Topology of the gauge condition and new confinement phases in non-abelian gauge theories

The gauge-fixing constraint in a gauge field theory is crucial for understanding both short-distance and long-distance behavior of non-abelian gauge field theories. We define what we call “non-propagating” gauge conditions such as the unitary gauge and “approximately non-propagating” or renormalizable gauge conditions, and study their topological properties. By first fixing the non-abelian part of the gauge ambiguity we find that SU(N) gauge theories can be written in the form of abelian gauge theories with N ? 1 fold multiplicity enriched with magnetic monopoles with certain magnetic charge combinations. Their electric chargesare governed by the instanton angle θ.If θ is continuously varied from 0 to 2π and a confinement mode is assumed for some θ, then at least one phase-transition must occur. We speculate on the possibility of new phases: e.g., “oblique confinement,” where $\text{θ ? π}$, and explain some peculiar features of this mode. In principle there may be infinitely many such modes, all separated by phase transition boundaries.     

Plaquette formulation and the Bianchi identity for lattice gauge theories

We extend Halpern's field-strength formulation and dual potentials (for continuum gauge theories) to abelian and non-abelian lattice gauge theories. New results include: (i) plaquette formulation of all lattice gauge theories, (ii) the strong coupling expansion is seen as (a) a perturbation in dual links or (b) a gradual restoration of the lattice Bianchi identity. To leading order in the strong coupling expansion the lattice Bianchi identity is completely ignored. Geometrical interpretation of the lattice Bianchi identity is presented along with a discussion of the “abelianization” of the non-abelian identity and its connection with gauge-invariant variables. For abelian theories we also show that the dual potential is Fourier conjugate to the Bianchi identity and that the Coulomb gas representation of these theories is easily obtained in this formulation.     

Composite description for quarks and leptons from a geometrical picture of complementarity

We propose a geometrical formulation of complementarity between the Higgs phase and the confining phase of a non-abelian gauge theory, introducing local “vierbein” fields to provide for a one-to-one mapping between the states of the two phases. We investigate the gauge structure when a subgroup of the global symmetry is also gauged. The formalism is used to build up a composite model based on a SU(4) of subcolor complementary to a Pati-Salam SU(4), and allowing for dynamical symmetry breaking of SU(2)_L ? U(1).     

Approaches to a non-abelian antisymmetric tensor gauge field theory

Two distinct attempts at constructing a theory of non-abelian antisymmetric tensor gauge fields (ATGF's) are considered. First, a recently proposed geometry of abelian ATGF's is reviewed and then generalized to the non-abelian case. The resulting geometric action is non-local and is invariant under non-local gauge transformations; in the local limit the action describes free fields. Lattice actions for both the abelian and non-abelian ATGF theories are also presented. In the second approach, a lattice action for non-abelian ATGF's is constructed using a plaquette variables that carry four internal indices. The continuum limit is also a non-interacting theory.     

A test for the gluon selfcoupling in the reactionse + e −→4 jets andZ 0→4 jets

We consider the reactionse ⁺ e ^?→γ^*→4 jets andZ ^o→4 jets with the 4 jets coming in two pairs of essentially back to back jets of high and low energy. We calculate the angular distribution of the low energy jet axis with respect to the high energy jet axis in QCD, in an abelian gluon model “QED” and a phase space model (PS). Using simple helicity arguments we show that our angular distribution is very sensitive to the triple gluon coupling in QCD. This is then confirmed by a complete calculation. Our correlation offers, therefore, a direct test for QCD as a non-abelian gauge theory.     

Cancellation of quadratically divergent mass corrections in globally supersymmetric spontaneously broken gauge theories

The one-loop quadratically divergent mass corrections in globally supersymmetric gauge theories with spontaneously broken abelian and non-abelian gauge symmetry are studied. Quadratically divergent mass corrections are found to persist in an abelian model with an ABJ anomaly. However, additional supermultiplets necessary to cancel the ABJ anomaly, turn out to be sufficient to eliminate the quadratic divergences as well, rendering the theory natural. Quadratic divergences are shown to vanish also in the case of an anomaly free model with spontaneously broken non-abelian gauge symmetry.     

