Analysis of π0 photoproduction at intermediate and high energies

Nuclear Physics B79 (1974) 431-460. North-Holland Publishing Company

ANALYSIS OF n o PHOTOPRODUCTION INTERMEDIATE

AT

AND HIGH ENERGIES

I.S. BARKER Daresbury Laboratory A. DONNACHIE and J.K. STORROW

University of Manchester and Daresbury Laboratory Received 4 March 1974 (Revised 28 May 1974) Abstract: An amplitude analysis ofn ° photoproduction is carried out in the t range -0.2 (GeV/c)~ /> t ~ -0.7 (GeV/c)2 using finite energy sum rules and fixed t dispersion relations and fitting all available data from 3 GeV upwards. All four FESRs are fitted in contrast to previous work where it was assumed that one of the unnatural parity amplitudes (/74) is zero and its FESR was ignored. Using the evidence of the' relevant FESR we find that it is not negligible, particularly at lower energies, but is found to have the energy dependence typical of an amplitude corresponding to exotic exchange, i.e. very steep energy dependence at the lower energies flattening out at higher energies. The fit is extrapolated below 3.0 GeV and experimental predictions given. In particular, predictions for possible double polarisation measurements are discussed. The amplitudes at 4 GeV are presented and they show that the dual absorption model gives the correct zero structure for the imaginary parts of spin-flip amplitudes but not for the no-flip amplitudes.

1. Introduction A number of single polarisation measurements of neutral pion photoproduction have become available in the last two years and it is possible that double polarisation measurements will shortly be undertaken. Thus at the m o m e n t we have data on four of the seven measurements needed to determine the amplitudes for the process, up to an overall phase which cannot be measured directly, and certain sign ambiguities, which require three further measurements [1 ]. While it is clearly not possible to carry out a model independent amplitude analysis as yet, it is possible to supplement these data with information provided by multipole analysis of lowenergy data via finite energy sum rules (FESRs) and/or fixed ~ dispersion relations (FTDRs). Various analyses [ 2 - 5 ] along these lines have been made. The analysis presented here is the first to use FTDRs to calculate the real parts of amplitudes and to demand a fit to the FESRs. Our analysis also differs from all previous ones

432

LS. Barker et al., 7r° photoproduction

in that we do not constrain the CGLN [6] amplitude A 3 to be zero. This amplitude corresponds to the exchange of the quantum numbers P ( - 1 ) J = C ( - 1 ) 1 = - 1 , which is a quark model second class exotic. Pomeron-Regge cuts produced (in an eikonal or absorptive picture) using an s-channel helicity conserving pomeron do not contribute to A 3, so that orthodox Regge theory predicts A 3 0. This is compatible with the existing high energy data, although not demanded by it. However the FESR for this amplitude implies that it is non-zero and for this reason we believe that it is essential to include A 3 in any analysis. We find that it makes a significant difference below 4 GeV. We also carried out fits with A 3 = 0 (not fitting the A 3 FESR, of course) for purposes of comparison. We fit rr° photoproduction data only, thus making no isospin decomposition in the t-channel, and restrict ourselves to the t-range 0.2 (GeV/c) 2 ~< - t ~< 0.7 (GeV/c) 2 since it is in this t range that good data for differential cross sections, polarised target asymmetry, recoil polarisation and polarised beam asymmetry exist. We fit at various values of t in this range independently and then smooth in t. It is intended to use this analysis as a basis of t-dependent fit over a larger t range in due course. For the time being we prefer to present the less model-dependent analysis alone. In sect. 2 we describe our method, giving the necessary kinematics, a detailed discussion on the use of FESRs plus FTDRs and their evaluation, the parametrisation we use, including the assumed trajectory functions, and a general discussion of the high energy data. In sect. 3 we discuss details of the fits, in sect. 4 we give our results and some predictions and in sect. 5 we present our conclusions. Cross-section and polarisation formulae are given in the appendix. =

2. Method 2.1. K i n e m a t i c s

We use the parity-conserving t-channel helicity amplitudes F i (i = 1...4) which are related to the usual CGLN [6] invariant amplitudes A i (i = 1 ...4) by FI =-AI

+ 2mA4

F 3 = 2mA 1 - tA 4 ,

,

F2 =AI + tA2 , F 4 =A 3 .

(2.1)

The amplitudes F i are convenient because of their simple crossing properties (F 4 crossing odd, F 1, F 2 and F 3 crossing even) and also because of their t-channel properties. F 1 and F 2 are respectively natural and unnatural parity t-channel amplitudes to all orders in s, and F 3 and F 4 respectively natural and unnatural parity tchannel amplitudes to leading order in s. We work in terms of the crossing anti-symmetric variable v-

s -

u

4m"

(2.2)

1.S. Barker et al., ~r° photoproduction

433

For some purposes it is convenient to think in terms of the s-channel helicity amplitudes which we label S1, $2, N and D * using an obvious modification of Wiik's notation [7]: In numerical work we use the exact crossing matrix** but the asymptotic crossing relation is useful in qualitative considerations of absorption and we give it here. "Fll

~2m

~

- V/Z-'{

2m-Sli

F2[ =-4__~ 0

x/S?

~

0

F3

t

2mx/-L-T -2m~/-L-T t

1

0

I

'

]

N[

I

~

(2.3)

D I I

F 4

0

-1

iS2]

We see immediately that F 1 and F 4 are mainly s-channel helicity flip and F 2 and F 3 non-flip.

2.2. Parametrisation For v > vc (which we take as corresponding to E,y = 1.5 GeV) we parametrise the imaginary part of the amplitudes as a sum of Regge terms i.e. Im Fi(v~,t) = ~ [J/ivaj-1 . 1

(2.4)

We then calculate Re Fi(v, t) by a FTDR, using a low-energy multipole analysis for v < vc, and fit the resulting Fi(v, t) to the data and the FESRs. F o r F 1 a n d F 3 we allow the w pole*** and w cut and f o r F 2 the B pole and o cut.

a 1 =a w = a

, .-

2

tt-mto) ~+1,

2~ a 2 = a B=a'tt" - m § ) + l ,

a 3 _- a w e = 3I a , t + 1 - - a , m t2o .

(2.5)

We should also of course include the B cut which has as its trajectory i

~

t

2

aBc =-~a t + 1 - - a m t o . However, as we cannot hope to separate B cut from other low-lying effects such as Regge-Regge cuts this is only one of the possibilities we try for a 4. 1 t . We have assumed a pomeron slope of ~-a in calculating the slopes of the cuts and

* $1, $2 are single flip amplitudes, N a non-flip and D a double-flip amplitude. ** Which can be obtained by combining the formulae of ref. [8] with eq. (2.1). *** Any p contribution is included in the to contribution.

