We study the resummation of self-energy diagrams into dressed propagators in the case of purely virtual particles and compare the results with those obtained for physical particles and ghosts. The three geometric series differ by infinitely many contact terms, which do not admit well-defined sums. The peak region, which is outside the convergence domain, can only be reached in the case of physical particles, thanks to analyticity. In the other cases, nonperturbative effects become important. To clarify the matter, we introduce the energy resolution $ΔE$ around the peak and argue that a "peak uncertainty" $ΔE\gtrsim ΔE_{\text{min}}\simeq Γ_{\text{f}}/2$ around energies $ E\simeq m_{\text{f}}$ expresses the impossibility to approach the fakeon too closely, $m_{\text{f}}$ being the fakeon mass and $Γ_{\text{f}}$ being the fakeon width. The introduction of $ΔE$ is also crucial to explain the observation of unstable long-lived particles, like the muon. Indeed, by the common energy-time uncertainty relation, such particles are also affected by ill-defined sums at $ΔE=0$, whenever we separate their observation from the observation of their decay products. We study the regime of large $Γ_{\text{f}}$, which applies to collider physics (and situations like the one of the $Z$ boson), and the regime of small $Γ_{\text{f}}$, which applies to quantum gravity (and situations like the one of the muon).
The search for purely virtual quanta has attracted interest in the past. We consider various proposals and compare them to the concept of fake particle, or "fakeon". In particular, the Feynman-Wheeler propagator, which amounts to using the Cauchy principal value inside Feynman diagrams, violates renormalizability, unitarity and stability, due to the coexistence of the prescriptions $\pm iε$. We contrast the Feynman, fakeon and Feynman-Wheeler prescriptions in ordinary as well as cut diagrams. The fakeon does not have the problems of the Feynman-Wheeler propagator and emerges as the correct con…
We give a simple proof of perturbative unitarity in gauge theories and quantum gravity using a special gauge that allows us to separate the physical poles of the free propagators, which are quantized by means of the Feynman prescription, from the poles that belong to the gauge-trivial sector, which are quantized by means of the fakeon prescription. The proof applies to renormalizable theories, including the ultraviolet complete theory of quantum gravity with fakeons formulated recently, as well as low-energy (nonrenormalizable) theories. We clarify a number of subtleties related to the study o…
The worldline formalism has previously been used for deriving compact master formulas for the one-loop N-photon amplitudes in both scalar and spinor QED, and in the vacuum as well as in a constant external field. For scalar QED, there is also an analogous master formula for the propagator dressed with N photons in the vacuum. Here, we extend this master formula to include a constant field. The two-photon case is worked out explicitly, yielding an integral representation for the Compton scattering cross section in the field suitable for numerical integration in the full range of electric and ma…
The infrared behaviour of the gauge-invariant dressed fermion propagator in massless QED_3 is discussed for three choices of dressing. It is found that only the propagator with the isotropic (in three Euclidean dimensions) choice of dressing is acceptable as the physical fermion propagator. It is explained that the negative anomalous dimension of this physical fermion does not contradict any field-theoretical requirement.
