Resonances in p̄p→ωπ0 with masses 1960 to 2410 MeV

24 May 2001

Physics Letters B 508 (2001) 6–16 www.elsevier.nl/locate/npe

Resonances in pp ¯ → ωπ 0 with masses 1960 to 2410 MeV A.V. Anisovich c , C.A. Baker a , C.J. Batty a , D.V. Bugg b , V.A. Nikonov c , A.V. Sarantsev c , V.V. Sarantsev c , B.S. Zou b,1 a Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UK b Queen Mary, London E1 4NS, UK c PNPI, Gatchina, St. Petersburg district, 188350, Russia

Received 27 March 2001; accepted 2 April 2001 Editor: L. Montanet

Abstract An analysis of Crystal Barrel data on pp ¯ → ωπ 0 is presented in the mass range 1960–2410 MeV. There is a strong and well defined I = 1, J P C = 3+− resonance with mass and width (M, Γ ) (2025 ± 15, 107 ± 20) MeV. The partial wave analysis finds further resonances with the following masses and widths: J P C = 4−− (2240 ± 25, 190 ± 50), J P C = 2−− (2240 ± 55, 330 ± 60), and J P C = 1+− (1960 ± 40, 210 ± 80) MeV. There is a further definite contribution from a 3+− state with mass ∼ 2245 MeV, but its mass and width are poorly determined. A contribution is observed from ρ3 (1985), whose mass and width is better determined elsewhere. There is a strong low mass contribution to J P C = 1−− , consistent with the earlier observation of ρ1 (2000).  2001 Elsevier Science B.V. All rights reserved.

We present a study of s-channel (formation) resonances with isospin I = 1 and charge conjugation C = −1 in the channel pp ¯ → ωπ 0 . The amplitude analysis follows the lines of classic analyses of πN and hyperon resonances. The data are obtained using the Crystal Barrel detector to study the 5γ final state, where ω → π 0 γ . These data complement those on the ωηπ 0 channel, which has the same isospin and C parity [1]. Data from the Crystal Barrel experiment on channels ωπ 0 , ωη and ωη at 600, 1200 and 1940 MeV/c have been reported previously [2]. The data were taken at LEAR, using a p¯ beam at nine momenta from 600 to 1940 MeV/c, corresponding to the mass range 1960–2410 MeV. A detailed

E-mail address: [email protected] (D.V. Bugg). 1 Now at the Institute of High Energy Physics, Beijing 100039,

China.

technical description of the detector is given by Aker et al. [3]. The p¯ beam interacts in a liquid hydrogen target 4.4 cm long at the centre of the detector. The incident beam is counted by a coincidence between a proportional counter and a silicon counter of 5 mm diameter situated ∼ 5 cm upstream of the target. A trigger for interactions is provided by two veto counters ∼ 20 cm downstream of the target. Charged particle detectors (a silicon vertex detector, a multiwire chamber and a jet drift chamber) cover 98% of the solid angle. They are used here as vetos to define a neutral final state. Photons are detected in a barrel of 1380 CsI crystals, again covering 98% of the solid angle; they provide an angular resolution of ±20 mrad in both polar and azimuthal angles. The γ detection remains efficient down to energies of 20 MeV, where a threshold is set to reject noise. The energy resolution E is given by E/E = 2.5%/E 1/4 , with E in GeV. The total photon energy from all CsI crystals

0370-2693/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 4 7 5 - 0

A.V. Anisovich et al. / Physics Letters B 508 (2001) 6–16

is summed on-line to provide a trigger for events with a total energy within 200 MeV of that expected for pp ¯ interactions [4]. With a typical p¯ beam intensity of 2 × 105 s−1 , the trigger rate for neutral final states is ∼ 60 s−1 . The off-line analysis selects events containing exactly 5 photon showers. These events are first fitted kinematically to 5γ , then π 0 π 0 γ and finally ωπ 0 . Almost the entire background arises from 3π 0 events after the loss of one photon. The effect of this background and others has been studied using a Monte Carlo simulation with GEANT. After tuning to minimise all backgrounds, the following cuts are selected: (i) Confidence Level (CL) for ωπ 0 > 10%; (ii) CL(ηπ 0 γ ) < 1% to reject ωη; (iii) CL(ωπ 0 ) higher than the confidence level of any other fitted channel, e.g., ωη ; such channels have much lower branching fractions than ωπ 0 ; (iv) CL(ωπ 0 ) > 0.8 × CL(5γ ), a minor refinement to check that the ω is well reconstructed; (v) to eliminate combinatoric problems where the γ from the decay of the ω is paired with the wrong (spectator) π 0 , the best solution is required to have a confidence level at least a factor 5 better than the alternative combination of π 0 and γ . Numbers of events, reconstruction efficiencies and estimated backgrounds are shown in Table 1. Fig. 1(a) shows the π 0 γ mass distribution at 1800 MeV/c, for events fitted to π 0 π 0 γ . The ω(782) signal has a mass within 1.5 MeV of the value of the

