Dimensional transmutation

Abstract: A systematic study is made of the relevant degrees of freedom and the dynamics of quantum chromodynamics (QCD). We find that the dynamical properties of QCD are, to a large extent, a consequence of the structure of the vacuum arising from the tunneling between degenerate, classically stable, vacuums, and that the relevant degrees of freedom can be taken to be the Euclidean path histories that can be used to calculate the tunneling in the semiclassical approximation. This nonperturbative vacuum structure appears well suited to the major features of QCD, i.e., the dimensional transmutation that determines the size of the hadrons and the strong-interaction coupling constant, the source of dynamical chiral symmetry breaking, and the mechanism responsible for quark confinement.

Abstract: The quantum-mechanical $D$-dimensional inverse square potential is analyzed using field-theoretic renormalization techniques. A solution is presented for both the bound-state and scattering sectors of the theory using cutoff and dimensional regularization. In the renormalized version of the theory, there is a strong-coupling regime where quantum-mechanical breaking of scale symmetry takes place through dimensional transmutation, with the creation of a single bound state and of an energy-dependent $s$-wave scattering matrix element.

Abstract: Recently, the author has constructed a series of four dimensional non-critical string theories with eight supercharges, dual to theories of light electric and magnetic charges, for which exact formulas for the central charge of the space-time supersymmetry algebra as a function of the world-sheet couplings were obtained. The basic idea was to generalize the old matrix model approach, replacing the simple matrix integrals by the four dimensional matrix path integrals of N=2 supersymmetric Yang-Mills theory, and the Kazakov critical points by the Argyres-Douglas critical points. In the present paper, we study qualitatively similar toy path integrals corresponding to the two dimensional N=2 supersymmetric non-linear sigma model with target space CP^n and twisted mass terms. This theory has some very strong similarities with N=2 super Yang-Mills, including the presence of critical points in the vicinity of which the large n expansion is IR divergent. The model being exactly solvable at large n, we can study non-BPS observables and give full proofs that double scaling limits exist and correspond to universal continuum limits. A complete characterization of the double scaled theories is given. We find evidence for dimensional transmutation of the string coupling in some non-critical string theories. We also identify en passant some non-BPS particles that become massless at the singularities in addition to the usual BPS states.