A fresh look at eta2(1645), eta2(1870), eta2(2030) and f2(1910) in pbar-p ->eta + 3pizero

There is a large discrepancy between results of Crystal Barrel and WA102 for the branching ratio R = BR[eta2(1870)->a2(1320)pi]/ BR[eta2(1870)->f2(1270)eta]. An extensive re-analysis of the Crystal Barrel data redetermines branching ratios for decays of eta2(1870), eta2(1645), eta2(2030) and f2(1910). This re-analysis confirms a small value for R of 1.60+-0.39, inconsistent with the value 32.6+-12.6 of WA102. The likely origin of the discrepancy is that the WA102 data contain a strong f2(1910)->a2-pi signal as well as eta2(1870). There is strong evidence that the eta2(1870) has resonant phase variation. A peak in f2(1270)a0(980) confirms closely the parameters of the a2(2255) resonance observed previously. A peak in eta2(2030)-pi is interpreted naturally in terms of pi2(2245) with reduced errors for mass and width M=2285+-20(stat)+-25(syst) MeV, Gamma=250+-20(stat)+-25(syst) MeV.


Introduction
The objective of this paper is to re-examine data from Crystal Barrel (CBAR) [1] [2] [3] concerning η 2 (1645), η 2 (1870), η 2 (2030) and f 2 (1910). One motivation is to study a large discrepancy for branching ratios of η 2 (1870). Earlier CBAR work found a branching ratio see Table 5 of Ref. [3]. This is much smaller than the value 20.4 ± 6.6 claimed by the WA102 collaboration in central production of ηππ [4]. Other branching ratios determined by the CBAR data are redetermined here. This re-analysis incorporates many further details of spectroscopy in this mass range which have appeared since the year 2000. A review of these earlier publications will set the scene and introduce the relevant resonances. For a comprehensive review of the CBAR work withp in flight see Ref. [5]. The first study of η 2 (1645) and η 2 (1870) in CBAR data was at two beam momenta: 1200 and 1940 MeV/c [1] [2]. The η 2 (1645) was observed decaying to a 2 (1320)π and a 0 (980)π. No other a 0 or a 2 appear in the data, so a 0 will be used hereafter as a shorthand for a 0 (980) and likewise a 2 for a 2 (1320).
There was also a strong f 2 (1270)η signal near its threshold ∼ 1810 MeV. It could not be explained as the high mass tail of η 2 (1645). The reason was that a single η 2 (1645) decaying to f 2 η would contain a large f 2 η signal in both numerator and denominator of the Breit-Wigner amplitude; cancellation between numerator and denominator cannot accomodate the large f 2 (1270)η signal. The data were fitted with the addition of the η 2 (1870) though the data did not rule out the possibility of a non-resonance threshold effect at that time. The majority of CBAR data withp in flight were taken in 1996. For η3π 0 , statistics were a factor 7 higher than earlier data at each of nine beam momenta ranging from 600 to 1940 MeV/c, i.e. an overall increase of statistics by a factor ∼ 30. There are typically 25-30K events at each beam momentum and a total of 213K events. Fig. 1 shows scatter plots from CBAR data in three ranges of ηππ mass. Fig. 1(a) shows the η 2 (1645) mass range; there is a vertical band due to a 2 π and also a peak in ππ near 1 GeV. The spin analysis ruled out f 0 (980), showing that the peak in ππ is due to the low mass tail of η 2 (1870) → f 2 (1270)η. Fig. 1(b) shows the η 2 (1870) mass range. The a 2 (1320) and f 2 (1270) bands appear of similar strength; however, the f 2 is somewhat broader and therefore stronger. There is clearly no large excess of a 2 decays in this mass interval. The branching ratio quoted by WA102 would require an η 2 (1870) → a 2 π signal a factor ∼ 3.5 larger than f 2 (1270)η when one allows for relative decay rates of a 2 → ηπ and f 2 (1270) → π 0 π 0 and for charge combinations. Fig. 1(c) shows the mass range of η 2 (2030); there is a peak where a 2 (1320) and f 2 (1270) bands cross. The a 2 π band is strong and f 2 (1270)η is weaker. Note also that weak a 0 (980) bands are visible in all three panels.
There is additional evidence for η 2 (2030) → f 2 (1270)η and a 2 π in further CBAR data for pp → ηπ 0 π 0 , see Fig. 11 of Ref. [6]. A distinctive feature of this state is its strong decay to [a 2 π] L=2 , where L is the orbital angular momentum in the decay.
Following the year 2000 publication of η3π 0 results, a combined analysis was made of 10 sets of data with isospin I = 0 and C-parity +1. Six of these were CBAR data forpp in flight → π 0 π 0 , ηη, ηη ′ , ηπ 0 π 0 , η ′ π 0 π 0 and ηηη [7]. The other four were high quality differential cross sections and polarisations forpp → π + π − from two experiments: Eisenhandler et al. [8] and PS172, Hasan et al. [9]. This analysis revealed two towers of resonances centred at ∼ 2000 and 2270 MeV, with all J P for qq states expected in this mass range. Most have been observed in at least three sets of data. Polarisation data provide a clean separation ofpp 3 P 2 and 3 F 2 states. The f 2 (1910) of the PDG [10] was confirmed and identified as dominantly 3 P 2 ; a neighbouring 3 F 2 state was observed at 2001 MeV. Further f 2 (2240) 3 P 2 and f 2 (2295) 3 F 2 states were also observed. In the new analysis reported here, the first three of these f 2 states play a significant role. The f 2 (1910) lies close to η 2 (1870) and is important for the discussion of WA102 data.
