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PREPRINT
BNL­E852­97­001
Study of the Reaction ú
\Gamma p ! jú
\Gamma p at 18 GeV=c
G. S. Adams, 8 T. Adams, 1 Z. Bar­Yam, 5 J. M. Bishop, 1 V. A. Bodyagin, 6 B. B. Brabson, 4
D. S. Brown, 7 N. M. Cason, 1 S. U. Chung, 2 R. R. Crittenden, 4 J. P. Cummings, 5 [*]
S. P. Denisov, 3 V. A. Dorofeev, 3 A. R. Dzierba, 4 J. P. Dowd, 5 P. Eugenio, 5 J. Gunter, 4
R. W. Hackenburg, 2 M. Hayek, 5 E. I. Ivanov, 1 I. A. Kachaev, 3 W. Kern, 5 E. King, 5
O. L. Kodolova, 6 V. L. Korotkikh, 6 M. A. Kostin, 6 J. Kuhn, 8 R. Lindenbusch, 4 V. V. Lipaev, 3
J .M. LoSecco, 1 J. J. Manak, 1 J. Napolitano, 8 M. Nozar, 8 C. Olchanski, 2 A. I. Ostrovidov, 6
T. K. Pedlar, 7 A. V. Popov, 3 D. R. Rust, 4 D. I. Ryabchikov, 3 A .H. Sanjari, 1 L. I. Sarycheva, 6
E. Scott, 4 K. K. Seth, 7 W. D. Shephard, 1 N. B. Sinev, 6 J. A. Smith, 8 P. T. Smith, 4
D. L. Stienike, 1 T. Sulanke, 4 S. A. Taegar, 1 S. Teige, 4 D. R. Thompson, 1 I. N. Vardanyan, 6
D. P. Weygand, 2 D. B. White, 8 H. J. Willutzki, 2 J. Wise, 7 M. Witkowski, 8 A. A. Yershov, 6
D. Zhao, 7
---The E852 Collaboration---
Version 1.0 ­ DRAFT ONLY ­ NOT FOR DISTRIBUTION
2 Brookhaven National Laboratory, Upton, Long Island, NY 11973, USA
3 Institute for High Energy Physics, Protvino, Russian Federation
4 Indiana University, Bloomington, IN 47405, USA
5 University of Massachusetts Dartmouth, North Dartmouth, MA 02747, USA
6 Moscow State University, Moscow, Russian Federation
7 Northwestern University, Evanston, IL 60208, USA
1 University of Notre Dame, Notre Dame, IN 46556, USA
8 Rensselaer Polytechnic Institute, Troy, NY 12180, USA
December 17, 1997

Abstract
Details of the analysis of the jú \Gamma system studied in the reaction ú \Gamma p ! jú \Gamma p
at 18 GeV=c are given. Separate analyses for the 2fl and ú + ú \Gamma ú 0 de­
cay modes of the j are consistent. The amplitude analysis indicates the
presence of interference between the a 2 (1320) and a J PC = 1 \Gamma+ wave
between 1.2 and 1.6 GeV=c 2 . The phase difference between these waves
shows phase motion not attributable solely to the a 2 (1320). The data
can be fitted by interference between the a 2 (1320) and an exotic 1 \Gamma+ res­
onance with M = (1370 \Sigma16 +50
\Gamma30 ) MeV=c 2 and \Gamma = (385 \Sigma40 +65
\Gamma105 )
MeV=c 2 . These results are compared with those of other experiments.
Typeset using REVT E X

I. INTRODUCTION
In a previous publication [1], we presented evidence for an exotic meson produced in the
reaction
ú \Gamma p ! jú \Gamma p (1)
at 18 GeV=c, with the decay mode j ! flfl. The purpose of this paper is to provide
details of that analysis, to discuss additional analyses of that data, and to provide a detailed
comparison of our results with those of other experiments. We also provide a comparison
of those results with data from our experiment on Reaction (1) but from the j ! ú + ú \Gamma ú 0
decay mode.
The jú system is particularly interesting in searching for exotic (or non­qq) mesons
because the system has spin (J), parity (P), and charge­conjugation (C) in the sequence
J PC = 0 ++ ; 1 \Gamma+ ; 2 ++ ; 3 \Gamma+ ::: for L = 0; 1; 2; 3; ::: . (Here L is the orbital angular momentum
of the jú system.) Hence a resonance with an jú decay mode with odd L is manifestly
exotic. a Having isospin I=1, such a resonance could not be a glueball (2g; 3g; : : :), but it
could be a hybrid (qqg) or a multiquark (qqqq) state.
A. Models
Properties of hybrids and multiquark mesons have been discussed in the framework of
various models [2--9]. Calculations based upon the MIT bag model predict [3,4] that an
I = 1, 1 \Gamma+ hybrid (qqg) will have a mass near 1.4 GeV=c 2 . On the other hand, the flux­tube
model [5,6] predicts the mass of the lowest­lying hybrid state to be around 1:8 GeV=c 2 .
Characteristics of bag­model S­wave multiquark states (which would have J P = 0 + , 1 + or
2 + ) have been discussed [7] but those for a 1 \Gamma state have not. Recently, an analysis of the
a A qq meson with orbital angular momentum ` and total spin s must have P = (\Gamma1) `+1 and C =
(\Gamma1) `+s . A resonance with J PC = 0 \Gamma\Gamma ; 0 +\Gamma ; 1 \Gamma+ ; 2 +\Gamma ; 3 \Gamma+ \Delta \Delta \Delta does not satisfy these conditions and
must be exotic.
1

multiquark hybrids has been carried out, based on the diquark cluster model [8]; this model
predicts a lowest­lying isovector J PC = 1 \Gamma+ state at 1.39 GeV with a very narrow width
(' 8 MeV). Ultimately, the most reliable estimate of the masses is expected to come from
lattice­gauge calculations; the latest mass estimates for a 1 \Gamma+ hybrid are in the range 1.7 to
2.1 GeV [9].
B. Previous Experiments
Production of jú events with J PC = 1 \Gamma+ has been observed in several experiments
[10--13]. Alde et al. [10], in a study of ú \Gamma p interactions at 100 GeV/c at CERN (the GAMS
experiment), claimed to observe a 1 \Gamma+ state in the jú 0 system at 1.4 GeV=c 2 produced via
unnatural parity exchange (the P 0 partial wave---the naming convention is discussed below).
This result was later brought into question [14]. Aoyagi et al. [11], in a ú \Gamma p experiment
at 6.3 GeV/c at KEK, observed a rather narrow enhancement in the jú \Gamma system at 1.3
GeV=c 2 in the natural parity exchange 1 \Gamma+ spectrum (P+ ). Beladidze et al. [12], in the VES
experiment at IHEP, (ú \Gamma N interactions at 37 GeV/c) also reported a P+ signal in the jú \Gamma
state, but their signal was broader and had a significantly different phase variation from
that of the KEK experiment. While the phase difference between the P+ and D+ waves
was independent of jú mass in the KEK analysis, that phase difference did show significant
mass dependence in the VES analysis. Finally, the Crystal Barrel experiment at CERN has
recently reported [13] on the presence of signals in NN annihilations in both the charged
and neutral jú systems. In this paper, we compare the results of our analysis with those of
references [11--13].
2

II. EXPERIMENTAL DETAILS
A. E852 Apparatus
Our data sample was collected in the first data run of E852 at the Alternating Gradi­
ent Synchrotron (AGS) at Brookhaven National Laboratory (BNL) with the Multi­Particle
Spectrometer (MPS) [15] augmented by additional detectors. A diagram of the experimental
apparatus is shown in Fig. 1. A tagged ú \Gamma beam of momentum 18 GeV=c was incident on
a one­foot long liquid hydrogen target at the center of the MPS magnet. The target was
surrounded by a four­layer cylindrical drift chamber (TCYL) [16] used to trigger on the
proton recoil of Reaction (1), and a 198­element cylindrical thallium­doped cesium iodide
array (CsI) [17] capable of rejecting events with wide­angle photons. The downstream half
of the magnet was equipped with six drift chamber modules (DC1­6) [18], each consisting of
seven planes, used for charged­particle tracking. Interspersed among these were three pro­
portional wire chambers (TPX1­3) to allow triggering on the multiplicity of forward tracks;
a window­frame lead scintillator photon veto counter (DEA) to ensure photon hermeticity; a
forward scintillation counter (CPVB) to veto forward charged tracks for neutral triggers; and
a window­frame scintillation counter (CPVC) to identify charged particles entering the DEA.
Beyond the magnet were a newly­built drift chamber (TDX4) consisting of two x­planes; two
scintillation counters (BV and EV) to veto non­interacting beam tracks and elastic scatters
respectively; and a 3045­element lead glass electromagnetic calorimeter (LGD) [19] to detect
forward photons. Further details are given elsewhere [20].
B. Trigger
The trigger for Reaction (1) required a recoil charged particle in the TCYL detector
and one charged particle traversing each of the first two TPX chambers. In addition, an
electronic algorithm coupling energy and position information in the LGD calorimeter [19]
(an ``effective­mass'' trigger) was utilized for the purpose of enhancing the fraction of j's
3

