c

Nuclear Physics B107 (1976) 189-210 © North-Holland Publishing Company

REAL PART OF THE K±p FORWARD SCATTERING AMPLITUDE AT 4.2, 7 AND 10 GeV/c P. BAILLON, Y. DI~CLAIS *, M. FERRO-LUZZI, P. JENNI **, J.-M. PERREAU, J. SI~GU1NOT * and T. YPSILANTIS CERN, Geneva Received 30 January 1976

The differential cross section of K-p and K+p elastic scattering has been measured at 4.2, 7 and 10 GeV/c m the very forward region of scattering angles. The measurements have been made at the CERN PS by means of multiwire proportional chambers and counters. The region of momentum transfers t is 0.001 < [tl < 0.10 GeV2 at the highest momentum and 0 001 < Itl < 0.03 GeV2 at the lowest. Over these regions the Coulomb and the nuclear amplitudes reach thetr maximum interference. We have used a parametrisation of the above amplitudes to determine the value of the real part of the nuclear forward scattering amplitude. A dispersion relation fit has then been performed using these and earlier measurements; the asymptotic behaviour of the K-+preal parts has been examined in the light of this fit.

1. Introduction ThIS 1S the description of a study of the differential cross section of K~p elastic scattering in the region of small forward angles at three incident momenta: 4.2, 7 and 10 GeV/c. The study is an extension of our earlier work at lower energies [ 1]. It has been done with the same technique and, apart from minor modifications, the same apparatus. The methods used to normalise, correct and fit the data are the same as in ref. [ 1]. The results of the two experiments can thus be considered as a homogeneous set of data relating to the forward scattering amplitude of K+p from 1 to 10 GeV/c. The determination of the forward scattering amplitude from the differential cross sections is made possible by three facts. One is that at small angles the Coulomb interaction plays a dominant and known role. The second is that the nuclear amplitude In the forward direction can be simply described experimentally by an exponential function of the m o m e n t u m transfer t. A third handle, important but not essential, is provided by the optical theorem which relates the total cross sec

* On leave from University of Caen, Caen, France. . . On leave from ETH, Zurich, Switzerland. 189

190

P. Baillon et al / K~p forward scattering

tion to the imaginary part of the nuclear amplitude at t = 0. The above information allows a straightforward description of the differential cross section near t = 0 in terms of a complex nuclear amplitude. The latter can then be determined from the data. We estimate the maximum systematical uncertainty on the normalisation of our differential cross sections to be in the region of 1 to 2%. This, together with the statistical precision, results in an absolute error of the order of +0.06 on the ratio of the real to imaginary part of the forward nuclear amplitude. Going one step further, we then combine all the values of the K±p real part obtained between 1 and 10 GeV/c to perform a fit based on the dispersion relations. We use the known total cross sections up to 200 GeV/c to calculate the dispersion integral and introduce an appropriate number of free parameters to find the conditions for which the measured and calculated real parts agree within the errors. A good fit is obtained with only two parameters, indicative of a correct parametrisation of the asymptotic behaviour of the total cross sections. The value of the K N Z - K N A combined coupling constant remains practically unchanged with respect to our earlier determination using only the low energy data. Concurrently with the above measurements we have acquired data on the forward differential cross section of other reactions, viz. pp at 4.2, 7 and 10 GeV/c, ~p at 4.2, 6, 8 and 10 GeV/c, K+d and K - d at 10 GeV/c. They will be reported later.

2. Apparatus The experiment has been performed at the CERN protonsynchrotron in the P16 beam of the East Hall between March and October 1975. The apparatus, shown in fig. 1, is essentially the same that was used in our earlier study of the forward scattering of K~p, n±p, ~p and K - d in the momentum range between 1 and 3 GeV/c [ 1]. For this reason we will restrict ourselves to a cursory description of the method referring the interested reader to ref. [ 1] for the details. Only the modifications of the set-up specifically required by the increase in momentum will be discussed at some length. The principle of the method IS that a measurement of the direction of a particle before and after passing through the hydrogen target is all that one needs for measuring the very forward elastic scattering reaction provided all the inelastic interactions are recognised and rejected. The measurement is achieved by a series of multiwire proportional chambers (MWPC) along the beam axis, the inelastic rejection by a combination of (a) veto boxes looking at the target and (b) analysis of the tracks traversing the chambers downstream of the target. The incident kaons are discriminated against the pions (~60 per negative K, " 2 0 per positive K on the average between 4 and 10 GeV/c) and the protons (~20

P. Badlon et al. / Kep forward scattering

Helium

191

Iron (hodron absorber)

Lead-sclnhllotor

Fig. 1. Lay-out of the apparatus. This is a plan view with the beam coming from the left. The section of the ~erenkov counters (CI to C3) is schematic and not to scale. In particular, it does not show the beam-transport elements present there' C1 and C2 are separated by a quadrupole, C2 is contained inside a second quadrupole and separated from Ca by a 2 m bending magnet, C3 is contained inside the last quadrupole. The remaining part is a scaled-up version of the set-up of ref [ 1 ]. The multiwire proportional chambers are marked W, the plastic scintillators S, the lead-scintillator sandwiches B The central veto-box B1 surrounding the hquid hydrogen target has the same shape and dimension as described in ref. [ 1 ]. Bags containing helium gas are Inserted between the chambers in order to reduce the multiple scattering.

per positive K) b y t h e t h r e s h o l d ~ e r e n k o v c o u n t e r s C1, C 2 a n d C 3 *. T h e t h r e e scintillators S 1 to S 3 d e f i n e a b e a m size o f 4 1 . 5 c m in d i a m e t e r at t h e t a r g e t w i t h an average divergence of-+3 m r a d . T h e i r o n c o l l i m a t o r screens the t a r g e t f r o m a b e a m h a l o w h i c h w o u l d p r o d u c e a h i g h a c c i d e n t a l rate a n d i n t e r a c t i o n s in the m a t e r i a l s u r r o u n d i n g t h e target. T h e m c x d e n t particle d i r e c t i o n is m e a s u r e d b y t w o MWPC's, W 1 a n d W 2, e a c h o n e h a v i n g an x- a n d y - p l a n e w i t h a 2 m m wire spacing, t h e d i r e c t i o n is m e a s u r e d w i t h a precision o f 4 0 . 2 m r a d . T h e k a o n flux at t h e t a r g e t was t y p i c a l l y ~ 5 0 0 p e r PS b u r s t o f ~ 3 0 0 msec (every 2.5 sec) w i t h a m o m e n t u m bite o f +0.7%. T h e s a t u r a t I o n o f o u r a c q u i s i t i o n rate o c c u r r e d at ~250 tuggers per burst. T h e t a r g e t Is a 25 c m c y h n d e r , 6 c m in & a m e t e r , s u r r o u n d e d b y a v e t o - b o x B !