In 2001, a combined analysis was also made of data on I = 1, C = +1 states [11]. The spectrum of states is less complete than for I = 0, C = +1, because of the lack of polarisation data. Nonetheless, an a 2 (2255) appeared clearly in three channels of data. Secondly, there is a less well identified π 2 (2245). These two states now appear in the analysis reported here. So, in summary, the picture has developed significantly since the earlier analysis of η3π 0 .
The η3π 0 channel may appear to be a complicated channel to analyse, because of the multiplicity of ηπ 0 and π 0 π 0 combinations. However, η 2 (1645) and η 2 (1870) were found here; also, once one knows the mass and width of η 2 (2030) from ηππ final states, it is easily detected in the present data via its strong decay to [a 2 π] L=2 , which has a very distinctive angular dependence. Interferences between channels provide intricate information on identifiable resonances, even though the angular correlations cannot be displayed because they are multi-dimensional. It is necessary to work from log likelihood and mass projections of ηπ, ππ, 3π and ηππ. A valuable check on the analysis is to introduce deliberate errors into angular dependence of amplitudes; genuine signals then drop to low values.
Two alternative starting points have been adopted. The first is the year 2000 analysis, which now improves. The second starts from the WA102 ratio for η 2 (1870) → a 2 π and f 2 (1270)η. This gives a considerably worse fit. After iterations, the two fits converge to a single solution except for minor points of ambiguity in a 0 π decays. No significantly different solutions have been found at any beam momentum despite searches from a variety of initial parameters.
The layout of the paper is as follows. Sections 2 and 3 go through technical details of the analysis procedure. It may be useful to glance first at figures of later sections, so as to appreciate the rationale for the steps discussed in Sections 2 and 3; the techniques need to be adapted to what is found in the data. Section 2 introduces the channels which are required and deals with formulae. These are well known from earlier literature, but need to be documented. The one point requiring special treatment is the opening of the f 2 (1270)η threshold, close to η 2 (1870). It is necessary to fold the width of f 2 (1270) into the phase space for the f 2 η final state appearing in the Breit-Wigner amplitude. Secondly, the way the spin dependence is treated in terms of partial waves is discussed.
Section 3 presents features of the data. Fig. 4 shows mass spectra for 3π, ππη, πη and ππ at one representative momentum, 1642 MeV/c; other momenta show similar features and Fig. 1 of Ref. [3] presents spectra at 1800 MeV/c. This section gives more detailed comparisons with data in further figures. Section 4 then presents essential results. Table 1 shows changes of log likelihood when each channel is dropped from the fit and all others are re-optimised. This Table identifies directly the important channels and their significance levels, and how they vary with beam momentum. Fig. 7 presents the cross sections for all reactions as a function of beam momentum; these lead to considerable insight into the physics. Peaks shown in Fig. 8 may be identified with a 2 (2255) and π 2 (2245).
A revised set of branching ratios is derived for η 2 (1645), η 2 (1870), η 2 (2030) and f 2 (1910). The strong decay modes change little, but there are some significant changes from the earlier publication in weak channels, for reasons which are understood. The important ratio for η 2 (1870) → a 2 (1320)π and f 2 (1270)η changes only slightly and remains completely inconsistent with the WA102 result.
Section 5 therefore re-examines WA102 mass projections for η 2 (1645) and η 2 (1870) → a 0 π, a 2 π and f 2 (1270)η. There is good agreement with CBAR data for η 2 (1645) → a 2 π and a 0 π and their ratio of intensities. There is also reasonable agreement for the line-shape of η 2 (1870) → f 2 (1270)η. So masses and widths of these states agree well between the two experiments. The evidence for the controversial f 2 (1870) → a 2 π signal rests on a small bump in the a 2 π mass spectrum in WA102 data. It now appears likely that some or all of this bump is instead due to f 2 (1910). This state has a strong decay to a 2 π and weaker decay to f 2 (1270)η. An earlier WA102 publication in fact claimed to observe J P = 2 + peaks near 1900 MeV in central production of a 2 (1320)π → ρππ and in f 2 (1270)ππ [12]. Section 6 presents evidence that η 2 (1870) has resonant phase variation. Section 7 summarises results and draws conclusions.

Methodology and formulae for fitting data
The data are fitted by the maximum likelihood method, i.e. fitting every individual event without binning. Log likelihood is normalised so that a change of 0.5 corresponds to a change in χ 2 of 1. For the high statistics available here, log likelihood follows the χ 2 distribution closely as the number of variables is varied. The following channels are fitted: Here σ stands for the ππ S-wave amplitude. An incoherent phase space background is also included. It arises from experimental cross-talk between the η3π 0 final state and other final states, e.g. 4π 0 and ηηπ 0 π 0 . This background is known accurately and is discussed in the earlier publication [3]; it is in the range 7.7-9.6%, increasing slowly with beam momentum. We can dispose of channels (4) and (5) quickly. Their ππη peaks are narrow and have no significant impact on other channels. The f 1 (1285) is fitted as decaying purely to a 0 π. Its mass and width need tuning by a few MeV to fit the height and width of the observed peak. The η(1440) is fitted with decays to a 0 π and ησ, with interference between them. There is also evidence for η(1440) → f 0 (980)η, discussed in Ref. [13] and confirmed in [14]. The η(1440) appears clearly only at low beam momenta up to 1200 MeV/c and the fit to it does not change significantly from that reported earlier.
Tests have been made for additional resonances produced inpp → X + π where X has quantum numbers J P = 0 − , 1 + , 3 + or 4 + . There is no significant evidence for any of these. The high spin states would be easily detectable from their strong angular dependence. Adding 0 − → a 0 π and f 0 (980)η does give a small improvement in log likelihood, typically 20, but this is because these amplitudes have no angular dependence and are prone to picking up noise; there is no discernable optimum in log likelihood as ππη mass is varied. The final fit omits 0 − states other than η(1440).

