Critical Look at β -Function Singularities at Large N

Abstract

We propose a self-consistency equation for the β-function for theories with a large number of flavours, N , that exploits all the available information in the Wilson-Fisher critical exponent, ω, truncated at a fixed order in 1/N . We show that singularities appearing in critical exponents do not necessarily imply singularities in the β-function. We apply our method to (non-)abelian gauge theory, where ω features a negative singularity. The singularities in the β-function and in the fermion mass anomalous dimension are simultaneously removed providing no hint for a UV fixed point in the large-N limit.Introduction.-There are indications that perturbative series in quantum field theory are, in general, asymptotic series with zero radius of convergence. In theories with a large number of flavour-like degrees of freedom, N , a re-organization of the perturbative expansion in powers of 1/N is convenient. It can be shown that at fixed order in 1/N expansion, the number of diagrams contributing grows only polynomially rather than factorially: convergent series are obtained that can be summed up within their radius of convergence.There is a vast literature on resummed results corresponding to the first few orders in 1/N expansion, mainly for RG functions obtained via direct diagram resummation or critical-point methods, see e.g. Refs .Since the perturbative series at fixed order in 1/N are convergent, singularities in the (generically complex) coupling are expected. Appearance of such singularities on the real-coupling axis seems to be true for all the d = 4 theories analyzed so far, thereby having a dramatic effect on RG flows. In particular, the appearance of singularities in the coefficients of the 1/N expansion for gauge and Yukawa β-functions have inspired speculations of a possible UV fixed point [23][24][25][26][27][28][29].More generally, the UV fate of gauge theories for which asymptotic freedom is lost has broad theoretical interest, and this is in fact the case of matter-dominated theories. There, a non-trivial zero of the β-function can be envisaged if the large-N resummation produces a contribution to β functions such that lim g→r β 1/N (g) = −∞, where r is the radius of convergence of the 1/N series. Near the singularity, the O(1/N ) contribution exceeds the leading-order result, and it is clear that a zero must emerge. Unfortunately, close to the radius of convergence the perturbative expansion in 1/N is broken, and higher-order cannot be neglected. Further shadow on the existence of the fixed point as a consistent conformal field theory is cast by studying anomalous dimensions of other operators in the vicinity of the β-function singularity: in the case of large-N QED truncated at O(1/N ), the anomalous dimension of the fermion mass diverges [1, 2], and it was recently pointed out that in the large-N QCD the anomalous dimension of the glueball operator breaks the unitarity bound [30]. Recently, the first lattice simulations to investigate the existence of possible fixed points appeared [31]. Even though th...

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