relative to ú 0 's in the sample. A total of 47 million triggers of this type were recorded.
C. Event Reconstruction and Selection
Of the 47 million triggers, 47,200 events were reconstructed which were consistent with
Reaction (1). These were selected by requiring:
1. two photons reconstructed in the LGD;
2. one forward track reconstructed in DC1­6;
3. one recoil track reconstructed in TCYL;
4. a common vertex, in a target fiducial volume, reconstructed from the charged tracks
and the beam track;
5. no photon energy deposited in the DEA detector or outside the fiducial volume of the
LGD;
6. a difference in azimuth of Ÿ 20 ffi between the reconstructed TCYL track and the missing
momentum vector;
7. Ÿ 20 MeV of energy deposit (beyond that associated with the TCYL track) in the CsI
detector to require a proton recoil (analysis was carried out with and without this cut);
and
8. a SQUAW [21] kinematic fit (requiring energy and momentum conservation) to Reac­
tion (1) with a confidence level ? 10%.
D. Experimental Acceptance
The experimental acceptance is determined by a Monte Carlo method. Events are gen­
erated [22] with isotropic angular distributions in the Gottfried­Jackson (GJ) frame. ( The
GJ frame is a rest frame of the jú \Gamma system in which the z­axis is in the direction of the
4

beam momentum, and the y­axis is in the direction of the vector cross­product of the target
and recoil momenta.) After adding detector simulation using GEANT [23], the Monte Carlo
event sample is subjected to the same event­selection cuts and run through the same analysis
as the data. A second method (called SAGEN) which did not use GEANT was employed as
well. This method also used SAGE as the event generator, but instead of using GEANT, a
geometrical acceptance was used. This second method allowed acceptances to be calculated
much more quickly. The amplitude analysis results discussed below obtained from the two
methods were found to be consistent with each other. The average acceptances shown in
Figs. 2­4 are based on the SAGEN method.
The average acceptance as a function of jú \Gamma effective mass is shown in Fig. 2. The
average acceptance decreases by about a factor of two over the effective mass region from 1.0
to 2.0 GeV=c 2 (Average acceptances are calculated for peripheral production and isotropic
decay.)
Shown in Figs. 3 and 4 is the acceptance as a function of cos` and of OE for various values of
the jú \Gamma effective mass. Here cos` and of OE are are the polar and azimuthal angles measured
in the GJ frame. The polar angle is the angle between the beam direction and the j direction
in this frame. The inefficiency in the backward direction corresponds to slow j's and fast
ú \Gamma 's in the lab. The slow j's lead to slow fl's which are often produced at wide angles and
thus miss the LGD. The acceptance in OE is relatively uniform.
Finally, shown in Fig. 5 is the average acceptance (GEANT­based) as a function of jt 0 j =
jt \Gamma t min j, where t is the the four­momentum­transfer between the initial­ and final­state
protons. The dramatic decrease in acceptance below about jt 0 j = 0:08(GeV=c) 2 is due to a
trigger requirement. In particular, since we require the presence of a recoil proton in TCYL,
the trigger cannot be satisfied if the proton stops in the hydrogen target.
5

E. Background Studies
Shown in Fig. 6 is the 2fl effective mass distribution for events in the a 2 (1320) mass
region. The central cross­hatched region in Fig. 6 shows the events which remain after
the SQUAW­based kinematic­fitting cuts. The width of the j peak (oe ú :03 GeV=c 2 ) is
consistent with that expected from Monte Carlo studies. Two methods have been used to
study the background in our sample. Method 1 used the shaded sidebands of Fig. 6 and
allows us to study the non­j background in the data.
In Fig. 7 is shown the missing­mass squared distribution for the data sample before kine­
matic fitting. The shaded area shows the events which remain after kinematic fitting. Recall
that one expects the distribution for good events for Reaction 1 to peak at the square of the
proton mass or at a value of 0:88 GeV=c 2 .
A scatterplot of the 2fl effective mass versus the missing­mass squared is shown in Fig. 8.
Background studies using our Method 2 takes as the background estimator a region sur­
rounding the central signal region seen here instead of using the sidebands of Fig. 6. In
this way, background events of both the non­j type and of the type with missing ú 0 's are
taken into account. (The background is estimated using the region included within the outer
elliptical area of Fig. 8 but not within the middle elliptical region.)
Shown in Fig. 9 is the effective­mass distribution of the background, estimated using
Method 1. In this figure are shown the effective­mass distribution for each side­band region
as well as the summed distribution for the background regions. Because the background
regions have different thresholds, one higher than the signal region and one lower than the
signal region, the histograms are shifted by an appropriate amount (so that their thresholds
match that of the signal region) before summing.
In Fig. 10 is shown the polar angular distribution of the background events from Method 1
in the a 2 (1320) effective­mass region. Both uncorrected and corrected angular distributions
are shown. This angular distribution is consistent with isotropy although there is some
tendency for the distribution to have an excess below the region cos ` ! \Gamma0:5.
6

III. GENERAL FEATURES OF THE DATA
The a 2 (1320) is the dominant feature of the jú \Gamma effective­mass spectrum shown in Fig. 11.
The background, which is shown shaded in the figure, is estimated from Method 2 above,
and is approximately 7% at 1.2 GeV=c 2 , falling to 1% at 1.3 GeV=c 2 .
The acceptance­corrected distribution of jt 0 j = jt \Gamma t min j is shown for jt 0 j ? 0:08(GeV=c) 2
in Fig. 12. (Our acceptance is quite low below 0.08 (GeV=c) 2 as discussed in section II D.)
Here t is the the four­momentum­transfer between the target proton and the outgoing proton
and t min is the minimum value for this quantity for a given jú effective mass. The shape of
this distribution is consistent with previous experiments and has been shown to be consistent
with natural­parity exchange production in Regge­pole phenomenology [24,25].
The acceptance­corrected b distribution of cos `, the cosine of the angle between the j
and the beam track in the GJ frame, is shown in Fig. 13 for various ranges of jú \Gamma effective
mass. The presence of a significant forward­backward asymmetry in cos ` is obvious.
The forward­backward asymmetry in cos ` is plotted as a function of jú effective mass
in Fig. 14. Here, the asymmetry is defined as (F \Gamma B)=(F + B) where F (B) is the number
of events in the mass bin with the j decaying forward (backward) in the GJ frame. For
this figure, the asymmetry was calculated for events in the higher­acceptance region with
j cos `j ! 0:8. c The asymmetry is large, statistically significant and mass dependent. Since
the presence of only even values of L would yield a symmetric distribution in cos `, the
observed asymmetry requires that odd­L partial waves be present and that they interfere
with even­L partial waves to describe the data.
The azimuthal angular distribution as a function of jú \Gamma effective mass is shown in Fig. 15.
The observed structure has a clear sin OE component which indicates the presence of m = 1
b For illustration purposes, the acceptance correction is calculated here for isotropic decay of the jú
system. The acceptance correction used in the amplitude analysis discussed below is based upon the
observed decay angular distribution.
c The asymmetry function was plotted for various ranges of the decay angle and the presence of a strong
asymmetry was noted in all cases.
7

natural­parity­exchange waves in the production process. (See the discussion in Section IV
below.)
Shown in Fig. 16 are the ú \Gamma p and jp effective­mass distributions for the data sample.
It is important to note that the absence of baryon isobar production is required for the
assumptions of our PWA to be valid. There is at most a very small amount of isobar pro­
duction in the region M(úp) ! 2:0 GeV=c 2 in Fig. 16a and none in Fig. 16b. The amplitude
analysis described in Section IV was checked to insure that isobar production did not effect
our results. This was done by redoing the analysis after requiring M(úp) ? 2:0 GeV=c 2 .
The results were not effected by this cut.
IV. PARTIAL­WAVE ANALYSIS
A. Procedure
A partial­wave analysis (PWA) [26,27] based on the extended maximum likelihood
method has been used to study the spin­parity structure of the jú \Gamma system. In Appendix
A, a brief description of the formalism is given as are the relationships between the the par­
tial wave amplitudes (assuming L Ÿ 2) and the moments of the angular distribution. The
technique of the extended maximum likelihood analysis is covered in Appendix B, where
the interplay of the experimental moments and the acceptance is described. (The experi­
mental acceptance is incorporated into the PWA by using the accepted Monte Carlo events
described above to calculate normalization integrals -- see ref. [26]).
The partial waves are parameterized in terms of the quantum numbers J PC as well as m,
the absolute value of the angular momentum projection, and the reflectivity ffl [28]. In our
naming convention, a letter indicates the angular momentum of the partial wave in standard
spectroscopic notation, while a subscript of 0 means m = 0, ffl = \Gamma1, and a subscript of +(\Gamma)
means m = 1, ffl = +1(\Gamma1). Thus, S 0 denotes the partial wave having J PC m ffl = 0 ++ 0 \Gamma , while
P \Gamma signifies 1 \Gamma+ 1 \Gamma , D+ means 2 ++ 1 + , and so on.
We consider partial waves with m Ÿ 1 in our analysis. This assumption is true, of course,
8