Based on the rainbow approximation of Dyson-Schwinger equation and the assumption that the inverse dressed quark propagator at finite chemical potential is analytic in the neighborhood of $μ=0$, a new method for obtaining the dressed quark propagator at finite chemical potential $μ$ from the one at zero chemical potential is developed. Using this method the dressed quark propagator at finite chemical potential can be obtained directly from the one at zero chemical potential without the necessity of numerically solving the corresponding coupled integral equations by iteration methods. A compari…
We study the propagator of a colored scalar particle in the background of a non-abelian gauge field using the worldline formalism. It is obtained by considering the open worldline of a scalar particle with extra degrees of freedom needed to take into account the color charge of the particle, which we choose to be in the fundamental representation of the gauge group. Specializing the external gauge field to be given by a sum of plane waves, i.e. a sum of external gluons, we produce a master formula for the scalar propagator with an arbitrary number of gluons directly attached to the scalar line…
In the first part of this series, we employed the second-order formalism and the ``symbol'' map to construct a particle path-integral representation of the electron propagator in a background electromagnetic field, suitable for open fermion-line calculations. Its main advantages are the avoidance of long products of Dirac matrices, and its ability to unify whole sets of Feynman diagrams related by permutation of photon legs along the fermion lines. We obtained a Bern-Kosower type master formula for the fermion propagator, dressed with $N$ photons, in terms of the ``$N$-photon kernel,'' where t…
We investigate the properties of a dressed electron which reduces, in a particular class of gauges, to the usual fermion. A one loop calculation of the propagator is presented. We show explicitly that an infra-red finite, multiplicative, mass shell renormalisation is possible for this dressed electron, or, equivalently, for the usual fermion in the abovementioned gauges. The results are in complete accord with previous conjectures.
We consider an electron which is electromagnetically dressed in such a way that it is both gauge invariant and that it has the associated electric and magnetic fields expected of a moving charge. We study the propagator of this dressed electron and, for small velocities, show explicitly at one loop that at the natural (on-shell) renormalisation point, $p_0=m$, ${\bold p}= m{\bold v}$, one can renormalise the propagator multiplicatively. Furthermore the renormalisation constants are infra-red finite. This shows that the dressing we use corresponds to a slowly moving, physical asymptotic field.
We revisit the problem of the gauge invariance in the Coleman-Weinberg model in which a $U(1)$ gauge symmetry is driven spontaneously broken by radiative corrections. It was noticed in previous work that masses in this model are not gauge invariant at one-loop order. In our analysis, we use the dressed propagators of scalars which include a resummation of the one-loop self-energy correction to the tree-level propagator. We calculate the one-loop self-energy correction to the vector meson using these dressed propagators. We find that the pole mass of the vector meson calculated using the dresse…
In the first-quantised worldline approach to quantum field theory, a long-standing problem has been to extend this formalism to amplitudes involving open fermion lines while maintaining the efficiency of the well-tested closed-loop case. In the present series of papers, we develop a suitable formalism for the case of quantum electrodynamics in vacuum (part one and two) and in a constant external electromagnetic field (part three), based on second-order fermions and the symbol map. We derive this formalism from standard field theory, but also give an alternative derivation intrinsic to the worl…
A condensate of $\pvec = 0$ partons in the perturbative vacuum gives rise to a term $\proptoδ^4(p)$ in the free PQCD propagators. The leading condensate contribution to the quark propagator can be exactly summed since there is a factor $δ^4(p)$ associated to each loop. We calculate the dressed quark propagator in the presence of either a gluon or a quark condensate, for a number of colors $N \to \infty$. The dressed quark propagator satisfies a Dyson-Schwinger type equation which can be exactly solved within our framework. In the case of a gluon condensate the dressed quark propagator has no p…
A theory of quantum gravity has been recently proposed by means of a novel quantization prescription, which is able to turn the poles of the free propagators that are due to the higher derivatives into fakeons. The classical Lagrangian contains the cosmological term, the Hilbert term, $\sqrt{-g}R_{μν}R^{μν}$ and $\sqrt{-g}R^{2}$. In this paper, we compute the one-loop renormalization of the theory and the absorptive part of the graviton self energy. The results illustrate the mechanism that makes renormalizability compatible with unitarity. The fakeons disentangle the real part of the self ene…
