.kNNALS OF PHYSICS: 42, 81-96 (1967)

K+-Deuterium Scattering

RAMESH CHAND

Department of Physics, Wayne Stale University, Detroit, Michigan @‘!~OC?

In the multiple scat.tering formalism for ZLVN system, low-energy K+-d elastic and ineIastic scattering cross sect,ions, as fllnctions of the Z = 0, s-wave ZiN scattering 1engt.h CYO , and the Kc-laboratory-momentum kL , are calcrl- lated. Using the experimental values for the Z = 1, s-wave KV scattering para- meters, al(O) = -0.29 F, T, = 0.5 F, we find that the limited experimental data on the K+-d charge-exchange scattering can be explained by 010 = -O.ll~~:~!F. It is also observed that, unlike t,he case for K+-p scattering, higher partial waves contribute significantly to the 6+-d scattering for K+-laboratory- momentum around 300 MeV/c or larger.

1NTROI)UCTION

The experimental data (1) on Ktproton scattering for Kf-laboratory- momentum k, in the range of 140-642 RleV/c strongly suggests that the I = 1, KN nuclear int,eract,ion is predominantly s-wave, w&h the dat,a fitted reasonablywell bycrl(O) = -0.20 & 0.015 F, and rl = 0.5 f 0.15 F, or alt.erna- tivcly, in t,erms of a purely repulsive core of radius I’,~ = 0.3 f 0.01 F.’ The scattering length al(O) and the effective range 1’1 are defined in terms of the s-wave KN phase shift a1 , by the general relation:

i 1.1 I

for isospin I and the center-of-mass (cm.) momentum p. Since, it is quite well known that the K’-p system does not form a bound state, the negative value of al( 0) implies the repulsive nature of the I = 1, s-wave KN nuclear potential (2).

Now, the question arises, what ran one say about the nature of the I = 0, s-wave KN nuclear potent.ial, whether it is attractive or repulsive, and also, can one determine t.he values of 1 he I = 0, s-wave, KN scattering parameters. For this purpose, the study of elastic and inelastic scattering of Kf-mesons on deuterium nuclei provides bhe simplest possible situation available to us.’ The

1 The s-wave LV phase shift ~r(Kx’) is related to the repulsive hard-core radius rCI in isospin state Z by the relation: rcl = --8,(K:V)/p, for K,V c.m. momentum p.

2 One might think that t)he strldy of lizo-p interactions are better suited than t,he K+-tl reactions for determining the correct value of LYE However, it turns out that, at low ener- gies, A$p processes provide reasonable discrimination between the varions Z?iV scattering solutions, and very little discrimination between the various values of ~10 . Hence, unless the K&p data is improved considerably, K&p interactions are not suitable for deter- mining the low-energy behavior of the I = 0, s-wave ZW interactions.

81

s2 CHAR’11

main reason for this is t’hat ~ hince dcuterium is n very loosely bound system of neutron and proton with the avcrnge n--l) separation large compared with the range of KN interaction ( N~,/M~), the c%lculntions 011 the K’-d system can bc regarded as fairly reliable. Also, some data on lowenergy Ii+-tl Fcnttering are available (3) at the present time and more dat,a are being obtained.

At low energies (16, s 300 RleV/c), t’he following processrs occu 1: fol K’-cl collisions :

K+ + d + K+ + d: e1a.st.k; (l.“a)

+ K’ + 72 + p: disint,egration; (Zb)

+ K” + p + p: charge-exchange. (1.3)

The thresholds for processes (1.2b) and (1.3~) are k, = 60 and 97 MeV/c respectively. The calculations on these processes in the impulse approximation have been performed by Ferreira (4), Gourdin et al. (5)) and Stenger et al. ( 6). However, as has been shown by Day et al. (7)) the use of impulse approximatjion in these processes can not be trusted, and that the effects of multiple scattering of K-mesons by nucleons are of considerable importance. In this paper, the multiple scattering calculations on the Kf-d system are performed in the zero- range boundary-condition formalism developed for the low-energy K--d int.cr- actions (a), using t.he 1 = 0, s-wave KN scattering length (~0 as an unl~lown input parameter. The calculations assume t,hat the Kf-meson is captured from t’he s-orbit.