in the limit of \Gammat = 0, since the nucleon helicities give rise to the states with m = 0 or m = \Sigma1
only. But this assumption can be dealt with---experimentally---since the moments H(LM)
with M = 3 or M = 4 can be checked, to see how important the states j` mi are in the data
with jmj – 2. This has been done with our data. The moments H(33), H(43) and H(44)
are all small in the a 2 (1320) region, and a fit including j22i shows only a very small amount
of this wave and is very broad. In particular, it does not affect our P+ wave!
We also assume that the production spin­density matrix has rank one. This assumption
is discussed in Appendix C.
Goodness­of­fit is determined by calculation of a ü 2 from comparison of the experimental
moments with those predicted by the results of the PWA fit. A systematic study has been
performed to determine the effect on goodness­of­fit of adding and subtracting partial waves
of J Ÿ 2 and M Ÿ 1. We find that although no significant structure is seen in the waves
of negative reflectivity (see below), their presence in the PWA fit results in a significant
improvement in goodness­of­fit compared to a fit which includes only the dominant positive­
reflectivity partial waves. We have also performed fits including partial waves with J = 3 and
J = 4. Contributions from these partial waves are found to be insignificant for M(jú \Gamma ) ! 1:8
GeV=c 2 . Thus, PWA fits shown or referred to in this letter include all partial waves with
J Ÿ 2 and m Ÿ 1 (i.e. S 0 , P 0 , P \Gamma , D 0 , D \Gamma , P+ , and D+ ). A non­interfering, isotropic
background term of fixed magnitude determined as described by Method 2 in Section II D
is used.
B. Results
The results of the PWA fit of 38,200 events in the range 0:98 ! M(jú \Gamma ) ! 1:82 GeV=c 2
and 0:10 ! jtj ! 0:95 GeV 2 are shown in Figs. 17 and 18. In Fig. 17 the acceptance­corrected
numbers of events predicted by the PWA fit for the D+ and P+ intensities and the phase
difference between these amplitudes, \Delta\Phi, are shown as a function of M(jú \Gamma ). (The smooth
curves shown in this figure are discussed below in Section V B. There are eight ambiguous
9

solutions in the fit [27,29,30]. These solutions are mathematically discrete but with equal
likelihoods -- that is, they predict exactly the same values for the angular observables. We
show the range of fitted values for these ambiguous solutions in the vertical rectangular bar
at each mass bin, and the maximum extent of their errors is shown as the error bar. d
The a 2 (1320) is clearly observed in the D+ partial wave (Fig. 17a). A broad peak is
seen in the P+ wave at about 1:4 GeV=c 2 (Fig. 17b). The phase difference \Delta\Phi increases
through the a 2 (1320) region, and then decreases above about 1.5 GeV=c 2 (Fig. 17c). This
phase behavior will allow us to study the nature of the P+ wave.
Shown in Fig. 18 are the fitted intensities for waves which are produced by negative­
reflectivity (or unnatural­parity) exchange. The predicted numbers of events for these waves
are generally small and are all consistent with zero above about 1.3 GeV=c 2 . Although there
is some non­zero contribution from the D \Gamma and (especially) the S 0 waves below this region,
the uncertainties and ambiguity ranges associated with these waves make it impossible to
do a definitive study of them to determine their nature.
It is important to note that the forward­backward asymmetry noted earlier is due to
interference in the natural­parity exchange sector rather than to the unnatural­parity ex­
change waves. This is illustrated in Fig. 19 which shows the predicted asymmetry separately
for the natural and unnatural parity exchange waves. It is clear that the asymmetry due
to the unnatural­parity waves is about an order of magnitude less than that due to the
natural­parity waves. Also shown in Fig. 19 is the comparison of the asymmetry present in
the data with that predicted by the fit. The fit clearly does an excellent job in representing
the data points.
The unnormalized spherical harmonic moments H(LM) and their prediction from the
PWA fit as a function of mass are shown in Fig. 20. Here H(LM) =
P N
i=1 Y M
L (` i ; OE i ) (N being
the number of events in a given bin of M(jú \Gamma )). The relationships between the moments and
the amplitudes is given in Appendix A, Eqs. A27. An examination of the H(30), H(32),
d These rectangular bars are quite small and thus not apparent for the D+ intensity, but are quite clear
for the P+ intensity and the phase difference distributions.
10

H(40) and H(42) moments along with a comparison with Eqs. A27 shows that the D+
amplitude dominates and demonstrates clearly that the P+ partial wave is required for the
PWA fit to describe the experimental moments. These moments cannot be described solely
by the combination of the D+ partial wave and experimental acceptance.
The change in \Gammalog(Likelihood) brought about by variation of the predicted number
of events for the P+ partial wave for the 1:30 ! M(jú \Gamma ) ! 1:34 GeV=c 2 bin is shown
for all the ambiguous solutions in Fig. 21. The number of predicted P+ events from the
PWA fit ranges from 330 events to 530 events with typical errors of 280 events. (A change
in \Gammalog(Likelihood) of 0.5 corresponds to one standard deviation.) For all solutions, the
liklihood function gets so bad below 100 events that the P+ wave is clearly required to fit
the data. Thus the observed variation in \Gammalog(Likelihood) further demonstrates that the P+
partial wave is required to describe our data.
C. Systematic Studies
PWA fits have been performed in various regions of jtj. A fit of 20,000 events in the range
0:0 ! jtj ! 0:25 GeV 2 yields results consistent with those of the fit shown. The a 2 (1320)
is still clearly observed in the D+ partial wave as is the structure in the P+ wave. In a fit
of 1000 events in the range 0:95 ! jtj ! 10:0 GeV 2 , no significant structure is seen in any
partial wave other than D+ , in which the a 2 (1320) is again observed. The lack of statistics
at large jtj precludes us from drawing further conclusions from this observation.
A PWA fit has been carried out excluding those events with jcos` GJ j ? 0:8 (the region in
which experimental acceptance is poorest). The results of this fit are consistent with those
shown above.
To test for \Delta and N \Lambda contamination, a fit has been done in which events with M(ú \Gamma p) !
2:0 GeV=c 2 are excluded. Again, this cut is seen to have no qualitative effect on the PWA
results.
Fits were also carried out on Monte Carlo events generated with a pure D+ wave to
11

determine whether structure in the P+ wave could be artificially induced by acceptance
effects, resolution, or statistical fluctuations. Shown in Fig. 22 are the results of such a fit.
We do find that a P+ wave can be induced by such effects. This `leakage' leads to a P+ wave
that: (1) mimics the generated D+ intensity (and in our case would therefore have the shape
of the a 2 (1320)); and (2) has a phase difference \Delta\Phi that is independent of mass. Neither
property is present in our study and we conclude that the P+ structure which we observe is
not due to `leakage'.
Fits have been performed allowing L = 3 and L = 4. We find that these waves are
negligible in the region below M(jú) of about 1.7­1.8 GeV=c 2 .
The data have been fit using different parametrizations of the background. The back­
ground has been set at fixed values determined from the two different background estimates
discussed previously. In another fit, the background has been set to zero. And finally, a
fit was performed allowing the background level to be a free parameter. Although the neg­
ative reflectivity waves do change somewhat for different treatments of the background e ,
the results for the D+ and P+ waves are very stable. In particular, the behavior of the
relative phase between the P+ and the D+ waves does not change substantively in any of
these checks.
D. Comparison with Previous Experiments
These results for the P+ and D+ intensities and their phase difference are quite consistent
with the VES results [12] as can be seen in Fig. 23. In particular, the behavior of the shape
of the phase difference is virtually identical to that reported by VES. f This is particularly
noteworthy since, as will be seen below, it is this phase difference which allows us to draw
e Since the background and the S 0 wave are both isotropic, the fitting program cannot distinguish easily
between them.
f The magnitude of the phase difference is shifted by about 20 ffi relative to that of VES. A produc­
tion phase shift would not be unexpected because of the differing energies and targets in the two
experiments.
12

conclusions regarding the nature of the P+ wave.
Our results are compared with those of the KEK experiment [11] in Fig. 24. In this case,
it is clear that the two results differ. The KEK results have a P­wave intensity which is
narrower (much like the a 2 (1320) shape) and a P­D phase difference which is constant --
results which are similar to those shown in our Monte Carlo study where the P+ wave is not
present in the sample but is induced by ``leakage''.
E. Comparison with j ! ú + ú \Gamma ú 0 Data Sample
A second data set in another topological class (with two additional charged particles in
the final state) has been used to study Reaction (1) with the decay mode j ! ú + ú \Gamma ú 0 .
This data sample has significantly different acceptance than the j ! 2fl sample and the
systematics are generally different as well.
Shown in Fig. 25 is the ú + ú \Gamma ú 0 effective mass distribution from this data set. There is
a clear j peak as well as a strong peak in the ! region. After kinematic fitting, a sample of
2,235 events was obtained which was consistent with Reaction 1 with j ! ú + ú \Gamma ú 0 and on
which an amplitude analysis was carried out. Fig. 26 shows the effective mass distribution
for this sample of events. As expected, the a 2 (1320) dominates this spectrum.
Although the data sample is very small for this topology, we have carried out an amplitude
analysis in order to compare with the primary j ! 2fl analysis. Results of the analysis are
shown in Fig. 27 where we compare the shapes of the P+ intensities for the two data sets
as well as the P+ \Gamma D+ phase differences. Despite the rather large statistical uncertainties,
there is remarkable agreement between these distributions.
V. MASS­DEPENDENT FIT
In an attempt to understand the nature of the P+ wave observed in our experiment,
we have carried out a mass­dependent fit to the results of the mass­independent amplitude
analysis. The fit has been carried out in the jú mass range from 1.1 to 1.6 GeV=c 2 . In this
13

fit, we have assumed that the D+ ­wave and the P+ ­wave decay amplitudes are resonant and
have used relativistic Breit­Wigner forms for these amplitudes.
A. Procedure
We shall use a shorthand notation w to stand for the jú mass, i.e. w = M(jú \Gamma ).
Representing the mass­dependent amplitudes for D+ and P+ as V ` (w) for ` = 2 and 1, we
may write
V ` (w) = e i OE ` \Delta ` (w) B ` (q) [a ` + b ` (w \Gamma w 0
` ) + c ` (w \Gamma w 0
` ) 2 ] 1=2 (2)
where q is the jú breakup momentum at mass w. Here OE ` is the production phase (mass
independent g ), associated with a wave `. The quantities \Delta ` (w) and B ` (q) are the standard
relativistic Breit­Wigner form and the barrier factor, respectively, and are given below. The
square­root factor has been introduced primarily to take into account possible deviations
from the standard Breit­Wigner form, away from the resonance mass (denoted by w 0
` ). The
overall normalization of a wave is governed by a ` , while the constants b ` and c ` allow for
deviations in the mass spectra from the Breit­Wigner form. The constants a ` , b ` and c ` are
all real, so that the square­root factor does not effect the rapidly varying phase implied by
the standard Breit­Wigner form. h
The barrier functions [31], which are real, are given by
g We have tried a linear dependence in mass for the phase; the fits did not require it.
h For a Breit­Wigner form with a constant width, the phase rises 90 degrees over one full width centered
at the resonance mass.
14

