Measurement of the invariant-mass spectrum for the two photons from the $\eta \to {\pi }^0\gamma \gamma$ decay

New results on the rare, doubly radiative decay $\eta \to {\pi }^0\gamma \gamma$ have been obtained from a revised analysis of the Crystal Ball experiment performed at the AGS. The analysis yields the first information on the dependence of the decay width, $\Gamma (\eta \to {\pi }^0\gamma \gamma )$, on the two-photon invariant mass squared, $m^2(\gamma \gamma )$. A re-evaluation of the branching ratio is also made, $\mathit{BR}(\eta \to {\pi }^0\gamma \gamma )=(2.21\pm 0.{24}_{\mathrm{stat}}\pm 0.{47}_{\mathrm{syst}})\times {10}^{-4}$; it implies that the decay width is $\Gamma (\eta \to {\pi }^0\gamma \gamma )=0.285\pm 0.{031}_{\mathrm{stat}}\pm 0.{061}_{\mathrm{syst}}$ eV. These results are close to predictions based on chiral perturbation theory with vector-meson dominance.

Figure 1

Comparison of the χPTh calculations of Refs. [6, 7] showing the dependence of the $\eta \to {\pi }^0\gamma \gamma$ decay width on $m^2(\gamma \gamma )$ (left) and on $m(\gamma \gamma )$ (right).

Figure 2

Distributions of the vertex $z$ coordinate obtained from the kinematic-fit output for the ${\pi }^{-}p\to {\pi }^0\gamma \gamma n\to 4\gamma n$ hypothesis: (a) LH${}_2$ data, (b) empty-target data, (c) MC simulation for ${\pi }^{-}p\to \eta n\to {\pi }^0\gamma \gamma n$, (d) MC for ${\pi }^{-}p\to \eta n\to 3{\pi }^{0}n$, (e) MC for ${\pi }^{-}p\to {\pi }^0{\pi }^{0}n$, and (f) MC for ${\pi }^{-}p\to \eta n\to \gamma \gamma n$.

Figure 3

Two-dimensional density distributions of the larger (top) and smaller (bottom) ${\pi }^0\gamma$ invariant mass versus the ${\pi }^0\gamma \gamma$ invariant mass for events selected as ${\pi }^{-}p\to {\pi }^0\gamma \gamma n\to 4\gamma n$ candidates: (a), (d) MC simulation for ${\pi }^{-}p\to {\pi }^0{\pi }^{0}n$, (b), (e) MC for ${\pi }^{-}p\to \eta n\to 3{\pi }^{0}n$, (c),(f) MC for ${\pi }^{-}p\to \eta n\to {\pi }^0\gamma \gamma n$. The lines in distributions (a), (b), and (c) depict the three cuts that are tested. These cuts are implemented by discarding all events for which $m_{\max }({\pi }^0\gamma )$ lies above the line. The line in distributions (d), (e), and (f) depicts the cut on $m_{\min }({\pi }^0\gamma )$. This cut requires discarding all events for which $m_{\min }({\pi }^0\gamma )$ lies below the line.

Figure 4

Two-dimensional density distributions of the effective cluster radius $R$ versus the cluster energy of the photons for events selected as ${\pi }^{-}p\to {\pi }^0\gamma \gamma n\to 4\gamma n$ candidates are shown for the two “direct” photons from η decay (top) and the two photons from the ${\pi }^0$ decay (bottom): (a),(c) MC simulation for ${\pi }^{-}p\to \eta n\to {\pi }^0\gamma \gamma n$, (b),(d) MC for ${\pi }^{-}p\to \eta n\to 3{\pi }^{0}n$. The lines in the distributions depict two R-cuts that we tested; they require discarding all events if they have a cluster with energy larger than 0.06 GeV which $R$ lies above the line.

Figure 5

Spectra that illustrate the $\eta \to {\pi }^0\gamma \gamma$ acceptance obtained when the events are selected at the 10% CL, with the vertex $z$ coordinate within $\pm 8$ cm, after applying cut no. 1 on the cluster radius $R$ and cut no. 1 on $m_{\max }({\pi }^0\gamma )$: (a) $m^2(\gamma \gamma )$ spectrum (dashed line) from $4\times {10}^5$ ${\pi }^{-}p\to \eta n\to {\pi }^0\gamma \gamma n$ events, with η decaying according to phase space, which were initially generated as an input to the MC simulation, plus the spectrum (solid line) finally obtained for these events after the MC simulation and analysis; (b) the $\eta \to {\pi }^0\gamma \gamma$ acceptance as a function of $m^2(\gamma \gamma )$, obtained as the ratio of the two spectra in (a); (c) comparison of the $m^2(\gamma \gamma )$ spectra obtained using the ${\pi }^{-}p\to \eta n\to {\pi }^0\gamma \gamma n\to 4\gamma n$ hypothesis (solid line) and the ${\pi }^{-}p\to {\pi }^0\gamma \gamma n\to 4\gamma n$ hypothesis (crosses).

Figure 6

The ${\pi }^0\gamma \gamma$ invariant mass spectra for the ${\pi }^{-}p\to {\pi }^0\gamma \gamma n$ candidates selected at the 10% CL, with the vertex $z$ coordinate within $\pm 5$ cm, after applying cut no. 2 on the cluster radius $R$, cut no. 1 on $m_{\max }({\pi }^0\gamma )$, and with $m^2(\gamma \gamma )$ between 0.027 and 0.04 GeV${}^2/c^4$: (a) MC simulation for $4\times {10}^5$ ${\pi }^{-}p\to \eta n\to {\pi }^0\gamma \gamma n$ events, where the $\eta \to {\pi }^0\gamma \gamma$ decay is simulated according to phase space; (b) MC for $4\times {10}^7$ ${\pi }^{-}p\to {\pi }^0{\pi }^{0}n$ events; (c) experimental LH${}_2$ candidates (solid line) and the $\eta \to 3{\pi }^0$ background (crosses) expected from the MC simulation; (d) the experimental spectrum remaining after subtraction of the $\eta \to 3{\pi }^0$ background fitted to a Gaussian for the expected signal plus a polynomial function for the background; the result of the fit to the Gaussian alone is superimposed as well.

Figure 7

Our results for $d\Gamma (\eta \to {\pi }^0\gamma \gamma )/{\mathit{dm}}^2(\gamma \gamma )$ (triangles) compared to χPTh predictions from Ref. [6]: the vector-meson contribution alone (solid line) and with the $a_0$ contribution added assuming either the constructive (dash line) or destructive (dot line) interference.

Figure 8

Same as Fig. 6 but for the full $m^2(\gamma \gamma )$ spectrum.