Di-Pion Correlations in Heavy Quarkonium

• The paper demonstrates that di-pion correlations exhibit a bump–dip structure near threshold, revealing interference and final-state interaction effects in quarkonium decays.
• It employs chiral unitary and coupled-channel methods to quantitatively model ππ interactions and reproduce spectral anomalies with high statistical agreement.
• The study highlights how multi-channel interference and non-perturbative QCD dynamics provide a unified framework for interpreting charmonium and bottomonium decay patterns.

Di-pion correlation in heavy quarkonium decays refers to the strong dynamical interplay between the two-pion subsystem emerging from transitions of charmonium ($\psi$) and bottomonium ($\Upsilon$) mesons. The correlations arise from final-state interaction (FSI) effects, channel couplings, and interference among multiple decay topologies, leading to pronounced anomalies and substructure in di-pion invariant mass spectra. These phenomena provide high-sensitivity probes of non-perturbative QCD dynamics, chiral symmetry breaking, and resonance formation in the meson-meson sector.

1. Experimental Features: Observation of Di-pion Substructure

Recent data from the BESIII Collaboration, representing the largest sample of $\psi(3686)$ decays ($2.7124\pm0.0014\times10^9$ events), revealed a distinct “bump–dip” feature near the di-pion mass threshold in $\psi(3686)\to J/\psi\,\pi^+\pi^-$ (Wang et al., 13 Nov 2025). The anomaly is characterized by:

• A sharp peak at $M_{\pi\pi}\simeq0.285$ GeV.
• A pronounced dip near $M_{\pi\pi}\simeq0.305$ GeV (threshold region $0.28-0.32$ GeV).
• Statistical significance: $\chi^2/\mathrm{d.o.f.}=1.02$ in the local fit region, strongly excluding purely phase-space or QCD multipole expansion explanations.

Similar near-threshold enhancements and intermediate-mass dips are consistently observed in bottomonium transitions, e.g., $\Upsilon(nS)\to\Upsilon(mS)\,\pi^+\pi^-$, and reflected in multi-channel Dalitz plot analyses (Baru et al., 2020).

2. Chiral Unitary and Coupled-Channel Formalism

Di-pion correlations are fundamentally rooted in the resummation of strong $\pi\pi$ final-state interactions, encoded using a chiral unitary approach (Wang et al., 13 Nov 2025, Surovtsev et al., 2016):

• The effective chiral Lagrangian for pseudoscalar-meson interactions is

$$\mathcal{L}_2 = \frac{f^2}{4}\langle \partial_\mu U\,\partial^\mu U^\dagger + \chi U^\dagger + \chi^\dagger U \rangle,$$

with $U=\exp(2i\Phi/f)$ and $f=0.093$ GeV.

• Projected onto $S$-wave, the coupled channel interaction kernel $V_{ij}(s)$ spans $\pi\pi$, $K\bar{K}$, and $\eta\eta$.
• Unitarisation is performed via the Bethe–Salpeter equation,

$T(s) = \frac{V(s)}{1-V(s)\,G(s)},$

with $G(s)$ regularized using either a cutoff ($q_{max}=0.6$ GeV) or dimensional subtraction for threshold matching.

• The approach successfully captures key resonances ($f_0(500)$ and $f_0(980)$).

In alternative formulations, analyticity and unitarity are imposed in the Surovtsev–Bydžovský et al. coupled-channel S-matrix formalism, combining $\pi\pi$, $K\bar{K}$, and $\eta\eta$ sectors on an eight-sheeted Riemann surface and expressing the multi-channel $T_{ij}(s)$ amplitudes (Surovtsev et al., 2016).

In bottomonium transitions, a dispersive Omnès formalism is utilized (Baru et al., 2020), with detailed inclusion of heavy-quark spin-symmetry (HQSS) constraints and point-like transitions between open- and hidden-bottom channels. Short-range couplings ($C_d, C_f$) and low-energy constants ($c_1, c_2$) are fitted to channel-specific data, while the coupled-channel $\pi\pi$-$K\bar{K}$ $T$-matrix is fully parameterized from Roy-equation results.

3. Correlation Function and Observables

The di-pion correlation function $C(k)$ is a central observable, encoding both FSI dynamics and spatial emission characteristics:

$$C(k) = 1 + \frac{|T(s)|^2}{\Phi(s) \times F_\text{source}(R)}$$

• $k$: relative momentum ($s=4(k^2 + m_\pi^2)$)
• $\Phi(s) = \frac{k}{8\pi\sqrt{s}}$: two-body phase-space factor
• $F_\text{source}(R)$: spatial source function (Gaussian model),

$$F_\text{source}(R) = 4\pi \int_0^\infty r^2 S_{12}(r) |j_0(kr) + \int \frac{d^3q}{(2\pi)^3} \frac{T(s) j_0(qr)}{s-(\omega_1+\omega_2)^2+i\epsilon}|^2 dr$$

with $$S_{12}(r) = \frac{1}{(R\sqrt{4\pi})^3} \exp\left(-\frac{r^2}{4R^2}\right)$$, $j_0(x)=\sin x/x$, $R=$ source radius.

