c

Nuclear Physics B99 (1975) 211-231 © North-Holland Publishing Company

A STUDY OF THE REACTION K+n ~ K+~r-p AT 2 - 3 GeV/c S.L. BAKER, S. BANERJEE, J.R. CAMPBELL, A.K.M.A. ISLAM, G. MAY and D.B. MILLER Imperial College, London J.E. ALLEN, P.V. MARCH, S.H. MORRIS, K.O'BRIEN and C.E. PEACH WestfieM College, London Received 22 January 1975 (Revised 4 August 1975)

Results are given from a study of 15 518 events of the reaction K+d --, K+n-pp. The K+n- spin density matrix and the constraints imposed on it by positivity have been studied. Analyses of K+n ~ K+~- elastic scattering have been carried out using methods developed by Estabrooks and Martin and Ochs and Wagner for the analogous case of nn scattering. Results are found to be in agreement with earlier Krr scattering studies using the reaction K+p --~K+rr-A ++ at much higher energies. The S-wave scattering length is found to be in agreement with the prediction of current algebra.

1. Introduction Recent trends in the analysis of particle scattering data have emphasised the great importance of model-independent amplitude analyses. So far, however, these [1 ] have only been carried out for nN elastic scattering at 6 GeV/c where a complete set of data is available. For many reactions (particularly those induced by kaons), the complete polarisation measurements are not likely to become available for many years. It is, therefore, of some interest to see just how far one can proceed with the analysis of the necessarily incomplete data which is available from, for ex~ ample, a bubble chamber experiment. To this end, we present here an analysis of the reaction K+n ~ K + n - p in which we have attempted to keep to a minimum the number of model-dependent assumptions. In sect. 2 we give some particulars of our experiment and, more especially, our data-processing and event selection methods. We also have tried to explain here how we have isolated a sample of the reaction K+n ~ K+n p from that for the reaction which is actually seen in the bubble chamber, namely K+d -+ K+Tr pp. Sect. 3 contains some remarks on the reaction K+n ~ K*(890)N and several comparisons are made with other experiments to show that a single pole model (unmodified one211

212

S.L. Baker et al. / K+n -+ K + n - p

Table 1 The total number of fitted events in our sample and the corresponding cross sections for various reactions Incident beam momentum

2.18 GeV/c

2.43 GeV/c

2.70 GeV/c

Reaction

No. of events

crosssection No. of (mb) events

crosssection No. of (rob) events

crosssection (rob)

K+d ~ K+rr-pp K+n---'K+rr-p K+n~K*°p K+n ~ K*+n K+p~K*+p K+n~K°rr°p

5457 5194 2553 394 762 1045

5446 2.9 -+ 0.2 5141 2.1 -+0.2 2152 0.95---0.1 344 1.8 ---0.2 636 1.7 +-0.1 1053

4615 2.6 +-0.2 4346 1.6 -+0.1 1616 0.76-+0.1 301 1.4 -+0.2 564 1.5 -+0.1 921

2.4 +- 0.2 1.3 +-0.1 0.70"--0.1 1.3 +-0.1 1.2 -+0.1

We do not give a cross section measurement for the reaction K+d ~ K+n-pp due to our scanning procedure. pion exchange) is inadequate in our energy range. Sect. 4 is an analysis of the Sand P-wave K+n - spin density matrix including the positivity constraints which in fact provide quite significant limits. It should be emphasised that this section involves no model-dependent assumptions at all. In the subsequent sections we have made several such assumptions in two separate attempts to extract K~ isospin I = 1 phase shifts using the methods developed by Estabrooks and Martin [4] and Ochs and Wagner [5], respectively, and applied by them to the reaction n - p -+ n+Tr-n. We have tried to be particularly careful in these sections to emphasise the physical assumptions involved. Recent determinations of the Kn phase shifts for Kn masses below approximately 1.1 GeV have been made by Bingham et al. [6] and Matison et al. [7] and comparisons of their results with ours are given. A review of earlier results on Kit phase shifts has been given by Trippe [8] and references to other experiments can be found there. Finally, in sect. 6, we give the results of a comparison of our data with the soft meson prediction of Weinberg [9] and Griffith [10] and summarise our results for Kn scattering.

