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).
Mass-Dependent Fits
To interpret the results of mass-independent PWA, mass-dependent chi2
fits of partial wave pairs were done. Resonant waves were parameterized
with relativistic Breit-Wigner dependencies including mass-dependent
widths.
1) Fit of 0-+ pi(1800) shape
With only 2 resonance candidates and 4 waves, a choice of what to fit
is very limited. In addition, the pi(1800) meson is significantly better
established than the pi2(1880) state. Therefore, the first step was to
fit only the pi(1800) Breit-Wigner shape in both a0eta and f0pi decay
channels.
2) Phase of pi(1800)->a0eta
The only way to prove the resonant behavior of pi(1800) in this reaction without
relying on an assumption about the nature of pi2(1880) state is to fit its
phase against supposedly non-resonant 2-+ a0eta wave. The later wave is parameterized
with a single parameter - its constant production phase: Parameters of pi(1800) state
were fixed from fit (1) above.
3) Fit of 2-+ pi2(1880)->a2eta
The intensity of 2-+ a2eta wave and its phase difference with pi(1800)->a0eta
wave were also fitted.
While this fit has a pretty good chi2/dof=1.1, the phase plot at low mass
may indicate that a more sophisticated parameterization may be necessary
to describe pi2(1880)->a2eta well below its nominal threshold.
4) Phase of pi(1800)->f0pi
Finally, attempts to fit the phase of the 0-+ pi(1800)->f0pi wave do not
give satisfactory results. As an example, the fits against the 2-+ pi2(1880)->a2eta
wave are shown below. In the first fit, all fitted parameters were set free.
In the second fit, masses and widths were fixed, and only normalizations and
production phases were allowed to vary.
Both fits have chid2/dof close to 4. There are at least 3 reasons for that.
pi(1800) branching ratio
A ratio of pi(1800)->f0pi to pi(1800)->a0eta decays was determined by integrating
the fitted Breit-Wigner shapes. The obtained value
is 0.48+-0.17. Another approach was to do a series of PWA fits in which decay amplitudes
of two 0-+ waves, a0(980)eta and f0(1500)pi, were tied together with a real coefficient
representing the ratio of their branching ratios. This coefficient became a
parameter of PWA fit.
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.
