4 research outputs found
A Feynman integral in Lifshitz-point and Lorentz-violating theories in R<sup>D</sup> ⨁ R<i><sup>m</sup></i>
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
Generalizing the O(N)-field theory to N-colored manifolds of arbitrary internal dimension D
We introduce a geometric generalization of the O(N)-field theory that
describes N-colored membranes with arbitrary dimension D. As the O(N)-model
reduces in the limit N->0 to self-avoiding polymers, the N-colored manifold
model leads to self-avoiding tethered membranes. In the other limit, for inner
dimension D->1, the manifold model reduces to the O(N)-field theory. We analyze
the scaling properties of the model at criticality by a one-loop perturbative
renormalization group analysis around an upper critical line. The freedom to
optimize with respect to the expansion point on this line allows us to obtain
the exponent \nu of standard field theory to much better precision that the
usual 1-loop calculations. Some other field theoretical techniques, such as the
large N limit and Hartree approximation, can also be applied to this model. By
comparison of low and high temperature expansions, we arrive at a conjecture
for the nature of droplets dominating the 3d-Ising model at criticality, which
is satisfied by our numerical results. We can also construct an appropriate
generalization that describes cubic anisotropy, by adding an interaction
between manifolds of the same color. The two parameter space includes a variety
of new phases and fixed points, some with Ising criticality, enabling us to
extract a remarkably precise value of 0.6315 for the exponent \nu in d=3. A
particular limit of the model with cubic anisotropy corresponds to the random
bond Ising problem; unlike the field theory formulation, we find a fixed point
describing this system at 1-loop order.Comment: 57 pages latex, 26 figures included in the tex