### Abstract

We illustrate a rigorous approach to express the totally symmetric isotropic tensors of arbitrary rank in the n-dimensional Euclidean space as a linear combination of products of Kronecker deltas. By making full use of the symmetries, one can greatly reduce the efforts to compute cumbersome angular integrals into straightforward combinatoric counts. This method is generalised into the cases in which such symmetries are present in subspaces. We further demonstrate the mechanism of the tensor-integral reduction that is widely used in various physics problems such as perturbative calculations of the gauge-field theory in which divergent integrals are regularised in d= 4-2ϵ space-time dimensions. The main derivation is given in the n-dimensional Euclidean space. The generalisation of the result to the Minkowski space is also discussed in order to provide graduate students and researchers with techniques of tensor-integral reduction for particle physics problems.

Original language | English |
---|---|

Article number | 025801 |

Journal | European Journal of Physics |

Volume | 38 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2017 Mar 1 |

### Fingerprint

### Keywords

- combinatorics
- Feynman integral
- isotropic tensor
- tensor angular integral
- tensor-integral reduction

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*European Journal of Physics*,

*38*(2), [025801]. https://doi.org/10.1088/1361-6404/aa54ce

**Combinatorics in tensor-integral reduction.** / Ee, June Haak; Jung, Dong Won; Kim, U. Rae; Lee, Jungil.

Research output: Contribution to journal › Article

*European Journal of Physics*, vol. 38, no. 2, 025801. https://doi.org/10.1088/1361-6404/aa54ce

}

TY - JOUR

T1 - Combinatorics in tensor-integral reduction

AU - Ee, June Haak

AU - Jung, Dong Won

AU - Kim, U. Rae

AU - Lee, Jungil

PY - 2017/3/1

Y1 - 2017/3/1

N2 - We illustrate a rigorous approach to express the totally symmetric isotropic tensors of arbitrary rank in the n-dimensional Euclidean space as a linear combination of products of Kronecker deltas. By making full use of the symmetries, one can greatly reduce the efforts to compute cumbersome angular integrals into straightforward combinatoric counts. This method is generalised into the cases in which such symmetries are present in subspaces. We further demonstrate the mechanism of the tensor-integral reduction that is widely used in various physics problems such as perturbative calculations of the gauge-field theory in which divergent integrals are regularised in d= 4-2ϵ space-time dimensions. The main derivation is given in the n-dimensional Euclidean space. The generalisation of the result to the Minkowski space is also discussed in order to provide graduate students and researchers with techniques of tensor-integral reduction for particle physics problems.

AB - We illustrate a rigorous approach to express the totally symmetric isotropic tensors of arbitrary rank in the n-dimensional Euclidean space as a linear combination of products of Kronecker deltas. By making full use of the symmetries, one can greatly reduce the efforts to compute cumbersome angular integrals into straightforward combinatoric counts. This method is generalised into the cases in which such symmetries are present in subspaces. We further demonstrate the mechanism of the tensor-integral reduction that is widely used in various physics problems such as perturbative calculations of the gauge-field theory in which divergent integrals are regularised in d= 4-2ϵ space-time dimensions. The main derivation is given in the n-dimensional Euclidean space. The generalisation of the result to the Minkowski space is also discussed in order to provide graduate students and researchers with techniques of tensor-integral reduction for particle physics problems.

KW - combinatorics

KW - Feynman integral

KW - isotropic tensor

KW - tensor angular integral

KW - tensor-integral reduction

UR - http://www.scopus.com/inward/record.url?scp=85010805100&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85010805100&partnerID=8YFLogxK

U2 - 10.1088/1361-6404/aa54ce

DO - 10.1088/1361-6404/aa54ce

M3 - Article

AN - SCOPUS:85010805100

VL - 38

JO - European Journal of Physics

JF - European Journal of Physics

SN - 0143-0807

IS - 2

M1 - 025801

ER -