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Table 1 Numbers of ωπ 0 events after background subtraction, reconstruction efficiency  and estimated backgrounds (BG). Column 2 shows the total energy available in the centre of mass system 

BG

(%)

(%)

9456

21.1

11

2050

42366

21.1

9

1050

2098

13139

20.7

11

1200

2149

37392

20.2

13

1350

2201

22813

19.6

15

1525

2263

14652

18.6

15

1642

2304

16108

18.1

14

1800

2360

15313

16.8

14

1940

2409

14753

15.8

13

Momentum

CM energy

(MeV/c)

(MeV)

600

1962

900

Events

Particle Data Group (PDG) [5]. This checks that the reconstruction energy is assessed correctly. The mass resolution has a standard deviation of 18 MeV for the ω and 11.5 MeV for the π 0 , whose mass distribution is shown in Fig. 1(b). Fig. 1(c) shows the invariant mass distribution of the spectator π 0 with the π 0 from the decay of the ω. There is a very small broad shoulder due to f2 (1270), consistent with what is expected from 3π 0 background; the f2 (1270) dominates the 3π 0 channel [6].

Fig. 1. Mass distribution of (a) π 0 γ pairs after a kinematic fit to π 0 π 0 γ , (b) γ γ pairs after a kinematic fit to 5γ , (c) π π pairs after the kinematic fit to ωπ 0 .

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A.V. Anisovich et al. / Physics Letters B 508 (2001) 6–16

Fig. 2. Angular distributions for ωπ 0 in the centre of mass frame as a function of production angle Θ; they are uncorrected for acceptance. Full histograms show the partial wave fit. Dotted curves show the acceptance. Dashed histograms show the fitted singlet differential cross section, multiplied by the acceptance.

Fig. 2 shows production angular distributions at all beam momenta; full histograms show the results of the partial wave analysis. Results are uncorrected for acceptance, which is included in the maximum likelihood fit. There are holes in the CsI detector around the beam both upstream and downstream, each 1% of the solid angle. In consequence, the acceptance for ωπ 0 falls off rapidly for centre of mass production angles with | cos Θ| > 0.85, as shown by the dotted histograms in Fig. 2. Angular distributions require waves up to orbital angular momentum 5 in both pp ¯ and ωπ channels. Integrated cross sections are shown

in Fig. 3 and compared with the partial wave fit. These integrated cross sections are corrected for acceptance; errors cover uncertainties in differential cross sections near cos Θ = 1. The partial wave analysis fits the full process of production and decay of the ω using relativistic tensor expressions for amplitudes; J P up to 5− are included. The decay angular distribution of the ω gives some information about its spin alignment between helicities 0 and ±1; however, a significant proportion of this spin information is carried away in the unmeasured polarisation of the photon. Background is estimated from

A.V. Anisovich et al. / Physics Letters B 508 (2001) 6–16

Fig. 3. Integrated cross sections for ωπ 0 , corrected for branching ratios of ω → π 0 γ and π 0 → γ γ and for acceptance. The curve shows the partial wave fit.

a Monte Carlo simulation of the competing channels (mostly 3π 0 ); it is included in the fit and adjusted at each beam momentum to the level of the estimated background. The partial wave analysis has been carried out (a) at all beam momenta separately in terms of partial wave amplitudes, (b) at all beam momenta simultaneously in terms of a sum of s-channel resonances. In Ref. [7], extensive data from many channels for I = 0, C = +1 have been fitted successfully in terms of s-channel resonances; that motivates the attempt to fit present data in terms of resonances following a broadly similar pattern. The parametrisation in terms of resonances introduces the important constraint of analyticity, since Breit–Wigner amplitudes are analytic functions of s. It smoothes out considerable fluctuations in phases found in analysis (a) at individual beam momenta. Partial wave amplitudes take the form √ ρpp ¯ ρωπ
gi exp(iφi )BL (q)B" (p) , f= (1) p Mi2 − s − iMi Γi i B" and BL are standard Blatt–Weisskopf centrifugal barrier factors [8] for production with orbital angular momentum " in the pp ¯ channel and decay with orbital angular momentum L to ωπ 0 ; p and q are centre of mass momenta in pp ¯ and meson channels,