[In principle, one should take into account the possibilihy that the capture of K+ meson may take place from the p (or higher) state. However, despite tile fact that in this case the multiple scattering calculations become very compli- cated, the inclusion of even the p-wave capt,ure introduces t,oo many additional unknown KN scattering parameters into the formalism, wit,h the result that the analysis of the low-energy k-+-d scattering data carmot be very meaningful. Besides, the experimental data on low-energy K’-p scattering is consistent with the assumption of s-wave KN interactions. iSevertheless, due to the fact that the tot,al cross section for a process is the sum of the va.rious part,ial-wave cross sec- tions, as we shall see later, t.he assumption of s-wave capture provides us with some information about the possible contributions from higher part.ial waves.]

In this paper, we ignore t.he spin-flip effects which intuitively are expected to be negligible, and also, neglect dcut’erium binding energy (except in the kine- matics). The two-body KN nuclear interact,ions in the study of K+-d system are assumed to be charge-independent with the result that Coulomb effects and the mass differences bet,ween particles in the same isospin multiplet are neglected.3

3 For the K--d syst)em, Dalitz and I [Ramesh Chand and R. H. I)alitz, Ann. Phys. 20, 1 (1962)] have found that the inclusion of (A Ta - K-1 and (IL-P) mass differences amount to an effect of order one percent.

iti-d SCATTERING 53

These assumptions are reasonable since me are int>erested in the Kf-laboratory- momentum range of 175-300 I\leV/cyclc, for which the c.m. energy is reasonably large relative to the mass differences, and yet not, high enough that the assump- tion of s-interactions for the KN processes ceases t’o be adequate. Since the values of kL which are of interest to us are not too high, the nucleon recoil effects in the multiple scatt*ering calculations are not expected to be too important, and hence arc neglected. Moreover, haking into account nucleon recoil cfkct’s col,,vectZy in the calculations is not easy, especially since at low energies, t)he cont,ributions due to the inclusion of Fermi momenta of the nucleons in deuterium can be expected to be qllite importantJ.

The cnlculat,ions on the K+-rl system using Fadeev-type multiple scat’tcring theory-with the two-body interactions being s-wave, nonlocal, separable poten- t8i:ds of the Ynmaguchi fornl-have been performed by Hetherington and Schick ( -9). It. is not clear whether Fadcev-t’ype calculations provide improvement over our zero-range bouncl;lry-~onditioIls formalism. In this paper, our main aim is ~tot to c*ompare the various theoretical models, but to analyze the existing experi- mcutnl data. However, an import,ant check on the reliability of any calculations is provided by the uuitarity c*ondition, where the sum of the individual cross sections is compared wit’11 t,he total cross section, obtained using optical theorem. Hetherington et al. do not. ralculat,e the K+-d charge-exchange and disintegrat,ion cross sections and henc*e, it is not possible to check the consistency of their c:~kulations. In our case, n-c find t’hat the unitarity is snt’isfied to wit)hin 7 5, which is quit,e remarkable in view of the fact that we make a large number of approximations especially in the cnlculations of charge-exchange and disintegra- t,iorl cross sect,ions. In adclit,ion, we obtain the important result, that out of the various k-+-c,! scat,tcring processes, charge-exchange scatt)ering is the most, suit- able for determining the correct’ value of (~0 .

In Sec*tion II, we give a brief description of the calculations, based on the model of Ref. 8. The results are discussed in Section III.

II. CALCULATIONS

A. KNN WAVEFUSCTION

The wavefunction of the KNN system can be written as

+(+)(r, R1, R,) = exp (zk.r)J/d(R)A,, + &, (Zla)

A,, = K+(pln2 - ,n&/d), @lb)

where bhe first term describes t,he initial K+-cl system for cm. momentum k, A” denotes the initial isospin wavefunction in which the symbols K+, p and n refer to the isospin wavefunctions of t’he corresponding particles and the labels 1 and 2 refer to the two nucleons; $d(R) represents the deuteron radial and spin wavefunction (the label SC refers to scattering) ; and r, RI , R2 refer to the posi-

s-l (‘HAN0

tions of the K.*-meson and the nucleons 1 and 2, respectively. E’or convenicIl(ac’, we choose the origin of our coordinate system at the cm. of the two nucleons, RI + R, = 0, and R = RI - Rz defines t,he relative separation of the two nu- cleans. Since as a result of interactions, Ohe final NN sysl,em can be both in :m