A study of 't Hooft and Wilson loops

An investigation is undertaken for 't Hooft loop operators in four-dimensional gauge theories. For the first time, a perimeter law is shown to be their behavior in weak coupling Wilson lattice (and continuum) non-abelian SU(N) gauge theories for all N. However, it is also argued that this perimeter law is poor criterion for quark confinement. Rather, it is suggested that non-leading long-distance behavior is what is crucial and relevant in distinguishing non-abelian from abelian (and hence confining from non-confining) theories. A new object, “the 't Hooft line”, is introduced to measure this non-leading behavior and is computed in strong coupling on the lattice. There, one finds magnetic screening characterized by a magnetic screening mass, m_s. It is shown to all orders in strong coupling that m_s is the glueball mass, a result which is expected to persist in weak coupling and in the continuum. Two further consequences of this work are that pure non-abelian gauge theories cannot be in a Higgs phase and that in such models that absence of massless physical particles implies confinement.Finally, non-leading behavior in Wilson loops is examined. The present picture of confinement suggests the absence of van der Waals forces in Yang-Mills theories.     

Gauge anomaly cancellation in chiral gauge theories

We consider chiral fermions interacting minimally with abelian and non-abelian gauge fields. Using a path integral approach and exploring the consequences of a mechanism of symmetry restoration, we show that the gauge anomaly has null expectation value in the vacuum for both cases (abelian and non-abelian). We argue that the same mechanism has no possibility to cancel the chiral anomaly, what eliminates competition between chiral and gauge symmetry at full quantum level. We also show that the insertion of the gauge anomaly in arbitrary gauge invariant correlators gives a null result, which points towards anomaly cancellation in the subspace of physical state vectors.     

Gauging of chiral bosonized actions

We present bosonic actions which are equivalent to various chiral fermion theories. For the case of one chiral fermion coupled to an abelian gauge field, we present two bosonized actions, one corresponding to regularizing in the vector conserving scheme and the other in the left-right scheme. We then propose an action for the non-abelian bosonization of Weyl fermions which is a WZW action coupled to a fixed curved background. The chiral WZW action is then coupled to non-abelian gauge fields. We derive the anomalies of the axial current (in the vector conserving scheme) and the left-right currents (in the left-right regularization scheme), both for the abelian and non-abelian bosonized actions. The expressions for the anomalies are identical to those derived in the corresponding fermionic theories.     

Spherically symmetric classical solutions in SU(2) gauge theory with a Higgs field

A consistent ansatz for time dependent classical solutions in an SU(2) gauge theory with a doublet Higgs field is presented. The (3+1)-dimensional field equations are reduced to those of an effective (1+1)-dimensional theory. This ansatz describes solutions which travel between topologically distinct classical vacua of the non-abelian gauge theory. The real time version of these solutions describes the creation and decay of the unstable static “sphaleron”, the imaginary time version describes a euclidean instanton.     

Non-abelian Gauge Symmetry for Fields in Phase Space: a Realization of the Seiberg-Witten Non-abelian Gauge Theory

The Seiberg-Witten formalism has been realized as an electrodynamics in phase space (associated to the Dirac equation written in phase space) and this fact is explored here with non-abelian gauge group. First, a physically heuristic presentation of the Seiberg-Witten approach is carried out for non-abelian gauge in order to guide the calculation procedures. These results are realized by starting with the Lagrangian density for the free Dirac field in phase space. Then a field strength is derived, where the non-abelian gauge group is the SU(2), corresponding to an isospin (non-abelian) field theory in phase space. An application to nucleon is then discussed.

     

Topological superfluid in a fermionic bilayer optical lattice

In this paper, a topological superfluid phase with Chern number ?? = ±1, possessing gapless edge states and non-Abelian anyonsis designed in a ?? = ±1 topological insulator proximity to ans-wave superfluid on an optical lattice with the effective gauge fieldand layer-dependent Zeeman field coupled to ultracold fermionic atoms’ pseudo spin. Wealso study its topological properties and calculate the phase stiffness by using therandom-phase-approximation approach. Finally we derive the temperature of theKosterlitz-Thouless transition by means of renormalized group theory. Owning to theexistence of non-Abelian anyons, this ?? = ±1 topological superfluid may be a possible candidate fortopological quantum computation.     