LS. Barker et al., ~r° photoproduction

434

tried various values of ~' between 0.9 and 1.0 GeV/c. The trajectories are held fixed in fitting since otherwise the procedure is rather unstable. It might be objected that the parametrisation of the Regge cut part should include powers of logarithms, i.e. v a-1 [In (v)]7 which would be present in realistic cut models. However, since 3' is unknown and the energy range is limited we do not feel that it is sensible to include 3' as a parameter. We therefore ignore this possibility. This enables us to evaluate the high energy integrals analytically. Further the main effect of these terms would be to break the normal phase energy relation and we do not satisfy this anyway as our amplitudes are a sum of several Regge terms. For the exotic amplitude F 4 we considered non-leading co i.e. (ctw - 1) and pomeron-B cut. These did not prove satisfactory and we allowed two unknown trajectories a 5 and ot6 to contribute, trying many possible values. We also experienced difficulty in simultaneously fitting the F 4 FESR and having Im F 4 continuous at v = vc so we modified the parametrisation of Im F 4 even further. However, this exercise was more in the nature of verifying that we could achieve continuity in v at v = v c without affecting either our fits or the extrapolation down to E~t = 2.5 GeV, than trying to find a more realistic parametrisation of the amplitude. For the other amplitudes we take the cut trajectories because they have some theoretical basis. Since experimentally 0tef f = (0.19 +-0.03) + (0.27 +--0.04)t we clearly need terms in addition to the poles in order to reduce the effective intercept and flatten the effective slope.

2.3. FESRs and FTDRs 2. 3.1. Theory We calculated the high energy real parts using FTDRs dividing the integral into two parts v < v c and v > vc. For v < vc we used multipole analyses to evaluate the integral (see subsect. 2.3.3) and for v > vc we substituted Im Fi(v, t) from eq. (2.4). We obtained, for a crossing even amplitude (i.e. F 1, F 2, F3). Re

c v' Im Fi(v'' t) 2 ; v'aJ BiVB F i(v, t) - - +2 . d r ' + ~ ~ji P v,f---p2 dv' ' rr ,) v'2 _ v2 • v2 _ v 2 vo

I

Vc

(2.6)

and, for the crossing odd amplitude (i.e. F4)

ReFi(v't)-v2

Biv -

v2

+2v f c lmFi(v"t) dv ' +2v -v-~---~ rr vo

-

~ji P fvc v'cq-l.S-~ ~ v - - _ v 2 dr'" (2.7)

The high-energy integration can be carried out explicitly and written in terms of a hypergeometric function. The coefficients of the Born terms, B i can be expressed in terms of e 2, g2 and the anomalous magnetic moment of the proton. The amplitudes F i are then fitted to the high-energy data and the FESRs. The FESRs can be written as

I.S. Barker et aL, ~o photoproduetion v

2 fc UBBi + ~ 3

.

u'

435

ai+l

, 2 PIiPc ~ Im Fi(u, t) du' = 7r - ~/ a / + l

(2.8)

~0 for a crossing-even amplitude and I/

C

2 f Im Fi(~,', t) du ' = 2 ~ ~/iUc~ Bi + ~ UO 7r j aj

(2.9)

for a crossing-odd amplitude. We do not use higher moment FESRs as they would give too much weight to the part of the integral just below u = uc where the multipoles are rather uncertain. Our approach is somewhat unusual in that we use both FESRs and FTDRs despite the fact that they incorporate similar assumptions i.e. artalyticity and high-energy behaviour. Our reasoning is as follows. An approach which only uses FTDRs tends in practice, to produce poor fits to the FESRs because the phase-energy relation is broken by violating the FESR. As an example consider a crossing odd amplitude with only one Regge term i.e. Im F i =/3u ~-1 for v > uc. Then

'[-

Re F i(u, t) =-~

Bi-

Ii' vo

,

Im F i ( v , t) du'

~ ~ J (2.10)

The first term on the right-hand side is a fixed pole at J = 0. Its residue i.e. the coefficient of u- 1 , is zero if the FESR is satisfied, the coefficient of u-3 vanishes if the second moment FESR is satisfied and so on. Thus breaking of the phase energy relation can be achieved by including right signature fixed poles which contribute to the FESR but not to the high energy imaginary part. We do not rule out the possibility of fixed poles at right signature points, particularly in F 4 where such an object would be at J = 0 and could easily give the dominant contribution at high energies, and indeed this is one of the possibilities we tried. It should be pointed out that theories of fixed poles do not lead us to expect them to be prominent in neutral pion photoproduction. The pseudovector model [9] has fixed poles only in charged pion photoproduction, and patton models [10] predict their existence in pion photoproduction if the pion were elementary, and then only in amplitudes whose Born terms arise from the charge coupling of nucleons i.e. F1, F 2 and F 3. In these amplitudes the right signature points are J = - 1 , - 3 ... and such low-lying objects would be difficult to detect with present data. Any approach which only uses one Regge contribution per amplitude (albeit an effective a) will be certain to prefer large fixed pole contributions as this is the only way in which the phase energy relation can be broken. This can be seen in fig. 4 of ref. [4]. The use of an effective a is also unreliable because then the real part at high energy will be wrong. This is why we prefer to use several a's in each

436

LS. Barker et aL, n o photoproduction

amplitude - it enables us to break the phase energy relation and also to fit the energy dependence of the cross section. It has been argued that a model which fits the data and satisfies FTDRs automatically satisfies FESRs [3,11 ]. However this is only true because arbitrary fixed pole contributions are included. Thus we feel that it is imperative to fit the FESRs and while it might be argued that by fitting the FESRs, the FTDRs are made redundant and that we can calculate the real parts using the phase energy relation, it should be noted that this is true only if all moment FESRs are fitted. Turning this argument around, calculating the real parts using the phase energy relation assumes that all moment FESRs are fitted and we have rejected this procedure because of uncertainties in the multipole analyses near v = v c. In fact it makes very little difference as we are discussing corrections to F 4 of O(v - 3 ) and to the other F ' s of O(v - 4 ) but we feel that using FTDRs makes for more reliable real parts, particularly for extrapolations to the 2 - 3 GeV energy range. 2.3.2. E v a l u a t i o n

We evaluate the high energy integral in the FTDR analytically using the result v 'a dr' _ v~_ 1 v7~- _- v 2 tan

2 p -n ,c

(-~) +

(@) 2 2vc~-1 ( °~+1 °~+3 /Vc] 2 ) 7 r ( a + l ) 2F1 I, 2 ' 2 •' \ v ] " (2.11)

