58 research outputs found
Two-loop renormalization-group analysis of critical behavior at m-axial Lifshitz points
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
. Utilizing dimensional regularization and minimal subtraction
of poles in 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
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 can be
computed numerically, and for special values of m analytically. The
expansions of the critical exponents , , ,
, the wave-vector exponent , and the correction-to-scaling
exponent are obtained to order . 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"
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
New explicit expressions are derived for the one-loop two-point Feynman
integral with arbitrary external momentum and masses and in D
dimensions. The results are given in terms of Appell functions, manifestly
symmetric with respect to the masses . 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
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 were presented for the critical
exponents , and
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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