### Abstract

*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 |
---|---|

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

^{D}⨁ R

*.*

^{m}*Mathematical Methods in the Applied Sciences*,

*41*(5), 2220-2246. https://doi.org/10.1002/mma.4763

}

^{D}⨁ R

*'*

^{m}*Mathematical Methods in the Applied Sciences*, vol. 41, no. 5, pp. 2220-2246. https://doi.org/10.1002/mma.4763

**A Feynman integral in Lifshitz-point and Lorentz-violating theories in R ^{D} ⨁ R^{m}.** / Paris, R. B.; Shpot, M. A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A Feynman integral in Lifshitz-point and Lorentz-violating theories in RD ⨁ Rm

AU - Paris, R. B.

AU - Shpot, M. A.

PY - 2018/3/30

Y1 - 2018/3/30

N2 - We evaluate a 1-loop, 2-point, massless Feynman integral ID,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum RD ⨁ Rm . Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of ID,m(p,q) in terms of powers of the variable X:=4p2/q4, where p=|p|, q=|q|, p Є RD, q Є Rm, and in terms of generalised hypergeometric functions 3F2(−X), when X<1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.

AB - We evaluate a 1-loop, 2-point, massless Feynman integral ID,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum RD ⨁ Rm . Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of ID,m(p,q) in terms of powers of the variable X:=4p2/q4, where p=|p|, q=|q|, p Є RD, q Є Rm, and in terms of generalised hypergeometric functions 3F2(−X), when X<1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.

U2 - 10.1002/mma.4763

DO - 10.1002/mma.4763

M3 - Article

VL - 41

SP - 2220

EP - 2246

JO - Mathematical Methods in the Applied Sciences

T2 - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

IS - 5

ER -

^{D}⨁ R

*. Mathematical Methods in the Applied Sciences. 2018 Mar 30;41(5):2220-2246. https://doi.org/10.1002/mma.4763*

^{m}