In evaluating the low-energy integrals and the FESRs we use the multipole analysis of Devenish et al. [ 12]. This is compatible with the analysis of Moorhouse and Oberlack [13] and this latter analysis gives errors on the multipoles which we use to estimate the errors on the FESRs. The error on the kth multipole M.,k -+ AMk, (o+ (o gives us an error in the contribution of that multipole to the F i FESR, C k" - A C k , and to convert these to an error on the F i FESR, A I (i) we assume

A I ( i ) = /k~=l (AC(i)) k 2 "

(2.12)

This implies that there is no correlation between the individual multipole errors and is clearly an upper bound on A I (i). We have checked that a recent analysis of these authors [14] using a slightly different resonance spectrum does not significantly change the input to our calculation. The most striking feature of the FESRs is that the F 4 FESR is definitely nonzero. To show why this is so we plot Im F 4 ( v , t) against v at t = - 0 . 2 (GeV/c) 2 (fig. 1). The Born term is more than cancelled by the P33(1236) and above this resonance Im F 4 is negative. We show the breakdown of the FESR into the individual resonance contributions in table 1, where we see that the non-zero FESR is due to

1.S. Barker et aL, 7r° photoproduction

437

t=-~ {~tg) 3

-100. ,'7

~ -20.0" I,A.~

E

-30-0. -40.0 Fig. 1. Im F4(v, t) at t = - 0.2 (GeV/c) 2 plotted against u (see text). constructive contributions from well-known resonances. If the Regge exponents are less than zero* then we can consider the F 4 FESR as a superconvergence relation. oo

(2.12)

B 4 +-~ f Im F4(v', t) du' = O, v0 and since B4 +2ic

ImF4(v',t)dv' <0

(2.13)

V0

(this is the FESR), we see that Im F 4 must change sign above E.y = 1.5 GeV. This has important consequences.

2.4. High-energy data Essentially we fit all available data from E. r = 3 GeV upwards, between t = - 0 . 2 (GeV/c) 2 and t = - 0 . 7 (GeV/c) 2. The set is do ~ - a t E . r = 6, 9, 12, 15 GeV,

(SLAC),

[15];

E. r = 4 GeV,

(Liverpool),

[16] ;

E 7 = 3, 4, 5, 5.8 GeV,

(DESY),

[171 .

* We will see later that this is demanded by the high-energy data.

L~ OO

Table 1 Resonance contributions to the F4 FESR (GeV) - 2 t=-0.2

t=-0.3

N(938) P33(1236) Pl1(1470) D13(1520) F15(1690) Others

4.14 -4.40 0.52 -0.77 -0.58 -0.16

Total

- 1 . 2 4 +- 0.50

-+ 0.35 -+ 0.25 +- 0.14 -+ 0.07 -+ 0.14

4.14 -4.49 0.52 -0.89 -0.43 -0.06

t=-0.4

+- 0.32 -+ 0.25 +- 0.15 -+ 0.05 -+ 0.09

- 1 . 2 0 +- 0.46

4.14 ~4.58 0.52 -1.01 -0.26 0.05

t=-0.5

+- 0.30 -+ 0.25 +- 0.16 +- 0.03 -+ 0.06

- 1 . 1 5 +- 0.43

4.14 -4.68 0.52 -1.13 -0.07 0.12

t=-0.6

-+ 0.27 +- 0.25 -+ 0.17 +- 0.01 +_0.08

- 1 . 1 0 +- 0.42

4.14 -4.77 0.52 -1.25 0.14 0.16

t=-0.7

+- 0.24 +- 0.25 -+ 0.18 -+ 0.02 +- 0.11

- 1 . 0 5 -+ 0.42

4.14 -4.86 0.52 -1.37 0.38 0.17

-+ 0.22 +- 0.25 +- 0.21 -+ 0.05 -+ 0.16

- 1 . 0 2 -+ 0.43

~o

~"

439

1.S. Barker et al., zr° photoproduction

Polarised photon asymmetry ~ at E. r = 4, 6, 10 GeV,

(SLAC),

[15];

E 7 = 3 GeV,

(CEA-MIT),

[18].

(Liverpool),

[ 19].

Polarised target asymmetry T at E7 = 4 GeV, Recoil polarisation at E.t = 2.9 + 5.33 Itl,

(M.I.T.-Tufts), [20].

All data were taken at t = - 0 . 2 , - 0 . 3 , - 0 . 4 , - 0 . 5 , - 0 . 6 , - 0 . 7 (GeV/c) 2, interpolated if necessary. The only data omitted were: (a) a measurement of T at 4 GeV from DESY [21] which appeared while work was in progress. These data do not affect our results as they are compatible with the Liverpool T data [19]. (b) a rather low statistics differential cross-section measurement [22] which was obtained as a by-product of a deuterium experiment. (c) a recent differential cross-section measurement from DESY at E. r = 4 GeV [23]. We have preferred to use the Liverpool data at this energy because of its appreciably lower systematic errors. The two measurements are consistent for - t < 0.5 but show some discrepancies for larger - t . There are no striking inconsistencies in this data set although the DESY crosssection data is rather higher than the trend of the SLAC data. The Liverpool crosssection data is consistent with the trend o f the latter. Another difficulty with the DESY cross-section data is that near the dip at t = 0.5 (GeV/c) 2 the t bins are very big and interpolation difficult. Accordingly we have placed very little weight on the data here. The only general points of interest in the cross section data are the rather fiat Oteff, aef f = (0.19+0.03) + (0.27 + 0 . 0 4 ) t ,

(2.14)

and the well-known dip, around t -~ - 0 . f i (GeV/c) 2. This is normally explained by a zero in the otherwise dominant F 1 amplitude at this point (or, to look from the s-channel, zeros in S 1 and $2). The main general point about the nucleon polarisation data is the question of whether R and T are equal. This has immediate consequences for the question o f whether F 4 = 0 .because (to leading order in s), da N

R dt

~

1

167r ( t _ 4 m 2)

Im(FIF~+(t_am2)F4F~)

'

(2.15)

LS. Barker et al., ~r° photoproduction

440

TdO

"d"7 ~

~ 1 Im (F1F~ - ( t - 4 m 2 ) F 4 F ~ ), 161r ( t _ 4 m 2)

(2.16)

and so we see that (R - T) da ~ ~

Im (F2F*).