9

respectively. We adopt a radius of 0.83 fm for the centrifugal barrier radius in all partial waves up to " = 4, as determined in Ref. [7]; as discussed below, this radius is increased to 1.1 fm for initial G states, " = 5. In the Breit–Wigner denominator, the widths of the resonances Γi are taken to be constant, in view of the large number of open channels. We shall, however, comment on the possible effects of thresholds for some partial waves. The factors ρ give explicitly the normalisation of cross sections for the phase space in pp ¯ and ωπ channels; the factor 1/p allows for the flux in the entrance channel. All partial waves may couple to orbital angular momentum J ± 1 in either the pp ¯ entrance channel or in the decay to ωπ . As examples, J P C = 1−− may couple to initial pp ¯ 3 S1 and 3 D1 partial waves, both of which decay to ωπ 0 with L = 1; the 1+− pp ¯ singlet state decays with L = 0 or 2. For a longlived resonance, one expects that multiple scattering through the resonance will lead to the same phase for both upper and lower L. We therefore fit the ratio of coupling constants in these coupled partial waves with a real ratio r = gJ +1 /gJ −1 ; from Eq. (1), r refers to the ratio of amplitudes excluding the centrifugal barrier factors. We have carried out an analysis of the ωπ 0 data alone, allowing masses and widths of resonances to adjust freely. We have also carried out a combined analysis with ωηπ 0 data of Ref. [1]. There are only minor differences between the two analyses, so we quote in Table 2 only results from the combined fit to both data sets. Fig. 4 shows the intensities of partial waves fitted to the ωπ 0 data in the combined fit. The filled circles show results of fits at individual beam momenta. They scatter somewhat around the overall fit, indicating the magnitude of systematic uncertainties when phases are set free at individual momenta. The small high partial waves are determined only at the higher momenta, so they are kept fixed when fitting single momenta. At the lowest beam momentum, 600 MeV/c, there is some correlation amongst low partial waves 1 P1 , 3 S1 and 3 D2 . Fig. 5 shows Argand diagrams for the amplitudes. Most partial waves are best determined by the ωπ 0 data, with three exceptions. The ωηπ 0 data play a strong role in determining the parameters of the ρ3 (3−− ) resonance at 2265 MeV and the ρ2 (2−− ) state at 2240 MeV and also the b1 (1+− ) state at

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A.V. Anisovich et al. / Physics Letters B 508 (2001) 6–16

Table 2 Resonance parameters from a combined fit to present data and ωηπ 0 . Values in parentheses are fixed from considerations discussed in the text. Column 4 shows changes in S = log likelihood when each resonance is omitted from the fit to ωπ 0 data and others are reoptimised. The final column shows the ratio rL of coupling constants gJ +1 /gJ −1 for decays to ωπ 0 JPC

Mass M

Width Γ

S

(MeV)

(MeV)

(ωπ 0 )

rL

b1 (1+− )

1960 ± 40

210 ± 80

289

0.7 ± 0.5

b1 (1+− )

2220 ± 55

300 ± 65

79

2.5 ± 1.2

b3 (3+− )

2025 ± 15

107 ± 20

1115

2.2 ± 0.7

b3 (3+− )

2245 ± 50

350+100 −50

346

1.2 ± 0.5

b5 (5+− )

(2500)

(225)

36

ρ1 (1−− )

1965 ± 30

165 ± 30

295

ρ2 (2−− )

1970 ± 35

170 ± 100

227

0.9 ± 1.2

ρ2 (2−− )

2240 ± 55

330 ± 60

296

0.9 ± 0.6

ρ3 (3−− )

(1985)

(170)

64

ρ3 (3−− )

2265 ± 20

170 ± 35

52

ρ4 (4−− )

2240 ± 25

190 ± 50

1159

ρ5 (5−− )

