Dimensional Georgi-Glashow Model Research Articles

(2+1)−d Georgi-Glashow model near critical temperature

We study correlations functions of magnetic vortices $V$ and Polyakov loop $P$ operators in the 2+1 dimensional Georgi-Glashow model in the vicinity of the deconfining phase transition. In this regime the (dimensionally reduced) model is mapped onto a free theory of two massive Majorana fermions. We utilize this fermionic representation to explicitly calculate the expectation values of $V$ and $P$ as well as their correlators. In particular we show that the $VV$ correlator is large, and thus the anomalous breaking of the magnetic $U(1)$ symmetry is order one effect in the near critical region. We also calculate the contribution of magnetic vortices to the entropy and the free energy of the system.

Creating a monopole in 4D gauge theories

The problem of defining the second quantized monopole creation operator in non-Abelian gauge theories is discussed and exemplified by the (3 + 1)-dimensional Georgi-Glashow model. We construct the “coherent state” operator M(x) that creates the Coulomb magnetic field in terms of the Dirac singular electromagnetic potential. Our calculation of the vacuum expectation value of this operator 〈M(x)〉 in the confining phase indicates that it is free from the singularity along the Dirac string and in the leading order of perturbation theory the 〈M(x)〉 vanishes as a power of the volume of the system. This supports the conception that inclusion of the nonperturbative effects introduces an effective infrared cutoff on the calculation providing the finiteness of vacuum expectation value 〈M(x)〉.

Magnetic monopoles in 4D: a perturbative calculation

We address the question of defining the second quantised monopole creation operator in the 3+1 dimensional Georgi-Glashow model, and calculating its expectation value in the confining phase. Our calculation is performed directly in the continuum theory within the framework of perturbation theory. We find that, although it is possible to define the "coherent state" operator M(x) that creates the Coulomb magnetic field, the dependence of this operator on the Dirac string does not disappear even in the nonabelian theory. This is due to the presence of the charged fields (W^{\pm}). We also set up the calculation of the expectation value of this operator in the confining phase and show that it is not singular along the Dirac string. We find that in the leading order of the perturbation theory the VEV vanishes as a power of the volume of the system. This is in accordance with our naive expectation. We expect that nonperturbative effects will introduce an effective infrared cutoff on the calculation making the VEV finite.

The Higgs field and the ultraviolet behaviour of the vortex operator in 2+1 dimensions

We calculate the change in the ultraviolet behaviour of the vortex operator due to the presence of dynamical Higgs field in both 2+1 dimensional QED and the 2+1 dimensional Georgi-Glashow model. We find that in the QED case the presence of the Higgs field leads at the one loop level to power like correction to the propagator of the vortex operator. On the other hand, in the Georgi-Glashow model, the adjoint Higgs at one loop has no affect on the vortex propagator. Thus, as long as the mass of the Higgs field is much larger than the gauge coupling constant, the ultraviolet behaviour of the vortex operator in the Georgi-Glashow model is independent of the Higgs mass.

Higgs- and quark-inspired modifications of the finite-temperature properties of the Polyakov model

(2+1)-dimensional Georgi-Glashow model, else called the Polyakov model, is explored at nonzero temperatures and in the regime when the Higgs boson is not infinitely heavy. The finiteness of the Higgs-boson mass leads to the appearance of the upper bound on the parameter of the weak-coupling approximation, necessary to maintain the stochasticity of the Higgs vacuum. The modification of the finite-temperature behavior of the model emerging due to the introduction of massless quarks is also discussed.

Center symmetry and abelian projection at finite temperature

At finite temperature, there is an apparent conflict between Abelian projection and critical universality. For example, should the deconfinement transition of an SU(2) gauge theory projected to U(1) lie in the Z(2) universality class of the parent SU(2) theory or in the U(1) universality class? I prove that the projected theory lies in the universality class of the parent gauge theory. The mechanism is shown to be non-local terms in the projected effective action involving Polyakov loops. I connect this to the recent work by Dunne et al. on the deconfinement transition in the 2+1 dimensional Georgi-Glashow model, leading to a discussion of the role of renormalization group fixed points in Abelian projection.

Fundamental matter and the deconfining phase transition in 2+1 D

We analyze the effect of heavy fundamentally charged particles on the finite temperature deconfining phase transition in the (2+1)-dimensional Georgi–Glashow model. We show that in the presence of fundamental matter the transition turns into a crossover. The near critical theory is mapped onto the 2-dimensional Ising model in an external magnetic field. Using this mapping we determine the width of the crossover region as well as the specific heat as a function of the fundamental mass.