(2.17)

Comparison of the R and T data is rather difficult because the experiments are at different energies. Ignoring this, then it was concluded by Deutch et al. [20] that R and T are not equal. However the question is further complicated because the comparison necessitates interpolation in t. We have done the comparison for our interpolated data andtfind that at the t-values where R and T are significantly different then the inequality [2] {R - T{ < 1 - 1C

(2.18)

is violated, with the obvious conclusion that it is impossible to say whether R and T are equal with any reliability. In practice the question is not a serious one since the comparison of R and T is not a very sensitive test of whether F 4 = 0 as the difference R - T measures the interference of two small amplitudes F 2 and F 4.

3. Details of fits Since the distinctive feature of our fit and our main difficulty in achieving a fit were due to the non-zero F 4 we discuss this feature first. We experienced great difficulty in achieving a simultaneous fit to the large F 4 FESR and the small unnatural parity cross section (1 - ~;)do/dt at 3 and 4 GeV. We note that this cannot be achieved by introducing a right signature pole at J = 0 which would ensure a fit to the FESR without contributing to the high energy imaginary part since it would give a large real part at high energy through the FTDR. We found that a fit could only be obtained by taking two Regge terms, a 5 and c~6, with c~5 just below J = 0 and rather flat (e.g. a Regge-Regge cut) and a6 very low-lying, around - 3 . From the point of view of Regge theory this is rather implausible. From a phenomenolog, cal stand-point it is rather more attractive as many exotic amplitudes which can be observed directly by cross section measurements (e.g. K - p backward scattering) [24] also show steep energy dependence below 5 GeV (da/dt ~ s - 8 ) with some evidence of a slower energy dependence (compatible with the theoretically expected Regge cuts) above this energy*. No theoretically plausible explanation for this behaviour has been given. In our fit we found it necessary to have the coefficients/354 and ~64 both positive in order to satisfy the FESR (superconvergence relation) and fit the data. Since * For recent reviews of exotic exchange see refs. [25,26].

LS. Barker et al., lr° photoproduction

441

below v = v c, Im F4(v, t) is negative we have a discontinuity in Im F4(v, t) at v = vc. We cannot achieve the necessary change of sign by taking/~64 < 0 as then we cannot fit the data, essentially because Im F 4 does not change sign quickly enough above v = vc. A very sharp change of sign is required and this could not be achieved by a Regge type parametrisation. We finally achieved a fit with a continuous Im F 4 by modifying the parametrisation to -1 -1 Im F4(v, t) = fl54 vas-1 +/364v(va6-2 + r(p 2 - a2) ~°:6 ) ,

(3.1)

with r = - 0 . 0 0 8 , a 2 = vc2 - 0 . 2 5 (GeV/c) 2 and a 6 = - 3 . 0 . This produces a sufficiently rapid change of sign. We do not claim that this is a realistic parametrisation of Im F4(v, t) but rather that a form can be found which fits the data, satisfies the FESR and is continuous in Im F4(v, t), and that the extrapolation to E. r = 2.5 GeV is unaffected by the modification to the amplitude. For the other amplitudes we experienced no difficulty in obtaining smooth imaginary parts at v = vc with the original parametrisation. We carried out fits at each t independently and then smoothed in t. To achieve smoothness we found it was sufficient to fit with increased weighting of the FESR except at t = - 0 . 5 and - 0 . 6 (GeV/c) 2 where we had to interpolate parameters by hand. The chi-squared increased very little on smoothing except at t = - 0 . 5 (GeV/c) 2 (see later). The parameters corresponding to our best smoothed fit are given in table 2. The best fits are shown in figs. 2 - 6 . These correspond to taking a 4 and a 5 to be Regge-Regge cuts (actually P' - co and w-co cuts respectively, although the distinction is unimportant),

Table 2 "Residues" of preferred fit t=-0.2

t=-0.3

t=-0.4

t=-0.5

t=-0.6

t=-0.7

311 Oat 341 ~

-5.47 0.66 -2.98

-3.92 1.22 -2.76

- 1.07 1.52 -3.51

.0.50 0.86 -2.92

1.90 1.04 -3.04

4.28 0.65 -3.77

3~2 3~2

-0.07 0.25

-0.09 0.25

-0.05 0.16

0.02 -0.18

0.10 -0.21

0.25 -0.18

313 333 34a

3.79 -1.46 0.09

3.41 -1.26 0.36

2.79 -1.36 0.81

1.10 -0.76 1.38

-0.30 -0.67 2.18

- 2.45 -0.71 3.46

354 /364

0.34 8.60

0.35 8.36

0.39 6.85

0.29 6.72

0.42 6.17

0.61 4.15

The underlined parameters are held f'txed. In calculating the amplitudes using these parameters the unit of energy is~taken at 1 GeV.

442

I.S. Barker et al., 7r° p h o t o p r o d u c t i o n 4' 2"

o

(GeV)

0;2

0;4

~

F

.'21,

-4.

3 016

0:8

2 1' F3

1

-10

0

0.2

0;4

0;8

0;6 2

0"5

0

0"2

0:4

0"6

0:8

(GeV)- 2 0;2

0;4

0;6

0;8

-1"

t

-2 F4

-1.0" It I--*- ('GeV/c) 2 Fig. 2. Fits to FESRs for (a) F1 ; (b) F 2 ; (c) F 3 ; (d) F4.

~4 = 2 -

rn2-mf 2 +0.5t,

(3.2)

~5 = 1 -

2m2~ + 0.5 t ,

(3.3)

a 6 = - 3.0,

(3.4)

which we found gave the best fit. We have examined the range within which we could vary the trajectories a5 and a 6 . Reasonable fits may be obtained if the intercept of the Regge-Regge cut is allowed to vary within +0.3 of the canonical value. However the J = 0 fixed pole alternative, although it fitted the data, could not be continued smoothly b e y o n d the dip, presumably because it would tend to dominate the cross section here, necessitating a rapid change in the other parameters in order to reproduce the observed 0~ef f. We found taking a 5 = ~to - 1 and a 5 = aBc gave slightly worse fits. The value a 6 = - 3 . 0 used for a6 is an approximate upper bound. Since there are uncanceUed fixed poles at J = - 2 , - 4 , etc. in this amplitude, because we do not fit the higher m o m e n t sum rules, we do not expect to be particu-

LS. Barker et aL, n o photoproduction

443

f 1.0

Ey= 4 GeV ILiverpool) 143

Ey=6 GeV CSLAC)

Oq f e~

.= ~ 0.1

f

e~

{

Ey= 9 GeV(SLAC)

)

Ey = 12GeV (SLAC)

[

Ey = I5 GeV CSLAC)

}

0

-o 0-1

{ ~

0.¸

0.01

o~2

0:4

o'.6

o:8

It I-'- (GeV/c) ~ Fig. 3. Fits to SLAC [15] and Liverpool [16] cross-section data.

larly sensitive to the exact value of Ot6 provided it is sufficiently low-lying, and lower values gave equally good fits. For the trajectory Ct4 w e found that smooth fits with comparable X2 could be obtained for a variety of assumptions. In particular a B cut and a fixed pole at J = - 1 gave comparable results.

