arXiv subjects

Bai-Long Hoid

Publications and source records attributed to Bai-Long Hoid.

18 recordsLinked to original sources

arXiv2026

Dispersive analysis of the $\boldsymbol{J/\psi \to \gamma \pi^0 \pi^0}$ process

We present a dispersive amplitude analysis of the low-energy $\pi^0\pi^0$ system in the radiative decay $J/\psi\to\gamma\pi^0\pi^0$, using the mass-independent BESIII $0^{++}$ and $2^{++}$ intensities, the total spectrum, and the measured $0^{++}-2^{++}$ $E1$ phase difference. The isoscalar $S$-wave is described by a coupled-channel $\pi\pi/K\bar K$ Muskhelishvili-Omn\`es representation, which implements the strong final-state interactions associated with the $f_0(500)$ and $f_0(980)$. The $D$-wave is treated with a single-channel $\pi\pi$ Muskhelishvili-Omn\`es representation, where we identify all kinematic constraints of helicity amplitudes before transforming them to the experimental $E1$, $M2$, and $E3$ multipoles. Smooth short-distance production of a pseudoscalar-meson pair is encoded in subtraction polynomials, while left-hand-cut effects are estimated and found to be numerically subleading. We identify the negative solution of the BESIII $0^{++}-2^{++}$ $E1$ phase ambiguity, after using the modulo-$\pi$ freedom of production amplitudes, as the phase solution compatible with unitarity constraints, showing that the measured phase information can be accommodated with the Omn\`es phase motion without requiring large additional phases. By normalizing the BESIII intensities with the extracted branching fraction, we fix the absolute scale of the fitted amplitudes, making them suitable as input for future dispersive studies of two-pion contributions to gravitational form factors.

hep-ph↗

arXiv2026

Dispersive analysis of the $\boldsymbol{\phi \to \gamma \pi^0 \pi^0}$ process

We present an analysis of the radiative decay $\phi \to \gamma \pi^0 \pi^0$ in a dispersive framework, where the two-pion subsystem undergoes strong final-state interactions that cover the $f_0(500)$ and $f_0(980)$ regions. We employ a coupled-channel Muskhelishvili-Omn\`es framework that allows for a consistent treatment of two scalar resonances and crossed-channel singularities induced by the Born and vector-meson exchanges. We explicitly verify the equivalence between the modified and standard Muskhelishvili-Omn\`es representations for vector-meson pole contributions when the isoscalar Omn\`es matrix is chosen asymptotically bounded, and we adopt the standard representation in decay kinematics. This yields, for the first time, a parameter-free dispersive prediction for the kaon Born rescattering, which provides a dominant contribution. To obtain a good fit to the KLOE and SND data, we employ a once-subtracted coupled-channel dispersion relation with heavier left-hand cut contributions and two unknown subtraction constants. The results demonstrate the consistency among the data for $\pi\pi$ scattering, $\gamma\gamma$ fusion, and $\phi $ radiative decay, thereby validating the underlying dispersive formalism and the input used for the hadronic Omn\`es matrix and left-hand cuts.

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arXiv2025

Improved Standard-Model predictions for $\eta^{(\prime)}\to \ell^+ \ell^-$

The rare decays $\eta^{(\prime)}\to\ell^+\ell^-$, $\ell\in\{e,\mu\}$, are highly suppressed in the Standard Model, both by their chirality structure and the required loop attaching the lepton line to the $\eta^{(\prime)}\to\gamma^*\gamma^*$ matrix element. The latter is described by a single scalar function, the transition form factor, which has recently been studied in great detail for $\eta^{(\prime)}$ in the context of the pseudoscalar-pole contributions to hadronic light-by-light scattering in the anomalous magnetic moment of the muon. Based on these results, we evaluate the corresponding prediction for the $\eta^{(\prime)}$ dilepton decays, supplemented by an improved evaluation of the asymptotic contributions including pseudoscalar mass effects. In particular, the dispersive representation for the $\eta^{(\prime)}$ transition form factors allows us, for the first time, to perform a robust evaluation of the imaginary parts due to subleading channels besides the dominant two-photon cut. Our final results are $\text{Br}[\eta\to e^+e^-]=5.37(4)(2)[4]\times 10^{-9}$, $\text{Br}[\eta\to \mu^+\mu^-]=4.54(4)(2)[4]\times 10^{-6}$, $\text{Br}[\eta'\to e^+e^-]=1.80(2)(3)[3]\times 10^{-10}$, and $\text{Br}[\eta'\to \mu^+\mu^-]=1.22(2)(2)[3]\times 10^{-7}$, where the errors refer to the uncertainty in the normalized branching fraction, the one propagated from $\text{Br}[\eta^{(\prime)}\to\gamma\gamma]$, and the total uncertainty, respectively. The branching fraction for $\eta\to\mu^+\mu^-$ exhibits a mild $1.6\sigma$ tension with experiment, and we explore the bounds that can be derived on physics beyond the Standard Model.