I = 0 and I = 1 st’ates, t’he scattered wavefunct,ion cm be written as

AC = Pdr, RI, Rdb + @lb-, RI, RdAlJ/dR), (2.21)

Al = (?.~)““K”plpz - k’+(plnz + n1 p2)/&. (2.2h)

In the limit of fixed scattering centers, each scattering gives rise to outgoing waves of appropriate amplitude with a0 and @I having the general form

al(r, RI, Rz) = Cr(R1, RB) exp(y ‘_’ i ,r 1

p’ ”

+ D I

(R1 , Rq) exp(ik I r - RZ I)

(2.3)

jr-RR,1 ’

wit,h the coeffirient,s C, and Dr to be determined by the boundary conclitions. The Fermi statistics for the nucleons requires

Co(R,, RI) = Do(R, , Rz), (2.h)

CI(%, RI) = -DI(% , Rd. (2.4b)

From now on, we shall abbreviate C1(R1 , R,) and D1(R1 , R2) by C1 and D, . Denoting byPoi and Pt the projection operators for isospin states 0 and 1 of the k’Ni system (label i on N refers to the ith nucleon), the boundary conditions for the KNN wavefunction ran be written as (8)

[ I r - & I J/(+)0-, RI , %)]PR~

= (010 PO' + a1 PI') d( , r t Ri I) ( I r - Ri I #+)(r, RI , R?))] (2.&L)

r-R,’

poi = $i(l - THINK), pli = fi(3 + T~*zK), ( 2..ib)

where t’he bar denotes an average over the direction of (r - Ri), and (Ye the energy-dependent (see expression 1.1) s-wave KN scattering length for rm. momentum p, corresponding to the Kf-laboratory momentum Ic, . The use of the boundary condition (2.5) for i = 1, together with the relations (2.4), give us

K+-d SCAT'FEKING I-

\ c .I

This completes the dctcrmination of our KNN wavefunction. However, we note that, even though the KN interactions have been specified by the boundary conditions (2.5), it is in fact possible to give an explicit expression for the KNN nuclear potential I’. The use of zero-range boundary conditjions implies that, one can USC pseudo-potential for describing the RNN interactions wit.h the correct expression given by

v= & [ / r - RI 1 h(r - RI) + [ r - Rz I 6 (r - Rz)I, ( 2.s )

where the 6’s are t’he three-dimensional delta-functions. IVow, using (2.1) and (2.8), we shall calculate t.he cross sections for t,he various low-energy K+-tl reactions.

B. k’+-d ELASTIC SCATTERING

The scattering nmplit’ude f( 0) for elastic scattering of K+-mesons from cm. momentum k to final c.m. momentum k’ (k’ek = k* cos 8) is given by (8)

f(e) = (wpE:r;/%Ic) ( c#q 1 v 1 +‘+Q, (2.9)

where

We = (/i” + ‘)llg’)1’2. El; = (ii2 + Illdye, (2.10a)

I< = WA + 81; ) (asob)

and & is the appropriate final-state wavefunction with form given by

~$f = exp (ik’*r&(R)&. ( 2.10r)

Defining

Ai = xll?ld(R)[j4K)“2Rl-1), (2.11)

where x1 is the deuteron spin wavefunction, expression (2.9) simplifies to (8)

For numerical evaluation of (2.12), we use the Hult#h& function for Q(R), given by Moravcsik (loj,

ud(R) = (0.850)“2 [exp (-0.232R) - exp (-1.202R], (3.13)

where R is measured in Fermis. This wavefuncbion is normalized to unity by

J’

m Q’(R) dR = 1. (2.14)

0

In terms of f(e), the expressions for elastic differential cross section and the total cross section (using optical theorem) are given by

dUel/dQ = Ifie) 12, (2.15)

mot = g Imf(O). (2.16)

C. K+-d CHARGE-EXCHANGE SCATTERING

For this process (and also for deuteron disintegration process), it is natural to choose a final-state wavefunction in which the N-N force is taken into account.4 Therefore, the appropriate final-state wavefunction leading to an I = 1, NN state can be written as

