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.

Brazil

Roberto Correa da Silveira

Colombia

Fernando E. Serna

Bruno El-Bennich

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.

Gastão Krein

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.

Caroline S.R. Costa

Adam Freese

Ian C. Cloët

Australia

Peter C. Tandy

Intrinsic glue and Wilson lines within dressed quarks

Based on the generalized quantum electrodynamics expression for the Podolsky propagator, which preserves gauge invariance for massive photons, we propose a model for the massive gluon propagator that reproduces well-known features of established strong-interaction models in the framework of the Dyson-Schwinger equation. By adjusting the Podolsky mass and the coupling strength we thus construct a model with simple analytical properties known from perturbative theory, yet well suited to describe a confining interaction. We obtain solutions of the Dyson-Schwinger equation for the quark at spacelike momenta on the real axis as well as on the complex plane and solving the bound-state problem with the Bethe-Salpeter equation yields masses and weak decay constants of the <math><mrow><mi>π</mi></mrow></math>, <math><mrow><mi>K</mi></mrow></math> and <math><msub><mi>η</mi><mi>c</mi></msub></math> in excellent agreement with experimental values, while the <math><mi>D</mi></math> and <math><msub><mi>D</mi><mi>s</mi></msub></math> are reasonably well described. The analytical simplicity of this effective interaction has the potential to be useful for phenomenological applications and may facilitate calculations in Minkowski space.

German Ramos-Zambrano

Podolsky propagator in the gap and bound-state equations

The ladder kernel of the Bethe–Salpeter equation is amended by introducing a different flavor dependence of the dressing functions in the heavy-quark sector. Compared with earlier work this allows for the simultaneous calculation of the mass spectrum and leptonic decay constants of light pseudoscalar mesons, the $$D_u$$ <math> <msub> <mi>D</mi> <mi>u</mi> </msub> </math> , $$D_s$$ <math> <msub> <mi>D</mi> <mi>s</mi> </msub> </math> , $$B_u$$ <math> <msub> <mi>B</mi> <mi>u</mi> </msub> </math> , $$B_s$$ <math> <msub> <mi>B</mi> <mi>s</mi> </msub> </math> and $$B_c$$ <math> <msub> <mi>B</mi> <mi>c</mi> </msub> </math> mesons and the heavy quarkonia $$\eta _c$$ <math> <msub> <mi>η</mi> <mi>c</mi> </msub> </math> and $$\eta _b$$ <math> <msub> <mi>η</mi> <mi>b</mi> </msub> </math> within the same framework at a physical pion mass. The corresponding Bethe–Salpeter amplitudes are projected onto the light front and we reconstruct the distribution amplitudes of the mesons in the full theory. A comparison with the first inverse moment of the heavy meson distribution amplitude in heavy quark effective theory is made.