The Schwinger-Dyson equation for the quark in the rainbow approximation at finite temperature (T) is solved numerically without introducing any ansatz for the dressed quark propagator. The dymanical quark mass-function and the wave-function renormalization are found to have non-trivial dependence on three-momentum, Matsubara-frequency and temperature. The critical temperature of the chiral phase transition (T_c) and the T-dependence of the quark condensate are highly affected by the wave-function renormalization. We found that T_c \simeq 155 MeV which is consistent with the result of the finit…
The Gluon and Ghost propagators in Landau gauge QCD are investigated using the Schwinger-Dyson equation approach. Working in Euclidean spacetime, we solve for these propagators using a selection of vertex inputs, initially for the ghost equation alone and then for both propagators simultaneously. The results are shown to be highly sensitive to the choices of vertices. We favor the infrared finite ghost solution from studying the ghost equation alone where we argue for a specific unique solution. In order to solve this simultaneously with the gluon using a dressed-one-loop truncation, we find t…
We elucidate constraints imposed by confinement and dynamical chiral symmetry breaking on the infrared behaviour of the dressed-quark and -gluon propagators, and dressed-quark-gluon vertex. In covariant gauges the dressing of the gluon propagator is completely specified by P(k^2):= 1/[1+Pi(k^2)], where Pi(k^2) is the vacuum polarisation. In the absence of particle-like singularities in the dressed-quark-gluon vertex, extant proposals for the dressed-gluon propagator that manifest P(k^2=0)=0 and Max[P(k^2)]~10 neither confine quarks nor break chiral symmetry dynamically. This class includes all…
We obtain in analytical form the dressed propagator of the Rarita-Schwinger field taking into account all spin components and discuss shortly its properties.
Quenched lattice-QCD data on the dressed-quark Schwinger function can be correlated with dressed-gluon data via a rainbow gap equation so long as that equation's kernel possesses enhancement at infrared momenta above that exhibited by the gluon alone. The required enhancement can be ascribed to a dressing of the quark-gluon vertex. The solutions of the rainbow gap equation exhibit dynamical chiral symmetry breaking and are consistent with confinement. The gap equation and related, symmetry-preserving ladder Bethe-Salpeter equation yield estimates for chiral and physical pion observables that s…
Recent lattice-QCD results for the dressed-gluon propagator are used within the quark Dyson-Schwinger equation to determine the gluon-quark vertex dressing necessary to reproduce the lattice-QCD results for the dressed-quark propagator. Both quenched and full QCD lattice simulations, for a range of low quark current masses, are analyzed. The chiral extrapolation is made through this continuum DSE form. Resulting chiral and physical pion observables are investigated.
In this article, we calculate the dressed quark propagator with the flat bottom potential in the framework of the rain-bow Schwinger-Dyson equation. Then based on the nonperturbative dressed quark propagator, we calculate the $π$ decay constant and the quark condensate. The $π$ decay constant is an important parameter in describing the interplay between dynamical symmetry breaking and confinement, while the quark condensate is an order parameter for dynamical chiral symmetry breaking. To implement confinement, we prove that the dressed quark propagator has no poles on the real timelike $p^2$ a…
We describe a loopwise perturbative truncation scheme for quantum transport equations in the Kadanoff-Baym formalism, which does not necessitate the use of the so-called Kadanoff-Baym or quasi-particle ansaetze for dressed propagators. This truncation scheme is used to study flavour effects in the context of Resonant Leptogenesis (RL), showing explicitly that, in the weakly-resonant regime, there exist two distinct and pertinent flavour effects in the heavy-neutrino sector: (i) the resonant mixing and (ii) the oscillations between different heavy-neutrino flavours. Moreover, we illustrate that…
In this article, we calculate the dressed quark propagator with the flat bottom potential in the framework of the rain-bow Schwinger-Dyson equation, which is determined by mean field approximation of the global colour model lagrangian. The dressed quark propagator exhibits a dynamical symmetry breaking phenomenon and gives a constituent quark mass about 392 MeV, which is close to the value of commonly used constituent quark mass in the chiral quark model. Then based on the dressed quark propagator, we calculate some properties of the vacuum, such as quark condensate, mixed quark condensate $g_…
We derive general and explicit expressions for the unrenormalized and renormalized dressed propagators of fermions in parity-nonconserving theories with inter-generation mixing. The mass eigenvalues, the corresponding mass counterterms, and the effect of inter-generation mixing on their determination are discussed. Invoking the Aoki-Hioki-Kawabe-Konuma-Muta renormalization conditions and employing a number of very useful relations from Matrix Algebra, we show explicitly that the renormalized dressed propagators satisfy important physical properties.
