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> &#8270; <italic>&#960;</italic>&#8239;&#8594;&#8239; <italic>&#960;&#960;</italic>

Five-point functions and five-body wave functions play an important role in many areas of nuclear and particle physics, e.g., in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><mo stretchy="false">→</mo><mn>3</mn></math> scattering processes, in the five-gluon vertex, or in the study of pentaquarks. In this work we consider the permutation group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>S</mi><mn>5</mn></msub></math> to facilitate the description of such objects. We work out the multiplets transforming under irreducible representations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>S</mi><mn>5</mn></msub></math> and provide compact formulas allowing one to cast the permutations of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mn>12345</mn></msub></math> into combinations with definite permutation symmetry. We also give the explicit expressions for the irreducible multiplet products. We consider several practical applications as examples: We arrange the four-momenta and Lorentz invariants of a five-point function into the multiplet structure, we work out the color tensors of the five-gluon vertex in the multiplet notation, and we discuss applications for five-body wave functions like those of pentaquarks.

Portugal

Raul D. Torres

Five-point functions and the permutation group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>S</mi><mn>5</mn></msub></math>

We extend the Bethe-Salpeter formalism to systems made of five valence particles. Restricting ourselves to two-body interactions, we derive the subtraction terms necessary to prevent overcounting. We solve the five-body Bethe-Salpeter equation numerically for a system of five scalar particles interacting by a scalar exchange boson. To make the calculations tractable, we implement properties of the permutation group <math altimg='si1.svg'><msub><mrow><mi>S</mi></mrow><mrow><mn>5</mn></mrow></msub></math> and construct an approximation based on intermediate two- and three-body poles. We extract the five-body ground and excited states along with the spectra obtained from the two-, three-, and four-body equations. In the limit of a massless exchange particle, the two-, three, four- and five-body states coexist within a certain range of the coupling strength, whereas for heavier exchange particles the five-body system becomes Borromean. Our study serves as a building block for the calculation of pentaquark properties using functional methods.

M.T. Peña

Five-body systems with Bethe-Salpeter equations

We present results for single axial-vector and scalar meson pole contributions to the hadronic light-by-light scattering (HLbL) part of the muon’s anomalous magnetic moment. In the dispersive approach to these quantities (in narrow width approximation) the central inputs are the corresponding space-like electromagnetic transition form factors. We determine these directly using a functional approach to QCD by Dyson–Schwinger and Bethe–Salpeter equations in the very same setup we used previously to determine pseudo-scalar meson exchange ( $$\pi $$ , $$\eta $$ and $$\eta '$$ ) as well as meson ( $$\pi $$ and ) box contributions. Particular care is taken to preserve gauge invariance and to comply with short distance constraints in both the form factors and the HLbL tensor. Our result for the contributions from a tower of axial-vector states including short distance constraints is $$a_\mu ^{\text {HLbL}}[\text {AV-tower+SDC}] = 24.8 \,(6.1) \times 10^{-11}$$ . For the combined contributions from $$f_0(980), a_0(980), f_0(1370)$$ and $$a_0(1450)$$ we find $$a_\mu ^{\text {HLbL}}[\text {scalar}] = -1.6 \,(5) \times 10^{-11}$$ .

Germany

Christian Fischer

Tim Haeuser

Oliver Regenfelder

Axial-vector and scalar contributions to hadronic light-by-light scattering

We present quantitative results for masses and the internal structure of four-quark states with two heavy quarks, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>Q</mi><msup><mi>Q</mi><mo>′</mo></msup><mover accent="true"><mi>q</mi><mo stretchy="false">¯</mo></mover><msup><mover accent="true"><mi>q</mi><mo stretchy="false">¯</mo></mover><mo>′</mo></msup></mrow></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><mo>,</mo><msup><mi>Q</mi><mo>′</mo></msup><mo>∈</mo><mo stretchy="false">{</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo stretchy="false">}</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>q</mi><mo>,</mo><msup><mi>q</mi><mo>′</mo></msup><mo>∈</mo><mo stretchy="false">{</mo><mi>u</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>s</mi><mo stretchy="false">}</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>∈</mo><msup><mn>1</mn><mo>+</mo></msup></math>. The composition of these states in terms of meson-meson and diquark-antidiquark pairs, extracted from a relativistic four-body Faddeev-Yakubowski equation, is dynamically determined from underlying QCD forces. We find states at energy levels in very good agreement with lattice QCD and, where available, with experimental states. Their internal structure, most notably between the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>b</mi><mi>c</mi></mrow></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>T</mi><mrow><mi>b</mi><mi>b</mi></mrow><mo>−</mo></msubsup></math>, show significant and sizeable variations.

Joshua Hoffer

Christian S. Fischer

Structure of open-flavor four-quark states in the charm and bottom region

We compute the bound state properties of three-dimensional scalar <math><msup><mi>ϕ</mi><mn>4</mn></msup></math> theory in the broken phase. To this end, we extend the recently developed technique of spectral Dyson-Schwinger equations to solve the Bethe-Salpeter equation and determine the bound state spectrum. We employ consistent truncations for the two-, three- and four-point functions of the theory that recover the scaling properties in the infinite coupling limit. Our result for the mass of the lowest-lying bound state in this limit agrees very well with lattice determinations.