2. Experimental details The data were collected in a 930 000 picture exposure of the British 1.5 m bubble chamber, filled with deuterium, at the Rutherford Laboratory. The electrostatically separated K + beam was tuned for three separate momenta - 2 . 2 , 2.45 and 2.7 GeV/c and the data sample corresponds to 3.8 events]/ab spread almost equally over the three momenta. The film has been scanned, measured twice on the Imperial College HPD and a final measurement made on a conventional film plane digitiser, all in the manner now considered standard for bubble chamber experiments. The mea-

S.L. Baker et aL/ K+n ~ K+n-p

213

k"d .-,,k*Trpp SLOWER

PROTON MOMENTUM DISTRIBUTION

lz%

LLJ UJ

i

0

0.1

i~_

0.2 MOMENTUM

0 "3

i

0~.

GeV/c

Fig. 1. Plot containing data from aU three momenta. The curve is the prediction o f the Hulthen wave function for the deuteron.

surements have been passed through the CERN programs MDT-GRIND-SLICE and the resulting numbers of events are given in table 1. For the large class of events with an odd number of charged outgoing tracks, we inserted a zero momentum proton in the kinematical fit with errors z2uPx = zSa°y = zS.Pz/1.37 = 30 MeV/e and allowed the fit routine to iterate from this starting position. This procedure has often been used in deuterium bubble chamber experiments and has been found [11] to give reliable results for reactions (such as this) where there is no missing neutral particle. It should be noted that in a four-constraint fit reaction at relatively low beam momenta, such as ours, there are essentially no ambiguity problems and events can be uniquely identified with a high degree of confidence. The momentum distribution of the slower of the two protons in the reaction K+d -+ K + n - p p is shown in fig. 1, and can be seen to be in agreement with the prediction of the deuteron wave function. It should be pointed out, however, that the event selection criteria used by our scanners discriminate strongly against the high m o m e n t u m tail o f this distribution. To obtain a sample of the reaction K+n -+ K+Tr-p it seems reasonable to cut this distribution at 300 MeV/c and consider all lower momentum protons as spectators

214

S.L. Bakeretal. / K + n ~ K + n p

Table 2 The extrapolated K+zr- t-channel helicity frame spherical harmonic moments given to ease the comparison of our data and those of other experiments K+n mass (GeV)

hi0

b20

0.65-0.8 0.8-0.85 0.85-0.875 0.875-0.9 0.9-0.925 0.925-0.975 0.975-1.025 1.025-1.1 1.1-1.2 1.2-1.35

0.197 ± 0.02 0.185 ± 0.02 0.163±0.02 0.018 ± 0.03 0.008 ± 0.02 -0.028 ± 0.01 -0.032 ± 0.02 0.015 +- 0.03 0.044 ± 0.03 0.097 _+0.01

-0.004 ± 0.03 0.031 ± 0.02 0.110±0.02 0.108 ± 0.01 0.107 ± 0.02 0.108 ± 0.02 0.102 -+ 0.02 0.081 ± 0.03 0.088 ± 0.03 0.119 ± 0.02

The normalisation is such that boo = 1 / x / ~ . t > -0.5 (GeV/c) 2.

The linear extrapolations used data with

in the sense o f the impulse a p p r o x i m a t i o n . We have checked that these p r o t o n s are isotropically distributed in the laboratory system. Absolute normalisation o f our cross section measurements has b e e n p e r f o r m e d [12] using about one third o f the present sample, on the basis of the n u m b e r o f identified r decays.

3. The reactions K+n -+ K * ( 8 9 0 ) N In this section we shall consider the set o f reactions K+p ~ K * p ~ - ~ K 07r+ ,

(1)

K+n-+ K*+n ~-~ K°Tr+ ,

(2)

K+n -+ K * 0 p [ - ~ K+Tr- ,

(3)

In each o f these we have defined the K* by a simple KTr effective mass selection from 0.84 to 0.94 GeV. We give the resultant cross sections in table 2 and show the charge exchange cross sections together w i t h similar results from o t h e r experiments in fig. 2. In this graph we have also plotted the available cross sections for the line-

