Resonances formed by pbar-p and decaying into pizero-pizero-eta for masses 1960 to 2410 MeV
A.V. Anisovich, C.A. Baker, C.J. Batty, D.V. Bugg, C.Hodd, J.Kisiel, V.A. Nikonov, A.V. Sarantsev, V.V. Sarantsev, I. Scott, B.S. Zou
aa r X i v : . [ h e p - e x ] S e p Resonances Formed by ¯ pp and Decaying into π π η for Masses 1960to 2410 MeV A.V. Anisovich c , C.A. Baker a , C.J. Batty a , D.V. Bugg b , C. Hodd b , J. Kisiel d , V.A. Nikonov c , A.V. Sarantsev c ,V.V. Sarantsev c , I. Scott b , B.S. Zou b a Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX,UK b Queen Mary and Westfield College, London E1 4NS, UK c PNPI, Gatchina, St. Petersburg district, 188350, Russia d University of Silesia, Katowice, Poland
Abstract
Data on ¯ pp annihilation in flight into π π η are presented for nine beam momenta 600 to 1940 MeV/c.The strongest four intermediate states are found to be f (1270) η , a (1320) π , ση and a (980) π . Partial waveanalysis is performed mainly to look for resonances formed by ¯ pp and decaying into π π η through theseintermediate states. There is evidence for the following s -channel I = 0 resonances : two 4 ++ resonanceswith mass and width (M, Γ) at (2044, 208) MeV and (2320 ±
30, 220 ±
30) MeV; three 2 ++ resonances at(2020 ±
50, 200 ±
70) MeV, (2240 ±
40, 170 ±
50) MeV and (2370 ±
50, 320 ±
50) MeV; two 3 ++ resonancesat (2000 ±
40, 250 ±
40) MeV and (2280 ±
30, 210 ±
30) MeV; a 1 ++ resonance at (2340 ±
40, 340 ±
40) MeV;and two 2 − + resonances at (2040 ±
40, 190 ±
40) MeV and (2300 ±
40, 270 ±
40) MeV.
PACS: 13.75Cs, 14.20GK, 14.40Keywords: mesons, resonances, annihilation
The Crystal Barrel detector is being used to make a systematic study of the mass region 1960 to 2410 MeV in¯ pp annihilation in flight at LEAR, with ¯ p beams of momenta 600 to 1940 MeV/c. The objective is to studyresonances in the formation process, i.e. the s -channel. Here we study data in ¯ pp → π π η for resonancesdecaying to a (1320) π , f (1270) η , f (1500) η , f (980) η , a (980) π and ση . We use σ to denote the broad ππ S-wave amplitude up to ∼ ηηπ have beenpresented elsewhere [2,3], and work is in progress on other channels such as 3 π and π π η ′ .From earlier work, it is known that the mass range we explore contains many resonances [4]; a detailed studyof ¯ pp → π − π + using a polarised target has provided much of the current evidence [5,6]. The f (2050) is wellknown, and from the quark model of meson resonances one expects that it will be accompanied by f and f resonances close-by in mass. We shall indeed provide evidence for these resonances and a further one withquantum numbers J P C = 2 − + and similar mass. At higher masses, towards the top of the LEAR range, therehas been evidence for f (2300) and f (2340) [4], and it is anticipated from the Veneziano model [7] that thereis likely to be a tower of resonances around this mass. We shall provide evidence for such states with quantumnumbers 4 + , 3 + , 2 + , 1 + and 2 − .These resonances are anticipated ¯ qq states. This mass range is also likely to contain glueballs with quantumnumbers 0 − + and 2 ++ , predicted in the mass range 2000–2400 MeV by various theoretical models [8,9,10].Hybrids may also be present. Decays of these exotic resonances to η and σ seem to be favoured in f (1500)decay [11], charmonium decay and J/ Ψ radiative decays [12]. Hence the ηf (1270) and ησ channels are ofparticular interest.The layout of this paper will be as follows. In section 2, the procedure for data processing and event selectionis outlined; the data are presented and their gross features are discussed. Section 3 gives the formalism usedfor the partial wave analysis. Section 4 gives the results for partial wave amplitudes. Then, in Section 5 we fitpartial waves to resonances. Finally, Section 6 provides a summary. Now at IHEP, Beijing 100039, China Experiment and Data Processing
The data were taken at LEAR by the Crystal Barrel Collaboration, using a trigger on neutral final states at ninebeam momenta from 600 to 1940 MeV/c. An average of 9 × triggers were taken at each momentum. Thedetector has been described fully in an earlier publication [13].A liquid hydrogen target 4.4 cm long is surrounded at increasing radii by a silicon vertex detector, a multiwirechamber for triggering, a jet drift chamber to detect charged particles and finally 1380 CsI crystals to detectphotons. The present data were taken with a trigger demanding a neutral final state. For this purpose, thesilicon vertex detector, multiwire chamber and jet drift chamber were used simply to veto charged particles.The barrel of CsI crystals covers 98% of 4 π solid angle. Crystals are 16 radiation lengths long and point towardsthe target. The angular resolution is ∼ ±
