Di-Pion Decays of Heavy Quarkonium in the Field Correlator Method

Mechanism of di-pion transitions $nS\to n^{\mathrm{\prime }}S\pi \pi (n=3,2;n^{\mathrm{\prime }}=2,1)$ in bottomonium and charmonium is studied with the use of the chiral string-breaking Lagrangian allowing for the emission of any number of $\pi (K,\eta ),$ and not containing fitting parameters. The transition amplitude contains two terms, $M=a-b$, where first term (a) refers to subsequent one-pion emission: $\mathrm{{\rm Y}}(nS)\to \pi B{\bar{B}}^{\ast }\to \pi \mathrm{{\rm Y}}(n^{\mathrm{\prime }}S)\pi$ and second term (b) refers to two-pion emission: $\Upsilon(nS)\to\pi\pi B\bar B\to\pi\pi\Upsilon(n’S)$. The one-parameter formula for the di-pion mass distribution is derived, $\frac{dw}{dq}\sim$(phase space) $\mathrm{\mid }\eta -x{\mathrm{\mid }}^2$, where $x=\frac{q^2-4m_{\pi }^2}{q_{max}^2-4m_{\pi }^2},$ $q^2\equiv M_{\pi \pi }^2$. The parameter $\eta$ dependent on the process is calculated, using SHO wave functions and imposing PCAC restrictions (Adler zero) on amplitudes a,b. The resulting di-pion mass distributions are in agreement with experimental data.