The projection onto the Light-Front of a Minkowski space Faddeev-Bethe-Salpeter equation model truncated at the valence level is applied to study the proton structure with constituent quarks. The ...dynamics of the model has built-in: (i) a bound diquark brought by a contact interaction, and (ii) the separation by ∼ΛQCD of the infrared and ultraviolet interaction regions. The model parameters are fine tuned to reproduce the proton Dirac electromagnetic form factor and mass. From that, the non-polarized longitudinal and transverse momentum distributions were computed. The results for the evolved non-polarized valence parton distributions suggest that: (i) the explicit consideration of the spin degree of freedom of both quark and diquark seems not relevant to it, and (ii) the comparison with the global fit from the NNPDF4.0 calls for higher Fock components of the wave function beyond the valence one.
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 confinement/deconfinement transition described the Polyakov–Nambu–Jona–Lasinio (PNJL) model is extended to be operative at zero temperature regime. In this study, the scalar and vector channel ...interaction strengths of the original PNJL model are modified by introducing a dependence on the traced Polyakov loop. In such a way the effective interactions depend on the quark phase and in turn provides a backreaction of the quarks to the gluonic sector, also at zero temperature. On general grounds from quantum chromodynamics this is an expected feature. The thermodynamics of the extended model (PNJL0) is studied in detail. It presents along with a suitable choice of the Polyakov potential, a first order confined/deconfined quark phase transition even at
T
=
0
. We also show that the vector channel plays an important role in order to allow
Φ
≠
0
solutions for the PNJL0 model. Furthermore, the sensitivity of the combined quarkyonic and deconfinement phases to the vector interaction strength and the proposed parametrization of the Polyakov-loop potential at
T
=
0
allowed to set a window for the bulk values of the relevant parameters.
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 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.
The bound state Bethe–Salpeter amplitude was expressed by Nakanishi using a two-dimensional integral representation, in terms of a smooth weight function g, which carries the detailed dynamical ...information. A similar, but one-dimensional, integral representation can be obtained for the Light-Front wave function in terms of the same weight function g. By using the generalized Stieltjes transform, we first obtain g in terms of the Light-Front wave function in the complex plane of its arguments. Next, a new integral equation for the Nakanishi weight function g is derived for a bound state case. It has the standard form g=Ng, where N is a two-dimensional integral operator. We give the prescription for obtaining the kernel N starting with the kernel K of the Bethe–Salpeter equation. The derivation is valid for any kernel given by an irreducible Feynman amplitude.
The soft gluon limit of the longitudinal part of the quark-gluon vertex is studied by resorting to non-perturbative approaches to quantum chromodynamics (QCD). Based on a Slavnov–Taylor identity ...(STI), the longitudinal form factors is expressed in terms of the quark-ghost kernel, the quark self energy and the quark wave function. An exact relation between the non-vanishing longitudinal form factors is derived for the soft gluon limit and explored to understand the behaviour of the vertex. Within a Ball–Chiu vertex, the form factor
λ
1
was analysed using recent lattice simulations for full QCD for the soft gluon limit. The lattice data shows that the gluon propagator resumes the momentum dependence of such component of the vertex. This connection is understood via a fully dressed one-loop Bethe–Salpeter equation. The behaviour of the remaining longitudinal form factors
λ
2
(
p
2
)
and
λ
3
(
p
2
)
is investigated combining both the information of lattice simulations and the derived relations based on the STI.
A model for the B±→π−π+π± decay amplitude is proposed to study the large CP violation observed at the high mass region of the Dalitz plane. A short distance b→u amplitude with the weak phase γ is ...considered together with the contribution of a hadronic charm loop and a s-wave DD¯→ππ rescattering. In the model, the χc0 appears as a narrow resonant state of the DD¯ system below threshold. It is introduced in an unitary two channel S-matrix model of the coupled DD¯ and ππ channels, where the χc0 complex pole in DD¯ channel shows its signature in the off-diagonal matrix element and in the associated DD¯→ππ transition amplitude. The strong phase of the resulting decay amplitude has a sharp sign change at the DD¯ threshold, changing the sign of the CP asymmetry, as it is observed in the data. We conclude that the hadronic charm loop and rescattering mechanism are relevant to the broadening of the CP asymmetry around the χc0 resonance in the ππ channel. This novel mechanism provides a possible interpretation for the CP asymmetry challenging experimental result presented by the LHCb collaboration for the B±→π−π+π± decay in the high mass region.
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.
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