The transverse charge density of the pion is extracted from a dispersive analysis of the <math altimg='si1.svg'><msup><mrow><mi>e</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><msup><mrow><mi>e</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'>+</mo></mrow></msup><msup><mrow><mi>&#960;</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo></mrow></msup></math> exclusive annihilation data. A logarithmic dispersion relation is used to compute the unknown phase of the timelike pion form factor from the modulus obtained from the annihilation cross section. The method is model-independent and permits quantitative uncertainty estimates. The density is obtained with few-percent accuracy down to <math altimg='si2.svg'><mi>b</mi><mo>&#8764;</mo><mn>0.1</mn></math> fm; at smaller distances it depends qualitatively on the assumed high-energy behavior of the timelike form factor. Implications for pion structure and the relevance of pQCD asymptotics are discussed.

Spain

Enrique Ruiz Arriola

Pablo Sanchez-Puertas

United States

Christian Weiss

Pion transverse charge density from <italic>e</italic> <sup>+</sup> <italic>e</italic> <sup>&#8722;</sup> annihilation data and logarithmic dispersion relations

We employ a dispersion relation that allows us to recover the phase of the electromagnetic form factor of the pion from its absolute value above threshold. Compared to alternative approaches building on the phase, this approach builds on experimental input directly accessible at colliders. Employing the precise datasets from the <math><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math> reaction, we obtain the phase of the electromagnetic form factor up to 2.5 GeV, well beyond standard dispersive approaches. In addition, we separate the isovector and isoscalar components, that allows us to extract the <math><mi>P</mi></math>-wave <math><mi>π</mi><mi>π</mi></math> phase shift. We also provide relevant results, including the radius of the form factor and bounds in the spacelike region. Last, but not least, the study assess potential systematic uncertainties from the interpolation method and potential zeros of the form factor.

Phase of the electromagnetic form factor of the pion

We perform a combined study of the two hadronic decays <math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math> and <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> using a detailed analysis of the semileptonic decays <math><msup><mi>D</mi><mo>+</mo></msup><mo>→</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math> (<math><mo>ℓ</mo><mo>=</mo><mi>e</mi></math>, <math><mrow><mi>μ</mi></mrow></math>) thanks to the high-statistics dataset provided by the BESIII Collaboration. We propose simple and suitable amplitude parametrizations of the studied reactions that shall be of interest to experimentalists for upcoming analyses. These new parametrizations are based on the naïve-factorization hypothesis and the description of the resulting matrix elements in terms of well-known hadronic form factors, with special emphasis on the <math><mi>K</mi><mi>π</mi></math> scalar and vector cases. Such form factors account for two-body final-state interactions which fulfill analyticity, unitarity, and chiral symmetry constraints. As a result of our study, we find that the <math><mi>P</mi></math>-wave contribution fits nicely within the naïve-factorization approach, whereas the <math><mi>S</mi></math>-wave contribution requires complex Wilson coefficients that hint for possibly genuine three-body nonfactorizable effects. Our hypothesis is further supported by the examination of <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> decays, where we achieve a description in overall good agreement with data.

R. Escribano

P. Masjuan

Combined study of hadronic <math><msup><mi>D</mi><mo>+</mo></msup><mo>→</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>+</mo></msup></math> and <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> decays by means of the analysis of semileptonic <math><msup><mi>D</mi><mo>+</mo></msup><mo>→</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math> decays

The baryonic form factor of charged pions is studied in detail in the Nambu–Jona-Lasinio model with constituent quarks, where the spontaneously broken chiral symmetry is the key dynamical ingredient guaranteeing the would-be Goldstone boson nature of the pseudoscalar mesons octet. In general, this form factor arises when the isospin symmetry is broken, which is the case if the <math><mi>u</mi></math> and <math><mi>d</mi></math> quark masses are split, as in the real world, or if the electromagnetic effects were taken into account. We obtain estimates for this basic property of the pion resulting from the quark mass splitting for a range of model parameters, and importantly, for different pion masses, going up to the values used in lattice QCD. We find very stable model results, with the mean square radius of <math><msup><mi>π</mi><mo>+</mo></msup></math> in the range <math><mrow><mo>(</mo><mn>0.05</mn><mi>–</mi><mn>0.07</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></math>. From charge conjugation, the baryonic form factor of <math><msup><mi>π</mi><mo>+</mo></msup></math> and <math><msup><mi>π</mi><mo>−</mo></msup></math> are equal and opposite. We also obtain the transverse-coordinate, relativistically invariant baryonic density of the charged pion. In <math><msup><mi>π</mi><mo>+</mo></msup></math>, the inner region carries a negative, and the outside—a positive baryon number density, both cancelling to zero, as obviously the pion carries no net baryon charge. We also carry out an analogous analysis for the kaon, where the effect is much larger due to the sizable <math><mi>s</mi></math> and <math><mi>u</mi><mo>/</mo><mi>d</mi></math> quark mass splitting. We discuss the prospects of lattice QCD measurements of the baryonic form factors of charged pions and kaons.

