The form factor <math altimg='si1.svg'><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn><mi>&#960;</mi></mrow></msub><mo stretchy='false'>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo stretchy='false'>)</mo></math> for the anomalous process <math altimg='si2.svg'><msup><mrow><mi>&#947;</mi></mrow><mrow><mo>&#8270;</mo></mrow></msup><mi>&#960;</mi><mo stretchy='false'>&#8594;</mo><mi>&#960;</mi><mi>&#960;</mi></math> is calculated in the isospin limit for several values of the light current-quark mass (i.e., the pion mass) using Dyson&#8211;Schwinger and Bethe&#8211;Salpeter equations. Beyond a quark interaction kernel representing gluon-mediated interactions, leading beyond-rainbow-ladder effects at low energies are incorporated by back-coupling pions as explicit degrees of freedom. Building upon an earlier calculation of the quark-photon vertex that captures the branch cut associated with the two-pion threshold and the <italic>&#961;</italic>-meson resonance, the form factor <math altimg='si1.svg'><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn><mi>&#960;</mi></mrow></msub><mo stretchy='false'>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo stretchy='false'>)</mo></math> is determined for timelike Mandelstam <italic>s</italic>. In particular, predictions are made for the kinematics relevant to the Primakoff reaction studied with COMPASS/AMBER at CERN.

Spain

Angel S. Miramontes

Austria

Gernot Eichmann

Reinhard Alkofer

Timelike form factor for the anomalous process <italic>&#947;</italic> <sup>&#8270;</sup> <italic>&#960;</italic>&#8239;&#8594;&#8239; <italic>&#960;&#960;</italic>

The gauge-boson, ghost and fermion propagators as well as the gauge-boson-ghost vertex function are studied for <math><mrow><mi>SU</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></math>, <math><mrow><mi>Sp</mi><mo>(</mo><mrow><mn>2</mn><mi>N</mi></mrow><mo>)</mo></mrow></math>, and <math><mrow><mi>SO</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></math> gauge groups. We solve a set of coupled Dyson-Schwinger equations in Landau gauge for a variable, fractional number <math><msub><mi>N</mi><mi>f</mi></msub></math> of massless fermions in the fundamental representation. For large <math><msub><mi>N</mi><mi>f</mi></msub></math> we find a phase transition from a chirally broken into a chirally symmetric phase that is consistent with the behavior expected inside the conformal window. Even in the presence of fermions the gauge-boson-ghost-vertex dressing remains small. In the conformal window this vertex shows the expected power law behavior. It does not assume its tree-level value in the far infrared, but the respective dressing function is a constant greater than 1.

United Kingdom

Fabian Zierler

Dependence of the Landau gauge ghost-gluon-vertex on the number of flavors

Abundant phenomenology suggests that strong decays from relatively low-excitation hadrons into other hadrons proceed by the creation of a light quark-antiquark pair with zero total angular momentum, the so called <math><mrow><mmultiscripts><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow></math> mechanism originating from a scalar bilinear. Yet the quantum chromodynamics (QCD) interaction is perturbatively mediated by gluons of spin one, and QCD presents a chirally symmetric Lagrangian. Such scalar decay term must be spontaneously generated upon breaking chiral symmetry. We attempt to reproduce this with the help of the quark-gluon vertex in Landau gauge, whose nonperturbative structure has been reasonably elucidated in the last years, and insertions of a uniform, constant chromoelectric field. This is akin to Schwinger pair production in quantum electrodynamics (QED), and we provide a comparison with its two field-insertions diagram. We find that, the symmetry being cylindrical, the adequate quantum numbers to discuss the production are rather <math><mrow><msub><mrow><mmultiscripts><mrow><mi>Σ</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mn>0</mn></mrow></msub></mrow></math>, <math><mrow><msub><mrow><mmultiscripts><mrow><mi>Σ</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn></mrow></msub></mrow></math>, and <math><mrow><msub><mrow><mmultiscripts><mrow><mi>Π</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mn>0</mn></mrow></msub></mrow></math> as in diatomic molecules, and we indeed find a sizeable contribution of the third decay mechanism, which may give a rationale for the <math><mrow><mmultiscripts><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow></math> phenomenology, as long as the momentum of the produced pair is at or below the scale of the bare or dynamically generated fermion mass. On the other hand, ultrarelativistic fermions are rather ejected with <math><mrow><msub><mrow><mmultiscripts><mrow><mi>Σ</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn></mrow></msub></mrow></math> quantum numbers. In QED, our results suggest that <math><mrow><msub><mrow><mmultiscripts><mrow><mi>Σ</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mn>0</mn></mrow></msub></mrow></math> dominates, whereas the constraint of producing a color singlet in QCD leads to <math><mrow><msub><mrow><mmultiscripts><mrow><mi>Π</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow><mrow><mn>0</mn></mrow></msub></mrow></math> dominance at sub-GeV momenta.

Felipe J. Llanes-Estrada

Alexandre Salas-Bernárdez

Spinning pairs: Supporting <math><mrow><mmultiscripts><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow></math> quark-pair creation from Landau-gauge Green’s functions

Particle production by ultra-strong electric fields is a well-studied research field. Nevertheless, despite repeated attempts to relate the production rate within the field to the formation time of a particle, the latter is still shrouded in mystery. We provide an interpretation of a particle distribution at finite times enabling us to isolate and, therefore, identify the relevant time scales regarding particle formation in quantum physics within and beyond perturbation theory.

