Inverse Gram Research Articles

Scalar one-loop four-point Feynman integrals with complex internal masses

Based on the method in Refs.~{\tt [D.~Kreimer, Z.\ Phys.\ C {\bf 54} (1992) 667} and {\tt Int.\ J.\ Mod.\ Phys.\ A {\bf 8} (1993) 1797]}, we present analytic results for scalar one-loop four-point Feynman integrals with complex internal masses. The results are not only valid for complex internal masses, but also for real internal mass cases. Different from the traditional approach proposed by G. 't Hooft and M. Veltman in the paper {\tt[Nucl.\ Phys.\ B {\bf 153} (1979) 365]}, this method can be extended to evaluate tensor integrals directly. Therefore, it may open a new approach to cure the inverse Gram determinant problem analytically. We then implement the results into a computer package which is {\tt ONELOOP4PT.CPP}. In numerical checks, one compares the program to {\tt LoopTools version} $2.12$ in both real and complex mass cases. We find a perfect agreement between the results generated from this work and {\tt LoopTools}.

Golem95C: A library for one-loop integrals with complex masses

We present a program for the numerical evaluation of scalar integrals and tensor form factors entering the calculation of one-loop amplitudes which supports the use of complex masses in the loop integrals. The program is built on an earlier version of the golem95 library, which performs the reduction to a certain set of basis integrals using a formalism where inverse Gram determinants can be avoided. It can be used to calculate one-loop amplitudes with arbitrary masses in an algebraic approach as well as in the context of unitarity-inspired numerical reconstruction of the integrand. Program summaryProgram title: golem95-1.2.0Catalogue identifier: AEEO_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEO_v2_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 182 492No. of bytes in distributed program, including test data, etc.: 950 549Distribution format: tar.gzProgramming language: Fortran95Computer: Any computer with a Fortran95 compilerOperating system: Linux, UnixRAM: RAM used per integral/form factor is insignificantClassification: 4.4, 11.1External routines: Some finite scalar integrals are called from OneLOop [1,2], the option to call them from LoopTools [3,4] is also implemented.Catalogue identifier of previous version: AEEO_v1_0Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 2317Does the new version supersede the previous version?: YesNature of problem: Evaluation of one-loop multi-leg integrals occurring in the calculation of next-to-leading order corrections to scattering amplitudes in elementary particle physics. In the presence of massive particles in the loop, propagators going on-shell can cause singularities which should be regulated to allow for a successful evaluation.Solution method: Complex masses can be used in the loop integrals to stand for a width of an unstable particle, regulating the singularities by moving the poles away from the real axis.Reasons for new version: The previous version was restricted to massless particles in the loop.Summary of revisions: Real and complex masses are supported, a general μ parameter for the renormalization scale is introduced, improvements in the caching system and the user interface.Running time: Depends on the nature of the problem. A single call to a rank 6 six-point form factor at a randomly chosen kinematic point, using complex masses, takes 0.06 seconds on an Intel Core 2 Q9450 2.66 GHz processor.

Complete reduction of one-loop tensor 5- and 6-point integrals

We perform a complete analytical reduction of general one-loop Feynman integrals with five and six external legs for tensors up to rank $R=3$ and 4, respectively. An elegant formalism with extensive use of signed minors is developed for the cancellation of inverse Gram determinants. The 6-point tensor functions of rank $R$ are expressed in terms of 5-point tensor functions of rank $R\ensuremath{-}1$, and the latter are reduced to scalar four-, three-, and two-point functions. The resulting compact formulas allow both for a study of analytical properties and for efficient numerical programming. They are implemented in Fortran and Mathematica.

Golem95: A numerical program to calculate one-loop tensor integrals with up to six external legs

We present a program for the numerical evaluation of form factors entering the calculation of one-loop amplitudes with up to six external legs. The program is written in Fortran95 and performs the reduction to a certain set of basis integrals numerically, using a formalism where inverse Gram determinants can be avoided. It can be used to calculate one-loop amplitudes with massless internal particles in a fast and numerically stable way. Program summaryProgram title: golem95_v1.0Catalogue identifier: AEEO_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEO_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 50 105No. of bytes in distributed program, including test data, etc.: 241 657Distribution format: tar.gzProgramming language: Fortran95Computer: Any computer with a Fortran95 compilerOperating system: Linux, UnixRAM: RAM used per form factor is insignificant, even for a rank six six-point form factorClassification: 4.4, 11.1External routines: Perl programming language (http://www.perl.com/)Nature of problem: Evaluation of one-loop multi-leg tensor integrals occurring in the calculation of next-to-leading order corrections to scattering amplitudes in elementary particle physics.Solution method: Tensor integrals are represented in terms of form factors and a set of basic building blocks (“basis integrals”). The reduction to the basis integrals is performed numerically, thus avoiding the generation of large algebraic expressions.Restrictions: The current version contains basis integrals for massless internal particles only. Basis integrals for massive internal particles will be included in a future version.Running time: Depends on the nature of the problem. A rank 6 six-point form factor at a randomly chosen kinematic point takes 0.13 seconds on an Intel Core 2 Q9450 2.66 GHz processor, without any optimisation. With compiler optimisation flag -O3 the same point takes 0.09 seconds. Timings for lower point form factors are: All form factors for five-point functions from rank 0 to rank 4: 0.04 s. All form factors for rank 5 five-point functions: 0.05 s. All form factors for four-point functions from rank 0 to rank 4: 0.01 s.

An algebraic/numerical formalism for one-loop multi-leg amplitudes

We present a formalism for the calculation of multi-particle one-loop amplitudes, valid for an arbitrary number N of external legs, and for massive as well as massless particles. A new method for the tensor reduction is suggested which naturally isolates infrared divergences by construction. We prove that for N>4, higher dimensional integrals can be avoided. We derive many useful relations which allow for algebraic simplifications of one-loop amplitudes. We introduce a form factor representation of tensor integrals which contains no inverse Gram determinants by choosing a convenient set of basis integrals. For the evaluation of these basis integrals we propose two methods: An evaluation based on the analytical representation, which is fast and accurate away from exceptional kinematical configurations, and a robust numerical one, based on multi-dimensional contour deformation. The formalism can be implemented straightforwardly into a computer program to calculate next-to-leading order corrections to multi-particle processes in a largely automated way.

Reduction of one-loop tensor 5-point integrals

A new method for the reduction of one-loop tensor 5-point integrals to related 4-point integrals is proposed. In contrast to the usual Passarino–Veltman reduction and other methods used in the literature, this reduction avoids the occurrence of inverse Gram determinants, which potentially cause severe numerical instabilities in practical calculations. Explicit results for the 5-point tensor coefficients are presented up to rank 4. The expressions for the reduction of the relevant 1-, 2-, 3-, and 4-point tensor coefficients to scalar integrals are also included; apart from these standard integrals no other integrals are needed.