58 research outputs found

    Two-loop renormalization-group analysis of critical behavior at m-axial Lifshitz points

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    We investigate the critical behavior that d-dimensional systems with short-range forces and a n-component order parameter exhibit at Lifshitz points whose wave-vector instability occurs in a m-dimensional isotropic subspace of Rd{\mathbb R}^d. Utilizing dimensional regularization and minimal subtraction of poles in d=4+m2−ϵd=4+{m\over 2}-\epsilon dimensions, we carry out a two-loop renormalization-group (RG) analysis of the field-theory models representing the corresponding universality classes. This gives the beta function βu(u)\beta_u(u) to third order, and the required renormalization factors as well as the associated RG exponent functions to second order, in u. The coefficients of these series are reduced to m-dependent expressions involving single integrals, which for general (not necessarily integer) values of m∈(0,8)m\in (0,8) can be computed numerically, and for special values of m analytically. The ϵ\epsilon expansions of the critical exponents ηl2\eta_{l2}, ηl4\eta_{l4}, νl2\nu_{l2}, νl4\nu_{l4}, the wave-vector exponent βq\beta_q, and the correction-to-scaling exponent are obtained to order ϵ2\epsilon^2. These are used to estimate their values for d=3. The obtained series expansions are shown to encompass both isotropic limits m=0 and m=d.Comment: 42 pages, 1 figure; to appear in Nuclear Physics B; footnote added, minor changes in v

    Reply to "Comment on Renormalization group picture of the Lifshitz critical behaviors"

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    We reply to a recent comment by Diehl and Shpot (cond-mat/0305131) criticizing a new approach to the Lifshitz critical behavior just presented (M. M. Leite Phys. Rev. B 67, 104415(2003)). We show that this approach is free of inconsistencies in the ultraviolet regime. We recall that the orthogonal approximation employed to solve arbitrary loop diagrams worked out at the criticized paper even at three-loop level is consistent with homogeneity for arbitrary loop momenta. We show that the criticism is incorrect.Comment: RevTex, 6 page

    A massive Feynman integral and some reduction relations for Appell functions

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    New explicit expressions are derived for the one-loop two-point Feynman integral with arbitrary external momentum and masses m12m_1^2 and m22m_2^2 in D dimensions. The results are given in terms of Appell functions, manifestly symmetric with respect to the masses mi2m_i^2. Equating our expressions with previously known results in terms of Gauss hypergeometric functions yields reduction relations for the involved Appell functions that are apparently new mathematical results.Comment: 19 pages. To appear in Journal of Mathematical Physic

    Lifshitz-point critical behaviour to O(ϵ2){\boldsymbol{O(\epsilon^2)}}

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    We comment on a recent letter by L. C. de Albuquerque and M. M. Leite (J. Phys. A: Math. Gen. 34 (2001) L327-L332), in which results to second order in ϵ=4−d+m2\epsilon=4-d+\frac{m}{2} were presented for the critical exponents νL2\nu_{{\mathrm{L}}2}, ηL2\eta_{{\mathrm{L}}2} and γL2\gamma_{{\mathrm{L}}2} of d-dimensional systems at m-axial Lifshitz points. We point out that their results are at variance with ours. The discrepancy is due to their incorrect computation of momentum-space integrals. Their speculation that the field-theoretic renormalization group approach, if performed in position space, might give results different from when it is performed in momentum space is refuted.Comment: Latex file, uses the included iop stylefiles; Uses the texdraw package to generate included figure
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