Matthias Diez

Germany

Christian Kohlfürst

Identifying time scales in particle production from fields

We explore the analytic structure of three-point functions using contour deformations. This method allows continuing calculations analytically from the spacelike to the timelike regime. We first elucidate the case of two-point functions with explicit explanations how to deform the integration contour and the cuts in the integrand to obtain the known cut structure of the integral. This is then applied to one-loop three-point integrals. We explicate individual conditions of the corresponding Landau analysis in terms of contour deformations. In particular, the emergence and position of singular points in the complex integration plane are relevant to determine the physical thresholds. As an exploratory demonstration of this method’s numerical implementation we apply it to a coupled system of functional equations for the propagator and the three-point vertex of <math><msup><mi>ϕ</mi><mn>3</mn></msup></math> theory. We demonstrate that under generic circumstances the three-point vertex function displays cuts which can be determined from modified Landau conditions.

Markus Q. Huber

Wolfgang J. Kern

Analytic structure of three-point functions from contour deformations

We study electromagnetic as well as strong isospin breaking effects in the isospin mass splittings of light pseudoscalar and vector mesons. To this end we employ a coupled system of quark Dyson-Schwinger and meson Bethe-Salpeter equations whose interaction kernels contain gluon, pion and photon exchange interactions. In bound states, QCD-induced isospin breaking is manifest on different levels. On the one hand, a different explicit up- and down-quark mass directly affects the propagators of the constituent quarks. On the other hand, it leads to different interaction kernels within the isospin multiplets. In addition, electromagnetic isospin breaking is induced via a photon exchange diagram. Using the kaon iso-doublet and the charged pion masses as input to determine the up, down and strange quark masses we find for the pion, kaon and rho meson mass splittings different patterns each. In particular, our results provide evidence that the effects from two sources of mass splittings, the different quark masses and the different quark charges, do not add up linearly.

Mexico

Ángel S. Miramontes

Christian S. Fischer

Hèlios Sanchis-Alepuz

Electromagnetic and strong isospin breaking in light meson masses

We calculate gluon and ghost propagators in Yang-Mills theory in linear covariant gauges. To that end, we utilize Nielsen identities with Landau gauge propagators and vertices as the starting point. We present and discuss numerical results for the gluon and ghost propagators for values of the gauge parameter <math><mn>0</mn><mo>&lt;</mo><mi>ξ</mi><mo>≤</mo><mn>5</mn></math>. Extrapolating the propagators to <math><mi>ξ</mi><mo>→</mo><mi>∞</mi></math>, we find the expected qualitative behavior. We provide arguments that our results are quantitatively reliable at least for values <math><mi>ξ</mi><mo>≲</mo><mn>1</mn><mo>/</mo><mn>2</mn></math> of the gauge-fixing parameter. It is shown that the correlation functions, and, in particular, the ghost propagator, change significantly with increasing gauge parameter. In turn, the ghost-gluon running coupling as well as the position of the zero crossing of the Schwinger function of the gluon propagator remain within the uncertainties of our calculation unchanged.

Martin Napetschnig

Jan M. Pawlowski

Yang-Mills propagators in linear covariant gauges from Nielsen identities

An exploratory study of the timelike pion electromagnetic form factor in a Poincaré-covariant bound state formalism in the isospin symmetric limit is presented. Starting from a quark interaction kernel representing gluon-intermediated interactions for valence-type quarks, nonvalence effects are included by introducing pions as explicit degrees of freedom. The two most important qualitative aspects are, in view of the presented study, the opening of the dominant <math><mi>ρ</mi></math>-meson decay channel and the presence of a multiparticle branch cut setting in which the two-pion threshold is crossed. Based on a recent respective computation of the quark-photon vertex, the pion electromagnetic form factor for spacelike and timelike kinematics is calculated. The obtained results for its absolute value and its phase compare favorably to the available experimental data, and they are analyzed in detail by comparing them to the expectations based on an isospin-symmetric version of a vector-meson dominance model.

Hèlios Sanchis Alepuz

Elucidating the effect of intermediate resonances in the quark interaction kernel on the timelike electromagnetic pion form factor

We derive general properties of the scale-dependent effective spectral dimensions of non-perturbative gauge boson propagators as they appear as solutions from different methods in Yang-Mills theories. In the ultraviolet and for short timescales the anomalous dimensions of the propagators lead to a slight decrease of the spectral dimension as compared to the one of a free propagator. Lowering the momentum scale, the spectral dimension decreases further. The class of propagators which display a maximum at Euclidean momenta, and thus violate positivity, always approaches a spectral dimension of one for large times. We also show that the longest time intervals are not related to the deep infrared but to the momentum scale defined by the position of the maximum.

Wolfgang Kern

Spectral dimension as a tool for analyzing nonperturbative propagators

We discuss an approach for accessing bound state properties, like mass and decay width, of a theory within the functional renormalization group approach. An important cornerstone is the dynamical hadronization technique for resonant interaction channels. The general framework is exemplified and put to work within the two-flavor quark-meson model. This model provides a low-energy description of the dynamics of two-flavor QCD with quark and hadronic degrees of freedom. We compare explicitly the respective results for correlation functions and observables with first principle QCD results in a quantitative manner. This allows us to estimate the validity range of low energy effective models. We also present first results for pole masses and decay widths. Next steps involving real-time formulations of the functional renormalization group are discussed.