The strength of the interaction of three pions and a photon, $\textit{F}$ is predicted by the axial anomaly in terms of the pion decay constant, a relation that is frequently used to constrain low-energy radiative processes involving pions, but only tested experimentally at the 10% level. Here, we present a new avenue to test this prediction, via a fit of a dispersive description of the $\textit{γ}$ → 3$\textit{π}$ amplitude to data for $\textit{e}$ $\textit{e}$ → 3$\textit{π}$. From the global fit to SND, CMD-2, and BaBar data we obtain $\textit{F}$ = 33.1(1.7) GeV , in agreement with the chiral prediction at the level of 5%. We also consider dispersive fits to the recent data by Belle II, in which case we observe tensions with the dispersive constraints, the width parameters of $\textit{ω}$ and $\textit{ϕ}$, and the chiral anomaly.

Switzerland

Martin Hoferichter

Germany

Bai-Long Hoid

Bastian Kubis

Extracting the chiral anomaly from $\textit{e}$

<p>Next to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mn>0</mn></msup></math> pole, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>η</mi><mo>′</mo></msup></math> intermediate states give rise to the leading singularities of the hadronic light-by-light tensor, resulting in sizable contributions to the anomalous magnetic moment of the muon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mi>μ</mi></msub></math>. The strength of the poles is determined by the respective transition form factors (TFFs) to two (virtual) photons. We present a calculation of these TFFs that implements a number of low- and high-energy constraints, including the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>η</mi><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">→</mo><mi>γ</mi><mi>γ</mi></math> decay widths, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>η</mi></mrow><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mi>γ</mi></mrow></math> spectra, chiral symmetry for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>η</mi></mrow><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">→</mo><mn>2</mn><mo stretchy="false">(</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">)</mo></mrow></math> amplitudes, vector-meson couplings, and asymptotic limits. Crucially, we investigate the role of the leading left-hand singularity generated by the exchange of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mn>2</mn></msub></math> tensor meson, yielding, for the first time, an estimate of the associated factorization-breaking corrections. Our final results, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>η</mi><mtext> </mtext><mrow><mi>pole</mi></mrow></mrow></msubsup><mo>=</mo><mn>14.7</mn><mo stretchy="false">(</mo><mn>9</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><msup><mrow><mi>η</mi></mrow><mrow><mo>′</mo></mrow></msup><mtext> </mtext><mrow><mi>pole</mi></mrow></mrow></msubsup><mo>=</mo><mn>13.5</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup></mrow></math>, conclude a dedicated effort to evaluate the pseudoscalar-pole contributions to hadronic light-by-light scattering using dispersion relations, amounting to a combined <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mtext>PS poles</mtext></mrow></msubsup><mo>=</mo><msubsup><mrow><mn>91.2</mn></mrow><mrow><mo>−</mo><mn>2.4</mn></mrow><mrow><mo>+</mo><mn>2.9</mn></mrow></msubsup><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup></mrow></math>.</p>

Simon Holz

Precision Evaluation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi></math>- and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>η</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math>-Pole Contributions to Hadronic Light-by-Light Scattering in the Anomalous Magnetic Moment of the Muon

The pseudoscalar-pole contributions to hadronic light-by-light scattering are determined by the respective transition form factors (TFFs) into two virtual photons. These TFFs constitute complicated functions of the photon virtualities that, in turn, can be reconstructed in a dispersive approach from their discontinuities. In this work, we present such an analysis for the $\textit{η}$ TFFs, implementing a number of constraints from both experiment and theory: normalizations from the $\textit{η}$ → $\textit{γγ}$ decay widths, unitarity constraints from the $\textit{η}$ → $\textit{π}$ $\textit{π}$ $\textit{γ}$ spectra, chiral symmetry for the $\textit{η}$ → 2($\textit{π}$ $\textit{π}$ ) amplitudes, vector-meson couplings, singly-virtual data from $\textit{e}$ $\textit{e}$ → $\textit{e}$ $\textit{e}$ $\textit{η}$ , and the asymptotic behavior predicted by the light-cone expansion. In particular, we account for the leading left-hand-cut singularity by including effects from the $\textit{a}$ resonance, necessitating the solution of an inhomogeneous Muskhelishvili-Omnès problem via a carefully chosen path deformation. The resulting TFFs allow us to evaluate the $\textit{η}$ -pole contributions to the anomalous magnetic moment of the muon, $$ {a}_{\mu}^{\upeta -\textrm{pole}} $$ = 14$\textit{.}$7(9)×10 and $$ {a}_{\mu}^{\upeta^{\prime }-\textrm{pole}} $$ = 13$\textit{.}$5(7)×10 , completing a dedicated program for the lowest-lying pseudoscalar intermediate states in a dispersive approach to hadronic light-by-light scattering, $$ {a}_{\mu}^{\textrm{PS}-\textrm{poles}} $$ = $$ {91.2}_{-2.4}^{+2.9} $$ × 10 .

