The exact decomposition of gauge variables in lattice Yang–Mills theory

In this Letter, we consider lattice versions of the decomposition of the Yang–Mills field a la Cho–Faddeev–Niemi, which was extended by Kondo, Shinohara and Murakami in the continuum formulation. For the SU(N)$\mathit{SU}(N)$ gauge group, we propose a set of defining equations for specifying the decomposition of the gauge link variable and solve them exactly without using the ansatz adopted in the previous studies for SU(2)$\mathit{SU}(2)$ and SU(3)$\mathit{SU}(3)$. As a result, we obtain the general form of the decomposition for SU(N)$\mathit{SU}(N)$ gauge link variables and confirm the previous results obtained for SU(2)$\mathit{SU}(2)$ and SU(3)$\mathit{SU}(3)$.     

Neutral ρ meson in a strong magnetic field in the SU(2) lattice gauge theory

The correlation functions of vector and pseudoscalar currents have been calculated in the external strong magnetic field in SU(2) lattice gluodynamics. The masses of the neutral ρ meson with different spin projections s = 0, ±1 to the axis parallel to the external magnetic field B have been calculated. The ρ meson mass with zero spin s = 0 decreases with the growth of the magnetic field and the ρ meson masses with s = ±1 increase with the magnetic field.     

On Painlevé/gauge theory correspondence

We elucidate the relation between Painlevé equations and four-dimensional rank one \(\mathcal {N} = 2\) theories by identifying the connection associated with Painlevé isomonodromic problems with the oper limit of the flat connection of the Hitchin system associated with gauge theories and by studying the corresponding renormalization group flow. Based on this correspondence, we provide long-distance expansions at various canonical rays for all Painlevé \(\tau \)-functions in terms of magnetic and dyonic Nekrasov partition functions for \(\mathcal {N} = 2\) SQCD and Argyres–Douglas theories at self-dual Omega background \(\epsilon _1 + \epsilon _2 = 0\) or equivalently in terms of \(c=1\) irregular conformal blocks.     

Laplace,Bäcklund and gauge transformations in the two-dimensional Toda lattice

The linear Bäcklund transformation (LBT) associated with the two-dimensional Toda lattice is shown to be equivalent to a sequence of Laplace transformations of a hyperbolic linear differential equation. When the Toda lattice is cut at a point, the corresponding Laplace invariant vanishes and the LBT can be integrated.     

Skeleton inequalities and the asymptotic nature of perturbation theory for Φ4-theories in two and three dimensions

We use the polymer representation of ⁴-quantum field theories to prove an infinite family of correlation inequalities, called skeleton inequalities, for the 2n-point Green's functions. As an application, we show that they imply that Feynman perturbation theory is asymptotic in less than four dimensions.     

Perturbation expansions and gauge choices in Judd—Ofelt theory

We examine the question of whether choices between different perturbation expansions or different gauges require the introduction of time-odd (and hence odd-rank) operators in Judd—Ofelt theory. We show that with a careful application of a Rayleigh—Schrödinger perturbation expansion in the length gauge only even-rank operators are required.     

Direct gauging of the Poincaré group V. Group scaling,classical gauge theory,and gravitational corrections

Homogeneous scaling of the group space of the Poincaré group,P ₁₀, is shown to induce scalings of all geometric quantities associated with the local action ofP ₁₀. The field equations for both the translation and the Lorentz rotation compensating fields reduce toO(1) equations if the scaling parameter is set equal to the general relativistic gravitational coupling constant 8Gc ^–4. Standard expansions of all field variables in power series in the scaling parameter give the following results. The zeroth-order field equations are exactly the classical field equations for matter fields on Minkowski space subject to local action of an internal symmetry group (classical gauge theory). The expansion process is shown to breakP ₁₀-gauge covariance of the theory, and hence solving the zeroth-order field equations imposes an implicit system ofP ₁₀-gauge conditions. Explicit systems of field equations are obtained for the first- and higher-order approximations. The first-order translation field equations are driven by the momentum-energy tensor of the matter and internal compensating fields in the zeroth order (classical gauge theory), while the first-order Lorentz rotation field equations are driven by the spin currents of the same classical gauge theory. Field equations for the first-order gravitational corrections to the matter fields and the gauge fields for the internal symmetry group are obtained. Direct Poincaré gauge theory is thus shown to satisfy the first two of the three-part acid test of any unified field theory. Satisfaction of the third part of the test, at least for finite neighborhoods, seems probable.     

