Non-perturbative constraints on the quark and ghost propagators

In QCD both the quark and ghost propagators are important for governing the non-perturbative dynamics of the theory. It turns out that the dynamical properties of the quark and ghost fields impose non-perturbative constraints on the analytic structure of these propagators. In this work we explicitly derive these constraints. In doing so we establish that the corresponding spectral densities include components which are multiples of discrete mass terms, and that the propagators are permitted to contain singular contributions involving derivatives of $\delta(p)$, both of which are particularly relevant in the context of confinement.Comment: 13 pages; v3: matches published versio

Spectral density constraints in quantum field theory

Determining the structure of spectral densities is important for understanding the behaviour of any quantum field theory (QFT). However, the exact calculation of these quantities often requires a full non-perturbative description of the theory, which for physically realistic theories such as quantum chromodynamics (QCD) is currently unknown. Nevertheless, it is possible to infer indirect information about these quantities. In this paper we demonstrate an approach for constraining the form of spectral densities associated with QFT propagators, which involves matching the short distance expansion of the spectral representation with the operator product expansion (OPE) of the propagators. As an application of this procedure we analyse the scalar propagator in $\phi^4$-theory and the quark propagator in QCD, and show that constraints are obtained on the spectral densities and the OPE condensates. In particular, it is demonstrated that the perturbative and non-perturbative contributions to the quark condensate in QCD can be decomposed, and that the non-perturbative contributions are related to the structure of the continuum component of the scalar spectral density.Comment: 14 pages; v2: references and additional discussion adde

Dyson-Schwinger equation constraints on the gluon propagator in BRST quantised QCD

The gluon propagator plays a central role in determining the dynamics of QCD. In this work we demonstrate for BRST quantised QCD that the Dyson-Schwinger equation imposes significant analytic constraints on the structure of this propagator. In particular, we find that these constraints control the appearance of massless components in the gluon spectral density.Comment: 8 pages; v2: matches published versio

Boundary terms in quantum field theory and the spin structure of QCD

Determining how boundary terms behave in a quantum field theory (QFT) is crucial for understanding the dynamics of the theory. Nevertheless, boundary terms are often neglected using classical-type arguments which are no longer justified in the full quantum theory. In this paper we address this problem by establishing a necessary and sufficient condition for arbitrary spatial boundary terms to vanish in a general QFT. As an application of this condition we examine the issue of whether the angular momentum operator in Quantum Chromodynamics (QCD) has a physically meaningful quark-gluon decomposition. Using this condition it appears as though this is not the case, and that it is in fact the non-perturbative QCD structure which prevents the possibility of such a decomposition.Comment: 14 pages; v2: references/discussion added and minor typos corrected, version published in Nucl.Phys.

Rigorous constraints on the matrix elements of the energy-momentum tensor

The structure of the matrix elements of the energy-momentum tensor play an important role in determining the properties of the form factors $A(q^{2})$, $B(q^{2})$ and $C(q^{2})$ which appear in the Lorentz covariant decomposition of the matrix elements. In this paper we apply a rigorous frame-independent distributional-matching approach to the matrix elements of the Poincar\'{e} generators in order to derive constraints on these form factors as $q \rightarrow 0$. In contrast to the literature, we explicitly demonstrate that the vanishing of the anomalous gravitomagnetic moment $B(0)$ and the condition $A(0)=1$ are independent of one another, and that these constraints are not related to the specific properties or conservation of the individual Poincar\'{e} generators themselves, but are in fact a consequence of the physical on-shell requirement of the states in the matrix elements and the manner in which these states transform under Poincar\'{e} transformations.Comment: 11 pages; v2: additional comments added, matches published versio

Non-perturbative insights into the spectral properties of QCD at finite temperature

In quantum field theories at finite temperature spectral functions describe how particle systems behave in the presence of a thermal medium. Although data from lattice simulations can in principle be used to determine spectral function characteristics, existing methods rely on the extraction of these quantities from temporal correlators, which requires one to circumvent an ill-posed inverse problem. In these proceedings we report on a recent approach that instead utilises the non-perturbative constraints imposed by field locality to extract spectral function information directly from spatial correlators. In particular, we focus on the application of this approach to lattice QCD data of the spatial pseudo-scalar meson correlator in the temperature range $220-960 \, \text{MeV}$, and outline why this data supports the conclusion that there exists a distinct pion state above the chiral pseudo-critical temperature $T_{\!\text{pc}}$.Comment: 8 pages, 2 figures; talk presented at the XVth Quark confinement and the Hadron Spectrum conference, 1-6 August 2022, Stavanger, Norwa

Pion spectral properties above the chiral crossover of QCD

Spectral functions encode a wealth of information about the dynamics of any given system, and the determination of their non-perturbative characteristics is a long-standing problem in quantum field theory. Whilst numerical simulations of lattice QCD provide ample data for various Euclidean correlation functions, the inversion required to extract spectral functions is an ill-posed problem. In this work, we pursue previously established constraints imposed by field locality at finite temperature $T$, namely that spectral functions possess a non-perturbative representation which generalises the well-known K\"{a}ll\'{e}n-Lehmann spectral form to $T>0$. Using this representation, we analyse lattice QCD data of the spatial pseudo-scalar correlator in the temperature range $220-960 \, \text{MeV}$, and obtain an analytic expression for the corresponding spectral function, with parameters fixed by the data. From the structure of this spectral function we find evidence for the existence of a distinct pion state above the chiral pseudo-critical temperature $T_{\text{pc}}$, and contributions from its first excitation, which gradually melt as the temperature increases. As a non-trivial test, we find that the extracted spectral function reproduces the corresponding temporal lattice correlator data for $T=220 \, \text{MeV}$.Comment: 19 pages, 3 figures; v3: references updated, matches published versio

Non-perturbative insights into the spectral properties of QCD at finite temperature

In quantum field theories at finite temperature spectral functions describe how particle systems behave in the presence of a thermal medium. Although data from lattice simulations can in principle be used to determine spectral function characteristics, existing methods rely on the extraction of these quantities from temporal correlators, which requires one to circumvent an illposed inverse problem. In these proceedings we report on a recent approach that instead utilises the non-perturbative constraints imposed by field locality to extract spectral function information directly from spatial correlators. In particular, we focus on the application of this approach to lattice QCD data of the spatial pseudo-scalar meson correlator in the temperature range 220−960 MeV, and outline why this data supports the conclusion that there exists a distinct pion state above the chiral pseudo-critical temperature Tpc