444

LS. Barker et al., ~r° p h o t o p r o d u c t i o n 10-

1'0-

Ey=3 GeV (,DESY)

%

1'0

Ev=4GeV r-.DESY) It.

.c

1"(3

,Z:,

Ev=5GeV (DESY) ~'C

Ev=5-8GeV{.DESY~

0.1-

o'-2

0'4

m.-,- (c~/c)

2 0:6

0'.8

Fig. 4. Fits to DESY [ 17] cross-section data. We do not quote any errors on the parameters since they are highly correlated. However in sect. 4 we show estimates of errors on the amplitudes corresponding to the scatter obtained in a variety of reasonable fits. We found that we needed all o f the parameters except/]42, although some were only required at certain t-values. In particular a large a 4 contribution was needed, presumably in order to get a reasonable O~eff, and it also gave important contributions to the FESRs. The w c contribution to F 1 was only essential for I t I > 0.5 (GeV/c) 2 and the answers for F 2 were rather unstable, presumably because of the smallness of this amplitude.

I.S. B a r k e r et al., ~r° p h o t o p r o d u c t i o n

445

1.0" Ey = 3 GeV

0.5-

0o

1.0~.

~

.,..~{._~

l

Ey = 4 GeV

0.5-

0"

1.0~.

I Ey=6 GeV 0-5-

0"

1.0~. , , , . ~ ~

}

Ey = 10 GeV

0.5'

o'.2

0'.4

0:6

o'-8

It I---(GeV/c) 2

Fig. 5. Fits to SLAC [15] and CEA-MIT [18] Z data, The only t-values at which the fits were less than satisfactory were t = - 0 . 4 (GeV/c) 2 and - 0 . 5 (GeV/c) 2. The problem at the former value was peculiar data, in particular the very large ~ value at 3 GeV (1.07 -+0.15). In our type o f parametrisation it is impossible to fit this simultaneously with the F 4 F E S R and the 4 GeV data (0.77 -+0.045). In fact even in our fits with F 4 = 0 we could not get satisfactory fits to ~ at 3 GeV. Whereas this difficulty was present in the fixed-t fits the difficulty at t = - 0 . 5 (GeV/c) 2 only appeared when we smoothed in t. Before smoothing the fit was exceptionally good. After smoothing our cross section tends

446

I.S. Barker et al., n ° p h o t o p r o d u c t i o n

1-0-

Ey = 4 GeV

0.5-

~-

0

0;4

0'2

0~6

0;8

1.0

-0.5-

-1,0 Itl

"~GeV/c) 2

(a)

1'0-

Ey = (2.9 + 5"33 It0GeV

0"5-

n"

0-,8

0

1~0

-0.5-

-1-

Itl-.-CGeV/c) 2 Cb) Fig. 6. Fits to (a) Liverpool [19] T data and (b) MIT-Tufts [20] R data. to be too big at E7 = 4 GeV and too small at 6, 9, 12 and 15 GeV i.e. our otcff is too low. However we cannot raise it and retain smoothness. Here again, we are suspicious of the data. The SLAC data was obtained in a single arm experiment with a brammstrahlung beam [ 15 ]. Apart from the difficulty of the subtraction of the Compton scattering, which is particularly important in the dip because of the small zr0 photoproduction cross section, this experiment has the following difficulties of interpretation.

1.S. Barker et al., 7r° photoproduction (a) The larger t bins cross section if the dip Liverpool data [ 16]. (b) Although d a / d t values but interpolated

447

(At = 0.04 GeV/c 2) will tend to give an overestimate of the is very close to - 0 . 5 (GeV/c) 2, which it appears to be in the is quoted at the above four energies, these are not measured values using d o / d t at many energies and assuming the form

-~t = F ( t ) ( s - m2)2c~-2.

(3.5)

In fact in the dip the higher energy points lie slightly below the best fit line [27] though not by enough to explain the discrepancy. On the other hand the Liverpool data might be too low in the dip. It is certainly lower than the DESY data [23] at this point. However we cannot expect a theoretical analysis to decide which experiments are wrong - it can only give indications of when experiments are incompatible. The only substantially different fit we found was in fact a Michigan model [28] type fit i.e. Im F 1 has a zero near t ~ - 0 . 5 GeV/c 2 but the pole residue (/311) was structureless. However, this fit did not give continuous imaginary parts at u = uc even for Im F 1 . Worden [2] commented on a discrepancy between the Michigan model fit to the high-energy data and the low-energy data, although in his model calculation it manifested itself in a poor fit to the FESRs. We also carried out some fits with F 4 = 0 (not fitting the F 4 FESR, of course) for purposes of comparison. The quality of the fits to the other data was very similar with the same difficulties in fitting the data at t = - 0 . 4 and - 0 . 5 (GeV/c) 2 as described earlier. The resulting s-channel amplitudes were very similar at E 7 = 4 GeV but diverged significantly at lower energies (see sect. 4). It is perhaps worth pointing out that even with F 4 = 0 we only have S 1 = S 2 to leading order in s. This leads to differences in R and T at finite energies e.g. 4% differences at 4 GeV seemed to be common. We did not feel that it was useful to obtain errors for this series of solutions since our main interest is in the fits including F 4.

4. Results

4.1. s-channel amplitudes The s-channel helicity amplitudes at E~r = 4 GeV are plotted in fig. 7. The amplitudes corresponding to the fits with F 4 = 0 are not sufficiently different to warrant plotting. The following features are worthy of comment. (a) To a good approximation S 1 = S 2. This simply reflects the fact that IF 1] >> IF41. (b) The analogous statement for non-flip and double-flip i.e. N--- - D is not universally true though the amplitudes shown retain some vestige of it, particularly in the imaginary parts.

448

LS. Barker et al., ~r° photoproduction

E,I=4GeV N (GeV)-2

S~ (GeV)-2 0"047

0.04]

0.02-

0"02.

0;3 "{ "[ 0;6

0.0

i

-

itl(GeV/c)204. I

0.02-

0-04 -

0-02-

0"02.

o

~ 0.6 fie- ~lt

" -~

i

I(GeV/c)2

t

0.04-

(GSe~)2 0-04 -

0;3 x

t

t

D -2 (GeV) 0"04]

t 0.6

°°21 t T t t"

0i6 I

0.02.