Treatment of partial waves
Most channels involve a two-stage processpp → X + π, X → Y + π. The orbital angular momentum in the decay to X + π will be denoted by ℓ and that in the subsequent decay to Y + π by L. Thepp initial state is a mixture of spin singlet and triplet partial waves. The total spin S is limited to S z = ±1 or 0 along the beam direction; S z = ±1 give identical angular distributions. A problem in the analysis is the absence of polarisation data. The consequence is that triplet contributions to X are not cleanly separated between J = ℓ, ℓ ± 1 and ℓ ± 2. It is therefore not possible to do a full partial wave analysis of both production and decay. This would become possible if data were available at a future date from a polarised target.   2 sketches the process of production and decay. The a 2 and recoil pion are drawn in thē pp rest frame. The decay of the a 2 is shown after a Lorentz transformation to the a 2 rest frame. One way of writing amplitudes is to use rotation matrices to express the initial state |J, J z > in terms of a linear combination quantised along the a 2 direction. This combination is invariant under a Lorentz boost to the rest frame of the a 2 [15]. Then the a 2 decay amplitude may be expressed in terms of the usual Legendre polynomials. The problem with this approach is that the rotation matrices at the first step depend on J.
This problem may be avoided by a procedure known as the Wick rotation. After a Lorentz boost to the rest frame of the a 2 , a rotation of axes is made through an angle −θ, the same as in the production process but with opposite sign. Rotation matrices then cancel between the first and second steps. The Wick rotation alters the angles of π and η in thepp rest frame, because of the Lorentz boost. It preserves the fact that the initial state is restricted to J z = 0 or 1. Since J is not known, it is however necessary to discard the angular dependence of the production process or parametrise it empirically. The way the programme is written, the Wick rotation is made for every channel and every event just once, and the required amplitudes are stored. This speeds up the analysis by a large factor. Two improvements of details over the earlier analysis are introduced. For singlet states, S z = 0. The processespp → X +π may go via emission of a pion with orbital angular momentum ℓ ≥ 1, because of the pseudoscalar nature of the pion. For low momenta of the spectator pion, several channels rise steeply, see Figs. 7 and 8 below. This is consistent with P-state production. If the η 2 π final states in reactions (7)-(9) are produced via P-state pion emission, the initial state is restricted to J P C = 2 −+ unless it is exotic (J P C = 3 −+ or 1 −+ ). It would be surprising if exotics couple topp and there is no evidence for such exotics in other CBAR data in flight. This leads to the useful restriction that the initial state is spin singlet, with S z = 0. A further point is thatpp → J P C = 2 −+ → [2 −+ + π] L=1 has Clebsch-Gordan coefficients such that the final state is purely where J ′ is the spin of the η 2 . This leads to a distinctive angular dependence for the whole amplitude describing both production and decay. It is helpful in isolating the processpp → π 1 (2245) → [η 2 (2030)π] L=1 . However, in addition we detect some significant production from initial spin triplet states, particularlȳ Interferences between all channels are included. However, several spin triplet states produced frompp may feed a single final state such as [f 2 π] L=1 . As a result, interferences between channels are not fully coherent. To accomodate this detail, each interference term is multiplied by a coherence factor which is allowed to optimise in the range ±2. There are also interferences between two a 2 π and two a 0 π combinations for each resonance in ηππ. These interferences are fully coherent for a single resonance.

The treatment of phase space
The η 2 (1870) lies close the the f 2 (1270)η threshold. The intensity of the f 2 η decay needs to be parametrised so as to include the line-shape of the f 2 into the available phase space. The formula for the general casepp → X + Z, where X and Z both have significant width, is given by Eq. (40) in Ref. [16]. This formula is used for channels (2) and (3),pp → f 2 (1270)a 0 and a 2 σ. For the simpler case of η 2 (1870) → f 2 (1270)η, it reduces to where p is the momentum of the η in the f 2 η rest frame; √ s and √ s 1 are the corresponding masses of f 2 η and f 2 . Also F F (s) is a form factor for η 2 (1870) → f 2 (1270)η. It is taken as a Gaussian exp(−αp 2 ), where α = 4.5 (GeV/c) −2 , corresponding to a radius of interaction 0.73 fm for the overlap of f 2 and η. From a wide range of CBAR and other data, α is known with an error of ±1.0 (GeV/c) −2 . In the range of the present data, results vary little over the range α = 3.5 − 5.5 (GeV/c) −2 . However, the exponential dependence may be an approximation.
Then the Breit-Wigner amplitude for where Γ 1 and Γ 2 are constants describing decays to (1) a 2 π and a 0 π, (2) f 2 η. Above 1.96 GeV, the Fermi function may be an approximation, but the line-shape of η 2 (1870) is falling fast there. In principle, this could lead to ambiguities in fitting the η 2 (2030), but we find that this state is produced in a different range of beam momenta, so in practice there is no problem. A further potential complication is that (M 2 − s) of the Breit-Wigner denominator should strictly be replaced by ( m(s) is the 'running mass, which makes the formula fully analytic [17]. At a sharp threshold, m(s) peaks strongly at the threshold. However, for a threshold as wide as f 2 η, its effect is small and can be absorbed into optimised values of M and Γ 2 . In our earlier publication on the present data in the year 2000, a sizable a 0 (980)σ amplitude was fitted. It accounted for ∼ 15% of the η3π 0 cross section, see Fig. 2(a) of Ref. [3]. It was fitted using the 1993 parametrisation of the σ amplitude by Zou and Bugg [18]. Since then, the understanding of the σ amplitude has improved greatly through the work of Caprini et al. [19], using the Roy equations. That work fits the ππ scattering length to (0.220±0.005)m −1 π and alters the s-dependence of the ππ amplitude below 600 MeV quite significantly. In addition, there is now evidence for significant coupling of the σ to KK: g 2 KK (σ)/g 2 ππ (σ) = 0.6 ± 0.15 [20]. We now use Eqs. (1)-(11) of [21] for the σ; they fit accurately the predictions of Colangelo et al. up to 1 GeV and fit better the inelasticity required above the KK threshold. The additional structure in the ππ amplitude is modest, but enough that the evidence for the a 0 (980)σ amplitude in present data almost disappears. The remaining signal is barely a 2 standard deviation effect. It is now omitted and systematic errors covering the possible signal will be included in branching ratios discussed below in subsection 4.2. The earlier a 0 (980)σ signal was perturbing the fit via interferences with a 0 (980) signals from decays of η 2 (1645), η 2 (1870) and η 2 (2030). A consequence is that there are now rather large changes to the branching ratios of the η 2 (1870) and η 2 (2030) to a 0 (980)π. The basic difficulty here is that the earlier broad σ amplitude had little structure and gave a rather flexible fit to the data. The branching ratios of η 2 (1645) and η 2 (1870) to f 2 (1270)η and a 2 (1320)π are more robust and change little.
