non-Abelian Gauge Field Theory Research Articles

Low-Energy Effective Theories of the 1/2 - Filled Hubbard Model in the Continuum Limit

It has been observed that when the gauge fields are present on the link, fermion propagation is possible in the weak coupling limit due to the dominance of the hopping term, which corresponds to the colour gauge interaction in the lattice QCD formulation. The production of low energy skyrmionic excitation at the fermionic site destroys the underlying antiferromagnetic order. In the continuum limit, the kinetic term in the lattice QCD corresponds to the rearrangement of the fermionic constituents through their propagation within the confined domain of the bound stateconfigurations of the interacting system which gives rise to a running coupling constant leading to asymptotic freedom. When one can assign a colour to a particular quantum number of a fermionic component in a limited state, it shows that QCD may be thought of as a generalised non-Abelian gauge field theory since these degrees of freedom play a part in the restricted area of the system and examines the continuous limit of the Hubbard-like model and the weak coupling limit that results from the abolition of the antiferromagnetic order and fermion propagation. This is equivalent to the non-Abelian color gauge field interaction. It is noted that the generalised spin fluctuation may be linked to the colour gauge field. This formalism's discovery of pseudoscalar Goldstone bosons associated with chiral symmetry breaking is in line with (3+1)D continuum QCD.

Generalized Yang-Mills theory under rotor mechanism

This paper follows the previous work on generalized abelian gauge field theory of higher-order derivatives under rotor model and extends the study to the most generalized non-abelian case. We find that the rotor mechanism from the abelian case applies nicely to the non-abelian case under the Lorentz gauge condition. Under the rotor mechanism, the gauge field transforms as Tμa→□nTμa. When the order of field derivative is n=0, this restores back to the original Yang-Mills action. Our work gives an extensive generalization of the Yang-Mills theory with higher-order field derivatives. We also compute the equation of motion and Noether's current of the generalized non-abelian gauge field theory. Finally, we study the dynamic instability issue of the theory by the Ostrogradsky construction and the analysis of the 00-component of the energy-momentum tensor.

Nonlinearity of the non-Abelian lattice gauge field theory according to the spectrum of Kolmogorov-Sinai entropy and complexity

Abstract The quark-gluon plasma is written by the non-Abelian gauge theory. The dynamics of the gauge SU(2) are given by the Hamiltonian function, which contains the quadratic part of the field strength tensor F μ v a {\rm{F}}_{\mu v}^{\rm{a}} expressed in Minkowski space. The homogeneous Yang-Mills equations are solved on lattice Nd considering classical approximation, which exhibits chaotic dynamics. We research the time-dependent entropy-energy relation, which can be shown by the energy spectrum of Kolmogorov-Sinai entropy and the spectra of the statistical complexity.

Monotonicity of solution to the dark monopole equations in non-Abelian gauge field theory

In this paper, we combine a two-step iterative shooting argument and a fixed-point theorem approach to establish the existence of the solution of the dark monopole equations in a recently formulated non-Abelian gauge field theory by Deglmann and Kneipp and discuss its qualitative properties. In addition, the energy-minimizing solution of the dark monopole equations also satisfies the monotonicity, thus settling an unsettled but characteristic issue left out in an existence theorem regarding such a solution and confirming evidence from earlier numerical simulations.

Higher-rank tensor non-Abelian field theory: Higher-moment or subdimensional polynomial global symmetry, algebraic variety, Noether's theorem, and gauging

With a view toward a fracton theory in condensed matter, we introduce a higher-moment polynomial degree-p global symmetry, acting on complex scalar/vector/tensor fields (e.g., ordinary or vector global symmetry for p$=0$ and p$=1$ respectively). We relate this higher-moment global symmetry of $n$-dimensional space, to a lower degree (either ordinary or higher-moment, e.g., degree-(p-$\ell$)) subdimensional or subsystem global symmetry on layers of $(n-\ell)$-submanifolds. These submanifolds are algebraic affine varieties (i.e., solutions of polynomials). The structure of layers of submanifolds as subvarieties can be studied via mathematical tools of embedding, foliation, and algebraic geometry. We also generalize Noether's theorem for this higher-moment polynomial global symmetry. We can promote the higher-moment global symmetry to a local symmetry, and derive a new family of higher-rank-m symmetric tensor gauge theory by gauging, with m = p$+1$. By further gauging a discrete $\mathbb{Z}_2^C$ charge conjugation (particle-hole) symmetry, we derive a new general class of rank-m tensor non-abelian gauge field theory (the gauge structure is non-commutative thus non-abelian but not an ordinary group): a hybrid class of (symmetric or non-symmetric) higher-rank-m tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879], interplaying between gapless and gapped sectors.

