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Physics[FeynmanIntegral][ExpandDimension] - expand a d-dimensional result for a Feynman integral keeping terms up to order 0 in the dimensional parameter FeynmanIntegral:-varepsilon

Physics[FeynmanIntegral][Series] - expand in series returning a result always up to the order specified

Calling Sequence

ExpandDimension(expression)

Series(expression, z, order)

Parameters

expression

-

any expression, equation, set, list or matrix of them, typically involving FeynmanIntegral:-varepsilon dimensional parameter

z

-

the expansion variable

order

-

(optional) a nonnegative integer specifying the order, so that the result is always up to Ozorder

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. Generally speaking, these integrals can be evaluated using dimensional regularization with the Evaluate command, whose output is corresponds to computing the integral in d-dimensions. In this context, ExpandDimension takes the dimension as d=42ϵ and expands around ϵ=0 keeping terms up to order Oϵ

• 

The expansion of sums is done by first analyzing the structure of the GAMMA poles of the summand and, depending on the case, splitting the sum into two parts, separating the part that is divergent when d=4. Then d is taken equal to 42ϵ and the expansion around ϵ=0 is done using Series.

• 

The Series command works as the series command but - say z is the series expansion variable - it always returns up to order, so a series structure up to Ozorder; for the difference with series, see the Examples section.

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:

An example departing from an interaction Lagrangian

LλφX3

LλφX3

(2)

A process with one incoming and one outgoing particle a 1-loop

FeynmanDiagramsL,incomingparticles=φ,outgoingparticles=φ,numberofloops=1,diagrams

%FeynmanIntegral98λ2DiracP__2+P__1π3E__1E__2P__1+p__22m__φ2+Physics:-FeynmanDiagrams:-εp__22m__φ2+Physics:-FeynmanDiagrams:-ε,p__2

(3)

To evaluate the integral, using dimensional regularization, computing the integral over the loop momentum in dimension d, you can use  Evaluate

Evaluate

98π1ϵλ2δP__2+P__1%sumΓϵ+nm__φ2ϵ2nP__12nΓn+1Γ2n+2,n=0..∞E__1E__2

(4)

This result contains the dimensional parameter ϵ. To expand the dimension of this result around d=42ϵ keeping terms up to order 0 in ϵ you can use

ExpandDimension

98λ2δP__2+P__1πE__1E__2ϵ−1+98λ2δP__2+P__12lnm__φ+γ%sumΓnP__12nΓn+1m__φn2Γ2n+2,n=1..∞+lnπE__1E__2π+Oϵ

(5)

Computing the integral without expanding and expand in a second step allows for better control and follow-up of the computation. Alternatively, you can compute the two steps in one go using the expanddimension option of Evaluate:

Evaluate,expanddimension

98λ2δP__2+P__1πE__1E__2ϵ−1+98λ2δP__2+P__12lnm__φ+γ%sumΓnP__12nΓn+1m__φn2Γ2n+2,n=1..∞+lnπE__1E__2π+Oϵ

(6)

The Series command has the same syntax as series but always return up to Ozorder. Consider for instance:

ⅇzGAMMAz

ⅇzΓz

(7)

Indicating the order as equal to 3, the output by series, however, can be relative; in this example it starts at 1z and goes up to Oz2:

series,z,3

z−1+γ+1+112π2+12γ2γ+12z+Oz2

(8)

The output of Series is up to Ozorder regardless of degree of the first term of the series

Series,z,3

z−1+γ+1+112π2+12γ2γ+12z+ζ33π2γ12γ36+π212+γ22γ2+16z2+Oz3

(9)

A different example, indicating the order equal to 1; here series default approach results in terms up to Oϵ2

seriesGAMMAϵ+n,ϵ,1assumingn::nonposint

1−1nΓ1nϵ−1+Ψ1n−1nΓ1n+π26−1nΨ11nΓ1n2+Ψ1n2Γ1n2−1nΓ1n+Ψ1n2−1nΓ1nϵ+Oϵ2

(10)

The same computation with output up to the indicated order, Oϵ

SeriesGAMMAϵ+n,ϵ,1assumingn::nonposint

1−1nΓ1nϵ−1+Ψ1n−1nΓ1n+Oϵ

(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][ExpandDimension] and Physics[FeynmanIntegral][Series] commands were introduced in Maple 2021.

• 

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