Effective interaction models of quantum chromodynamics, based on quark degrees of freedom, have been successfully employed to compute the properties of a large array of ground and excited meson and baryon states, along with their electromagnetic form factors, distribution functions, and thermal behavior. Amongst them, the contact-interaction model, while nonrenormalizable, implements confinement, satisfies Lorentz covariance, and correctly describes chiral symmetry and its dynamical breaking pattern. Original studies focused on the light hadron sector in the isospin limit and were thereafter extended to heavy mesons and baryons. The strong effective couplings, as well as infrared and ultraviolet regulators, are flavor-dependent model parameters adjusted to reproduce hadronic observables. In contrast, in this study we combine SU(4) flavor-symmetry breaking couplings, obtained from one-loop vacuum polarization amplitudes in the presence of background constituent quark currents, with the contact-interaction model. This allows us to reduce the number of mass-dimensioned parameters and to consistently relate all flavored couplings to a single mass scale, while the masses and weak decay constants of the pions, kaons, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mi>s</mi></msub></math> mesons are in good agreement with average reference values. Allowing for realistic isospin breaking, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>/</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>=</mo><mn>1.7</mn></mrow></math>, in conjunction with the effect of the flavored couplings, leads to a mass splitting, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>m</mi></mrow><mrow><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></msub><mo>−</mo><msub><mrow><mi>m</mi></mrow><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></msub><mo>≈</mo><mn>0.3</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, that agrees with lattice QCD values. For the kaons, the mass difference is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>m</mi></mrow><mrow><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></msub><mo>−</mo><msub><mrow><mi>m</mi></mrow><mrow><msup><mrow><mi>K</mi></mrow><mrow><mo>±</mo></mrow></msup></mrow></msub><mo>≈</mo><mn>2.3</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, whereas <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><msup><mi>D</mi><mo>±</mo></msup></msub><mo>−</mo><msub><mi>m</mi><msup><mi>D</mi><mn>0</mn></msup></msub><mo>≈</mo><mn>0.5</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub></math> is 6% lighter than the experimental mass.

Brazil

Fábio L. Braghin

Bruno El-Bennich

Colombia

Fernando E. Serna

Charge dependence of mesons with flavored contact-interaction couplings

The correct description of the hadron’s structure requires understanding how quarks and gluons form the observable hadrons and how they are distributed within them. Two key nonperturbative quantities encapsulate this information: parton distribution functions (PDFs) and fragmentation functions (FFs). The former define a probabilistic light-front momentum distribution of partons within a hadron, whereas the latter describe the hadronization process of high-energy partons. Computing these functions analytically poses significant challenges, as it demands models that accurately incorporate the nonperturbative infrared dynamics of quantum chromodynamics (QCD). In this work, we compute the pion PDF and its elementary and full FFs using the Curci-Ferrari (CF) model. This model enables the exploration of nonperturbative QCD effects by introducing a gluon-mass scale within the Landau gauge QCD Lagrangian. The two-point quark and gluon correlation functions derived from the CF model agree well with lattice-QCD results and reproduce the pion decay constant in the chiral limit, consistent with chiral perturbation theory. The resulting pion PDF and FFs computed with the CF quark propagator and pion Bethe-Salpeter amplitude are in good qualitative and quantitative agreement with those obtained using the Qin-Chang model, a benchmark approach to nonperturbative QCD. These findings support the broader applicability of the Curci-Ferrari model in hadron phenomenology.

Gustavo B. Bopsin

Gastão Krein

Roberto C. da Silveira

Parton distribution and fragmentation functions with massive gluons

An exploratory nonperturbative calculation of the quark propagator in light-cone gauge is motivated by distribution amplitudes whose definition implies a Wilson line. The latter serves to preserve the gauge invariance of the hadronic amplitudes and becomes trivial in light-cone gauge. To that end, we explore the corresponding Dyson-Schwinger equation in its leading truncation and with a dressed vertex derived from a Ward identity in light-cone gauge. The quark’s mass and wave renormalization functions are found to depend on the orientation of the quark momentum relative to the lightlike four-vector below 1 GeV, which expresses the light-cone gauge dependence of the propagator, while a third, complex-valued amplitude exhibits little dependence on that orientation and vanishes in the quark’s rest frame.