215

S.L. Baker et al. / K+n -. K+~r-p

CHARGE-EXCHANGE CROSS-SECTIONS

o

?~

~ k*n---k'p

c:)

k ÷

',

k-

'2 •

p

,

,

,

,

i

,

,

,,

GeV/c lab

Fig. 2. The cross section for the reaction K n K p and the line-reversed reaction K-p -+ --*0 K n plotted vervus beam momentum. The lines are the result of the power law fits whose parameters are g~venin the box. •

•

•

+

-->

* 0

reversed reaction K - p ~ [ * 0 n. Any model based on exchange-degenerate trajectories must predict equal cross sections for these two reactions at least in the highenergy limit. In fact the cross sections can be seen to be almost equal at momenta as low as those of this experiment. This is in marked contrast to the reactions K+n -+ K0p and K - p -+ [ 0 n where the two cross sections differ by a factor of two in our energy range. In the following sections we shall be particularly concerned with the (K+Tr- ) subsystem in reaction (3)• It is therefore important to establish at the outset whether there is significant production of competing reactions of the type K+n ~ K+N * ~rr-p . To this end we present, in fig. 3, the Dalitz plot for the K+Tr-p final state as seen in our experiment. This shows clearly that even the overall data sample for this final state is completely dominated by the (K+rr-)p reaction and indeed by K*(890)

S.L. Baker et al. / K+n -+ K+Tr-p

216

3 " 0 - -

=.

2.0

(.D

..

;'.

•

•

B

!

m

".;.i:

I:

-,

- ,~

÷

1"0--

• ~!

,

:.;,

.. : : ,

,,.

"~;v~'.~ :'4,"

, , ,

I

,"."

,

" ....

,."...:

:,;..'.

' '

. .;,.,,,.,

..~ . ~,

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2

, , ,

,' ',.

I , ,

1.5

0.5

.,

•

:."

. "

:

.

:: .....

.

'

,'..,':

,,

2.5

,

~,~'

~

I

3.5

M ( P ~-) GeV

400~ '°°F

200~__ •6

0"8

1"0

M

(K+.rf -) Fig.

3a, b.

1"2

GeV

1"4

,

,,

,

,

217

S.L. Baker et al. / K+n --, K+~r-p

200

15C

I0C

50

1.0

1.2

1"4 M (p,n'-)

1"6

18

GeV

Fig. 3. The Dalitz plot for the reaction K+n ~ K+n-p and the K+Tr- and ~r-p mass plots. Note the almost complete absence of N*. Events with a spectator proton of momentum less than 0.3 GeV/e have been plotted using data from all three beam momenta.

production. Moreover, the energy dependence of the K*(890) cross section, shown in fig. 2, is consistent with that expected for pion exchange right down to 2 GeV/c. The comparative absence of baryon resonance effects permits us to proceed to an analysis of the Krr subsystem with confidence. Using the three reactions (1), (2) and (3) we have performed a t-channel isospin decomposition and extracted the I t = 0 and I t = 1 differential cross sections which are shown in fig. 4. The interference terms between the I t = 0 and I t = 1 amplitudes have opposite signs in reactions (1) and (2) and so can be eliminated.

4. K+~ - density matrix analysis We show, in fig. 5, the s-channel helicity system spherical harmonic moments, alm, for the K+Tr- system up to masses of 1.3 GeV. These are normalised in such a way that /max I d2° - ~ ~ a l m R e Y ~ ( O , fb), dmK,rdt 1= 0 m = 0

ao0 =

218

S.L. Baker et al. / K+n ~ K+Tr-p

K + N~

K~890)

N

0

¢ i, i

I}, ! i,

;[

}

I

{

II

i-.L

! ii

p.

I

i

t~ "0

I

_

.

"2

_

I

13

Ii --i~ I I o

-'1

t

I:0 t

I =1 t

-'2 ( GeV/c)a -'3

Fig. 4. The t-channel isospin 0 and 1 differential cross sections separated out as explained in sect. 3. Since there are no significant differences between the different incident momenta we have combined all three in this plot. The weighted average beam momentum is 2.54 GeV/c.