+f’(k’, q> = exp (zk’*r)dN(q, RI&, (2.17)

where k’ and q denote the kaon cm. momentum and the nucleon momentum in the NN rest frame. Due to expression (2.2b) for Al , the charge-exchange cross section is related to the cross section for the sum of all the processes leading to an I = 1, NN state, by the relation

l&T,.,. 2 da’ cl0 =32ir

(2.15)

In order to obtain the differential cross section, we first note that the density of final states is given by

pr = dk d3q/(27$ dEf , (2.19a)

where the final energy Ef can be written as

Ef N 2,mN + ‘dk’ + kf2/4mrf + q2/mN ) (2.19b)

4 Due to OLW neglect of spin-flip effects and the assumption of s-interactions, the final N-N states are limited to triplet states of odd parity.

h+--d SCATTERING s’7

treating the ltuon relntivistic~ally nntl the nucleons nonrelativisticnlly. Since :I complete calculation for this process [md also for process (1.2b)] is very lengthy, ~vlrc shall obt,ain a11 estimalc of the charge-exchange cross section, using the following approximate sum rule:

&J:q, R’)&$(q, R) d”qjWr)” = ,‘; [6(R - R’) - 6(R + R’)]. ( ‘) “0 ), -.-

[There are two main approximations which are involved here. Firstly, WC assume that the kaon momentum spectrum peaks at a value corresponding to a fret KN csollision at the same laboratory momentum, and hence approxjmatc the energy-conservation b-fun&on by fixing q in it corresponding t,o t,his pe:& value of k’. Secondly, we extend the sum over final momenta q beyond the limit provided by energy conservation. The error involved in this (‘ase will be much larger than for thr cleuteron disintegration case because of the nonzcro orbital angular momentum in the final p-p system w&h the result that the peal< in the Q distribution will occur for much larger values and hence a large error results in the use of sum rule due to &ending the sum over q b(byond t.he limit provided by 6(,Y - Ef). The use of the sum rules is discaussetl in c*onsidernble d&ail on pages 451 and 453 of Ref. 8. Wrc may mention here that t’he cnlrula- tions can be performed on the charge exchange and disintjrgration processes in a stJraight,forward manner without t#he use of the approximnle sum rules. How- ever, in this case t,he c&:ulntions bccomc quite involved and numerical cvalu:~- lion of the results on the computer prohibitively expensive.]

The use of the sum rule [Eq. (2.20)] lends to t,he following expression for the chnrge-exehangc differenM cross section (8)

I& = 11 - J+Pl f sin kR sin k’R TR- - k’R 1 ,

(3.21)

(2.22b)

(2.22(S)

( 2.22(l)

(2.22e)

S8 CHAi'iin

c = 4,4” - A2 co2 8, B = 1;’ + 2w,E’ + 3&, (2.22f)

E’ = dm,? + P/4, A = w+d, (2.22g)

where for k’ we have used the approximation that it corresponds to a free h'N collision, leading to an outgoing lcaon at angle 0 in the K+-d c.m. system.

D. K+-d DISINTEGRATION SCATTERING

For processes (l.Zb), the final n-p system cm be in both t,he I = 0, and I = 1 states. From charge-independence, it follows that the disintegration cross- section corresponding to I = 1, N-N state of the system, is given by

(2.23j

where da,,,./dQ is given by (2.21). Denoting by dcii,/dil the disintegration cross section for the I = 0 configuration of the NN system, the expression for the total disintegration differential cross section can be written as

dudis -= dQ

(2.24)

In order to calculate dgii,/dQ, the appropriate final-state wavefunction can be written as5

+?(k’, q) = exp (ik’*r)#k(q, R)Ao . (2.2.5)

To obtain the expression for d~~is/dfl, we shall use the following approximate sum rule [see parenthetical statement following Eq. (2.20)] for the 1 = 0, N-N wavefunctions :

&d&h+(R) + j &,dq, R’)&;(q, R) d3q/(2a)”

= 4; [6(R -R’) + 6(R + R’)].

This leads to

(2.26)

(2.27)

where da,,/dCl is given by (2.15), and the expression for dm’/dQ can be written as (8)

s

m da’/dQ = 2[ d(Rj[g+ JI + g- Jz - gJ-I dR, (2.28)

0

5 Due to our assumption of s-interactions and the neglect of spin-flip effects, the final N-N states are limited to triplet states of even parity.

