Study of Kπ scattering using the reactions K±p → K±π+nand K±p → K±π−Δ ++at 13 GeV

Nuclear Physics B133 (1978) 4 9 0 - 5 2 4 © North-Holland Publishing C o m p a n y

STUDY OF Kn SCATTERING

USING THE REACTIONS

K-+p --, K-+n+n A N D K-+p --, K+-n - A ++ A T 13 G e V / c * P. E S T A B R O O K S

McGill University, Montreal, Canada R.K. CARNEGIE

Carleton University, Ottawa, Canada A.D. MARTIN

University of Durham, Durham City, England W.M. DUNWOODIE,

T.A. LASINSKI and D.W.G.S. LEITH

Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305 Received 29 August 1977

An elastic Kn partial-wave analysis is presented. It is based on high statistics data for the reactions K+-p --, K+-~r+n and K-+p --. K e n - A + + at 13 GeV obtained in a spectrometer experiment performed at SLAC. For each reaction, a t-dependent parametrization of the production amplitudes provides information on b o t h the Krr mass dependence of the production mechanisms and on Krt scattering. Knowledge of the t-dependence then allows a calculation of the Krr partial-wave amplitudes for Krr masses from 0.7 to 1.9 GeV. The results of such analyses using data for (i) the neutral recoil reactions, (ii) the 2x++ recoil reactions, and (iii) b o t h neutron and A++ recoil reactions simultaneously, are presented. Besides the leading JP = 1 - , 2 +, and 3 - resonances at MK~r = 0.896, 1.434, and 1.78 GeV, there is evidence in two of the four possible partial-wave solutions for a broad P-wave resonant-like structure in the region of 1700 MeV. The I = 1 S-wave magnitude rises slowly and smoothly to a m a x i m u m near 1400 MeV, but then decreases rapidly between 1400 and 1600 MeV. This structure is strongly indicative of an S-wave resonance near 1450 MeV. The charge-two KTr reaction is dominated by S-wave scattering with a total cross section decreasing from 4 m b at 0.9 GeV to 2 mb at 1.5 GeV. Both the I = 1 Swave below 1400 MeV and the I = 3 S-wave are well described by an effective range parametrization.

* Work supported by the National Research Council of Canada and the Energy Research and Development Administration o f the United States. 490

P. Estabrooks et al. / K+-p --, K+-n+n, K%r - A ++

491

1. Introduction In this paper we describe a Krr partial-wave analysis using the high statistics SLAC 13 GeV/c spectrometer data [1] on the four reactions K-p ~ K-n+n ,

(1)

K+p -* K+n+n ,

(2)

K+p -+ K+~r-A ++ ,

(3)

K p - + K - n - A ++.

(4)

For each reaction, examination o f the t-dependence at fixed Kn mass, MKTr, of the spherical harmonic moments, YLM, of the Kn angular distribution enables us to isolate the rr exchange contribution and also to develop an adequate parametrization of the m o m e n t u m transfer dependence of the other possible exchanges [2,3]. Armed with this information, we can then perform meaningful Kn partial-wave analyses of the data on reactions (1) to (4) in single wide t bins ( - t ' < 0.15 GeV 2 for reactions (1), (2), and (4) and - t ' < 0.2 GeV 2 for reaction (3)). Although the behavior of the data as a function o f momentum transfer is very different for the neutron and delta recoil reactions, we find that this procedure enables us to use the Kn moments in the physical region to calculate Kn partial-wave amplitudes which are independent of the nature o f the particle recoiling against the Krr system. We can therefore perform joint Kn partial-wave analyses of both K -+n+n and K +-rr-2x physical region moments, which take into account the different momentum transfer structure of the two reactions while requiring a common Kn scattering amplitude. It is this analysis which provides our best determination of Kn scattering. In sect. 2 we discuss the experimental data selection and present Kn mass spectra and unnormalized M = 0 moments for all 4 reactions at small momentum transfer. In sect. 3 we investigate the t-dependence of both neutron and A ++ recoil reactions. This makes possible the partial wave analysis of physical region moments which is presented in sect. 4 for both neutron and 2~++ recoil reactions, first individually and then jointly. The I = 3 Kn scattering obtained directly from the analyses o f reactions (2) and (4) is found to be dominated by the S-wave. The analysis of reactions (1) and (3) yields a unique partial-wave solution forMK~ < 1.46 GeV which exhibits the 1 - K * ( 8 9 0 ) and 2+K*(1435) resonances and a n / = ½ S-wave cross section that rises smoothly to a maximum near 1400 MeV. However, for higher Kn mass we find several possible solutions which we classify in terms of the zero structure of the scattering amplitude. Each solution contains a broad spin-parity 3 - resonance [4] ; the solutions differ in the structure of the non-leading partial waves. In sect. 5 we present Argand plots of the Kn partial-wave amplitudes for all four solutions and discuss the implications of this analysis on the nature of the natural spin-parity strange meson spectrum. All four solutions indicate the existence of a 0 + K meson which differs from and lies at a higher mass (~1425 MeV) than the traditional K

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P. Estabrooks et al. / K+p -+ Ken+n, K+rr-A ++

meson previously discussed in the literature. In addition two of the solutions imply the existence of a l - K * meson near 1675 MeV.

2. The experiment The experiment was performed at SLAC using 13 GeV rf separated K -+ beams incident on a 1 m hydrogen target. A forward wire spark chamber spectrometer [5] was used to detect the outgoing K and rr. Events corresponding to the neutron recoil reactions (1) and (2) and the A ++ recoil reactions (3) and (4) were selected by requiring that the missing mass opposite the Krr system lie in the range 0.75 < MM < 1.05 GeV for the neutron reactions and 1.0 < MM < 1.4 GeV for the delta reactions. In addition for the two neutron recoil reactions (1) and (2), a direct experimental subtraction was made to correct the experimental angular distributions for events associated with the small K - p -+ KrrA ° background occurring in this neutron missing mass interval [6]. In the case of the A ++ recoil reactions [1], the experimental missing mass spectrum from 1.0 to 1.6 GeV was fitted to a Breit-Wigner A resonance plus a polynomial background. The Krr cross sections were then corrected to correspond to the A ++ resonance contribution only. A multicell (~erenkov counter was used to provide K/rr identification for the doubly charged K~r systems in reactions (2) and (4). For reactions (1) and (3), positive K/rr identification was not necessary. Instead events that were ambiguous with a forward K °, A, or in the case of reaction (1), the reactions K - p ~ ~A, 0A, were explicitly rejected. In the case of reaction (1), events for which the rrn invariant mass was less than 2.0 GeV were also rejected. The experimental data samples contain 51 000 and 14 000 events respectively for reactions (1) and (2) and 104 000 and 26 000 for reactions (3) and (4). In addition, very large samples of K -+ -+ rr±Tr+rrbeam decays were obtained simultaneously with the KTr data and provided a direct measurement of the K+/K - relative normalization which is known to +2%. A maximum likelihood fitting procedure is used to correct the observed KTr data for the effects of the spectrometer acceptance, event selection criteria, and other factors. This yields acceptance corrected reaction cross sections and the spherical harmonic moments, YLM, of the Krr angular distribution as a function of Krr mass and four momentum transfer, t' ( - t tmin). The resulting KTr mass spectra and the t-channel unnormalized M = 0 moments are presented in figs. I - 4 as functions of Krr mass for the small momentum transfer region for each of the four reactions. The momentum transfer cuts are It'l < 0.15 GeV 2 for reactions (1), (2) and (4) and It I < 0.2 GeV 2 for reaction (3). The errors shown are the statistical errors only. The K+rr+n cross section is substantial; it rises smoothly from threshold to a broad maximum near 1.6 GeV. The K - a - A ++ cross section exhibits a noticeably different KTr mass dependence and appears to remain roughly constant over this mass range. This apparently different mass variation in fact reflects the different neutron and

P. Estabrooks et al. / K+-p -~ K+-Tr+n, K±~r--A ++ 3000

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I.I 1.3 m (K-v -+)

(GeV)

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_1

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Fig. 1. The ( u n n o r m a l i z e d ) t - c h a n n e l m o m e n t s of the K - n + a n g u l a r d i s t r i b u t i o n in the t' range 0 < - t ' < 0.15 G e V 2 as a f u n c t i o n OfMK7 r.

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Fig. 2. The ( u n n o r m a l i z e d ) t - c h a n n e l m o m e n t s o f the K+Tr + d i s t r i b u t i o n in the t' r a n g e 0 < t' < 0.15 G e V 2 as a f u n c t i o n OfMK7 r.

/ K+-p -+ K+-n+n,

1). Estabrooks et al.

494

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Fig. 3. T h e ( u n n o r m a l i z e d ) t - c h a n n e l m o m e n t s o f t h e K+~r - a n g u l a r d i s t r i b u t i o n in t h e t' r a n g e 0 < - t ' < 0 . 2 G e V 2 as a f u n c t i o n o f M K n . T h e d a t a s h o w n h a v e n o t b e e n c o r r e c t e d f o r t h e n o n A ++ b a c k g r o u n d in t h e missing m a s s c u t 1.0 < MM < 1.4 G e V .

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(GeV)

Fig. 4. T h e ( u n n o r m a l i z e d ) t - c h a n n e l m o m e n t s o f t h e K - ~ r - a n g u l a r d i s t r i b u t i o n in t h e t' r a n g e 0 < - t ' < 0 . 1 5 G e V 2 as a f u n c t i o n o f M K n . T h e d a t a h a v e b e e n c o r r e c t e d f o r t h e n o n - A ++ b a c k g r o u n d in t h e missing m a s s c u t 1.0 < MM < 1.4 G e V .