Our analysis will assume that only S- and P-waves are present so that all m o m e n t s with 1 > 2 should be zero. We note that this is true up to 1.1 G e V apart f r o m a possible effect in the l = 3 m o m e n t s presumably due to P-D interference in the K * ( 8 9 0 ) region. We neglect this. We therefore have five measured m o m e n t s , R e P 0 S = ½~

al0,

RePl0 = ~ x/~a21x/~l~ ,

Re PlS = 1 ~

all,

P l - 1 = - - V " ~ a22V/4~4 ,

P00 - - P l l = N/~a20V/~-4 s- , to d e t e r m i n e a h e r m i t i a n positive-definite 4 X 4 density matrix,

(4a)

S.L. Baker et al. / K+n ~ K+~-p

K*n

219

) (K*~-)P a

alO

II

0.1

o, tf+++t++++ +t+%+++t o

+

t++H+++ +~++~l+~l~+~+

_~+++t

++~+

-0"1 0"1

a20

a21

+ + -,+÷4+~++++~+++t--[t_-[ _H_+ +~t

0-2

+tH

,

0.1

-0.1

-0.2

0-1

a30

a22

0 -0.1 -0.

-0.2 0"6

0"8

I'0

1"2

M (K*~-)

0"6

0'8

I'0

1"2

(GeV)

Fig. 5. The s-channel helicity system spherical harmonic moments, aim, for K+~T- masses up to 1.3 GeV/c. All events with t > - 0 . 5 (GeV/c) 2 were used in these plots.

220

S.L. Baker et al. / K + n ~ K+~r-p

~..o

C a

o

O

,.Y:, ,C. M

0

0"7 o

0q

0"9

~(k'.l

0"9 M(k*n")

1.1 GeV

1.1

eev

Fig. 6. The projection on to the axes of the four-dimensional physical region (shown shaded), allowed by positivity for the unmeasured s-channel helicity frame density matrix elements in K+n - mass bins and for t' < 0.1 (GeV/c) 2.

Pll

PlO

Pl-1

PlS 1

pro

p00

-pTo

p0s/

Pl-I

--PlO

Pll

--PlS /

p?s

p~s

-pTs

pssJ

(4b)

Thus there are four unmeasured real numbers (namely P l l , Im Pl0, Im P0S and Im PlS) which can have any value so long as the matrix is positive definite for which a necessary and sufficient condition is that all four principal'upper-left sub-determinants are positive, e.g. the first two conditions are Pl 1 ~ 0 and P00Pl 1 - IPl012 ~ 0. This in practice, turns out to be a severe restriction on the allowed region for the four unmeasured parameters. Projections o f this four-dimensional volume on to the four axes are shown in fig. 6. Statistical errors on the domain boundaries are only given for the P l l plot. For further details of the method, see ref. [13]. We note that similar but very much stronger limits have been obtained by Grayer et al. [13] in the reaction r r - p -~ 7r+zr-n at 17 GeV/c. Encouraged b y the fact that one is able to limit significantly the value of the

S.L. Baker et aL / K+n ~ K+lr-p

221

NATURAL PARITY EXCHANGE EIGENVALUE

o "T

~._y=

o

,

0.7

0"9 M(k~n'')

1.1 GeV

1"3

,

0"7

,

,

0"9

I

,

1"1 M(k"rt') GeV

,

,

i

1-3

Fig. 7. The allowed ranges for the four s-channel helicity frame density matrix cigenvalue in K+~r- mass bins and for t' < 0.1 (GeV/c) 2. The allowed ranges for the three unnatural parity exchange eigenvalues are indicated by horizontal, vertical and diagonal shading (in increasing order of magnitude).

measured elements, we next proceeded to investigate the allowed ranges for the eigenvalues of the density matrix. This was done by generating, at random, values of the four unmeasured parameters, checking to see if the resultant matrix was positive definite and if so calculating the eigenvalues with a standard program. The resulting ranges are shown in fig. 7. One eigenvalue corresponds to natural parity exchange and is equal to P l l +/91-1" The other three, which are ordered by their magnitude, correspond to unnatural parity, exchange but one has to be zero due to parity conservation and in fact the smallest eigenvalue is limited to very small values indeed. The size of the allowed range piesumably is an indication of the errors in our results. Here we just point out (but see sect. 6 and 7) that the second smallest eigenvalue is always consistent with zero though the allowed range at the K* mass is much wider than the corresponding range at the p mass in the experiment of Grayer et al. It is true however, that the rank of the density matrix is always consistent with two, as would be the case if the nucleon flip and non-flip unnatural parity exchange amplitudes were coherent.