B 0 (q) = 1
B 1 (q) =
Ÿ z
z + 1
– 1=2
B 2 (q) =
''
z 2
(z \Gamma 3) 2 + 9z
# 1=2
B 3 (q) =
''
z 3
z(z \Gamma 15) 2 + 9(2z \Gamma 5)
# 1=2
B 4 (q) =
''
z 4
(z 2 \Gamma 45z + 105) 2 + 25z(2z \Gamma 21) 2
# 1=2
(3)
where z = (q=q R
) 2 and q R
= 0:1973 GeV/c corresponding to 1 fermi. Note that B ` (p) ) 1
as q ) 1.
The relativistic Breit­Wigner functions can be written
\Delta ` (w) =
''
\Gamma 0
`
\Gamma ` (w)
#
e i ffi ` (w) sin ffi ` (w) (4)
where \Gamma 0
` is the nominal width (mass independent) and \Gamma ` (w) is the mass­dependent width
given by
\Gamma ` (w) = \Gamma 0
`
/
w 0
`
w
!/
q
q 0
`
! ''
B ` (q)
B ` (q 0
` )
# 2
(5)
where q 0
` is the breakup momentum evaluated at w = w 0
` . The mass­dependent phase shift
ffi ` (w) is given by
cot ffi ` (w) =
''
w 0
`
\Gamma ` (w)
# 2
4 1 \Gamma
/
w
w 0
`
! 2
3
5 (6)
and the overall phase for the `­wave amplitude is
\Phi ` = OE ` + ffi ` (w): (7)
Since we are dealing with two waves P+ and the D+ , we can only measure OE = OE 2 \Gamma OE 1 .
Then the phase difference being measured experimentally, corresponds to
\Delta\Phi = \Phi 2 \Gamma \Phi 1 = OE + ffi 2 (w) \Gamma ffi 1 (w) (8)
15

Finally, the experimental mass distribution for each wave ` is given by
doe `
dw
= jV ` (w)j 2 pq (9)
where pq is the phase­space factor for which p is the breakup momentum in the overall center­
of­mass frame of the jú \Gamma system (or of the final­state proton) in Reaction ((1)). Since the
problem here is for a given
p s, all other relevant factors, including that of the beam flux,
have been obsorbed into the amplitude itself, i.e. the constants a ` , b ` and c ` .
The input quantities to the fit included, in each mass bin: the P+ ­wave intensity; the
D+ ­wave intensity; and the phase difference \Delta\Phi (the relevant formulas are given in (8) and
(9)). Each of these quantities was taken with its error (including correlations) from the
result of the amplitude analysis. One can view this fit as a test of the hypothesis that the
correlation between the fitted P­wave intensity and its phase (as a function of mass) can be
fit with a resonant Breit­Wigner amplitude.
We find that the fit does not improve significantly when the P+ wave is modified from
the Breit­Wigner form, and hence set b 1 and c 1 = 0 for the final fit. We also note that the
magnitude of the quantities b 2 and c 2 in the final fit correspond to a small deviation of the
D+ ­wave intensity of the order of 1%.
B. Results
Results of the fit are shown as the smooth curves in Fig. 17a, b, and c. The mass and
width of the J PC = 2 ++ state (Fig. 17a) are (1317 \Sigma1 \Sigma2) MeV=c 2 and (127 \Sigma2 \Sigma2) MeV=c 2
respectively [32]. (The first error given is statistical and the second is systematic.) The mass
and width of the J PC = 1 \Gamma+ state as shown in Fig. 17b are (1370 \Sigma16 +50
\Gamma30
) MeV=c 2 and (385
\Sigma40 +65
\Gamma105 ) MeV=c 2 respectively. Shown in Fig. 17d are the Breit­Wigner phase dependences
for the a 2 (1320) (line 1) and the P+ waves (line 2); the fitted D+ \GammaP + production phase
difference (line 3); and the fitted D+ \GammaP + phase difference (line 4). (Line 4, which is identical
to the fitted curve shown in Fig. 17c, is obtained as line 1 \Gamma line 2 + line 3.)
16

The systematic errors have been determined from consideration of the range of solutions
possible because of the ambiguous solutions in the PWA. Since there are 8 ambiguous solu­
tions per mass bin and we are fitting over 12 mass bins, it is clearly impossible to try all 8 12
possible combinations. Instead, we have fit some 10 5 combinations where the values to be
fitted in each mass bin have been chosen at random from among the 8 ambiguous PWA solu­
tions. The resulting fits generally clump into a group with reasonable values of ü 2 =dof(Ÿ 2)
and into a group with poor values. The systematic errors on the mass and width given above
are taken from the extremes observed for the solutions with reasonable values of ü 2 =dof .
The central values quoted above are taken from a fit which uses the average values of the
input parameters in each bin.
In order to determine the sensitivity of our result to the exact function being used to fit
the D­wave intensity distribution we have redone the fit using two other hypotheses. First
we have performed a fit in which the mass­dependent amplitude is given by Eq.(2), but with
b 2 = c 2 = 0. Second, we have taken b 2 = c 2 = 0 in Eq.(2) and also replaced the Blatt­
Weisskopf barrier functions for each wave by the factor q ` . Although the resulting fits are
poorer in quality, we find that the parameters of the fit do not change significantly compared
to the systematic uncertainty described above.
The fit to the resonance hypothesis has a ü 2 =dof of 1.49. The fact that the production
phase difference can be fit by a mass­independent constant (of 0.6 rad) is consistent with
Regge­pole phenomenology i in the absence of final­state interactions. If one fits the data to a
non­resonant (constant phase) P+ wave, and also postulates a Gaussian intensity distribution
for the P+ wave, one obtains a very poor fit with a ü 2 =dof of 7.08. Finally if one allows a mass­
dependent production phase, a ü 2 =dof of 1.55 is obtained for the non­resonant hypothesis ---
but the production phase must have a very rapid variation with mass. j Such a phase
variation cannot be excluded, but is not expected for any known model. Note that for this
i The signature factor and the residue functions are at most t­dependent (not mass dependent) (see
ref. [24]).
j The fit requires a linear production phase difference with a slope of ­4.3 rad/GeV.
17

non­resonant hypothesis one must have a separate hypothesis for the observed structure in
the P+ intensity --- a structure which is explained naturally by the resonance hypothesis.
Our fitted parameters are compared with values recently reported by the Crystal Barrel
experiment [13] in Table I. That experiment reports that a J PC = 1 \Gamma+ resonance in the jú
channel is required to fit their data in the annihilation channel pn ! ú \Gamma ú 0 j. Their fitted
parameters are very consistent with those determined from our mass­dependent analysis. The
fact that similar properties are determined from these very different reactions lends further
credence to the hypothesis that the results obtained are due to physical effects rather than
to some artifact of the analysis.
VI. SUMMARY AND CONCLUSIONS
In this paper, we have discussed the details of the amplitude analysis of data from Reac­
tion 1. Interference between D­wave and P­wave amplitudes produced with natural parity
exchange is required in order to explain the data. Using this interference, we have shown
that the P­wave phase has a rapid variation with mass and that this phase variation coupled
with the fitted P­wave intensity distribution is well­fitted by a Breit­Wigner resonance with
mass and width of (1370 \Sigma16 +50
\Gamma30
) MeV=c 2 and (385 \Sigma40 +65
\Gamma105
) MeV=c 2 respectively. We
conclude that there is credible evidence for the production of a J PC = 1 \Gamma+ exotic meson.
ACKNOWLEDGMENTS
We would like to express our deep appreciation to the members of the MPS group.
Without their outstanding efforts, the results presented here could not have been obtained.
We would also like to acknowledge the invaluable assistance of the staffs of the AGS and
BNL, and of the various collaborating institutions. This research was supported in part
by the National Science Foundation, the US Department of Energy, and the Russian State
Committee for Science and Technology.
18