P. Estabrooks et al. / K+-p -+ Ken+n, K+lr-~ ++

495

A ++ recoil m o m e n t u m transfer dependence and particularly the rapid increase in tmin with Kn mass that occurs for the A ++ reaction. The fits to the K+n+n and those to the K - n - k ++ data were performed with L, M ~< 2. Separate fits indicated that higher L moments were not required to describe these data. Indeed the L = 1 and L = 2 moments in figs. 2 and 4 are quite small indicating that the I = 23-Krr system is predominantly S-wave. For the K - n + n and K+rr.-A ++ data, the maximum L value used in the fits increased from L = 2 to L = 6 with increasing Kn mass, as indicated in figs. 1 and 3, with M ~< 2 for all masses. L ~< 4 moments are required to describe the K*(1435) region, and the L ~< 6 moments are necessary above 1.6 GeV. The prominent features of the K - n + and K % r distributions are due to the 1 - K * ( 8 9 0 ) and 2+K*(1435) mesons. In addition, as reported previously [4], these data have been used to demonstrate the presence of a broad spin-parity 3 - K * resonance at ~1.78 GeV.

3. IOn production

Both neutron and zX recoil reactions are historic sources of information on Kn phase shifts [7,8]. Although the n exchange contribution to the A recoil reactions does not vanish at t = 0 as it must do in the neutron recoil reactions, this advantage is counterbalanced by several disadvantages: (i) the difficulty, in the absence o f a direct measurement o f the recoil system, o f cleanly isolating the broad A resonance from the background in the missing mass spectrum; (ii) kinematic reflections from the diffractive Q production reactions, expected [1] to be important at low MK,r; (iii) the rapid increase of tmi n with increasingMKn in the 2x reactions. At M K , = 1.5 GeV, tmin = - 0 . 0 6 GeV 2 for Plab = 13 GeV/c, so that, in a t' region 0 < - t ' < 0.2 GeV 2 , one can see only about 20% of the n exchange. (iv) Moreover, the A recoil reactions are intrinsically more complicated than the neutron ones, due to the doubling of possible recoil helicity states. We discuss the Knn production mechanisms in subsect. 3.1 and KnA production in subsect. 3.2. 3.1. K n production in K p -+ Krrn

The extraction of Kn scattering amplitudes from these data requires isolating the n exchange contribution to the production amplitude. Data at 4 GeV [9] for the line reversed reactions K - p -+ K* (890) n , K+n -+ K*(890) p

(5)

have provided valuable information about the K* (890) production mechanisms. In ref. [10] it was shown that these two reactions could be simply described in terms of strongly exchange degenerate n-B and A 2-p Regge exchanges and 'cuts' (non-

496

P. Estabrooks et al. / K ± p ~ K+-n+n, K±~r- A ++

evasive contributions, absorptive corrections ...) which have simple t-channel structure. We have shown [2] that for K*(890) and K*(1435) production in reaction (1) at 13 GeV, a good description of the momentum transfer dependence of the data is also provided by this simple exchange model. Moreover, the MK,, dependence of the ratios of non-pi/pi exchanges was seen to be ahnost identical to that observed in nN -~ pN compared with nN -+ fN [11 ]. For each Kn mass interval, we parametrize the t dependence of the amplitudes L~,_+ for production of a Kn state with angular momentum L, t-channel helicity X, by natural (+) or unnatural ( - ) parity exchange by

x/-t L o = gL t12

t '

1

/-1- ~ g r v ~

+ 1)7c

LI+=-~gLX/~+I) L~,± = 0 ,

3'C+2

X~> 2 ,

7A[t'l

, (6)

where { ~-i} refers to an incident K - or K ÷. For partial-wave analysis, we are only concerned with small t, e.g., the It[ < 0.2 GeV a region. The simplified parametrization of eq. (6) is a good small It I appro~:imation to the more general description used in ref. [2]. The relation o f g L to the Kn scattering amplitudes, aL, is given by n

MKrr aLeo(t /'z2)

gL = t ' ~ 0 ( 2 ) ~ -

(7)

where the subscript on c)¢ refers to the net Kn charge, the superscript to the recoil particle. The Kn scattering amplitudes are written * as aL =N/'~+

1 e L sin 8L eiSL ,

(8a)

in the elastic region, and a L = la L [ e i~L ,

(8b)

in the inelastic region. Requiring purely elastic scattering corresponds to setting eL = 1. The normalization constant, c~g, is determined by requiring the K - n + Pwave in the 900 MeV region to be an elastic Breit-Wigner resonance. The K ÷ normalization constant, c~ ~, is then fixed by the relative K - and K + experimental normalizations and the value of cry. We have also calculated cK~ from the absolute 1 3 For K - n + scattering, we use a L =- a~ + ~a L where the superscripts refer to twice the Kn isospin and the a~,I are given by eq. (8) for elastic scattering. The isospin Clebsch-Gordan coefficient of 2 has been absorbed into the definition of c~ O.

497

P. Estabrooks et al. / K+-p ~ Kin+n, K ! n - A ++ 0.94~ I

I

~ , ~

750

MK~rL-- 1 . 0 GeV I

K- p.--.~ K-gr ÷n

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(b)

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represent the t-dependent amplitude parametrization described in sect. 3 of the text. experimental normalization and the Chew-Low equation [12]. This gives a value some 10% larger than that obtained from the requirement that the K*(890) be elastic. The parametrization of eq. (6) provides a good description of the t dependence of the m o m e n t s , N ( Y L M ) , of the K - n + angular distribution in all MK~ bins. In fig. 5a both the data and the results o f the t-dependent fits are shown for - t ' < 0.2 GeV 2 for the 0.94 < MK~r < 1.0 GeV bin. We have checked that performing the fits for - t ' < 0.3 GeV 2 does not significantly change the values of the parameters.

498

P

Estabrooks et al.

K±p ~ K+-Tr+n, K±Tr-~ ++ K -+ p

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Fig. 6. The MKzr dependence of the parameters of eqs. (6) and (9) to (12) describing the t-dependence of (a) the K~rn, and (b) the K~rA production amplitudes. The open circles of fig. 6a (b) refer to K+~r+(K-zr-) production, the solid points to K-~r+(K+Tr - ) production. The curves are the polynomial fits used in the single t bin analyses o f sect. 4.

In fig. 6a, the values of the production parameters * b, 7 c , 7A, describing the tdependenc e are plotted as functions of K - n + mass. For the K+n + reaction, only the S-wave is significantly non-zero and so b is the only reliably determined parameter. Its values are represented by the open circles in fig. 6 and are seen to be in agreement with the K - n ÷ results (solid points). We note that the p-A2 and cut contributions become increasingly important with decreasing KTr mass. In the course of describing the t-dependence of the Kn production using eqs. (6), we also obtain results for the Kn scattering amplitudes. The I = 3 S-wave phase shifts, 6 3, obtained by imposing elastic unitarity in the fits to the t-dependence of the

* These parameters were obtained from fits in which e S o f eq. (8) was left as a free parameter while ep was set equal to h 0 .

I

499

P. Estabrooks et al. / K+p ~ K+-Tr+n, K+-n - A ++

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data. The open circles were obtained by extrapolating the production amplitudes to t = u2, the solid points are from the single t bin analysis. The curves represent effective range fits with the parameter values of table la.

K+Tr+ m o m e n t s are shown as open circles in fig. 7a. The exotic P- and D-waves were f o u n d to be consistent with zero, 6 3 = 6 3 = 0. The I = ½ phase shifts, 6 ~ and 6 ~, in the elastic region, MK_Tr+ < 1.3 G e V , are shown by the open circles in fig. 8a.

P. Estabrooks et al. / K+-p ~ K+-*r+n,K+-+r-&++

500 180

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(GeV)

Fig. 8. The I = I Kzr phase shifts in the elastic region calculated from (a) K - p --, K-n+n, or (b) K+p -+ K+zr--A÷+ data. Open circles represent results from extrapolating to t = #2, solid points come from the single t bin analysis. The input values of 6 ~ were taken from the effective range curve of fig. 7. The curves through 6~ represent Breit-Wigner fits to the points from 0.8 to 1.0 GeV, yielding the resonance parameters shown in table lb. The curves through 61 represent effective range fits (for M < 1.2 GeV) yielding the parameters listed in table 1c.

3.2. K?r production in Kp -+ K g A ++ Isolation of the ~Texchange contributions to the A ++ recoil reactions (3) and (4) is complicated by the larger number of possible helicity states of the A. We use L~A~v to represent the amplitude for angular m o m e n t u m L, helicity X, K~T production by natural (+) or unnatural ( - ) parity exchange with A helicity o f ~A and w i t h N = + corresponding to proton helicity of +½. We have shown in ref. [3]that the momentum transfer and K* helicity structure of K*(890) and K*(1435) production in reaction (3) can be well described by: (i) strongly exchange degenerate ?r-B Regge exchanges coupling only to the t-channel amplitudes L°+ (and L _0 l _ ) , . (ii) EXD p-A2 1- in the contributions coupling to the t-channel spin flip amplitudes L ~+ and L 1+ - , as expected from the Stodolsky-Sakurai model [13] of p ratio L ~++= x/3-L t] + exchange; (iii) non-factorizing 'cut' contributions to the s-channel net helicity nonflip amplitudes (S)L ~+ and (S)L ~_. For K* (890) and K* (1435) production, the MK7r dependence of the ratios of non-pi/pi exchanges was seen to be almost identical to that observed in gN ~ lrzrN and in KN --' KzrN. For each MK, , interval, we parametrize the ~-B contributions to the (t-channel)

P. Estabrooks et al. / K±p --, K±*r+n,K+-¢r--A++

501

amplitudes as X/~A

L°I+(zr'B) : g L

- MN) 2 - t

tl 2 - t

'

(9)

while the p-A2 contributions are given by L1 + ~

1

1+

~

'

~

t_t.p-A2) =-~--~L 3+(p-A2) =X/2 7A(--t ) x / L ( L + 1 ) g L ,

(10)

where gL is given by eq. (12) below. The factor of X/3 is motivated by the StodolskySakurai model. In addition to the pole exchange contributions of eqs. (9) and (10), all t-channel amplitudes also have contributions obtained by crossing the s-channel cut contributions, parametrized as (S)Lll- = ~