Andrés Gómez

Jan Horak

Jan M. Pawlowski

Jonas Wessely

Nicolas Wink

Bound states from the spectral Bethe-Salpeter equation

We discuss the spectrum and the internal composition of ground and excited four-quark states in the charm and bottom energy region. To this end we extend previous calculations within the framework of the relativistic four-body Faddeev-Yakubovsky equation to include quantum numbers with <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>0</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup><mo>,</mo><msup><mn>0</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup><mo>,</mo><msup><mn>1</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup><mo>,</mo><msup><mn>1</mn><mrow><mo>+</mo><mo>−</mo></mrow></msup></math> and <math><msup><mn>1</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math> and study their internal composition in terms of heavy-light meson pairs, hadroquarkonia and diquark-antidiquark clusters. We observe similar patterns in the charm and bottom energy region with different compositions of the four-quark states depending on <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup></math> quantum numbers. Most notably, we find that all states with <math><mi>C</mi><mo>·</mo><mi>P</mi><mo>=</mo><mo>+</mo><mn>1</mn></math> are dominated by heavy-light meson contributions, whereas for axial-vector states with <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>+</mo><mo>−</mo></mrow></msup></math> including the <math><msub><mi>Z</mi><mi>c</mi></msub><mo>(</mo><mn>3900</mn><mo>)</mo></math> we find a much more complicated picture depending on the flavor content. We systematically compare our results for the spectrum with existing experimental results and provide predictions for future analyses.

Hidden-flavor four-quark states in the charm and bottom region

We explore a new method to calculate the valence light-front wave function of a system of two interacting particles, which is based on contour deformations combined with analytic continuation methods to project the Bethe-Salpeter wave function onto the light front. In this proof-of-concept study, we solve the Bethe-Salpeter equation for a scalar model and find excellent agreement between the light-front wave functions obtained with contour deformations and those obtained with the Nakanishi method frequently employed in the literature. The contour-deformation method is also able to handle extensions of the scalar model that mimic certain features of QCD such as unequal masses and complex singularities. In principle the method is suitable for computing parton distributions on the light front such as parton distribution functions, transverse momentum distributions and generalized parton distributions in the future.

Eduardo Ferreira

Alfred Stadler

Going to the light front with contour deformations

A longstanding question in QCD is the origin of the mass gap in the Yang-Mills sector of QCD, i.e., QCD without quarks. In Landau gauge QCD this mass gap, and hence confinement, is encoded in a mass gap of the gluon propagator, which is found both in lattice simulations and with functional approaches. While functional methods are well suited to unravel the mechanism behind the generation of the mass gap, a fully satisfactory answer has not yet been found. In this work we solve the coupled Dyson-Schwinger equations for the ghost propagator, gluon propagator and three-gluon vertex. We corroborate the findings of earlier works, namely that the mass gap generation is tied to the longitudinal projection of the gluon self-energy, which acts as an effective mass term in the equations. Because an explicit mass term is in conflict with gauge invariance, this leaves two possible scenarios: If it is viewed as an artifact, only the scaling solution survives; if it is dynamical, gauge invariance can only be preserved if there are longitudinal massless poles in either of the vertices. We find that there is indeed a massless pole in the ghost-gluon vertex, however in our approximation with the assumption of complete infrared dominance of the ghost this pole is only present for the scaling solution. We also put forward a possible mechanism that may reconcile the scaling solution, with an infrared dominance of the ghost, with the decoupling solutions based on longitudinal poles in the three-gluon vertex as seen in the PT-BFM scheme.

João M. Silva

Mass generation in Landau-gauge Yang-Mills theory

Models proposed to explain recently discovered heavy-light four-quark states already assume certain internal structures, i.e., the (anti)quark constituents are grouped into diquark/antidiquark clusters, heavy-meson/light-meson clusters (hadrocharmonium) or heavy-light meson molecules. We propose and use an approach to four-quark states based on Dyson-Schwinger and Bethe-Salpeter equations that has the potential to discriminate between these models. We study the masses of heavy-light <math><mrow><mi>c</mi><mi>q</mi><mover><mrow><mi>q</mi></mrow><mrow><mo>¯</mo></mrow></mover><mover><mrow><mi>c</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> and <math><mi>c</mi><mi>c</mi><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>q</mi><mo>¯</mo></mover></math> four-quark states with <math><mi>q</mi><mo>=</mo><mi>u</mi></math>, <math><mrow><mi>d</mi></mrow></math>, <math><mrow><mi>s</mi></mrow></math> and quantum numbers <math><mrow><mi>I</mi><mo>(</mo><msup><mrow><mi>J</mi></mrow><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>)</mo><mo>=</mo><mn>0</mn><mo>(</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>+</mo><mo>+</mo></mrow></msup><mo>)</mo><mo>,</mo><mn>1</mn><mo>(</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>+</mo><mo>−</mo></mrow></msup><mo>)</mo><mo>,</mo><mn>0</mn><mo>(</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo><mo>+</mo></mrow></msup><mo>)</mo></mrow></math> and <math><mrow><mn>1</mn><mo>(</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msup><mo>)</mo><mo>,</mo><mn>0</mn><mo>(</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msup><mo>)</mo><mo>,</mo><mn>1</mn><mo>(</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow></math>. We identify the dominant components of the ground states with these quantum numbers and suggest candidates for corresponding experimental states. Most notably, we find strong heavy-light meson-meson and negligible diquark-antidiquark components in all <math><mi>c</mi><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>c</mi><mo>¯</mo></mover></math> states, whereas for <math><mi>c</mi><mi>c</mi><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>q</mi><mo>¯</mo></mover></math> states diquarks are present. A potential caveat in the <math><mi>I</mi><mo>=</mo><mn>0</mn></math> channels is the necessary but costly inclusion of <math><mi>c</mi><mover><mi>c</mi><mo>¯</mo></mover></math> components which is relegated to future work.