* The main discrimination against pions Is provided by the 4.5m CI counter. Its inefficiency has been measured to be less than one part over 10 s. Of the other two, one is set to count the kaons, the other is set to see pions for the negative, kaons for the positive beam. C2 and Ca are inside the last two beam quadrupoles, separated by a bending magnet This has the effect that 8-rays emitted by protons in C2 are unhkely to reach C3 and simulate a K signature.

192

P. Baillon et al. / K+-pforward scattermg

consisting of lead-scintillator sandwiches (~3 radiation lengths). Differently from the earlier version of ref. [1], the downstream end of B 1 was fitted with a scintillator having a 10 cm diameter hole centered on the beam axis. So as to further reduce the unprotected forward hemisphere, where charged and neutral particles could pass unobserved, we have extended B 1 in the outgoing branch by Introducing two additional lead-scintillator sandwiches ~3 radiation lengths each. These counters (B 2 and B 3 in fig. 1) are squares with holes in the center subtending 140 mrad the first, 40 mrad the second. The outgoing branch is defined by the five MWPC's W3 to W7. With these we perform the following operations: (a) the scattered particle direction is reconstructed to an accuracy of t 0 . 1 5 mrad by a least-square procedure *, (b) the presence of a kink along this trajectory, indicative of a decay, is detected with a high efficiency and reliability, (c) the presence of more than one track in the forward direction is noted and used to reject the event. Finally, the iron wall (120 to 160 cm depending on the incident momentum setting) together with the S4 scintillator and the W8 chamber have the purpose of detecting the muons among the outgoing particles. These may come from the direct decay of the beam kaons along the target (which give a geometrical reconstruction undistinguishable from that of genuine scatters) or from decays outside the target region of either the beam particles or the scattered particles. In all cases we use this Information to reject the event. Notice that, contrary to the procedure followed in [1 ], we did not introduce a hardware cut-off on the scattering angle. This would have required a "beam counter" of critical dimensions and of uncertain effect on the selected angular distribution, the beam divergence is comparable to the angles of Interest for the experiment and the gain in acquisition rate introduced by the beam rejection would have been marginal. Therefore our trigger definition was simply C 1 • C 2 • C 3 • S 1 • S 2 • $3, where the ~erenkov counters Ct are m coincidence or antlcoincldence according to the chosen particle. All the other Information (the B-counters and their individual elements plus the S4 counter) was recorded on a pattern unit and transferred together with the chamber-coordinates on magnetic tape.

3. Data analysis The reconstruction of the events follows exactly the same hnes of ref. [ 1 ]. The information from the five chambers downstream of the target is sufficiently redundant to allow an unambiguous Identification and reconstruction of single straight tracks. This information was combined with the antlselectlon provided by the Bcounters and S 4 to arrive at the angular distribution of elastic scattering over the region of angles from 0 to 40 mrad. In fig. 2 we show an example of the resolution * Notice that the geometrical error of the reconstruction is generally smaller than the uncertainty due to the multiple scattering; the latter is discussed In sect. 3.

P Badlon et al / K’p forward scattermg

004

4m-

_z

193

_

s q s 0 IO GeVlc

-100

1500 0 IO s q d 0 30

z.

GsVlc

‘a F I,

I

1

-40

-20

I

I

I

0

I 20

I

I

I

40

Zvertex (cm)

Fig. 2 Positron along the beam axis (z) of the reconstructed interaction vertex of elastic scattering% The center of the target is at z = 0. The interval of z covered by the target IS indrcated by the shaded region The histograms represent the normahsed difference between full and empty target events in two intervals of transferred momentum q for K’p at 7 GeV/c The upper graph, covering the angular range from 5.7 to 14.3 mrad, corresponds to

on the mteractton vertex. empty-target events subtracted here the full-target. has the of removing the interin the other than and (b) decays occurring the target m rts neighbourhood. We a looser on the position than drd in [ 11; slightly reduces statrstrcal srgmficance low angles full/empty ratio down) but a possible error. The hmtt to reahstrc measurement the elastic drstrrbutron is the vicinity 1 mrad; this angle multrple scattering and the target subtraction meanmgless. Fig. shows the

P. Batllon et al. / K+-pforward scattermg

194

1,0

E A

v

i

K.°

o

q3

r

..,..-

..///!

05

v

/ / /

oi o

/ /-~--H

effect

2 I

I

I

0.1

0 2

0 3

I/p,8

0.4

(GeV/c)-~

Fig. 3. Multiple scattering angles (inclusive of the geometrical resolution) measured by means of the "unscattered" particles (see text). (0) is the root mean square space-angle, P and 3 are respectively the momentum and velocity of the incident (and scattered) particle. The solid lines show that the measurements with full and empty-target are in agreement with the expected linear dependence of (0) on l/P3. The dashed line shows the multiple scattering angle expected from 25 cm of liqmd hydrogen.