1 (s)l 1

1

43+ = ~ g L X / ~

+ 1) 7C ,

(11)

to the t-channel. In general, the cut contributions to the two different s-channel amplitudes need not be related by the x/3 ofeq. (11), but we found that the data at small [t'l could not determine the ratio of the two cuts so we chose to relate them as in eq. (11). The simplified parametrization of eqs. ( 9 ) - ( 1 1 ) is a good small It l approximation to the more general description used in ref. [3]. The KTr scattering amplitudes, aL, of eq. (8) are related to gL by _ cvl& MK1r aL ebL(t-u2) g L - 't'O(2) N/q

(12)

As in the neutron recoil reactions, C~o~ is determined by requiring the K+Tr- P-wave in the 900 MeV region to be an elastic Breit-Wigner resonance; c~# is then calculated from qZo~ and the relative K - and K + experimental normalizations. The parametrization of eqs. ( 9 ) - ( 1 2 ) provides a good description of the small-t behavior of the moments,N(YLM), of the K+Tr- angular distribution in allMK, bins, as demonstrated, for example, in fig. 5b for the 0.94MKn < 1.0 GeV bin. In fact, we shall find that all the bL of eq. (12) cannot be determined from the small t' data, so we set bF = b D = bp --- b. In fig. 6b, the values of the production parameters * bs, b, 7¢, 7A describing the t-dependence are plotted as functions of K+Tr- mass. For the K - n - reaction, only the S-wave is significantly non-zero so bs, shown by the open circles in fig. 6b, is the only reliably determined parameter. As in the KTrn case, these fits also produce Kn scattering amplitudes. The values of 6~ and 61, 6~ for MR+ n- < 1.3 GeV, resulting from fits with elastic unitarity imposed, are shown by the open circles in figs. 7b and 8b, respectively.

~' These parameters were obtained from fits in which eS of eq. (8) was left as a free parameter while ep was set equal to 1.0.

502

P. Estabrooks et al. / K i p -~ K+-w+n, K+-Tr--A++

4. I~n partial waves from single t bin analyses A detailed study of the Kn partial waves as a function of Kn mass can be carried out by using the experimental Kn moments evaluated in small KTr mass intervals for a single wide momentum transfer bin in the n exchange dominated small t region. The procedure used to extract the Kn partial-wave amplitudes from these single wide t bin Kn moments is presented in subsect. 4.1. The results o f separate analyses of 3.,+ 7r+ and K - n - scattering are given in subsect. 4.2. Results from the p u r e l y / = ~r,_ independent analyses o f the K-Tr + and K+n - data in the elastic region (MK, < 1.3 GeV) are presented in subsect. 4.3. In subsect. 4.4, after a discussion of the discrete ambiguity problem that occurs at higher Kn mass, we present the results of the separate K - n + and K+Tr- partial-wave analyses in the inelastic region,MK~ > 1.3 GeV. Finally in subsect. 4.5, we present the results of a simultaneous partial-wave analysis of both the neutron and delta recoil data, allowing for the different t-dependences o f the two reactions. 4.1. M e t h o d o f analysis

Consideration o f all the moments of the Kn angular distribution in a broad t bin permits a more detailed investigation o f the MK7r structure of the Kn partial-wave amplitudes. The M = 0 Kn moments to be used in the following analyses have been presented for reactions ( 1 - 4 ) in figs. 1 - 4 respectively. The m o m e n t u m transfer cuts used are [t'l < 0.2 GeV 2 for K+p ~ K+Tr-A ++ and It'l'< 0.15 GeV 2 for the other reactions. The calculation of the Kn partial waves from these t-averaged moments is straightforward since the t-dependence of each amplitude is known. The analytic form of the t-dependence is given by eqs. (6) and ( 9 ) - ( 1 2 ) . In the preceding section the values of the parameters contained in these expressions were determined at different values of Kn mass by independent fits to the observed t-dependence of the moments. In this section, we assume that the Kn mass dependence of these t structure parameters is given by low order polynomial fits to the independently determined points shown in fig. 6. Thus the actual values o f the t structure parameters used in the subsequent single t bin analyses correspond to the curves * shown infig. 6. We have verified that the results obtained for the KTr partial waves are unaffected by this smoothing of the mass dependence of these parameters describing the t structure of the moments. This was done by repeating the charge-zero KTr analyses of the previous section with the parameters bs, b, 7 c , and 7A fixed at values corresponding to the smooth curves in fig. 6. We found no significant differences between the * Since b and b S for the 4 ++ recoil reactions are not very well determined at high MKn, we assume that b and b S have constant values of 0.5 and 1.5 respectively for reaction (3) for MK~r > 1.5 GeV. These values are consistent with those obtained in the t-dependent fits of subsect. 3.2.

503

P. Estabrooks et al. / K+-p ~ Kin+n, K+-n- A ++

resulting phase shifts and those obtained in the previous section. This indicates that the correlation between small changes in the values of the parameters describing the production processes and the quantities which describe Kn scattering, the partialwave amplitudes, is quite small. The Kn moments in the t bin 0 < - t ' < T can now be expressed in terms of the t , partial-wave amplitudes, aL, and t integrals, ( L x L u ) , as demonstrated, for example, in the appendix * of ref. [14], where o t

,

(LxLu) -

f dt'L'x(t')L~(t') -T

(13)

a L, a~ T

In each MKn interval, the (L'xL~) can be calculated explicitly and the ar are then determined from a least squares fit to the Kn moments. 4.2. K+n + and K - n -

results

For K+n + and K - n - , we assume elastic unitarity so that las [ is just [sin 6 3 I. The negative sign of 6~ is selected in the K - n + and K+n - partial-wave analyses by requiring that the I = ½ S-wave be elastic in the low MKn region. The results for 6~ are shown as the solid points in fig. 7a and b and can be seen to be in excellent agreement with those (open circles) obtained from the t-dependent fits o f the previous section. Here also, the exotic P- and D-wave phase shifts are always smaller than three degrees for MK~ < 1.8 GeV. The K+Tr+ (K-Tr-) total cross section can b~ calculated at each MKn value using these phase shifts; the cross section decreases from 4.8 -+ 0.7 (4.4 -+ 0.4) mb at 0.9 GeV to 2.4 -+ 0.3 (1.8 -+ 0.2) mb at 1.6 GeV. We have fitted these results for 63 to an effective range form 1 q2L+l cot 62 r = ~L2/+l~r L21q Z •

(14)

We find that 8~ is well described by eq. (14) with the low-energy Kn parameters shown in table la; there is good agreement between the parameters calculated from the K+n+n phase shifts and those from the K - n - A ++ analysis. These parameters yield the curves shown in fig. 7. 4.3. Elastic K - n + and If+n - results

In K - n + and K+n - scattering, the I = ~1 D-wave is known to be inelastic [15]. The I = ½ S- and P-waves could also become inelastic for MKn > 1.3 GeV. Therefore

* The conversion from the density matrices of ref. [14] to the notation used here is given by pL'L = Re[aL,a~(L,xL~)].

504

P. Estabrooks et al. I K i p -~ Ken+n, K+-n--A++

Table 1 The1 values of the parameters describing the MK~r dependence of (a) the I = 3 S-wave, (b) the I P-wave for 0.8 < MKTr < 1 GeV, (c) the I = 21 S-wave for MKn < 1.2 GeV and (d) the D-wave magnitude in the mass range 1.25 < MK~r < 1.6 GeV Parameter

Neutron recoil only

(a) a 3 (GeV -1) r~(GeV - l )

A ++ recoil only

-1.03 + 0.I0 -0.94 +- 0.5

(b) M R (MeV) I'(MeV) R (GeV - 1 )

896.0 +- 0.7 50.5 +- 1.1 6.9 -+ 2.7

(c) a 1 (GeV -1) rsr(GeV-1)

2.4 -+ 0.1 -1.7 +-0.3

(d) x M R (MeV) F (MeV) R (GeV -1)

0.51 1431 88 2.0

-0.92 -+ 0.08 -1.30 +- 0.4 895.3 +_0.3 52.5 -+ 0.4 3.7 -+ 0.5 2.28 -+ 0.05 -2.2 -+0.1

-+ 0.03 +- 5 ±9 ± 0.9

0.48 1439 109 3.3

+ 0.03 _+4 -+ 8 -+ 1.0

Combined analysis -1.00 +- 0.05 -1.76 +_0.3 895.7 +_0.3 52.9 _+0.4 4.3 _+0.4 2.39 +- 0.04 -1.76+_0.10 0.49 1434 98 2.4

_+0.02 _+2 -+ 5 -+ 0.6

The errors reflect the systematic, as well as the statistical, uncertainties. we only impose elastic unitarity for MK~r < 1.2 GeV. The I = 3 p. and D-wave phase shifts are assumed to be zero while the I = 3 S-wave is given by the effective range curves * o f fig. 7. In the region f r o m 1.2 to 1.3 GeV, we require the P- and I = ~- Swaves to be elastic but allow inelasticity in the D-wave. Since the P-wave is small (and not well d e t e r m i n e d ) in this mass region, forcing it to be elastic is n o t overly restrictive. Thus, the imposition o f elastic unitarity for S- and P-waves is, in practice, simply a prescription for determining the overall phase. The results for 81 and ~5~ in the region 0.7 < M K ~ r < 1.30 G e V are shown by the solid points in fig. 8. T h e y are in good agreement with the ( e x t r a p o l a t e d in t) results (open circles) o f the previous section. The curves on the/5 ~ plot represent Breit-Wigner fits to the phase shifts ap

] i6~

~ / 3 - sin ~ ve

(_~__R)3

MRF = m~

- M~.