The σ → ππ amplitude is needed only up to ∼ 1100 MeV. Its form for elastic scattering is known quite precisely. In elastic scattering, it is parametrised in the form N(s)/D(s), where the numerator contains an Adler zero just below threshold, making the amplitude weak at low momenta. In some production processes with large momentum transfers, e.g. J/Ψ → ωσ, the numerator needs to be replaced with a constant in order to reproduce a broad peak in the mass range 450-500 MeV, produced by a pole in D(s). In the present data, there is no evidence for this behaviour, so the amplitude is taken to be that of elastic scattering.

Fits to data
In order to demonstrate the existence of η 2 (1645), η 2 (1870) and η 2 (2030), it is necessary to fit data at all beam momenta and show that various selections of events require the presence of all three, with consistent masses, widths and ratios of decay amplitudes at all beam momenta. The earliest publication studied data at just two beam momenta, 1200 and 1940 MeV/c [2]. It was immediately obvious that these two momenta required two resonances η 2 (1645) and η 2 (1870), produced with considerably different relative intensities at the two momenta. The η 2 (1870) has a strong decay to f 2 (1270)η and the η 2 (1645) does not. The picture developed further when data at all nine beam momenta were available. A further η 2 (2030) was required, with a distinctive decay to [a 2 π] L=2 [3]. Here we shall not repeat this lengthy story, but refer the reader to the original publications. The conclusion from all beam momenta combined is that just these three states are sufficient to fit all the data, with consistent decay amplitudes at all beam momenta. Here we simply illustrate the quality of fits to mass projections. Fig. 4 shows mass projections for 3π, ππη, πη and ππ at a beam momentum of 1642 MeV/c. Points with errors are data; fits are shown as histograms. In (a), there is a high mass peak due to π 2 (1670)η, π 2 → f 2 (1270)π. In remaining panels there are peaks due to f 1 (1285), a 0 (980) and a 2 (1320) and a shoulder due to f 2 (1270).
In earlier work, there was some evidence for η 2 (1870) → [ησ] L=2 . There remains a small improvement in log likelihood when this channel is included. This improvement is however typically 30, which is less than for almost all other channels. Furthermore, its inclusion increases errors on other channels, i.e. fits become less stable including it. The problem is that there is no narrow signature of the σ → ππ S-wave, so it tends to absorb any noise in the data. This decay is now omitted. Table 1 shows changes in log likelihood when each channel is omitted from the fit and all other channels are re-optimised. One sees immediately the significance level of each channel and its  Table 1: Changes in log likelihood when each channel is removed from the fit and others are re-optimised.

Results
dependence on beam momentum. We now wish to draw conclusions from peaks observed in some channels of data. Fig. 7 shows cross sections v. beam momentum. The absolute normalisation is taken from cross sections determined in Ref. [3] for the whole η3π 0 data; they are uncorrected for branching fractions of η and π 0 → γγ and therefore correspond directly to the number of events collected. The integrated cross section varies little with beam momentum, with a small (10%) enhancement near 2270 MeV; it is shown in Fig. 2 of Ref. [3]. On Fig. 7, smooth curves for individual channels (2)-(12) are drawn through the data within one standard deviation. Errors are typically 5−10% and some examples will be displayed with errors in Fig. 8. Cross sections for several channels expand at low beam momenta 900 and 600 MeV/c to follow the 1/v dependence of the total cross section, where v is the beam velocity in the centre of mass.
In Fig. 7(b), there is a strong peak in the channel f 2 a 0 near 2250 MeV and a broader peak in η 2 (2030)π at slightly higher mass. There is a possible peak in η 2 (1645)π in the same mass range. There is also a peak in f 2 (1910)π near 2150 MeV. The channel π 2 (1670)η peaks strongly at high mass. There is evidence for production of f 2 (2240)π at the highest two beam momenta. Other channels show only weak structure, except that a 2 σ peaks at low masses.
The question arises how to interpret the peaks. Are they due to resonances? Here it is necessary to take care over details in the formulae. The amplitude for a resonance such as The factor k in the denominator is the momentum of thep in thepp centre of mass; this is the flux factor for the incident beam. The phase space factor for ρ(pp, s) in the numerator is k/ √ s, multiplied by a centrifugal barrier factor. We choose to make comparisons with data by accounting explicitly for the intensity factors k/ √ s forpp and 1/k 2 from the flux factor; the two together give a factor 1/k √ s = 1/v, where v is relativistic velocity. We multiply cross sections of  Fig. 8(a), there is a peak closely resembling the a 2 (2255), reported in the earlier combined analysis of I = 1, C = +1 CBAR data with M = 2255 ± 20 MeV and Γ = 230 ± 15 MeV; it was observed as a clear peak in data for pp → f 2 (1270)π, see panel (ℓ) of Fig. 38 of Ref. [5]. It was observed in both 3 F 2 and 3 P 2 decays to f 2 (1270)π, with an amplitude ratio −2.13 ± 0.20 favouring coupling ofpp to 3 F 2 . The dashed curve of Fig. 8(a) shows the remaining phase space factor calculated with this ratio for 3 F 2 and 3 P 2 production and with S-wave decay to f 2 a 0 . The rise of this curve with mass fails to fit the data. Any other ratio of 3 P 2 and 3 F 2 also fails to fit the data. Adding P-state f 2 a 0 decays makes the dashed curve peak even higher in mass.