Roberto Correa da Silveira

Quark gap equation in light-cone gauge

The elementary fragmentation function that describes the process <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>q</mi><mo>→</mo><mi>π</mi></mrow></math> is predicted by applying crossing and charge symmetry to the cut diagram of the pion valence quark distribution function. This elementary probability distribution defines the ladder-kernel of a quark jet fragmentation equation, which is solved self-consistently to obtain the full pion fragmentation function. The hadronization into a pion employs the complete Poincaré invariant Bethe-Salpeter wave function, though the overwhelming contribution to the fragmentation function is due to the leading Bethe-Salpeter amplitude. Compared to a Nambu–Jona–Lasinio model prediction, the fragmentation function we obtain is enhanced in the range <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>z</mi><mo>≲</mo><mn>0.8</mn></mrow></math> but otherwise in good qualitative agreement. The full pion fragmentation function is overall greater than the elementary fragmentation function below <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>z</mi><mo>≲</mo><mn>0.6</mn></mrow></math>.

Pion fragmentation functions from a quark-jet model in a functional approach

The leading Fock-state light-front wave functions of heavy quarkonia and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>D</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi></math> mesons are obtained from the projections of their Bethe-Salpeter wave functions on the light front. We compute their leading-twist time-reversal even transverse momentum distributions and parton distribution functions. Mirroring the behavior of parton distribution amplitudes, the support of both distributions is increasingly narrower and shifted toward larger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> as a function of the meson mass. The dependence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi mathvariant="bold-italic">k</mi></mrow><mrow><mo>⊥</mo></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></math> of the transverse distributions does not factorize into separate functions, and their falloff with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="bold-italic">k</mi><mo>⊥</mo><mn>2</mn></msubsup></math> is much slower than that of light mesons.

Parton distribution functions and transverse momentum dependence of heavy mesons

We study the gauge dependence of the quark propagator in quantum chromodynamics by solving the gap equation with a nonperturbative quark-gluon vertex which is constrained by longitudinal and transverse Slavnov-Taylor identities, the discrete charge conjugation and parity symmetries and which is free of kinematic singularities in the limit of equal incoming and outgoing quark momenta. We employ gluon propagators in renormalizable <math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>ξ</mi></mrow></msub></mrow></math> gauges obtained in lattice QCD studies. We report the dependence of the nonperturbative quark propagator on the gauge parameter, in particular we observe an increase, proportional to the gauge-fixing parameter, of the mass function in the infrared domain, whereas the wave renormalization decreases within the range <math><mn>0</mn><mo>≤</mo><mi>ξ</mi><mo>≤</mo><mn>1</mn></math> considered here. The chiral quark condensate reveals a mild gauge dependence in the region of <math><mi>ξ</mi></math> investigated. We comment on how to build and improve upon this exploratory study in future in conjunction with generalized gauge covariance relations for QCD.

José Roberto Lessa

United States

Mexico

Adnan Bashir

Portugal

Orlando Oliveira

Gauge dependence of the quark gap equation: An exploratory study

We calculate the strong decay couplings for <math><mi>ρ</mi><mo>→</mo><mi>π</mi><mi>π</mi></math>, <math><mi>ϕ</mi><mo>→</mo><mi>K</mi><mi>K</mi></math>, <math><msup><mi>K</mi><mo>*</mo></msup><mo>→</mo><mi>K</mi><mi>π</mi></math>, and <math><msup><mi>D</mi><mo>*</mo></msup><mo>→</mo><mi>D</mi><mi>π</mi></math> in a unified and consistent approach based on the impulse approximation, nonperturbative solutions of the quark-gap equation and the Poincaré invariant Bethe-Salpeter amplitudes of vector and pseudoscalar mesons. In particular, we obtain the coupling <math><msub><mi>g</mi><mrow><msup><mi>D</mi><mo>*</mo></msup><mi>D</mi><mi>π</mi></mrow></msub><mo>=</mo><mn>17.2</mn><msubsup><mn>4</mn><mrow><mo>−</mo><mn>2.30</mn></mrow><mrow><mo>+</mo><mn>3.06</mn></mrow></msubsup></math> in very good agreement with the experimental value by CLEO, which corresponds to a strong effective coupling between heavy vector and pseudoscalar mesons to the pion of <math><mover><mi>g</mi><mo>^</mo></mover><mo>=</mo><mn>0.5</mn><msubsup><mn>8</mn><mrow><mo>−</mo><mn>0.08</mn></mrow><mrow><mo>+</mo><mn>0.10</mn></mrow></msubsup></math>.