222

S.L. Baker et a L / K+n ~ K+Tr-p

B1NGHAMET

AL.

~, MAT[ SON E I A L. *

I ,

0

I

THIS EXPT.

,A ,I I

blo

i I , L/

i/

I

0

6

l~l

'7 'f

b

0

20

?,: 0

iT, o

0.7

I

I f ,

0.8

I

,

i

0.9 M(k÷rr-) GeV

i

1.0,

Fig. 8. The extrapolated K+n- t-channel helicity frame spherical harmonic moments, blo and b2o. The normalisation is as specified in sect. 4 for the alrn moments. The extrapolation used data with t > -0.5 (GeV/c) 2.

5. K~r phase shifts To date, Kn phase shifts have been extracted using the classical procedure of Goebel [2] and Chew and Low [3] in which the t-channel helicity (GottfriedJackson) frame K+Tr spherical harmonic moments, him , are evaluated in several tbins and extrapolated to the pion exchange pole to obtain on-shell K+Tr- moments. This is the method employed, for example, by Matison et al. [7] and Bingham et al. [6], both studies using the reaction K+p --> K + n - A ++ at respectively 12 GeV/c and at various momenta between 2.5 and 12.7 GeV/c (but mainly above 5 GeV/e). It would clearly be desirable to confront these determinations by others obtained from a completely different reaction such as the one we have studied, namely K+n --, K + n - p . This three-particle final state raises a quite different set of problems and biases from those involved in the previously cited analyses. The model-independent analysis presented in sect. 4 shows that in our case, the (K+rr - ) system demonstrates features which require the presence of other exchange processes in ad-

223

S.L. Baker et aL / K+n ~ K+Tr-p

K+n -.--~K '" rr- p FOR VARIOUS t.1-12 GeV

I

,

,

f-

[

--i--

K+-n"- MASS BINS I

} '---t--

O975-1"025 GeV I

'_~_--F----t-'

.9" o

'

I

r-,

,

1.025- 1.1GeV

I

i

t

I

I

I

0-925- 0975 GeV

0"90 - 0"925GeV

I

; ~__~

!

An 00 e,..o

[

'

1.2 -1'35GeV

--b

L_ - ~ . . . . I

'

f-

I

r

co,'~ ek.c:, °

[

0"875-0"9 GeV

0"es-0-a7sGevl

(-

_~__--~ __I__- . [ -

co cb

r

I

o

0"1

i

I -I- !- r

r--i-

t

-e0-0asG-~v

0"3 t'

I

[

0 ' 6 5 - 0'80 GeV

0'1 (GeV/c)a

• I

0'3

1 Fig. 9. Some plots showing the t-dependence (or lack of it) of the expression Pos/(Poo + ~PSS). in various K+~r- mass bins, evaluated using the s-channel helicity frame density matrix elements.

dition to one-pion exchange. Chew-Low extrapolation if attempted, thus leads, not surprisingly, to on-shell moments significantly different from those previously reported as can be seen in fig. 8. Instead of using the extrapolation method, we have, therefore, employed techniques which have been developed in connection with the analogous 7rn problem by Estabrooks and Martin [4] and Ochs and Wagner [5]. These have the advantage of also using the moments with rn > 0. They do, necessarily, of course, make model-dependent assumptions as will be indicated below. In this connection, it is interesting to note the behaviour of P0s/(P00 + ~Pss) 1 which, in fig. 9, is plotted as a function of t' for various Kzr mass bins. For dominance of the unnatural parity exchange, helicity-zero amplitudes by pion exchange this quantity should be independent of t. It can be seen that this is reasonable for t' < 0.3 (GeV/c) 2 and K+n - masses less than 1.1 GeV. This is a particularly noteworthy observation, since here, unlike in the mr analogue, B exchange is, a p r i o r i , allowed.

