This analysis was done on a data sample obtained by E852 experiment at Brookhaven in 1995.

- 2-photon invariant mass (6 entries per event): Postscript
- Momentum distributions: pi+, pi-, pi0, eta
- Missing mass: Postscript
- t-dependence: Postscript
- Confidence level of the SQUAW kinematic fitter for the main hypothesis: Postscript
- Confidence level for competing hypotheses ("anti-cut"): Postscript.

Plot labels correspond to a number of different particles tried in the event.

Only events above the default 10^{-4}confidence level cut are histogrammed.

- Missing mass: Postscript
- t-dependence: Postscript
- Confidence level for the main hypothesis: Postscript
- Dalitz plot for all events: Postscript
- Dalitz plot for 1.8-1.9 GeV: Postscript
- Invariant mass distributions and Gottfried-Jackson angles: Postscript

(predicted Monte Carlo distributions from the final PWA fit are overlayed).

In the PWA fits below, additional partial waves were tried and rejected.

- 50 MeV bin: GIF
- 100 MeV bin: GIF

Both 1-dimensional scans of the likelihood point to the value of mass around 1480-1485 MeV. The conservative estimate for the largest error (corresponding to 0.5-change in the likelihood) is 25 MeV at most.

The best value of width is 110 MeV for likelihood fits in a single 50 MeV bin
centered at pi(1800), and 130 MeV for a single 100 MeV bin. Taking into
account the asymmeteric shape of the 100 MeV plot, we conclude that a fair
estimate of the f0(1500) width is 120+50-30 MeV.

Because both pi2(1880)->a2eta and pi(1800)->f0(1500)pi decays are near or even below nominal threshold, a double integration was used at each MINUIT step to account for available phase space for such decays. The first integration was over the Breit-Wigner shape of pi(1800) or pi2(1880) within the 50-Mev bin used in PWA. The second integration was over the available Breit-Wigner shape of their decay products (a2, a0 or f0).

- Fit with independent poles in a0eta and f0pi: Postscript
- Fit with a single shared pole in a0eta and f0pi: Postscript

- Phase difference of pi(1800)->a0eta vs non-resonant 2-+ a0eta wave: Postscript

- Fit of pi2(1880)->a2eta wave: Postscript

- Fit with all free parameters: Postscript
- Fit with fixed masses and widths: Postscript

- While the pi(1800)->f0(1500)pi decay is very close to nominal threshold, there are other known and kinematically-favored decay modes like a0(980)eta, f0(980)pi, sigma/pi, etc. Near-threshold decay with many other open channels is likely to behave in accordance with a sophisticated K-matrix description rather that as a typical single-channel Breit-Wigner.
- Unfortunatelly, the interference of the f0pi wave with other waves are not very well measured in our case. There are 4 contributions which are completely isotropic in all angles: both 0-+ waves, their interference term, and an isotropic background wave. They differ only over the etaetapi Dalitz plot. As one can see from this Dalitz plot for events in a 100 MeV bin near the pi(1800) peak, the available phase space is so limited that there is no clear separation between the a0(980) and f0(1500) horizontal/vertical bands, their intersection region (to measure their interference), and empty areas (to normalize the background level to). Therefore, maximum likelihood fit is likely to have difficulty separating contributions from the a0eta/f0pi interference term and from non-interfering background. And badly-measured interference term is the primary reason for erratic phase difference between partial waves.
- Finally, our model assumes that background is isotropic over the Dalitz plot. In fact, it is likely to be anisotropic because detector is likely to produce more lower-momentum "fake" eta mesons than the higher-momentum ones. Due to this model limitation, a likelihood fit may have difficulty determining the right level of isotropic background and compensating for this with incorrect magnitude of the interference term.

- Likelihood vs 0-+ f0(1500)pi / a0(980)eta ratio: GIF

The plot above shows change in the likelihood in the 50 MeV bin at pi(1800) mass (which is 1870 MeV). From the change in the likelihood, the ratio of 2 waves seems to be R = 0.40 +- 0.15 which is consistent with the value from Breit-Wigner integration.

It is unclear why this value is so different from R=0.08 determined by VES, and
R=0.03 found by Crystal Barrel. It is unlikely to be caused by differencies in PWA.
While the f0pi wave was found at 48% level of the a0eta one, it constitutes only
about 14% from the total intensity. To be consistent with VES, it should be no more
than 2%-3%. The mass plot (plot (d)) indicates that the amount
of f0(1500) in the raw invariant mass spectrum is closer to 14% rather than to 3%.
That's what PWA fit finds at the end as well. It is not obvious at the moment what is
the cause of different etaeta mass spectra in E852 and VES.