Classical gauge theories as dynamical systems—Regularity and chaos

In this review we present the salient features of dynamical chaos in classical gauge theories with spatially homogeneous fields. The chaotic behaviour displayed by both abelian and non-abelian gauge theories and the effect of the Higgs term in both cases are discussed. The role of the Chern-Simons term in these theories is examined in detail. Whereas, in the abelian case, the pure Chern-Simons-Higgs system is integrable, addition of the Maxwell term renders the system chaotic. In contrast, the non-abelian Chern-Simons-Higgs system is chaotic both in the presence and the absence of the Yang-Mills term. We support our conclusions with numerical studies on plots of phase trajectories and Lyapunov exponents. Analytical tests of integrability such as the Painlevé criterion are carried out for these theories. The role of the various terms in the Hamiltonians for the abelian Higgs, Yang-Mills-Higgs and Yang-Mills-Chern-Simons-Higgs systems with spatially homogeneous fields, in determining the nature of order-disorder transitions is highlighted, and the effects are shown to be counter-intuitive in the last-named system.     

Parity anomalies in gauge theories in 2 + 1 dimensions

It is shown that the introduction of massless fermions in an abelian gauge theory in 2 + 1 dimensions does not lead to any parity anomaly despite a non-commutativity of limits in the structure function of the odd part of the vacuum polarisation tensor. However, a parity anomaly does exist in non-abelian theories due to a conflict between gauge invariance under large gauge transformations and the parity symmetry.     

How chiral are type-II superstrings?

We study the type-II superstrings in four dimensions by studying vacua where massless chiral multiplets transform as complex representations of the non-abelian gauge group. We show that the gauge group can only be SU(3) and that such fields transform as 3 of SU(3). However, attempts to obtain the theory with N=1 supergravity fail. It turns out that the “different” constructions via asymmetric orbifolds give the same massless spectrum with necessarily N=2 supergravity.     

Abelian Chern-Simons theory, Stokes’ theorem, and generalized connections

Generalized connections and their calculus have been developed in the context of quantum gravity. Here we apply them to abelian Chern-Simons theory. We derive the expectation values of holonomies in U(1) Chern-Simons theory using Stokes’ theorem, flux operators and generalized connections. A framing of the holonomy loops arises in our construction, and we show how, by choosing natural framings, the resulting expectation values nevertheless define a functional over gauge invariant cylindrical functions.The abelian theory considered in the present article is the test case for our method. It can also be applied to the non-abelian theory. Results will be reported in a companion article.     

On the random vector potential model in two dimensions

The random vector potential model describes massless fermions coupled to a quenched random gauge field. We study its abelian and non-abelian versions. The abelian version can be completely solved using bosonization. We analyse the non-abelian model using its supersymmetric formulation and show, by a perturbative renormalization group computation, that it is asymptotically free at large distances. We also show that all the quenched chiral current correlation functions can be computed exactly, without using the replica trick or the supersymmetric formulation, but using an exact expression for the effective action for any sample of the random gauge field. These chiral correlation functions are purely algebraic.     

On spin three self interactions

In this paper we start the construction of a self interacting massless spin three field theory. The first order self interaction is constructed. The free field gauge invariance is modified by a term of first order in the coupling constant. Finally the abelian gauge transformation of the free Lagrangian is shown to become non-abelian.     

A computation of Tr(−1)F in supersymmetric gauge theories with matter

We introduce “multi-twisted” boundary conditions (MTBC) for gauge theories in a finite volume. MTBC can be applied for a class of gauge theories with non-trivial centers and allow the existence of some matter multiplets which transform non-trivially under the center group. The Witten index for the supersymmetric extension of the gauge theory can be calculated when global abelian symmetries exist and turns out to be non-zero and independent of the matter representations.     

Vector field effective action in the open superstring theory

Using a path integral technique we find a closed expression (to all orders in α′) for the abelian, constant field strenght limit of the (tree) effective action for the massless vector field in the open superstring theory. The result is a modification of the Born-Infeld action found in the Bose string theory case. One-loop correction to the effective action is computed and shown to be finite if the gauge group is SO(32). It is demonstrated how the on-shell superstring scattering amplitudes can be calculated in the path-integral approach. We determine the leading (O(α') and O(α′²)) terms in the full non-abelian effective action starting from the known results for the 3-point and 4-point amplitudes. We find that because of the equivalence theorem the coefficients of some of the invariant structures in the effective lagrangian cannot be fixed from the S-matrix. In the path integral approach this ambiguity manifests itself as a 2d renormalization scheme (and Weyl gauge choice) ambiguity. We also discuss the leading terms in the gravitational effective actions in the closed (super) string theories and point out that whether or not the R² terms form the “Gauss-Bonnet” combination depends on choice of a renormalization (massless exchange subtraction) scheme.