Exact One Loop Running Couplings in the Standard Model

Taking the dominant couplings in the standard model to be the quartic scalar coupling, the Yukawa coupling of the top quark, and the SU(3) gauge coupling, we consider their associated running couplings to one loop order. Despite the non-linear nature of the differential equations governing these functions, we show that they can be solved exactly. The nature of these solutions is discussed and their singularity structure is examined. It is shown that for a sufficiently small Higgs mass, the quartic scalar coupling decreases with increasing energy scale and becomes negative, indicative of vacuum instability. This behavior changes for a Higgs mass greater than 168 GeV, beyond which this couplant increases with increasing energy scales and becomes singular prior to the ultraviolet (UV) pole of the Yukawa coupling. Upper and lower bounds on the Higgs mass corresponding to new physics at the TeV scale are obtained and compare favourably with the numerical results of the one-loop and two-loop analyses with inclusion of electroweak couplings.Comment: 5 pages, LaTeX, additional references and further discussion in this version. Accepted for publication in Canadian Journal of Physic

Spectrum of non-Hermitian heavy tailed random matrices

Let (X_{jk})_{j,k>=1} be i.i.d. complex random variables such that |X_{jk}| is in the domain of attraction of an alpha-stable law, with 0< alpha <2. Our main result is a heavy tailed counterpart of Girko's circular law. Namely, under some additional smoothness assumptions on the law of X_{jk}, we prove that there exists a deterministic sequence a_n ~ n^{1/alpha} and a probability measure mu_alpha on C depending only on alpha such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix a_n^{-1} (X_{jk})_{1<=j,k<=n} converges weakly to mu_alpha as n tends to infinity. Our approach combines Aldous & Steele's objective method with Girko's Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of mu_alpha. In contrast with the Hermitian case, we find that mu_alpha is not heavy tailed.Comment: Expanded version of a paper published in Communications in Mathematical Physics 307, 513-560 (2011