Dispersion relation for hadronic light-by-light scattering: $\textit{η}$ and $\textit{η}$

We discuss a novel and simple parametrisation of the pion vector form factor that transparently connects spacelike and timelike regions of the momentum transfer <math altimg='si1.svg'><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></math>. Our parametrisation employs the framework of conformal mapping and respects the known analyticity properties of the form factor, accounting explicitly for the <math altimg='si2.svg'><mi>&#961;</mi><mo stretchy='false'>(</mo><mn>770</mn><mo stretchy='false'>)</mo></math>-meson pole. The parametrisation manifestly fulfils the normalisation condition at <math altimg='si3.svg'><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>0</mn></math> as well as further restrictions at the pion production threshold and in the limit <math altimg='si4.svg'><mo stretchy='false'>|</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy='false'>|</mo><mo stretchy='false'>&#8594;</mo><mo>&#8734;</mo></math>. In contrast to the widely used Omn&#232;s parametrisation, our approach does not use the pion&#8211;pion scattering phase shift as input. We confront the parametrisation with experimental data from <italic>&#960;H</italic> scattering and <math altimg='si5.svg'><msup><mrow><mi>&#964;</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo></mrow></msup><mo stretchy='false'>&#8594;</mo><msup><mrow><mi>&#960;</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo></mrow></msup><msup><mrow><mi>&#960;</mi></mrow><mrow><mn>0</mn></mrow></msup><mi>&#957;</mi></math> decay. We already find a good description of the data with only five free parameters, which include the pole mass and decay width of the <math altimg='si2.svg'><mi>&#961;</mi><mo stretchy='false'>(</mo><mn>770</mn><mo stretchy='false'>)</mo></math>.

United Kingdom

Matthew Kirk

France

Méril Reboud

Danny van Dyk

A simple parametrisation of the pion form factor

The polarizabilities of light pseudoscalar mesons can be extracted from differential cross sections for Compton scattering near threshold. While this has been accomplished for charged pions employing Primakoff reactions, a corresponding measurement for kaons will be affected by the presence of the $$K^*(892)$$ resonance not too far from threshold. We propose a method to extend the energy range serviceable for this purpose by reconstructing the $$K^*(892)$$ contribution model-independently from its $$K\pi $$ intermediate state, using dispersion theory. We point out that, in contrast to the charged-pion analog, there is likely no strong hierarchy between sum and difference of electric and magnetic dipole polarizabilities; we discuss the sensitivity to disentangling both by improved experimental angular coverage.

Dominik Stamen

Jan Dammann

Yannis Korte

Polarizabilities from kaon Compton scattering

We study the effects of anomalous thresholds on the non-local form factors describing the hadronization of the light-quark contribution to B → (P, V)γ * transitions. Starting from a comprehensive discussion of anomalous thresholds in the triangle loop function for different mass configurations, we detail how the dispersion relation for ππ intermediate states is affected by contour deformations mandated by the anomalous branch points. Phenomenological estimates of the size of the anomalous contributions to the form factors are provided with couplings determined from measured branching fractions and Dalitz plot distributions. Our key finding is that anomalous effects are suppressed on the ρ(770) resonance, while off-peak the effects can become as large as   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (10%) of the full (light-quark-loop induced) non-local form factors. We comment on future generalizations towards higher intermediate states and the charm loop, outlining how the dispersive framework established in this work could help improve the non-local form factors needed as input for a robust interpretation of B → (P, V)ℓ + ℓ  −  decays.

Simon Mutke

Anomalous thresholds in B → (P, V)γ * form factors

A reliable determination of the pole parameters and residues of nucleon resonances is notoriously challenging, given the required analytic continuation into the complex plane. We provide a comprehensive analysis of such resonance parameters accessible with Roy–Steiner equations for pion–nucleon scattering—a set of partial-wave dispersion relations that combines the constraints from analyticity, unitarity, and crossing symmetry—most prominently of the <math><mi>Δ</mi><mo>(</mo><mn>1232</mn><mo>)</mo></math> resonance. Further, we study the Roper, <math><mi>N</mi><mo>(</mo><mn>1440</mn><mo>)</mo></math>, resonance, which lies beyond the strict domain of validity, in comparison to Padé approximants, comment on the role of subthreshold singularities in the S-wave, and determine the residues of the <math><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>500</mn><mo>)</mo></math>, <math><mi>ρ</mi><mo>(</mo><mn>770</mn><mo>)</mo></math>, and <math><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>980</mn><mo>)</mo></math> resonances in the t-channel process <math><mi>π</mi><mi>π</mi><mo>→</mo><mover><mrow><mi>N</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>N</mi></math>. The latter allows us to test—for the first time fully model independently in terms of the respective residues—universality of the <math><mi>ρ</mi><mo>(</mo><mn>770</mn><mo>)</mo></math> couplings and the Goldberger–Treiman relation expected if the scalars behaved as dilatons, in both cases revealing large deviations from the narrow-resonance limit.

