Low-temperature renormalization group for the Lifshitz point

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

“…The possession of information on the behaviour of the integral I D,m (p, q) near the boundary lines of the critical dimensions d u (m) = 1 2 m + 4 and d (m) = 1 2 m + 2 (see Figure 1) has allowed the achievement of several important results. (1) The agreement between the large-n expansion and the dimensional expansion near d u (m) for the correlation critical exponents L2 and L4 has been explicitly shown for generic m at d u (m) − d ∶= ′ → 0 in Shpot et al 7 (2) A similar consistency check between these exponents at large n and at low dimensions d = d (m) + with → 0 (obtained in Grest and Sak 15 ) has been performed in Shpot et al 6, section 6 (3) A consideration of the large-n expansions of L2 and L4 at m = 1 near the critical dimensions d u (1) = 4.5 and d (1) = 2.5, combined with their knowledge at certain other points of the (m, d) plane, has suggested schematic plots for the O(1∕n) coefficients of these exponents as functions of the space dimensionality d; see Shpot et al 6, figure 3 Both of these curves appeared to show nontrivial nonmonotonic behaviour. *Another calculation that allowed the breaking of the O(m) symmetry can be found in Inayat-Hussain and Buckingham 14 for a very specific case with m = d.…”

“…We now demonstrate that the representation (64) in terms of the Horn function H 4 is equivalent to the expression in (15) involving the imaginary part of a Gauss hypergeometric function. From Prudnikov et al 42, (6.8.1.17) we have…”

“…) whose domain of convergence is given by |t| + | | < 1. ** However, the value = −1, which would be needed to match the Gauss function in (15) with that of (65), is forbidden by the last condition. To avoid this difficulty we apply the transformation (39) to analytically continue the 2 F 1 function in (66).…”

“…Inspection of Figure 1 shows that the lower boundary d = 1 2 m + 2 is encountered for D in the range 0 < D < 2, whereas the upper boundary d u = 1 2 m + 4 is encountered in the range 0 < D < 4. In the particular cases D = 1 and D = 3, where we use the representations given in (15) and (16), ** Until this point, the present calculation has gone in parallel with that of Shpot and Pogány. 27, p546 The formulas 27, (5.6) and that at the bottom of Shpot and Pogány 27, p546 are appropriate for this region of the variables on matching the notation.…”

Critical behavior atm-axial Lifshitz points: Field-theory analysis and ε-expansion results

“…Other investigations employed the dimensionality expansion about the lower critical dimension 31 d * (m)ϭ2ϩm/2 for n у3, or the 1/n expansion. 2,32,33 Unfortunately, the ⑀-expansion results to order ⑀ 2 one group of authors 1,28,29 obtained for the correlation exponents l2 and l4 and the wave-vector exponent ␤ q are in conflict with those of Sak and Grest 30 for the cases mϭ2 and mϭ6.…”