Without any analytical assumption we solve the ladder QED2+1 in Minkowski space. Further, we transform Greens functions to the Temporal Euclidean space, wherein we show that in the special case of ladder QED2+1 the solution is fully equivalent to the Minkowski one.
The obtained complex fermion propagator exhibits confinement because of the lack of the pole at the real timelike $q^2$ axis.
Without any analytical assumption we solve the ladder QED2+1 in Minkowski space. Obtained complex fermion propagator exhibits confinement in the sense that it has no pole. Further, we transform Greens functions to the Temporal Euclidean space, wherein we show that in the special case of ladder QED2+1 the solution is fully equivalent to the Minkowski one. Obvious invalidity of Wick rotation is briefly discussed. The infrared value of the dynamical mass is compared with other known approaches, e.g. with the standard Euclidean calculation.
The electromagnetic form factors calculated through Euclidean Bethe-Salpeter amplitude and through the light-front wave function are compared with the one found using the Bethe-Salpeter amplitude in Minkowski space. The form factor expressed through the Euclidean Bethe-Salpeter amplitude (both within and without static approximation) considerably differs from the Minkowski one, whereas form factor found in the light-front approach is almost indistinguishable from it.
We shall discuss about some analytic properties of the high-energy parton-parton (and hadron-hadron) scattering amplitudes in gauge theories, when going from Minkowskian to Euclidean theory, and we shall see how they can be related to the still unsolved problem of the s-dependence of the total cross-section.
In a recent work we described the form taken by the Minkowski-space gluon propagator in the presence of an A^2 condensate. This condensate has been seen to play an important role in calculations which make use of the operator product expansion. The latter work has led to the publication of a large number of papers which discuss how the A^2 condensate could play a role in a gauge-invariant formulation. In the present work we discuss the properties of both the Euclidean-space and Minkowski-space gluon propagator. In the case of the Euclidean-space propagator we can make contact with the results …
We give out a simple way to connect the parton distribution functions defined in Minkowskian space and the nonperturbative QCD methods grounded in Euclidean space (e.g., lattice QCD(LQCD), Dyson-Schwinger (DS) equations, functional renormalization group (FRG) approach) in this work. We combine the MIT bag model with the DS equation approach to calculate the longitudinal distribution function, transverse distribution function and scalar distribution function in a proton at renormalization point $μ= 2\,\text{GeV}$. We look then insight into the dressed effects on the axial, tensor and scalar cha…
We describe the development of Analytic Perturbation Theory (APT) in QCD, called Fractional APT (FAPT), which has been suggested to apply the renormalization group evolution and QCD factorization technique in the framework of APT.
Without any analytical assumption we solve the ladder QED2+1 in Minkowski space. Further, we transform Greens functions to the Temporal Euclidean space, wherein we show that in the special case of ladder QED2+1 the solution is fully equivalent to the Minkowski one. QCD quark gap equation is solved in the framework of Temporal Euclidean space as well. In both models, the obtained complex fermion propagators exhibits confinement, since it does not satisfies Khallen-Lehmann representation.
The challenge to obtain from the Euclidean Bethe--Salpeter amplitude the amplitude in Minkowski is solved by resorting to un-Wick rotating the Euclidean homogeneous integral equation. The results obtained with this new practical method for the amputated Bethe--Salpeter amplitude for a two-boson bound state reveals a rich analytic structure of this amplitude, which can be traced back to the Minkowski space Bethe--Salpeter equation using the Nakanishi integral representation. The method can be extended to small rotation angles bringing the Euclidean solution closer to the Minkowski one and could…
I make some simple observations about the calculation of weighted averages over energy of Minkowski space spectral densities from weighted averages over time of Euclidean space correlation functions, measured in latice simulations. The correlator of two vector currents is used as an example, where it appears that a determination of a weighted average of the spectral function near the rho pole at the five per cent level is possible from lattice simulations.