By differentiating the dressed quark propagator with respect to a variable background field, the linear response of the dressed quark propagator in the presence of the background field can be obtained. From this general method, using the vector background field as an illustration, we derive a general formula for the four-quark condensate $<{\tilde 0}|:{\bar q}(0)γ_μq(0){\bar q}(0)γ_μq(0):|{\tilde 0}>$. This formula contains the corresponding fully dressed vector vertex and it is shown that factorization for $<{\tilde 0}|:{\bar q}(0)γ_μq(0){\bar q}(0)γ_μq(0):| {\tilde 0}>$ holds only when the d…
The propagators of unstable particles are considered in the framework of the convolution representation. Spectral function was found for a special case when the propagator of scalar unstable particle has Breight-Wigner form. The expressions for the dressed propagators of unstable vector and spinor fields are derived in an analytical way for this case. We got the propagators in modified Breit-Wigner forms which correspond to the complex-mass definition.
The quark-gluon vertex in Landau gauge QCD is investigated in the Dyson-Schwinger approach. The aim is to obtain a fully self-consistent solution of the quark propagator and the quark-gluon vertex Dyson-Schwinger equation (DSE) in the vacuum using the gluon propagator as input from other calculations. The truncation scheme used in an earlier study is systematically improved and the first decisive step towards a full solution is presented. First results for the quark propagator with a fully dressed quark-gluon vertex in a truncation that incorporates all twelve vertex tensor-structures are show…
Extending work by Gies and Karbstein on the Euler-Heisenberg Lagrangian, it has recently been shown that the one-loop propagator of a charged scalar particle in a constant electromagnetic field has a one-particle reducible contribution in addition to the well-studied irreducible one. Here we further generalize this result to the spinor case, and find the same relation between the reducible term, the tree-level propagator and the one-loop Euler-Heisenberg Lagrangian as in the scalar case. Our demonstration uses a novel worldline path integral representation of the photon-dressed spinor propagat…
Propagators of unstable particles are considered in the spectral representation which naturally follows from the concept of continuous mass. The spectral functions are found with the help of the most general formal and physical assumptions. Dressed propagators of unstable scalar, vector, and spinor fields are derived in an analytical way for a variant of parameter space. The structure of the propagators is in a correspondence with the complex mass scheme.
We construct the gauge invariant three-photon decay amplitude of particle-antiparticle bound states modeled by the Dyson-Schwinger and Bethe-Salpeter equations. Application to the quark-antiquark ($q\bar{q}$) bound states is emphasized. An essential aspect of our formulation is that it applies to any underlying quantum field theoretic model of the $q\bar{q}$ system, and not just to models, like exact QCD, where the quark self-energy $Σ$ couples to the electromagnetic field solely via dressed quark propagators. In this way, applications to effective field theories and other QCD motivated models…
I discuss a degree of freedom in formulating perturbation theory that is often neglected: the in- and out-states need not be empty. The inclusion of (free) particles in the asymptotic states modifies the on-shell prescription of the free propagator. This affects loop contributions -- but the modified expansion is {\it a priori} as justified as the standard one with Feynman prescription. It is possible to dress the quark propagator to all orders with zero-momentum gluons from the perturbative ground state. The dressed quark has no pole and thus cannot appear as an external particle in the S-mat…
Ghosts versus Unstable Particles in Quantum Field Theory
Departamento de Física de Partículas, Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela（圣地亚哥-德孔波斯特拉大学加利西亚高能物理研究所粒子物理系）
Motivated by the gluon condensate in QCD we study the perturbative expansion of a gauge theory in the presence of gauge bosons of vanishing momentum, in the specific case of an abelian theory. The background is characterised by a dimensionful parameter $Λ$ affecting only the on-shell prescription of the free (abelian) gluon propagator. When summed to all orders in $gΛ$ the modification is equivalent to evaluating standard Green functions in a pure gauge field with an imaginary gauge parameter $\propto Λ$. We show how to calculate the corresponding dressed Green functions, which are Poincare an…