20 mrad in both polar angle and azimuth. The detection efficiency ishigh for photons down to energies below 20 MeV. The energy resolution ∆ E is given by ∆ E/E = 0 . /E / ,where E is in GeV.The incident ¯ p beam was pure and monoenergetic with momentum spread ∆ p/p < . P.Si. ¯ V identifying interactions in the target. The beam intensity was typically 2 × ¯ p /s and at times was twice this.The interaction rate in the target (excluding ¯ pp elastic scattering, where the forward ¯ p generally counted in theveto counter) was typically 3KHz. Of this, ∼ −
2% consisted of neutral final states, so the trigger rate forall-neutral events was 20–60 Hz. In order to filter out events which obviously fail to conserve energy, the totalenergy in the CsI crystals was summed on-line [14]; a fast trigger rejected those events with total energy falling ∼
200 MeV or more below that of ¯ pp annihilation.The absolute normalisation is derived from beam counts P.Si , target length and density, the number ofdetected events and a Monte Carlo simulation of reconstruction efficiency in the CsI barrel. Details of thisnormalisation are given in a paper on the π π final state [15]. A dependence of the reconstruction efficiency onbeam rate is observed, and the normalisation has to be obtained from an extrapolation to zero beam rate. Thenormalisation uncertainty is estimated as ±
3% from 1800 to 1050 MeV/c and increases to ±
6% at 900 and 600MeV/c. Data at 1940 MeV/c were taken in separate, earlier runs, and have an estimated uncertainty of ± ± .
4% from the target length,common to all momenta.
A large number of alternative prescriptions have been examined for selecting events. At high momenta, oneof the problems is that photons from π decay sometimes merge into a single shower. Conversely, one showersometimes splits into a primary shower and a nearby secondary shower, caused by Compton scattering. Theprobability that this occurs is ∼
10% per photon. In early studies, an attempt was made to salvage ηπ π eventsfrom 5 γ or 7 γ final states. However, the gain in statistics was small ( ∼ µb )600 20385 26.3 71 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . pp → π π η with η → γγ .2ata are fitted kinematically to a large number of physics channels: 43 for (4 − γ . In order to assessbranching ratios to every channel and cross-talk between them, we generate at least 20,000 Monte Carlo eventsfor every one of the 43 fitted channels, using GEANT. In the first approximation, events fitting the correctchannel determine the reconstruction efficiency ǫ i in each channel. Events fitting the wrong channel estimatethe probability of cross-talk x ij between channels i and j . More exactly, we solve a set of 43 x 43 simultaneousequations containing on the left-hand side the observed number of fitted data events D i , and on the right-handside reconstruction efficiencies and true numbers of created events N i in every channel and terms allowing forcross-talk x ij between channels: D i = ǫ i N i + X j = i x ij N j . (1)The solution is constrained so that the numbers of real events, N i , in every channel are positive or zero.Figure 1: Cross section for ¯ pp → π π η with η → γγ .This procedure is carried out for a variety of confidence levels (1, 5, 10, 20%) and using a wide variety ofselection procedures. A choice is then made, optimising the ratio of signal to background. We find that thisratio is not very sensitive to confidence level over the range 5–20% for ηπ π events.Among 6 − π , 3 π , π π η and π π ω with ω → π γ . Therelative branching ratios for these channels are roughly 1.1 : 1 : 0.4 : 0.4 at 1800 MeV/c. To select the π π η channel, we demand exactly 6 photons satisfying a 7C kinematic fit with confidence level > π with confidence level > .
01% are rejected, and also those few events fitting π π η ′ , π ηη , π ηη ′ and 3 η with confidence level larger than that for π π η . The Monte Carlo simulation shows that the worst backgroundsarise from ωπ π , ( ω → π γ ) when one photon is lost, and from 4 π events when two photons are lost. Residualbackgrounds from these two processes are 1.5% and 0.8% respectively at 1800 MeV/c. Including other smallbackgrounds, the total is 3 . ± .
