Stueckelberg Formalism Research Articles

Pole-skipping for massive fields and the Stueckelberg formalism

Pole-skipping refers to the special phenomenon that the pole and the zero of a retarded two-point Green’s function coincide at certain points in momentum space. We study the pole-skipping phenomenon in holographic Green’s functions of boundary operators that are dual to massive p-form fields and the dRGT massive gravitational fields in the AdS black hole background. Pole-skipping points for these systems are computed using the near horizon method. The relation between the pole-skipping points of massive fields and their massless counterparts is revealed. In particular, as the field mass m is varied from zero to non-zero, the pole-skipping phenomenon undergoes an abrupt change with doubled pole-skipping points found in the massive case. This arises from the breaking of gauge invariance due to the mass term and the consequent appearance of more degrees of freedom. We recover the gauge invariance using the Stueckelberg formalism by introducing auxiliary dynamical fields. The extra pole-skipping points are identified to be associated with the Stueckelberg fields. We also observe that, as the mass varies, some pole-skipping points of the wave number q may move from a non-physical region with complex q to a physical region with real q.

Feebly coupled vector boson dark matter in effective theory

A model of dark matter (DM) that communicates with the Standard Model (SM) exclusively through suppressed dimension five operator is discussed. The SM is augmented with a symmetry U(1)X ⊗ Z2, where U(1)X is gauged and broken spontaneously by a very heavy decoupled scalar. The massive U(1)X vector boson (Xμ) is stabilized being odd under unbroken Z2 and therefore may contribute as the DM component of the universe. Dark sector field strength tensor Xμν couples to the SM hypercharge tensor Bμν via the presence of a heavier Z2 odd real scalar Φ, i.e. 1/Λ XμνBμνΦ, with Λ being a scale of new physics. The freeze-in production of the vector boson dark matter feebly coupled to the SM is advocated in this analysis. Limitations of the so-called UV freeze-in mechanism that emerge when the maximum reheat temperature TRH drops down close to the scale of DM mass are discussed. The parameter space of the model consistent with the observed DM abundance is determined. The model easily and naturally avoids both direct and indirect DM searches. Possibility for detection at the Large Hadron Collider (LHC) is also considered. A Stueckelberg formulation of the model is derived.

Spin-2 Twisted Duality in (A)dS

Starting from the dual Lagrangians recently obtained for (partially) massless spin-2 fields in the Stueckelberg formulation, we write the equations of motion for (partially) massless gravitons in (A)dS in the form of twisted-duality relations. In both cases, the latter admit a smooth flat limit. In the massless case, this limit reproduces the gravitational twisted-duality relations previously known for Minkowski spacetime. In the partially-massless case, our twisted-duality relations preserve the number of degrees of freedom in the flat limit, in the sense that they split into a decoupled pair of dualities for spin-1 and spin-2 fields. Our results apply to spacetimes of any dimension greater than three. In four dimensions, the twisted-duality relations for partially massless fields that appeared in the literature are recovered by gauging away the Stueckelberg field.

Consistent deformations of free massive field theories in the Stueckelberg formulation

Cohomological techniques within the Batalin-Vilkovisky (BV) extension of the Becchi-Rouet-Stora-Tyutin (BRST) formalism have proved invaluable for classifying consistent deformations of gauge theories. In this work we investigate the application of this idea to massive field theories in the Stueckelberg formulation. Starting with a collection of free massive vectors, we show that the cohomological method reproduces the cubic and quartic vertices of massive Yang-Mills theory. In the same way, taking a Fierz-Pauli graviton on a maximally symmetric space as the starting point, we are able to recover the consistent cubic vertices of nonlinear massive gravity. The formalism further sheds light on the characterization of Stueckelberg gauge theories, by demonstrating for instance that the gauge algebra of such models is necessarily Abelian and that they admit a Born-Infeld-like formulation in which the action is simply a combination of the gauge-invariant structures of the free theory.