The full curve shows the result of multiplying the dashed curve by the line-shape of a 2 (2255), assuming a Breit-Wigner resonance of constant width. It is remarkably close to the data. We regard this as further confirmation of the a 2 (2255). A small improvement is possible by reducing the mass by 10 MeV to 2245 MeV and decreasing the width to 225 MeV, but these changes are within the errors of the earlier determination and also within errors of present data.  Figure 8: Production of (a) f 2 (1270)a 0 compared to phase space for S-wave production (dashed curve), a fit to f 2 (2255) (full curve) and a fit with parameters of this state optimised (dotted curve); (b) η 2 (2030)π compared to phase space for P-state production (dashed curve), a fit using the line-shape of π 2 (2245) (full curve) and a fit with parameters of this state optimised (dotted). Fig. 8(b) shows results for the channel η 2 (2030)π. Production of this final state is dominantly (∼ 74%) withpp helicity 0, suggesting production viapp → π 2 (2245): The full amplitude D for this process is rather distinctive. Suppose it is written in fixed axes in thepp centre of mass. The initial state is spin singlet, helicity 0. Suppose the spectator pion of Eq. (18) is produced at angle θ of Fig. 2 and with azimuthal angle φ around the beam direction. Let the momentum of the decay pion in Eq. (18) be q. Let the decay angle of the a 2 (1320) in Eq. (20) after the Wick rotation be α, with azimuthal angle β, using the same axes as for Eq. 18. Then = i √ 2qB 1 (q) sin θ sin α cos α sin(φ − β).
Here B 1 (q) is the centrifugal barrier factor for L = 1 decay. The amplitude for the decay a 2 (2030) → f 2 (1270)η has an identical form. Amplitudes for decays of a 2 (2030) → [a 0 (980)π] L=2 take a similar form with angles α and β those of the decay pion. The amplitude for decays to a 2 (2030) → [a 2 (1320)π] L=2 involves a combination of spin 2 of the a 2 (1320) with L = 2 to make spin 2 of the a 2 (2030). This amplitude may be constructed along the same lines using Clebsch-Gordan coefficients or using tensor algebra. Because of the additional angular dependence on L = 2 in the decay to [a 2 (1320)π] L=2 , it is particulary distinctive and stands out clearly in the data. From earlier work, the parameters of π 2 (2245) have sizable errors, M = 2245 ± 60 MeV, Γ = 320 +100 −40 MeV [11]. The full curve of Fig. 8(b) shows a fit with these parameters; the width is clearly too large and the mass somewhat too low.
The data determine relative phases of strong channels with errors of ∼ ±15 • at every momentum. The phase variation of f 2 a 0 and a 2 (2030)π channels agree within these errors. If f 2 a 0 is resonant, then so is a 2 (2030)π and this can only be explained in terms of some combination of π 2 (2245) and a 2 (2255). There are further triplet states a 3 (2275) and a 1 (2245) in this vicinity, but they can only contribute to decays to [η 2 (2030)π] ℓ=2 ; this is inconsistent with the observed line-shape of the peak and can only make a small contribution to the data. Systematic errors assigned to the mass and width of π 2 (2245) cover any weak contribution from this source.

Other channels
The production of η 2 (1875)π is distinctively different to that of η 2 (2030)π. Production from the initialpp system with helicity ±1 is strongly dominant, requiring an initial triplet state. However, no conclusion can be drawn from the slowly varying cross section in Fig. 7(a). A minor correction to Ref. [3] is that the η 2 (1875)π signal plotted there in Fig. 2(a) was multiplied by a factor 2 to make it clearly visible; the intensity recorded there is close to the present analysis.
Production of f 2 (1910)π is roughly equal from initial states with helicity 0 and 1, but shows significant slow variation with mass. This is possible from initial triplet states. The peak near 2150 MeV may indicate production from the initial state a 2 (2175).
Production of π 2 (1670)η is dominantly viapp helicity 1, (i.e. spin triplet) but again fluctuates smoothly with mass by more than a factor 2. No firm conclusion can be drawn from the variation of cross section with mass.
A final point is that data are available at 900 MeV/c and were included in the earlier analysis [3]. However, there are small cross sections for several channels and these are difficult to determine with confidence. This beam momentum has been studied, but is discarded from the present analysis because of substantial systematic errors in weak channels. There are also low statistics at 600 MeV/c, and this momentum is omitted for the same reason.

Branching fractions
Fortunately, the branching fractions of η 2 (1645), η 2 (1870) and η 2 (2030) are not sensitive to the question whether or not their production goes directly via resonances in thepp channel. If such resonances are involved, all decay modes of these channels pick up the same phase variation from the production process. The dominant decay of η 2 (2030) is to [a 2 π] L=2 ; that for η 2 (1870) is to f 2 (1270)η and that for η 2 (1645) is to a 2 π. Branching fractions will be quoted with respect to these dominant channels. Some of the smaller branching fractions have changed significantly since the analysis of the year 2000.