3% at 1800 MeV/c. For lower beam momenta, the background increases slightly.At 600 MeV, the total is 4 . ± . ωπ π (1.7%), 4 π (0.9%) and ωω (0.9%).Table 1 summarises numbers of selected events, the reconstruction efficiency and cross sections. Statistics at 600MeV/c are lower because most data were taken without the threshold cut on total energy in the trigger. Thecross sections for the ηπ π channel are also shown in Fig.1. There are clear enhancements at low mass andaround 2200–2300 MeV. Note that for a constant amplitude the cross sections should decrease steadily as theenergy increases, see equn (23) below.Fig. 2 shows the confidence level (CL) distribution for data of beam momentum at 1.2 GeV/c. The slightpeak at high confidence level arises from events where all particles emerge close to the beam direction, withthe result that the vertex is poorly defined. We apply no cut on the coordinate of the vertex along the beamdirection, so as to avoid biasing the data selection. The rise at low confidence levels is followed accurately downto 10% by the Monte Carlo simulation; it arises from overlapping showers in the CsI detectors.In order to illustrate the cleanliness of the η signal, we have made an additonal fit to π π γγ . Fig. 3 then showsthe mass distribution of γγ pairs in the vicinity of the η peak for CL ( π π γγ ) > . CL ( π π η ) > . η peak is well centred at the correct mass, 547.5 MeV and the backgroundunder the η signal is compatible with that expected from the Monte Carlo simulation.3igure 2: Distribution of confidence level for ¯ pp → π π η events at beam momentum 1.2 GeV/c.Figure 3: Mass distribution of γγ pairs for CL ( π π γγ ) > . CL ( π π η ) > . pp → π π η at incident beam momenta 0 . − .
94 GeV/c.5 .2 Features of the Data
Fig. 4 shows Dalitz plots at the nine available momenta and Figs. 5 and 6 projections on to ηπ mass and ππ mass. The most prominent feature of the Dalitz plot consists of a diagonal band due to f (1270) π . Thereare weaker horizontal and vertical bands due to a (980) π and a (1320) π . The f (1270) η signal grows withrespect to a (1320) π as the beam momentum rises; this is a natural consequence of the increasing phase spacefor f (1270) η , whose threshold is at 1820 MeV. Very weak peaks are visible in the ππ mass projection of Fig. 6due to f (1500) η and f (980) η . In addition, there is some slowly varying contribution covering the whole Dalitzplots; it may come from the broad σ , i.e., f (400 − f (1270) and a (1320) by a few MeV from PDG values in order to achieve the optimum fits. Thisis because our main aim is to fit the production and decay angular distributions of these resonances.Figure 5: Data and fit (histogram) of invariant mass spectra for π π (1 entry/event).Figs. 7 and 8 show differences on the Dalitz plot between fit and data. There are small systematic discrepanciesat the extreme right-hand edge of the Dalitz plot near an ηπ mass of 1450 MeV. This discrepancy may be due to a (1450) or a (1660) or ˆ ρ (1405). The effect is small and cannot be analysed unambiguously into partial waves.Fits including these components have almost no effect on the main components of the fit, with the exception of ησ , which covers the whole Dalitz plot and can absorb other small, ill-defined contributions.Figs. 9 and 10 show production angular distributions (after acceptance correction) for events lying in the f (1270) mass band (1275 ±
100 MeV) and for events lying in the a (1320) mass band (1320 ±
50 MeV). It isimmediately obvious that high orbital angular momenta are involved for both f η and a π at the higher beammomenta. The histograms show results of the partial wave fit described below. For the π π η final state, possible ¯ pp initial singlet states are 0 − + , 2 − + , 4 − + etc; for ¯ pp spin triplet, allowedstates are 1 ++ , 2 ++ , 3 ++ , 4 ++ , 5 ++ etc. For our case with center-of-mass energies below 2.41 GeV, only 0 − + ,6igure 6: Data and fit (histogram) of invariant mass spectra for π η (2 entries/event).7igure 7: Difference between Dalitz plots of fit and data where fit > data.2 − + , 1 ++ , 2 ++ , 3 ++ and 4 ++ are expected to be significant [6] and this has