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Physics[FeynmanIntegral][ToAbstractRepresentation] - rewrite in abstract form a 1 loop Feynman integral expressed in standard form; the abstract form is used when performing the integral's tensor reduction

Physics[FeynmanIntegral][FromAbstractRepresentation] - rewrite back into standard form a given 1 loop Feynman integral expressed in abstract form

Calling Sequence






any expression, equation, set, list or matrix of them, typically involving Feynman tensor integrals



When computing the scattering matrix S for a particle process (momentum representation, see FeynmanDiagrams) the result, at one or more loops, contains Feynman integrals. Depending on the fields entering the interaction Lagrangian, the numerator of the integrand of such an integral may involve the loop momentum integration variable (one or a product of them) with free spacetime indices. That is the case of a tensor Feynman integral. Generally speaking, tensor integrals are computed by first reducing them to scalar integrals. In this context, ToAbstractRepresentation rewrites 1-loop Feynman integrals in an abstract form, convenient for performing the integral's tensor reduction, that in the context of the FeynmanIntegral package can be performed using the TensorReduce command.


The FromAbstractRepresentation command reverses the operation performed by ToAbstractRepresentation.






To remain closer to textbook notation, display the imaginary unit with a lowercase i


The simplest case of a massive φ field, the integral containing two propagators and one external momentum P1μ to which corresponds the mass m1.


p__1μμp__12m__φ2+εp__1P__12m__12+εⅆp__1 4


Before reducing this tensor integral to a linear combination of scalar integrals, it is convenient to represent the integral in abstract form, implemented as follows:




In this output we see the integral has 2 propagators, the first one has 0 external momentum (i.e. none) and mass mφ2. The second propagator has external momentum P1 to which corresponds the mass m1. Finally the loop momentum integration variable is p1 and the last operand, in this example equal to 0 means there are no contracted powers of p1, the loop integration variable. To retrieve the non-abstract form from the abstract one you can use


p__1μμp__12m__φ2+εp__1P__12m__12+εⅆp__1 4


The reduction of this tensor integral to a linear combination of scalar Feynman integrals uses this rewriting internally and results in


p__1μμp__12m__φ2+εp__1P__12m__12+εⅆp__1 4=P__1μμm__12m__φ2P__1·P__11p__12m__φ2+εp__1P__12m__12+εⅆp__1 4+1p__12m__φ2+εⅆp__1 41p__1P__12m__12+εⅆp__1 42P__1·P__1


The TensorReduce command can optionally return intermediate steps of the reduction process, from 1 to 7. Steps 2, 3 and 4 return a result using this abstract representation. For example,


* Partial match of 'step' against keyword 'outputstep'




P__1μp__1μμp__12m__φ2+εp__1P__12m__12+εⅆp__1 4=P__1μC1P__1μμ



* Partial match of 'step' against keyword 'outputstep'




1p__12m__φ2+εⅆp__1 42+1p__1P__12m__12+εⅆp__1 42+P__1·P__1m__12+m__φ21p__12m__φ2+εp__1P__12m__12+εⅆp__1 42=C1P__1·P__1


Back to the reduction process, note that, by design, TensorReduce does not evaluate the integrals so that one can follow the computational process clearly. The evaluation can be performed next by passing this result to Evaluate




Note also that Evaluate automatically calls TensorReduce that in turn uses ToAbstractRepresentation and FromAbstractRepresentation to perform the reduction of tensor integrals when that is the case. So, passing the Feynman integral directly to Evaluate results in the same process all in one go




See Also

Dgamma, Evaluate, FeynmanDiagrams, FeynmanIntegral[Overview], Parametrize, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Setup, TensorReduce



[1] Smirnov, V.A., Feynman Integral Calculus. Springer, 2006.


[2] Weinberg, S., The Quantum Theory Of Fields. Cambridge University Press, 2005.


[3] Bogoliubov, N.N., and Shirkov, D.V. Quantum Fields. Benjamin Cummings, 1982.