In practical fits, $C(k)$ peaks at $\simeq1.6$ near $k\simeq0.18$ GeV for $R=1$ fm, and damps toward unity at higher $k$. Variations in $R$ modulate the correlation, consistent with femtoscopic and source-size analyses (Wang et al., 13 Nov 2025). The coupled-channel formalism enables generalizations to all intermediate states.

Alternative definitions employ the normalized di-pion mass spectrum as a proxy for the correlation function:

$$C(m_{\pi\pi}) = \frac{1}{\Gamma_\text{tot}} \frac{d\Gamma}{dm_{\pi\pi}}$$

reflecting FSI dynamics via $|F(s)|^2$.

4. Interference Effects and Channel Dynamics

The observed di-pion spectral features are not resonant phenomena but arise from interference among OZI-suppressed decay topologies and coupled-channel FSI (Wang et al., 13 Nov 2025, Surovtsev et al., 2016):

• Decay amplitudes are linear combinations of $T_{i1}(s)$ ($i=1,2,3$: $\pi\pi$, $K\bar{K}$, $\eta\eta$), e.g.,

$$F_{\psi(2S)\to J/\psi\,\pi\pi}(s) = d_1(s)\,T_{11}(s) + d_2(s)\,T_{21}(s) + d_3(s)\,T_{31}(s)$$

where coefficients are polynomials in $s$ and encode tree-level and loop-induced contributions.

• The expansion,

$$|F|^2 = \sum_{i=1}^3 |c_i\,T_{i1}|^2 + 2\,\Re \sum_{i<j}(c_i\,T_{i1})(c_j\,T_{j1})^*$$

demonstrates constructive interference near threshold and destructive interference at intermediate masses, producing bell-shaped peaks and dips in the invariant mass spectrum.

• Phenomenological fits indicate $V_1=(4.23\pm0.04)\times10^4$, $V_2=(1.09\pm0.01)\times10^9$, phase $\phi=(0.048\pm0.001)$ radians for $\psi(3686)$ decays, with robust statistical agreement to the BESIII data.

These mechanisms underlie spectrum shaping in both charmonium and bottomonium transitions, with channel-dependent features (e.g., dips at $0.45$–$0.7$ GeV and near $1$ GeV in $\Upsilon(4S,5S)\to\Upsilon(1S)\pi\pi$).

5. Numerical Fits and Comparative Analysis

Global fits to heavy quarkonium decay data across multiple collaborations (ARGUS, CLEO, CUSB, Crystal Ball, Belle, BaBar, Mark II–III, DM2, BES II) yield $\chi^2/\mathrm{ndf}\approx1.24$ (Surovtsev et al., 2016). The coupled-channel models, with a small number of parameters, reproduce not only the sharp threshold enhancements but also broad dips and spectral distortions:

• The chiral-unitary model produces the full inclusive $\pi^+\pi^-$ spectrum up to $M_{\pi\pi}\sim0.9$ GeV, consistent with BESIII anomaly extension (Wang et al., 13 Nov 2025).
• Dalitz-plot fits in $\Upsilon(10860)\to\pi^+\pi^-\Upsilon(nS)$ exploit short-range B-meson interactions, HQSS-mandated relative couplings, and dispersive treatments of FSI (Baru et al., 2020).
• For $\Upsilon(1S,2S,3S)$ final states, the fitted low-energy constants are channel-dependent but highly correlated, reflecting underlying dynamical constraints.
• Switching off resonant channels confirms that spectrum features result from multi-channel interference and not from isolated resonant structures.

6. Physical Interpretation, Outlook, and Broader Implications

The observed di-pion correlations and related anomalies affirm the central role of strong FSI, chiral dynamics, and unitarity in shaping heavy quarkonium decay spectra:

• The bump–dip structure near threshold is a nonresonant interference effect, not attributable to exotic states, but to the interplay of OZI-suppressed topologies feeding $\pi\pi$ FSI (Wang et al., 13 Nov 2025).
• The chiral-unitary and coupled-channel approaches provide a unified description across both $\psi$ and $\Upsilon$ families, consistent with universality of di-pion correlations (Surovtsev et al., 2016).
• Future femtoscopic measurements, analyzing $C(k)$ dependence on emission source size $R$, collision system, or decay channel, can further constrain non-perturbative sector dynamics. Upcoming data from BESIII, Belle II, and PANDA are poised to test predictions related to threshold cusps, source radii, and channel mixing effects.
• The strong agreement between dispersive, HQSS-constrained models and experiment in $\Upsilon(10860)\to\pi^+\pi^-\Upsilon(nS)$ decays provides substantive support for the molecular interpretation of $Z_b(10610)$ and $Z_b(10650)$ as B-meson bound/resonant states (Baru et al., 2020).
• The correlation formalism opens a route for quantitative studies of femtoscopy in heavy-quarkonium environments, deepening the understanding of correlated mesonic matter and non-perturbative QCD.

In summary, di-pion correlations in heavy quarkonium decays stand as a sensitive, theoretically rigorous probe of QCD at low energies, with practical implications for particle spectroscopy, source imaging, and the dynamical structure of meson-meson interactions across multiple quarkonium systems.