224

S.L. Baker et aL / K+n ~ K+Tr-p

5.1. The method o f Estabrooks and Martin

The five measured K+n - density matrix parameters, which are listed in sect. 4, may be explicitly expanded in terms of the helicity amplitudes TX~ by ReP0 S = Re(A.~_AS~ +,t0 ,tS*'~ RePlS =V/~ Re(A~_A S* + A ; _ A S * ) , O 0 0 - P l l = IAO 1 2 - 1 ~(IA.H+ 12 + I A ~ I 2 ) + I A 0 _ I 2 - 1 ( I a + _ 1 2 + l z + _ 1 2 ) , RePl 0 =,v/~21Re(A,~_A+* + A O A + * ) , /91_ 1 = ~(LA~_+12 - Ih++l 2 + Ih++_[2 - Ih+_ 12), where A~# = V~.~(Tal/3-+ Tff~1) and A0~S = T~a~S.The subscripts on the helicity amplitudes specify nucleon flip and non-flip and the superscripts the KTr helicity state. The essential simplification made by Estabrooks and Martin [4] in their analysis of zrn scattering is to neglect unnatural parity non-flip (A 1) exchange. Thus: AO = 0,

A++ = 0 ,

AS++ = 0 .

As pointed out at the end of sect. 4, our eigenvalue analysis is also consistent with such an assumption. In our case, the evidence is perhaps not too strong, but since a small A 1 exchange amplitude, only occurs quadratically in the above set of equations (there is no n-A 1 interference term), its neglect is probably not too serious. If the overall phase is fixed by defining A °_ to be real, there are now six real parameters (A0_, ReAS_, ImAS+_, ReA+_, ImA+_ and IA++[2 + IA+_ [2) and six equations (the above five plus the trace condition) so we may solve the equations explicitly. The above argument is applicable to both the s-channel, where it is true to order 1/s, and the t-channel helicity frame moments, where it is exactly true. In an attempt to include the effects of absorption we have determined the s-channel frame amplitudes since there is reason to expect that s-channel helicity is approximately conserved in absorptive processes. We then linearly extrapolated the ratio AS+ - - //A 0+ - (using all data with - t < 0.5 (GeV/e) 2) to the pion-exchange pole. A m e r i t of this procedure of extrapolating ratios of amplitudes is that it avoids effects due to the Chew-Low boundary, which is not uniquely defined in a reaction with a virtual neutron target. To extract the Knl = ~ S-wave phase shift, 61/2, we have that A S_ = -~ sin 61/2 exp(i6s1/2) + ½ sin 63/2 exp(i83/2),

Ao

81/2 exp(i81/2),

neglecting the I = ~ P-wave. The I = ~, S-wave phase shift, 83/2 has been taken to be negative and of magnitude given by o3/2 = ~4rrsin263/2 = 1.8 mb

(see ref. [6] )

225

XL. Baker et al. / K’n -+ K+rr-p

Table 3 Thel = i S-wave Kn scattering phase shift as determined __ ._. ~___ tPhase shifts (deg) K7-r mass DOWN UP GeV) 0.65-0.80 0.80-0.85 0.85-0.875 0.875-0.90 0.90-0.92s 0.925-0.975 0.975-1.025 1.025-1.10 1.10-1.20

14.1* 4 14.7 + 2 34.7 + 2 33.5 YI 6 44.5 f 6 45.1 f 6 54.2* 3 50.9 f 2 53.7 -? 10

Where no value is given in the right-hand multiples of 180”.

53.5 70.3 131.2 6.1

column

M(K+fl-)

by the fist

method

described

in sect. 5

f 2 f 15 f. 29 f 4

the solution

is unique

apart from addition

of

GeV

Fig. 10. The on-shell phase shift fitted assuming elastic unitarity using the Estabrooks and Martin method. We have also plotted the results of the earlier experiments. For clarity, only DOWN solution branches are given except in the case of the present analysis where two points from the UP solution are shown and are distinguished by being marked UP.

S.L. Baker et al. / K+n ~ K+~r-p

226

PHASE COHERENCETEST 0 = PHASE ( A'~

1.6

/ AO~_)

o

E-M

METHOD

÷

O-W

METHOD

1.4 1'2 1'0 r~ LIJ D

O'B 0"6 0.4 0"2 0 -0'2 0,6

' 0'~B~[ 'T~'~'~--' 0 112 M(K %r-)

G,~V

Fig. 11. The relative phase of the helicity-zero and one P-wave helicity amplitudes as determined in the analysis using the Estabrooks and Martin m e t h o d . The actual phases plotted are for the schannel amplitudes in the interval t > - 0 . 1 (GeV/c) 2. The P-wave amplitudes are too small below 0.8 GeV to permit a meaningful phase determination.

