Momentum-space Propagator Research Articles

Position-space gluon propagator from quenched lattice QCD

We report novel lattice QCD results for the position-space gluon propagator in Landau gauge from quenched simulations. Using standard Wilson action, we computed gluon propagator in position space with a detailed treatment of hypercubic errors. Gluon propagator is scrutinized in position-space, discussing on the usefulness of the long-distance behavior of gluon propagator for constraining the gluon spectral function or the deep infrared running of the gluon propagator in momentum-space.

Gribov horizon in Noncommutative QED

It is known that Noncommutative QED (NCQED) exhibits Gribov ambiguities in the Landau gauge. These ambiguities are related to zero modes of the Faddeev-Popov operator and arise in the ghost propagator when it has a pole. In this work, we establish a positive Faddeev-Popov operator for NCQED and the condition for the ghost propagator not to have poles, the so-called Gribov no-pole condition. This condition is implemented in the path integral and allows for the calculation of the photon propagator in momentum space, which is dependent on the squared non-commutativity parameter. In the commutative limit, the standard QED is recovered.

The short-distance behavior of an Abelian Proca model based on a one-parameter extension of the covariant Heisenberg algebra

Recently, a one-parameter extension of the covariant Heisenberg algebra with the extension parameter [Formula: see text] [Formula: see text] is a non-negative constant parameter which has a dimension of [Formula: see text] in a [Formula: see text]-dimensional globally flat spacetime has been presented which is a covariant generalization of the Kempf–Mangano algebra [see G. P. de Brito, P. I. C. Caneda, Y. M. P. Gomes, J. T. Guaitolini Junior and V. Nikoofard, Adv. High Energy Phys. 2017, 4768341 (2017) and A. Kempf and G. Mangano, Phys. Rev. D 55, 7909 (1997)]. The Abelian Proca model is reformulated from the viewpoint of the above one-parameter extension of the covariant Heisenberg algebra. It is shown that the free space solutions of the above modified Proca model describe two massive vector particles with different effective masses [Formula: see text] where [Formula: see text] is the characteristic length scale in our model. In addition, the Feynman propagator in momentum space for the modified Abelian Proca model is calculated analytically. Our numerical estimations show that the maximum value of [Formula: see text] in a four-dimensional spacetime is near the electroweak length scale, i.e. [Formula: see text]. We show that in the infrared/large-distance domain, the modified Proca model behaves like an Abelian massive Lee–Wick model which has been presented by Accioly and his co-workers in A. Accioly, J. Helayel-Neto, G. Correia, G. Brito, J. de Almeida and W. Herdy, Phys. Rev. D 93, 105042 (2016). The short-distance behavior of the modified Proca model is studied in the massless limit and the explicit forms of the inhomogeneous infinite derivative Maxwell equation and the infinite derivative Poisson equation are obtained. Finally, note that in the low-energy limit [Formula: see text], the results of this paper are compatible with the results of the usual Proca model.

Extracting topological information from momentum space propagators

A new topological invariant quantity, sensitive to the analytic structure of both fermionic and bosonic propagators, is proposed. The gauge invariance of our construct is guaranteed for at least small gauge transformations. A generalization compatible with the presence of complex poles is introduced and applied to the classification of propagators typically emerging from non-perturbative considerations. We present partial evidence that the topological number can be used to detect chiral symmetry breaking or deconfinement.

Maximally-localized position, Euclidean path-integral, and thermodynamics in GUP quantum mechanics

In dealing with quantum mechanics at very high energies, it is essential to adapt to a quasiposition representation using the maximally-localized states because of the generalized uncertainty principle. In this paper, we look at maximally-localized states as eigenstates of the operator ξ=X+iβP that we refer to as the maximally-localized position. We calculate the overlap between maximally-localized states and show that the identity operator can be expressed in terms of the maximally-localized states. Furthermore, we show that the maximally-localized position is diagonal in momentum-space and that the maximally-localized position and its adjoint satisfy commutation and anti-commutation relations reminiscent of the harmonic oscillator commutation and anti-commutation relations. As application, we use the maximally-localized position in developing the Euclidean path-integral and introduce the compact form of the propagator for maximal localization. The free particle momentum-space propagator and the propagator for maximal localization are analytically evaluated up to quadratic-order in β. Finally, we obtain a path-integral expression for the partition function of a thermodynamic system using the maximally-localized states. The partition function of a gas of noninteracting particles is evaluated. At temperatures exceeding the Planck energy, we obtain the gas’ maximum internal energy N∕2β and recover the zero heat capacity of an ideal gas.

