First-principles extraction of the light-quark mass ratio from the isospin-violating decay η′→3π

Gross, Treiman, and Wilczek (1979) asked how the up–down quark-mass difference $m_d-m_u$ could be determined from isospin-violating observables, proposing three probes: the electromagnetic–quark-mass decomposition of the pion mass splitting, the ratio of $\eta'\to\pi^0\pi^+\pi^-$ to $\eta'\to\eta\pi^+\pi^-$ decay widths, and the small vector form factor in $\Sigma^-\to\Lambda e\bar\nu$. Two probes have since been settled: ab initio lattice QCD+QED now determines the mass decomposition and the ratios $m_u/m_d$ and $Q$ to percent precision (excluding $m_u=0$), and hyperon data plus lattice calculations pin down $\Sigma$–$\Lambda$ mixing. The $\eta'\to 3\pi$ route remains blocked on theory: BESIII has measured both decay modes in full amplitude analyses, but no controlled representation of the amplitude exists that could convert the measurement into a quark-mass ratio — the most advanced calculation, a unitarized $U(3)$ chiral treatment with pairwise rescattering only, predates the amplitude analyses and performed no quark-mass extraction. What is needed is a controlled representation of the amplitude that converts the measured $\eta'\to 3\pi$ dynamics into a quark-mass ratio. The problem is to construct that representation, extract $(m_d-m_u)/(m_s-\hat m)$ from the measured $\eta'\to 3\pi$ dynamics with a quantified uncertainty budget, and test its consistency with lattice-QCD+QED determinations.

Background

Isospin is the approximate $SU(2)$ symmetry that treats the up ($u$) and down ($d$) quarks as two states of one particle; it is broken both by the small difference $m_d-m_u$ between their masses and by electromagnetism (the $u$ and $d$ carry different electric charges). Because quarks are confined, $m_d-m_u$ cannot be measured directly: it must be inferred from its effects on hadron masses and decay amplitudes. A standard invariant for this purpose is the double ratio $Q^2=(m_s^2-\hat m^2)/(m_d^2-m_u^2)$, with $\hat m=(m_u+m_d)/2$ and $m_s$ the strange-quark mass, which is insensitive to the renormalization convention for light-quark masses; the value of $m_u$ itself matters because $m_u=0$ is one historical solution to the strong CP problem (why quantum chromodynamics, QCD, appears to conserve the combined charge-conjugation–parity symmetry despite an allowed symmetry-breaking term).

Gross, Treiman, and Wilczek (1979) asked how $m_d-m_u$ could be pinned down from isospin-violating observables at a time when the needed hadronic matrix elements could not be computed. Using partially conserved axial-vector current (PCAC) relations — exact chiral-symmetry statements relating divergences of axial-vector currents to quark-mass insertions — they proposed three probes: (i) the decomposition of the pion mass splitting $M_{\pi^+}-M_{\pi^0}$ into electromagnetic and quark-mass parts, where the then-current electromagnetic estimate disagreed with experiment; (ii) the isospin-violating decay $\eta'\to 3\pi$, whose rate relative to $\eta'\to\eta\pi\pi$ they argued measures the $\eta'$–$\pi^0$ mixing generated by $m_d-m_u$ through anomalous gluonic matrix elements; and (iii) the small vector form factor $g_V$ (the coefficient of the vector-current contribution to the decay amplitude) in the semileptonic decay $\Sigma^-\to\Lambda e\bar\nu$, fed by $\Sigma$–$\Lambda$ mixing. They were explicit that the matrix elements of the gluonic anomaly operator (tied to the axial $U(1)_A$ anomaly of QCD, which makes the $\eta'$ much heavier than the octet mesons) could not then be computed or measured independently.

Later developments define what remains. First, chiral perturbation theory (χPT) — the systematic low-energy effective field theory of QCD built on the chiral $SU(3)$ symmetry of the light quarks — became the standard tool, extended to $U(3)$ (including the singlet $\eta'$) via large-$N_c$ counting, where $N_c$ is the number of QCD colors; in this framework $\eta'\to 3\pi$ is an isospin-violating transition (isospin 0 to isospin 1) driven at leading order by $m_d-m_u$. Second, ab initio lattice QCD — the nonperturbative evaluation of QCD by discretizing space-time and simulating it numerically — supplemented with quantum electrodynamics (QCD+QED) now determines the electromagnetic pion splitting, the violation of Dashen's theorem (the statement that, in the chiral limit, the electromagnetic self-energy differences of the charged and neutral pions and kaons are equal), and the quark-mass ratios themselves: $m_u/m_d$ and $Q$ are fixed at the percent level, and $m_u=0$ is excluded at more than $24\sigma$. Third, the experiments the 1979 paper called very demanding have been done: $\eta'\to\pi^+\pi^-\pi^0$ was observed by the CLEO experiment in 2009, and both $\eta'\to 3\pi$ modes have since been measured by the BESIII experiment (at the Beijing Electron–Positron Collider), including a full amplitude analysis.

What has not been done is the theoretical conversion. The specific extraction mechanism proposed in 1979 was refuted by Borasoy, Meißner, and Nißler (2006), who showed that its two underlying assumptions are not justified. The most advanced existing calculation of $\eta'\to 3\pi$ — the unitarized $U(3)$ chiral coupled-channels analysis of Borasoy and Nißler (2005), which iterates the pairwise two-particle rescattering in an integral (Bethe–Salpeter) equation — predates the amplitude analyses, fits the then-available decay data, and performs no quark-mass extraction. For the mature analogue $\eta\to 3\pi$, Khuri–Treiman-type dispersive analyses — representations that reconstruct a three-body decay amplitude from analyticity and unitarity by resumming the pairwise final-state interactions — now extract $Q$ successfully, but the corresponding representation for $\eta'\to 3\pi$, which lies above the $\eta\pi$ threshold and involves the poorly convergent $U(3)$ chiral expansion of the anomalous singlet sector, is stated by the 2022 review of Gan, Kubis, Passemar, and Tulin to require serious theoretical advances: no sophisticated theoretical analysis of the latest $\eta'\to 3\pi$ experimental results exists to date.

Problem Statement

Construct a controlled representation of the isospin-violating decays $\eta'\to\pi^+\pi^-\pi^0$ and $\eta'\to\pi^0\pi^0\pi^0$ that is based on analyticity and two-body unitarity, resums the two-body final-state interactions ($\pi\pi$ rescattering, including the $\rho(770)$ vector-meson resonance, and $\eta\pi$ intermediate states), and contains the anomalous $\eta'$ and its mixing with $\eta$ and $\pi^0$ in the $U(3)$ singlet sector, with electromagnetic corrections and $\eta$–$\pi^0$ mixing treated at the same order as the $m_d-m_u$ insertion. Use this representation to extract the light-quark-mass ratio $(m_d-m_u)/(m_s-\hat m)$, with $\hat m=(m_u+m_d)/2$, from the measured BESIII branching fractions and Dalitz distributions (the event distribution over the two-pion invariant-mass variables of the three-body final state) of $\eta'\to 3\pi$, with a complete uncertainty budget. The extraction may equivalently target the double ratio $Q^2=(m_s^2-\hat m^2)/(m_d^2-m_u^2)$, which fixes $(m_d-m_u)/(m_s-\hat m)$ once the independently known ratio $m_s/\hat m$ is supplied; either way, state the resulting consistency or quantified tension with the lattice-QCD+QED determinations ($m_u/m_d\approx 0.46$–$0.49$, $Q\approx 22$–$23$).

References

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