A modification of Faddeev–Popov approach free from Gribov ambiguity

We propose a modified version of the Faddeev–Popov (FP) quantization approach for non-Abelian gauge field theory to avoid Gribov ambiguity. We show that by means of introducing a new method of inserting the correct identity into the Yang–Mills generating functional and considering the identity generated by an integral through a subgroup of the gauge group, the problem of Gribov ambiguity can be removed naturally. Meanwhile by handling the absolute value of the FP determinant with the method introduced by Williams and collaborators, we lift the Jacobian determinant together with the absolute value and obtain a local Lagrangian. The new Lagrangian will have a nilpotent symmetry which can be viewed as an analog of the Becchi–Rouet–Stora–Tyutin symmetry.

Non-Abelian gauged fracton matter field theory: Sigma models, superfluids, and vortices

By gauging a higher-moment polynomial degree global symmetry and a discrete charge conjugation (i.e., particle-hole) symmetry coupled to matter fields (two symmetries mutually non-commutative), we derive a new class of higher-rank tensor non-abelian gauge field theory with dynamically gauged fractonic matter fields: Non-abelian gauged fractons interact with a hybrid class of higher-rank (symmetric or generic non-symmetric) tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879, 1911.01804]. We also apply a quantum phase transition similar to that between insulator v.s. superfluid/superconductivity (U(1) symmetry disordered phase described by a topological gauge theory or a disordered Sigma model v.s. U(1) global/gauge symmetry-breaking ordered phase described by a Sigma model with a U(1) target space underlying Goldstone modes): We can regard our tensor gauge theories as disordered phases, and we transient to their new ordered phases by deriving new Sigma models in continuum field theories. While one low energy theory is captured by degrees of freedom of rotor or scalar modes, another side of low energy theory has vortices and superfluids -- we explore non-abelian vortices (two types of vortices mutually interacting non-commutatively beyond an ordinary group structure) and their Cauchy-Riemann relation.

From Slavnov—Taylor Identities to the Renormalization of Gauge Theories

An important, and highly non-trivial, problem is proving the renormalizability and unitarity of quantized non-Abelian gauge theories. Lee and Zinn-Justin have given the first proof of the renormalizability of non-Abelian gauge theories in the spontaneously broken phase. An essential ingredient in the proof has been the observation, by Slavnov and Taylor, of a non-linear, non-local symmetry of the quantized theory, a direct consequence of Faddeev and Popov’s quantization procedure. After the introduction of non-physical fermions to represent the Faddeev-Popov determinant, this symmetry has led to the Becchi-Rouet-Stora-Tyutin fermionic symmetry of the quantized action and, finally, to the resulting Zinn-Justin equation, which makes it possible to solve the renormalization and unitarity problems in their full generality. For an elementary introduction to the discussion of quantum non-Abelian gauge field theories in the spirit of the article, see, for example, L. D. Faddeev, “Faddeev-Popov ghosts,” Scholarpedia 4 (4), 7389 (2009); A. A. Slavnov, “Slavnov-Taylor identities,” Scholarpedia 3 (10), 7119 (2008); C. M. Becchi and C. Imbimbo, “Becchi-Rouet-Stora-Tyutin symmetry,” Scholarpedia 3 (10), 7135 (2008); J. Zinn-Justin, “Zinn-Justin equation,” Scholarpedia 4 (1), 7120 (2009).

SU(2) non-Abelian gauge field theory in one dimension on digital quantum computers

An improved mapping of one-dimensional SU(2) non-Abelian gauge theory onto qubit degrees of freedom is presented. This new mapping allows for a reduced unphysical Hilbert space. Insensitivity to interactions within this unphysical space is exploited to design more efficient quantum circuits. Local gauge symmetry is used to analytically incorporate the angular momentum alignment, leading to qubit registers encoding the total angular momentum on each link. The results of a multi-plaquette calculation on IBM's quantum hardware are presented.