0.02]

0'04

0.04J

Fig. 7. s-channel helicity amplitudes at E7 = 4 GeV. Open circles are real parts. Solid circles imaginary parts. (c) Both Im S 1 and Im S 2 have zeros in the t ~- - 0 . 5 , - 0 . 6 (GeV/c) 2 region. However the real parts do not have zeros here. This is precisely what is expected from the dual absorption model (DAM) [29]. (d) Im N seems to have no zero in the t range considered, although the situations for Jtl > 0.5 GeV/c 2 is unclear. However, it certainly has no zero near t ~ - 0 . 2 which is where the DAM would imply that it should. Regge cut models which produce cuts by convoluting poles with the pomeron would not agree with the DAM here because of the high spins involved. This is because we would expect the w to be evasive i.e. have an extra factor of t in its coupling to N and this affects the relative weight of pole and cut in the final amplitude [28].

I.S. Barker et al., n ° photoproduetion

449

Argyres et al. [4] fitted the high energy data with a zero, in fact with Im N o: J o ( b x / _ t). However their fit to the zeroth moment FESR for the amplitude which is asymptotically proportional to N is very poor. It can clearly be seen in theirfig. 4that the FESR has no zero for Itl < 1 GeV/c 2 whereas their fit does. They obtain better fits to the higher moment FESRs which do havethe zero. This simply reflects the well-known [2,4] fact that the higher mass resonance contributions all have a zero at the appropriate t-value whereas the nucleon Born term and the/x(1236) contributions do not. Worden [2] calculated FESRs for a similar amplitude (fig. 3 of his paper); isolating the ¢r° photoproduction case there is no evidence of geometrical zero. The absence (or presence) of a zero in the FESR by itself does not prove or disprove anything about the absence (or presence) of a zero in the imaginary part at high energy since fixed poles at right signature points can contribute to the former but not the latter [2]. To test this we introduced fixed poles at J = - 1 into F 1 and F 3 in the place of the low-lying cut a 4 and found an acceptable fit. However the zero structure of Im N was not changed and so we concluded that Im N does not have the zero predicted by DAM. On this point the situation in charged pion photoproduction is confused [2,3,5]. Hontebeyrie et al. [3] claim that the data require a zero in Im N as demanded by DAM whereas Barbour and Moorhouse claim that the data are definitely incompatible with this zero. They also claim that the data on neutral pion photoproduction are insensitive to the presence or otherwise of a zero in Im N. However they do not fit the FESRs and as we have seen this is where the zero hypothesis has difficulty. 4. 2. t-channel identifications Since we only identify Regge terms by trajectory function we expect some difficulties of interpretation, particularly for low-lying objects in the J-plane. This is particularly true for our a4 terms as we stated in subsect. 2.2. The fact that the coefficients of t, ~ turn out to be much bigger compared to the B pole residue than would be expected from an absorption model confirms our misgivings over considering the fourth contribution as the B cut. Presumably other effects contribute, such as w - P ' cuts. As stated above we also found that at4 could be replaced by a fixed pole at J = - 1 in F 1 and F 3. In the same vein although we have identified the higher lying object in F 4 as a Regge-Regge cut other possibilities are not ruled out. 4.3. Comparison with absorptive cut models We have already compared our results with the DAM in subsect. 4.1 ; this did not require us to make any explicit separation between pole and cut contributions to the amplitudes. Comparison with other cut models necessitates such a separation and, as we have indicated above, there are difficulties in identifying such contributions on the basis of their energy dependence alone. Moreover, since we only param-

450

LS. Barker et al., ~r° photoproduction

etrise the imaginary parts of our amplitudes, the separation of the real parts associated with the individual contributions to the amplitudes requires additional and not necessarily unique assumptions. The simplest such assumption is to attribute to each contribution the appropriate Regge pole phase. The discussion of subsect. 2.3.1 shows that such a procedure is consistent to O(u - 3 ) in the crossing odd amplitude F4, and to O(v - 4 ) in the crossing even amplitudes. If we make such a separation, then we can only identify the trajectory a 3 (which we have previously described as the w cut) with the contribution from the tip of an absorptive w-pomeron cut. If those parts of the cut which lie lower in the J plane are important, then their effects will be included in those of the lower lying trajectories a2, a 4 and a 5 . Of course other J plane singularities may be represented by these trajectories. With such caveats in mind we may compare our results with absorptive cut models. We shall concentrate on the two single flip amplitudes S 1 and S 2, which dominate the cross section. Compared with nN elastic scattering, cuts seem to be more important in determining the structure of flip amplitudes. First we compare with the old Argonne [30] and Michigan models [31 ]. An examination of the residues ~11 and/~13 shows that the co pole exhibits a zero i n F 1 between t = - 0 . 4 and - 0 . 5 (GeV/c) 2 and in F 3 between t = - 0 . 5 and - 0 . 6 (GeV/c) 2. Apart from the small separation of the zeros this is what we would expect from an Argonne model with nonsense wrong signature zeros in the pole residues. The trajectory a 3 shows the features we would expect of an absorptive cut. The residues 1331 and ~33 are comparatively fiat and structureless, and their signs are such that in S 1 and S 2 a 3 interferes destructively with ct1 at small I tl. Of course at larger Itl the interference becomes increasingly constructive. Away from the dip region, the magnitude of the ct3 contribution is smaller by a factor two or three than that of a 1 . This again is an Argonne-like feature; there is no over-absorption and the amplitude structure is mainly determined by NWSZ in the pole contributions rather by cut-pole interference as in the Michigan model. It is known however that neither the Argonne nor the Michigan model is adequate to describe the structure revealed in the amplitude analysis of the lrN system [32]. In particular the fact that the I t = 1 nonflip amplitude has real and imaginary part zeros at substantially different t-values constitutes a serious difficulty for the older absorption models. This is easily understood once it is realised that in such models the dominant contribution to the cut discontinuity is located near its branch point [33 ], and hence for small I tl is near to the pole in the J plane. It follows that up to a sign cut and pole phases are similar and thus such cuts cannot separate the zeros. A variety of modified absorption models have been introduced to overcome this defect [ 3 4 - 3 6 ] . Here we pick out two which are illustrative of the general features of such models. In the model of Hartley and Kane [34], the phase of the absorbing amplitude is modified so as to introduce a large negative real to imaginary part ratio. This is done * The trajectory a 6 is too low-lying to be included consistently in this analysis.