ACCOUNTING FOR THE FINITENESS OF THE HIGGS–BOSON MASS IN THE 3D GEORGI–GLASHOW MODEL

(2 + 1)-dimensional Georgi–Glashow model is explored in the regime when the Higgs boson is not infinitely heavy, but its mass is of the same order of magnitude as the mass of the W-boson. In the weak-coupling limit, the Debye mass of the dual photon and the expression for the monopole potential are found. The cumulant expansion applied to the average over the Higgs field is checked to be convergent for the known data on the monopole fugacity. These results are further generalized to the SU (N)-case. In particular, it is found that the requirement of convergence of the cumulant expansion establishes a certain upper bound on the number of colors. This bound, expressed in terms of the parameter of the weak-coupling approximation, allows the number of colors to be large enough. Finally, the string tension and the coupling constant of the so-called rigidity term of the confining string are found at arbitrary number of colors.

Confining strings and RG flow in the (2+1)-dimensional Georgi-Glashow model and its SU(3)-generalization

Confining strings and RG flow at finite temperature are investigated in the (2+1)-dimensional Georgi-Glashow model. This is done in the limit when the electric coupling constant is much larger than the square root of mass of the Higgs field, but much smaller than the vacuum expectation value of this field. The modification of the Debye mass of the dual photon with respect to the case when it is considered to be negligibly small compared to the Higgs mass, is found. Analogous modifications of the potential of monopole densities and string coupling constants are found. At finite temperature, the mass of the Higgs field scales according to a novel RG equation. It is checked that in the limit when the original theory is reduced by the RG flow to the 2D XY model, the so-evolved Higgs mass is still much smaller than the squared electric coupling constant. The SU(3)-theory of confining strings is also discussed within the same approximations.

Chicken or the egg; or who ordered the chiral phase transition?

We draw an analogy between the deconfining transition in the (2+1)-dimensional Georgi-Glashow model and the chiral phase transition in (3+1)-dimensional QCD. Based on the detailed analysis of the former we suggest that the chiral symmetry restoration in QCD at high temperature is driven by the thermal ensemble of baryons and antibaryons. The chiral symmetry is restored when roughly half of the volume is occupied by the baryons. Surprisingly enough, even though baryons are rather heavy, a crude estimate for the critical temperature gives ${T}_{c}=180 \mathrm{MeV}.$ In this scenario the binding of the instantons is not the cause but rather a consequence of the chiral symmetry restoration.

Instanton molecules at high temperature - the Georgi-Glashow model and beyond

We show that correlators of some local operators in gauge theories are sensitive to the presence of the instantons even at high temperature where the latter are bound into instanton-anti-instanton "molecules". We calculate correlation functions of such operators in the deconfined phase of the 2+1 dimensional Georgi-Glashow model and discuss analogous quantities in the chirally symmetric phase of QCD. We clarify the mechanism by which the instanton-anti-instanton molecules contribute to the anomaly of axial U(1) at high temperature.

Deconfining phase transition in 2+1 D: the Georgi-Glashow model

We analyze the finite temperature deconfining phase transition in 2+1 dimensional Georgi-Glashow model. We show explicitly that the transition is due to the restoration of the magnetic $Z_2$ symmetry and that it is in the Ising universality class. We find that neglecting effects of the charged $W$ bosons leads to incorrect predictions for the value of the critical temperature and the universality class of the transition, as well as for various correlation functions in the high temperature phase. We derive the effective action for the Polyakov loop in the high temperature phase and calculate the correlation functions of magnetic vortex operators.

Monopole-antimonopole condensation in the interpolating Georgi-Glashow model

We study the three dimensional Georgi-Glashow model (which interpolates smoothly between pure U(1) and SU(2) limits) using a constrained cooling which preserves 't Hooft-Polyakov monopoles. We find that the monopole-antimonopole condensation gives an area law for the Wilson loops. The monopole contribution to the string tension is close to the Monte Carlo value in the intermediate region.

Critical Higgs mass for the (2 + 1) dimensional Georgi-Glashow model

The ground state of a (2+1) dimensional Georgi-Glashow model is discussed in a dilute monopole gas approximation. In addition to the Coulomb interaction of the latter, we include a Higgs force which is attractive irrespective of the sign of monopole charges. For (Higgs masses) −1 comparable to the mean free distance of monopoles we find a phase transition which is presumably of first order. Strings of monopoles dominate the clustering behaviour near the critical Higgs mass.

Physical mechanism for quark confinement via pseudoparticles

We investigate in detail the physical mechanism whereby confinement is achieved in the (2 + 1) -dimensional Georgi-Glashow model through a medium of pseudoparticles. The arguments are semiclassical and avoid using the path-integral formalism. We show that vacuum fluctuation produces circular vortexlike electric field lines, which change the classical electric field between two fixed charges into a stringlike confinement field. We also show that when the magnitude of the fixed charges is sufficiently large the string splits longitudinally into two halves, which move away from each other. Finally, we comment on how confinement may be achieved in Yang-Mills theory in 3 + 1 dimensions.