Background Gauge Field Research Articles

Effects of the Chemical Potential in two-dimensional Quantum Field Theories

In this talk we study the effects of a nonzero chemical potential in ( 1 + 1 ) dimensions quantum field models at finite temperature. We start by considering massless fermions in an abelian gauge field background and calculate the n -point amplitudes using the real time formalism. Our calculation shows that the chiral anomaly is unaffected by the presence of a chemical potential at finite temperature. We also find that retarded amplitudes vanish. We then consider the imaginary time formalism and find that the two- and three-point functions vanish, this result being consistent with the real time calculations.

The universal extra dimensional model with [formula omitted] extra-space

We propose a new universal extra dimensional model that is defined on a six-dimensional spacetime which has a two-sphere orbifold S 2 / Z 2 as an extra-space. We specify our model by choosing the gauge symmetry as SU ( 3 ) × SU ( 2 ) × U ( 1 ) Y × U ( 1 ) X , introducing field contents in six dimensions as their zero modes correspond to the standard model particles, and determining a boundary condition of these fields on orbifold S 2 / Z 2 . A background gauge field that belongs to U ( 1 ) X is introduced there, which is necessary to obtain massless chiral fermions in four-dimensional spacetime. We then analyze Kaluza–Klein (KK) mode expansion of the fields in our model and derive the mass spectrum of the KK particles. We find that the lightest KK particles are the 1st KK particles of massless gauge bosons at tree level. We also discuss the KK parity of the KK modes in our model and confirm the stability of the lightest KK particle which is important for dark matter physics.

Noncommutative 𝒩 = 1 super Yang-Mills, the Seiberg-Witten map and UV divergences

Classically, the dual under the Seiberg-Witten map of noncommutative U(N), {\cal N}=1 super Yang-Mills theory is a field theory with ordinary gauge symmetry whose fields carry, however, a \theta-deformed nonlinear realisation of the {\cal N}=1 supersymmetry algebra in four dimensions. For the latter theory we work out at one-loop and first order in the noncommutative parameter matrix \theta^{\mu\nu} the UV divergent part of its effective action in the background-field gauge, and, for N>=2, we show that for finite values of N the gauge sector fails to be renormalisable; however, in the large N limit the full theory is renormalisable, in keeping with the expectations raised by the quantum behaviour of the theory's noncommutative classical dual. We also obtain --for N>=3, the case with N=2 being trivial-- the UV divergent part of the effective action of the SU(N) noncommutative theory in the enveloping-algebra formalism that is obtained from the previous ordinary U(N) theory by removing the U(1) degrees of freedom. This noncommutative SU(N) theory is also renormalisable.

Dimensional reduction and vacuum structure of quiver gauge theory

We describe the structure of the vacuum states of quiver gauge theories obtained via dimensional reduction over homogeneous spaces, in the explicit example of SU(3)-equivariant dimensional reduction of Yang-Mills-Dirac theory on manifolds of the form M x CP(2). We pay particular attention to the role of topology of background gauge fields on the internal coset spaces, in this case U(1) magnetic monopoles and SU(2) instantons on CP(2). The reduction of Yang-Mills theory induces a quiver gauge theory involving coupled Yang-Mills-Higgs systems on M with a Higgs potential leading to dynamical symmetry breaking. The criterion for a ground state of the Higgs potential can be written as the vanishing of a non-abelian Yang-Mills flux on the quiver diagram, regarded as a lattice with group elements attached to the links. The reduction of SU(3)-symmetric fermions yields Dirac fermions on M transforming under the low-energy gauge group with Yukawa couplings. The fermionic zero modes on CP(2) yield exactly massless chiral fermions on M, though there is a unique choice of spin(c) structure on CP(2) for which some of the zero modes can acquire masses through Yukawa interactions. We work out the spontaneous symmetry breaking patterns and determine the complete physical particle spectrum in a number of explicit examples, some of which possess quantum number assignments qualitatively analogous to the manner in which vector bosons, quarks and leptons acquire masses in the standard model.

