system of differential equations for this integral and all its sub-topologies to a form linear in $ϵ$, where the ${ϵ}^{0}$ part is strictly lower triangular. This system is easily solved order by order in the dimensional regularization parameter $ϵ$. This is an example of an elliptic multiscale integral involving several elliptic subtopologies. Our methods are applicable to similar problems.
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the iterated structure of the all order result for the two loop sunrise integral
Journal of Mathematical Physics, 2016Co-Authors: Luise Adams, Christian Bogner, Stefan WeinzierlAbstract:We present a method to compute the Laurent Expansion of the two-loop sunrise integral with equal non-zero masses to arbitrary order in the dimensional regularisation e. This is done by introducing a class of functions (generalisations of multiple polylogarithms to include the elliptic case) and by showing that all integrations can be carried out within this class of functions.
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resolution of singularities for multi loop integrals
Computer Physics Communications, 2008Co-Authors: Christian Bogner, Stefan WeinzierlAbstract:We report on a program for the numerical evaluation of divergent multi-loop integrals. The program is based on iterated sector decomposition. We improve the original algorithm of Binoth and Heinrich such that the program is guaranteed to terminate. The program can be used to compute numerically the Laurent Expansion of divergent multi-loop integrals regulated by dimensional regularisation. The symbolic and the numerical steps of the algorithm are combined into one program.