A general comment is that data determine ratios of amplitudes rather than ratios of intensites. That is, log likelihood has a parabolic minimum as a function of the amplitude ratio. The procedure is therefore to fit data using, for example, the magnitude of the η 2 (1645) → a 2 π amplitude as one variable and the ratio of amplitudes η 2 (1645) → a 0 π/η 2 (1645) → a 2 π as the second. This makes it easy to investigate fluctations with beam momentum. The average of each ratio of amplitudes is found by weighting values at each beam momentum with the total number of a 2 (1645) events summed over both channels. The same procedure is used for η 2 (1870), η 2 (2030) and f 2 π channels like f 2 (1910)π. This gives more stable results than trying to determine errors of ratios at each momentum and using them in the weighting procedure; errors fluctuate somewhat from momentum to momentum because of large errors for phases fitted to weak channels. Having found the average ratios of amplitudes, a second pass is made through the fit, fixing these ratios at all momenta. Then the ratio of intensities is obtained from the total number of events fitted to each channel, integrated over all beam momenta.
In evaluating branching ratios, it is necessary to make use of Eqs. (13)- (15). Here there is a dilemma. It is convenient to parametrise the Breit-Wigner denominator as far as possible with constant widths for each decay channel, as is conventional in the Particle Data Tables. Our procedure is to modify the numerator of Eq. (15) to Γ 2 F (s), so as to agree with the denominator. We then evaluate branching ratios at each beam momentum, and average over momenta to determine Γ 1 /Γ 2 . This procedure converges within errors after one iteration. The same procedure is used to evaluate the effects of L = 2 centrifugal barriers on widths for The acceptance for strong channels f 2 (1270)η and [a 2 π] L=2 varies rather strongly with beam momentum. Numerical results are shown in Table 2, normalised to 1 at the highest beam momentum 1940 MeV/c; they are evaluated using the Monte Carlo simulation of the detector. One sees a large variation of the intensity ratio R for some channels. This variation almost disappears at 1940 MeV/c, where all decay channels are nearly fully open. We shall tabulate branching ratios R corrected to this momentum. Above this momentum, results may be affected by errors in the Fermi function adopted in Eq. (14), so this is close to the optimum compromise. Errors in branching ratios due to these uncertainties are included in errors quoted in Table 3 below. We regard this procedure as an improvment on the work of Ref. [3], where branching ratios were evaluated purely from geometric acceptance without the effects of centrifugal barriers or the form factor F F (s) of Eq. (13)  Coming to technicalities, it will be necessary to correct the number of observed a 2 (1320)π and f 2 (1270)η events in π 0 π 0 η for unobserved decays. The f 2 (1270) has a branching ratio of 0.848/3 to π 0 π 0 [10]. The amplitude for the I = 1 component ofpp → πX, X → f 2 η is given by In (23), X stands for the amplitude of X → f 2 η; the brackets in the last line identify pions coming from f 2 (1270). For X → a 2 π, the decay amplitude of X is 1/3(π + a − 2 − π 0 a 0 2 + π − a + 2 ), where the minus sign for π 0 a 0 2 can be absorbed into the fitted phase for this channel. In the π 0 a 0 2 final state, what is actually observed is i.e. six combinations. The coherent sum of all combinations is fitted to the data. Table 3 shows in column 3 branching fractions from the previous analysis [3] for comparison purposes. The next column lists what is fitted now. The final column corrects this for all charge states and for the branching fractions of a 2 (1320) → πη (14.5%) and a 0 (980)π to πη. This last branching fraction is taken to be the value used by WA102, (86%) for easy comparison with their results. This is close to the value adopted by the PDG [10]. If their value is adopted, a 0 (980) branching fractions increase by a factor 1.015 ± 0.021. Results in columns 4 and 5 supercede the earlier results.
Branching ratios of decays to a 0 (980)π final states are unstable. They depend somewhat on whether or not the a 0 (980)σ channel is included in the fit. The basic difficulty is that the σ amplitude is broad and gives rise to interferences all over the 4-body phase space. The a 0 σ channel is therefore not well determined. In the absence of definite evidence that it is needed, we omit it. All a 0 signals are weak and their phases with respect to dominant decays have quite large errors. There are also strong correlations between couplings of η 2 (1645), η 2 (1870) and η 2 (2030) to a 0 π. In the earlier analysis, a 0 π decays interfered with a 0 σ, giving apparently small but unreliable errors.
From the present analysis, the ratio for η 2 (1645) lies close to the WA102 result 0.077 ± 0.016; the a 0 (980) signal is clearly visible as a peak in their raw data [4]. For this reason, the weighted mean of these two values has been adopted and fixed for η 2 (1645). This helps stabilise fits to a 0 π. However, there are large correlations between η 2 (1870) and η 2 (2030) decays to a 0 (980)π. Together with uncertainties due to possible contributions from a 0 (980)σ, the result is that branching fractions of η 2 (1870) → a 0 π and η 2 (2030)π can both vary freely over the range 0.1 to 0.85 in present data. Errors on values given in Ref. [3] for these two channels need to be increased substantially to take account of these systematic errors. New estimates are given in Table 3 . These uncertainties are not significantly correlated with branching fractions to f 2 (1270)η and a 2 π. Branching ratios between f 2 (1270)η and a 2 (1320)π decay modes are mostly quite stable, because these decays are strong. The ratio of decays of η 2 (1870) to f 2 (1270)η and a 2 (1320)π remains stable. So does the ratio of decays of η 2 (2030) to f 2 (1270)η and [a 2 π] L=2 , because these are conspicuous decays.
A mistake has been located in the branching fraction of η 2 (2030) to [a 2 π] L=0 reported in Ref. [3]. The corrected value is given in entry 6 of Table 3. This decay is much weaker than that to [a 2 π] L=2 .