K+*l SCATTERING 89

where

(2.29a)

(2.29h)

(229c)

In the next section, we discuss the results of the numerical evaluat.ions of (2.15), (2.16), (2.21), and (2.24).

III. RESULTS

A. ELASTIC SCATTERING

Expression (2.12) for f(O) has been evaluated numerically as a function of cos 0 for several values of the scattering length 010 . The differential cross sections so obt#ained are presented in Fig. 1. Despit’e our assumption of s-interactions and the neglect of mass differences between particles belonging to the same isospin mult,iplet, these angular distributions fall off rapidly in the backward direction, reflecting the fact that the probability for the deuteron to survive decreases as the momentum transfer from the Kf-meson to deuteron increases (as cos o decreases). At Dhe present time, there is no experimental data on the low-energy

FIG. 1. The elastic differential cross sections du,l/dg vs. cos 0 as a function of the scatter- ing length (~0; 010 = -0.11 F for curve a; -0.05 F for curve h; 0.01 F for curve c; and 0.07 F for curve d. The calculations assume that CY* (0) = -0.29 F, and ri = 0.5 F.

90 C-HAND

elastic scattering distribuCons, probably due IO the dificulty of distinguishiIlg elastically scattered deuterons from the protons c*oming from the disintegration processes. Nevertheless, since t,here is very little tliscriminntion between thcsc various curves, low energy elastic scat,tcrin g angular dkkrihutions arc not vtary suitable for determining the correct value of CQ . We may note that even though the elastic angular distributions have been cnlculnt~ed using the approximate KNN wavefunction, the forward peaking of these distributions justifies t,o some extent the correctness of our calculations.

The total elastic cross sections (~~1 and the tot,al cross sections ctot for the sum of all the K+-d processes (obtained using optical theorem) as a function of o(o are given in Table I for several values of k, . These cross sections decrease with the increase of CQ reflecting t’he fact that there is const’ructive interference (since CQ is negative) between the I = 0 and I = 1, KN scatkerings. The fact that the

TABLE I

COMPARISON OF THE SUM OF THE INDIVIDUAL CROSS SECTIONS FOR THE K+-tl SCATTERING PI~OCESSES"

:MeV/c) Scattering

lenpth w @;F!

de1 b-h) u~..~. (mb) (T,I,~ (mb) gtot (optical)

btj)

175 -0.11 18.40 0.76 5.55 23.81 24.72 175 -0.07 17.15 1.13 5.36 22.88 23.64

175 -0.03 15.91 1.56 5.19 22.08 22.G7 175 0.01 14.73 2.06 5.10 21.48 21.89

175 0.05 13.60 2.64 5.06 21.07 21.29 230 -0.11 13.98 0.95 9.57 23.29 24.50 230 -0.07 13.01 1.40 9.13 22.48 23.54 230 -0.03 12.04 1.93 8.73 21.82 22.70

230 0.01 11.12 2.54 8.12 21.37 22.08 230 0.05 10.24 3.23 8.19 21.14 21.G6 275 -0.11 11.41 1.09 12.16 23.20 24.66

275 -0.07 10.60 1.59 11.5G 22.45 23.75 275 -0.03 9.81 2.18 11.00 21.86 22.99 275 0.01 9.05 2.85 10.55 21.50 22.4.5

275 0.05 8.33 3.60 10.20 21.37 22.13

330 -0.11 9.17 1.24 14.73 23.34 25.14 330 -0.07 8.51 1.79 13.96 22.64 24.2G 330 -0.03 7.87 2.44 13.25 22.11 23.56

330 0.01 7.26 3.17 12.67 21.82 23.09

330 0.05 6.68 3.98 12.20 21.76 22.86

* Elastic-scattering cross section c,, , charge-exchange-scattering cross section s~.~. , and disintegration cross section (~,n~. The total cross section qot is obtained using the op- tical theorem. These cross sections are functions of the scattering length 010 and the K+- laboratory momentum kr. The calcldations assume that w(0) = -0.29 F and r1 = 0.5 F.

h-+-d SCATTERING 91

decrease with increasing o(o is murh more rapid for uel than for ntot implies that the former can be used to determine CQ , if experimental data with reasonably small errors become available. The rapid decrease of elastic cross sections with increasing lc, implies t,hat t,he probability for the deuteron ho survive the collision, decreases with increasing energy. The energy dependence of ntot is rather small.

