Pion Bethe-Salpeter Research Articles

QCD anomalies in electromagnetic processes: A solution to the γ→3π puzzle

In this work, the γ→3π form factor is calculated within the Dyson-Schwinger equations framework using a contact interaction model within the so-called modified rainbow ladder truncation. The present calculation takes into account the pseudovector component in the pion Bethe-Salpeter amplitude (BSA) and π−π scattering effects, producing a γ→3π anomaly that is 1+6Rπ2 larger than the low energy prediction. Here Rπ is the relative ratio of the pseudovector and pseudoscalar components in the pion BSA; with our parameters’ input, this correction raises the γ→3π anomaly by around 10%. The main outcome of this work is the unveiling of the origin of such a correction, which could be a possible explanation of the discrepancy between the existing experimental data and the low energy prediction. Moreover, it is highlighted how the magnitude of the anomaly is affected in effective theories that require an irremovable ultraviolet cutoff. We find that for both the anomalous processes π→2γ and γ→3π, the missing contribution to the anomaly can be compensated by the additional structures related with the quark anomalous magnetic moment. Published by the American Physical Society 2024

Unpolarized transverse-momentum dependent distribution functions of a quark in a pion with Minkowskian dynamics

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%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim 6\\%$$\\end{document}. 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>m2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\ extbf{k}_\\perp |^2> m^2$$\\end{document}, with the quark mass ∼ΛQCD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim {\\varLambda }_{QCD}$$\\end{document}.

Pion inspired by QCD: Nakanishi and light-front integral representations

The pion structure in Minkowski space is explored using the Nakanishi integral representation. A general framework is developed for the pion Bethe-Salpeter amplitude based on the K\"allen-Lehmann representation of the dressed quarks with an ansatz for the pseudoscalar $\ensuremath{\pi}\ensuremath{-}q\overline{q}$ vertex fulfilling the axial Ward-Takahashi identity. The Nakanishi weight functions are derived for the scalar amplitudes associated with the decomposition of the pion Bethe-Salpeter amplitude in different operator bases in the Dirac spinor space in terms of the involved spectral densities. The approach is applied to an analytical model of the pion Bethe-Salpeter amplitude, which is combined with Landau gauge lattice QCD results for the quark running mass at spacelike momentum. From the Nakanishi integral representation several pion observables were calculated, such as the decay constant, the spin decomposition of the valence probabilities, longitudinal and transverse momentum distributions from the valence component of the light-front wave function.

A Nakanishi-based model illustrating the covariant extension of the pion GPD overlap representation and its ambiguities

A systematic approach for the model building of Generalized Parton Distributions (GPDs), based on their overlap representation within the DGLAP kinematic region and a further covariant extension to the ERBL one, is applied to the valence-quark pion's case, using light-front wave functions inspired by the Nakanishi representation of the pion Bethe–Salpeter amplitudes (BSA). This simple but fruitful pion GPD model illustrates the general model building technique and, in addition, allows for the ambiguities related to the covariant extension, grounded on the Double Distribution (DD) representation, to be constrained by requiring a soft-pion theorem to be properly observed.

Minkowski space pion model inspired by lattice QCD running quark mass

Abstract The pion structure in Minkowski space is described in terms of an analytic model of the Bethe–Salpeter amplitude combined with Euclidean Lattice QCD results. The model is physically motivated to take into account the running quark mass, which is fitted to Lattice QCD data. The pion pseudoscalar vertex is associated to the quark mass function, as dictated by dynamical chiral symmetry breaking requirements in the limit of vanishing current quark mass. The quark propagator is analyzed in terms of a spectral representation, and it shows a violation of the positivity constraints. The integral representation of the pion Bethe–Salpeter amplitude is also built. The pion space-like electromagnetic form factor is calculated with a quark electromagnetic current, which satisfies the Ward–Takahashi identity to ensure current conservation. The results for the form factor and weak decay constant are found to be consistent with the experimental data.

Pion generalized parton distributions within a fully covariant constituent quark model

We extend the investigation of the Generalized Parton Distribution for a charged pion within a fully covariant constituent quark model, in two respects: (i) calculating the tensor distribution and (ii) adding the treatment of the evolution, needed for achieving a meaningful comparison with both the experimental parton distribution and the lattice evaluation of the so-called generalized form factors. Distinct features of our phenomenological covariant quark model are: (i) a 4D Ansatz for the pion Bethe-Salpeter amplitude, to be used in the Mandelstam formula for matrix elements of the relevant current operators, and (ii) only two parameters, namely a quark mass assumed to hold $m_q=~220$ MeV and a free parameter fixed through the value of the pion decay constant. The possibility of increasing the dynamical content of our covariant constituent quark model is briefly discussed in the context of the Nakanishi integral representation of the Bethe-Salpeter amplitude.

