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


eta pi+pi-pi0 pi- event sample (~170k events)
eta eta pi- event sample (~4.4k events)

Partial-Wave Analysis (PWA)

In addition to Postscript files, PWA results are given in a raw form as generated by maximum likelihood fits (xvf files). These are ascii files which can be plotted with a custom E852 software.

Final PWA fit with 4 waves and background
"Events" plots are wave intensities predicted by PWA (i.e., acceptance corrected). "Phase" plots are phase difference between the first and the second partial waves. This PWA fit is used in Figs.2,3 in the paper. The found level of isotropic background is from 5% to 10%. The shapes and phases of the partial waves are discussed in a section on mass-dependent fits below.

Fit with f0(1520) instead of f0(1480) Conclusions of the paper do not change if the PDG parameters for f0(1500) are used instead of values fitted from these data.

Fit without f0 wave Significant change in the likelihood and leakage of the pi(1800) peak into background wave confirms that f0pi partial wave is required.

Fit with 60 MeV bins instead of 50 MeV Results are not sensitive to the bin size.

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

Fit with f0(1300) instead of f0(1500) Fit with additional f2(1270)pi wave Fit with 4 additional 2-+ waves Fit with additional 1++ wave Fit without the 2-+ waves and with many 1++ waves Fit with many additional 2-+ and 1++ waves

Parameters of the f0(1500) isobar

The mass and width of the f0(1500) state are not firmly established. Therefore, we treated them as unknown to be determined from a maxium likelihood fit of this event sample. To do this, decay amplitudes of the 0-+ f0(1500)pi wave were generated with varying values of either mass or width. The likelihood was obtained by doing PWA in a single 50 MeV bin centered at 1870 MeV (the nominal mass of pi(1800) in mass-dependent fits), as well as in a broader 100 MeV bin. 1-dimensional scans of the likelihood are given as a function of one parameter when another parameter is fixed at its best value.

1) Mass

Likelihood vs f0 mass:

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.

2) Width

Likelihood vs f0 width:

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.

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.

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).

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.