The Large-N approach of O(N) Models

• 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 has a low accuracy.
• 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 dimension of 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.