A Feynman integral in Lifshitz‐point and Lorentz‐violating theories in

Abstract

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|, bold-italicp∈RD, bold-italicq∈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.

Isotropic Lifshitz scaling in four dimensions

“…does not show any pathology either in the infrared or in the ultraviolet region [20]. One easily realizes that the relevant region corresponds to the quadrilateral displayed in Fig higher order corrections and in particular by the effective change in the dimensions induced by a non-vanishing anomalous dimension that has been neglected so far.…”

Indications of isotropic Lifshitz points in four dimensions

“…The set of six flow equations (18), (24 -28), represents the output of the specific truncation made on the effective action in Eq. (13).…”

“…After the determination of a line of fixed point for the simplified set of flow equations, we include the effects of the longitudinal fluctuations and analyze the full set, Eqs. (18), (24 -28). One can easily check that now α 2 and W π do not show a compensating behavior such that their product, J = W π α 2 , is t-independent, as observed when the longitudinal fluctuations are neglected.…”

“…Therefore, we solve numerically the flow equations (18), (24 -28) for the parameters α 2 , m 2 σ , W π , W σ , Z π and Z σ , with the 'time' t = ln(K UV /k) running from t = 0 to large positive values (infrared region) and, in particular, we monitor the t-dependence of the dimensionless stiffness J, as the approaching of a fixed point requires an asymptotically constant J. The numerical analysis is more easily performed by using the minimal infrared regulator introduced before, R = W π k 4 , which does not depend on the propagator momentum.…”

“…The nature of the LP is essentially established by the three parameters (d, m, N), where N indicates the number of fields and, in particular, its properties are crucially related to the parameter of anisotropy m. So, for instance, the analysis in [1] indicates a m-dependent upper critical dimension d u (m) = 4 + m/2 while, as indicated in [5], one expects for the lower critical dimension of the O(N) theory: d l (m) = 2 + m/2. At the same time it is clear that even the techniques selected to analyze the problem must be adapted to the specific value of m. In fact, the difficulties encountered in handling the propagators for generic m and d (see [18]), made the calculation of the O( 2 ) contribution in the -expansion a very difficult task which was eventually solved in [19][20][21]. Analogous difficulties appeared in the computation of the critical properties at large N with the relative O(1/N) corrections [22].…”