On a renormalizable class of gauge fixings for the gauge invariant operator [formula omitted

The dimension two gauge invariant non-local operator Amin2, obtained through the minimization of ∫d4xA2 along the gauge orbit, allows to introduce a non-local gauge invariant configuration Aμh which can be employed to built up a class of Euclidean massive Yang–Mills models useful to investigate non-perturbative infrared effects of confining theories. A fully local setup for both Amin2 and Aμh can be achieved, resulting in a local and BRST invariant action which shares similarities with the Stueckelberg formalism. Though, unlike the case of the Stueckelberg action, the use of Amin2 gives rise to an all orders renormalizable action, a feature which will be illustrated by means of a class of covariant gauge fixings which, as much as ’t Hooft’s Rζ-gauge of spontaneously broken gauge theories, provide a mass for the Stueckelberg field.

Neutrino oscillations in Stueckelberg semiclassical relativistic dynamics

The Stueckelberg formulation of a manifestly covariant relativistic classical and quantum mechanics is briefly reviewed and it is shown that in this framework a simple (semiclassical) model exists for the description of neutrino oscillations. The model is shown to be consistent with the field equations and the Lorentz force (developed here without and with spin by canonical methods) for Glashow-Salam-Weinberg type non-Abelian fields interacting with the leptons. We discuss a possible fundamental mechanism, in the context of a relativistic theory of spin for (first quantized) quantum mechanical systems, for CP violation. The model also predicts a possibly small "pull back,", i.e., early arrival of a neutrino beam, for which the neutrino motion is almost everywhere within the light cone, a result which may emerge from future long baseline experiments designed to investigate neutrino transit times with significantly higher accuracy than presently available.

Variant [formula omitted] supersymmetric non-Abelian Proca–Stueckelberg formalism in four dimensions

We present a new (variant) formulation of N=1 supersymmetric compensator mechanism for an arbitrary non-Abelian group in four dimensions. We call this ‘variant supersymmetric non-Abelian Proca–Stueckelberg formalism’. Our field content is economical, consisting only of the two multiplets: (i) A non-Abelian vector multiplet (AμI,λI,CμνρI) and (ii) a compensator tensor multiplet (BμνI,χI,φI). The index I is for the adjoint representation of a non-Abelian gauge group. The CμνρI is originally an auxiliary field Hodge-dual to the conventional auxiliary field DI. The φI and BμνI are compensator fields absorbed respectively into the longitudinal components of AμI and CμνρI which become massive. After the absorption, CμνρI becomes no longer auxiliary, but starts propagating as a massive scalar field. We fix all non-trivial cubic interactions in the total Lagrangian, and quadratic interactions in all field equations. The superpartner fermion χI acquires a Dirac mass shared with the gaugino λI. As an independent confirmation, we give the superspace reformulation of the component results.

Helicity decomposition of ghost-free massive gravity

We perform a helicity decomposition in the full Lagrangian of the class of Massive Gravity theories previously proven to be free of the sixth (ghost) degree of freedom via a Hamiltonian analysis. We demonstrate, both with and without the use of nonlinear field redefinitions, that the scale at which the first interactions of the helicity-zero mode come in is $\Lambda_3=(M_Pl m^2)^{1/3}$, and that this is the same scale at which helicity-zero perturbation theory breaks down. We show that the number of propagating helicity modes remains five in the full nonlinear theory with sources. We clarify recent misconceptions in the literature advocating the existence of either a ghost or a breakdown of perturbation theory at the significantly lower energy scales, $\Lambda_5=(M_Pl m^4)^{1/5}$ or $\Lambda_4=(M_Pl m^3)^{1/4}$, which arose because relevant terms in those calculations were overlooked. As an interesting byproduct of our analysis, we show that it is possible to derive the Stueckelberg formalism from the helicity decomposition, without ever invoking diffeomorphism invariance, just from a simple requirement that the kinetic terms of the helicity-two, -one and -zero modes are diagonalized.

Spectral function of a fermion coupled with a massive vector boson at finite temperature in a gauge invariant formalism

We investigate spectral properties of a fermion coupled with a massive gauge boson with a mass $m$ at finite temperature ($T$) in the perturbation theory. The massive gauge boson is introduced as a $U(1)$ gauge boson in the Stueckelberg formalism with a gauge parameter $\ensuremath{\alpha}$. We find that the fermion spectral function has a three-peak structure for $T\ensuremath{\sim}m$ irrespective of the choice of the gauge parameter, while it tends to have one faint peak at the origin and two peaks corresponding to the normal fermion and antiplasmino excitations familiar in QED in the hard thermal loop approximation for $T\ensuremath{\gg}m$. We show that our formalism successfully describe the fermion spectral function in the whole $T$ region with the correct high-$T$ limit except for the faint peak at the origin, although some care is needed for choice of the gauge parameter for $T\ensuremath{\gg}m$. We clarify that for $T\ensuremath{\sim}m$, the fermion pole is almost independent of the gauge parameter in the one-loop order, while for $T\ensuremath{\gg}m$, the one-loop analysis is valid only for $\ensuremath{\alpha}\ensuremath{\ll}1/g$ where $g$ is the fermion-boson coupling constant, implying that the one-loop analysis cannot be valid for large gauge parameters as in the unitary gauge.