been confirmed in our analysis;4 − + has been tried, but is not significant. The corresponding ¯ pp states with total angular momentum J, orbitalangular momentum L and total spin angular momentum S in the usual contracted form S +1 L J are: S for0 − + , D for 2 − + , P for 1 ++ , P or F for 2 ++ , F for 3 ++ , and F or H for 4 ++ .Let us choose the reaction rest frame with the z axis along the ¯ p beam direction. Then the squared modulusof the total transition amplitude is the following [16]: I = | A − + + A − + | + | A M =11 ++ + A M =13 ++ | + | A M = − ++ + A M = − ++ | + | A M =02 ++ + A M =04 ++ | + | A M =12 ++ + A M =14 ++ | + | A M = − ++ + A M = − ++ | (2)+2 Re [( A M =12 ++ + A M =14 ++ )( A M =11 ++ + A M =13 ++ ) ∗ − ( A M = − ++ + A M = − ++ )( A M = − ++ + A M = − ++ ) ∗ ]where M is the spin projection on the z-axis in the initial state. The absence of M=0 for 1 ++ and 3 ++ is due tothe vanishing of the Clebsch-Gordon (CG) coefficient ( J = 2 n + 1 , M J = 0 | L = 2 n + 1 , M L = 0; S = 1 , M S = 0)with n as an integer. The relative minus sign for the interference term of ( even ) ++ and ( odd ) ++ partial waveswith M=1 and M=-1 is also due to a property of CG coefficients.Each partial wave amplitude A J PC includes contributions from various intermediate states (n), i.e., A J PC = X n C n A J PC → n (3)where C n are free complex parameters to fit the data. In the present analysis, only f (1270) η , a (1320) π , a (980) π , ση , f (980) η and f (1500) η intermediate states are considered. Amplitudes A J PC → n are constructedfrom relativistic Lorentz covariant tensors, Breit-Wigner functions and Blatt-Weisskopf barrier factors [17]. Theamplitudes used for f (1500) η and f (1270) η intermediate states in our final fit are the following: A − + → f η = G f , (4)8igure 8: Difference between Dalitz plots of fit and data where fit < data. A − + → f η = T αβ ˜ t (2) αβ B ( k ) G f , (5) A − + → f η = φ αβ (0)˜ t (2) αβ B ( k ) G f , (6) A − + → f η ( l =0) = φ αβ (0) T αβ G f , (7) A − + → f η ( l =2) = φ αβ (0)˜ t (2) αγ T γβ B ( k ) G f , (8) A M ++ → f η = φ α ( M )˜ t (1) α B ( k ) G f , (9) A M ++ → f η ( l =1) = φ α ( M )˜ t (1) β T αβ B ( k ) G f , (10) A M ++ → f η ( l =3) = φ α ( M )˜ t (3) αβγ T βγ B ( k ) G f , (11) A M ++ → f η ( l =1) = φ µα ( M ) ǫ αβγδ P β ˜ t (1) γ T µδ B ( k ) G f , (12) A M ++ → f η ( l =3) = φ µα ( M ) ǫ αβγδ P β ˜ t (3) µνγ T νδ B ( k ) G f , (13) A M ++ → f η = φ αβγ ( M ) k α k β k γ B ( k ) G f , (14) A M ++ → f η ( l =1) = φ αβγ ( M )˜ t (1) α T βγ B ( k ) G f , (15) A M ++ → f η ( l =3) = φ αβγ ( M )˜ t (3) αβδ T δγ B ( k ) G f , (16) A M ++ → f η ( l =3) = φ µνλα ( M ) k µ k ν ǫ αβγδ P β k γ T δλ B ( k ) G f (17)where k µ is the four-momentum of the η , G f = ( M f − s ππ − iM f Γ f ) − and G f = ( M f − s ππ − iM f Γ f ) − are Breit-Wigner propagators for f and f . T µν is a rank-2 tensor for f and is formed by the four-momentum(p) of f and its break-up four-momentum (q) as T µν = [ q µ q ν −
13 ( g µν − p µ p ν s ππ ) q ] B ( q ) . (18)9igure 9: Data and fit (histogram) of angular distribution dσ/d cosθ η for M ππ between 1175 and 1375 MeV (1entry/event). 10igure 10: Data and fit (histogram) of angular distribution dσ/d cosθ π for M ηπ between 1270 and 1370 MeV (2entries/event). 11he Blatt-Weisskopf barrier factors B l ( k ) with a radius of 1 fm, the rank- l tensors ˜ t ( l ) δ ··· δ l for pure l -wave orbitalangular momentum of the ηf , system, and the spin-J wave functions φ δ ··· δ J ( M ) are standard as given in [17].For f (980) η and ση intermediate states, the formulae are the same as for f (1500) η except for a different G f for which we take the parameterization of Ref. [18], i.e., G f (980) = 1 M R − s ππ − ig π p − m π /s ππ − ig K p − m K /s ππ (19)with M R = 0 .
99 GeV, g π = 0 .
117 GeV , g K = 0 .
273 GeV , m π = 0 .
135 GeV and m K = 0 .
496 GeV; G σ = 1 + C s ππ M σ − s ππ − iM σ (Γ ( s ππ ) + Γ ( s ππ )) , (20)where C is a complex constant to be fitted by the data,Γ ( s ) = G p − m π /s p − m π /M σ · ( s − m π / M σ − m π / e − ( s − M σ ) / β (21)Γ ( s ) = G p − m π /s exp (Λ( s − s )) · exp (Λ( s − M σ )) p − m π /M σ (22)with M σ = 1 .