sured rms space angle (0) due to the apparent multiple scattering plotted against 1/P3 for the momenta and reactions of this experiment. The expected multiple scattering due to the hydrogen target alone is shown by the dotted line; it can be seen that the quadratic sum of this and the empty-target angle reproduces rather well the full-target measurement. As a result of the off-line analysis we arrive at the number of events summarlsed in table 1. Here we list in detail how many lncldent tracks were reconstructed and what is the final number of useful events in the various regions of Interest of the differential cross section. Notice that the large amount of events in the "nonuseful" category are in fact quite useful because they mainly consist of unscattered beam tracks which help an a large variety of checks and calculations needed for the correctaons discussed below. The angular distributions have been normallsed on the basis of the reconstructed incident tracks using an effective target length of 25.2 cm, a hydrogen density of 0.0707 g/cm 3 and the muon contamination (fu - 1) hsted in table 2. The latter has been calculated with a Monte-Carlo program simulating the Ku2 decays beyond the last K-defining ~erenkov accepted by the S 1 , S 2 and S 3 counters. The following corrections have been applied to the accepted angular distributions: (1) Acceptance. Has been calculated with a Monte-Carlo program which takes into account the geometrical characteristics of the chambers and the boundaries introduced by the various B-counters. Fig. 4 shows the fraction of accepted tracks at each incident momentum P as a function of the transferred momentum q -~ P sm 0. In the fits to the differential cross sections discussed m sect. 4 we will only use data with an acceptance of more than 90%.

P Batllon et al / K~p forward scattermg

195

Table 1 Summary of the statistics Reaction and momentum (GeV/c)

Flux ( 10 6 )

Accepted events (103) q > 0.01 GeV/c

0 1 > q > 0.025 GeV/c (F-MT)

K-p

4.2

F MT

7.60 6.16

87.3 31.1

8.51 3.44

4.26

K-p

7.0

F MT

10.02 7.64

191.8 56.7

11.52 4.25

5.94

K-p

10.0

F MT

10.70 7.76

323.6 101.8

12.40 5.11

5 35

K+p

4.2

F MT

8.52 6 56

94.6 32.5

8.80 3.71

3.98

K+p

7.0

F MT

10.38 7.81

191.2 58.1

11.92 4.61

5.80

K+p

10.0

F MT

11.73 8.07

350.8 105.3

14.05 5.48

6.09

For each reaction and incident momentum we list the total flux (1.e. the number of reconstructed incident tracks) and the number of events passing the various acceptance criteria (geometry, veto-boxes, no muon-filter signal, etc.). All accepted events have a minimum transferred momentum q of 0.01 GeV/c; the number of events in the "interference region" are listed in the last two columns, separated between full target (F), empty target (MT) and normalised full-minus-empty (F-MT) events. (ii) Absorption. Both incident and scattered particles suffer interactions while traversing the hydrogen and the material of the chambers. We correct for those interactions which result in the non-acceptance of the event through the reconstruction chain by using the known absorption cross sections. The correction factors are listed in table 2. (iu) Decays. We assume that all decays are detected by our apparatus and rejected. Therefore, using the known lifetimes to calculate the decay rate, we correct the results with the factors listed in table 2. (iv) Transmission o f the hadrons through the iron wall. This results in the event being wrongly taken for a muon whereas in reality we detect the end products of a hadronic shower. This effect (see table 2) has been measured directly by using the apparent transmission of the "unscattered" beam tracks (defined as those which are only affected by multiple scattering) appropriately corrected for the muon contamination of the beam and the fraction of decays occurring downstream of W7. (v) Delta rays. A number of elastic scatterlngs are rejected at the reconstruction

P. Batllon et al. / K±p forward scattering

196 Table 2 Correction factors Reaction and Momentum (GeV/c)

fd

ftt

ach

aH

fx 2

f8

fT

K-p

4.2

F MT

1.202 1.202

1.011 1.011

1.009 1.009

1.027 1 000

1.01 1.01

1.052 1.021

1 080 1.080

K-p

7.0

F MT

1.116 1 116

1.011 1.011

1.009 1.009

1.025 1 000

1.01 1.01

1.032 1.018

1.049 1.049

K-p

10.0

F MT

1.080 1.080

1.009 1.009

1.009 1.009

1.024 1 000

1.01 1.01

1.1396 1.053

1.050 1.050

K+p

4.2

F MT

1.202 1 202

1.012 1.012

1.008 1.008

1o018 1.000

1.01 1.01

1.053 1.020

1.140 1.140

K+p

7.0

F MT

1.116 1.116

1.012 1.012

1.008 1.008

1.018 1.000

1.01 1.01

1.031 1.017

1.080 1 080'

K+p

10.0

F MT

1.080 1.080

1.008 1 008

1.008 1.008

1.018 1.000

1.01 1.01

1.068 1.030

1.079 1.079

For each reaction, incident momentum and target state (F for full, MT for empty), we list the factors used to correct the differential cross section following the procedure described in detail in ref. [ 1]. The various entries compensate fd for the loss of events due to K-decays, f~ for the presence of muons in the incident beam, ach and a H for the absorption of the incident and the scattered particles when going through the chambers and the hydrogen, f×2 for the X2 cut of the geometrical reconstruction,f~ for the loss of events due to d-ray emission, fT for the transmission of the hadron shower through the muon filter. stage because either the incident or the scattered pamcle knocks out an electron during the traversal of the apparatus; the electron may have sufficient energy to generate a signal in one of the B-counters or, when emitted in the forward region, to simulate a two-track event in the chambers. We have corrected for this effect by combining a direct measurement (using the "unscattered" beam particles to determine the fraction which Is accompanied by a B-counter signal) with a MonteCarlo simulation which calculates the number of ~i-rays traversing the chambers without crossing the B-counters. The latter effect turns out to be neghglble with respect to the former at our energies and for the geometrical configuration of the apparatus. The correction factors are listed in table 2. Notice that with this procedure we also correct for the effect of accidentals in the B-counters. An indirect verification of some of the above corrections (vlz. those for absorption (il), decay (ill) and 8-rays (v)) is provided by the "unscattered" beam particles defined above. It can be seen that these three effects are responsible for the attenuation of the beam from incoming to "unscattered" outgoing. Any deviation of the measured from the expected attenuation would indicate either a miscalculation of

P Baillon et al / K+-pforward scattering

ell

197

r,O

2GeV/c I

I0

7GeWc '~

c 0

~. 0,5 o

t

I

f

I0

ell 0

I0 GeV/c I

OI

Transferred

I

I

02,

/ t I

03

0,4

momentum, q (GeV/c)

Fig. 4. Geometrical acceptance of the apparatus as a function ofq -~ P sin 0 where P and 0 are respectwely the incident momentum and the scattering angle of the detected particle Notice that in the very forward direction q is practically the momentum of the recoil proton in Kp scattering. The acceptance has been calculated with a Monte-Carlo program the above three effects or a systematic defect in the geometrical reconstruction of the tracks. We find agreement to better than 0.5% at all momenta and reactions. The corrected differential cross sections are hsted in table 3 and shown in figs. 5 and 6. A comparison with the earher work in this region [2] shows important deviations in absolute value, it also emphasises the lack of data in the crucial low-t region of the angular distribution of these experiments.