Topological aspects of fermions on a honeycomb lattice

We formulate a model of relativistic fermions moving in two Euclidean dimensions based on a tight-binding model of graphene. The eigenvalue spectrum of the resulting Dirac operator is solved numerically in smooth U(1) gauge field backgrounds carrying an integer-valued topological charge Q, and it is demonstrated that the resulting number of zero-eigenvalue modes is in accord with the Atiyah-Singer index theorem applied to two continuum flavors. A bilinear but gauge non-invariant chirality operator appropriate for distinguishing the topological zero modes is identified. When this operator is used to calculate Q, it is found that the maximum topological charge capable of being measured in this fashion scales with the perimeter of the lattice. Some concluding remarks compare these results to what is known for staggered lattice fermions.

Inner brane: A D3-brane in Nappi-Witten space from an inner automorphism

Wess-Zumino-Witten (WZW) models are abstract conformal field theories with an infinite-dimensional symmetry which accounts for their integrability, and at the same time they have a sigma-model description of closed-string propagation on group manifolds which, in turn, endows the models with an intuitive geometric meaning. We exploit this dual algebraic and geometric property of WZW models to construct an explicit example of a field-dependent reflection matrix for open strings in the Nappi-Witten model. Demanding the momentum outflow at the boundary to be zero determines a certain combination of the left and right chiral currents at the boundary. This same reflection matrix is obtained algebraically from an inner automorphism, giving rise to a space-filling D-brane. Half of the infinite-dimensional affine Kac-Moody symmetry present in the closed-string theory is preserved by this unique combination of the left and the right chiral currents. The operator-product expansions of these boundary currents are computed explicitly and they are shown to obey the same current algebra as those of the closed-string chiral currents. Different choices of the inner automorphisms correspond to different background gauge field configurations. Only those B-field configurations, and the corresponding D-branes, that preserve the diagonal part of the infinite-dimensional chiral algebras are allowed. In this way the existence of the D-branes in curved spaces is further constrained by the underlying symmetry of the ambient spacetime.

Holographic melting of heavy baryons in plasma with gluon condensation

We propose a Dirac-Born-Infeld (DBI) D5 vertex brane plus Nc fundamental strings configuration to describe a baryon probe in strongly coupled gauge theory with gluon condensation at finite temperature via AdS/CFT correspondence. We investigate properties of this configuration in a dilaton deformed AdS5 × S5 background, in which IIB string theory is dual to super Yang-Mills theory in a state with a constant self-dual gauge field (Fmn = F*mn) background. We find that for most values of temperature T and gluon condensation parameter q (q = π2⟨FmnFmn⟩), there always exists a screening length Ls. The relation Ls ∼ 1/T has been checked. We give the q dependence of Ls. We calculate the boost velocity v(v = −tanh η) and angular velocity ω dependence of Ls for a baryon probe, and obtain Ls = L0*(1−v2)1/4 for large v and Ls ∼ ω−1, which are consistent with those dependence relations in the point brane plus strings case, and find that the usual relations have been modified by q. We also calculate the mass of baryon and find T dependence of baryon mass.

Automatic generation of Feynman rules in the Schrödinger functional

We provide an algorithm to generate vertices for the Schrödinger functional with an abelian background gauge field. The background field has a non-trivial color structure, therefore we mainly focus on a manipulation of the color matrix part. We propose how to implement the algorithm especially in python code. By using python outputs produced by the code, we also show how to write a numerical expression of vertices in the time-momentum as well as the coordinate space into a Feynman diagram calculation code. As examples of the applications of the algorithm, we provide some one-loop results, ratios of the Λ parameters between the plaquette gauge action and the improved gauge actions composed from six-link loops (rectangular, chair and parallelogram), the determination of the O( a) boundary counter term to this order, and the perturbative cutoff effects of the step scaling function of the Schrödinger functional coupling constant.

Non-Abelian structures in compactifications of M-theory on seven-manifolds with SU(3) structure

We study M-theory compactified on a specific class of seven-dimensional manifolds with SU(3) structure. The manifolds can be viewed as a fibration of an arbitrary Calabi-Yau threefold over a circle, with a U-duality twist around the circle. In some cases we find that in the four dimensional low energy effective theory a (broken) non-Abelian gauge group appears. Furthermore, such compactifications are shown to be dual to previously analyzed compactifications of the heterotic string on K3 x T^2, with background gauge field fluxes on the T^2.