CFT adapted gauge invariant formulation of massive arbitrary spin fields in AdS

Using Poincaré parametrization of AdS space, we study massive totally symmetric arbitrary spin fields in AdS space of dimension greater than or equal to four. CFT adapted gauge invariant formulation for such fields is developed. Gauge symmetries are realized by using Stueckelberg formulation of massive fields. We demonstrate that the mass parameter, curvature and radial coordinate contributions to the gauge transformation and Lagrangian of the AdS massive fields can be expressed in terms of ladder operators. Three representations for the Lagrangian are discussed. Realization of the global AdS symmetries in the conformal algebra basis is obtained. Modified de Donder gauge leading to simple gauge fixed Lagrangian is found. The modified de Donder gauge leads to decoupled equations of motion which can easily be solved in terms of the Bessel function. New simple representation for gauge invariant Lagrangian of massive (A)dS field in arbitrary coordinates is obtained. Light-cone gauge Lagrangian of massive AdS field is also presented.

CFT adapted gauge invariant formulation of arbitrary spin fields in AdS and modified de Donder gauge

Using Poincaré parametrization of AdS space, we study totally symmetric arbitrary spin massless fields in AdS space of dimension greater than or equal to four. CFT adapted gauge invariant formulation for such fields is developed. Gauge symmetries are realized similarly to the ones of Stueckelberg formulation of massive fields. We demonstrate that the curvature and radial coordinate contributions to the gauge transformation and Lagrangian of the AdS fields can be expressed in terms of ladder operators. Realization of the global AdS symmetries in the conformal algebra basis is obtained. Modified de Donder gauge leading to simple gauge fixed Lagrangian is found. The modified de Donder gauge leads to decoupled equations of motion which can easily be solved in terms of the Bessel function. Interrelations between our approach to the massless AdS fields and the Stueckelberg approach to massive fields in flat space are discussed.

GAUGE-INVARIANT REFORMULATION OF THE VECTOR SCHWINGER MODEL WITH A PHOTON MASS TERM AND ITS HAMILTONIAN, PATH INTEGRAL AND BRST FORMULATIONS

Using the Stueckelberg formalism, we construct a gauge-invariant version of the vector Schwinger model (VSM) with a photon mass term studied by one of us recently. This model describes two-dimensional massive electrodynamics with massless fermions, where the left-handed and right-handed fermions are coupled to the electromagnetic field with equal couplings. This model describing the 2D massive electrodynamics becomes gauge-noninvariant (GNI). This is in contrast to the case of the massless VSM which is a gauge-invariant (GI) theory (as a consequence of demanding the regularization for the theory to be GI). In this work we first construct a GI theory corresponding to this model describing the 2D massive electrodynamics, using the Stueckelberg formalism and then we recover the physical contents of the original GNI theory studied earlier, under some special gauge choice. We then study the Hamiltonian, path integral and BRST formulations of this GI theory under appropriate gauge-fixing. The theory presents a new class of models in the 2D quantum electrodynamics with massless fermions but with a photon mass term.

VECTOR SCHWINGER MODEL WITH A PHOTON MASS TERM: GAUGE-INVARIANT REFORMULATION, OPERATOR SOLUTIONS AND HAMILTONIAN AND PATH INTEGRAL FORMULATIONS

We consider the vector Schwinger model (VSM) describing two-dimensional electrodynamics with massless fermions, where the left-handed and right-handed fermions are coupled to the electromagnetic field with equal couplings, with a mass term for the U(1) gauge field and then study its operator solutions and the Hamiltonian and path integral formulations. We emphasize here that although the VSM has been studied in the literature rather widely but only without a photon mass term (which was a consequence of demanding the regularization for the VSM to be gauge-invariant (GI)). The VSM with a photon mass term is seen to be a gauge-noninvariant (GNI) theory. Using the standard Stueckelberg formalism we then construct a GI theory corresponding to the proposed GNI model. From this reformulated GI theory, we further recover the physical contents of the proposed GNI theory under a very special gauge choice. The theory proposed and studied here presents a new class of models in the two-dimensional quantum electrodynamics with massless fermions but with a photon mass term.