4. Real parts and dispersion relations The expressions used to fit the differential cross sections are those described in ref. [1J. Briefly, we write the cross secUon in terms of the Coulomb ( f c ) and the nuclear (fN) amplitudes,

do dt

_

Ifc +fNI 2

(1)

198

P. Badlon et al. / K~p forward scattering

Table 3 Differential cross sections K - p in (a), K+p in (b) (a) K - p Incident m o m e n t u m (GeV/c)

10

7

4.2

t

do/dt

do/dt

do/dt

(GeV 2 )

(mb/GeV2 )

(mb/GeV 2 )

(mb/GeV 2 )

125 44 ± 10.63 77.25 ± 8.60 47.94 ± 7 33 43.66 ± 6.30 39.70 ± 5.48 31.70 ± 4.75 28.54 ± 4.37 32.16 ± 4.05 27.74 ± 3.54 26 78 ± 3.32 26.93 ± 3.10 23.77± 2.89 24.36 ± 2.68 24.53 ± 2.65 20.82 ± 2.31 23.86 ± 2.34 23.09 ± 2.02 23.76 ± 1.87 21.49 ± 1.86 22.74 ± 1.91 22.89 ± 1.78 23.94 ± 1.61 21.33 ± 1.54 24.67 ± 1.63 22.74 ± 1.50 21.74 -+ 1.41 22.12 ± 1 39 20.34± 1 3 3 17 62 ± 1.27 17.79 ± 1.26 17.85 ± 1.14 19.10 ± 1.27 17.34 ± 1.14 17.30± 1.13 17.32± 1.11 17.13± 1.16 15.79 ± 1.01 16.78 ± 1.04 15.27 + 0.97

192.09 ± 13.47 115.26 ± 10.35 67.85 ± 8.25 72.58 ± 7.11 58.08 ± 6.15 39.76 ± 5.36 44.09 ± 4.93 39.97 ± 4.33 37.70 ± 3.87 44.06 ± 3.58 32.37 ± 3.45 32.32 ± 3.24 33.55 ± 2.88 27.03 ± 2.68 19.81 ± 2.57 20.87 ± 2.47 21.75 ± 2.25 26.08 ± 2.16 26.74 ± 2.09 25.07 ± 1.96 24.31 ± 1.93 24.65 ± 1.78 26.75 +- 1.83 26.30 ± 1.75 22 23 ± 1.59 24.08 ± 1.56 23.35 ± 1.52 20.75 ± 1.44 21.61± 1.38 21.38 ± 1.29 22.63 ± 1.33 23.05 ± 1.30 19.57 ± 1.24 19 70 ± 1.18 20.60 ± 1 22 20.14± 1.17 21 8 2 ± 1 1 9 18.97 ± 1.09 18.44 ± 1.08 16.10 ± 1.01

O.O01O6 0.00141 0.00181 0.00226 0.00276 0.00331 0.00391 0.00456 0.00525 0.00600 0.00680 0.00765 0.00855 0.00950 0.01050 0.01155 0.01265 0.01380 0.01500 0.01624 0.01754 0.01889 0.02029 0.02173 0.02323 0 02478 0.02637 0.02802 0.02971 0.03146 0.03325 0.03509 0.03699 0.03893 0.04092 0.04296 0.04505 0.04719 0.04938 0.05162 0.05391

340.78 ± 196.44 ± 86.98 ± 87.67 ± 75.85± 45.77 ± 45.01 ± 53 70 ± 41.08± 37.64 ± 35.94 ± 44.68 ± 40.93 ± 40.36 ± 33.20± 34.90 ± 33.33± 34.31± 32 14 ± 29.66 ± 28.95 ± 29.84 ±

24.15 17.39 13 20 10.85 8.75 8.00 7 01 6.26 5.65 5.09 4.76 4 49 4.08 4.07 3.57 3.40 2.92 2.91 2.86 2.67 2.56 2.68

P Baillon et a l / l~p forward scattering

199

Table 3 (continued) (a) K - p (cont') Incident m o m e n t u m (GeV/c)

10

7

4.2

t

do/dt

do/dt

do/dt

(GeV 2)

(mb/GeV 2)

(mb/GeV 2)

(mb/GeV 2)

0.05626 0.05863 0.06107 0.06355 0.06609 0.06867 0.07130 0.07398 0.07671 0.07949 0.08232 0.08519 0.08812 009109 0.09411 0.09718

14 29 -+ 15.67 -+ 16.73 -+ 16.23 -+ 14.30 -+ 13.61 -+ 14.77 +13 33 +13.21 +12.85 +12.60 +11.22-+ 13.22 -+ 11.77± 12.09 +10 24 -+

0.96 0.96 0.93 0.93 0.87 0.86 0 86 0.83 0.80 0.78 0.74 0.72 0.80 0.71 0.72 0.66

17.36 -+ 1.04 15.75 -+ 1.01

(b) K+p 0.00106 0.00141 0.00181 0.00226 0.00276 0.00331 0.00391 0.00456 0.00525 0.00600 0.00680 0.00765 0.00855 0.00950 0.01050 0.01155 001265 0.01380 0.01500 0.01624 0.01754 0.01889