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arXiv2025

Modified L\"uscher zeta-function and the modified effective range expansion in the presence of a long-range force

An efficient numerical algoritm is proposed for the calculation of the modified L\"uscher zeta-function in the presence of a long-range force. Using the formalism developed in Ref.~\cite{Bubna:2024izx} for the analysis of synthetic data on the finite-volume energy levels in a toy model, it is demonstrated that, in contrast to the standard L\"uscher approach, the truncation of the higher partial waves has very little effect on the final result. Furthermore, the regularization and renormalization of the modified L\"uscher zeta-function is discussed in detail, as well as the problems arising within the cutoff regularization. It is shown that, using the renormalization scheme proposed in the present paper, one obtains modified effective range expansion parameters of natural size in all partial waves.

hep-lat↗

arXiv2025

Extracting the chiral anomaly from $e^+e^-\to 3\pi$

The strength of the interaction of three pions and a photon, $F_{3\pi}$ 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 $\gamma^*\to3\pi$ amplitude to data for $e^+e^-\to 3\pi$. From the global fit to SND, CMD-2, and BaBar data we obtain $F_{3\pi}=33.1(1.7)\,\text{GeV}^{-3}$, 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 $\omega$ and $\phi$, and the chiral anomaly.

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arXiv2025

Kaon Physics: A Cornerstone for Future Discoveries

The kaon physics programme, long heralded as a cutting-edge frontier by the European Strategy for Particle Physics, continues to stand at the intersection of discovery and innovation in high-energy physics (HEP). With its unparalleled capacity to explore new physics at the multi-TeV scale, kaon research is poised to unveil phenomena that could reshape our understanding of the Universe. This document highlights the compelling physics case, with emphasis on exciting new opportunities for advancing kaon physics not only in Europe but also on a global stage. As an important player in the future of HEP, the kaon programme promises to drive transformative breakthroughs, inviting exploration at the forefront of scientific discovery.

hep-ph↗

arXiv2024

Dispersion relation for hadronic light-by-light scattering: $\eta$ and $\eta'$ poles

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 $\eta^{(\prime)}$ TFFs, implementing a number of constraints from both experiment and theory: normalizations from the $\eta^{(\prime)}\to\gamma\gamma$ decay widths, unitarity constraints from the $\eta^{(\prime)}\to\pi^+\pi^-\gamma$ spectra, chiral symmetry for the $\eta^{(\prime)}\to 2(\pi^+\pi^-)$ amplitudes, vector-meson couplings, singly-virtual data from $e^+e^-\to e^+e^-\eta^{(\prime)}$, 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 $a_2$ resonance, necessitating the solution of an inhomogeneous Muskhelishvili-Omn\`es problem via a carefully chosen path deformation. The resulting TFFs allow us to evaluate the $\eta^{(\prime)}$-pole contributions to the anomalous magnetic moment of the muon, $a_\mu^{\eta\text{-pole}}=14.7(9)\times 10^{-11}$ and $a_\mu^{\eta'\text{-pole}}=13.5(7)\times 10^{-11}$, completing a dedicated program for the lowest-lying pseudoscalar intermediate states in a dispersive approach to hadronic light-by-light scattering, $a_\mu^{\text{PS-poles}}=91.2^{+2.9}_{-2.4}\times 10^{-11}$.

hep-ph↗

arXiv2024

Precision evaluation of the $\eta$- and $\eta'$-pole contributions to hadronic light-by-light scattering in the anomalous magnetic moment of the muon