-/M

RF

'

1-1 + ( q R R ) 2 q

from 0.8 to 1.0 GeV. The resulting resonance parameters are listed in table 1 b; the K+rr - and K - n + results are in good agreement. The K* widths q u o t e d have been c o r r e c t e d for the small effects of-the e x p e r i m e n t a l mass resolution, which is estimated to be +--5MeV. * These are the exotic phase shift values which were also used in the t-dependent K-lr + and K+~r- analyses of the previous section.

P. Estabrooks et al. / K+-p ~ Kin+n, K ± , - A ++

505

The curves on the S-wave plots represent the results of effective range fits of the form of eq. (14) to 6~. The resulting low-energy KTr parameters, listed in table lc, again demonstrate the excellent agreement between the K-•r + and K+Tr- results. We note that, at e v e r y M K , value, there is another possible value of the S-wave magnitude and phase relative to 6 ~ but it is in such violent disagreement with unirarity (except right at the P-wave resonance) that there can be no question that it is unphysical. In addition, it is of course impossible, as in any energy-independent phase shift analysis, to rule out rapid 180 ° changes in the phase shifts between any two neighbouring mass bins. 4.4. K - T r + a n d K+Tr - p a r t i a l w a v e s in the inelastic region

In mass regions where the KTr partial waves may be inelastic, it is no longer sensibl~ to impose elastic unitarity. Without accurate information on the total cross section or on inelastic channels, the data determine only the magnitudes and relative phases of the different partial waves. Fortunately, in non-exotic channels such as K-Tr+ or K+Tr- , the existence of resonances in the leading partial waves, coupled with unitarity and continuity requirements, provides strong constraints on this phase. There is also a problem of discrete ambiguities * in the inelastic region. This is most readily apparent as an indeterminacy in the signs of the imaginary parts of the (Barrelet) zeros [16], z i , of the scattering amplitude and, in fact, the signs o f Im zi provide a convenient [17] means of distinguishing and classifying the various solutions for the partial waves. Moreover, only when one o f the lm z i approaches zero is it possible to change from one solution to another. In the Krr case, since elastic unitarity allows only one solution up to about 1.3 GeV, the possibility of discrete ambiguities does not arise until one of the Im z i approaches zero. From fig. 9 where the real and imaginary parts of the zeros have been plotted (for one choice of sign for Im zi), it is apparent that this cannot occur before M K . ~ 1.45 GeV. However, in the region from 1.4 to 1.5 GeV, both Im zl and Im z2 could change sign so that although below 1.4 GeV there is only one possible solution, above 1.5 GeV there are four which can be specified by the signs of Im z 1 and Im z 2 , as indicated in table 2 ** In figs. 10 and 11 we present the magnitudes and phases of the K-Tr + and K+Trpartial waves for all four solutions. Although the solution is unique for MKn < 1.5 GeV, for clarity we show each solution in the mass interval 1.3 to 1.9 GeV, and include the magnitudes and phases for the full mass range with solution A. The Sand P-partial waves below 1.3 GeV were calculated from the 1 = ~ and I = 3 phase shifts shown in figs. 7 and 8. In this MK~r region the imposition of elastic unitarity determines the overall phase. Above 1.3 GeV, the data determine only the relative * We truncate the partial-wave expansions at L = 3. Small non-zero higher partial waves lead to the usual continuum ambiguity patches around our solutions. ~'* We have chosen to label the Kn solutions in the same way as the nn solutions of ref. [ 17 ].

P. Estabrooks et aL / K+-p ~ K%r+n, K+-~-& ++

506

I

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Fig. 9. The (complex) zeros of the non-exotic KTr scattering a m p l i t u d e as calculated from th( partial waves of figs. 12 and 13, resulting from the simultaneous analysis of K+Tr- and K - n + data. The solid points (open circles) represent the real (imaginary) parts of the zeroes. Either l m z l and/or lm z 2 could change sign at MK7r ~ 1.5 GeV.

Table 2 The definition of the four possible K - w + partial-wave solutions according to the signs of the l m z i at MK~r = 1.6 GeV; the existence of the K*(1435) resonance requires Im z 2 < 0 below about 1.5 GeV and the K*(1780) resonance requires Im z 3 < 0 everywhere Solution

Im z 1 (1.6)

Im z 2 (1.6)

m

-

--

B

+

-

C

+

+ +

D

ssutu q a e o l~ l u q l oz.ts~tldtuo aA~ "AOBI gL I j o qlpI.A~ p u e A o D R L ' r ssetu j o [~,] a a u e u -osoJ e j o l e q l o l p u o d s o ~ o a ol u o s o q o uooq s e q a s e q d O^eA~-::I o q l ' s a s o d a n d ~ u ! l l o l d .Io~ .d~b o l 0A.Ill~10a p o s n s e o t u oJe s o s e q d a q l A o D 9"[ OAOqV "ltl0p.IA0 S.I SUO.II31~OJ [.IO0OJ ellOp p u e u o J l n o u UOOA~lOq lUOtUOOJ~e p o o ~ 0 q l 'U.Ie~ OOUO 'ojOqA~ p I aIqul Ut. pols!l s J o l o t u e J ~ d oou~uosoJ ~u!PlO.t£ ' ( | I p u e 0 1 "s~.tJ u o S0AJrlO) LtlJOJ JO!.~.t/~A -1.tO~8 e O1 AOD 9" [ o l gg' I t u o J j s o p n l ! u ~ e t u OAeA~-(I oql ~ u ! l l U £ q p a u ! e l q o s e ~ s.tq,L "os~qd Jou~!AA-1toJ8 aAea~-(I ( o l n l o s q u ) a q l o l aa!luIO~ UA~OqS OJ~ s o s e q d aaet~ -el p u e -d '-S oql 'AOD 9' I > u:x//g > ~. I Jo::I "SOAI~A~Ie!lJ~d luaJoJJ.Ip o q l JO soseqd

"0 < ~z tu I '0 < lz tu I , q uo.tlnlos u!'.0 < ~z tu I '0 > Iz m I 'D uo!lnlos u! '.0 > ~z m I '0 < [z tu I ' 8 uo!lnlos u ! ' 0 > ~z m I pu~ 0 > lz m I 'V uoBnI os u! al!qA~ '0 > ~Z m I aA~q suo!lnlos lI~ 'AOD g'I ~^oqv "I~g!lu~p! ~ suo!lnlos snoj ii ~ 'A~D F'I lnoq~3 ~aoia 8 "AaI~ ~LI qlp!A~ 'AaD 8L'I sstup, jo o3ueuoso~ 13~uBuasasdox aasn~ otil Aq u~oqs sonleA oql OA~q O1 Uosoqo S.I A~ 'AOD 9"1 < u>I/4t aoj '.OAth30q~ £q u~oqs oseqd ~ou~!AA-1to:tfl aql aAeq 03..LIasoq3 s.[ ( ] 0 'AoO 9"I > ")I/4~ > g ' I Jod "Alpel!un o.tlSelO £q pox!j s! oseqd [iI~.tOAO oql 'AO o g'[ A~OlOfl 'p[ alqel u! pols.q s.mlatueaed oql ql!m IOVl ol slt.J Iou~l.A4t-~,.tg~flluaso~doa (1¢ puel qt, I aoj SOA.Ii'loOqI "SOA13AXIg!l.ted +u ,'Sl oql JO sosgqd pug sopnl!u~etu aq~L "0I "~H (AaO) +x-'~Ix',l ~°i

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508

P. E s t a b r o o k s et al. / K ± p -+ K +~r+n, K ± ~ - A ++

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MK,-~- (GeV) Fig. l l. The magnitudes and phases of the K+~ - partial waves, the curves for la D i and q)D represent Breit-Wigner fits to laD I with the parameters listed in table ld. The curve for (~F represents the phase associated with a resonance of mass 1.78 GeV and width 175 MeV. For M K n < 1.3 GeV, the overall phase is fixed by elastic unitarity. For 1.3 < MKn < 1.6 GeV, (MKn > t.6 GeV) the phases are measured relative to @D (q~F) chosen to have the Breit-Wigner phase shown by the curve.

value the choice of overall phase is arbitrary and all values provide identical values of dOKn/d~.

4.5. Simultaneous neutron/delta analyses The results of the previous subsections indicate good agreement between Kn partial waves calculated from the neutron recoil reactions (1) and (2) and those obtained using the delta recoil reactions (3) and (4). Moreover, it is straightforward to apply the analysis techniques of subsect. 4.1 to a combined neutron/delta analysis. As before, we express the Kn moments in terms of the partial-wave amplitudes and the

P. Estabrooks et al. / K+-p ~ Ken+n, K+-~r-~ ++

509

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Fig. 12. (a) The I = 3 S-wave Krr phase shift obtained from a simulatenous analysis o f single t bin K - T r - and K+n + data. The curve represents an effective range fit with the parameters listed in table la. (b) The I = I S- and P-wave KTr phase shifts obtained from the simultaneous K+~rand K - n + analysis o f subsect. 4.5 o f the text. The input values of 63 were taken from the effective range curve of fig. 12a. The curve through 6 ~ represents an effective range fit (for MK~r < 1.2 GeV) yielding the parameters in table lc, while the parameters of the Breit-Wigner fit to 6 can be found in table lb.

r

,

t a v e r a g e s ( L x L u) o f eq. ( 1 3 ) . N o w , h o w e v e r , t h e a L are c o m m o n t o t h e n e u t r o n a n d d e l t a recoil r e a c t i o n s w h i l e t h e (L'xL*~) are c a l c u l a t e d s e p a r a t e l y for e a c h r e a c t i o n . T h e I = ~ S-wave p h a s e s h i f t s o b t a i n e d f r o m t h e c o m b i n e d K + n + / K - n - a n a l y s i s are p l o t t e d in fig. 12a. A s b e f o r e , w e f i n d t h a t t h e I = ~ P- a n d D - w a v e s are negligible ( a l w a y s less t h a n 3 ° ) a n d t h a t t h e M K , r d e p e n d e n c e o f ~53 is well d e s c r i b e d b y t h e

510

P. Estabrooks et al. /K+-p -* Ken+n, Ken--A ++

effective range formula of eq. (14) with the parameters listed in table la. This fit is represented by the curve on fig. 12a and also serves to specify the 63 input in the elastic K + n - / K - n + analysis. The I = ½ S- and P-wave phase shifts o b t a i n e d f r o m the j o i n t K + n - / K - n + analysis are shown in fig. 12b for MK~ < 1.3 GeV. Once again, we see a steady increase with MK~ of 6~/z, which is well described by an effective range f o r m w i t h the parameters o f table lc. Apart f r o m the single MK~ bin at 895 MeV, we find no evidence for the up solution for 6~. It is, o f course, n o t possible to rule out sudden 180 ° changes in any phase shift. The K * ( 8 9 0 ) resonance parameters resulting from a Breit-Wigner fit to 6~ for 0.8 < M K ~ < 1.0 GeV are listed in table lb.