067 GeV, G = 1 .
378 GeV, β = 0 . G = 0 . . − and s = 2 . .For a π and a π intermediate states, the formulae are similar to those for f η and f η , but need symmetrizationfor two pions. The Breit-Wigner propagators for a , a , f (1500) and f assume constant widths. The massesand widths (M, Γ) for a and f (1500) are fixed to be (0 . . . a and f are adjusted to fit the data. Based on these formulae, the data at each momentum are fitted bythe maximum likelihood method.It is possible that the process ¯ pp → ηπ π is driven, at least partially, by t -channel Regge exchanges. Evenso, by Watson’s theorem, each partial wave will acquire the phase variation of any s -channel resonance which ispresent; that is, amplitudes will contain singularities due to both s - and t -channel poles. Our strategy will beto express T matrices for individual partial waves T L,J as sums over s -channel resonances. The formulae we useare σ J PC → n ( s ) = N k n sk i | A J PC → n ( s ) | , (23) A J PC → n ( s ) = X j B L ( k i )Λ nj B l ( k n ) M nj − s − iM nj Γ nj , (24)where s = M pp = M ππη , N is the normalization constant, k i and k n are the center-of-mass momenta of initialstate and channel n respectively; B L and B l are barrier factors for the initial state and state n respectively; Λ nj are complex fitting parameters; M nj and Γ nj are masses and widths for resonances to be fitted. This prescriptionbuilds in the required threshold behaviour in each partial wave. By using a sum of resonances, we satisfy theconstraint of analyticity. The fit is shown as histograms in Figs. 5-6 for the mass spectra. It is obviously not perfect as regards broad, slowlyvarying components in the ππ projection of Fig. 6. However, since we are mainly interested in scanning the largercomponents from f (1270) η , ση , a π and a (980) π intermediate states, we ignore those smaller contributions forthe present study.The intensities of dominant partial waves are displayed in Fig. 11, and we shall discuss a fit to them below.The data points with error bars shown in Figs. 11 and 12 are our final fitted results for the partial wave crosssections σ J PC → n at each momentum for ¯ pp → π π η with η → γγ . Small waves are displayed in Fig. 12. Partialwaves with less significant contribution than those in Fig.12 are dropped from our final fit. Table 2 shows the12igure 11: Cross sections for partial waves making the largest contributions to ¯ pp → π π η with η → γγ . Fordiagrams with two components, the first label corresponds to the bigger component. The curves are the fit tothe data points in the figure and the relative phases between components.13igure 12: Cross sections for partial waves included in the final fit but giving smaller contributions than thosein Fig. 11 to ¯ pp → π π η with η → γγ . J P C
Mass (MeV) Width (MeV) Γ ¯ pp Γ f η Γ tot · ¯ pp Γ a π Γ tot · ¯ pp Γ ση Γ tot · ¯ pp Γ a π Γ tot · ++ . ± .
14 5 . ± . ++ ±
30 220 ±
30 1 . ± . . ± . ++ ±
40 250 ±
40 0 . ± .
08 0 . ± . . ± . ++ ±
30 210 ±
30 1 . ± . . ± . . ± . ++ ±
50 200 ±
70 2 . ± . . ± . ++ ±
40 170 ±
50 2 . ± . . ± . ++ ±
50 320 ±
50 0 . ± .
64 16 ± ++ ∼ ∼ ++ ±
40 340 ±
40 0 . ± . ±
30 0 . ± . − + ±
30 150 ±
30 1 . ± . ± ± − + ±
40 190 ±
40 3 . ± . . ± . . ± . − + ±
40 270 ±
40 2 . ± . . ± . . ± . f (2050) are fixed at PDG values, and the status of the 0 − state at 2140 MeV is questionable,as discussed in the text. The f (1700) is beyond the accessible mass range. All states have I = 0, G = +1.14asses and widths of resonances included in the fit. Errors cover the range of values observed in a large varietyof fits. The f (1700) is below the range of masses accessible here, so its parameters are only approximate.The relative phases of the partial waves at each momentum are shown in Fig.13. Since there is no interferencebetween spin singlet and spin triplet, or between M=0 and M=1 for spin triplet, there will be one overall phaseundetermined for each M of spin triplet and for spin singlet. Hence we can only determine relative phases fromour partial wave analysis. For spin singlet (0 − and 2 − ), the phases are relative to the partial wave of 2 − → f η with L=0. For spin triplet with M=0, the phases are shown relative to 4 + → a π with L=3. For spin tripletwith | M | = 1, the phases are relative to 4 + → f η with L=3. J P = 4 + For 4 ++ , a peak around 2090 MeV is clear for all 4 ++ channels. It can be fitted by a Breit-Wigner amplitudewith the mass and width fixed to the PDG values for the well established 4 + resonance f (2050). The shift ofthe peak position to 2090 MeV is due to the centrifugal barrier factors for both initial and final states. Its decaysinto f η and a π appear with comparable strength in the ηπ π channel.In addition to the f (2050), there is clearly another 4 ++ peak around 2 .