APPENDIX A: PARTIAL­WAVE FORMULAS
In this appendix, the angular distributions are derived for the jú \Gamma system produced in
Reaction (1). The distributions are given both in terms of the moments and the amplitudes
in the reflectivity basis. For a system consisting of S, P and D waves, explicit formulas for
the moments as functions of the partial waves are also given.
In the Gottfried­Jackson (GJ) frame, the amplitudes may be expanded in terms of the
partial waves for the jú \Gamma system:
U
k(\Omega\Gamma =
X
`m
V `mk A `m (A1)
where V `mk stands for the production amplitude for a state j`mi and k represents the spin
degrees of freedom for the initial and final nucleons (k = 1; 2 for spin­flip and spin­nonflip
amplitudes). A `m is the decay amplitude given by
A `m
(\Omega\Gamma =
s
2` + 1
4ú D ` \Lambda
m0 (OE; `; 0) = Y m
`
(\Omega\Gamma (A2)
where the
angles\Omega = (`; OE) describe the direction of the j in the GJ frame. It is noted,
in passing, that the small d­function implicit in (A2) is related to the associated Legendre
polynomial via
d `
m0 (`) = (\Gamma) m
v u u t (` \Gamma m)!
(` +m)! P m
` (cos `) (A3)
The angular distribution is given by
I(\Omega\Gamma =
X
k
jU
k(\Omega\Gamma j 2 (A4)
It should be emphasized that the nucleon helicities are external entities and the summation
on k is applied to the absolute square of the amplitudes. A complete study of the jú \Gamma
system requires four variables: M(jú \Gamma ), \Gammat and the two angles in \Omega\Gamma The distribution (A4)
is therefore to be applied to a given bin of M(jú \Gamma ) and of \Gammat.
19

The angular distribution may be expanded in terms of the moments H(LM) via
I(\Omega\Gamma =
X
LM
` 2L + 1
4ú
'
H(LM)D L \Lambda
M0 (OE; `; 0) (A5)
where
H(LM) =
X
`m
` 0 m 0
/
2` 0 + 1
2` + 1
! 1=2
ae `` 0
mm 0
(` 0 m 0 LM j`m)(` 0 0L0j`0) (A6)
where ae is the spin­density matrix given by
ae `` 0
mm 0
=
X
k
V `mk V \Lambda
` 0 m 0 k (A7)
It is seen that the moments H(LM) are measurable quantities since
H(LM) =
Z
d\Omega I(\Omega\Gamma D L
M0 (OE; `; 0) (A8)
The normalization integral is
H(00) =
Z
d\Omega I(\Omega\Gamma (A9)
The symmetry relations for the H's are well­known. From the hermiticity of ae, one gets
H \Lambda (LM) = (\Gamma) M H(L \Gamma M) (A10)
and, from parity conservation in the production process, one finds
H(LM) = (\Gamma) M H(L \Gamma M) (A11)
These show that the H's are real.
Parity conservation in the production process can be treated with the reflection operator
which preserves all the relevant momenta in the S­matrix and act directly on the rest states
of the particles involved. It is important to remember that the coordinate system is always
20

defined with the y­axis along the production normal. In this case the reflection operator is
simply the parity operator followed by a rotation by ú around the y­axis.
The eigenstates of this reflection operator are
jffl`mi = `(m)
n
j`mi \Gamma ffl(\Gamma) m j` \Gamma mi
o
(A12)
where
`(m) = 1
p
2
; m ? 0
= 1
2 ; m = 0
= 0; m ! 0
(A13)
For positive reflectivity, the m = 0 states are not allowed, i.e.
jffl`0i = 0; if ffl = + (A14)
The reflectivity quantum number ffl has been defined so that it coincides with the naturality
of the exchanged particle in Reaction (1). One can prove this by noting that the meson
production vertex is in reality a time­reversed process in which a state of arbitrary spin­
parity decays into a pion (the beam) and a particle of a given naturality (the exchanged
particle)
J j J ! s js + ú (A15)
where j's stand for intrinsic parities. The helicity­coupling amplitude F J for this decay [33]
is
A J
p (M) / F J
– D J \Lambda
M – (OE p ; ` p ; 0) (A16)
where – is the helicity of the exchanged particle and the subscript p stands for the `produc­
tion' variables. M is the z­component of spin J in a J rest frame. From parity conservation
in the decay, one finds
21

F J
– = \GammaF J
\Gamma–
(A17)
where one has used the relationships j J = (\Gamma) J (true for two­pseudoscalar systems) and
j s = (\Gamma) s (natural­parity exchange). The formula shows that the helicity­coupling amplitude
F J is zero if – is zero. Since angular momentum is conserved, its decay into two spinless
particles in the final state cannot have M = 0 along the beam direction (the GJ rest system),
i.e. the D J ­function is zero unless M = –, if ` p = OE p = 0. Finally, one may identify J with
` and M with m, which proves (A14).
The modified D­functions in the reflectivity basis are given by
ffl D ` \Lambda
m0 (OE; `; 0) = `(m)
h
D ` \Lambda
m0 (OE; `; 0) \Gamma ffl(\Gamma) m D ` \Lambda
\Gammam0 (OE; `; 0)
i
(A18)
It is seen that the modified D­functions are real if ffl = \Gamma1 and imaginary if ffl = +1:
(\Gamma) D ` \Lambda
m0 (OE; `; 0) = 2`(m)d `
m0 (`) cos mOE
(+) D ` \Lambda
m0 (OE; `; 0) = 2i`(m)d `
m0 (`) sin mOE
(A19)
The overall amplitude in the reflectivity basis is now
ffl U
k(\Omega\Gamma =
X
`m
ffl V `mk
ffl A `m (A20)
where
ffl A `m =
s
2` + 1
4ú
ffl D ` \Lambda
m0 (OE; `; 0) (A21)
and the resulting angular distribution is
I(\Omega\Gamma =
X
fflk
j ffl U
k(\Omega\Gamma j 2 (A22)
It is seen that the sum involves four non­interfering terms for ffl = \Sigma and k = 1; 2. The absence
of the interfering terms of different reflectivities is a direct consequence of parity conservation
in the production process. Following convention, we use the notation for partial amplitudes
via
22

[ ` ] 0 = (\Gamma) V `0 ; [ ` ] \Gamma = (\Gamma) V `1 ; [ ` ] + = (+) V `1 (A23)
where [ ` ] stands for the partial waves S, P , D, F and G for ` =0, 1, 2, 3 and 4.
Consider an example where the maximum ` is 2. One sees that there are in general twelve
non­zero experimental moments:
H(00); H(10); H(11); H(20); H(21); H(22)
H(30); H(31); H(32); H(40); H(41); H(42)
(A24)
while the partial waves [ ` ] are, for unnatural­parity exchange,
S 0 ; P 0 ; P \Gamma ; D 0 ; D \Gamma (A25)
and, for natural­parity exchange,
P+ ; D+ (A26)
One wave in each naturality can be set be real (S 0 and P+ , for example), so that there are
again twelve real parameters (to be determined). It is helpful to write down the moments
explicitly in terms of the partial waves:
23

H(00) = S 2
0 + P 2
0 + P 2
\Gamma +D 2
0 +D 2
\Gamma + P 2
+ +D 2
+
H(10) = 1
p
3
S 0 P 0 + 2
p
15
P 0 D 0 + 1
p
5
(P \Gamma D \Gamma + P+D+ )
H(11) = 1
p
6
S 0 P \Gamma + 1
p
10
P 0 D \Gamma \Gamma 1
p
30
P \Gamma D 0
H(20) = 1
p
5
S 0 D 0 + 2
5 P 2
0 \Gamma 1
5 (P 2
\Gamma + P 2
+ ) + 2
7 D 2
0 + 1
7 (D 2
\Gamma +D 2
+ )
H(21) = 1
p
10
S 0 D \Gamma + 1
5
s
3
2 P 0 P \Gamma + 1
7
p
2
D 0 D \Gamma
H(22) = 1
5
s
3
2 (P 2
\Gamma
\Gamma P 2
+ ) + 1
7
s
3
2 (D 2
\Gamma
\Gamma D 2
+ )
H(30) = 3
7
p
5
(
p
3P 0 D 0 \Gamma P \Gamma D \Gamma \Gamma P+D+ )
H(31) = 1
7
s
3
5 (2P 0 D \Gamma +
p
3P \Gamma D 0 )
H(32) = 1
7
s
3
2 (P \Gamma D \Gamma \Gamma P+D+ )
H(40) = 2
7 D 2
0 \Gamma 4
21 (D 2
\Gamma +D 2
+ )
H(41) = 1
7
s
5
3 D 0 D \Gamma
H(42) =
p
10
21 (D 2
\Gamma
\Gamma D 2
+ )
(A27)
APPENDIX B: MAXIMUM­LIKELIHOOD ANALYSIS
This appendix is devoted to an exposition of the experimental moments, the acceptance
moments and the acceptance­corrected (or `true') moments and the relationships among
them. Finally, the extended likelihood functions are given as functions of the `true' and
acceptance moments.
One may determine directly the experimental moments (unnormalized) as follows:
H x (LM) =
n
X
i
D L
M 0 (OE i ; ` i ; 0) (B1)
where the sum is over a given number n of experimental data in a mass bin. But this is
given by, from (A8),
24