Poland

Wojciech Broniowski

Baryonic form factors of the pion and kaon in a chiral quark model

In this work we investigate the prospect of observing new-physics signatures via CP violation in η (′) → π 0 μ + μ  −  and η′ → ημ + μ  −  decays at the REDTOP experiment. We make use of the SMEFT to parametrise the new-physics CP-violating effects and find that the projected REDTOP statistics are not competitive with respect to nEDM experiments. This reasserts the η → μ + μ  −  process as the most promising channel to find CP-violation at this experimental facility.

Rafel Escribano

Emilio Royo

New-physics signatures via CP violation in η (′) → π 0 μ + μ  −  and η′ → ημ + μ  −  decays

The baryon form factor of charged pions arises since isospin symmetry is broken. We obtain estimates for this basic property in two phenomenological ways: from simple constituent quark models with unequal up and down quark masses, and from fitting to <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></math> data. All our methods yield a positive <math><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup></math> baryon mean square radius of <math><msup><mrow><mo>(</mo><mn>0.03</mn><mo>−</mo><mn>0.04</mn><mspace width="0.25em"></mspace><mtext>fm</mtext><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math>. Hence, a picture emerges where the outer region has a net baryon, and the inner region a net antibaryon density, both compensating each other such that the total baryon number is zero. For <math><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></math> the effect is opposite.

Baryonic content of the pion

In this work we study the axial contributions to the hadronic light-by-light piece of the muon anomalous magnetic moment. We point out some theoretical ambiguities in previous estimates, and opt to perform a new evaluation using resonace chiral theory, that is free of them. As a result, we obtain <math><msubsup><mi>a</mi><mi>μ</mi><mrow><mtext>HLbL</mtext><mo>;</mo><mi>A</mi></mrow></msubsup><mo>=</mo><mspace></mspace><mrow><mo>(</mo><mn>0.</mn><msubsup><mn>8</mn><mrow><mo>−</mo><mn>0.8</mn></mrow><mrow><mo>+</mo><mn>3.5</mn></mrow></msubsup><mo>)</mo></mrow><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>11</mn></mrow></msup></math>, that might suggest a smaller value than most recent calculations, underlining the relevance of the off-shell prescription and the need for future work along this direction. Further, we find that our results depend critically on the asymptotic behavior of the form factors, and as such, emphasizes the relevance of future experiments at large photon virtualities. In addition, we present general results regarding the involved axial form factors description, comprehensively examining (and relating) the current approaches, that shall be of general interest.

Mexico

Pablo Roig

Axial-vector exchange contribution to the hadronic light-by-light piece of the muon anomalous magnetic moment

In this study, we investigate the imprints of CP violation in certain η muonic decays that could arise within the Standard Model effective field theory. In particular, we study the sensitivities that could be reached at REDTOP, a proposed η facility. After estimating the bounds that the neutron EDM places, we find still viable to discover signals of CP violation measuring the polarization of muons in η → μ + μ − decays, with a single effective operator as its plausible source.

Czechia

CP violation in η muonic decays

In this study, we revisit and complete the full next-to-leading order corrections to pseudoscalar double-Dalitz decays within the soft-photon approximation. Comparing to the previous study, we find small differences, which are nevertheless relevant for extracting information about the pseudoscalar transition form factors. Concerning the latter, these processes could offer the opportunity to test them—for the first time—in their double-virtual regime.

Karol Kampf

Jiři Novotný

Radiative corrections to double-Dalitz decays revisited

In this work, we calculate the branching ratios for the η η ′ → ℓ ¯ ℓ $$ \eta \left(\eta^{\prime}\right)\to \overline{\ell}\ell $$ decays, where ℓ = e,μ . These processes have tiny rates in the Standard Model due to spin flip, loop and electromagnetic suppressions, which could make them sensitive to New Physics effects. In order to provide a reliable input for the Standard Model, we exploit the general analytical properties of the amplitude. For that purpose, we invoke the machinery of Canterbury approximants, which provides a systematic description of the underlying hadronic physics in a data-driven fashion. Given the current experimental discrepancies, we discuss in detail the role of the resonant region and comment on the reliability of χPT calculations. Finally, we discuss the kind of new physics which we think would be relevant to account for them.