I.S. Barker et al., 7r° p h o t o p r o d u c t i o n

451

by adding a peripheral component to the pomeron with a radius which increases logarithmically with t,. Crossing symmetry then determines the appropriate real part*. In the model of Ringland et al. [35], the phase of the absorbed amplitude is changed by multiplying the constant part of the signature factor by i. All such models give similar results for vector exchanges. Absorption is increased in the imaginary part and is reduced in the real part, moving the imaginary part zero to smaller Itl values. Worden [33], has shown that the cut discontinuity in such models is no longer dominated by the region near the branch point. The tight connection between the phase of the pole and of the cut is then broken by those parts of the cut which lie lower in the J plane. In our analysis we could regard the trajectory 54 as representing such lower-lying parts of the.absorptive cut. In view of the small size of the 52 contributions compared with those of 54, this is more plausible than regarding 54 as an absorptive B cut. We find that 54 gives contributions to S 1 and S 2 which are roughly equal in magnitude to those from 53, and are approximately ~ 7r out of phase. However the signs of the 54 residues found in our fit are such that the sum of the 53 and 54 contributions are predominantly real. Thus the real parts of the single flip amplitudes are more strongly absorbed than the imaginary parts, and the real part zeros occur at smaller Itl than those of the imaginary parts. This is the converse of the situation found in the 1 t = 1 no flip amplitude in 7rN scattering, and disagrees with the expectations from cut models designed to reproduce such a zero structure. In the case of the i factor model of Ringland et al. [35], it is easy to see how this contradiction arises. The model would predict a rotation of the phase of the usual absorptive cut by ~rr. In our case the effect of the 54 contribution is to rotate the net cut contribution through --~lr. That we do not agree with such modified cut models should not be surprising since Worden has shown [33] that they do not satisfy FESRs, whereas our analysis does. Finally we consider the smaller amplitudes N and D. Here the situation is somewhat simpler since it turns out that the low-lying trajectories are unimportant for Itl ~< 0.5 (GeV/c) 2. In both N and D the effective pole 51 interferes destructively with the effective cut 53 . From the previous discussion it is clear that if the pole dominates the cut at small Itl, we will expect to see real and imaginary part zeros close to the zeros of the pole residues - a typical Argonne feature. This occurs in N near Itl = 0.5 (GeV/c) 2. However the cut and pole contributions are more nearly equal in D - essentially because the unnatural parity cut (given by the residue/332) and the natural parity cut (given by/331 and/333 ) have the same sign in D and opposite sign in N. Because of the slight phase difference (modulo 7r) between 51 and 53 and the small contributions of other trajectories, this leads to a small amplitude * In the analysis of Collins and Fitton [28], the real part arises from including f exchange in the absorbing amplitude.

452

LS. Barker et al., n ° photoproduction

of no well defined zero structure. The enhancement of the cut in the amplitude of higher helicity flip is somewhat at variance with conventional ideas about absorption; however it must be remembered that this is only a small effect in a small amplitude. 4.4. E x t r a p o l a t i o n to l o w e r energy

One of the objects of this analysis was to obtain reliable extrapolations to lower energy and to this end we retained F 4. Thus the first question to be answered is what difference this makes. In fig. 8 we show the different predictions for ~ at 2.5 GeV from the fits with and without F 4 . This difference is certainly measurable although for lower values of ol6 it is reduced. We also give our predictions for the quantities G and H which can be measured in double polarisation experiments with a linearly polarised beam on a polarised target (fig. 9). These quantities are defined in the appendix. To leading order in s, G is very sensitive to whether or not F 4 is present as it interferes predominantly with the large amplitude F 1 . From this point of view H is not as interesting a quantity as it measures the interference o f f 4 with F 3. Also shown in fig. 9 are the predictions for the fit with F 4 = 0. Both G and H are different since F 2 is changed because of the assumption that this amplitude contains all of the unnatural parity contributions. The complete double polarisation formulae are given in the appendix for both linearly and circularly polarised beams (assuming no recoil polarisation measurement). Here we have tried to identify the more interesting measurements. 1'0-

~" E¥=2.5 GeV 0

0

I I

o 0

o

0.5.

0~3

016

Itl (GeV[c)~

0.5-

].o.

Fig. 8. Predictions for l~ at 2.5 GeV. The open circles are a fit with F 4 = 0.

453

[.S. Barker et aL, ~r° photoproduction

(a) Ey = 2.5 GeV G 1.0-

1.0

0.5

0'5"

o

o

o

01'3

0-6

0.6 i

)tl (GeV/~) ~

It I(GeV/c')

2

o o

-0"5

-0-5-

tt

-1.0

-1.0"

(b) Ey=4 GeV G

1.0-

1.0-

0'5-

0"5"

o

o

0;3 I -0'5'

-1'0"

?

0;6 I o

t

~

~

0.6

Itl .

}tl CG~V/c) ~

"1" o (G eV/c)z -0"5"

o

I

t

t

-1.0"

Fig. 9. Predictions for G and H at (a) E7 = 2.5 GeV and (b) E, r = 4 GeV. The open circles are a fit with/74 = 0.

The two solutions with or without F 4 can give different extrapolations of the differential cross section. The solution with F4, because of the sharp increase in unnatural parity as we go lower in energy, predicts that the dip is washed out at the lower energies, whereas the solution with F 4 = 0 predicts that the dip shape remains the same. Data exist a t E,r = 2.0 GeV which indicate that the dip does change shape. Unfortunately we cannot reliably extrapolate our solution so far down in energy so we give the two predictions for do/dr at E 7 = 2.5 GeV (fig. 10). In fact although the change in the shape of d a / d t is common to all fits which include F 4,

454

I.S. Barker et al., ~r° p h o t o p r o d u c t i o n

3"0-

o

Ey:2-5GeV

t 2.0-

O

1.5-

~

° 3

ttxo

10. 13.8-

O

O

13,6"

<1 o!3

o~

,,(Gev/J

Fig. 10. Predictions for d a / d t at E,y = 2.5 GeV. The open, circles are a fit with F4 = 0.

the energy at which the change takes place depends on the value of a 6. If a 6 is very low-lying, the change in shape takes place at lower energies. This is because the size of this contribution is mainly determined by the FESRs as opposed to the high-energy data. Fig. 10 corresponds to o~6 = - 3 . Finally to support our contention that F 4 has the typical energy dependence of an exotic amplitude we show IF412 plotted at a fixed t = - 0 . 2 (GeV/c) 2 in fig. 11. The precise values of IF 4 [2 obviously depend on the value chosen for a 6 . However the flattening off at high energy is a common feature.

