We showed that the conventional expansion
in powers of the field for
the critical potential
of 3-dimensional O(N) models in the
large-N limit, does not converge for values of
larger than some critical value.
We found numerical evidence for
conjugated branch points in the complex plane (see figure below).
Padé approximants [L+3/L] for the critical potential
apparently converge at large
This allows high-precision
calculation of the fixed point in a more
suitable set of coordinates.
We argue that the singularities are
generic and not an artifact of the large-N limit.
Ignoring these singularities
may lead to inaccurate approximations.
Figure 9: Real and imaginary parts of the roots of the denominator
(filled squares) and numerator (crosses) of
a [26/23] Padé approximant for the critical potential.
More details in:
Y. Meurice, Complex Singularities of the Critical Potential in the
Large-N limit, hep-th/0208181, Phys. Rev. D (in press).
Work in Progress, Plans:
We have studied the critical properties (fixed points, exponents) of O(N)
hierarchical non linear sigma models at values of N ranging from 1
to 130. This work involved J. J. Godina, B. Oktay and L. Li. We are
presently attempting to determine the first four coefficients of the
1/N expansion in order to get some idea about the Borel summability
of the series. Preliminary results seem to favor the Borel summability.
We have observed that in a system of coordinates where
the unstable fixed point can be approximated by polynomials, the procedure
which consists in considering bare potential truncated at order
We are planning to investigate if similar problems
appear near tricritical fixed points. In particular,
reconsidering the RG flows in
a larger space of bare parameters may affect the generic
the intersections of hypersurface of various codimensions and help us
finding a more general realization of spontaneous breaking of
scale invariance with a
dynamical generation of mass as in the Bardeen-Moshe-Bander mechanism.