Next to the $\pi^0$ pole, $\eta$ and $\eta'$ 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 $a_\mu$. 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 $\eta^{(\prime)}\to\gamma\gamma$ decay widths, $\eta^{(\prime)}\to\pi^+\pi^-\gamma$ spectra, chiral symmetry for the $\eta^{(\prime)}\to 2(\pi^+\pi^-)$ amplitudes, vector-meson couplings, and asymptotic limits. Crucially, we investigate the role of the leading left-hand singularity generated by the exchange of the $a_2$ tensor meson, yielding, for the first time, an estimate of the associated factorization-breaking corrections. Our final results, $a_\mu^{\eta\text{-pole}}=14.7(9)\times 10^{-11}$ and $a_\mu^{\eta'\text{-pole}}=13.5(7)\times 10^{-11}$, conclude a dedicated effort to evaluate the pseudoscalar-pole contributions to hadronic light-by-light scattering using dispersion relations, amounting to a combined $a_\mu^{\text{PS-poles}}=91.2^{+2.9}_{-2.4}\times 10^{-11}$.

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arXiv2024

Improved Standard-Model prediction for ${K_L\to \ell^+\ell^-}$

We present a comprehensive calculation of the $K_L\toγ^*γ^*$ form factor in dispersion theory, using input from the leptonic decays $K_L\to\ell^+\ell^-γ$, $K_L\to \ell_1^+\ell_1^-\ell_2^+\ell_2^-$, the hadronic mode $K_L\to π^+π^-γ$, the normalization $K_L\toγγ$, and the matching to asymptotic constraints. As key result we obtain an improved determination of the long-distance contribution to $K_L\to\ell^+\ell^-$, leading to the Standard-Model predictions $\text{Br}[K_L\toμ^+μ^-]=7.44^{+0.41}_{-0.34}\times 10^{-9}$, $\text{Br}[K_L\to e^+e^-]=8.46(37)\times 10^{-12}$, and more stringent limits on physics beyond the Standard Model. We provide a detailed breakdown of the current uncertainty, and delineate how future experiments and the interplay with lattice QCD could help further improve the precision.

hep-ph↗

arXiv2023

Comparing phenomenological estimates of dilepton decays of pseudoscalar mesons with lattice QCD

Dilepton decays of pseudoscalar mesons have been drawing particular interest, thanks to their sensitivity to both the QCD dynamics at low energy and also signals beyond the Standard Model. In this context, we shortly review our recent study on an improved Standard-Model prediction for the rare decay $π^0\to e^+e^-$, and compare it with the first determination on the lattice that predicted also the $π^0\to γγ$ decay width as a byproduct. In addition, we discuss our recent work on $K_L\to\ell^+\ell^-$ decays and its connection to lattice QCD. We comment on the current uncertainty estimates and discuss how they could be improved profiting from future experiments and progress in lattice QCD.

hep-lat↗

arXiv2023

A phenomenological estimate of isospin breaking in hadronic vacuum polarization

Puzzles in the determination of the hadronic-vacuum-polarization contribution currently impede a conclusive interpretation of the precision measurement of the anomalous magnetic moment of the muon at the Fermilab experiment. One such puzzle concerns tensions between evaluations in lattice QCD and using $e^+e^-\to\text{hadrons}$ cross-section data. In lattice QCD, the dominant isospin-symmetric part and isospin-breaking (IB) corrections are calculated separately, with very different systematic effects. Identifying these two pieces in a data-driven approach provides an opportunity to compare them individually and trace back the source of the discrepancy. Here, we estimate the IB component of the lattice-QCD calculations from phenomenology, based on a comprehensive study of exclusive contributions that can be enhanced via infrared singularities, threshold effects, or hadronic resonances, including, for the first time, in the $e^+e^-\to3π$ channel. We observe sizable cancellations among different channels, with a sum that even suggests a slightly larger result for the QED correction than obtained in lattice QCD. We conclude that the tensions between lattice QCD and $e^+e^-$ data therefore cannot be explained by the IB contributions in the lattice-QCD calculations.

hep-ph↗

arXiv2023

Isospin-breaking effects in the three-pion contribution to hadronic vacuum polarization