Infrared asymptotic freedom of a hierarchical φ 3 6 lattice theory

For the weakly coupled lattice ₃ ⁶ theory in a hierarchical model approximation a nonperturbative renormalization group analysis in the spirit of Gawedzki and Kupiainen is performed to study the flow of the effective actions. We deduce a domain of attraction to the tricritical (Gaussian) fixed point. The two relevant coupling constants of the problem are controlled by analytic continuation to complex domains, tracing their images under the renormalization group iterations.     

On the infrared scaling solution of SU(N) Yang–Mills theories in the maximally Abelian gauge

An improved method for extracting infrared exponents from functional equations is presented. The generalizations introduced allow for an analysis of quite complicated systems such as Yang–Mills theory in the maximally Abelian gauge. Assuming the absence of cancellations in the appropriately renormalized integrals the only consistent scaling solution yields an infrared enhanced diagonal gluon propagator in support of the Abelian dominance hypothesis. This is explicitly shown for SU(2) and subsequently verified for SU(N), where additional interactions exist. We also derive the most infrared divergent scaling solution possible for vertex functions in terms of the propagators’ infrared exponents. We provide general conditions for the existence of a scaling solution for a given system and comment on the cases of linear covariant gauges and ghost–anti-ghost symmetric gauges.     

Ising gauge theory in 3.999… dimensions

We numerically study Ising gauge theories in non-integer dimensions below four dimensions using fractals. We find indications that the first-order transition of the d = 4 theory becomes second order for d = 4 − ϵ for arbitrarily small non-zero ϵ. This suggests that the upper critical dimension of abelian gauge theories is four.     

On the scaling behavior of the chiral phase transition in QCD in finite and infinite volume

We study the scaling behavior of the two-flavor chiral phase transition using an effective quark–meson model. We investigate the transition between infinite-volume and finite-volume scaling behavior when the system is placed in a finite box. We can estimate effects that the finite volume and the explicit symmetry breaking by the current quark masses have on the scaling behavior which is observed in full QCD lattice simulations. The model allows us to explore large quark masses as well as the chiral limit in a wide range of volumes, and extract information about the scaling regimes. In particular, we find large scaling deviations for physical pion masses and significant finite-volume effects for pion masses that are used in current lattice simulations.     

A note on the ‘one-link’ integral of U(N) lattice gauge theory

We calculate the one-link U(N) integral in closed form by a direct method, i.e., polar decomposition and integration over agular variables. The result agrees with the known solution of the Brower-Nauenberg equation, at least forN4.Notice that in Reference [7] a direct integration was given forN=2 andN=3 However, the identification with Equation (2) was not given explicity, but on the basis on the assumed uniqueness of the solution.     

Representation of an interaction in the formalism of operator BRST quantization for the standard electroweak theory in the Rξ gauge

An operator BRST quantization of the spontaneously broken SU(2)×U(1) electroweak theory is carried out in the R gauge. A representation is found for the interaction.Leninist Communist Youth League, Tomsk Pedagogical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 79–83, July, 1992.     

A finite size scaling analysis of the O(4)-invariant lattice δ4 theory

The critical behaviour of the three- and four-dimensional N=4 vector model is investigated by means of a Monte Carlo simulation on lattices with size between 4³ and 16³, and between 4⁴ and 12⁴, respectively. For obtaining information about some critical properties of the model, we use a method due to Binder which is based on the theory of finite size scaling. For the three-dimensional model we get estimates of the critical exponents ν and η which are compatible with estimates obtained from the ϵ-expansion. In four dimensions we study for two different values of the bare self-coupling λ (λ=1 in our normalization, and λ=∞) the scaling behaviour of some Green function ratios at the phase boundary. In both cases we find compatibility with the “predicted” scaling behaviour at the gaussian fixed point. This is another independent numerical hint that the continuum limit of the four-dimensional O(4)-invariant lattice δ⁴-model is a free field theory.     

Chern–Simons theory in the temporal gauge and knot invariants through the universal quantum R-matrix

In temporal gauge A₀=0$A_0=0$ the 3d Chern–Simons theory acquires quadratic action and an ultralocal propagator. This directly implies a 2d R-matrix representation for the correlators of Wilson lines (knot invariants), where only the crossing points of the contours projection on the xy plane contribute. Though the theory is quadratic, P-exponents remain non-trivial operators and R  -factors are easier to guess then derive. We show that the topological invariants arise if additional flag structure R³⊃R²⊃R¹${\mathbb{R}}^3\supset {\mathbb{R}}^2\supset {\mathbb{R}}^1$ (xy plane and a y line in it) is introduced, R is the universal quantum R  -matrix and turning points contribute the “enhancement” factors qρ$q^{\rho }$.