4-VECTOR FIELDS AND INDEFINITE METRICS

We generalize the Stueckelberg formalism in the (1/2, 1/2) representation of the Lorentz Group, see Ref. 1–2. We analize the problem of the mass generation and of the indefinite metrics from the modern viewpoints, see Ref. 3–4. Some relations to other modern-physics models are found.

Thermal gauge boson masses of the electroweak theory in the broken phase

Thermal effects in the broken phase of the electroweak theory are studied in the strongly interacting Higgs boson limit. In that limit and at tree level the bosonic sector of the theory is a gauged non-linear sigma model. The associated one-loop thermal effective action for soft fields is then computed by using the background field method together with the Stueckelberg formalism. This effective action describes thermal corrections to the masses of the gauge bosons W, Z and the photon. It is the proper generalization of the hard thermal effective action of a Yang-Mills theory when there is a Higgs mechanism for a heavy Higgs particle.

Corrections to oblique parameters induced by anomalous vector boson couplings

We study quadratically divergent radiative corrections to the oblique parameters at LEP1 induced by non-standard vector boson self-couplings. We work in the Stueckelberg formalism and regulate the divergences through a gauge-invariant higher derivative scheme. Using consistency arguments together with the data we find a limit on the anomalous magnetic moment Delta kappa of the W-boson, |Delta kappa| <= 0.26.

A U(1) Gauge Theory for Antisymmetric Tensor Fields

We show that a U(1) gauge theory defined in the configuration space for closed p-branes yields the gauge theory of a massless rank-(p + 1) antisymmetric tensor field and the Stueckelberg formalism for a massive vector field.

Integrating out the standard Higgs field in the path integral

We integrate out the Higgs boson in the electroweak standard model at one loop and construct a low-energy effective Lagrangian assuming that the Higgs mass is much larger than the gauge-boson masses. Instead of applying diagrammatical techniques, we integrate out the Higgs boson directly in the path integral, which turns out to be much simpler. By using the background-field method and the Stueckelberg formalism, we directly find a manifestly gauge-invariant result. The heavy-Higgs effects on fermionic couplings are derived, too. At one loop the log M H terms of the heavy-Higgs limit of the electroweak standard model coincide with the UV-divergent terms in the gauged non-linear σ-model, but vertex functions differ in addition by finite constant terms. Finally, the leading Higgs effects to some physical processes are calculated from the effective Lagrangian.

Deriving nondecoupling effects of heavy fields from the path integral: A heavy Higgs field in an SU(2) gauge theory.

We describe a method to remove non-decoupling heavy fields from a quantized field theory and to construct a low-energy one-loop effective Lagrangian by integrating out the heavy degrees of freedom in the path integral. We apply this method to the Higgs boson in a spontaneously broken SU(2) gauge theory (gauged linear sigma-model). In this context, the background-field method is generalized to the non-linear representation of the Higgs sector by applying (a generalization of) the Stueckelberg formalism. The (background) gauge-invariant renormalization is discussed. At one loop the log M_H-terms of the heavy-Higgs limit of this model coincide with the UV-divergent terms of the corresponding gauged non-linear sigma-model, but vertex functions differ in addition by finite (constant) terms in both models. These terms are also derived by our method. Diagrammatic calculations of some vertex functions are presented as consistency check.

Classical strings subjected to the Lorentz force in loop space.

Interactions of strings with local tensor fields are discussed in terms of the Lorentz force in loop space, the space of all loops in space-time, at the level of classical mechanics. We consider the Nambu-Got\ifmmode \bar{o}\else \={o}\fi{} string interacting with a U(1) gauge field on the loop space and derive a necessary condition to close the algebra that consists of constraints in the system. Examining this condition, we find at the classical level that (a) strings interact with second-rank antisymmetric tensor fields, (b) only tensionless strings interact with massive vector fields in the Stueckelberg formalism, and (c) strings do not interact with massive third-rank tensor fields in a generalized Stueckelberg formalism.