Exact effective action for(1+1)-dimensional fermions in an Abelian background at finite temperature and chemical potential

In this paper we study the effects of a nonzero chemical potential in $(1+1)$-dimensional quantum field models at finite temperature. We particularly consider massless fermions in an Abelian gauge field background and calculate the effective action by evaluating the $n$-point functions. We find that the structure of the amplitudes corresponds to a generalization of the structure noted earlier in a calculation without a chemical potential (the associated integrals carry the dependence on the chemical potential). Our calculation shows that the chiral anomaly is unaffected by the presence of a chemical potential at finite temperature. However, unlike in the absence of a chemical potential, odd point functions do not vanish. We trace this to the fact that in the presence of a chemical potential the generalized charge conjugation symmetry of the theory allows for such amplitudes. In fact, we find that all the even point functions are even functions of $\ensuremath{\mu}$, while the odd point functions are odd functions of $\ensuremath{\mu}$ which is consistent with this generalized charge conjugation symmetry. We show that the origin of the structure of the amplitudes is best seen from a formulation of the theory in terms of left- and right-handed spinors. The calculations are also much simpler in this formulation and it clarifies many other aspects of the theory.

Heat kernel expansion in the covariant perturbation theory

Working within the framework of the covariant perturbation theory, we obtain the coincidence limit of the heat kernel of an elliptic second order differential operator that is applicable to a large class of quantum field theories. The basis of tensor invariants of the curvatures of a gravity and gauge field background, to the second order, is derived, and the form factors acting on them are obtained in two integral representations. The results are verified by the functional trace operation, by the short proper time (Schwinger–DeWitt) expansions, as well as by the computation of the Green function for the two-dimensional scalar field model.

Flux Hamiltonians, Lie algebras, and root lattices with minuscule decorations

We study a family of Hamiltonians of fermions hopping on a set of lattices in the presence of a background gauge field. The lattices are constructed by decorating the root lattices of various Lie algebras with their minuscule representations. The Hamiltonians are, in momentum space, themselves elements of the Lie algebras in these same representations. We describe various interesting aspects of the spectra, which exhibit a family resemblance to the Dirac spectrum, and in many cases are able to relate them to known facts about the relevant Lie algebras. Interestingly, various realizable lattices such as the kagomé and pyrochlore can be given this Lie algebraic interpretation, and the particular flux Hamiltonians arise as mean-field Hamiltonians for spin-1/2 Heisenberg models on these lattices.

Particle dynamics and emergent gravity

The emergent gravity proposal is examined within the framework of non-commutative QED/gravity correspondence from particle dynamics point of view. In particular it is demonstrated how a non-commutative U(1) background gauge field can be interpreted as a curved background, as far as the quantum and classical dynamics of a charged particle are concerned.

Polarization tensor of charged gluons in color magnetic background field at finite temperature

We calculate the polarization tensor of charged gluons in a Abelian homogeneous magnetic background field at finite temperature in one loop order Lorentz background field gauge in full generality. Thereby we first determine the ten independent tensor structures. For the calculation of the corresponding form factors we use the Schwinger representation and represent form factors as double parametric integrals and a sum resulting from the Matsubara formalism used. The integrands are given explicitly in terms of hyperbolic trigonometric functions. Like in the case of neutral gluons, the polarization tensor is not transversal. Out of the tensor structures, seven are transversal and three are not. The nontransversal part follows explicitly from our calculations.

The Laplacian on homogeneous spaces

The solution of the eigenvalue problem of the Laplacian on a general homogeneous space G∕H is given. Here, G is a compact, semisimple Lie group, H is a closed subgroup of G, and the rank of H is equal to the rank of G. It is shown that the multiplicity of the lowest eigenvalue of the Laplacian on G∕H is just the degeneracy of the lowest Landau level for a particle moving on G∕H in the presence of the background gauge field. Moreover, the eigenspace of the lowest eigenvalue of the Laplacian on G∕H is, up to a sign, equal to the G-equivariant index of the Kostant’s Dirac operator on G∕H.