We present a model for the quark-antiquark interaction formulated in Minkowski space using the Covariant Spectator Theory. The quark propagators are dressed with the same kernel that describes the interaction between different quarks. By applying the axial-vector Ward-Takahashi identity we show that our model satisfies the Adler-zero constraint imposed by chiral symmetry. For this model, our Minkowski-space results of the dressed quark mass function are compared to lattice QCD data obtained in Euclidean space. The mass function is then used in the calculation of the electromagnetic pion form f…
We shortly review different methods to obtain the scattering solutions of the Bethe-Salpeter equation in Minkowski space. We emphasize the possibility to obtain the zero energy observables in terms of the Euclidean scattering amplitude.
We propose a method to reconstruct the Bethe-Salpeter amplitude in Minkowski space given the Euclidean Bethe-Salpeter amplitude -- or alternatively the Light-Front wave function -- as input. The method is based on the numerical inversion of the Nakanishi integral representation and computing the corresponding weight function. This inversion procedure is, in general, rather unstable, and we propose several ways to considerably reduce the instabilities. In terms of the Nakanishi weight function, one can easily compute the BS amplitude, the LF wave function and the electromagnetic form factor. Th…
We determine the gluonic spectral function $SU(3)$ Yang-Mills theory as well as we found fermionic spectral functions in the strong quenched QED where we found new solutions. Our novel technique provides solutions with the usual branch cut for propagators while not showing any pole structure at the first Riemann sheet (identical with entire complex plane) of complex square of momentum. Implications and further utilizations are briefly adressed for QCD and Standard model calculations.
We give a short introduction to the Analytic Perturbation Theory (APT) in QCD, describe its problems and suggest as a tool for their resolution the Fractional APT (FAPT). We also describe shortly how to treat heavy-quark thresholds in FAPT. As an applications of this technique we discuss (i) the pion form factor calculation in the Euclidean FAPT and (ii) the Higgs boson decay $H^0\to b\bar{b}$ in Minkowskian FAPT. We conclude with comparison of both approaches, Euclidean and Minkowskian FAPT.
We analyze the analytic structure of the Covariant Spectator Theory (CST) contribution to the self-energy amplitude for a scalar particle in a ϕ^2 χ-theory. To this end we derive dispersion relations in 1+1 and in 3+1 dimensional Minkowski space. The divergent loop integrals in 3+1 dimensions are regularized using dimensional regularization. We find that the CST dispersion relations exhibit, in addition to the usual right-hand branch cut, also a left-hand cut. The origin of this "spectator" left-hand cut can be understood in the context of scattering for a scalar ϕ^2 χ^2-type theory. If the in…
We consider the correspondence between solutions of non-gravitational field theories formulated in Euclidean space-time and Minkowski space-time. Infinitely many "Euclidean" spaces can be obtained from M4 via a group of transformations in which the Wick rotation is a special case. We then discuss how the solutions of gauge field theories formulated in these "Euclidean" spaces have a one-to-one correspondence with the solutions of field theories formulated in Minkowski space-time, provided we avoid the one-point compactification into S4 = E4 U infinity.
Some results obtained by a new method for solving the Bethe-Salpeter equation are presented. The method is valid for any kernel given by irreducible Feynman graphs. The Bethe-Salpeter amplitude, both in Minkowski and in Euclidean spaces, and the binding energy for ladder + cross-ladder kernel are found. We calculate also the corresponding electromagnetic form factor.
We investigate the properties of quark mass functions in quantum chromodynamics calculated by the Schwinger-Dyson equation in the strong coupling region, in which the loop integration is performed in Minkowski space. The calculated results are compared with those obtained by integration in Euclidean space.
We investigate the properties of fermion mass functions in quantum electrodynamics calculated by the Schwinger-Dyson equation in the strong coupling region, in which the loop integration is performed in Minkowski space. The calculated results without the instantaneous exchange approximation are compared with those obtained by integration in Euclidean space.