112.19 -+ 10.61 88.71 +- 8.28 48.99 +- 6.87 46.05-+ 5 65 39.99 -+ 4.83 33.00-+ 4.47 21.42 -+ 3 93 23.87 +- 3.49 28.71 + 3.28 21.40 + - 2.95 23 47-+ 2.72 22.42 +- 2.48 19.15 -+ 2.24 18.10 +- 2.15 15.91 -+ 1.92 17.28-+ 1.87 16.92 -+ 1.79 19 42 + 1 70 18.55 +- 1.63 16.81 +- 1.55

177.69 +- 14.59 115.25 -+ 10.78 86.77-+ 8 51 70.79 +- 7.01 48 49-+ 6.12 46 94 +- 5.14 41.23_+ 4 67 29.95 _+ 3.83 36 08-+ 3.59 23 10-+ 3.34 24 1 9 -+ 3.26 20.31 -+ 2.73 24.18 _+ 2.69 20.75 -+ 2.52 23.59-+ 2 27 15.65 -+ 2.09 24 38 +- 1.98 18.22 -+ 1.90 17.13 -+ 1.77 18.60-+ 1.70 17.15 -+ 1.57

336.52 -+ 24.54 192.23 -+ 17.93 101.83 -+ 13.38 96.29 -+ 10.50 70.44 +- 8.93 49.73 -+ 7.49 40.87-+ 6.57 39.00-+ 5 73 24.46 +- 5.27 26 90-+ 4 72 24 81 -+ 4.28 29.21-+ 3.80 26.56 +- 3.51 24.63 -+ 3 18 23.35 -+ 3.05 22.76 -+ 2 72 22.58-+ 2 78 20.63-+ 2 52 18.23 +- 2.27 19.14-+ 2.29 18 88-+ 2.14 18.62-+ 2.08

200

P. Batllon et al. / K~p forward scattering

Table 3 (continued) (b) K+p (cont') Incident m o m e n t u m (GeV/c)

10

7

4.2

t

do/dt

do/dt

do/dt

(GeV 2)

( m b / G e V 2)

( m b / G e V 2)

( m b / G e V 2)

0.02029 002173 0.02323 0.02478 0 02637 0.02802 0.02971 0.03146 0.03325 0.03509 0.03699 0.03893 0.04092 0.04296 0.04505 0.04719 0.04938 0.05162 0.05391 005625 0.05863 0.06107 0.06355 0.06609 0.06867 0.07130 0.07398 0.07671 0.07949 0.08232 0.08519 0.08812 0.09109 0.09411 0.09718

14.37 ± 13.49 ± 15 10 ± 12.82 ± 14 46 ± 16.23 ± 13 1 6 ± 12,86 ± 13.98 ± 14.87 ± 12.08 ± 12 88 ± 14.35 ± 12.35 ± 13.29 ± 12.66 ± 11.11± 11.92± 10.49 ± 1157± 11.24 ± 10 57 ± 11 3 0 ± 10.26 ± 12.03 ± 10.26± 10.84 ± 10.73± 9.98 ± 7 86 ± 931± 9.88 ± 9.17 ± 10.06 ± 9.11±

18.91 +18.16 -+ 15.72-+ 16.93-+ 15.01 +13.93+16.69 ± 13 70 +16.33± 15.40± 14.93± 13.71± 13.76 ± 12.93 -+ 12.55 ± 13.55 ± 13.22 ± 1210± 11.57± 12.50 ± 14.08 ±

1.41 1.42 1.38 1.25 1.24 1.18 1.14 1 06 1.05 1.06 0.97 0 95 0.98 0.92 0 92 0.87 0.86 085 0.78 0.81 0.80 0.77 0.76 0.73 0.78 0.71 0.70 0.70 0.66 0 61 0.67 0.65 0.64 0.63 060

1.55 1.46 1.42 1.41 1 29 1.21 1.23 1.16 1.19 1.10 1.11 1.04 1.07 1 01 0.99 0.98 0.95 094 0.85 0.90 0.93

The intervals in m o m e n t u m transfer t correspond to equaUy spaced values of q = x/~ The values listed for da/dt have been corrected for the various effects discussed in the text. Notice that they are mcluswe of multiple scattering (see fig 3). We list only values on which we have applied acceptance corrections smaller than 10%; the lower limits o f t have been chosen so as to gave a multiple scattering and resolution effect equal for the three incident m o m e n t a .

0.0010-0.02 0.0014-0.06 0.0018-0.10 0 . 0 0 1 0 - 0 02 0.0014-0.06 0 0018-0.10

K-p K-p K-p K+p K+p K+p

25.4 23.5 22.6 -17.1 17.2 17.2

ot (mb) ~04 :t 0.4 ± 0.3 ± 0.3 ± 0.3 -+ 0.3

[7 4] 8.9 + 8.8 ± [4.8] 6.5 ± 6.1 ± 0.8 0.4

0.7 0.4

b (GeV -2 ) 0.10 0.09 0.03 0.38 -0.36 -0.22

c~

-+ 0.04 -+ 0.04 ± 0.05 ± 0.04 ± 0.04 ± 0.04

27 69 48 13 41 57

×2

22 42 55 22 42 55

N

0 43 0.60 -0.27 - 1 10 -1.75 -1.53

D (fro) -+ 0.17 ± 0.27 ± 0.46 ± 0.12 ± 0.19 ± 0.28

4.30 6.63 9.11 2.90 4.86 6.94

A (fm)

For each reaction and m o m e n t u m we hst the range of t covered by the data, the total cross section o t preferred by the fit (starting from the external value entered with a ±2% error), the slope b of the nuclear amplitude, the ratio c~ of the real to the imaginary part of the nuclear amplitude near t = 0, the value of the x 2 and the n u m b e r N of the data points used in the fit. Quantities m brackets have been kept fixed T h e slope at 4 2 GeV/c has been taken, for K p, from the m e a s u r e m e n t s at larger angles of the high statistics bubble chamber exposure o f K - p at 4.2 GeV/c (private c o m m u m c a t i o n from R. Hemingway, CERN), for l(+p, from an interpolation of various experiments m the region from 1 to 10 GeV/c (see ref. [1] for details) T h e last two columns give the real part o f the nuclear a m p h t u d e (D) in the lab system corresponding to the fitted ~ and the imaginary part (A) given b y the optical theorem on the basis of the fitted o t.

