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Physics[FeynmanIntegral][TensorBasis] - compute a basis of tensor structures from a given list of external momentum and another one with free spacetime indices

Calling Sequence

TensorBasis(list_of_external_momenta, list_of_spacetime_indices)

TensorBasis(list_of_external_momenta, list_of_spacetime_indices, symmetrize = ..)

Parameters

list_of_external_momenta

-

a list of external momenta, which by convention in the FeynmanIntegral package are written as P__n where n is an integer

list_of_spacetime_indices

-

a list of spacetime indices, that could be covariant or contravariant (preceded by )

symmetrize = ..

-

(optional) the right-hand side can be true (default) or false, to symmetrize the products of external momenta that appear in the returned basis

Description

• 

TensorBasis receives a list of external momenta, which by convention in the FeynmanIntegral package are written as P__n where n is an integer, and a list of spacetime indices, which by default are represented by greek letters (to change the kind of letter see Setup) and returns a tensor basis onto which one can expand a tensorial structure with as many indices as in list_of_spacetime_indices.

• 

The tensor basis returned is constructed by taking the multiple-Cartesian product of the list of external momenta, and the metric gμ,ν, as many times as the number of indices in the list of spacetime indices, and discarding permutations.

• 

The tensor basis is returned symmetrized, e.g. if a product of two tensors P1μP2ν appears in the basis, then the output contains P1μP2ν+P2μP1ν. To receive the tensor basis non-symmetrized pass the optional argument symmetrize = false

• 

These tensor basis are relevant in the context of the Passarino-Veltman approach for the reduction of tensor to scalar Feynman integrals implemented in the TensorReduce command.

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 is that of a single external momentum and only one spacetime index

TensorBasisP__1,μ

P__1μ

(2)

This basis allows for expressing the following tensor Feynman integral as a linear combination of the elements of the basis

%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

(3)

TensorReduce,step=1

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

%FeynmanIntegralp__1~mup__12m__φ2+Physics:-FeynmanDiagrams:-εp__1P__12m__12+Physics:-FeynmanDiagrams:-ε,p__1=C1P__1~mu

(4)

opening the way for the reduction process

=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)

and ultimately leading to its symbolic computation by evaluating the scalar FeynmanIntegrals above

=Evaluate

%FeynmanIntegralp__1~mup__12m__φ2+Physics:-FeynmanDiagrams:-εp__1P__12m__12+Physics:-FeynmanDiagrams:-ε,p__1=12P__1~mum__12m__φ2%.P__1,P__1π2ϵ%sum%sumΓn+n__1+1m__φ2ϵ2n2n__1P__12n__1m__12+m__φ2nΓϵ+n+n__1Γ2n__1+n+2Γ1+n,n__1=0..∞,n=0..∞π2ϵm__φ22ϵΓ1+ϵ+π2ϵm__122ϵΓ1+ϵ%.P__1,P__1

(6)

The case of two spacetime indices already results in a basis even when there are no external momenta

TensorBasis,μ,ν

gμ,ν

(7)

Products of the metric are introduced when the number of indices makes that necessary

TensorBasis,μ,ν,α,β

gμ,νgα,β+gα,μgβ,ν+gα,νgβ,μ

(8)

The non-symmetrized form of this basis

TensorBasis,μ,ν,α,β,symmetrize=false

gμ,νgα,β

(9)

Two more realistic examples

TensorBasisP__1,P__2,P__3,μ,ν

gμ,ν,P__1μP__1ν,P__1μP__2ν+P__1νP__2μ,P__1μP__3ν+P__1νP__3μ,P__2μP__2ν,P__2μP__3ν+P__2νP__3μ,P__3μP__3ν

(10)

TensorBasisP__1,μ,ν,α

gμ,νP__1α+gα,νP__1μ+gα,μP__1ν,P__1μP__1νP__1α

(11)

See Also

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

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][TensorBasis] command was introduced in Maple 2021.

• 

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