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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

FromAbstractRepresentation(expression)

ToAbstractRepresentation(expression)

Parameters

expression

-

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

Description

• 

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.

Examples

withPhysics:

withFeynmanIntegral

Evaluate,ExpandDimension,FromAbstractRepresentation,Parametrize,Series,SumLookup,TensorBasis,TensorReduce,ToAbstractRepresentation,ε,ϵ

(1)

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

interfaceimaginaryunit=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.

%FeynmanIntegralp__1`~mu`p__12m__phi2+iεp__1P__12m__12+iε,p__1

%FeynmanIntegralp__1~mup__12m__φ2+Physics:-FeynmanDiagrams:-εp__1P__12m__12+Physics:-FeynmanDiagrams:-ε,p__1

(2)

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:

ToAbstractRepresentation

𝕋μμ2,0,m__φ2,P__1,m__12,p__1,0

(3)

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 −1P1 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

FromAbstractRepresentation

%FeynmanIntegralp__1~mup__12m__φ2+Physics:-FeynmanDiagrams:-εp__1P__12m__12+Physics:-FeynmanDiagrams:-ε,p__1

(4)

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

=TensorReduce

%FeynmanIntegralp__1~mup__12m__φ2+Physics:-FeynmanDiagrams:-εp__1P__12m__12+Physics:-FeynmanDiagrams:-ε,p__1=12P__1~mum__12m__φ2%.P__1,P__1%FeynmanIntegral1p__12m__φ2+Physics:-FeynmanDiagrams:-εp__1P__12m__12+Physics:-FeynmanDiagrams:-ε,p__1+%FeynmanIntegral1p__12m__φ2+Physics:-FeynmanDiagrams:-ε,p__1%FeynmanIntegral1p__1P__12m__12+Physics:-FeynmanDiagrams:-ε,p__1%.P__1,P__1

(5)

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,

TensorReduce,step=2

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

P__1μ𝕋~mu2,0,m__φ2,P__1,m__12,p__1,0=P__1μC1P__1~mu

(6)

FromAbstractRepresentation

P__1μ%FeynmanIntegralp__1~mup__12m__φ2+IPhysics:-FeynmanDiagrams:-εp__1P__12m__12+IPhysics:-FeynmanDiagrams:-ε,p__1=P__1μC1P__1~mu

(7)

TensorReduce,step=4

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

12𝕋[]1,0,m__φ2,p__1,0+12𝕋[]1,P__1,m__12,p__1,0+12%.P__1,P__1m__12+m__φ2𝕋[]2,0,m__φ2,P__1,m__12,p__1,0=C1%.P__1,P__1

(8)

FromAbstractRepresentation

12%FeynmanIntegral1p__12m__φ2+IPhysics:-FeynmanDiagrams:-ε,p__1+12%FeynmanIntegral1p__1P__12m__12+IPhysics:-FeynmanDiagrams:-ε,p__1+12%.P__1,P__1m__12+m__φ2%FeynmanIntegral1p__12m__φ2+IPhysics:-FeynmanDiagrams:-εp__1P__12m__12+IPhysics:-FeynmanDiagrams:-ε,p__1=C1%.P__1,P__1

(9)

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

Evaluaterhs

P__1μμm__12m__φ2%.P__1,P__1π2ϵ%sum%sumP__12n__1m__12+m__φ2nm__φ2n__12ϵ2nΓn+n__1+1Γϵ+n+n__1Γ2n__1+n+2Γ1+n,n__1=0..∞,n=0..∞π2ϵm__φ22ϵΓ1+ϵ+π2ϵm__122ϵΓ1+ϵ2%.P__1,P__1

(10)

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

Evaluate

P__1μμm__12m__φ2%.P__1,P__1π2ϵ%sum%sumP__12n__1m__12+m__φ2nm__φ2n__12ϵ2nΓn+n__1+1Γϵ+n+n__1Γ2n__1+n+2Γ1+n,n__1=0..∞,n=0..∞π2ϵm__φ22ϵΓ1+ϵ+π2ϵm__122ϵΓ1+ϵ2%.P__1,P__1

(11)

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

References

  

[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.

Compatibility

• 

The Physics[FeynmanIntegral][ToAbstractRepresentation] and Physics[FeynmanIntegral][FromAbstractRepresentation] commands were introduced in Maple 2021.

• 

For more information on Maple 2021 changes, see Updates in Maple 2021.