2295 ± 75

250 ± 95

33

0.36 ± 0.05

2220 MeV. Secondly, the ωηπ 0 data determine contributions from J P C = 1−− at high mass; these make little contribution to ωπ 0 . Column 4 of Table 2 shows the effect on log likelihood when each resonance is dropped from the fit to ωπ 0 data and others are reoptimised. Our definition of log likelihood is such that it changes by 0.5 for a one standard deviation change in one parameter. Our experience elsewhere is that changes in log likelihood > 40 definitely require the presence of a component; allowing for the number of degrees of freedom, this change is equivalent to a 6.5 standard deviation effect statistically. We first outline the general conclusions of the analysis before coming to details. There is a requirement in each of the partial waves 1+− , 3+− and 3−− for two resonances in the neighbourhood of 1950– 2030 MeV and 2200–2280 MeV. The lower of the two b3 (3+− ) states at 2025 MeV makes a particularly large and well defined contribution. There is also a particularly strong 3 G4 ≡ 4−− resonance in the high mass region, Fig. 4(g). In addition, for J P C = 2−− , there is a

strong indication for a state at ∼ 2240 MeV, which appears clearly also in ωηπ 0 data; there is tentative evidence for a lower ρ2 (2−− ) state at 1970 ± 35 MeV, but its width is poorly defined. For J P C = 1−− , there is a strong requirement for a low mass contribution consistent with a resonance at ∼ 1965 MeV; ωηπ 0 data require further 1−− contributions at higher mass, but any such contributions in ωπ 0 are weak. A detail is that we include a small b5 (5+− ) contribution with a fixed mass of 2500 MeV, roughly where it is expected on the basis of a Regge trajectory. It has only a small effect on parameters of resonances fitted to other partial waves, but makes a significant improvement to the overall fit. Some resonance parameters are not well determined by ωπ 0 data alone, and others show peculiar features which we now discuss. The first point concerns only singlet states and is largely decoupled from triplet states. The dashed histograms of Fig. 2 show the angular distribution fitted to singlet states after multiplying by the acceptance, for comparison with full histograms. In the momentum range 600–1200 MeV/c, the singlet angular distribution varies rapidly. The fits shown on Fig. 2 do not quite fit the peak near cos Θ = 0 at 1050 and 1200 MeV/c. To achieve the present fit, it is essential to have a very rapid variation with mass in both the 1 P1 and 1 F3 partial waves. This demands a particularly narrow width for the lower b3 (3+− ) resonance, namely, 107 ± 20 MeV. The essential difficulty in the analysis lies in separating different triplet states, particularly in establishing the content of the 1−− sector. Because there are no data concerning the polarisation of the proton or antiproton, there is little separation between 3 S1 and 3 D1 amplitudes (and likewise between 3 D3 and 3 G3 ). For the ρ1 resonance at 1965 MeV, there are two alternative solutions with almost the same mass and width but differing in rpp ¯ value. The solution given in Table 2 is the better of the two solutions. This ambiguity is confined to the 1−− sector and has little effect on other partial waves. Because of the " = 2 centrifugal barrier in the 3 D1 channel, it also has only small effects even in the 1−− sector. The third point concerns a major surprise: there is a very large and well determined 3 G4 contribution peaking at ∼ 2260 MeV, see Fig. 4(d). At high beam momenta it is the dominant contribution. It decays strongly to ωπ 0 with L = 3 but also with a well

A.V. Anisovich et al. / Physics Letters B 508 (2001) 6–16

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Fig. 4. Intensities of partial waves fitted to ωπ 0 data. Full curves are for decays with L = J − 1 and dashed curves for decays with L = J + 1. Full and open circles show results of the analyses of individual momenta for decays with, respectively, L = J − 1 and L = J + 1.

defined L = 5 component. Because of the strong " = 4 centrifugal barrier in the pp ¯ channel, the fitted mass is lower, at 2240 ± 25 MeV. This is a surprisingly low mass. The GAMS collaboration has reported a 5−− state at 2330 ± 35 MeV. One expects tensor and spin– orbit splitting to be small for G states, hence that the ρ5 (5−− ) and ρ4 (4−− ) states should lie close together. The 3 G4 peak lies even lower than triplet F states, which have been observed in earlier analyses of 3π 0

data [6] and ηπ 0 and η π 0 [9]. That for the 3 G4 state in present data definitely appears to lie significantly lower. The wave function of the 3 G4 state is peaked strongly at large radii. The radius we have adopted for the centrifugal barrier from Ref. [7] is mainly sensitive to the dominant triplet F states in that analysis. It is plausible that, for high partial waves, the radius should be proportional to orbital angular momentum.