We investigate how quantum dynamics affects the propagation of a scalar field in Lorentzian AdS. We work in momentum space, in which the propagator admits two spectral representations (denoted "conformal" and "momentum") in addition to a closed-form one, and all have a simple split structure. Focusing on scalar bubbles, we compute the imaginary part of the self-energy $ {\rm Im} Π$ in the three representations, which involves the evaluation of seemingly very different objects. We explicitly prove their equivalence in any dimension, and derive some elementary and asymptotic properties of $ {\rm…
We propose an extension of the minimal-substitution prescription for coupling the electromagnetic field to hadronic systems with internal structure. The resulting rules of extended substitution necessarily distinguish between couplings to scalar and Dirac particles. Moreover, they allow for the incorporation of electromagnetic form factors for virtual photons in an effective phenomenological framework. Applied to pions and nucleons, assumed to be fully dressed to all orders, the resulting dressed currents are shown to be locally gauge invariant. Moreover, half-on-shell expressions of (hadron p…
It is argued why thermodynamic approximations in terms of dressed propagators may, at larger coupling strength, be better behaved than perturbative results, and why in hot QCD the hard thermal loop approximation of the thermodynamic potential cannot be expected to work close to the phase transition.
The dependence of the dressed quark propagator on the quark chemical potential is investigated in various models based on the Dyson-Schwinger equations. We find that the critical chemical potential of the chiral symmetry restoration transition is strongly dependent on the nature of the interaction kernel in the infrared region.
We discuss the far-from-equilibrium evolution of $ϕ^3$-theory in $1+1$ dimensions with the temporal functional renormalisation group \cite{Gasenzer:2007za, Gasenzer:2010rq}. In particular, we show that this manifestly causal approach leads to novel one-loop exact equations for fully dressed correlation functions. Within this setup, we numerically compute the dynamical propagator. Its behaviour suggests self-similarity far from equilibrium in a restricted momentum regime. We discuss the scaling exponents for our solution, as well as the numerical satisfaction of energy and particle number conse…
A loop or coupling expansion of a so-called n-particle irreducible (nPI) generating functional provides a well-defined approximation scheme in terms of self-consistently dressed propagators and n-point vertices. A self-consistently complete description determines the functional for arbitrarily high n to a given order in the expansion. We point out an equivalence hierarchy for nPI effective actions, which allows one to obtain a self-consistently complete result in practice. The method is applied to a SU(N) gauge theory with fermions up to four-loop or O(g^6) corrections. For non-equilibrium we …
Complex mass poles, or ghost poles, are present in the Hartree-Fock solution of the Schwinger-Dyson equation for the nucleon propagator in renormalizable models with Yukawa-type meson-nucleon couplings, as shown many years ago by Brown, Puff, and Wilets (BPW). These ghosts violate basic theorems of quantum field theory and their origin is related to the ultraviolet behavior of the model interactions. Recently, Krein et.al, proved that the ghosts disappear when vertex corrections are included in a self-consistent way, softening the interaction sufficiently in the ultraviolet region. In previous…
The complete spin structure of the pion-nucleon-loop contribution to the delta self-energy and dressed propagator is calculated in vacuum, with the most general form of the pion-nucleon-delta vertex. The imaginary parts of the ten Lorentz-scalar coefficients are calculated in closed form, while the real parts are obtained numerically from a dispersion relation. The effect of the pion-nucleon-delta coupling constants and form-factor on the pion-nucleon phase-shift in the spin-3/2 isospin-3/2 channel is studied.