In earlier work, the error on the mean was taken as the statistical error divided by √ N , where N is the number of beam momenta. However, it is now clear that systematic errors are somewhat larger than this. The systematic errors are estimated from (i) variation of results with the ingredients included in the fit, particularly the number of interferences included between channels; (ii) variations with masses and widths of η 2 (1645), η 2 (1870), η 2 (2030), f 2 (1920), f 2 (2001) and f 2 (2240), (iii) possible contribution from f 2 (2293), (iv) uncertainties in the background from other final states. Ultimately, systematic errors dominate for all branching ratios, particularly for decays to the weak a 0 π channels.

Fits to WA102 data
The WA102 collaboration measured central production of ππη and produced separate sets of data for η → γγ and π + π − π 0 [4]. These data have been read from their graphs and refitted.
Their approach was to use the K-matrix for η 2 (1645) and η 2 (1870). If a single amplitude is  Figure 9: Fits to WA102 data for central production of a 2 π, a 0 π and f 2 (1270)η with J P C = 2 −+ . The first column shows data for η → γγ and the second data for η → π + π − π 0 . Full curves are fits without η 2 (2030) and dashed curves with it. used in this approach, the amplitudes for η 2 (1645) and η 2 (1870) each loop round the Argand circle once; continuity of the amplitude then requires a zero between them. The dip due to this zero played a significant part in fitting the data.
The basic assumption of the K-matrix approach is that resonances combine in a production process in an identical way to elastic scattering. This is a pure assumption. In elastic scattering, ingoing and outgoing waves for all coupled channels must sum asymptotically to unit intensity. However, in production processes considered here, the amplitude is only a tiny fraction of thepp total cross section. There is no obvious reason why 2-body unitarity should apply in the same way as for elastic scattering. The K-matrix approach has been tested on four sets of experimental data in Ref. [22]; it failed seriously in every case. If it is used, phases of η 2 (1645) and η 2 (1870) need to be fitted freely, since final states may rescatter between one another, generating phases which need to be fitted arbitrarily. Our analysis therefore uses the isobar model. This makes no assumption about the effects of unitarity and allows separate phases for η 2 (1645) and η 2 (1870).
Refitted results are displayed on Fig. 9. Data for η → γγ and η → π + π − π 0 have been fitted simultaneously, including decays to all of a 0 (980)π, f 2 (1870)η and a 2 (1320)π. They have been fitted with and without η 2 (2030). That component was not known at the time of the WA102 analysis. Full curves on Fig. 9 show fits without η 2 (2030) and dashed curves the fits including it. With it, the total χ 2 improves from 124.5 to 95.8. The reason is obvious: it provides extra freedom in fitting small defects in the mass region above 2 GeV. This improvement in χ 2 of 28.7 needs to be balanced against the fact that there are three extra complex coupling constants for the three channels of η 2 (2030), i.e. 6 extra fitting parameters. The improvement in χ 2 is 3.6 standard deviations. It is debatable whether or not η 2 (2030) is really present. Conclusions will be drawn here from fits without it; these fits are more secure.
A significant point is that the K-matrix zero between the 1645 and 1870 MeV peaks made both of them narrower and pushed the peaks apart. In the WA102 fit, the mass of η 2 (1645) was 1605 ± 12 MeV for η → γγ data and 1619 ± 11 MeV for η → 3π. These are to be compared with the CBAR determination of 1645±6(stat)±20(syst) MeV. In the isobar model, the dip between the two resonances can be filled in by interferences between them. Table 5 shows fitted masses and widths for η 2 (1645) and η 2 (1870) for two cases. In the first, column 2, the parameters of η 2 (1870) are fitted freely.
The width of the η 2 (1870) tends to run away to a large value. In Fig. 9(d), the width of the η 2 (1870) peak in decays to f 2 (1270)η is sensitive to scatter in the points near the peak. The second fit (column 3) is made adding to χ 2 a contribution given by the CBAR masses and widths with statistical and systematic errors combined in quadrature. This extra constraint stabilises the fit and gives mass and width for η 2 (1870) closer to the CBAR values. In our opinion, the third column is the more reliable, bearing in mind that the addition of the η 2 (2030) increases the uncertainties from WA102 data even further. The explicit re-arrangement of Eq. (15) for where M and Γ refer to values for η 2 (1870) in the last two lines of Table 6; F (s) is given by Eq. (14). The CBAR data are fitted using Eqs. (14) and (26). Numerical values in the denominator are such that |f | 2 integrated over s reproduces the branching fractions for a 2 π, a 0 π and f 2 (1270)η in Table 3. The line-shape of η 2 (1870) is shown in Fig. 10 and is very close to that of a Breit-Wigner resonance of constant width. It peaks at 1792 MeV. This is related to the opening of the  Table 5: Masses and widths fitted to WA102 data without any constraint (column 2) and with a penalty function given by errors on CBAR masses and widths (column 3); Column 4 shows CBAR values of masses and widths. The remaining question is whether there is really any significant a 2 (1320) signal in WA102 data due to η 2 (1870) → a 2 (1320)π. In a 1997 publication [12], WA102 claimed to observe η 2 (1645) → a 2 (1320)π in central production of 4π, with a small shoulder at high mass which could be η 2 (1870). They reported a strong 2 + signal in a 2 (1320)π → ρππ at 1900 MeV and a broad 2 + peak in f 2 ππ near 2000 MeV. The integrated 2 + signal was considerably stronger than that for J P = 2 −+ . However, further data reported in the year 2000 on central production of 4π were interpreted in terms of 2 −+ → a 2 (1320)π → 4π, produced only with J z = ±1 [23]. The 2 ++ → f 2 (1270)ππ was found again but no 2 ++ → a 2 (1320)π. They make no comment on why this change from the 1997 work occurs. In central production via Pomeron exchange, there is no obvious reason why 2 ++ should not be produced with J z = 0, ±1 and ±2. Our view is that the data really need to be fitted with all allowed values of J z for both J P = 2 + and 2 − . It would be valuable if the Compass collaboration could check this point.