Invertible propagator for plane wave illumination of forward-scattering structures

Propagation of directed waves in forward-scattering media is considered. It is assumed that the evolution of the wave field is governed by the standard parabolic wave equation. An efficient one-step momentum-space propagator, suitable for a tilted plane wave illumination of extended objects, is derived. It is expressed in terms of a propagation operator that transforms (the complex exponential of) a linogram of the illuminated object into a set of its diffraction patterns. The invertibility of the propagator is demonstrated, which permits a multiple-shot scatter correction to be performed, and makes the solution especially attractive for either projective or tomographic imaging. As an example, high-resolution tomograms are obtained in numerical simulations implemented for a synthetic phantom, with both refractive and absorptive inclusions.

Euclidean path integral formalism in deformed space with minimum measurable length

We study time-evolution at the quantum level by developing the Euclidean path-integral approach for the general case where there exists a minimum measurable length. We derive an expression for the momentum-space propagator which turns out to be consistent with recently developed β-canonical transformation. We also construct the propagator for maximal localization which corresponds to the amplitude that a state which is maximally localized at location ξ′ propagates to a state which is maximally localized at location ξ″ in a given time. Our expression for the momentum-space propagator and the propagator for maximal localization is valid for any form of time-independent Hamiltonian. The nonrelativistic free particle, particle in a linear potential, and the harmonic oscillator are discussed as examples.

Spectral decomposition of the ghost propagator and a necessary condition for confinement

In this article we present exact calculations that substantiate a clear picture relating the confining force of QCD to the zero-modes of the Faddeev-Popov (FP) operator $\mathcal{M}(A) = - \partial \cdot D(A)$. This is done in two steps. First we calculate the spectral decomposition of the FP operator and show that the ghost propagator $\mathcal{G}(k; A) = \langle \vec{k}| \mathcal{M}^{-1}(A) | \vec{k} \rangle$ in an external gauge potential $A$ is enhanced at low $k$ in Fourier space for configurations $A$ on the Gribov horizon. This results from the new formula in the low-$k$ regime $\mathcal{G}^{ab}(k,A) = \delta^{ab} \lambda_{|\vec{k}|}^{-1}(gA)$, where $\lambda_{|\vec{k}|}(gA)$ is the eigenvalue of the FP operator that emerges from $\lambda_{|\vec{k}|}(0) = \vec{k}^2$ at $A$ = 0. Next we derive a strict inequality signaling the divergence of the color-Coulomb potential at low momentum $k$ namely, $\widetilde{\mathcal{V}}(k) \geq k^2 G^2(k)$ for $k \to 0$, where $\widetilde{\mathcal{V}}(k)$ is the Fourier transform of the color-Coulomb potential $\mathcal{V}(r)$ and $G(k)$ is the ghost propagator in momentum space. The first result holds in the Landau and Coulomb gauges, whereas the second holds in the Coulomb gauge only. We propose a new numerical lattice gauge fixing that should be closer to the present analytic approach than other numerical gauges.

The perturbative scalar massless propagator in Schwarzschild spacetime

A short derivation of the weak gravitational field approximation to the scalar massless propagator in Schwarzschild spacetime obtained by Paszko using the path-integral approach is given. The contribution from the direct coupling of the quantum field to the scalar curvature is explicitly included. The propagator complies with Hadamard's pattern, and the vacuum state is consistent with the perturbative version of the Boulware vacuum. The momentum space propagator is computed for massless or massive particles to the same perturbative order. The renormalized value of 〈ϕ2(x)〉 for the massless case is reproduced.

The vector meson mass in the large N limit of QCD

The vector meson mass is computed as a function of quark mass in the large N limit of QCD. We use continuum reduction and directly compute the vector meson propagator in momentum space. Quark momentum is inserted using the quenched momentum prescription.