The I = ½, P-wave phase shift has been represented by a Breit-Wigner formula tan

61/2 -

mRr m 2 _ m27r '

where F = F 0 [q/qR ] 3mR/mKn, mR = 0.8966 GeV and F 0 = 0.0517 GeV. We tried various forms for the Breit-Wigner formula and found that our results for the S-wave are practically independent of this. All of these parametrisations are essentially identical to those made by Matison et al. [7] and Bingham et al. [6]. The results for the I = ~ S-wave phase shift are given in table 3 and fig. 10 where we have also plotted the results o f refs. [6,7]. A notable feature o f this solution is the well-known UP-DOWN ambiguity in the neighbourhood o f the K*(890). In the present analysis this takes the fortT1 that for any amplitude solutions which leads to an

S.L. Baker et al. / K+n -* K+n p

227

Table 4 The I = ~ S-wave KTr scattering phase shift as determined by the first m e t h o d described in sect. 5 K+Tr mass (GeV)

0.70 0.74 0.74 0.77 0.77-0.80 0.80 0.83 0.83-0.86 0.86-0.89 0.89-0.92 0.92 0.95 0.95-1.00 1.00-1.05 1.05-1.10 1.10-1.20

Phase shift (dcg) DOWN

UP

6.2 ± 2 17.1+- 2 18.1 * 2 28.6 + 2 59.8 ± 4 34.7 +- 3 33.3 ± 3 28.9 ± 3 45.9 ± 3 57.0 ± 3 66.4 ± 4 71.9 ± 4

108.9 ± 4 158.7 -+ 4

Where no value is given in the right-hand column the solution is unique apart from addition of multiples of 180 ° .

K 11'

PHASE

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228

S.L. Baker et aL / K+n ~ K+Tr-p

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AMPLITUDES

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S-wave phase shift 61/2, there will exist another ambiguous solution with an S-wave phase shift given b y 180 ° - 51/2 and the opposite sign for Im A ; _ . The possibility could therefore have arisen o f resolving the ambiguity by demanding continuity in the energy dependence o f the relative phase between the two unnatural parity exchange P-wave amplitudes, A 0_ and A ; _ . However, as can be seen in fig. 11, our solution is consistent with zero phase difference between the two P-wave amplitudes, A sign measurement, therefore, is not significant. This so-called phase coherence property is expected in several absorption models. It is true, for example, in the model of Williams [14]. The analysis of Matison et al. favoured the DOWN solution as a result o f their including the measured absolute magnitude of the Kn cross section in their analysis. We have not done this for our analysis because of the uncertainties in extracting real neutron target cross sections from deuteron data. We expect these could have introduced a systematic normalisatiop error of the order of a few per cent.

S.L. Baker et al. / K+n --, K+n-p

229

5.2. The method o f Ochs and Wagner As can be seen in fig. 5, the density matrix element p 1-1 is practically always consistent with zero. Ochs and Wagner [5] pointed out that (for our case of S- and P-waves only) this implies that IA+_ I= hi+_ I and incorporated this condition into their analysis procedure for nn phase shifts. Further, to investigate any possible rapid mass variation, they used t-channel helicity frame moments averaged over small but physical values of - t rather than extrapolated moments which would have required larger mass bins. We proceeded in this way to calculate the ratio A S+_~..+_/a 0 and were then able to estimate 61/2 exactly as in the previous section. The results are given in table 4 and fig. 12. These were calculated using all events with t > - 0 . 1 5 (GeV/e) 2. They can easily be seen to be in agreement with the previous analysis. In particular, we find the same UP-DOWN ambiguity for the I = S-wave phase shift. Finally, we have plotted, in fig. 13, the ratio of the magnitudes of the helicityone and -zero, P-wave, unnatural parity exchange amplitudes. It can be seen that the relative importance of the helicity-one amplitude decreases with increasing K+rr - mass. In the Williams ansorption model [14], this ratio is predicted to be independent of the Kn mass.