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_< ~8o° Irr

.

0°

0.8

.

.

1.0

.

1.2

1.4

1.6

1.8

1..a

1.6

1.8

14

1.6

1.8

14

MK~ (GeV)

Fig. 13. The magnitude and phases of the Kit partial waves calculated in a joint K + n - / K - n +" analysis. The curves for la D I and OD represent the Breit-Wigner fits to laD [ with the parameters listed in table ld. The curves for OF represent a Breit-Wigner description of an F-wave resonance of mass 1.78 GeV, width 175 MeV. Below 1.3 GeV, the overall phase is fixed by elastic unitarity. For 1.3 < MK7r < 1.6 GeV (MK1r > 1.6 GeV) OD (OF) is chosen to have the Breit-Wigner phase shown by the curve. Below about 1.5 GeV, all four solutions are identical. Above 1.5 GeV, the four solutions are classified according to their Barrelet zeros.

The

(b)

-10.0÷0.5 -14.5+1.5 -16.5 i 1.5 -17.0±0.6 -18.0+1.2 -18.5+1.0 -18.2+0.7 -21.4 + 0.5 -21.1+0.5 -22.4 + 0.5

63 S

21.0 27.2 40.7 36.8 40.4 38.0 38.1 39.1

+ + + + ~ • + +

3.0 2.8 0.9 1.4 1.6 2.3 i.i 1.2

63 r

6.0 i0.0 14.0 23.9 28.3 39.5 49.6 70.6

+ + + + + ~ + +

0.9 0.7 0.5 0.5 0.5 0.5 0.6 1.1

shifts

3 6D

0.895 0.905 0.915 0.925 0.935 0.945 0.955 0.980

region.

1.22 1.26 1.30 1.34 1.38 1.44 1.52 1.60 1.72

MK~

39.6+1.5 34.7 + 1 . 3 33.5 ± 1 . 7 37.2 ± 1 . 8 40.9 ± 1 . 7 42.8 + 1 . 7 46.9 ± 1 . 6 48.4 + 0 . 7

in the elastic

-0.7±0.2 -0.6±0.2 -1.3 ± 0.2 -1.2±0.2 -1.3 + 0.2

shifts.

-0.8+0.2 -0.2+0.3 -0.9 ± 0.3 -1.1±0.3 -0.6 + 0.3

I = 1/2 S and P wave phase

0.73 0.82 0.90 0.94 0.98 1.02 1.06 i.i0 1.14 1.18

MK ~

I = 3/2 S, P, and D wave phase

0.730 0.780 0.820 0.845 0.855 0.865 0.875 0.885

The

(a)

3

96.2 108.3 122.4 134.6 140.4 147.4 150.9 157.0

± ± ± ± -+ ± + +

3.1 1.3 0.8 0.7 0.6 0.6 0.7 0.3

-23.5±0.6 -23.9±0.7 -23.8 + 0.7 -26.4+0.7 -25.0±0.9 -27.4+0.6 -26.7+0.7 -25.6 ± 1.0 -22.8±1.0

3 6S

53

1.020 1.060 i.i00 1.140 1.180 1.220 1.260 1.300

53.9 57.8 60.2 61.2 66.3 69.3 71.7 79.3

-0.6+0.3 -0.6±0.3 -0.I ± 0.4 -0.7±0.4 -0.3±0.4 -1.2±0.3 -2.4+0.3 -2.1 ± 0.4 -2.4+0.4

The K~ p a r t i a l waves of Fig. 12 and 13, as c a l c u l a t e d from the joint fits to the n e u t r o n and delta recoil reactions.

Table

+ ± + ± + ± + +

0.8 0.8 0.9 0.9 2.0 1.4 4.8 1.8

162.9 165.2 167.5 168.5 173.0 170.8 166.7 176.3

-1.5±0.2 -1.3±0.3 -1.7 ± 0.3 -2.5±0.3 -1.5±0.4 -2.3±0.3 -1.5±0.3 -1.4 + 0.4 -0.5±0.2

63

_+ ± -+ ± ± _+ + +

0.4 0.4 0.5 0.6 1.8 1.6 4.2 8.8 L~

i+

I+

0. 0. 0. 0. -6. -6. -3. -7. -i0. -6. -ii. 13. 2. -18.

1.14 1.18 1.22 1.26 1.30 1.34 1.38 1.42 1.46 1.50 1.56 1.65 1.75 1.85

8rot

-16. -16. -6. 0. -27.

MK~

1.50 1.56 1.65 1.75 1.85

Solution B

0ro t

MK~

0.63 0.56 0.27 0.14 0.36

0.87 0.93 0.96 0.98 1.01 1.05 1.06 1.00 0.84 0.52 0.40 0.33 0.39 0.48

0.01 0.01 0.01 0.03 0.02 0.02 0.02 0.01 0.01 0.03 0.02 0.02 0.02 0.03

+ ± ± ± +

0.01 0.01 0.03 0.04 0.03

[asl

± ± ± + ± ± 4 ± ± ± _+ ± + ±

las[

~s

± + + ± ± + -+ ± ± ± ± + ± ±

i. 2. i. 5. 2. 3. 2. i. i. 2. 3. 8. 4. 6.

161. + 2. 183. ± 3. 190. +i0. 97. ± 7. 122. ±i0.

73. 78. 81. 83. 99. 104. 109. 126. 142. 146. 164. 143. 131. 153.

~s

is c h o s e n as the r e f e r e n c e wave.

0.16 0.22 0.50 0.53 0.52

0.35 0.21 0.28 0.40 0.13 0.20 0.19 0.20 0.21 0.30 0.32 0.22 0.23 0.28

0.02 0.05 0.05 0.12 0.03 0.05 0.03 0.02 0.01 0.07 0.05 0.03 0.02 0.04

± ± + ± ±

0.07 0.05 0.02 0.02 0.03

lapl

~ + + ± ± ± ± ± ± ± + ± + +

lapI ± i. i 2. ± 2. + 4. ± 9. ±i0. 4 8. + 7. + 8. + 7. ± 5. +17. +ii. ±i0.

103. ±18. 114. +ii. iii. + 8. 144. ± 4. 187. + 8.

~p

169. 173. 171. 157. i~7. 164. 154. 163. 169. 209. 222. 177. 160. 189.

~P

0.58 0.50 0.30 0.23 0.33

0.01 0.09 0.04 0.01 0.35 0.40 0.75 1.02 1.02 0.71 0.58 0.48 0.44 0.48

0.02 0.09 0.06 0.14 0.02 0.04 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02

± ± + ± ±

0.03 0.04 0.02 0.02 0.03

laDl

± ± + 4 ± + -+ + + ± ± ± ± ±

laD[ i. i. 2. 6.

+D

± 7. _+ 4. ± 7.

± ± _+ ±

138. 152. 141. ±i0. 138. ± 8. 165. + 7.

0. 5. 2. i. 14. 22. 39. 74. 115. 138. 152. 135. 148. 183.

~D

0.09 0.ii 0.28 0.43 0.43

0.08 0.12 0.29 0.43 0.43

0.04 0.04 0.03 0.02 0.03

laFl + 0.04 + 0.03 ± 0.03 + 0.03 ± 0.03

± + _+ ± ±

laFl

23 21 20 20 20 21 30 40 36 17 ii ii 15 12

o to t (mb)

Otot (mb) ~F 22 9. ± 23. 14 26. ± 12. 14 30. 16 70. 15 127.

8. ± 25. 34. ± 13. 30. 70. 127.

~F

K~ partial w a v e s in the i n e l a s t i c region. All four s o l u t i o n s are i d e n t i c a l b e l o w MK~ = 1.5 GeV. The second column is the phase rotation, ~ , r e q u i r e d to give the o v e r a l l p h a s e shown in Fig. 15. Note that for rot 1.28 < MK~ < 1.60 the D wave is c h o s e n as the r e f e r e n c e wave and that in the r e g i o n MK~ > 1.6 GeV, the F wave

Solution A

(c)

Table 3, continued

12. -3. 8. i0. -12.

0r°t

0rot

18. 10. 13. 9. -8.

MK71

1.50 1.56 1.65 1.75 1.85

Solution D

1.50 1.56 1.65 1.75 1.85

MKn

Solution C

0.45 0.24 0.23 0.38 0.58

0.64 0.59 0.55 0.56 0.61

+ + ± + +

0.01 0.01 0.02 0.02 0.02

0.03 0.03 0.03 0.03 0.03

[as]

± ± i i +

[as]

127. 121. 92. 91. 123.

114. 122. 117. 127. 152.

-+ 4 . i10. +13. + 3. + 5.

~S

i. 2. 6. 3. 3.

i. 2. 6. 3. 3.

± ~ ± + ~

~S

+ ± -+ +±

+s

114. 122. 117. 127. 152.