Renormalized effective actions in radially symmetric backgrounds: Exact calculations versus approximation methods

Our previously developed calculational method (the partial-wave cutoff method) is employed to evaluate explicitly scalar one-loop effective actions in a class of radially symmetric background gauge fields. Our method proves to be particularly effective when it is used in conjunction with a systematic WKB series for the large partial-wave contribution to the effective action. By comparing these numerically exact calculations against the predictions based on the large-mass expansion and derivative expansion, we discuss the validity ranges of the latter approximation methods.

On gauge invariance of Wilsonian effective actions in (supersymmetric) gauge theories

We give a detailed critical discussion of the properties of Wilsonian effective actions Γ μ , defined by integrating out all modes above a given scale μ . In particular, we provide a precise and relatively convenient prescription how to implement the infrared cutoff μ in any loop integral that is manifestly Lorentz invariant and also preserves global linear symmetries such as e.g. supersymmetry. We discuss the issue of gauge invariance of effective actions in general and in particular when using background field gauge. Our prescription for the IR cutoff (as any such prescription) breaks the gauge symmetry. Using our prescription, we have explicitly computed, at one loop, many terms of the Wilsonian effective action for general gauge theories, involving bosonic and fermionic matter fields of arbitrary masses and in arbitrary representations, exhibiting the non-gauge invariant (as well as the gauge invariant) terms. However, for supersymmetric gauge theories all non-gauge invariant terms cancel within each supermultiplet. This is strong evidence that in supersymmetric gauge theories this indeed defines a Lorentz, susy and gauge invariant Wilsonian effective action. As a byproduct, we obtain the explicit one-loop Wilsonian couplings for all higher-derivative terms ∼ FD 2 n F in the effective action of arbitrary supersymmetric gauge theories.

Lattice chirality and the decoupling of mirror fermions

We show, using exact lattice chirality, that partition functions of lattice gauge theories with vectorlike fermion representations can be split into ``light and ``mirror parts, such that the ``light and ``mirror representations are chiral. The splitting of the full partition function into ``light and ``mirror is well defined only if the two sectors are separately anomaly free. We show that only then is the generating functional, and hence the spectrum, of the mirror theory a smooth function of the gauge field background. This explains how ideas to use additional non-gauge, high-scale mirror-sector dynamics to decouple the mirror fermions without breaking the gauge symmetry?for example, in symmetric phases at strong mirror Yukawa coupling?are forced to respect the anomaly-free condition when combined with the exact lattice chiral symmetry. Our results are also useful in explaining a paradox posed by a recent numerical study of the mirror-fermion spectrum in a toy would-be-anomalous two-dimensional theory. In passing, we prove some general properties of the partition functions of arbitrary chiral theories on the lattice that should be of interest for further studies in this field.

Neutral gluon polarization tensor in a color magnetic background at finite temperature

In the framework of SU(2) gluodynamics, we derive the tensor structure of the neutral gluon polarization tensor in an Abelian homogeneous magnetic field at finite temperature and calculate it in the one-loop approximation in the Lorentz background field gauge. The imaginary time formalism and the Schwinger operator method are used. The latter is extended to the finite temperature case. The polarization tensor turns out to be nontransversal. It can be written as a sum of ten tensor structures with corresponding form factors. Seven tensor structures are transversal; three are not. We represent the form factors in terms of double-parametric integrals and the temperature sum which can be computed numerically. As applications we calculate the Debye mass and the magnetic mass of neutral gluons in the background field at high temperature. A comparison with the results of other authors is done.

Publisher’s Note: Mass zeros of the fermionic determinant in four-dimensional quantum electrodynamics [Phys. Rev. D75, 065002 (2007)

The Euclidean fermionic determinant in four-dimensional quantum electrodynamics is considered as a function of the fermionic mass for a class of $O(2)\times O(3)$ symmetric background gauge fields. These fields result in a determinant free of all cutoffs. Consider the one-loop effective action, the logarithm of the determinant, and subtract off the renormalization dependent second-order term. Suppose the small-mass behavior of this remainder is fully determined by the chiral anomaly. Then either the remainder vanishes at least once as the fermionic mass is varied in the interval $0 < m < \infty$ or it reduces to its fourth-order value in which case the new remainder, obtained after subtracting the fourth-order term, vanishes at least once. Which possibility is chosen depends on the sign of simple integrals involving the field strength tensor and its dual.