6. Conclusions We have found that the I = ½ S-wave phase shifts are as plotted in figs. 10 and 12. These are found to be consistent with those reported in refs. [6,7]. We feel that it cannot be too strongly stressed that the analyses described have used data which are qualitatively different from those used in these earlier analyses. We have used a different reaction and low beam momenta so that the amplitude structure is quite different. The agreement observed is thus most encouraging and makes it much more convincing that the phase shifts are in fact essentially those that have been reported. We have also measured the relative phase of the unnatural parity exchange Pwave helicity-zero and -one amplitudes and find that the phase coherence property is well satisfied by our solutions. We should like to remark that although we have included K+n - data for masses greater than 1.1 GeV in several places in this paper, we have only done so for completeness and there must be several doubts about the utility of the high-mass data. Among these are our assumption of elastic unitarity, our disregard of the D-wave which is obviously not acceptable near the K*(1420) resonance, and the severe kinematic limit (Chew-Low boundary) imposed on the minimum value of t at the higher mass by our low beam momenta. Finally, in order to compare our data with the soft meson prediction [9,10], we have plotted in fig. 14, for the five lowest energy bins of the analysis described in sect. 5, the real part of the I = ½, S-wave K amplitude versus the pion lab energy COL,

S.L. Baker et al. / K+n ~ K+Tr-p

230

/

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where m2 coL =

- m2 _ m2 2m K

The variable col is used here since this plot displays the symmetry of the forward dispersion relation with respect to the s- and u-channels (for example, ref. [15 ] ). It can be seen that agreement is satisfactory, (apart, perhaps, from a departure at the lowest energy point, which we do not regard as significant). A fit to the 12 points shown yielded a 1/2 = 0.12 -+ 0.03 m -1 to be compared with a theoretical prediction of 0.16 + 0.02 m -1 . We should like to acknowledge a helpful communication with Dr. P. Estabrooks and several conversations with Professor I. Butterworth. We are grateful to our scanning and measuring staff and acknowledge financial support from the Science Research Council.

S.L. Baker et al. / K+n ~ K+~r-p

2 31

References [ 1] [2[ [3] [4] [5] [6]

[7] [8] [9] [10] [l 1] [12] [ 13 ]

[14] [ 15] [16]

F. ttalzen and C. Michael, Phys. Letters 36B (1971) 367. C. Goebel, Phys. Rev. Letters 1 (1958) 337. G. Chew and F. Low, Phys. Rev. 113 (1959) 1640. P. Estabrooks and A.D. Martin, Phys. Letters 41B (1972) 350. W. Ochs and F. Wagner, Phys. Letters 44B (1973) 271. H. Bingham, W. Dunwoodie, D. Drijard, D. Linglin, Y. Goldschmidt Clermont, F. Muller, T. Trippe, F. Grard, P. Herqute, J. Naisse, R. Windmolders, E. Colton, P. Schlein and W. Slater, International K + Collaboration, Nucl. Phys. B41 (1972) 1. M.J. Matison, A. Barbaro-Galtieri, M. Alston-Garnjost, S.M. Flatte, J.H. Friedman, G2R. Lynch, M.S. Rabin and F.T. Solmitz, Phys. Rev. D9 (1974) 1872. T.G. Trippe, Recent experimental studies of the Kn interactions, ANL/HEP 72 08, Vol. 1 (1971) 6. S. Weinberg, Phys. Rev. Letters 17 (1966) 616. R.W. Griffith, Phys. Rev. 176 (1968) 1705. J.R. Campbell, W.T. Morton and P.J. Negus, Nucl. Instr. 73 (1969) 93. G. May, Imperial College Ph.D thesis, available as Rutherford Laboratory report HEP/T/44. G. Grayer, B. ttyams, C. Jones, P. Weilhammer, W. Blum, H. Dietl, W. Koch, E. Lorentz, G. Lutjens, W. Manner, J. Meissburger, W. Ochs and U. Stierlin, Nucl. Phys. B50 (1972) 29. P.K. Williams, Phys. Rev. D1 (1970) 1312. A.D. Martin and T.D. Spearman, Elementary particle theory (North-Holland, Amsterdam, 1970) p. 295. J.L. Peterson, Phys. Reports 2 (1971) 155.