I discuss various situations in which perturbative expansions are used in Yang-Mills theories with asymptotic freedom and establish the limits of its applicability.
This work was carried out in 1985. It was published in Russian in Yad. Fiz. 44, 498 (1986) [English translation Sov. J. Nucl. Phys. 44, 321 (1986)]. None of these publications are available on-line. Submitting this paper to ArXiv will make it accessi
ble. *** A simple method is developed that makes it possible to determine the $k$-loop coefficient of the $beta$-function if the operator product expansion for certain polarization operators in the $(k -1)$ loop is known. The calculation of the two-loop coefficient of the Gell-Mann-Low function becomes trivial -- it reduces to a few algebraic operations on already known expressions. As examples, spinor, scalar, and supersymmetric electrodynamics are considered. Although the respective results for $beta^{(2)}$ are known in the literature, both the method of calculation and certain points pertaining to the construction of the operator product expansion are new.
The chiral symmetry of QCD shows up in the linear Weyl--Wigner mode at short Euclidean distances or at high temperatures. On the other hand, low-lying hadronic states exhibit the nonlinear Nambu--Goldstone mode. An interesting question was raised as
to whether the linear realization of the chiral symmetry is asymptotically restored for highly excited states. We address it in a number of ways. On the phenomenological side we argue that to the extent the meson Regge trajectories are observed to be linear and equidistant, the Weyl--Wigner mode is not realized. This picture is supported by quasiclassical arguments implying that the quark spin interactions in high excitations are weak, the trajectories are linear, and there is no chiral symmetry restoration. Then we use the string/gauge duality. In the top-down Sakai--Sugimoto construction the nonlinear realization of the chiral symmetry is built in. In the bottom-up AdS/QCD construction by Erlich et al., and Karch et al. the situation is more ambiguous. However, in this approach linearity and equidistance of the Regge trajectories can be naturally implemented, with the chiral symmetry in the Nambu--Goldstone mode. Asymptotic chiral symmetry restoration might be possible if a nonlinearity (convergence) of the Regge trajectories in an intermediate window of $n,J$, beyond the explored domain, takes place. This would signal the failure of the quasiclassical picture.
We study the worldvolume dynamics of BPS domain walls in N=1 SQCD with N_f=N flavors, and exhibit an enhancement of supersymmetry for the reduced moduli space associated with broken flavor symmetries. We provide an explicit construction of the worldv
olume superalgebra which corresponds to an N=2 Kahler sigma model in 2+1D deformed by a potential, given by the norm squared of a U(1) Killing vector, resulting from the flavor symmetries broken by unequal quark masses. This framework leads to a worldvolume description of novel two-wall junction configurations, which are 1/4-BPS objects, but nonetheless preserve two supercharges when viewed as kinks on the wall worldvolume.
We study the multiplicity of BPS domain walls in N=1 super Yang-Mills theory, by passing to a weakly coupled Higgs phase through the addition of fundamental matter. The number of domain walls connecting two specified vacuum states is then determined
via the Witten index of the induced worldvolume theory, which is invariant under the deformation to the Higgs phase. The worldvolume theory is a sigma model with a Grassmanian target space which arises as the coset associated with the global symmetries broken by the wall solution. Imposing a suitable infrared regulator, the result is found to agree with recent work of Acharya and Vafa in which the walls were realized as wrapped D4-branes in IIA string theory.