32 GeV in 4 + → f η in the M=1partial wave. This resonance may be identified with f (2300) of the PDG, observed earlier in many analysesof ¯ pp → π − π + . The mass, width and phase with respect to f (2050) are adjusted freely. The mass optimisesat M = 2320 ±
30 MeV and the width at Γ = 220 ±
30 MeV. These agree closely with earlier determinationquoted by the PDG, and also with recent VES data on ηπ + π − in the πA reaction [19]. The latter find M =2330 ± stat ) ± syst ) MeV, Γ = 225 ± ±
40 MeV. They also observe this resonance in ωω data [20]. The f (2300) is also observed in our data on ¯ pp → π π [15], with a slightly lower mass of 2295 MeV. The f (2300)resonance acts as a valuable interferometer, determining the phases of 3 + , 2 + and 1 + amplitudes over the massrange 2150–2400 MeV.From the M = 1 and M = 0 amplitudes for 4 + , we reconstruct the linear combinations for F and H .Their intensities are shown in Fig. 14 for f (1270) η and a (1320) π channels. The f (2050) resonance is almostpurely F . The a π channel is fed mostly by f (2050) with a possible weak contribution from f (2320); the H contribution to a π is barely significant. In contrast, the f η channel is fed by both f (2050) and f (2320)and the latter has a strong H component. This is in agreement with the analysis of ¯ pp → π − π + by Hasanand Bugg [6]; their Fig. 3 shows a strong H component in f (2320). The VES collaboration [19] finds that f (2320) decays dominantly to f η , in agreement with present results. J P = 3 + For J P C = 3 ++ , there are significant enhancements at low mass ( M ≃ a (980) π and f (1270) η with L = 1. At high mass ( M ≃ f (1270) η decays with both L = 1 and L = 3 decays. Fitted masses and widths are given in Table 2. There are no earlier listings of these resonancesby the PDG. The observed phase with respect to f (2050) and f (2300) shown in Fig. 13 obviously requires thepresence of at least one 3 + resonance, and is poorly fitted without two. The Argand diagram is shown in fig. 15. J P = 2 + For 2 ++ , there is a peak in f (1270) η at ∼ a (1320) π . At high massaround 2300 MeV, there is a strong peak in the a (1320) π channel. In f (1270) η , there is a further peak at ∼ ◦ phase advance.We find that the fit is poor without three resonances. The lowest peak fits naturally to a resonance with M = 2020 ±
50 MeV, Γ = 200 ±
70 MeV. Our data on ¯ pp → π π independently find a resonance at 2020 MeV[15], and the analysis of Hasan and Bugg [6] of data on ¯ pp → π − π + likewise finds an f resonance at 1996 MeV.We have tried an alternative fit using instead f (1920) observed by both GAMS [21] and VES [22] collaborations.The 2 ++ → a π partial wave can be reproduced equally well with this assignment, but the 2 ++ → f η partialwave is seriously underfitted by a factor 3 at 2050 MeV, ruling out a fit by f (1920) only.15igure 13: Relative phases (data points with error bars) obtained from the partial wave analysis and used forthe fit (curves) to get Argand plots together with masses and widths of the resonances. The phases for 2 + and4 + with M=0 are relative to 4 + → a π with L=3 and M=0; the phases for 1 + , 3 + and 4 + with M=1 are relativeto 4 + → f η with L=3 and M=1; the phases for 0 − and 2 − are relative to 2 − → f η with L=0.16igure 14: Contributions to (a) f (1270) η , (b) a (1320) π from F (black squares) and H (open triangles).The proximity of this resonance to f (2050) suggests that it may be identified as the ¯ qq F state expectednear this mass. Because 2 + amplitudes with M = 1 are negligible, P and F amplitudes have the same s -dependence; F is the larger by a factor 1.44. This strong coupling of f (2030) to ¯ pp F also suggestsidentification with ¯ qq F : high L in ¯ qq is likely to be associated with high L in decay channels, because of thepeaking of wave functions at large r .At higher masses, a fit with a single resonance, shown by the dashed curve in Fig. 16, is much poorer thanwith two separate resonances. The peak at 2240 MeV in f (1270) η has a mass compatible with ξ (2230) observedin J/ Ψ radiative decays [23], but has a larger width of about 170 MeV. This resonance may be interpreted asthe n = 4 ¯ qq P state, in the sequence f (1270), f (1565), f (1920), f (2240). The