H x (LM) =
Z
d\Omega j(\Omega\Gamma
I(\Omega\Gamma D L
M0 (OE; `; 0) (B2)
where
j(\Omega\Gamma represents the finite acceptance of the apparatus, and it includes software cuts,
if any. From (A5), one finds that
H x (LM) =
X
L 0 M 0
H(L 0 M 0 ) \Psi x (LM L 0 M 0 ) (B3)
where
\Psi x (LM L 0 M 0 ) =
/
2L 0 + 1
4ú
! Z
d\Omega j(\Omega\Gamma D L
M0 (OE; `; 0) D L 0 \Lambda
M 0 0 (OE; `; 0) (B4)
Note that the \Psi's have a simple normalization
\Psi x (LM L 0 M 0 ) = ffi LL 0 ffi MM 0
(B5)
in the limit
j(\Omega\Gamma = 1. The integral (B4) can be calculated using a sample of `accepted' MC
events. Let N x be the number of accepted MC events, out of a total of N raw MC events.
Then, the integral is
\Psi x (LM L 0 M 0 ) =
/
2L 0 + 1
4ú
!
1
N
Nx
X
i
D L
M0 (OE i ; ` i ; 0) D L 0 \Lambda
M 0 0 (OE i ; ` i ; 0) (B6)
The equation (B3) shows that one can predict the experimentally measurable moments (B1),
given a set fHg and \Psi's; this provides one a means of assessing the goodness of fit by forming
a ü 2 based on the set fH x g.
There exists an alternative method of determining \Psi's. For the purpose, one expands
the acceptance function
j(\Omega\Gamma in terms of the orthonormal D­functions, as follows:
j(\Omega\Gamma =
X
LM
(2L + 1)¸(LM) D L \Lambda
M0 (OE; `; 0) (B7)
where ¸(LM) is given by
¸(LM) = 1
4ú
Z
d\Omega j(\Omega\Gamma D L
M0 (OE; `; 0) (B8)
25

The complex conjugate is, from the defining formula above,
¸ \Lambda (LM) = (\Gamma) M ¸(L \GammaM ) (B9)
so that the acceptance function can be made explicitly real
j(\Omega\Gamma =
X
LM
(2L + 1)Ü(M)Re
n
¸(LM)D L \Lambda
M0 (OE; `; 0)
o
(B10)
where
Ü(M) = 2; M ? 0;
= 1; M = 0;
= 0; M ! 0
(B11)
One sees that Ü(M) defined in equation (A13) = 4` 2 (M ).
A set of ¸(LM) specifies completely the acceptance in the problem. The normalization
for the acceptance function has be chosen so that a perfect acceptance is given by
j(\Omega\Gamma = 1
and ¸(LM) = ffi L0 ffi M0 . The ¸(LM)'s can be measured experimentally using the accepted MC
events
¸(LM) = 1
4úN
Nx
X
i
D L
M0 (OE i ; ` i ; 0) (B12)
Finally, substituting (B7) into (B4), one finds
\Psi x (LM L 0 M 0 ) =
X
L 00 M 00
(2L 00 + 1)¸ \Lambda (L 00 M 00 )(LML 00 M 00 jL 0 M 0 )(L0L 00 0jL 0 0) (B13)
This formula shows an important aspect of the ¸(LM) technique of representing acceptance.
Although (B8) involves a sum in which L and M could be extended to infinity for an arbitrary
acceptance, there is a cutoff if the set fHg has maxima Lm and Mm [see (B3)]. The formula
above demonstrates that L 00 Ÿ 2Lm and jM 00 j Ÿ 2Mm .
In a partial­wave analysis, it is usually best to take a set of the partial waves, [`] 0 , [`] \Gamma and
[`] + , as unknown parameters to be determined in an extended maximum­likelihood fit. Since
26

there is an absolute scale in an extended maximum­likelihood fit, one has then in possession
the predicted numbers of events for all the partial waves, corrected for finite acceptance and
angular distributions. The partial waves in turn give rise to a set of predicted moments fHg.
But the H(00) is not 1 but the total predicted number of events from the fit, i.e. one should
be using the unnormalized moments. It is possible to choose H's as unknowns in the fit, but
the two sets of H's should be the same ideally---this affords one an effective way of assessing
self­consistency between the chosen moments and the partial waves.
For completeness, a short comment is given of the extended likelihood functions. The
likelihood function for finding `n' events of a given bin with a finite acceptance
j(\Omega\Gamma is defined
as a product of the probabilities,
L /
Ÿ ¯
n n
n!
e \Gamma ¯ n
– n
Y
i
''
I(\Omega i )
R
I(\Omega\Gamma j(\Omega\Gamma
d\Omega
#
(B14)
where the first bracket is the Poisson probability for `n' events. This is the so­called extended
likelihood function, in the sense that the Poisson distribution for `n' itself is included in the
likelihood function. Note that the expectation value ¯
n for n is given by
¯ n /
Z
I(\Omega\Gamma
j(\Omega\Gamma
d\Omega (B15)
The likelihood function L can now be written, dropping the factors depending on n alone,
L /
'' n
Y
i
I(\Omega i )
#
exp
Ÿ
\Gamma
Z
I(\Omega\Gamma j(\Omega\Gamma
d\Omega
–
The `log' of the likelihood function now has the form,
lnL /
n
X
i
lnI(\Omega i ) \Gamma
Z
d\Omega j(\Omega\Gamma I(\Omega\Gamma (B16)
which can be recast in terms of the ¸(LM)'s
lnL /
n
X
i
lnI(\Omega i ) \Gamma
X
LM
(2L + 1) H(LM) ¸ \Lambda (LM)
/
n
X
i
lnI(\Omega i ) \Gamma X
LM
(2L + 1) Ü(M)H(LM)Re¸(LM)
(B17)
27

H(LM)'s may be used directly as parameters in the fit or may be given as functions of the
partial waves. It is interesting to note that the ¸(LM)'s for L ? Lm and jM j ? Mm are not
needed in the likelihood fit. Note also that only the real part of the ¸(LM)'s are used in the
fit.
It should be borne in mind that a set of the moments fHg may not always be expressed
in terms of the partial waves. This is clear if one examines the formulas (A27). Consider,
for example, an angular distribution in which H(10) is the only non­zero moment. But this
moment is given by a set of interference terms involving even­odd partial waves. So at least
one term cannot be zero---for example, the interference term involving S­ and P­waves. But
then neither H(00) nor H(20) can be zero, since both S­ and P­waves are non­zero. One
must conclude then that a ü 2 based on the set fH x g may not necessarily be zero identically.
APPENDIX C: RANK OF THE DENSITY MATRIX
An assumption needed for the partial­wave analysis is that the density matrix has rank 1,
i.e. the spin amplitudes do not depend on the nucleon helicities. Our justification, so far, has
been that the fitted partial waves are very reasonable, that these waves can be fitted with
a very simple mass­dependent formula, that Pomeron­exchange amplitudes are in general
independent of nucleon helicities, and so on: : : .
The purpose of this Appendix is to point out that, under a simple model for mass
dependence of the partial waves, it is possible to prove that the spin density matrix has
rank 1. Suppose that one has found a satisfactory fit under a rank­1 assumption. One can
then show that, even if the problem involves both spin­nonflip and spin­flip at the nucleon
vertex---i.e. it appears to be a rank­2 problem---the spin density matrix in reality has a rank
of 1. Although this note is based on the results of our jú \Gamma analysis, the derivation does not
depend on the decay channels; the conclusions apply equally well to any decay channel, e.g.
the 1 \Gamma+ state at 1.6 GeV coupling to úae (a further discussion on this point is given at the
end of Section 3).
28

This note relies on some technicalities generally well known, and so they have been
presented without attribution. The reader may wish to consult a number of preprints and/or
papers, which deal with them in some detail [26,28,33--35].
1. Partial Waves Produced via Natural­parity Exchange
Consider our jú \Gamma system produced via natural­parity exchange. It consists of just two
waves D+ and P+ in the a 2 (1320) region. Without loss of generality, the decay amplitudes
[27] can be considered real, i.e.
A D
(\Omega\Gamma =
s
5
4ú
p
2 d 2
10 (`) sin OE = \Gamma
s
5
4ú
p
3 sin ` cos ` sin OE
A P
(\Omega\Gamma =
s
3
4ú
p
2 d 1
10 (`) sin OE = \Gamma
s
3
4ú sin ` sin OE
(C1)
Since one deals with the partial waves produced only by natural­parity exchange, one can
drop the subscript `+' from the waves, and the angular distribution is simply given by
I(\Omega\Gamma / jD A D
(\Omega\Gamma + P A
P(\Omega\Gamma
j 2
/
` 3
4ú
' fi fi fi
p
5D cos ` + P
fi fi fi 2
sin 2 ` sin 2 OE
/
` 3
4ú
' h
5jDj 2 cos 2 ` + 2
p
5 !fD \Lambda Pg cos ` + jP j 2
i
sin 2 ` sin 2 OE
(C2)
The integration over the angles can be carried out easily, to obtain
Z
I(\Omega\Gamma
d\Omega / jDj 2 + jP j 2 (C3)
as expected.
The spin density matrix is given by
I(\Omega\Gamma / jD A
D(\Omega\Gamma
+ P A
P(\Omega\Gamma
j 2 =
X
k;k 0
ae k;k 0 A k A \Lambda
k 0 (C4)
where fk; k 0 g = f1; 2g and `1' (`2') corresponds to D (P ). From this definition, one sees that
ae =
0
@ jDj 2 D P \Lambda
D \Lambda P jP j 2
1
A (C5)
29