Spain

Jacobo Ruiz de Elvira

Ulf-G. Meißner

Nucleon resonance parameters from Roy–Steiner equations

Based on recent progress in the systematic analysis of C and CP violation in the light-meson sector, we calculate the C-odd transition amplitudes η → π 0 ℓ + ℓ  −  and η  ′  → ηℓ + ℓ  − . Focusing on long-distance contributions driven by the lowest-lying hadronic intermediate states, we work out the correlations between these beyond-the-Standard-Model signals and the Dalitz-plot asymmetries in η → π 0 π + π  −  and η  ′   → ηπ + π  − , using dispersion theory.

Hakan Akdag

Andreas Wirzba

Correlations of C and CP violation in η → π 0 ℓ + ℓ − and η ′ → ηℓ + ℓ −

In recent years, a persistent tension between phenomenological and lattice QCD determinations of the pion–nucleon σ-term <math><msub><mrow><mi>σ</mi></mrow><mrow><mi>π</mi><mi>N</mi></mrow></msub></math> has developed. In particular, lattice-QCD calculations have matured to the point that isospin-violating effects need to be included. Here, we point out that the standard conventions adopted in both fields are incompatible, with the data-driven extraction based on the charged-pion mass, but lattice-QCD conventions relying on the mass of the neutral pion to define the isospin limit. The corresponding correction amounts to <math><mi>Δ</mi><msub><mrow><mi>σ</mi></mrow><mrow><mi>π</mi><mi>N</mi></mrow></msub><mo>=</mo><mn>3.1</mn><mo>(</mo><mn>5</mn><mo>)</mo><mspace width="0.2em"></mspace><mtext>MeV</mtext></math> when evaluated in chiral perturbation theory with low-energy constants determined from a Roy–Steiner analysis of pion–nucleon scattering as well as <math><msub><mrow><mi>σ</mi></mrow><mrow><mi>π</mi><mi>N</mi></mrow></msub></math> itself. It reduces the tension with lattice QCD, and should be included in the comparison to phenomenological determinations. We also update the extraction from pionic atoms accounting for the latest measurement of the width of pionic hydrogen, <math><msub><mrow><mi>σ</mi></mrow><mrow><mi>π</mi><mi>N</mi></mrow></msub><mo>=</mo><mn>59.0</mn><mo>(</mo><mn>3.5</mn><mo>)</mo><mspace width="0.2em"></mspace><mtext>MeV</mtext></math>, and provide the corresponding set of scalar couplings of the nucleon.

On the role of isospin violation in the pion–nucleon -term

We perform a theoretical analysis of the semileptonic decays <math><mrow><msup><mrow><mi>η</mi></mrow><mrow><mo>(</mo><mo>′</mo><mo>)</mo></mrow></msup><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mo>ℓ</mo></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mo>ℓ</mo></mrow><mrow><mo>−</mo></mrow></msup></mrow></math> and <math><msup><mi>η</mi><mo>′</mo></msup><mo>→</mo><mi>η</mi><msup><mo>ℓ</mo><mo>+</mo></msup><msup><mo>ℓ</mo><mo>−</mo></msup></math>, where <math><mo>ℓ</mo><mo>=</mo><mi>e</mi></math>, <math><mrow><mi>μ</mi></mrow></math>, via a charge-conjugation-conserving two-photon mechanism. The underlying form factors are modeled using vector-meson dominance, phenomenological input, and U(3) flavor symmetry. We consider both a monopole and a dipole model, the latter tailored such that the expected high-energy behavior is ensured. Furthermore, we benchmark the effect of <math><mi>S</mi></math>-wave rescattering contributions to the decays. We infer significant effects of the form factors neglected in the literature so far, still finding branching ratios of the various decays well below the current experimental upper limits.

Hannah Schäfer

Marvin Zanke

Semileptonic decays <math><mrow><msup><mrow><mi>η</mi></mrow><mrow><mo>(</mo><mo>′</mo><mo>)</mo></mrow></msup><mo>→</mo><msup><mrow><mi>π</mi></mrow><mn>0</mn></msup><msup><mrow><mo>ℓ</mo></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mo>ℓ</mo></mrow><mrow><mo>−</mo></mrow></msup></mrow></math> and <math><mrow><msup><mrow><mi>η</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>→</mo><mi>η</mi><msup><mrow><mo>ℓ</mo></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mo>ℓ</mo></mrow><mrow><mo>−</mo></mrow></msup></mrow></math> in the standard model