f (2370) finds a naturalexplanation as n = 2 ¯ qq F , i.e. the radial excitation of f (2020). Its strong L = 3 decay supports thisinterpretation. In present data, both f (2240) and f (2370) appear in both P and F , suggesting mixingbetween these states.The two peaks around 2020 MeV and 2370 MeV have masses and widths compatible with f (2010) and f (2340) listed by the PDG [4]. However, those observations were in the φφ channel and could be differentresonances, e.g. ¯ ss . We also remark that the peak in the φφ data of Etkin et al. [24] actually appears at ∼ φφ phase space which leads to a pole at much lower mass, 2020 MeV, in theK-matrix fit to their data. J P = 1 + For J P C = 1 ++ → a (1320) π and a (980) π , there is a peak at the lowest masses. This suggests a resonance closeto or below the ¯ pp threshold. However, as discussed below, the phase variation of the 1 + amplitude providesevidence for a resonance around 2340 MeV. The phase variation shown in Fig. 15 obviously requires resonantactivity in the mass range 2000–2400 MeV. J P = 2 − Partial waves with quantum numbers 2 − + and 0 − + correspond to ¯ pp singlet states, and therefore there is nointerference with even parity (triplet) partial waves. For 2 − + there is a strong peak in f (1270) η at ∼ a (1320) π at similar mass. There is evidence for a further peak at ∼ f (2240) (full curve) and without (dashed).lower peak is well fitted by a resonance with M = 2040 ±
40 MeV. The almost 360 ◦ phase advance observed onthe Argand diagram points strongly towards the presence of two resonances, the second at 2300 ±
40 MeV. ThePDG does not list any I = 0 J P C = 2 − + resonance in this mass range. A possible I = 1 partner is listed in theform of π (2100). J P = 0 − For 0 − + , there is a broad, slowly varying intensity with evidence for a strong peak superimposed at M ∼ − + object used in describing J/ Ψ radiativedecays to ρρ , ωω , K ∗ ¯ K ∗ , φφ and ηππ [25]. The peak at 2140 MeV may correspond to a narrow resonance.However, it is observed in the ησ channel, which contributes across the entire Dalitz plot. This contributionmight absorb weak components not presently fitted to the data, for example due to a (1450), a (1660), ˆ ρ (1405)or further resonances in the production process around 2 GeV. In view of this possibility, the interpretation interms of a resonance is ambiguous. Unfortunately, the relative phase with respect to 2 − is not well determined,so the phase variation cannot be used for independent evidence of resonant activity. To get more precise values for masses and widths for resonances, we use interfering sums of the Breit-Wigneramplitudes to fit the partial wave cross sections in Fig.11 and the relative phases of the partial waves in Fig.13simultaneously. The fit is shown in Figs. 11-13 as full curves.Besides the obvious resonances mentioned in the previous section, we need another 1 ++ resonance at about2340 MeV with width ∼
340 MeV. Without it, we cannot describe the relative phase between 1 ++ and 4 ++ partial waves; also we would need the lower 1 ++ resonance to be very narrow ( <
50 MeV) in order to explainthe sharply decreasing 1 ++ partial wave cross section. In our present fit with two 1 ++ resonances, the f (2340)amplitude interferes destructively with the tail of the lower 1 ++ resonance and causes the sharply decreasingcross section with a broad dip around 2340 MeV. The phase motion caused by this f (2340) can be seen clearlyin the Argand plots for 1 ++ partial waves of Fig.15.In Table 2, the branching ratios are calculated at the resonance masses and are corrected for their unseendecay modes, except for a (980) where Γ a π = Γ a π → ηππ .19or an ordinary q ¯ q state, the relative ratio f η / a π is expected to be smaller than 0.64. This allows for the36% component of ¯ ss in the η . The centrifugal barrier and phase space will further suppress f η . Most ofthe branching ratios in Table 2 are in qualitative agreement with what is expected for ¯ qq states. However, the f (2230) has an anomalously strong branching ratio to f (1270) η compared with a (1320) π .For the well-established f (2050), only 44% of its branching ratios are listed in the Particle Data Tables [4],in which ππ has a branching