4.2 7.0 10.0 4.2 7.0 10 0

Range of t (GeV 2 )

Reaction and m o m e n t u m (GeV/c)

Table 4 Results of the fit to the differential cross sections

O

I

~

0002

]

t

f (GeV2)

I I 0006

0102

OiO'3

I I 00<)8

'

4 2 GeV/c

O;OI I

I %1 0004

"\\\

I

K-p

0

\\

\

h

[

~

I

IO /

O002

\

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I

1

I

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_P

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J

I

I

I Y

I

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÷

008

OOIO

O06

]

7 GeV/c

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0

I

0,004

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~

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5° k,

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II

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::

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t

\\

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,

I ~%1

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,

J

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K-p

,

,

,

01008

I

'

I

,

,

0 OIO

006

I0 GeV/c

O~ | (GeV2)

0I

,

Fig. 5. Differential cross sections do/dt for K - p elastic scattering as a function of the m o m e n t u m transfer t at the three m o m e n t a of this experiment. At each m o m e n t u m we show the full t-range of the measurement (resets) and, on a larger t-scale, the "interference region". The thick sohd curves represent the result of the fit described m the text. Also shown, on the graphs of the interference region: the contribution due to the Coulomb amplitude (dash-dotted), the absolute value of the interference (dashed), the nuclear part of the cross section (thin straight line). The optical point is shown by the square at t = 0. Only the full circles have been used in the fit.

IO

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b

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~ IO001

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IG

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;

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1

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0008

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4 2 GeV/c

0004 0006 1' (GeVz)

50

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500 i

0002

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10000

0

-

I

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- x\

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006

.

]

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0 010

I

01012

.

008 :

1

-----YYi--

004

7 GeV/c

(GeV 2)

0 006

I

002

K÷p

i

Fig. 6. Same as in fig. 5 but for K+p.

I

/

p

~oc ~

50O

iuu,.

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~

0 002

i

0

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0 04

(GeV z )

01006 I

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0 012

l

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204

P. Badlon et al. / K~p forward scattermg -0 15

....

~""l

........

K+p

z

-0

I

I0 GeVlc

'

8~

' ' '''"I

E 7 v g 6 --

i '

20

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4

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I

tO-3

I I I~ I~'~-J~

lO-Z

[

LL,

....

I

m

0

iO -i

I' (GeV z)

Fig. 7. Example of the dependence of c~= Re fN(0)/Im fN(0) on the range of t chosen for the fit to the differential cross sections. The horizontal bars show the t-interval, the vertical bars give the diagonal error assigned to c~by the fitting procedure. The chosen interval and value of a is represented by the thicker lines and the full circle. The curve (to be read on the right scale) gives the expected increase in da/dt due to multiple scattering as a function of t These data refer to K+p at 10 GeV/c.

with

fc = -Q ~ ot fN - 4 h ~

F(t) exp(iQS),

(or + i) e x p ( - a b t ) ,

(2)

(3)

where Q = +1, depending on the charge of the incident particle, ~ is the velocity of the incident particle, F(t) is the Coulomb form factor of the proton taken equal to (1 + (t/0.71)) - 4 and 6 is the phase of the Coulomb amphtude taken equal to - ( l n (RZt) + 0.5772)/(137/3) with R = 1.48 fm. Notice that, if we introduce the usual real and Imaginary parts D and A defined by fN = (D + iA) x/-n/P, the optical theorem takes the form

a t = 47rheA/P,

(4)

where P is the lab m o m e n t u m of the incident kaon and a t is the total cross section. The two u n k n o w n quantities to be determined by the fit are the ratio 0t of the real to the imaginary part of the forward nuclear scattering amplitude (c~ = D / A ) and the slope b of the nuclear amphtude. The values of a t have been taken from a smooth interpolation of the results available in the literature (an exhaustive list of these references can be found m

P Batllon et al / K+-pforward scattering

205

ref. [1 ]) after checking that these values are in good agreement with our own estimate. We did this by means o f an analysis of the attenuation through the target of the beam particles. An error of -+2% has been arbitrarily assigned to the external measurement of a t so as to account for a possible systematic discrepancy between the normahsations. The value of a t with ItS error has been used as a constraint of the fit. The effect of multiple scattering has been folded into the calculated values of the differential cross section before performing the fit to the measurements. The average multiple scattering angles have been taken for each value of t according to the measurements shown in fig. 3. The results of the fits are collected in table 4; they are also shown by the curves in figs. 5 and 6. Notice that we hst only the conclusions of an exhaustive series o f tests in which we investigated the dependence of the parameters on a variety of conditions. As an example, fig. 7 shows the variation of a as a function o f t observed for K+p at 10 GeV/c. As it can be seen on this figure, the choice o f the t-interval is not crucial for the determination of a. On the other hand, we have tried to eliminate the influence of possible subjective factors by imposing the lower limit for each m o m e n t u m setting and reaction at a value of t corresponding to a predetermined multiple scattering and geometrical resolution. The upper limit was taken where the acceptance correction is equal to 10%. The final choice is indicated on the table. Additional checks have been done by fitting the values of the differential cross sections derwed without the use of the muon detector (i.e. the iron wall and the S4 counter). These cross sections, albeit less precise statistically because the full/ e m p t y ratio is considerably smaller, would indicate If a systematic bias had unaccountedly infiltrated our definition of a muon or the related corrections. No such bias was found. A further set of tests has been made in order to ascertain that the parametrisation o f the nuclear amphtude in terms o f eq. (3) is appropriate. We find that the data do not reqmre additional parameters (as for example a change in slope or in a as a function of t) and are well described by the above parametrisatlon. Finally, a variable scale factor was introduced as an addmonal free parameter to test for a possible normahsatlon error, here again we do not find indications of systematic errors. Fag. 8 shows a display of the real-to-imaginary ratios as a function of the lab momentum together with earher measurements [ 1 - 3 ] . The curve shown on this figure is the result o f the dispersion relation discussed below. Notice that the agreement with the earlier bubble chamber points [2] is far from good. The Carnegie et al. values at 10.4 and 14 GeV/c [3] are instead in good agreement with our calculation. Passing now to the expectatmns from dispersion relations, we have extended our previous analysis [ 1 ] to the new measurements. We have preferred not to introduce the measurements o f a from other sources partly because of the obvious disagreements visible m fig. 8 and partly because our normahsation and procedures are con-

E

0

?!