It is of interest to compare our results on a,1 and utot with the values obtained by Fadeev-type muhiple-scattering calculations of Hetheringion et ~2. (9). For 01~ = -0.01 E’ and k, = 230 lJIeV/c, Hetherington et al. obtain (T,, = 9.7 mb, ctot = 16.9 nib, as compared with our values of uCl = 12.3 mb, and utot = 21.9 mb. This shows t,hat our multiple-scatt,ering calculations result. in about 25 % higher values of k’+% elastic and total cross se&ions than those by Hetherington et al.

B. CHARGE-EXCHANGE SCATTERING

The charge-exchange-scattering angular distributions as a function of ~0 are given in Fig. 2, for two values of k,,. We first note that these distributions vanish in the forward direction due to the fact that we ignore spin-flip effects and hence forward charge-exchange scatkering would have t’he final two protons in a 3R1- state, which is forbidden for them by the Pauli principle. As 0 increases, da,.,./dQ increases, the fall for large values of 0 being clue to the variation in the phase space as calculated with our mean value of k’( 0). The large variations of these angular distributions with a0 suggest that, charge-exchange scattering is quite

t

-0.8 -0.4 Q 0, 0.1

FIG. 2. The charge-exchange differential cross sections du,Jcwt vs. cos 6’ as a function of the scattering length 010. The calculations asslnne that (YI(O) = - 0.20 F, and 1’1 = 0.5 F.

92 CH.4~1)

KL Experimental (~0 (Fermi)

WV/c) Value -0.13 -0.11 -0.09 -0.07 -0.05 -0.03 -0.01 0.01

175 - 0.61 0.76 0.93 1.13 1.34 1.5G 1.80 2.OG 230 0.9-o:r co 7 0.76 0.95 1.17 1.40 1.66 1.93 2.23 2.54 275 - 0.88 1.09 1.33 1.59 1.8i 2.18 2.50 2.85 330 2.8 f 0.5 1.01 1.24 1.50 1.79 2.10 3.44 2.79 3.17

n Total cross sections (in mb) for the processes KT + d + K0 + p + p were calculated as a function of the scattering length aa and the Ktlaboratory momentum k~ . The calculations assume that 01~(0) = -0.29 F, and I 1 = 0.5 F. Comparison is with results of Ilef. 3.

suitable for determining the correct value of cyo . Unfortunately, in the energy range (150 5 k, 5 300 MeV/c) where our calculations are expected to be reasonable, there is no experimental data available 011 da,.,./dfL

The total charge-exchange cross sections re.e. , obtained by integrating the angular distributions, together with t’he limited experimental data at low ener- gies, are presented in Table II. AS is intuitively quite obvious, these cross sec- tions increase with energy due t’o the increase of phase space.6 However, our calculations for ne.e. near the threshold momentum of 97 -IIeV/c arc certainly incorrect. Near threshold, the matrix element for charge exchange must be proportional to p, since the final nucleons can only be in a relative p-state, and the charge-exchange cross section must rise from threshold like (E - &,)‘. The increase of gc.e. with 010 is dictated by the term (01~ - 011)’ in expression (2.21)) since cq is negative. The comparison with the experimental data at 830 MeV/c shows that at this energy, 010 = -0.11%~ F. Unfortunately, due to large uncertainties in the experimentBal cross sections and the unreliability of our calculations beyond kL = 300 ;\IeV/c (mainly due t’o our assumption of s-interactions), one cannot make reliable estimate of the values of the scatter- ing length (~~(0) and the effective range I’,, . Nevertheless, the trend of u~.~. with energy shows that the correct value of so(O) will most probably be negattive, establishing the repulsive nature of the I = 0, s-wave KN nuclear potential.7 In the limit of zero range, the use of the above value of a0 for comparing the

6 At low energies, near the threshold, our calculation of n’c.e. (and also of udib) is expected to be au overestimate due to the inadequacy of the sum rule used.

7 It is well known that the negative value of ~(0) can imply two thiugs: (i) the repulsive nat,ure of the KN nuclear potential, or (ii) the existence of a KN bound state with attrac- tive nuclear potential [see (Z)]. S ince there is no experimentally known KN bound state, the negative value of 010 (0) establishes the repulsive nature of the KN nuclear potential.