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A.V. Anisovich et al. / Physics Letters B 508 (2001) 6–16

Fig. 5. Argand diagrams for fitted partial waves; all move anti-clockwise with increasing beam momentum. Crosses show beam momenta starting at 600 MeV/c.

For G states, we have tried increasing the radius of the centrifugal barrier to 1.1 fm. This improves the fit substantially (by ∼ 100 in log likelihood) and is

therefore used for the final fit. However, it has little effect on the fitted mass; it improves the determination of the width.

A.V. Anisovich et al. / Physics Letters B 508 (2001) 6–16

Partial wave analyses at individual momenta confirm the strong 3 G4 peak at 2260 MeV. Results are shown on Fig. 4(d) as filled circles for L = 3 decays and as open circles for L = 5 decays. In these analyses, errors are larger, particularly for phases, which show a large scatter. In view of the surprising result for 3 G4 , we have searched extensively for alternative solutions, but can find none which differ from the present solution, except for the minor variation noted above in the 1−− sector. We have examined possible cross-talk between 3 G4 and 3 D2 , but it does not appear to explain the requirement for a large 3 G4 component. The amplitude analysis has been repeated without data at | cos Θ| > 0.85, but gives essentially the same results; it is, therefore, unlikely that the large 3 G4 intensity is connected with acceptance variations for | cos Θ| > 0.85. Having discussed 3 G4 , we now turn to other partial waves. One expects the 3 G4 state to be accompanied by 3 G5 and 3 G3 resonances. Signals for both are very small compared to ρ4 (4−− ), see Fig. 4(e) and the dashed curve of Fig. 4(c). The ρ5 optimises at 2295 ± 75 MeV with Γ = 250 ± 95 MeV; errors are large because of the small amplitude. The mass lies close to that reported by GAMS [10] in ωπ , namely, 2330 ± 35 MeV; they omit the centrifugal barrier factors. For J P = 3− , at least two resonances are required. The lower one optimises at 2013 ± 30 MeV with Γ = 165 ± 35 MeV, but makes a fairly weak contribution, see Fig. 4(c). We have reported previously [7] a combined analysis of data on ηπ 0 π 0 , π 0 π 0 , ηη, ηη and π − π + . In the π − π + data, a very strong ρ3 appears at 1981 ± 14 MeV with Γ = 180 ± 35 MeV. This determination is very secure because of extensive data on the π − π + final state, including polarisation, at 100 MeV/c steps down to a p¯ beam momentum of 360 MeV/c (a mass of 1910 MeV). For our final fit, we adopt the weighted means from present data and π − π + , namely, M = 1987 MeV, Γ = 170 MeV. The good mass determination for this 3−− state from π − π + data provides a useful interferometer at low momenta for determining other triplet partial waves here. It helps in establishing that the phase variation expected of resonances really is present. In Ref. [7], the well established f4 (2050) makes a large contribution to all channels; it interferes with ρ3 (1985)