02

04

~

OI

L

,

,

,

i

i

I0

i RI

, , , , f

Lab

,

momentum

K p

Ref

[I]

(GeV/c)

I0

=

o Ref~ [2] o , d

•

i

,,+

l

i-- I f'

I00

[3]

• Th~s experiment (8adlon ef at.1976)

(b)

"E

E

,,_z

08

06

04

02

02

I

'

'

'

I-

0

Lab

. . . .

1

,

,,,

+

.

.

.

I0

/i 7

,

.

m o m e n t u m (GeV/c)

,

,

. . . .

-

I00

[

Fig. 8. Values of a = Re f N ( 0 ) / I m / N ( 0 ) for K - p (in a) and K+p (in b) as a function of the incident lab momentum. The full circles are from this experiment, the full triangles from our earlier experiment [ 1 I, the empty circles from ref. •2,3 [. The curves show the error corridor of the dispersion relation fit (with two parameters) discussed in the text. Only the full circles and triangles have been used in this fit.

(a)

I:E

I

"3

06

08

m

-

t-,J o

P. Badlon et al. / K~p forward scattering

207

slstent. Thus we have used only the 9 points for K - p and the 6 points for K+p from this experiment and from ref. [1]. The dispersion relation, of the once-subtracted type, has the form D+(co) = D+(0) + g 2 R ~ c co/[coy(co - COy)] + CO(I1 +12 + 1 3 ) '

(5)

where D+(CO) is the K+p real part at a total lab energy COof the incident K, g2 is the effective ~-A coupling constant squared and divided by 4zr, COy is the value of CO in the middle between the Z and the A pole and R = [(rn A - rnp) z - m2]/4rn 2 is the kanematlcal factor of the A pole. The dispersion integral has been divided into three parts: 11 over the unphysical region from the An to the gd'q threshold, 12 over the physical region from the KN and KN thresholds up to the highest energy (200 GeV) at which the K~p total cross sections have been measured, 13 over the region from the highest value at which the cross sections are known up to _+oo.For the calculation o f I 2 we have introduced the same smooth Interpolation of the total cross sections as in ref. [1]. For 11 we have used the low energy KN amplitudes of the zero-effective-range (ZRA) analysis of ref. [4]. The same considerations are valid here as in ref. [1 ] concerning the use of other low energy solutxons: effectiverange (ERA) solutions gave the same results except for a consistently larger value of the Z-A coupling constant. To calculate 13, where the cross sections are not known. we have used an inverse power dependence of the type CO-0.6 for the difference between the K+p and K - p total cross sections, as observed experimentally for co between 10 and 200 GeV [5]. The sum of the cross sections has been approximated to a constant equal to the measured value at 200 GeV. This is a different procedure from the one followed in our earlier fit of the region 1 < CO< 3 GeV [1], where we found that taking equal and constant total cross sections beyond 200 GeV was a satisfactory approximation. The wider range of COof the present analysis reqmres corrective terms amounting to the above parametrisatlon. In order to perform the fit we use a "discrepancy" function A defined as follows: A(CO) = [D(CO)- CO(I1 +I2 +I3)] (CO- COY)" At each value of co where a measurement of D exists we calculate a "measured" value of A. We then fit a polynomial expansmn m COto the measured A's. N A(CO) = ~ anCOn

(6)

(7)

where N is determined by the minimum number of coefficients required for a good fit. The coefficients are related to the unknown quantities of eq. (5): the K+p real part at zero energy D+(0) and the coupling constant squared g~. Correction factors arising from a possible inadequate calculation of the dispersion integral would appear under the form of terms in CO2 and higher. Fig. 9 shows the A versus co plot with the "measured" values of A and the result of the two-parameter fit. As shown In table 5, there is no need of terms higher than CO. The calculated value of a± as a function of P for the two-parameter fit is shown

P. Baillon et al. / K~p forward scattering

208

D

5

K

p

P

K +

A

(.9

v

-5 -I0

-5

0

5

I0

o.1 (GeV)

Fig. 9. Discrepancy function h (defined in eq. (6) o f the text) as a function o f the total lab energy to o f the incident K. T h e energy to is defined positive for K +, negative for K - . The points correspond to t h e real parts measured in this e x p e r i m e n t (full circles) and in ref [ 1 ] (triangles). Notice that the increase in error with increasing to is only due to the definition of A and n o t to a reduced accuracy on the m e a s u r e m e n t of the real part. The central shaded area represents the unphysical region b e t w e e n t h e K - p and K+p thresholds. The straight line is the result of t h e two-parameter fit described in the text.

Table 5 Results of the dispersion relation fit Order of the expansion

1

2

3

X2 ND

15.5 13

13.6 12

12.6 11

ao a1 a2 a3

- 0 . 2 3 -+ 0.03 - 0 . 1 1 - + 0.02

- 0 . 1 9 -+ 0.04 - 0 . 1 1 -+0.02 - 0 . 0 1 7 -+ 0.012

-0.20 -0.10 -0.014 -0.002

g~,

18.6 -+ 2.6

15.2 +_ 3.6

15.7 _+ 3.6

± 0.04 ± 002 +- 0 013 +- 0 002

The data used are the real parts of this e x p e r i m e n t (6 pomts) and those of ref [ 1 ] (9 points). We list the ×2 o f the fit, the n u m b e r o f degrees of freedom (ND), the coefficients o f the expansion (7) of the discrepancy function and the value of the square of the c o u p h n g c o n s t a n t at the combined I;-A pole divided b y 4~r.