K+-d SCATTERING 93

calculat’ions with the data at 330 RIeV/c shows that the higher partial waves beyond the s-wave must contribute quite significantly to the K+-d charge exchange cross sections, a situation quite different from t’he K+-p scattering case. This conclusion agrees with that reached by Warnock et al. (11)) using a quite different type of calculation.

C. DISINTEGRATION SCATTERING

The deuteron disint,egration angular distributions as a function of CX” are presented in Fig. 3 for two values of k, . These angular distributions fall rapidly in the forward direction, reflecting the fact that the probability for deut’eron disintegration decreases as the momentum transfer from the I!?-meson to t.he deuteron decreases. This result is easily unclerstood in the impulse spproxima- tion: forward scattering without spin flip leaves the final nucleons in the 3S1- state, with the scattering amplitude proportional to J $f+(R)#d(R) rl”R, which vanishes due to the orthogonality of the wave functions. Again there is no experimental data on the lowenergy Kf-d disintegration processes, probably due to the difficulty of separating these events from the elastic events. Sever- theless, since there is poor discrimination between these curves as a function of

0.2 t

kL= 230 MeV/c \

FJG. 3. The deuteron-disintegration differential cross sections dad;,/dR vs. cos 0 as a flmction of the scattering length LYY~ ; 010 = -0.11 F for curve a; -0.05 F for curve b; 0.01 F for curve c; and 0.07 F for curve d. The calcldat,ions assume that w(O) = -0.29 F, and T, = 0.5 F.

94 CHAND

CQ , disintegration angular distributions are not very suitnblc for determining the correct value of CQ .

The total disintegration cross sections cdis as a function of a0 and I<, are given in Table I. As is quite obvious, these disintegration cross sections increase monot’onically with lc, . The decrease of c<tis with the increase of (Ye is ,similar to the case for uei , due to constructive interference between the I = 0, and I = I, KN scattering amplitudes.

An important check on the consistency of our approximate calculations is provided by unitarity when we compare t’he sum of the individual cross sections for elastic, charge-exchange, and disintegration scatterings with t,he total cross section given by optical theorem. This has been done in Table I. The t,wo esti- mates of the total cross sections agree to within 7 % in the entire energy range of 1755300 MeV/c. This is quite remarkable considering the necessarily large number of approximations [especially in the calculations of deuteron-breakup cross sections (c~.~. and cdis)]. Due to t’he use of approximate sum rules, our calculations provide overestimates of u~.~. and odis ; nevertheless, the intuitive correctness of the shapes for the various angular distributions suggest that this unitarity check cannot be just a coincidence. In fact, in t’he limit of small scat#ter- ing amplitudes (a0 , crl + 0), we find that our multiple-scattering results reduce to those appropriate to impulse approximation. We also find that at energies below the threshold for inelastic scattering processes, unitarity is sat#isfied to within 15 % ( utot ‘v gel). This of course implies t,hat our calculations of individual cross sections above the threshold for cdis and (T,,.,. cannot be t’oo bad either, since the unitarity condition is satisfied to within few percent at energies both below and above the threshold for inelastic scat!terings. We must not forget, however, that at low energies such as those below t)he threshold for inelastic scatterings, our calculations are not reliable due to charge dependent effects arising from the (K+, K’), (,n, p) mass differences, and the K+-p Coulomb interactions. We may also mention here that even though in the calculations of KNN wavefunction, the nucleons are regarded as fixed, yet our calculations for the individual cross sections take into account the appropriate kinematical factors. Hence if the calculations were carried out consistently treating the nucleons fixed then the unitarity check would not be too significant because then our wavefunction would solve the Schrbdinger equation appropriate to that situation, with boundary conditions which are consistent with probability con-

servation. Since our calculations have been modified to take into accourlt ap- proximately a number of kinematical features appropriate to the actual situn- tions, this unitarity check is by no means t,rivial or accidental, and, in fact, to some extent it justifies the essential correctness of our calculations. We may mention that, although in our formalism we have neglected the mass differences between particles belonging to the same isospin multiplet, we do use the correct

K+-d SCATTEKTNG 95

values of the masses in t,he kinematical factors for the charge-exchange and disintegration processes.