One can work out the eigenvalues of this 2 \Theta 2 matrix:
– = fjDj 2 + jP j 2 ; 0g (C6)
One of the two allowed eigenvalues is zero, i.e. the rank of this matrix is 1. This is the
`rank­1' assumption one makes to carry out the partial­wave analysis and is valid for a given
mass bin.
Suppose now that the rank is 2, i.e.
I(\Omega\Gamma / jD 1 A
D(\Omega\Gamma + P 1 A
P(\Omega\Gamma
j 2 + jD 2 A
D(\Omega\Gamma + P 2 A
P(\Omega\Gamma
j 2 (C7)
where subscripts 1 and 2 stand for spin­nonflip and spin­flip amplitudes at the nucleon vertex
for reaction (1). Comparing (C2) and (C7), one finds immediately
jDj 2 = jD 1 j 2 + jD 2 j 2
jP j 2 = jP 1 j 2 + jP 2 j 2
!fP \Lambda Dg = !fP \Lambda
1 D 1 g + !fP \Lambda
2 D 2 g
(C8)
Let w be the effective mass of the jú \Gamma system. If the mass dependence is included
explicitly in the formula, one should write, in the case of rank 1,
doe(w; \Omega\Gamma
dwd\Omega
/ jD(w) A
D(\Omega\Gamma + P (w) A
P(\Omega\Gamma
j 2 pq (C9)
where p is the breakup momentum of the jú \Gamma system in the overall CM system and q is
the breakup momentum of j in the jú \Gamma rest frame. Note that both p and q depend on w.
Note also that the w dependence of the partial waves D and P are given in the formula.
Obviously, a similar expression could be written down for the case of rank 2.
One is now ready to make the one crucial assumption for a mass­dependent analysis of
the D and P waves: one assumes that two resonances---in D and P waves, respectively---are
produced in both spin­nonflip and spin­flip amplitudes. One may then write, for the rank­1
case,
30

D(w) = a e i ff e i ffi a sin ffi a
P (w) = b e i ffi b sin ffi b
(C10)
where a, b and the production phase ff are all real and independent of the jú \Gamma mass. In
addition, one can set a – 0 and b – 0 without loss of generality. ffi a and ffi b are the phase­
shifts corresponding to the resonances and are highly mass dependent. In its generic form,
the Breit­Wigner formula is given by the usual expression
cot ffi = w 2
0 \Gamma w 2
w 0 \Gamma 0
(C11)
where w 0 and \Gamma 0 are the standard resonance parameters. In this note, the width is considered
independent of w. Likewise, the barrier factor dependence for D and P is ignored. k
The formulas (C10) are generalized to the case of rank 2, as follows:
D 1 (w) = a 1 e i ff 1 e i ffi a sin ffi a
P 1 (w) = b 1 e i ffi b sin ffi b
D 2 (w) = a 2 e i ff 2 e i ffi a sin ffi a
P 2 (w) = b 2 e i ffi b sin ffi b
(C12)
Once again, a i , b i and ff i are real, a i – 0 and b i – 0, and independent of w. One finds, using
(C8),
a 2 = a 2
1 + a 2
2
b 2 = b 2
1 + b 2
2
ab cos(ff + ffi a \Gamma ffi b ) = a 1 b 1 cos(ff 1 + ffi a \Gamma ffi b ) + a 2 b 2 cos(ff 2 + ffi a \Gamma ffi b )
(C13)
k Although simplified formulas are used in this note, the results given here do not change even when
correct formulas are used. Note that, to go over to a correct formulation for each wave, one needs to
substitute the absolute value of the Breit­Wigner formula as follows:
sin ffi(w) ! B(q)
Ÿ \Gamma 0
\Gamma(w)
–
sin ffi(w)
where B(q) is the barrier factor and \Gamma(w) is the mass­dependent width. It should be noted that the
correction factors are all real, by definition.
31

A plot of cos(ff + ffi a \Gamma ffi b ) as a function of w is shown in Fig. 28 for three values of ff, i.e.
0 ffi , 45 ffi and 90 ffi . The resonance parameters for a and b of 1.0 and 0.151 are taken from the
mass dependent fit of Section VB as are the resonant masses and widths. l The normalized
absolute squares of the Breit­Wigner forms are given in Fig. 29, as is the `normalized' inter­
ference term. The same quantities, as they appear in Ref. [1], are shown in Fig. 30. This
figure shows how important the interference term is compared to the P­wave term. Note
also how rapidly the interference term varies as a function of w in the a 2 (1320) region. This
term, of course, is intimately related to the asymmetry in the Jackson angle and vanishes
when integrated over the angle, i.e. it does not contribute to the mass spectrum [see (C2)
and (C3)]. Fig. 31 shows the contour plot of the intensity distribution in w vs. cos `; note
the variation of the asymmetry as a function w.
For the last equation in (C13) to be true for any mass, the coefficient of cos(ffi a \Gamma ffi b ) or
sin(ffi a \Gamma ffi b ) on the left­hand side must be equal to that on the right­hand side, so that
ab cos ff = a 1 b 1 cos ff 1 + a 2 b 2 cos ff 2
ab sin ff = a 1 b 1 sin ff 1 + a 2 b 2 sin ff 2
(C14)
Taking the sum of the squares of the two formulas above and introducing the first two
equations of (C13), one obtains:
2a 1 b 1 a 2 b 2 cos ff 1 cos ff 2 + 2a 1 b 1 a 2 b 2 sin ff 1 sin ff 2
=a 2
1 b 2
2 + a 2
2 b 2
1
=a 2
1 b 2
2 (cos 2 ff 1 + sin 2 ff 1 ) + a 2
2 b 2
1 (cos 2 ff 2 + sin 2 ff 2 )
(C15)
which is recast into
0 = (a 1 b 2 cos ff 1 \Gamma a 2 b 1 cos ff 2 ) 2 + (a 1 b 2 sin ff 1 \Gamma a 2 b 1 sin ff 2 ) 2 (C16)
It is clear that each term must be set to zero, so that
l The value of ff as determined from this fit is 37:46 ffi ; for the purpose of illustration, one may consider
ff = 45 ffi close enough.
32

` a 1
b 1
'
cos ff 1 =
` a 2
b 2
'
cos ff 2
` a 1
b 1
'
sin ff 1 =
` a 2
b 2
'
sin ff 2
(C17)
Placing these back into (C14), one deduces that
` a
b
'
cos ff =
` a 1
b 1
'
cos ff 1 =
` a 2
b 2
'
cos ff 2
` a
b
'
sin ff =
` a 1
b 1
'
sin ff 1 =
` a 2
b 2
'
sin ff 2
(C18)
One may take---alternately---the sum of the squares of the two formulas above, or a division
of the second by the first, and obtain (remembering that the a's and b's are non­negative
real quantites),
a
b
= a 1
b 1
= a 2
b 2
tan ff = tan ff 1 = tan ff 2
(C19)
The last equation above demands that ff 1 and ff 2 are determined (up to \Sigmaú), but they have
to satisfy (C18). It is therefore clear that one must set ff = ff 1 = ff 2 . Next, one introduces
two new real variables x – 0 and y – 0, given by
x = a 1
a
= b 1
b
y = a 2
a
= b 2
b
(C20)
with the constraint x 2 + y 2 = 1.
Now one can prove that the case of rank 2 is reduced to that of rank 1. Indeed, one sees
immediately that
0
@ D 1
P 1
1
A = x
0
@ D
P
1
A and
0
@ D 2
P 2
1
A = y
0
@ D
P
1
A (C21)
and (C7) becomes identical to (C2).
33

2. Discussion
It is shown in this Appendix that the problem of two resonances in D+ and P+ in the jú \Gamma
system in (1) is---effectively---a rank­1 problem. For this to be true, the following conditions
have to be met:
(a) There exist two distinct resonances with different masses and/or widths. Note that the
crucial step, from (C13) to (C14), depends on that fact that ffi a \Gamma ffi b is non­zero and is
mass dependent.
(b) There exists a satisfactory rank­1 fit with two resonances in a given mass region, in
which each amplitude for D+ or P+ has the following general form
M k (w; \Omega\Gamma = r k e i ff k e i ffi k (w) f k (w) A
k(\Omega\Gamma (C22)
where k = f1; 2g and `1' (`2') corresponds to D+ (P+ ). ffi k (w) is the Breit­Wigner
phase and highly mass dependent, while r k and ff k are mass independent in the fit.
Of course, one of the two ff k 's can be set to zero without loss of generality, so that
there are three independent parameters, e.g. r 1 , r 2 and ff 1 (these were denoted a, b
and ff, respectively, in the previous section). f k (w) contains the absolute value of the
Breit­Wigner form, plus any other mass­dependent factors introduced in the model.
A
k(\Omega\Gamma carries the information about the rotational property of a partial wave k.
(c) The same two D+ and P+ resonances are produced in both spin­nonflip and spin­flip
amplitudes, with the same general form as given above---but with arbitrary r k 's and
ff k 's for each spin­nonflip and spin­flip amplitude. It has been shown in this appendix
that only one set of r k 's and ff k 's, i.e. r 1 , r 2 and ff 1 , is required for both spin­nonflip and
spin­flip amplitudes. (This is indeed a remarkable result; the rank­2 problem entails a
set of six parameters, but it has been shown that the set is reduced to that consisting
of just three.) Therefore, the distribution function in both w
and\Omega is given by
34

doe(w; \Omega\Gamma
dwd\Omega
/ j
X
k
M k (w; \Omega\Gamma j 2 pq (C23)
independent of the nucleon helicities.
In another words, the spin density matrix has rank 1. The key ingredients for this result
are that both spin­nonflip and spin­flip amplitudes harbor two resonances in D+ and P+ and
that the production phase is mass­independent. It should be emphasized that the derivation
given in this note does not depend on the existence of a good mass fit; it merely states that
any fit with a mass­independent production phase is necessarily a rank­1 fit. Of course, the
point is moot, if there exists no satisfactory fit in this model.
35