0.49 0.60 0.55 0.51 0.36

0.17 0.20 0.20 0.23 0.30

+ ± * ~ +

0,07 0.05 0.03 0.02 0.04

laP[

0.07 0.05 0.03 0.02 0.04

± + ± + +

0.06 0.04 0.02 0.02 0.03

[ap[

+ ± ± ± _+

laPI

0.17 0.20 0.20 0.23 0.30

~P

58. 70. 99. 132. 159.

~p

75. 83. 98. 119. 115.

± ± ± + ±

9. 5. 7. 4. 8.

418. +i0. -+16. ± 9. i 8.

~P

75. +18. 83. ~i0. 98. +16. 119. + 9. 115. ± 8.

0.64 0.42 0.12 0.13 0.27

0.67 0.48 0.31 0.29 0.29

~ + + * ÷

0.03 0.04 0.02 0.01 0.02

laDl

0.03 0.04 0.02 0.01 0.02

± + -+ ~ ±

0.03 0.04 0.05 0.02 0.03

[aDI

~ i + + _+

laD]

0.67 0.48 0.31 0.29 0,29

~D

138 152. 174. 90. 113.

~D

138. 152. 141. 139. 149.

~D

÷ 8. + 6. + 8.

±12. _+11. + 7.

+_ 8. ± 6. ± 8.

138. 152. 141. 139. 149.

0.08 0.11 0.31 0.42 0.42

0.09 0.12 0.28 0.41 0.43

± + + + ~

0.04 0.03 0.03 0.02 0.03

+ + ÷ + +

0.04 0.03 0.03 0.02 0.03

[aFI

i i + ± +

0.04 0.03 0.03 0.02 0.03

[aF[

laFl

0.09 0.12 0.28 0.41 0,43

~F

2. 17 30. 70. 127.

i 24. i 23.

~F

7. i 23. 29. + 12. 30. 70. 127.

~F

18 16 14 16 16

°t°t(mb)

20 17 15 18 16

O t o t (mb)

18 16 14 16 16

°t°t (mb)

7. i 23. 29, + 12. 30. 70, 127.

514

P. Estabrooks et al. / K+-p ~ K+-1r+n, K+-Tr-~ ++ I

I

1

~

1.0

I

I

I

. . . .

,.,_o_,_e--,

Io l

•

. J. . . . . .

0

t

I

I

I

I •

180"

o_o_O

~ o*

t

I

~111

I IP

......

_

q~

I

I

I

I

I

I

I

I

I

I

I

I

I

IopI

1.o

"

B

I

I

I

_

Oo _A.. *-

J

0.8

I

CP

I 1.2

l

÷'

I 1.6

'

I

-

MK,ir(GeV)

Fig. 14. The magnitudes and phases of the S- and P- partial waves of solution B as calculated from the combined K-Tr+n and K+Tr-~x++ single t bin analysis of subsect. 4.5. The choice of overall phase in the inelastic region is identical to that of fig. 13. The solid S-wave curves corres3 pond to effective range fits to 6S1 and ~S" The dashed curves above 1.2 GeV are extrapolations of these curves and the dashed-dotted curve represents the I = 23-S-wave contribution to la S I. The P-wave curves correspond to a Breit-Wigner resonance fit to 6p for 0.8 < M K n < 1.0 GeV and the extrapolation of this fit outside that MK7r region.

I n t h e inelastic region, MK~r > 1.3 G e V , t h e r e is still n o t a u n i q u e s o l u t i o n for the partial-wave m a g n i t u d e s a n d relative p h a s e s for KTr masses a b o v e 1.48 G e V . We ther~ fore p r e s e n t , in fig. 13 a n d table 3, all f o u r possible s o l u t i o n s for t h e K n partial waves f o r MK~r > 1.48 G e V , as well as t h e u n i q u e s o l u t i o n for l o w e r masses. As b e f o r e , we c h o o s e t h e overall p h a s e in t h e region 1.3
P. Estabrooks et al. / K+-p -, Ken+n, K+n

A ++

515

fitting the D-wave magnitude to a Breit-Wigner form in the region 1.25 < M K . < 1.60 GeV. These parameters are listed in table 1 d. For plotting purposes for M K , > 1.6 GeV, ~F is chosen to have the phase associated with a resonance of mass 1780 MeV and width 175 MeV [1], as shown by the curves in fig. 13. Taking these values for the mass and width of the 3 - resonance, we calculate an elasticity o f (19 + 2)%. We emphasize once again that for each value on MKn the choice of overall phase is arbitrary. A comparison of all the partial wave results presented in figs. 7, 8, 10, 11, 12, and 13 and the results listed in table 1 illustrates the consistency o f the separate K - l r +, K+Tr- , and combined analyses. We also found that the magnitude of X2 obtained from the individual fits to determine the partial waves at each K~r mass was similar for the separate neutron recoil and delta recoil analyses, and that in the combined analysis fits the contribution to X2 from the two different input data sets was comparable. It should also be pointed out that at high KTr mass the statistics of the neutron and delta recoil data are similar. Fig. 13 shows not only the presence of resonances in the leading 1 - , 2 + and 3 partial waves but also interesting structure in some of the non-leading partial-wave amplitudes. For instance, in solutions B and D there is a bump in lap I, coupled with a fairly rapid increase of ~p, in the 1.6 to 1.8 GeV region. Furthermore, in all four solutions, there is a rapid increase in [as l just above 1.4 GeV. This effect is more obvious in fig. 14 where we compare the S-wave magnitude and phase of solution B with the extrapolation (dashed line) of the effective range fits. Fig. 14 also compares the P-wave magnitude and phase with the extrapolation to higher masses of the K*(890) Breit-Wigner fits. These effects and their interpretation will be discussed at greater length in sect. 5.

5. Discussion of results Before discussing the new features o f KTr scattering resulting from our analysis, we first compare our results with those of other experiments. In subsect. 5.1 we discuss the I = ~ K~r scattering results, in 5.2 we compare results for I = ]l KTr scattering in the elastic region, and in 5.3 we discuss briefly the spin-parity 2+K*(1435) resonance. In subsect. 5.4, we present Argand diagrams for the four possible KTr solutions and discuss their structure in terms of resonances in the non-leading partial waves. We find evidence for a new I - 1 S-wave resonance and also, in two of the solutions, for a new 1-KTr resonance. In subsect. 5.5, we compare Klr and 7rTrscattering and in 5.6 we discuss possible methods for solving the ambiguity problem. 5.1. I = 3 K~r s c a t t e r i n g

We have already remarked that charge two Kn scattering is dominated by the Swave, with 83 and 8 3 always less than three degrees. The I = 3 S-wave phase shift,

516

P. Estabrooks et al. / K +p ~ K +-Tr+n, K +-~r--A ++

63, is well described by an effective range form and corresponds to a total cross section decreasing from 4 mb at 0.9 GeV to 2 mb at 1.5 GeV. This calculation depends on the assumption that the S-wave is elastic. If some inelasticity were allowed, the cross section would be larger. Our phase shifts are somewhat larger but, in general, consistent with previous measurements [7]. These previous measurements, which individually have large statistical and systematic uncertainties, are usually collectively represented as corresponding to a constant total cross section of 1.8 mb. Since our I = 3 normalization is extremely well known from (i) the existence of the elastic P-wave K*(890) resonance and (ii) the directly measured relative K+/K - experimental normalization, our results should be much more reliable. 5.2. KTr s c a t t e r i n g in t h e elastic r e g i o n

Our results for the I = ~1 S-wave phase shift, 8~, in the elastic region (MK~ < 1.3 GeV) are in good agreement with, but of much higher statistical significance than, other [8,18] experimental results. We note that our determination of the neutral K* (890) mass and width agrees with the currently accepted values quoted in ref. [15]. It is amusing to note that our value of a ~ / a 3 ~ - 2 . 4 agrees with the current algebra prediction [19] o f - 2 , although each individual scattering length is approximately twice as large as expected from current algebra. Scattering length comparisons are complicated by (i) the difficulty of extracting the 7r exchange signal from the large background in reactions (1) to (4) in the threshold region below 800 MeV, and (ii) by the large current algebra predictions for the effective range parameters, rs. 5.3. T h e K * ( 1 4 3 5 )

resonance

In contrast with the good agreement of our phase shifts with other results, our best estimate of the mass of the neutral 2 ÷ resonance is some 15 MeV higher than the Particle Data Group [20]. This effect, which is also noted in another recent KTr analysis [18], can be attributed to the fact that most previous determinations of the K*(1420) mass came from fits to Kn mass spectra in which the background was para metrized as a low-order polynomial. It is apparent from fig. 13 that for MK,, < 1.4 GeV, the S-wave is at least as large a contribution to the Kn cross section as the Dwave is, and is a slowly varying function Of MK~. However, the rapid decrease (in all four solutions) of la s I just above 1.4 GeV results in a cross section which peaks at a Klr mass b e l o w the D-wave resonance. We conclude that our result, 1434 + 2 MeV, based on fits to the extracted D-wave amplitude, yields a more reliable measurement of the resonance parameter values; it follows that the J P = 2 + Kn resonance should more properly be designated the K* (1435).

517

P. Estabrooks et al. / K+-p --, K+-n+n, K+-Tr-&++ 5.4. Argand diagrams and non-leading resonances

The K~ partial-wave results of the preceding section, as well as demonstrating the existence of the leading 1 - , 2 + and 3 - resonances, exhibit interesting new structure in the 0 + wave and also, for solutions B and D, in the P-wave. These effects are perhaps more readily apparent in the Argand diagrams of fig. 15. The points for MKn > 1.48 GeV shown in fig. 15 were obtained from a joint partial-wave analysis of the K+n - and K 7r+ data in overlapping 120 MeV bins at 40 MeV intervals. The results of this overlapping bin analysis are given in table 4. In each MKn interval above 1.3 GeV, the overall phase was chosen by requiring approximate Breit-Wigner behavior of the leading partial waves as well as satisfying unitarity and continuity. The amount of rotation from the initial Breit-Wigner phases is listed in tables 3 and 4. The I = ~S-wave was calculated by subtracting from the full K+Tr / K - n + S-wave the elastic I = ~ contribution given by the effective range fits shown in fig. 12. Fig. 15 clearly shows the circular motion of the S-wave in the Argand diagrams

SOLUTION •

SOLUTIONB /"

SOLUTION C

SOLUTIOND

I.I

,t

p

1.5

I . ~ ~ 5~

1F .