Isospin-breaking (IB) effects are required for an evaluation of hadronic vacuum polarization at subpercent precision. While the dominant contributions arise from the $e^+e^-\toπ^+π^-$ channel, also IB in the subleading channels can become relevant for a detailed understanding, e.g., of the comparison to lattice QCD. Here, we provide such an analysis for $e^+e^-\to 3π$ by extending our dispersive description of the process, including estimates of final-state radiation (FSR) and $ρ$-$ω$ mixing. In particular, we develop a formalism to capture the leading infrared-enhanced effects in terms of a correction factor $η_{3π}$ that generalizes the analog treatment of virtual and final-state photons in the $2π$ case. The global fit to the $e^+e^-\to 3π$ data base, subject to constraints from analyticity, unitarity, and the chiral anomaly, gives $a_μ^{3π}|_{\leq 1.8\,\text{GeV}}=45.91(53)\times 10^{-10}$ for the total $3π$ contribution to the anomalous magnetic moment of the muon, of which $a_μ^\text{FSR}[3π]=0.51(1)\times 10^{-10}$ and $a_μ^{ρ\text{-}ω}[3π]=-2.68(70)\times 10^{-10}$ can be ascribed to IB. We argue that the resulting cancellation with $ρ$-$ω$ mixing in $e^+e^-\to 2π$ can be understood from a narrow-resonance picture, and provide updated values for the vacuum-polarization-subtracted vector-meson parameters $M_ω=782.70(3)\,\text{MeV}$, $M_ϕ=1019.21(2)\,\text{MeV}$, $Γ_ω=8.71(3)\,\text{MeV}$, and $Γ_ϕ=4.27(1)\,\text{MeV}$.

hep-ph↗

arXiv2022

Chiral extrapolation of hadronic vacuum polarization and isospin-breaking corrections

By far the biggest contribution to hadronic vacuum polarization (HVP) arises from the two-pion channel. Its quark-mass dependence can be evaluated by combining dispersion relations with chiral perturbation theory, providing guidance on the functional form of chiral extrapolations, or even interpolations around the physical point. In addition, the approach allows one to estimate in a controlled way the isospin-breaking (IB) corrections that arise from the pion mass difference. As an application, we present an updated estimate of phenomenological expectations for electromagnetic and strong IB corrections to the HVP contribution to the anomalous magnetic moment of the muon. In particular, we include IB effects in the $\bar K K$ channel, which are enhanced due to the proximity of the $\bar K K$ threshold and the $ϕ$ resonance. The resulting estimates make it unlikely that the current tension between lattice-QCD and data-driven evaluations of the HVP contribution is caused by IB corrections.

hep-ph↗

arXiv2022

Improved Standard-Model prediction for $π^0\to e^+e^-$

We present an improved Standard-Model (SM) prediction for the dilepton decay of the neutral pion. The loop amplitude is determined by the pion transition form factor for $π^0\toγ^*γ^*$, for which we employ a dispersive representation that incorporates both space-like and time-like data as well as short-distance constraints. The resulting SM branching fraction, $\text{BR}[π^0\to e^+e^-]=6.25(3)\times 10^{-8}$ , sharpens constraints on physics beyond the SM, including pseudoscalar and axial-vector mediators.

hep-ph↗

arXiv2020

Hadronic vacuum polarization and vector-meson resonance parameters from ${e^+e^-\toπ^0γ}$

We study the reaction $e^+e^-\toπ^0γ$ based on a dispersive representation of the underlying $π^0\toγγ^*$ transition form factor. As a first application, we evaluate the contribution of the $π^0γ$ channel to the hadronic-vacuum-polarization correction to the anomalous magnetic moment of the muon. We find $a_μ^{π^0γ}\big|_{\leq 1.35\,\text{GeV}}=43.8(6)\times 10^{-11}$, in line with evaluations from the direct integration of the data. Second, our fit determines the resonance parameters of $ω$ and $ϕ$. We observe good agreement with the $e^+e^-\to3π$ channel, explaining a previous tension in the $ω$ mass between $π^0γ$ and $3π$ by an unphysical phase in the fit function. Combining both channels we find $\bar M_ω=782.736(24)\,\text{MeV}$ and $\bar M_ϕ=1019.457(20)\,\text{MeV}$ for the masses including vacuum-polarization corrections. The $ϕ$ mass agrees perfectly with the PDG average, which is dominated by determinations from the $\bar K K$ channel, demonstrating consistency with $3π$ and $π^0γ$. For the $ω$ mass, our result is consistent but more precise, exacerbating tensions with the $ω$ mass extracted via isospin-breaking effects from the $2π$ channel.