Solutions to the minimization problem arising in a dark monopole model in gauge field theory

We prove the existence of dark monopole solutions in a recently formulated Yang–Mills–Higgs theory model with technical features similar to the classical monopole problems. The solutions are obtained as energy-minimizing static spherically symmetric field configurations of unit topological charge. We overcome the difficulty of recovering the full set of boundary conditions by a regularization method which may be applied to other more complicated problems concerning monopoles and dyons in non-Abelian gauge field theories. Furthermore we show in a critical coupling situation that an explicit BPS solution may be used to provide energy estimates for non-BPS monopole solutions. Besides, in the limit of infinite Higgs coupling parameter, although no explicit construction is available, we establish an existence and uniqueness result for a monopole solution and obtain its energy bounds.

Domain Wall Solitons Arising in Classical Gauge Field Theories

Domain wall solitons are basic constructs realizing phase transitions in various field-theoretical models and are solutions to some nonlinear ordinary differential equations descending from the corresponding full sets of governing equations in higher dimensions. In this paper, we present a series of domain wall solitons arising in several classical gauge field theory models. In the context of the Abelian gauge field theory, we unveil the surprising result that the solutions may explicitly be constructed, which enriches our knowledge on integrability of the planar Liouville type equations in their one-dimensional limits. In the context of the non-Abelian gauge field theory, we obtain some existence theorems for domain wall solutions arising in the electroweak type theories by developing some methods of calculus of variations formulated as direct and constrained minimization problems over a weighted Sobolev space.

Gauge field theories with Lorentz-violating operators of arbitrary dimension

The classification of Lorentz- and CPT-violating operators in nonabelian gauge field theories is performed. We construct all gauge-invariant terms describing propagation and interaction in the action for fermions and gauge fields. Restrictions to the abelian, Lorentz-invariant, and isotropic limits are presented. We provide two illustrative applications of the results to quantum electrodynamics and quantum chromodynamics. First constraints on nonlinear Lorentz-violating effects in electrodynamics are obtained using data from experiments on photon-photon scattering, and corrections from nonminimal Lorentz and CPT violation to the cross section for deep inelastic scattering are derived.

Stable, finite energy density solutions in the effective theory of non-abelian gauge fields

We consider the gauge fixed partition function of pure SU(Nc) gauge theory in axial gauge following the field strength formalism. We integrate over 3(Nc2−1) field strengths using the Bianchi identities and obtain an effective action of the remaining 3(Nc2−1) field strengths in momentum space. We obtain the static solutions of the equations of motion (EOM) of the effective theory. The solutions exhibit Gaussian nature in the z component of momentum and are proportional to delta functions of the remaining components of momentum. The solutions render a stable, finite energy density in the strong coupling region and the parameters of the solutions are proportional to fourth root of the gluon condensate, which acts as a natural mass scale in the low energy phase of the theory.

A possibility to describe models of massive non-Abelian gauge fields in the framework of a renormalizable theory

Renormalizable theory of massive nonabelian gauge fields, which does not require the existence of observable scalar fields is proposed.

Pure gauge spin-orbit couplings

Planar systems with a general linear spin-orbit interaction (SOI) that can be cast in the form of a non-Abelian pure gauge field are investigated using the language of non-Abelian gauge field theory. A special class of these fields that, though a $2\times2$ matrix, are Abelian are seen to emerge and their general form is given. It is shown that the unitary transformation that gauges away these fields induces at the same time a rotation on the wavefunction about a fixed axis but with a space-dependent angle, both of which being characteristics of the SOI involved.The experimentally important case of equal-strength Rashba and Dresselhaus SOI (R+D SOI) is shown to fall within this special class of Abelian gauge fields, and the phenomenon of Persistent Spin Helix (PSH) that emerges in the presence of this latter SOI in a plane is shown to fit naturally within the general formalism developed. The general formalism is also extended to the case of a particle confined to a ring. It is shown that the Hamiltonian on a ring in the presence of equal-strength R+D SOI is unitarily equivalent to that of a particle subject to only a spin-independent but $\theta$-dependent potential with the unitary transformation relating the two being again the space-dependent rotation operator characteristic of R+D SOI.