7

1.5

1.7 1.5

Fig. 15. Argand plots for solutions A-D from a combined K+n-/K-Tr+ partial-wave analysis. The points for MKn > 1.48 GeV, plotted at 40 MeV intervals, were calculated using data in overlapping 120 MeV MKrr bins. At each point, the overall phase was chosen, as far as possible, to satisfy unitarity, continuity, and approximately Breit-Wignerbehavior for the leading 2+ and 3 resonances. The diameters of the unitarity circles are 2 x / ~ 1. The I = 1 S-wave was obtained by subtracting from the full S-wave amplitude the I = 3 S-wave amplitude calculated from the effective range curve of fig. 12a.

- 6 -i0 -i0 17 12 ii 5 - 8 -15 -18

1.50 1.54 1.58 1.62 1.66 1.70 1.74 1.78 1.82 1.86

1.50 1.54 1.58 1.62 1.66 1.70 1.74 1-.78 1.82 1.86

MK~

-16 -19 -13 - 3 - 6 6 0 - 8 -19 -28

0rot

Solution B

8rot

MK~

Solution A

0.69 0.53 0.35 0.29 0.22 0.i0 0.15 0.21 0.34 0.33

0.57 0.49 0.40 0.36 0.30 0.33 0.40 0.40 0.46 0.48

+ 0.01 -+ 0.02 _+ 0.03 +_ 0.02 + 0.02 + 0.04 + 0.03 -+ 0.03 + 0.03 + 0.04

lasl

-+ 0.01 -+ 0.02 + 0.03 -+ 0.01 +- 0.02 + 0.02 + 0.02 +- 0.02 -+ 0.03 + 0.03

Ias I

164 172 176 170 191 169 104 i01 116 121

149 155 159 134 148 130 130 139 148 151

2 2 4 6 7 5 3 2 2 4

+ 2 + 2 -+ 5 +i0 +ii +20 +- 7 +i0 -+ 3 +- 5

¢S

-+ -+ +-+ -+ + + -+ + -+

~S

0.17 0.34 0.45 0.48 0.45 0.55 0.55 0.53 0.48 0.46

0.30 0.20 0.23 0.19 0.22 0.23 0.25 0.24 0.23 0.22

0.04 0.03 0.04 0.01 0.02 0.01 0.02 0.02 0.03 0.03

-+ 0.06 -+ 0.03 +- 0.02 -+ 0.02 -+ 0.03 + 0.02 _+ 0.02 + 0.02 +- 0.02 + 0.03

laP I

++ -+ -+ + -+ + +_ -+ +_

ImP I

103 93 98 94 116 121 140 159 176 195

210 205 213 151 180 155 157 168 173 167

+ + -+ -+ -+ -+ -+ _+ +_ -+

CP

-+ + + -+ + + + -+ _+ ±

~P

15 4 4 7 7 5 4 3 4 6

4 7 8 17 14 12 9 7 8 ll

0.77 0.50 0.30 0.31 0.29 0.25 0.24 0.24 0.25 0.31

0.82 0.62 0.46 0.51 0.45 0.48 0.43 0.43 0.40 0.38

0.01 0.01 0.02 0.01 0.02 0.01 0.02 0.02 0.02 0.02

+- 0.02 + 0.03 +- 0.02 + 0.02 + 0.02 -+ 0.02 + 0.02 _+ 0.02 _+ 0.03 + 0.02

IaDI

+-+ -+ + -+ + + -+ -+ +

laD I

138 148 154 131 140 131 135 147 154 169

CD

138 148 154 124 139 133 144 162 174 178

CD

+ + _+ + ++ +

-+ _+ -+ -+ -+ +_+

7 7 7 7 6 5 8

5 6 4 4 4 6 5

0.12 0.16 0.21 0.29 0.33 0.37 0.42 0.45 0.46 0.41

0.12 0.16 0.21 0.29 0.31 0.37 0.42 0.45 0.46 0.46

-+ 0.03 + 0.03 -+ 0.03 + 0.03 -+ 0.02 + 0.02 -+ 0.02 +- 0.02 + 0.02 -+ 0.03

IaFI

-+ 0.03 -+ 0.03 -+ 0.03 -+ 0.03 -+ 0.02 -+ 0.02 -+ 0.02 + 0.02 -+ 0.02 -+ 0.03

IaF [

30 + i0 38 -+ 7 16 +- 8 24 32 45 65 90 113 130

~F

31 -+ 6 43 -+ 5 35 + 6 24 32 45 65 90 113 130

CF

The K~ p a r t i a l w a v e s in the mass region a b o v e 1.48 GeV, as c a l c u l a t e d from the joint fits to the neutron and delta recoil r e a c t i o n s using data in o v e r l a p p i n g 120 M e V bins. Note that for 1.28 < MK~ < 1.60 the D wave is chosen as the r e f e r e n c e w a v e and that in the r e g i o n MK~ > 1.6 GeV, the F w a v e is chosen as the reference wave.

Table 4

>

I

+

oo

12 2 - 7 12 7 16 i0 - 2 -13 -13

1.50 1.54 1.58 1.62 1.66 1.70 1.74 1.78 1.82 1.86

erot

18 12 9 17 2 16 i0 2 5 - 8

MK~

1.50 1.54 1.58 1.62 1.66 1.70 1.74 1.78 1.82 1.86

Solution D

0rot

MK~

Solution C

0.46 0.25 0.21 0.25 0.24 0.33 0.36 0.37 0.50 0.54

0.69 0.59 0.56 0.54 0.53 0.56 0.56 0.53 0.59 0.58

0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02

± ± ± + + + + + + ±

0.01 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.03 0.03

lasl

± i ± + ± ± ± ± + ±

lasl

122 123 129 74 97 79 91 106 119 131

ii0 122 132 iii 120 116 124 140 149 156

i 1 3 5 4 3 2 2 2 3

± 4 + 6 + 8 ill ± 9 + 4 + 3 + 2 + 2 + 3

¢S

+ _+ ± _+ + + ± ± ++

*S

0.58 0.65 0.55 0.58 0.52 0.54 0.53 0.48 0.42 0.28

0~22 0.26 0.13 0.25 0.19 0.23 0.24 0.22 0.24 0.24

0.06 0.04 0.01 0.04 0.02 0.02 0.02 0.02 0.03 0.04

+ ± ± ± + + ~ i + +

0.04 0.03 0.03 0.02 0.02 0.01 0.02 0.02 0.02 0.03

ImP I

± ± ± ± ± ± + ~ ± +

laP I

55 67 78 80 105 106 130 152 165 168

64 72 129 75 106 99 122 136 136 122

-+ _+ + ± ~ + ÷ + i +

~P 8 5 6 5 5 4 3 3 5 8

+ 9 + 4 ± 15 ± i0 + 12 + 8 ± 8 ± 7 + 8 + 7

*P

0.74 0.38 0.26 0.08 0.ii 0.06 0.13 0.15 0.21 0.25

0.76 0.51 0.39 0.30 0.28 0.27 0.31 0.30 0.29 0.27

0.02 0.03 0.01 0.02 0.02 0.01 0.01 0.01 0.01 0.02

+ ± ± + + ± ± ± + +

0.02 0.03 0.02 0.05 0.04 0.02 0.02 0.02 0.02 0.03

laDI

± i + + ± -+ -+ + ± -+

laDl

138 148 154 162 180 79 99 113 115 112

~D

138 148 154 135 144 130 135 153 157 144

*D

+ ii ± lO ± 27 ± 9 + 8 + 7 + 5

i 7 i 7 _+ 6 + 4 ± 4 + 6 ± 7

0.07 0.16 0.21 0.31 0.33 0.38 0.41 0.45 0.44 0.40

0.07 0.14 0.15 0.26 0.30 0.35 0.41 0.44 0.44 0.40

0.04 0.03 0.02 0.03 0.02 0.02 0.02 0.02 0.02 0.03

± ± ± _$ _+ ± + ± ± +

0.04 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.03

IaFI

i +_ ± i ± ± _+ ± ± ±

laFl

12 + 23 34 + i0 1 ± 12 24 32 45 65 90 113 130

CF

13 + 20 35 + 7 22 _+ 9 24 32 45 65 90 113 130

*F

~,~+ i >+ +

'+ m+

.~

.~

520

P. Estabrooks et al. / K±p ~ K+-n+n, K ! n - A ++

for all four solutions. Moreover, in all solutions, the speed is greatest in the region of 1.4 to 1.5 GeV. Although the distance between adjacent points on the Argand diagrams is obviously very sensitive to the choice of overall phase at each point, it seems extremely likely that almost all reasonable phase choices will exhibit resonant S-wave behavior similar to that of fig. 15. We therefore conclude that, in all four solutions, there is an S-wave rseonance with a mass of 1.4 to 1.45 GeV, width of 2 0 0 - 3 0 0 MeV, and branching ratio to Krr ofabout 0.8, 0.9, 0.5 or 0.8 for solution A, B, C or D, respectively as listed in table 5. Historically, the 0 ÷ K meson has been associated with the point where ~ ~ finally reaches 90 °. We find that this point occurs at a mass of about 1.35 GeV. It is difficult to quote a width for this so-called resonance due to the existence of the relatively narrow 0 ÷ resonance above 1400 MeV. The 'width' calculated from the 45 ° point at 900 MeV would be about 900 MeV. Fig. 15 also shows, in solutions B and D, a resonance-like circular motion of the 1- partial wave. As we have just discussed with reference to the S-wave, the details of this behavior, in particular the speed, are sensitive to the choice of overall phase. We estimate that this effect could be due to a P-wave resonance with a mass of 1.65 GeV, width of 2 5 0 - 3 0 0 MeV, and branching ratio to KTr of 25 to 30%. In view of the present lack of understanding of the problems of resonance definition and the determination of resonance pole parameters, we eschew a more quantitative determination of these S- and P-wave resonance parameters. An additional way of displaying the structure in these partial waves is shown in fig. 16 where for solution B we have plotted the projections of the real and imaginary parts of the partial-wave amplitudes along with their Argand diagrams normalized to unitarity circles of unit diameter. It is important, when attempting to infer resonance behavior from these plots, to remember that non-resonant backgrounds can severely affect the mass dependence of the partial waves.

