Actual solutions of the Bethe-Salpeter equation for a two-fermion bound system are becoming available directly in Minkowski space, by virtue of a novel technique, based on the so-called Nakanishi ...integral representation of the Bethe-Salpeter amplitude and improved by expressing the relevant momenta through light-front components, i.e. k±=k0±k3. We solve a crucial problem that widens the applicability of the method to real situations by providing an analytically exact treatment of the singularities plaguing the two-fermion problem in Minkowski space, irrespective of the complexity of the irreducible Bethe-Salpeter kernel. This paves the way for feasible numerical investigations of relativistic composite systems, with any spin degrees of freedom. We present a thorough comparison with existing numerical results, evaluated in both Minkowski and Euclidean space, fully corroborating our analytical treatment, as well as fresh light-front amplitudes illustrating the potentiality of non perturbative calculations performed directly in Minkowski space.
The unpolarized twist-2 (leading) and twist-3 (subleading), T-even, transverse-momentum dependent quark distributions in the pion are evaluated for the first time by using the actual solution of a ...dynamical equation in Minkowski space. The adopted theoretical framework is based on the 4D homogeneous Bethe–Salpeter integral equation with an interaction kernel given by a one-gluon exchange, featuring an extended quark-gluon vertex. The masses of quark and gluon as well as the interaction-vertex scale have been chosen in a range suggested by lattice-QCD calculations, and calibrated to reproduce both pion mass and decay constant. The sum rules to be fulfilled by the transverse-momentum dependent distributions are carefully investigated, particularly the leading-twist one, that has to match the collinear parton distribution function, and hence can be scrutinized in terms of existing data as well as theoretical predictions. Noteworthy, the joint use of the Fock expansion of the pion state facilitates an in-depth analysis of the content of the pion Bethe–Salpeter amplitude, allowing to calculate the gluon contribution to the quark average longitudinal fraction, that results to be
∼
6
%
. The current analysis highlights the role of the gluon exchanges through quantitative analysis of collinear and transverse-momentum distributions, showing, e.g. for both leading and subleading-twists, an early departure from the widely adopted exponential fall-off, for
|
k
⊥
|
2
>
m
2
, with the quark mass
∼
Λ
QCD
.
The quantitative investigation of the scalar Bethe–Salpeter equation in Minkowski space, within the ladder-approximation framework, is extended to include the excited states. This study has been ...carried out for an interacting system composed by two massive bosons exchanging a massive scalar, by adopting (i) the Nakanishi integral representation of the Bethe–Salpeter amplitude, and (ii) the formally exact projection onto the null plane. Our analysis, on one hand, confirms the reliability of the method already applied to the ground state and, on the other one, extends the investigation from the valence distribution in momentum space to the corresponding quantity in the impact-parameter space, pointing out some relevant features, like (i) the equivalence between Minkowski and Euclidean transverse-momentum amplitudes, and (ii) the leading exponential fall-off of the valence wave function in the impact-parameter space.
The ladder Bethe-Salpeter equation of a bound (1/2)+ system, composed by a fermion and a scalar boson, is solved in Minkowski space, for the first time. The formal tools are the same already ...successfully adopted for two-scalar and two-fermion systems, namely the Nakanishi integral representation of the Bethe-Salpeter amplitude and the light-front projection of the fulfilled equation. Numerical results are presented and discussed for two interaction kernels: (i) a massive scalar exchange and (ii) a massive vector exchange, illustrating both the correlation between binding energies and the interaction coupling constants, as well as the valence content of the interacting state, through the valence probabilities and the light-front momentum distributions. In the case of the scalar exchange, an interesting side effect, to be ascribed to the repulsion generated by the small components of the Dirac spinor, is pointed out, while for the vector exchange the manifestation of the helicity conservation opens new interesting questions to be addressed within a fully nonperturbative framework, as well as the onset of a scale-invariant regime.
The Bethe-Salpeter equation for a pseudoscalar bound-system, with i) a ladder kernel with massive gluons, ii) dynamically-dressed quark mass function and iii) an extended quark-gluon vertex, is ...solved in Minkowski space by using the Nakanishi integral representation of the Bethe-Salpeter amplitude. The quark dressing is implemented through a phenomenological ansatz, which was tuned by lattice QCD calculations of the quark running mass. The latter were also used for assigning the range of the gluon mass and the parameter featuring the extended color density. This framework allows to investigate the gluon dynamics that manifest itself in the quark dressing, quark-gluon vertex and the binding, directly in the physical space. We present the first results for low-density pseudoscalar systems in order to elucidate the onset of the interplay between the above mentioned gluonic phenomena, and we discuss both static and dynamical quantities, like valence longitudinal and transverse distributions.
A dynamical model is applied to the study of the pion valence light-front (LF) wave function, obtained from the actual solution of the Bethe-Salpeter equation in Minkowski space, resorting to the ...Nakanishi integral representation. The kernel is simplified to a ladder approximation containing constituent quarks, an effective massive gluon exchange, and the scale of the extended quark-gluon interaction vertex. These three input parameters carry the infrared scale ΛQCD and are fine-tuned to reproduce the pion weak decay constant, within a range suggested by lattice calculations. Besides fπ, we present and discuss other interesting quantities on the null-plane like (i) the valence probability, (ii) the dynamical functions depending upon the longitudinal or the transverse components of the LF momentum, represented by LF-momentum distributions and distribution amplitudes, and (iii) the probability densities both in the LF-momentum space and the 3D space given by the Cartesian product of the covariant Ioffe-time and transverse coordinates, in order to perform an analysis of the dynamical features in a complementary way. The proposed analysis of the Minkowskian dynamics inside the pion, though carried out at the initial stage, qualifies the Nakanishi integral representation as an appealing effective tool, with still unexplored potentialities to be exploited for addressing correlations between dynamics and observable properties.
The pion electromagnetic form factor has been calculated for the first time from the solutions of the Bethe-Salpeter equation, obtained directly in Minkowski space by using the Nakanishi integral ...representation of the Bethe-Salpeter amplitude. The one-gluon exchange kernel contains as inputs the quark and gluon masses, as well as a scale parameter featuring the extended quark-gluon vertex. The range of variability of these parameters is suggested by lattice calculations. After presenting a very successful comparison with the existing data in the whole range of the momentum transfer, we show how inserting the light-front (LF) formalism in our approach allows to achieve further interesting results, like the evaluation of the LF valence form factor and a first investigation of the end-points effects.
For the first time, a phenomenological analysis of the experimental electromagnetic form factors of the nucleon, both in the timelike and spacelike regions, is performed by taking into account the ...effects of nonvalence components in the nucleon state, within a light-front framework. Our model, based on suitable ansatzes for the nucleon Bethe–Salpeter amplitude and a microscopic version of the well-known Vector Meson Dominance model, has only four adjusted parameters (determined by the spacelike data with χ2/datum∼1.7), and yields a nice description of the experimental electromagnetic form factors in the physical region in the range −30(GeV/c)2<q2<20(GeV/c)2, except for the neutron one in the timelike region. Valuable information can be gained in the timelike region on possible missing Vector Mesons around q2∼4.5(GeV/c)2 and q2∼8.0(GeV/c)2.
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