ratio of (17 ± . pp → ππ , the ratio Γ ¯ pp Γ ππ Γ tot was reported to be (2 . ∼ . × − . Using this information, we can get the branching ratios of f (2050) to¯ pp , a π and f η to be (1 . ± . ± . ± . The f (2044), f (2000), f (2020) and η (2040) cluster closely into a tower of resonances, as anticipated in theVeneziano model. Likewise the f (2320), f (2280), f (2370), f (2340) and η (2300) show indications of clusteringinto a tower at the higher masses.The f (1920) originally discovered by both GAMS and VES has recently been confirmed in further VES datawith increased statistics, decaying to ωω [20]. There is also a strong f (1270) η signal in VES ηπ + π − data.Together with the f (2020) we observe here, f (2240) and f (2370), this tentatively completes the identificationof the ¯ qq I = 0 P and F states expected in this mass range.We conclude with some speculative suggestions of a scheme which concerns mixing of ¯ qq states with the 2 + glueball expected in this mass range. In our data on ¯ pp → ηηπ [2], there is evidence for a further broad f (1980)decaying to ηη , with mass M = 1980 ±
50 MeV, Γ = 500 ±
100 MeV. Its effects are seen clearly down to massesof ∼ + resonance in 4 π final states in central production [27].Such a broad state was predicted by Bugg and Zou [25]. It may be interpreted as a mixed state formed fromthe 2 + glueball, expected at ∼ − . qq states. Anisovich et al. [28] have argued that thismixing will lead to a broad state, accumulating the widths of nearby ¯ qq states and making them narrower. The f (1920) and f (2240) are indeed somwhat narrower this is usual for resonances in this mass range. Mixing witha glueball provides a natural explanation of the anomalous decays of f (2020) and f (2340) to φφ , observed byEtkin et al. [24].The glueball may be small, with radius ∼ . qq P states rather than ¯ qq F , whose wave functions are stronglylocalised at large r . The preferential decays of f (1920) and f (2240) to f (1270) η , despite its smaller phasespace than a (1320) π , may be a further indication of mixing with the 2 + glueball. In summary, we have observed a new decay mode ηππ for f (2050). In addition, we have evidence for 7 newor poorly established resonances in the energy range from 1.96 to 2.41 GeV, i.e., f (2320), f (2000), f (2280), f (2240), f (2340), η (2040) and η (2300). They appear to cluster into two towers of resonances around 2000–2050 MeV and 2300 MeV. Results are broadly consistent with earlier evidence for f (2300), f (2020) and f (2340). We thank the Crystal Barrel Collaboration for allowing use of the data. We also thank the technical staff of theLEAR machine group and of all the participating institutions for their invaluable contributions to the success ofthe experiment. We acknowledge financial support from the the British Particle Physics and Astronomy ResearchCouncil (PPARC). The St.Petersburg group thanks INTAS for financial support, contract RFBR 95-0267, andalso PPARC for financial assistance for collaborative work.
References [1] A. Anisovich et al.,
Study of ¯ pp → π π η from 600 to 1940 MeV/c .202] A. Anisovich et al., Study of the process ¯ pp → ηηπ from 1350 to 1940 MeV/c , Phys. Lett. B449 (1999) 145.[3] A. Anisovich et al., Observation of f (1770) → ηη in ¯ pp → ηηπ reactions from 600 to 1940 MeV/c , Phys.Lett. B 449 (1999) 154.[4] Particle Data Group, C. Caso et al., Euro. Phys. J. C3 (1998) 1.[5] A.Hasan et al., Nucl. Phys. B378 (1992) 3.[6] A.Hasan and D.V.Bugg, Phys. Lett. B334 (1994) 215.[7] G. Veneziano, Nu. Cim. 57A (1968) 190.[8] G.Bali et al. (UKQCD), Phys. Lett. B307 (1993) 378; H.Chen, J.Sexton, A.Vaccarino and D.Weingarten,Nucl. Phys. B (Proc. Suppl.) 34 (1994) 357.[9] V.A.Novikov, M.A.Shifman, A.I.Vainshtein and V.I.Zakhnov, Nucl. Phys. B 191 (1981) 301.[10] J.Y.Cui, J.M.Wu and H.Y.Jin, Phys. Lett. B424 (1998) 381.[11] V.V. Anisovich et al., Phys. Lett. B323 (1994) 233; C. Amlser et al., Phys. Lett. B355 (995) 425.[12] D.V.Bugg et al., Phys. Lett.
B353 (1995) 378.[13] E. Aker et al., Nucl. Instr. A321 (1992) 69.[14] C.A. Baker, N.P Hessey, C.N. Pinder and C.J. Batty, Nucl. Instr. and Methods in Phys. Res. A394 (1997)180.[15] A. Anisovich et al., ¯ pp → π π from 600 to 1940 MeV/cfrom 600 to 1940 MeV/c