REFERENCES
\Lambda Present address: Rensselaer Polytechnic Institute, Troy, NY 12180, USA.
[1] D.R. Thompson et al., Phys. Rev. Lett. 79, 1630 (1997).
[2] N. Isgur and J. Paton, Phys. Rev. D 31, 2910 (1985).
[3] M. Chanowitz and S. Sharpe, Nucl. Phys. B 222, 211 (1983).
[4] T. Barnes et al., Nucl. Phys. B 224, 241 (1983).
[5] F.E. Close and P.R. Page, Nucl. Phys. B 443, 233 (1995).
[6] T. Barnes et al., Phys. Rev. D 52, 5242 (1995).
[7] R.L. Jaffe, Phys. Rev. D 15, 267 (1977).
[8] Y. Uehara et al., Nucl. Phys. A 606, 357 (1996).
[9] P. Lacock, C. Michael, P. Boyle and P. Rowland, Phys. Rev. D 54, 6997 (1996); C.
Bernard et al., Nucl. Phys. B (Proc. Suppl.) 53, 228 (1997).
[10] D. Alde et al., Phys. Lett. B 205, 397 (1988).
[11] H. Aoyagi et al., Phys. Lett. B 314, 246 (1993).
[12] G.M. Beladidze et al., Phys. Lett. B 313, 276 (1993).
[13] W. D¨unnweber et al., ``Exotic jú State in pn and pp Annihilation at Rest into júú'',
presented at the Hadron97 Conference, 1997 (to be published); A. Abele et al., ``Exotic
jú State in pd Annihilation at Rest into ú \Gamma ú 0 jp spectator '', submitted to Phys. Lett. B.
[14] Y.D. Prokoshkin and S.A. Sadovsky, Physics of Atomic Nuclei 58, 606 (1995).
[15] S. Ozaki, ``Abbreviated Description of the MPS'', Brookhaven MPS note 40, unpublished
(1978).
[16] Z. Bar­Yam et al., Nucl. Instr. & Meth. A 386, 253 (1997).
36

[17] T. Adams et al., Nucl. Instr. & Meth. A 368, 617 (1996).
[18] S.E. Eiseman et al., Nucl. Instr. & Meth. 217, 140 (1983).
[19] R.R. Crittenden et al., Nucl. Instr. & Meth. A 387, 377 (1997).
[20] S. Teige et al., Proceedings of the Fifth International Conference on Calorimetry in High
Energy Physics, eds. Howard A. Gordon and Doris Rueger (World Scientific, Signapore,
1995) 161. See also S. Teige et al., ``Properties of the a 0 Meson'', submitted to Phys.
Rev.
[21] O.I. Dahl et al., ``SQUAW kinematic fitting program'', Univ. of California, Berkeley
Group A programming note P­126, unpublished (1968).
[22] J. Friedman, ``SAGE, A General System for Monte Carlo Event Generation with Pre­
ferred Phase Space Density Distributions'', Univ. of California, Berkeley Group A pro­
gramming note P­189, unpublished (1971).
[23] ``GEANT Detector Description and Simulation Tool'', CERN program Library Long
Writeups Q123, unpublished (1993).
[24] A.C. Irving and R.P. Worden, Phys. Rep. 34, 117 (1977).
[25] E.J. Sacharidis, Lett. Nuovo Cimento 25, 193 (1979).
[26] S.U. Chung, ``Formulas for Partial­Wave Analysis'', Brookhaven BNL­QGS­93­05, un­
published (1993).
[27] S.U. Chung, ``Amplitude Analysis for Two­pseudoscalar Systems'', Brookhaven BNL­
QGS­97­041 (1997); to be published in Phys. Rev. D.
[28] The naturality of the exchanged particle is given by the reflectivity of the wave. See S.U.
Chung and T.L. Trueman, Phys. Rev. D 11, 633 (1975).
[29] S.A. Sadovsky, ``On the Ambiguities in the Partial­Wave Analysis of ú \Gamma p ! jú 0 n Re­
action'', Inst. for High Energy Physics IHEP­91­75, unpublished (1991).
37

[30] E. Barrelet, Nuovo Cimento A 8, 331 (1972).
[31] F. v. Hippel and C. Quigg, Phys. Rev.5, 624 (1972).
[32] This mass and width are consistent with accepted values for the a 2 (1320) (R.M. Barnett
et al., Phys. Rev. D 54, 1 (1996)) when our resolution is taken into account.
[33] S. U. Chung, `Spin Formalisms,' CERN Yellow Report CERN 71­8 (1971).
[34] S. U. Chung, Phys. Rev. D 48, 1225 (1993).
[35] S. U. Chung et al., Annalen der Physik 4, 404 (1995).
38

FIGURES
FIG. 1. Experimental layout for E852.
FIG. 2. Average acceptance vs. effective mass.
FIG. 3. Acceptance vs. cos` for different effective mass regions.
FIG. 4. Acceptance vs. OE for different effective mass regions.
FIG. 5. Acceptance vs. jt 0 j.
FIG. 6. Two­photon effective mass distribution.
FIG. 7. Missing­massed squared distribution.
FIG. 8. Missing­mass squared vs. two­photon mass.
FIG. 9. Effective­mass distribution of the background.
FIG. 10. Angular distribution of the background.
FIG. 11. jú \Gamma effective mass distribution.
FIG. 12. Distribution of jt 0 j = jt \Gamma t min j, where t is the four­momentum transfer.
FIG. 13. Distributions of the cosine of the decay angle in the GJ frame for various effective
mass selections.
39

FIG. 14. Forward­backward asymmetry as a function of effective mass. The asymmetry
= (F \Gamma B)=(F + B) where F(B) is the number of events for which the j decays in the forward
(backward) hemisphere in the GJ frame.
FIG. 15. Distributions of the Trieman­Yang angle OE in the GJ frame for various effective mass
selections.
FIG. 16. Effective mass distributions for: a.) the ú \Gamma p; and the b.) jp systems.
FIG. 17. Results of the partial wave amplitude analysis. Shown are a) the fitted intensity
distributions for the D+ and b) the P+ partial waves, and c) their phase difference \Delta\Phi . The
range of values for the eight ambiguous solutions is shown by the central bar and the extent of the
maximum error is shown by the error bars. Also shown as curves in a), b), and c) are the results of
the mass dependent analysis described in the text. The lines in d) correspond to (1) the fitted D+
Breit­Wigner phase, (2) the fitted P+ Breit­Wigner phase, (3) the fitted relative production phase
OE, and (4) the overall phase difference \Delta\Phi.
FIG. 18. Results of the partial wave amplitude analysis. Shown are the fitted intensity distri­
butions for the waves produced by unnatural­parity exchange.
FIG. 19. Forward­backward asymmetry as a function of effective mass. Shown are: the total
asymmetry in the data (closed circles); the predicted asymmetry from the PWA fit (open squares);
the prediction of the fit for that part of the asymmetry due to natural­parity exchange (filled
squares); and the prediction of the fit for that part of the asymmetry due to the unnatural­parity
exchange waves (open circles).
FIG. 20. Experimental moments (open circles) shown with the predicted moments (open trian­
gles) from the amplitude analysis.
FIG. 21. Value of the log likelihood as a function of the number of P+ events in the PWA fit
for all 8 ambiguous solutions.
40

FIG. 22. Fitted P+ intensity and P+ \GammaD + phase difference for the Monte Carlo sample generated
with a pure D+ sample of a 2 (1320) events.
FIG. 23. Comparison of the results of this amplitude analysis with the VES experiment.
FIG. 24. Comparison of the results of this amplitude analysis with those of the KEK experiment.
FIG. 25. The ú + ú \Gamma ú 0 effective mass distribution for events with the topology of three forward
charged tracks, one recoil charged track, and two photon clusters consistent with a ú 0 .
FIG. 26. The jú \Gamma effective mass distribution for the j ! ú + ú \Gamma ú 0 event sample.
FIG. 27. Comparison of the results of the amplitude analysis for the j ! ú + ú \Gamma ú 0 and the
j ! 2fl samples.
FIG. 28. cos(ff + ffi a \Gamma ffi b ) as a function of w from 1.2 to 1.6 GeV for ff = 0 ffi (\Pi), ff = 45 ffi (+) and
ff = 90 ffi ( ).
FIG. 29. sin 2 ffi a (\Pi), sin 2 ffi a (+) and sin ffi a sin ffi b cos(ff + ffi a \Gamma ffi b ) ( ) as a function of w from 1.2
to 1.6 GeV, using ff = 45 ffi .
FIG. 30. a 2 sin 2 ffi a (\Pi), b 2 sin 2 ffi a (+) and 2 a b sin ffi a sin ffi b cos(ff + ffi a \Gamma ffi b ) ( ) as a function of
w from 1.2 to 1.6 GeV, where one has assumed that a = 1:0, b = 0:20 and ff = 45 ffi .
FIG. 31. Angular distribution in cos ` as a function of w from 1.2 to 1.6 GeV, where one has
assumed that a = 1:0, b = 0:151 and ff = 37:46 ffi .
41

TABLES
TABLE I. Comparison of the Results of E852 and the Crystal Barrel
Mass MeV=c 2 Width MeV=c 2
E852 1370 \Sigma16 +50
\Gamma30
385 \Sigma40 +65
\Gamma105
Crystal Barrel 1400 \Sigma20 \Sigma20 310 \Sigma50 +50
\Gamma30
42