hep-ph↗

arXiv2019

Three-pion contribution to hadronic vacuum polarization

We address the contribution of the $3π$ channel to hadronic vacuum polarization (HVP) using a dispersive representation of the $e^+e^-\to 3π$ amplitude. This channel gives the second-largest individual contribution to the total HVP integral in the anomalous magnetic moment of the muon $(g-2)_μ$, both to its absolute value and uncertainty. It is largely dominated by the narrow resonances $ω$ and $ϕ$, but not to the extent that the off-peak regions were negligible, so that at the level of accuracy relevant for $(g-2)_μ$ an analysis of the available data as model independent as possible becomes critical. Here, we provide such an analysis based on a global fit function using analyticity and unitarity of the underlying $γ^*\to3π$ amplitude and its normalization from a chiral low-energy theorem, which, in particular, allows us to check the internal consistency of the various $e^+e^-\to 3π$ data sets. Overall, we obtain $a_μ^{3π}|_{\leq 1.8\,\text{GeV}}=46.2(6)(6)\times 10^{-10}$ as our best estimate for the total $3π$ contribution consistent with all (low-energy) constraints from QCD. In combination with a recent dispersive analysis imposing the same constraints on the $2π$ channel below $1\,\text{GeV}$, this covers nearly $80\%$ of the total HVP contribution, leading to $a_μ^\text{HVP}=692.3(3.3)\times 10^{-10}$ when the remainder is taken from the literature, and thus reaffirming the $(g-2)_μ$ anomaly at the level of at least $3.4σ$. As side products, we find for the vacuum-polarization-subtracted masses $M_ω=782.63(3)(1)\,\text{MeV}$ and $M_ϕ=1019.20(2)(1)\,\text{MeV}$, confirming the tension to the $ω$ mass as extracted from the $2π$ channel.

hep-ph↗

arXiv2018

Dispersion relation for hadronic light-by-light scattering: pion pole

The pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon $(g-2)_μ$ is fully determined by the doubly-virtual pion transition form factor. Although this crucial input quantity is, in principle, directly accessible in experiment, a complete measurement covering all kinematic regions relevant for $(g-2)_μ$ is not realistic in the foreseeable future. Here, we report in detail on a reconstruction from available data, both space- and time-like, using a dispersive representation that accounts for all the low-lying singularities, reproduces the correct high- and low-energy limits, and proves convenient for the evaluation of the $(g-2)_μ$ loop integral. We concentrate on the systematics of the fit to $e^+e^-\to 3π$ data, which are key in constraining the isoscalar dependence, as well as the matching to the asymptotic limits. In particular, we provide a detailed account of the pion transition form factor at low energies in the time- and space-like region, including the error estimates underlying our final result for the pion-pole contribution, $a_μ^{π^0\text{-pole}}=62.6^{+3.0}_{-2.5}\times 10^{-11}$, and demonstrate how forthcoming singly-virtual measurements will further reduce its uncertainty.

hep-ph↗

arXiv2018

Pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon

The $π^0$ pole constitutes the lowest-lying singularity of the hadronic light-by-light (HLbL) tensor, and thus provides the leading contribution in a dispersive approach to HLbL scattering in the anomalous magnetic moment of the muon $(g-2)_μ$. It is unambiguously defined in terms of the doubly-virtual pion transition form factor, which in principle can be accessed in its entirety by experiment. We demonstrate that, in the absence of a direct measurement, the full space-like doubly-virtual form factor can be reconstructed very accurately based on existing data for $e^+e^-\to 3π$, $e^+e^-\to e^+e^-π^0$, and the $π^0\toγγ$ decay width. We derive a representation that incorporates all the low-lying singularities of the form factor, matches correctly onto the asymptotic behavior expected from perturbative QCD, and is suitable for the evaluation of the $(g-2)_μ$ loop integral. The resulting value, $a_μ^{π^0\text{-pole}}=62.6^{+3.0}_{-2.5}\times 10^{-11}$, for the first time, represents a complete data-driven determination of the pion-pole contribution with fully controlled uncertainty estimates. In particular, we show that already improved singly-virtual measurements alone would allow one to further reduce the uncertainty in $a_μ^{π^0\text{-pole}}$.

hep-ph↗