### Abstract

We evaluate a 1-loop, 2-point, massless Feynman integral

*I*_{D,m}(*p*,*q*) relevant for perturbative field theoretic calculations in strongly anisotropic*d*=*D*+*m*dimensional spaces given by the direct sum R^{D}⨁ R^{m}. Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions*D*and*m*. We obtain series expansions of*I*_{D,m}(*p*,*q*) in terms of powers of the variable*X*:=4*p*^{2}/*q*^{4}, where*p*=|**|,***p**q*=|**|,***q***Є R***p*^{D}, q Є R*, and in terms of generalised hypergeometric functions*^{m}_{3}*F*_{2}(−*X*), when*X*<1. These are subsequently analytically continued to the complementary region*X*≥1. The asymptotic expansion in inverse powers of*X*^{1/2}is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.Original language | English |
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Pages (from-to) | 2220-2246 |

Number of pages | 26 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 41 |

Issue number | 5 |

Early online date | 7 Feb 2018 |

DOIs | |

Publication status | Published - 30 Mar 2018 |

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### Cite this

Paris, R. B., & Shpot, M. A. (2018). A Feynman integral in Lifshitz-point and Lorentz-violating theories in R

^{D}⨁ R*.*^{m}*Mathematical Methods in the Applied Sciences*,*41*(5), 2220-2246. https://doi.org/10.1002/mma.4763