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in the π − π + channel, requiring a resonant phase variation. The ρ3 (1985) interferes in present data and ωηπ 0 and requires a resonant phase variation in turn for the 2−− and 1−− partial waves. The tails of these resonance at high mass require further resonant phase variation at the top of the available mass range, as one sees from the Argand diagrams of Fig. 5. A second 3 D3 state is to be expected around a mass of 2250 MeV. There is indeed evidence for this 3 D3 state in π − π + data of Ref. [7] at a mass of 2210 ± 40 MeV. This resonance appears in Fig. 4(c) and also more strongly in ωηπ 0 data of Ref. [1]. If we fit those data together with present data using a single state, it optimises at 2265 ± 20 MeV with Γ = 170 ± 35 MeV. There is a significant component coupling to pp ¯ 3 G3 ; the r parameter optimises at 2.5, but with a large error, ±1.1. From the quark model, one expects both a 3 D3 and a 3 G3 state in this upper mass range, the 3 G3 state accompanying the GAMS 3 G5 resonance. There was evidence in Ref. [7] for a separate 3 G3 state at ∼ 2300 MeV. There is the clear possibility that the present fit averages over unresolved 3 D3 and 3 G3 states. In Ref. [7], we have found that all I = 0, C = +1 states lie close to straight line trajectories of s = M 2 v. radial excitation with a mean slope 1.143 ± 0.013 GeV2 . From the well defined ρ3 (1987), this ansatz predicts the upper 3 D3 state at 2255 ± 15 MeV. For J P C = 2−− , there is a quite well defined state at 2240 ± 55 MeV in the region of the dip in Fig. 4(b); it appears also in the ωηπ 0 data of Ref. [1]. One expects a lower 3 D2 state close to the observed 3 D3 state, i.e., around 1980 MeV. Quite a large signal fits naturally around 2 GeV, as shown in Fig. 4(b). However, there is considerable cross-talk with the 3 S1 signal in this low mass range. The ωηπ 0 data and ωπ 0 together give a determination of its mass, M = 1970 ± 35 MeV, but its width is poorly determined. For J P C = 1−− , the data require a strong contribution at low mass, Fig. 4(a). It could be due to strong annihilation near the pp ¯ threshold, but is also consistent with a ρ1 at 2000 ± 30 MeV found in our earlier fits to pp ¯ → π − π + [7]. The rapid phase variation between 600 and 900 MeV/c in the Argand diagram of Fig. 5 favours a resonance interpretation. From the combined fit to ωηπ 0 and present data, a resonance of mass 1965 ± 30 MeV and width 165 ± 30 MeV is found. This width is somewhat smaller than in fitting

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A.V. Anisovich et al. / Physics Letters B 508 (2001) 6–16

(a)

(b)

Fig. 6. A comparison of M 2 for resonances with straight line trajectories against radial excitation number n; the slope of 1.143 GeV2 is fixed to the same value as in Ref. [7].

π − π + data, but the error from Ref. [7] was large. The π − π + data are available at ∼ 100 MeV/c steps down to 360 MeV/c and provide the more reliable determination of the mass. Further ρ1 states are expected at higher masses, e.g., ρ1 (2150) [5]. Such higher mass states have been tried in the fit to present data, but do not improve log likelihood significantly. They do, however, appear in ωηπ 0 data [1]. We now turn to singlet states b3 (3+− ) and b1 (1+− ). Both partial waves are very large at low momenta, The 1 F state is well defined with M = 2025 ± 15 MeV; it 3 has a particularly narrow width, Γ = 107 ± 20 MeV, for reasons discussed above. It is the most secure state in the entire analysis. The " = 3 centrifugal barrier in the pp ¯ channel pushes the peak in its intensity up to 2050 MeV in Fig. 4(g). We now compare this 1 F3 state with triplet F states in the same mass range. From Refs. [6] and [9], a 3 F4 state is observed at a mean mass of 2015 MeV, a 3 F3 state at 2070 MeV and a 3 F2 state at 2035 MeV. The centroid of these results (weighted by their multiplicity) is at 2038 MeV; this agrees with the mass of the 1 F3 state observed here within the error. There could be some small spin–spin splitting. There is a definite requirement for a second b3 (3+− ) state at higher mass, though no peak is visible in Fig. 4(g). The phase variations observed in the last two

panels of Fig. 5 require two states. Without it, log likelihood is worse by 346, a very large amount (> 20 σ for 5 degrees of freedom). However, its mass and width are poorly determined. This is associated with the problem of fitting the rapid variation in the singlet states around a beam momentum of 1200 MeV/c. If left free, the mass drifts down to ∼ 2245 MeV, but with only a small change in log likelihood. There is stronger evidence for this upper b3 (3+− ) state in ωηπ 0 data of Ref. [1]. The lower b1 (1+− ) state makes a large contribution in Fig. 4(f). It lies at the bottom end of the available mass range: with M = 1960 ± 40 MeV, Γ = 210 ± 80 MeV. From the fitted mass of this state and the well known mass of b1 (1235), the slope of a straight line trajectory of s = M 2 against excitation number is 1.165 ± 0.080 GeV2 , in satisfactory agreement with Ref. [7]. The next radial excitation is expected at about 2225 MeV. The Argand diagram in Fig. 5 shows two distinct resonances. The ωηπ 0 data of Ref. [1] demand strongly the presence of a state with mass 2220 ± 55 MeV and a width of 300 ± 65 MeV. There are strong interferences between the two b1 states in Fig. 4(f). In Ref. [7], it was observed that I = 0, C = +1 resonances were consistent for all quantum numbers with straight-line trajectories of M 2 against radial excitation number n. Fig. 6 shows a similar comparison with