Investigations in1+1dimensional latticeϕ4theory

In this work we perform a detailed numerical analysis of ($1+1$) dimensional lattice ${\ensuremath{\phi}}^{4}$ theory. We explore the phase diagram of the theory with two different parametrizations. We find that symmetry breaking occurs only with a negative mass-squared term in the Hamiltonian. The renormalized mass ${m}_{R}$ and the field renormalization constant $Z$ are calculated from both coordinate space and momentum space propagators in the broken symmetry phase. The critical coupling for the phase transition and the critical exponents associated with ${m}_{R}$, $Z$ and the order parameter are extracted using a finite-size scaling analysis of the data for several volumes. The scaling behavior of $Z$ has the interesting consequence that $⟨{\ensuremath{\phi}}_{R}⟩$ does not scale in $1+1$ dimensions. We also calculate the renormalized coupling constant ${\ensuremath{\lambda}}_{R}$ in the broken symmetry phase. The ratio ${\ensuremath{\lambda}}_{R}/{m}_{R}^{2}$ does not scale and appears to reach a value independent of the bare parameters in the critical region in the infinite volume limit.

Numerical study of the enlarged O(5) symmetry of the 3D antiferromagnetic RP2 spin model

We investigate by means of Monte Carlo simulation and finite-size scaling analysis the critical properties of the three-dimensional O(5) non-linear σ model and of the antiferromagnetic RP2 model, both of them regularized on a lattice. High accuracy estimates are obtained for the critical exponents, universal dimensionless quantities and critical couplings. It is concluded that both models belong to the same universality class, provided that rather non-standard identifications are made for the momentum-space propagator of the RP2 model. We have also investigated the phase diagram of the RP2 model extended by a second-neighbor interaction. A rich phase diagram is found, where most of the phase transitions are of the first order.

Fat link irrelevant clover overlap quark propagator

FLIC overlap fermions are a variant of the standard (Wilson) overlap action, with the FLIC (Fat Link Irrelevant Clover) action as the overlap kernel rather than the Wilson action. The structure of the FLIC overlap fermion propagator in momentum space is studied, and a comparison against previous studies of the Wilson overlap propagator in quenched QCD is performed. To explore the scaling properties of the propagator for the two actions, numerical calculations are performed in Landau Gauge across three lattices with different lattice spacing $a$ and similar physical volumes. We find that at light quark masses the acti ons agree in both the infrared and the ultraviolet, but at heavier masses some disagreement in the ultraviolet appears. This is attributed to the two action s having different discretisation errors with the FLIC overlap providing superior performance in this regime. Both actions scale reasonably, but some scaling violations are observed.

Phase structure and gauge boson propagator in the radially active 3D compact Abelian Higgs model

Unfreezing the radial degree of freedom, we study the three-dimensional Abelian Higgs model with compact gauge field and fundamentally charged matter. For small quartic Higgs self coupling and finite gauge coupling the model possesses a first order transition from the confined/symmetric phase to the deconfined/Higgs phase separated at some hopping parameter $\kappa_c$. Latent heat and surface tension are obtained in the first order regime. At larger quartic coupling the first order transition ceases to exist, and the behavior becomes similar to that known from the London limit. These observations are complemented by a study of the photon propagator in Landau gauge in the two different regimes. The problems afflicting the gauge fixing procedure are carefully investigated. We propose an improved gauge fixing algorithm which uses a finite subgroup in a preselection/preconditioning stage. The computational gain in the expensive confinement region is a speed-up factor around 10. The propagator in momentum space has a non-zero anomalous dimension in the confined phase whereas it vanishes in the Higgs phase. As far as the gauge boson propagator is concerned, we find that the radially active Higgs field provides qualitatively no new effect compared to the radially frozen Higgs field studied before.

Erratum

This paper proves that it is possible to build a Lagrangian for quantum electrodynamics which makes it explicit that the photon mass is eventually set to zero in the physical part on observational ground. Gauge independence is achieved upon considering the joint effect of gauge-averaging term and ghost fields. It remains possible to obtain a counterterm Lagrangian where the only non-gauge-invariant term is proportional to the squared divergence of the potential, while the photon propagator in momentum space falls off like 1 over (k-squared) at large k, which indeed agrees with perturbative renormalizability. The resulting radiative corrections to the Coulomb potential in QED are also shown to be gauge-independent. The experience acquired with quantum electrodynamics is used to investigate properties and problems of the extension of such ideas to non-Abelian gauge theories.

Non-perturbative singularity structure of gluon and quark propagators

A gauge invariant, non-local observable is constructed in lattice pure gauge theory, which is identical to the gluon propagator in a particular gauge. The transfer matrix formalism is used to show that this correlator decays exponentially with eigenvalues of the Hamiltonian. This implies a gauge invariant singularity structure of the propagator in momentum space, permitting a non-perturbative definition of a parton mass. The relation to gauge fixing and the extension to matter fields are discussed.