P Baillon et al. / K±p forward scattering

209

Table 6 Real parts expected from the 2-parameter dispersion relation fit of table 5 Incident momentum (GeV/c)

a

a+

D_ (fro)

D+ (fm)

0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.5 3.0 4.0 5.0 7.5 10 15 20 50 100 150

0.55 0.34 0.07 0.06 0.17 0.27 0.25 0.19 0.13 0.10 0.12 0.11 0.12 0.087 0.079 0.061 0.053 0.039 0.025 0.025 0.032 0.053 0.057 0.060

-0.74 -0.58 -0.51 -0.52 -0.53 -0.54 -0.54 -0.52 -0.51 -0.50 -0.49 -0.49 -0.48 -0.45 - 0 42 -0.37 -0.32 -0.25 -0.21 -0.16 -0.13 -0.046 0.005 0.018

0.88 0.70 0.13 0.11 0.28 0.47 0.51 0.42 0.30 0.23 0.29 0 28 0.31 0.25 0.26 0.25 0.26 0.28 0.23 0.33 0.54 2.2 4.7 7.5

-0.39 -0.38 -0.40 -0.46 -0.51 -0.55 -0.58 -0.60 -0.63 -0.65 -0.67 -0.69 -0.72 -0.80 -0.88 -1.0 -1.1 -1.3 -1.5 -1.7 -1.8 -1.6 0.40 2.2

We hst the calculated values of the ratio a± of the real to the imaginary part of the forward scattering amplitude for K±p respectively and the corresponding values D± of the real part in the lab system. The imaginary parts (not listed) correspond to the smooth interpolation of the total cross-section measurements listed in ref. [ 1 ]. Notice that the ~ and D values at the upper range of momenta are somewhat dependent on the assumptions made about the behavlour of the total cross sections beyond 200 GeV/c (see text). The values below 0.9 GeV/c axe the same as those listed m ref. [ 1 ]. by the curve in fig. 8 up to the highest m o m e n t u m ( ~ 1 5 0 G e V / c ) where the prediction can stdl be considered reliable. Three c o m m e n t s are w o r t h making: (a) The value o f g 2 has decreased from 22.0 + 2.6 o f our earlier analysis to 18.6 + 2.6. It is n o w in even b e t t e r agreement with the 19.2 e x p e c t e d f r o m SU(3) with pseudo-vector coupling. (b) The b e h a v l o u r o f ct b e y o n d 10 G e V / c is quite different from the one predicted by our earlier t w o - p a r a m e t e r fit. The discrepancy is mainly notable b e y o n d 20 G e V / c where the present fit predicts values o f ct approaching zero m contrast with the earlier sharply decreasing values. This is n o t surprising in view o f the wider

210

P. Batllon et aL/ K~p forward scattermg

m o m e n t u m range of the present analysis. It also appears that the preliminary measurements of a in the uppermost region of momenta [3] are in good agreement with our present calculation. Finally, we agree qu:te well with the calculations of ref. [6]. (c) The second-order term a 2 (see table 5) is small and compatible with zero. On the other hand, if we use this value to calculate the exponent of the total crosssection difference beyond 200 GeV, we find 0.3 -+ 0.1 to be compared with 0.6 used to calculate 13. Notice also that the absence of third-order terms in eq. (7) imphes that the assumption of a constant sum of the K+p and K - p total cross sections beyond 200 GeV xn calculating 13 :s supported by the fit. Table 6 hsts the values of D and tx of the preferred fit at selected momenta over the range from 0.9 to 150 GeV/c.

References [1] P. Baillon, C. Bricman, M. Ferro-Luzzl, P. Jenni, J.M. Perreau, R.D. Tripp, T. Ypsilantl~, Y. D~clais and J. S6guinot, Yellow Report CERN 75-10 (1975), Nucl. Phys. BI05 (1976) 365. P. Jenni, P. Baillon, C. Bricman, M. Ferro-Luzzi, J.M. Perreau, R.D. Tnpp, T. Ypsilantis, Y. D6claxsand J. S6guinot, Nucl. Phys. B94 (1975) 1, Nucl. Phys. B105 (1976) 1. [2] Th.H.J. Bellm, H. De Jonge, R.W Meyer, A.G. Tenner, P. Vons and H. Winzeler, Nuovo Cimento Letters 3 (1970) 389 (K+p at 5 GeV/c); Th.H.J. BeUm, H. De Jonge, E. De Lijser, P. Manuel, R.W. Meyer, A.G. Tenner, P. Vons, H. Voorthuis, P. Heinen, H.G.J,M. Tiecke, R.T. Van de WaUeand H. Wmzeler, Phys. Letters 33B (1970) 438 (K-p at 4 2 GeV/c); J.R. Campbell, V.T. Cocconi, P. Duinker, H. De Groot, J.D. Hansen, A. Head, J.M. Howie, G. Kellner and D.R O. Morrison, Nucl. Phys. B64 (1973) 1 (K-p at 10 GeV/c); R.W. Meijer, Ph.D. Thesis, Zeeman Laboratory, Univ. of Amsterdam (1973), unpublished (K-p at 14.3 GeV/c). [3] R.K. Carnegie, R.J. Cashmore, M. Davler, D.W.G.S. Leith, F. Richard, P. Schacht, P. Walden and S.H. Williams,Phys. Letters 59B (1975) 308 (K±p at 10.4 and 14 GeV/c); C. Ankenbrandt, M. Atac, R. Brown, S. Ecklund, P.J. Gollon, J. Lach, J. MaeLachlan, A. Roberts, L.A. Fajardo, R. Majka, J.N Marx, P. Nemethy, J. Sandweiss, A. Schiz and A J. Slaughter, Fermilab-Conf-75/61-Exp (August 1975), unpublished (K±p at 70, 100, 125 and 150 GeV/c). [4] B. Martin and M. Sakitt, Phys. Rev. 183 (1969) 1345. [5] A.S. Carroll et al., to be published in Phys. Letters (quoted by A.M. Wetherell in: Invited talk at the EPS Int. Conf. on high-energy physics, Palermo, June 1975). [6] R.E. Hendrick and B. Lautrup, Phys. Rev. D l l (1975) 529.