A.V. Anisovich et al. / Physics Letters B 508 (2001) 6–16

resonances from the present analysis, using the same slope of 1.143 GeV2 per radial excitation. Within the sizeable errors, results are consistent with this simple picture. In plotting the 3 D1 trajectory, we have followed the conventional assumption that ρ(1700) is a 3 D state, because of its close proximity to ρ (1690). 1 3 A general remark is that it may appear surprising that masses of states at the bottom of the available mass range, 1950–2000 MeV, can be determined with any accuracy. There are two points helping with precision. Firstly, the accurately known 3 D3 state at 1987 MeV provides an interferometer for other triplet states. Secondly, partial wave amplitudes except for 3 S go to zero at the pp ¯ threshold. The phases of these 1 partial waves show a rapid variation at the lowest beam momenta. These variations are clearly visible in the Argand diagrams of Fig. 5 for 3 D2 , 3 D3 and 1 P1 when one continues the curves to the origin, marked by the intersection of the dotted lines. It is this rapid variation which requires the resonances and helps determine their positions rather well. A second general comment is that it is possible that thresholds for decays of resonances to other channels may perturb the pole positions. The opening of a threshold has the effect of suppressing the upper side of a resonance, because of the additional full width. We conjecture that this effect might be connected with the low mass of the 3 G4 state. However, it is not obvious what threshold might be responsible. We can eliminate a2 (1320)ω and b1 (1235)η thresholds, because of their low branching ratios observed in the ωηπ 0 data; also f2 (1270)ρ and h1 (1170)π are related to the former two under the assumption of ideal mixing and can be eliminated. For the 3 G4 state, we have tried including in the Breit–Wigner denominator the full s-dependence of the decay width to ωπ , using the centrifugal barrier factors for decays with L = 3 and 5. This does improve the fit by ∼ 100 in log likelihood but increases the fitted mass only by 10 MeV. The ¯ → ωπ amplitude demands a very strong 3 G4 pp large Γ (pp), ¯ increasing rapidly with s because of the centrifugal barrier. However, its absolute magnitude relative to Γ (ωπ) is not known. This Γpp ¯ (s) will have the effect of suppressing to some degree the upper side of the resonance and could help explain the apparently low mass. A third general comment is that the partial wave analysis would be greatly improved if polarisation data

15

were available. Much of the polarisation information from the ω is carried away by the photon, though the decay angular distribution for ω → π 0 γ does provide some information. Polarisation data from the use of a polarised proton target would help greatly. Firstly, this information would separate helicity states and hence L = J + 1 from L = J − 1. The absence of this information for present data makes it difficult to separate 3 D1 from 3 S1 and likewise 3 G3 from 3 D3 . Secondly P dσ/dΩ depends on the imaginary part of interferences between partial waves, while dσ/dΩ itself contains the real parts of interferences. The polarisation data therefore help greatly in defining relative phases between partial waves and providing accurate determinations of masses and widths of resonances. A measurement with a polarised target appears straightforward and would make a major contribution to completing the accurate determination of resonances in this mass range. In summary, we are confident of the presence of a strong 3 G4 state with a pole position at M = 2240 ± 25 MeV, though its mass is inexplicably low compared with well defined 3 F states observed in Refs. [6,7,9]. A ρ5 at 2295 ± 75 MeV is weak, but consistent within the error with earlier GAMS data after allowing for the effect of the centrifugal barriers. There is a strong, well defined narrow b3 (3+− ) state at 2025 MeV, close to the centroid of triplet F states. A second b3 state is required at ∼ 2245 MeV, but its mass and width are poorly determined. Present data also require a strong b1 (1+− ) state at 1960. There is definite evidence for a 3 D2 state at 2240 MeV and tentative evidence for a lower one at 1970 MeV. A low mass J P C = 1−− peak is consistent with an earlier observation of ρ1 (2000) in Ref. [7].

Acknowledgement

We thank the Crystal Barrel Collaboration for use of the data. We acknowledge financial support from the British Particle Physics and Astronomy Research Council (PPARC). We wish to thank Prof. V.V. Anisovich for helpful discussions. The St. Petersburg group wishes to acknowledge financial support from PPARC and INTAS grant RFBR 95-0267.

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