Pion dissociation and Levinson’s theorem in hot PNJL quark matter

Pion dissociation by the Mott effect in quark plasma is described within the generalized Beth-Uhlenbeck approach on the basis of the PNJL model which allows for a unified description of bound, resonant and scattering states. As a first approximation, we utilize the Breit-Wigner ansatz for the spectral function and clarify its relation to the complex mass pole solution of the pion Bethe-Salpeter equation. Application of the Levinson theorem proves that describing the pion Mott dissociation solely by means of spectral broadening of the pion bound state beyond T_Mott leaves out a significant aspect. Thus we acknowledge the importance of the continuum of scattering states and show its role for the thermodynamics of pion dissociation.

Exploring the Pion Phenomenology Within a Fully Covariant Constituent Quark Model

An overview of the vector and tensor Generalized Parton Distributions for a charged pion is presented. Such observables, belonging to the set of quantities fundamental for a detailed investigation of the hadronic inner dynamics, have been evaluated within a fully covariant Constituent Quark Model, based on a proper Ansatz of the pion Bethe-Salpeter Amplitude and the Mandelstam formula for matrix elements of operators acting on relativistic composite systems. Given the very encouraging results already obtained for the vector distribution of a charged pion, the model has been extended to the tensor one, and some preliminary calculations will be illustrated.

Volume behavior of quark condensate, pion mass, and decay constant from Dyson-Schwinger equations

We solve the coupled system of Dyson-Schwinger and Bethe-Salpeter equations for the quark propagator and the pion Bethe-Salpeter amplitude on a finite volume. To this end we use a truncation scheme that includes pion cloud effects in the quark propagator and light mesons. We study volume effects in the quark condensate, the pion mass and the pion decay constant and compare to corresponding results in other approaches. In general we find large effects for volumes below V=(1.8 fm)^4.

Frame dependence of the pair contribution to the pion electromagnetic form factor in a light-front approach

The frame dependence of the pair-term contribution to the electromagnetic form factor of the pion is studied within the Light Front approach. A symmetric ansatz for the pion Bethe-Salpeter amplitude with a pseudo scalar coupling of the constituent to the pion field is used. In this model, the pair term vanishes for the Drell-Yan condition, while it is dominant for momentum transfer along the light-front direction.

Analytic proof that the quark model complies with partially conserved axial current theorems

The Weinberg theorem, the Adler self-consistency zero, the Goldberger and Treiman relation, and the Gell-Mann, Oakes, and Renner relation are proved analytically in full detail for quark models. These proofs are independent of the particular quark-quark interaction, and they are displayed with Feynman diagrams in a compact notation. I assume the ladder truncation, which is natural in the quark model, and also detail the diagrams that must be included in each relation. Off mass shell and finite size effects are included in the quark-antiquark pion Bethe-Salpeter vertices. The axial and vector Ward identities, for the quark propagator and for the ladder, exactly cancel any model dependence.

Goldstone boson's valence-quark distribution

Dynamical chiral symmetry breaking (DCSB) is one of the keystones of low-energy hadronic phenomena. Dyson-Schwinger equations provide a model-independent quark-level understanding and correlate that with the behaviour of the pion's Bethe-Salpeter amplitude. This amplitude is a core element in the calculation of pion observables and combined with the dressed-quark Schwinger function required by DCSB it yields a valence-quark distribution function for the pion that behaves as (1 – x ) 2 for x ∼ 1, in accordance with perturbative analyses. This behaviour can be verified at contemporary experimental facilities.

Pseudovector components of the pion,π0→γγ,andFπ(q2)

As a consequence of dynamical chiral symmetry breaking the pion Bethe-Salpeter amplitude necessarily contains terms proportional to ${\ensuremath{\gamma}}_{5}\ensuremath{\gamma}\ensuremath{\cdot}P$ and ${\ensuremath{\gamma}}_{5}\ensuremath{\gamma}\ensuremath{\cdot}k,$ where $k$ is the relative and $P$ the total momentum of the constituents. These terms are essential for the preservation of low-energy theorems, such as the Gell-Mann--Oakes-Renner relation and those describing anomalous decays of the pion, and to obtaining an electromagnetic pion form factor that falls as ${1/q}^{2}$ for large ${q}^{2},$ up to calculable $\mathrm{ln}{q}^{2}$ corrections. In a simple model, which correlates low- and high-energy pion observables, we find ${q}^{2}{F}_{\ensuremath{\pi}}{(q}^{2})\ensuremath{\sim}0.12--0.19{\mathrm{GeV}}^{2}$ for ${q}^{2}\ensuremath{\gtrsim}10{\mathrm{GeV}}^{2}.$

Pion mass and decay constant

Independent of assumptions about the form of the quark-antiquark scattering kernel we derive the explicit relation between the pion Bethe-Salpeter amplitude, Gamma_pi, and the quark propagator in the chiral limit; Gamma_pi necessarily involves a non-negligible gamma_5 gamma.P term (P is the pion four-momentum). We also obtain exact expressions for the pion decay constant, f_pi, and mass, both of which depend on Gamma_pi; and demonstrate the equivalence between f_pi and the pion Bethe-Salpeter normalisation constant in the chiral limit. We stress the importance of preserving the axial-vector Ward-Takahashi identity in any study of the pion itself, and in any study whose goal is a unified understanding of the properties of the pion and other hadronic bound states.