5. Conclusions The most interesting points which have emerged from our analysis are as follows. (a) The zero structure of the imaginary parts of the amplitudes. Here the DAM seems to work for the spin flip amplitude but not the no-flip amplitude. This could be because of the high spins involved in the process which causes the Regge pole contribution to be evasive. Barbour and Moorhouse [5] found the same feature in charged pion photoproduction. The absence of the zero is independent of whether F 4 is included. (b) F 4 is not negligible, particularly below 4 GeV. Its energy dependence is typical of an exotic amplitude. As far as we know no plausible explanation of this effect has ever been put forward.

I.S. B a r k e r et al., T ° p h o t o p r o d u c t i o n

455

~eVJ • z

1o

15 ~o

'?,

Lt.'~"

"1 2.5

go ido 15!o Ey (,GeV)

Fig. 11. Plot of IF412 against E.r at t = -0.2 (GeV) 2. We feel that these features were unambiguously determined by our analysis. The quantities which are least well determined by our analysis are the phases of the unnatural parity amplitudes F 2 and F 4 which are hardly constrained at all by the high-energy data. On the other hand IF2[ and lEa1 should be reliably determined. The most unambiguous test of our assignment at the unnatural parity amplitudes is to measure G and H. We wish to thank Jim Norem and Alan Hufton for communicating the Liverpool

LS. Barker et al., 7r° photoproduction

456

cross-section data prior to publication, John Rutherfoord, Tony Osborne and Bob Anderson for information about the various experiments in which they were involved, and the referee for suggesting the inclusion of subsect. 4.3.

Appendix. Some kinematic relations In this appendix we define our amplitude normalisation and the phases of the double polarisation asymmetries. Following the Basel convention for the lab coordinate system we have for the cross section for polarised photon incident on a polarised target d--t do (P, PT,~b,Po) = ~do unpolarised Itl --PT cos2~bZ

(A.1)

+ P x ( - P T sin 2¢H+P®F) - P y ( - T + P T cos 2~bR) - P z ( - P T sin 2~bG +POE)} , /

where P is the polarisation of the target, PT is the transverse polarisation of the beam at an angle 4~to the reaction plane and P~ is the degree of right circular polarisation. We note that for the best measurement of G and H the photon should be polarised at 45 ° to the reaction plane. The quantity R coincides with the recoil polarisation of the proton. The definitions of E, F, G and H coincide with those of Worden [2]. In terms of our s-channel amplitudes: do= ISI[ 2 + IS2[ 2 + IN[2 + IDj2 dt

(A.2)

d o I~ = 2 Re ($2S ~ - D N * )

(A.3)

d o T -- 2 Im (SIN* - S2D* ) dt

(A.4)

d--?-°R. = 2 Im (DS~ - N S { ) dt

(A.5)

do G --- - 2 Im (SIS ~ +ND*)

(A.6)

d ° H = - 2 Im (SID* +S2N* )

(A.7)

dOE = 2 Re (SzD* +S1N* ) dt

(A,8)

do~ dt

(A.9)

dt

dt

dt

= IS212 - IS1 12 - IDI2 + Igl2 "

Table A. 1 s-channel amplitudes Net helicity flip n

This paper

Wiik ref. [7]

Walker ref. [37]

Worden ref. [2]

Gault et al. ref. [28]

Argyres et al. ref. [4]

Hontebeyrie e t a l , ref. [3]

Barbour and Moorhouse ref. [51

0

~v

~v

~2

~'2

r+~_

M0

no

~o

1

$1

F1

H1

H4

TI_

M~

H_ 1

HI

1

$2

F2

//4

H1

T+I+

M1

Hl

H_ 1

%

2

D

D

H3

Ha

TI_+

M2

H2

H2

¢)

Relative normalisation

(s - m 2)

(s - m 2)

1

1

(s - m 2)

,/2.s

x/~(s

- m 2)

3x/~(s

- m 2)

- 8 x/"~

,¢~

,,7

It is only possible to assign phases to the relative normalisations when the amplitudes are related to the invariant amplitudes rather than to experimental quantities. Thus the phases o f Wiik's amplitudes a n d those of Gault et al. relative to the amplitudes used in this work are undefined. On the other hand, when low energy i n p u t is used and as in the work o f Worden and Argyres et al. the phases are k n o w n and are related ultimately to the k n o w n phases o f the Born terms. Our t-channel amplitudes are easily related to those o f other a u t h o r s by the use o f their definition in terms of the CGLN amplitudes A i. T h e reader is warned o f discrepancies o f sign a n d factors of t and t - / ~ 2 b e t w e e n our definitions of the F i and those of other authors.

4:-

LS. Barkeret al., 7r° photoproduction

458

In terms of the t-channel amplitudes we have asymptotically:

do(l),

= 3 - ~ ( t - 4 m 2 ) - I {-tIFl12+lF312~(t-4m2)(If212-tlf412)}'

d--t

(A.10): d ° ( T ~ = l@~Xf-S~(t-4m2) -1 Im (F1F ~ ~(t-4m2)F4F~}

dt ~R ]

= 1--~ ( t - 4m2)-1 Im

(AA1)

{tF4(2mF ~ -F~)+F2(-tF ~ + 2 m F ~ ) } , (A.12)

do F -1 Re -d-t ( H ) = 1-~ x / Z T ( t - 4m2)-1 ( I m ) { F 2 ( 2 m F ~ - F ~ ) + F4(-tF ~ + 2mF~)}. (A.13) Since empirically the unnatural parity amplitudes are small it is clear that single polarisation measurements give little phase information on F 2 and F 4. Assuming F 1 is the dominant natural parity amplitude G (or E) gives information on F 4 away from the forward direction and H (or F) on F 2. In table A.1 we relate our s-channel amplitudes to those of other authors to enable comparisons to be made more easily.

Note added in proof

Our model for F 4 would predict a rapid increase in the unnatural parity cross section as the beam energy falls below 3 GeV followed by a decrease somewhere above 1.5 GeV as the zero in F 4 is approached. This behaviour seems to be confirmed by polarised beam data between 1.26 and 2.5 GeV obtained by the Glasgow-LiverpoolSheffield Group (submitted to the 17th Int. Conf. on high energy physics, London, July 1974, paper 993). These show the asymmetry IS at first deepening (it becomes negative in the dip between E~ = 1.58 and 1.92 GeV) and then falling in as the beam energy decreases further.

References [1] G.R. Goldstein, J.F. Owens, J.P. Rutherfoord and M.J. Moravesik, Nucl. Phys. B, to be published: [2] R. Worden, Nuel. Phys. B37 (1972) 253.

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