Table 5 Resonance parameters for the non-leading I = ~ K~r partial waves; these parameters have been estimated by ascribing circular motion in the Argand diagrams of fig. 15 to resonances Partial wave

Solution

A

B

C

D

0÷

MR (MeV) F R (MeV) x MR (MeV) I?R (MeV) x M R (MeV) F R (MeV) x

1400 250 0.8

1400 225 0.5

<0.05

1450 325 0.9 1650 250 0.3

<0.05

1450 250 0.8 1650 300 0.25

<0.02

<0.02

<0.02

<0.1

1-

2+

P. Estabrooks et al. I K+-p -+ K-¢r+ +n, K±zr-A ++

,'/'//~ ::~ f

1.0

521

t l~s /.

",,

il

05

0.5 Re S

K~ Moss

(GeV)

K'~ ELASTIC S--WAVE AMPLITUDE

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0 ~`

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. ~

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'

0 ~

,

Re'P

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15 (GeV)

K'~ ELASTIC P-WAVE AMPLITUDE

-+ +1.5

+ ( KTr Mass

KTr Mass

(GeV)

(GeV)

0.61 Im D

0.6

Im F

0.3

-0.3

•

0 ~

'

i

Re D

T

1.0

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1.5 (GeV)

K~ ELASTIC D- WAVE AMPLITUDE

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1.5 (GeV)

K~ ELASTIC F--WAVE AMPLITUDE

2 + -"~

+ 1.5 ,

Krr

~

\t.5

4.

/ + Mass (GeV)

Krr M a s s

(GeV)

Fig. 16. Argand diagrams and projections of the real and imaginary parts of the Kn partial waves of solution B obtained from a combined K+~r-/K-n + partial-wave analysis. For MK7r < 1.3 GeV, the points are calculated from the phase shift results of fig. 12b (and table 3b). The points for MK~ > 1.3 GeV are the same as those of fig. 15. The unitarity circles are chosen to have unit diameter. The S-wave plotted is the I = I S-wave obtained by subtracting from the full S-wave amplitude the I = 3 amplitude obtained from the effective range curve of fig. 12a.

5.5. Comparison of Kn a n d nn scattering The similarity b e t w e e n the Kn partial waves o f fig. 15 and the nzr partial waves (see, for e x a m p l e , fig. 6 o f ref. [ 1 7 ] ) is truly remarkable: (i) b o t h Kn and nTr s y s t e m s have leading P-, D- and F-wave resonances; (ii) b o t h s y s t e m s s h o w evidence o f S-wave resonance behavior in the region o f the

P. Estabrooks et al.

522

I

I

1000

N ~

500

\

o > (~ 100

I

/Kep

~ K±Tr+n, K+-~r- A ++

I

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. . . . . . .

. . . . . . . .

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--:------

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0.8

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1.2MK~r(GeV)

-----:--.-

?

1.6

Fig. 17. Predictions for the L < 4 moments of the K0rr 0 angular distribution in the reaction K - p ~ KOrtOn at 13 GeV/c in the region It'L < 0.15 GeV 2, as calculated from our Kn partialwave results. The L > 3 moments are the same for all four solutions and equal to one half the corresponding K - p ~ K - n + n moments o f fig. 1. The shaded bands shown for each of the four solutions represent the range of values obtained if the overall phase of fig. 15 is changed by -+10 ° Below 1.46 GeV the overall phase is fixed by the imposition of elastic unitarity. leading D-wave resonances; (iii) b o t h s y s t e m s i n d i c a t e a P - w a v e r e s o n a n c e in t h e r e g i o n o f t h e l e a d i n g ( h i g h l y inelastic) F-wave state for solutions B and D; (iv) b o t h s y s t e m s h a v e large S - w a v e b a c k g r o u n d s in b o t h e x o t i c a n d n o n - e x o t i c

P. Estabrooks et al. / K+-p ~ K+n +n, Ken--A ++

523

charge states, while little (non-resonant) background is present in the L > 0 partial waves. It is indeed pleasing that the detailed dynamical behavior of the Kn and 7rn systems is so consistent with the naivest considerations of symmetry ideas. 5. 6. R e s o l u t i o n o f ambiguities

It is important to determine which of the KTr partial-wave solutions is the physical one. Unfortunately, as discussed in sect. 4, all four solutions give rise to the same Kn differential cross section. Moreover, none of the four solutions strongly violates unitarity. We also expect that, as in the 7r+Tr- case, analyticity in the form of fixed-u dispersion relations will severely constrain the overall phase but not rule out any of the possible solutions. Data on inelastic channels could help to rule out solutions. For instance, observation of a large P-wave cross section (of the order of 1.5 rob) in K~lr or some other inelastic channel in the mass region around 1.6 GeV would eliminate solutions A and C from the realm of possibilities. On the other hand, nonobservation of such a large P-wave cross section would only be useful in resolving the ambiguity if all inelastic channels had been observed. Even if such considerations do not distinguish between solutions, they will certainly provide strong constraints on the overall phase. Another possible way of resolving the ambiguities would be to measure the K%r ° final state produced via n exchange. These states have relative I = ~ and I = ~ couplings which are different from those in K~+Tr~ . Fig. 17 shows the predictions of each of the four Klr solutions for the differential cross section and unnormalized M ; 0, L ~< 3 moments for the reaction K - p -+ K°Tr°n at 13 GeV/c with [t'l < 0.15 GeV 2. We illustrate the sensitivity of the K°Tr° predictions to the choice of overall phase by plotting bands of values corresponding to variations of-+10 ° from the phases as plotted in fig. 15. It is apparent from fig. 16 that an accurate measurement o f K ° g ° production for 1.4 < MKn < 2 GeV could distinguish between solutions. One might also hope that, in view of the striking similarity between the nTr and Krr partial waves, resolution of the ambiguity problem in nlr, for instance by measurements of ~oao production, would, via SU(3), also resolve the Kn ambiguity.

6. Summary and conclusions Data on reactions (1) to (4) have allowed us to calculate the KTr partial waves for both the neutral and doubly charged K~ systems. The observed agreement between the K~ partial waves calculated using only the neutron recoil reactions (1) and (2) and those calculated using only the A ++ recoil reactions (3) and (4) indicates not only that the relative experimental K+/K - normalization is correct but also that we understand the production mechanisms at least well enough to extract the physical Kn scattering amplitudes. A joint analysis of neutron and A ++ recoil reactions clearly

524

P. Estabrooks et al. / K+-p ~ K+~r+n, K+-~--A ++

provides the best estimate o f the Kzr partial waves. We find evidence for a j R = 0 + resonance at 1425 MeV with a w i d t h o f about 250 MeV and a Kzr branching fraction of b e t w e e n 0.5 and 0.9, depending u p o n the solution. This reopens the question o f the status o f the 0 + nonet. We also find, in two solutions, evidence for a 1 - resonance at about 1650 MeV; this is presumably the SU(3) partner o f the p ' ( 1 6 0 0 ) . Accurate measurements o f K°zr ° p r o d u c t i o n for K e mass b e t w e e n 1.4 and 2.0 GeV should make it possible to establish which o f the four possible partial-wave solutions is the physical one.

References [ 1 ] W.M. Dunwoodie et al., to be published. [2] P. Estabrooks et al., Phys. Lett. 60B (1976) 473. [3] P. Estabrooks et al., Nucl. Phys. B106 (1976) 61. [4] G. Brandenburg et al., Phys. Lett. 60B (1976) 478. [5] G. Brandenburg et al., to be published. [6] G. Brandenburg et al., Phys. Lett. 59B (1975) 405. [7] D. Linglin et al., Nucl. Phys. B57 (1973) 64; A.M. Bakker et al., Nucl. Phys. B24 (1970) 211. [8] M.J. Matison et al., Phys, Rev. D9 (1974) 1872; H.H. Bingham et al., Nuel. Phys. B41 (1972) 1; R. Mercer et al., Nucl. Phys. B32 (1971) 381. [9] A.B. Wicklund et al., Argonne National Laboratory report; contributed paper to the 17th Int. Conf. on high-energy physics, London, 1974. [10] P. Estabrooks and A.D. Martin, Nucl. Phys. B102 (1976) 537. [11] A.D. Martin and C. Michael, Nucl. Phys. B84 (1975) 83. [12] C. Goebel, Phys. Rev. Lett. 1 (1958) 337; G.F. Chew and F. Low, Phys. Rev. 113 (1959) 1640. [13] L. Stodolsky and J.J. Sakurai, Phys. Rev. Lett. 11 (1963) 90. [14] G. Grayer et al., Nucl. Phys. B75 (1974) 189. [15] Particle Data Group, Rev. Mod. Phys. 48 (1976) 51. [16] E. Barrelet, Nuovo Cim. 8A (1972) 331. [17] P. Estabrooks and A.D. Martin, Nucl. Phys. B95 (1975) 322. [18] M.G. Bowler et al., Oxford University preprint Ref. 4•66 (1977). [19] J.L. Petersen, Phys. Reports 2 (1971) 155.