Strong two-meson decays of light and charmed vector mesons

We report on the first calculation of the longitudinal and transverse light front distribution amplitudes of the <math><msup><mi>D</mi><mo>*</mo></msup></math> and <math><msubsup><mi>D</mi><mi>s</mi><mo>*</mo></msubsup></math> mesons and their first four moments. As a by-product, we also obtain these distribution amplitudes for the <math><mi>ρ</mi></math>, <math><mi>ϕ</mi></math>, <math><msup><mi>K</mi><mo>*</mo></msup></math>, and <math><mi>J</mi><mo>/</mo><mi>Ψ</mi></math> mesons and confirm a prediction of lattice QCD for the vector kaon: while the longitudinal distribution amplitude is almost symmetric, the transverse one is oblique implying that the strange quark carries more momentum.

<math><msup><mi>D</mi><mo>*</mo></msup></math> and <math><msubsup><mi>D</mi><mi>s</mi><mo>*</mo></msubsup></math> distribution amplitudes from Bethe-Salpeter wave functions

We extend earlier studies of transverse Ward-Fradkin-Green-Takahashi identities in QED, their usefulness to constrain the transverse fermion-boson vertex and their importance for multiplicative renormalizability, to the equivalent gauge identities in QCD. To this end, we consider transverse Slavnov-Taylor identities that constrain the transverse quark-gluon vertex and derive its eight associated scalar form factors. The complete vertex can be expressed in terms of the quark’s mass and wave-renormalization functions, the ghost-dressing function, the quark-ghost scattering amplitude and a set of eight form factors. The latter parametrize the hitherto unknown nonlocal tensor structure in the transverse Slavnov-Taylor identity which arises from the Fourier transform of a four-point function involving a Wilson line in coordinate space. We determine the functional form of these eight form factors with the constraints provided by the Bashir-Bermudez vertex and study the effects of this novel vertex on the quark in the Dyson-Schwinger equation using lattice QCD input for the gluon and ghost propagators. We observe significant dynamical chiral symmetry breaking and a mass gap that leads to a constituent mass of the order of 500 MeV for the light quarks. The flavor dependence of the mass and wave-renormalization functions as well as their analytic behavior on the complex momentum plane is studied and as an application we calculate the quark condensate and the pion’s weak decay constant in the chiral limit. Both are in very good agreement with their reference values.

Luis Albino

Eduardo Rojas

Fernando Serna

Roberto Silveira

The impact of transverse Slavnov-Taylor identities on dynamical chiral symmetry breaking

We construct a quark target model (QTM) to incorporate intrinsic glue into effective low-energy models of QCD, which often contain only quark degrees of freedom. This method guarantees the gauge invariance of observables order by order in the strong coupling. The quark and gluon PDFs for the dressed quarks are obtained in the QTM at leading order. We demonstrate gauge invariance of the results by comparing both covariant and light cone gauges, with the former including an explicit Wilson line contribution. A key finding is that in covariant gauges the Wilson line can carry a significant amount of the light cone momentum. With coupling strength <math><mrow><msub><mi>α</mi><mi>s</mi></msub><mo>=</mo><mn>0.5</mn></mrow></math> and dressed quark mass <math><mrow><msub><mi>M</mi><mi>q</mi></msub><mo>=</mo><mn>0.4</mn><mspace width="4pt"></mspace><mi>GeV</mi></mrow></math>, we find quark and gluon momentum fractions of <math><mrow><msub><mrow><mo>〈</mo><mi>x</mi><mo>〉</mo></mrow><mi>q</mi></msub><mo>=</mo><mn>0.81</mn></mrow></math> and <math><mrow><msub><mrow><mo>〈</mo><mi>x</mi><mo>〉</mo></mrow><mi>g</mi></msub><mo>=</mo><mn>0.19</mn></mrow></math>, where the Wilson line contribution to the quark momentum fraction is <math><mrow><mo>−</mo><mn>0.18</mn></mrow></math>. We use the on-shell renormalization scheme and find that at one loop this Wilson line contribution does not depend on the covariant gauge but does vanish in light cone gauge as expected. This result demonstrates that it is crucial to account for Wilson line contributions when calculating quantum correlation functions in covariant gauges. We also consider the impact of a gluon mass using the gauge invariant formalism proposed by Cornwall, and combine these QTM results with two quark-level models to obtain quark and gluon PDFs for the pion.