We formulate the covariant equations for quark-antiquark bound states in Minkowski space in the framework of the Covariant Spectator Theory. The quark propagators are dressed with the same kernel that describes the interaction between different quarks. We show that these equations are charge-conjugation invariant, and that in the chiral limit of vanishing bare quark mass, a massless pseudoscalar bound state is produced in a Nambu-Jona-Lasinio (NJL) mechanism, which is associated with the Goldstone boson of spontaneous chiral symmetry breaking. In this introductory paper, we test the formalism …
The fermion self-energy is calculated from the rainbow-ladder truncation of the Dyson-Schwinger equation (DSE) in quantum electrodynamics (QED) for spacelike momenta and in the complex momentum plane close to the timelike region, both using Pauli-Villars regularization. Specifically, the DSE is solved in the complex momentum plane by rotating either the energy component of the four-momentum or the magnitude of Euclidean four-momentum to reach the timelike region in Minkowski space. The coupling constant is appropriately chosen to ensure the singularities of the fermion propagator are located i…
Mechanism for generation of vector meson resonances is studied in the framework of QCD Dyson-Schwinger equations, which are defined and actually solved in Minkowski space. It is suggested that the timelike pion form factor is generated by the interference of the background quark loops together with the resonant structure predominantly created in the photon-quark-antiquark vertex. It is suggested that QCD Green's functions involving quark fields are oscillating for timelike arguments, which makes the interference effect among various QCD Green's functions quite strong and important for the corr…
Using the Green function integral representation the Dyson-Schwinger equations are solved directly in Minkowski space. Essential ideas of the spectral techniques are discussed and applied on two renormalizable models: the Yukawa theory with massive pseudoscalar meson and conventional spinor QED. Within the momentum subtraction procedure, the appropriate renormalization is performed analytically which leads to the usual dispersion formulation. The electron propagator obtained in this frame is compared with the solution of Euclidean Dyson-Schwinger equation and with the perturbation theory resul…
The propagator of confined quarks is calculated for timelike momenta by transforming Minkowski Greens functions to the Temporal Euclidean space. Based on the framework of the Schwinger-Dyson equations the QCD quark propagator is obtained in two approximations which differ by assuming behaviour of gluon propagator. In both studied cases we get universal result for the light quarks: The quark mass function becomes complex bellow expected perturbative threshold, the obtained absolute value of the infrared mass is $M\simeq Λ_{QCD} $ with the infrared phase $\simeq {π\over 2}$. Permanent confinemen…
The problem of a relativistic bound-state system consisting of two scalar bosons interacting through the exchange of another scalar boson, in 2+1 space-time dimensions, has been studied. The Bethe-Salpeter equation (BSE) was solved by adopting the Nakanishi integral representation (NIR) and the Light-Front projection. The NIR allows us to solve the BSE in Minkowski space, which is a big and important challenge, since most of non-perturbative calculations are done in Euclidean space, e.g. Lattice and Schwinger-Dyson calculations. We have in this work adopted an interaction kernel containing the…
We present the solution of Schwinger-Dyson and Bethe-Salpeter equation (BSE) for a light flawour non-singlet spinless meson in Minkowski space. The equations are solved in momentum space without the use of the auxiliary Euclidean space. To exhibit that Minkowski space non perturbative calculations are actually possible is the main purpose of presented paper. According to confinement, the quark propagator is regular in momentum space, which allows us to look for the BSE solution directly in the Minkowski momentum space. We find the Minkowski space solution for confining theories is not only num…
We propose a model for the quark-antiquark interaction in Minkowski space using the Covariant Spectator Theory. We show that with an equal-weighted scalar-pseudoscalar structure for the confining part of our interaction kernel the axial-vector Ward-Takahashi identity is preserved and our model complies with the Adler-zero constraint for pi-pi-scattering imposed by chiral symmetry.
A new method for solving the Bethe-Salpeter equation is developed. It allows to find the Bethe-Salpeter amplitudes both in Minkowski and in Euclidean spaces and, as a by product, the light-front wave function. The method is valid for any kernel given by irreducible Feynman graphs. Bethe-Salpeter and Light-Front equations for scalar particles with ladder + cross-ladder kernel are solved.
The scalar three-body Bethe-Salpeter equation, with zero-range interaction, is solved in Minkowski space by direct integration of the four-dimensional integral equation. The singularities appearing in the propagators are treated properly by standard analytical and numerical methods, without relying on any ansatz or assumption. The results for the binding energies and transverse amplitudes are compared with the results computed in Euclidean space. A fair agreement between the calculations is found.