In the CBAR data analysed here, there is clear evidence for f 2 (1910) and f 2 (2001) decaying to a 2 (1320)π and f 2 (1270)η. The a 2 π decay dominates. This is readily understood from the fact that the L = 1 centrifugal barrier inhibits decay to f 2 (1270)η. We suggest that the small bump claimed by WA102 in a 2 (1320)π at 1860 MeV is due to f 2 (1910). We have fitted WA102 data using PDG parameters M = 1903, Γ = 196 MeV for f 2 (1910) instead of η 2 (1870). The fit, shown in Fig. 11, gives a slightly improved description of the data (by 11 in χ 2 ), but we are unable to go back to the original data and check the J P analysis. Figure 11: Fits to WA102 data with η 2 (1870) → a 2 π replaced by f 2 (1910) → a 2 π: (a) η → γγ, (b) η → π + π − π 0 .

mass (GeV) Events/40 MeV mass (GeV)
It remains an interesting question why η 2 (1870) has a fairly large f 2 (1270)η S-wave decay. The dispersive term m(s) of Eq. (17) leads to attraction in this channel near threshold [17]. It necessarily favours this decay mode. There is an isospin partner π 2 (1880) for η 2 (1870). It too has a strong decay mode to a 2 (1320)η [24]. 6 Evidence that the η 2 (1870) is resonant Several checks have been made that the η 2 (1870) has resonant phase variation. The first check is to remove the phase variation by replacing the amplitude by its modulus and refitting the data. At all beam momenta, this leads to a highly significant worsening of log likelihood. Column 2 of Table 5 shows the changes against beam momentum. The definition of log likelihood is such that a change of 0.5 in log likelihood should correspond to a one standard deviation change.
We have already remarked that fluctuations in Table 4 are a factor 1.4 above statistics. A more extensive examination of fluctuations in branching ratios shows they are in some cases up to a factor 2 above statistics, and that has already been taken into account in errors quoted in Table  3. Adopting this as a general rule leads to the conclusion that changes listed in Table 3 can be equated to changes in χ 2 . They average to 11.6 standard deviations per momentum, i.e. 28σ in total. )  1050  43 31  1200  81 98  1350  106 120  1525  91 47  1642  95 55  1800  195 127  1940 199 86 Table 6: Changes in log likelihood when the amplitudes for η 2 (1870) (i) are replaced by their modulus (no phase variation), (ii) use a denominator A − m(s) − iMΓ(s), see text.
The second check is that including the dispersive term of Eq. (16) into the Breit-Wigner denominator has little effect on log likelihood after minor alterations to fitted mass and width, well within errors quoted in Table 6. The dispersive term peaks at the f 2 (1270)η threshold, but with a large full width of ∼ 300 MeV. Numerically, it is easily absorbed into small shifts of fitted parameters.
There remains the possibility that the associated cusp in the real part of the amplitude could explain the phase variation without a resonance. The third check is to replace the Breit-Wigner denominator of Eq. (15) by This removes the resonance by changing M 2 − s to a constant A; it is the term in s which drives the real part of the amplitude through zero on resonance. Fig. 12 shows the Argand diagram of the amplitude fitted to WA102 data. [The vertical scale is no longer limited to 1 at the peak, because the imaginary part of the amplitude no longer needs to reach 1 in the absence of a resonance.] Re f Im f Figure 12: The Argand diagram for a fit to WA102 data using Eq. (26).
The imaginary part of the amplitude is forced by the data to peak at ∼ 1810 MeV, as before. The real part is positive everywhere. It turns out that the data do not force the constant A to go negative; instead it optimises close to +0.5. Then fits to η3π 0 data are worse than for a resonance by the amounts shown in column 3 of Table 6. At the lowest three beam momenta, where the η 2 (1870) is strongest, results are similar to column 2. At higher momenta, the changes drop by roughly a factor 2. The reason for this drop has been traced to the fact that the phase variation in Fig. 12 adds a degree of freedom in the fit compared with column 2, where there is no phase variation at all. However, the significance levels in column 3 still average 9 standard devations per momentum. If any phenomenologist wishes to develop a more complete dynamical model, the data are publicly available from the authors. Meanwhile, the evidence for resonant behaviour appears to be strong.

Conclusions
This work confirms that the η 2 (1870) has a branching fraction to f 2 (1270)η comparable with that to a 2 (1320)π, in agreement with our earlier analysis. It is not possible to fit these data with the large branching fraction found by WA102. Results of the two experiments agree well for the mass and width of η 2 (1645) and the branching fraction of its decays to a 0 (980)η. They also agree quite well for the mass and width of η 2 (1870) from decays to f 2 (1270)η. Small branching fractions reported in Table 2, particularly for decays to a 0 π, have changed significantly from earlier values for a complex of reasons which are understood.
A further result from the present analysis is that there is evidence for the presence of a 2 (2255) → f 2 (1270)a 0 (980) with parameters close to those of Ref. [11]. It is the third set of data in which it has been observed, the others being πη and 3π 0 . There is also evidence that the channel [η 2 (2030)π] L=1 is produced via π 2 (2245); its mass and width are determined better by present data than by earlier analyses: M = 2285±20(stat)±25(syst) MeV, Γ = 250±20(stat)±25(syst) MeV. This is the third channel in which it has been observed.
Further data on η3π 0 from a transversely polarised target would be very valuable. In such data, there are interferences between singlet and triplet states. It is likely that such information would allow a complete spin-parity analysis of the data, improving further on the present analysis.

Acknowledgements
We thank the Crystal Barrel Collaboration for the use of the data.