De Sitter brane gravity: from close-up to panorama

We find explicitly the induced graviton propagator on de Sitter branes embedded in various five-dimensional spacetimes; de Sitter branes in AdS and Minkowski space are particular cases. By studying the structure of the momentum-space propagator, we are able to extract interesting physics, much of which is qualitatively different from that of flat branes. We find that (1) there can be a set of graviton-like particles which mediate brane gravity at different scales; (2) localized gravity can exist even on de Sitter branes in Minkowski space; (3) Kaluza–Klein modes also contribute to conventional 4D gravity for de Sitter branes in AdS; and (4) Newton's constant can vary considerably with scale. We comment on the implications for the effective cosmological constant.

Quark propagator in the Landau gauge

The Landau gauge quark propagator in momentum space is investigated using the O(a)-improved Sheikholeslami--Wohlert (SW) quark action with a tree-level mean-field improved coefficient c_sw. We have studied the unimproved definition of the quark propagator, as well as two different tree-level O(a)-improved propagators. The ultraviolet behavior of the free lattice propagator is studied for each of these in order to establish which of them provides the most reliable description of the quark propagator up to the medium momentum regime. A general method of tree-level correction is introduced. This exploits asymptotic freedom and removes much of the trivial lattice artifacts at medium to high momenta. We obtain results for the quark propagator which are qualitatively similar to those typically used in quark models. A simple extrapolation of the infrared quark mass M(p^2=0) to the chiral limit gives (298+/-8+/-30) MeV, which is consistent with phenomenological expectations.

Lattice study of the gluon propagator in momentum space.

We consider pure glue QCD at $\ensuremath{\beta}=5.7$, $\ensuremath{\beta}=6.0$, and $\ensuremath{\beta}=6.3$. We evaluate the gluon propagator both in time at zero three-momentum and in momentum space. From the former quantity we obtain evidence for a dynamically generated effective mass, which at $\ensuremath{\beta}=6.0$ and $\ensuremath{\beta}=6.3$ increases with the time separation of the sources, in agreement with earlier results. The momentum space propagator $G(k)$ provides further evidence for mass generation. In particular, at $\ensuremath{\beta}=6.0$, for $300 \mathrm{MeV}\ensuremath{\lesssim}k\ensuremath{\lesssim}1GeV$, the propagator $G(k)$ can be fit to a continuum formula proposed by Gribov and others, which contains a mass scale $b$, presumably related to the hadronization mass scale. For higher momenta Gribov's model no longer provides a good fit, as $G(k)$ tends rather to follow an inverse $\mathrm{power}\mathrm{law}\ensuremath{\approx}\frac{1}{{k}^{2+\ensuremath{\gamma}}}$. The results at $\ensuremath{\beta}=6.3$ are consistent with those at $\ensuremath{\beta}=6.0$, but only the high momentum region is accessible on this lattice. We find $b$ in the range of 300 to 400 MeV and $\ensuremath{\gamma}$ about 0.7. Fits to particle + ghost expressions are also possible, often resulting in low values for ${\ensuremath{\chi}}_{\mathrm{DF}}^{2}$, but the parameters are very poorly determined. On the other hand, at $\ensuremath{\beta}=5.7$ (where we can only study momenta up to 1 GeV) $G(k)$ is best fit to a simple massive boson propagator with mass $m$. We argue that such a discrepancy may be related to a lack of scaling for low momenta at $\ensuremath{\beta}=5.7$. From our results, the study of correlation functions in momentum space looks promising, especially because the data points in Fourier space turn out to be much less correlated than in real space.

SPACE-TIME PROPAGATION OF CONFINED GLUONS

We assume that there is gluon condensate in the zero-momentum mode in the QCD ground state. A lowest-order calculation in terms of a condensate order parameter leads to a dynamical mass for gluons via the Schwinger mechanism and a gluon propagator with no on-mass-shell singularities — that is, the gluon is a "nonpropagating mode" in the gluon condensate. We transform our momentum-space propagator into coordinate space and find that the propagator has essentially the same delta-function light-cone singularities as the free propagator. However, in contrast to a theory without confinement, we show that the propagator exhibits exponential decay, both for time-like and space-like propagation. In this manner, we obtain a space-time characterization of the confinement phenomenon in terms of an order parameter of the condensate.