Discovery of the first Yang–Mills gauge particle — The gluon

This article “Discovery of the First Yang–Mills Gauge Particle — The Gluon” is dedicated to Professor Chen Ning Yang. The Gluon is the first Yang–Mills non-Abelian gauge particle discovered experimentally. The Yang–Mills non-Abelian gauge field theory was proposed by Yang and Mills in 1954, sixty years ago. The experimental discovery of the first Yang–Mills non-Abelian gauge particle — the gluon — in the spring of 1979 is summarized, together with some of the subsequent developments, including the role of the gluon in the recent discovery of the Higgs particle.

Black magic session of concordance: Regge mass spectrum from Casson’s invariant

Recently, there had been a great deal of interest in obtaining and describing all kinds of knots in links in hydrodynamics, electrodynamics, non-Abelian gauge field theories and gravity. Although knots and links are observables of the Chern–Simons (CS) functional, the dynamical conditions for their generation lie outside the scope of the CS theory. The nontriviality of dynamical generation of knotted structures is caused by the fact that the complements of all knots/links, say, in S3are 3-manifolds which have positive, negative or zero curvature. The ability to curve the ambient space is thus far attributed to masses. The mass theorem of general relativity requires the ambient 3-manifolds to be of nonnegative curvature. Recently, we established that, in the absence of boundaries, complements of dynamically generated knots/links are represented by 3-manifolds of nonnegative curvature. This fact opens the possibility to discuss masses in terms of dynamically generated knotted/linked structures. The key tool is the notion of knot/link concordance. The concept of concordance is a specialization of the concept of cobordism to knots and links. The logic of implementation of the concordance concept to physical masses results in new interpretation of Casson’s surgery formula in terms of the Regge trajectories. The latest thoroughly examined Chew–Frautschi (CF) plots associated with these trajectories demonstrate that the hadron mass spectrum for both mesons and baryons is nicely described by the data on the corresponding CF plots. The physics behind Casson’s surgery formula is similar but not identical to that described purely phenomenologically by Keith Moffatt in 1990. The developed topological treatment is fully consistent with available rigorous mathematical and experimentally observed results related to physics of hadrons.

Slavnov-Taylor identities, non-commutative gauge theories and infrared divergences

Abstract In this work we clarify some properties of the one-loop IR divergences in nonAbelian gauge field theories on non-commutative 4-dimensional Moyal space. Additionally, we derive the tree-level Slavnov-Taylor identities relating the two, three and four point functions, and verify their consistency with the divergent one-loop level results. We also discuss the special case of two dimensions.

Multiple Vortices in the Aharony–Bergman–Jafferis–Maldacena Model

Vortices in non-Abelian gauge field theory play important roles in confinement mechanism and are governed by systems of nonlinear elliptic equations of complicated structures. In this paper, we present a series of existence and uniqueness theorems for multiple vortex solutions of the BPS vortex equations, arising in the dual-layered Chern–Simons field theory developed by Aharony, Bergman, Jafferis, and Maldacena, over \({\mathbb{R}^2}\) and on a doubly periodic domain. In the full-plane setting, we show that the solution realizing a prescribed distribution of vortices exists and is unique. In the compact setting, we show that a solution realizing n prescribed vortices exists over a doubly periodic domain \({\Omega}\) if and only if the condition $$n < \frac{\lambda |\Omega|}{2 \pi}$$holds, where \({\lambda >0 }\) is the Higgs coupling constant. In this case, if a solution exists, it must be unique. Our methods are based on calculus of variations.

Topological Quantum Liquids with Quaternion Non-Abelian Statistics

Noncollinear magnetic order is typically characterized by a tetrad ground state manifold (GSM) of three perpendicular vectors or nematic directors. We study three types of tetrad orders in two spatial dimensions, whose GSMs are SO(3) = S(3)/Z(2), S(3)/Z(4), and S(3)/Q(8), respectively. Q(8) denotes the non-Abelian quaternion group with eight elements. We demonstrate that after quantum disordering these three types of tetrad orders, the systems enter fully gapped liquid phases described by Z(2), Z(4), and non-Abelian quaternion gauge field theories, respectively. The latter case realizes Kitaev's non-Abelian toric code in terms of a rather simple spin-1 SU(2) quantum magnet. This non-Abelian topological phase possesses a 22-fold ground state degeneracy on the torus arising from the 22 representations of the Drinfeld double of Q(8).