Study of transition form factors of the lightest pseudoscalars

TL;DR

This work computes the transition form factors (TFFs) of the lightest neutral pseudoscalar mesons ($\pi^0$, $\eta$, $\eta'$) within a $U(3)$ resonance chiral theory framework, incorporating mixing, vector resonances, and heavy multiplets to describe both timelike and spacelike regions. By fitting to a wide array of data—including time-like singly-virtual and space-like singly- and doubly-virtual TFFs, $P\to\gamma\gamma$, cross sections for $e^+e^-\to P\gamma$, Dalitz decays, and LQCD results—the authors tightly constrain the unknown couplings and produce robust $F_{P\gamma^*\gamma^*}(q_1^2,q_2^2)$. They then evaluate the pseudoscalar pole contributions to hadronic light-by-light scattering in the muon anomalous magnetic moment, obtaining $a_\mu^{\pi^0+\eta+\eta'} = (92.8\text{--}93.4)\times 10^{-11}$ with a sizeable portion coming from the asymptotic high-energy behavior. The results, consistent with several phenomenological analyses and in some tension with LQCD, emphasize the importance of precise high- and low-energy TFF constraints and motivate future measurements of related processes to refine the pseudoscalar TFFs and the HLbL contribution to $a_\mu$.

Abstract

In this paper, we study the transition form factors of the lightest pseudoscalar mesons, $π^0$, $η$, and $η'$, within the framework of resonance chiral theory. Our analysis is performed based on the data of time-like and space-like singly-virtual and space-like doubly-virtual form factors, as well as the relevant cross sections and latest invariant mass spectra of $e^+e^-$ pair for the process of $P\toγe^+ e^-$. The transition form factors of these pseudoscalars are obtained. Also, we evaluate their contributions to the light-by-light part of the anomalous magnetic moment of the muon. Our two Fits give similar results, where Fit-A gives $a_μ^{π^0 }=(61.6\pm 1.8)\times10^{-11}$, $a_μ^{η}=(15.2\pm1.7)\times10^{-11}$, $a_μ^{η'}=(16.0\pm 1.2)\times10^{-11}$, and the total contribution of neutral pseudo-scalar meson poles is $a_μ^{π^0+η+η'}=(92.8\pm2.9)\times10^{-11}$.

Figures (9)

• Figure 1: Feynman diagrams of $P\to \gamma^*\gamma^*$. The double line represents vector resonance.
• Figure 2: Normalized form factors $|F_{P}(q^2)|$ and $e^+e^-$ invariant mass spectra of $P\to \gamma e^{+}e^{-}$ and $\eta'\to\omega e^+e^-$. Fits A and B get almost the same results.
• Figure 3: Fit to the $e^+e^-\to \pi^0\gamma$ cross-section. The experimental datasets are from SND Achasov:2000zdAchasov:2003edSND:2016drmAchasov:2018ujw, CMD-2 CMD-2:2004ahv.
• Figure 4: Fit to the cross-sections of $e^+e^-\to \eta\gamma, \eta'\gamma$. The datasets are taken from SND Achasov:2000zdAchasov:2006dvAchasov:2013eliSND:2024qaq, CMD-2 CMD-2:2004ahvCMD-2:2001dnv.
• Figure 5: Fit to the Space-like singly-virtual and doubly-virtual TFFs. The experimental datasets are sourced from BaBar BaBar:2009rrjBaBar:2011nrp, Belle Belle:2012wwz, LEP L3:1997ocz, CLEO CLEO:1997fho, CELLO CELLO:1990klc and BESIII Redmer:2019zzr. The LQCD datasets are sourced from BMW Gerardin:2023naa, ETM ExtendedTwistedMass:2022ofmExtendedTwistedMass:2023hin and Lin's work Lin:2024khg. Here we use points with same color and different shapes for different data of CLEO CLEO:1997fho.