Electromagnetic from factors of charged and neutral kaons

The charged and neutral kaon form factors are calculated as a phenomenological application of model QCD Dyson-Schwinger equations. The results are compared with the pion form factor calculated in the same framework and yield F K ± ( Q 2) > F π ± ( Q 2) on Q 2 ϵ [0, 3] GeV 2; and a negative charge radius for the neutral kaon. These results are sensitive to the difference between the kaon and pion Bethe-Salpeter amplitude and the u- and s-quark propagation characteristics.

Pion Form Factor

We give the normalized leading asymptotic Q2 dependence of the pion form factor in quantum chromodynamics: Fπ(Q2)→Q2→±∞-2fπ2/bQ2ln|Q2|, where fπ is the pion decay constant and b=(11-2/3Nf)/16π2. Up to non-leading-logarithmic corrections, this is equivalent to Fπ(Q2)→Q2→±∞8παs(Q2)fπ2/(-Q2). These results are obtained by solving the light-cone pion Bethe-Salpeter equation in quantum chromodynamics to leading-logarithmic accuracy.

Parton model and the Bethe-Salpeter wave function

The quark-parton model is discussed from the point of view that the quark partons are the quanta created by the Fourier transforms of the quark fields at constant ${x}^{+}$. We argue using this picture that, for a given behavior of a hadron's deep-inelastic structure function ${W}_{2}(x)$ as $x\ensuremath{\rightarrow}1$, the Drell-Yan-West relation provides a lower bound on the behavior of its form factor ${F}_{1}({Q}^{2})$ as ${Q}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$. The connection between the parton and Bethe-Salpeter descriptions of hadron structure is described and used to translate known information about the pion's Bethe-Salpeter wave function into information about the amplitude for a pion to consist of precisely two quark partons. We find that the two-parton contributions to $\ensuremath{\nu}{W}_{2}(x)$ and $F({Q}^{2})$ behave roughly like $\ensuremath{\nu}{W}_{2}\ensuremath{\propto}{(1\ensuremath{-}x)}^{2}$ for small ($1\ensuremath{-}x$) and $F({Q}^{2})\ensuremath{\propto}{({Q}^{2})}^{\ensuremath{-}1}$ for large ${Q}^{2}$, respectively.

Light-cone behavior of the pion Bethe-Salpeter wave function in the ladder model

The Bethe-Salpeter wave function $\ensuremath{\chi}({q}^{\ensuremath{\nu}}+{P}^{\ensuremath{\nu}}, {q}^{\ensuremath{\nu}})$ for two spin-\textonehalf{} quarks bound by the exchange of a scalar meson is examined in the ladder model. We seek the behavior of $\ensuremath{\chi}$ as the squared momentum, ${(q+P)}^{2}$, on one leg becomes infinite while the squared momentum, ${q}^{2}$, on the other leg remains fixed. This behavior is investigated by making a Wick rotation, expanding $\ensuremath{\chi}$ in partial-wave amplitudes ${\ensuremath{\chi}}_{J}^{i}({q}^{2})$ of the group O(4), and then looking for the rightmost poles of ${\ensuremath{\chi}}_{J}^{i}({q}^{2})$ in the complex $J$ plane. Our results verify (in the ladder model) the useful hypothesis that the locations of these poles are independent of ${q}^{2}$ and can thus be computed in the ${q}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit by using conformal invariance.

Large-Q2behavior of the pion electromagnetic form factor

We study the large-${Q}^{2}$ behavior of the electromagnetic form factor of the pion, which is viewed as a quark-antiquark bound state in a (nongauge) quantum field theory. When the pion's Bethe-Salpeter wave function is expanded in O(4) partial waves, it is found that the information needed about the partial-wave amplitudes is their scaling behavior at large momentum and the locations of their poles in the complex $J$ plane. This information is determined by using the operator-product expansion, conformal invariance at short distances, and a regularity property that holds at least in the ladder model. The resulting behavior of the form factor is roughly $F({Q}^{2})\ensuremath{\sim}{({Q}^{2})}^{\ensuremath{-}1}$, with corrections due to anomalous dimensions.