IV. CONCLUSIONS

The comparison of our c*alculated cross sections for charge-exchange scattering with t,hr experimental data at low- energies leads to the following result)s. The I = 0, s-wave KN nuclear potential is repulsive wit,h the experimental data at 230 AIcV/c fitted by LY,, = -0.11’:::: IT, and quite contrary to the case for K ‘-p scattering, higher partial waves beyond s-wave (sontribute quite signifi- wltly to t,he K+-cl cross sections for k, 2 300 RIeV/c.

[After the completion of t’hese calculst~ions, it has come to our nt’tention that the use of the Cwrent illgebrns to meson-nucleon scat#tering gives t,he sum rule: ~~~~(01 = l i[c~(O) - CY,,(O)], where a&O) is the zero-energy s-wave TN scattering lengt,h in the I = “2 state. [See A. I’. Balachnndran, ,\I. Gunzik and E’. Nicodemi, “Current, Algebras and Meson-Nucleon Scattering Lengths,” Syracuse University Preprint 1206.-SU-63 (August 1966).] al(O) ‘v -0.29 F (1) and u~!~(O) ‘v -0.11 F (1%‘) requires 0(0(o) ‘u -0.07 F, which is consistent with our value, 010 = -O.ll?~:~~ F in the zero-range limit.]

Also, since among all t)he low-energy k’+%l cross sections, charge-exchnnge- scnt,tcring (sross sections provide the best, means of determining the correct value of CQ, it, is very desirable that detailed experiments be undertaken 011 K'-d

charge-exchange scattering for 1. LL in the range of 150P300 -1IeV,/c. We may note t,hnt, since among t#he various lowenergy k-+-cl processes the 1 = 0 s-wave KN scattering plays the maximum role in thP k-+-cl chharge-exrhangc scattering, it is no surprise that the charge-exchange-watterin g Cross sections provide the best means of determining the c~~uwt value of (Y” .

Regarding the inaclequacics of olw multiple-sc:nttt~rillg formalism, they are dis- cussed in considerable detail in Ref. 8, and hence we will not discuss them here. However, an important checak OII t,he consistency of the calculations is provided by unitnritjy, which in our vase is satisfied to within seven percent.

ACKKOWLEDGMENT

It is a pleasure to thank the st,nff of the IBM 7Oi4 computing center of Wayne State University for their cooperation during nlunerical computations.

RECEIVED: June 26, 1966.

REFERENCES

1. 8. GOLDHABER, W. CHIKOWHKY, G. GOLI)HABER,W. LEE, T. O'~%.~LL~R.\N, T. STUBBS, G. PJERROU, II. STORK, AND H. TICHO,~~L “Proceedings of the 1962 Annual Inter- national Conference on High-Energy Physics,” p. 356. CERN, Geneva, 1962.

2. H. A. BETHE AND I’. MORRISON, “Elementary Nuclear Theory”, p. 54. Wiley, New York, 1956.

96 CHAND

3. W. CHINOWSHY, G. G~LNIABER, S. GOLDIIABEK, W. LEE, T. ~'J~.\LLoH.~s, T. SWBB~, TV. SLATER, D. STORK, AND I-I. TKEJO, in I’roceedillgs of the 1960 Annual Internntiold Conference on High Energy Physics,” 11. 451. Interscience, New York, 1960.

4. E. M. FERREIRA,P~~S. Rezt. 116, 1727 (1959). 5. M. GOURDIN A~~~ A. MAR~IX, Nuouo Cimento 11, 670 (1959). 6. V. STENGER,~. LEE, D. STORK, H. TIVHO, G. (:OLDIUBER, .IND S. GOLDHIIBER, Phgs.

Rev. 134, Bllll (1964). i'. T. DAY, G. SNOW, AND J. SUCHER,~~~UOUO C'i,,lento 14, G37 (1959). 8. R.IMESH CH~KD, Ann. Phys. 22,438 (1963). 9. J. HETHERINGT-ON BND L. SHICK, Phys. Rev. 138, B1411 (1965).

10. M. MORA~CSIK, Xucl. Phys. 1, 113 (1958). 11. R. WARNOCK AND G. FRYE, Phys. Rev. 138, B947 (1965). id. S. DEBENEDETTI, “Nuclear Interactions,” p. 465. Wiley, Kew York, 1964.