A faddeev calculation of k+d elastic scattering

Nuclear Physics @ North-Holland

A402 (1983) 462476 Publishing Company

A FADDEEV

CALCULATION

OF K+d ELASTIC SCATTERING

J. SAE;TUDO Departamento de Fisica Teorica, Facultad de Ciencias, Zaragoza, Spain Received (Revised

8 December 16 February

1982 1983)

Abstract: Faddeev equations are used to analyze the available data for ibw-energy in order to determine the isosinglet K+N S- and P-wave scattering lengths. been properly taken into account.

elastic Kfd scattering Coulomb effects have

1. Introduction In recent years there has been an increasing interest in the KN interaction at low and intermediate energies I). The reason for this interest lies in the fact that kaon beams are more and more used in the formation of hypernuclei ‘) and in the study of reaction dynamics and nuclear structure 3z4). For the first kind of experiments K- beams are used, whereas K+ is preferred for the second kind. It is evident that a good knowledge of the two-body systems is crucial to obtain reliable information from such experiments 4). A reasonable knowledge of the K’N interaction in the T = 1 isospin channel has been obtained from K+p scattering experiments 576). Information on the T = 0 channel, instead, cannot be drawn directly, but through its contribution to K+d scattering. At high energies the impulse approximation is justified to deal with the K’d system, but at low energies, when multiple scattering effects become more important

7), it ceases

to be a good method

for extracting

information

on any of

the two-body systems. Aiming to improve the current knowledge of the T = 0 K’N interaction energies, we have applied, in a preceding paper ‘), the Faddeev formalism

at low to the

analysis of the existing data on K’d elastic scattering. Such analysis confirmed that the T = 0 K’N interaction is considerably less important than the T = 1 channel. Our analysis, however, did not take account of the two-body interaction in waves of angular momentum higher than zero nor the Coulomb repulsion. The effect of these interactions although small, might be important for the determination of the T = 0 scattering length. For this reason, they have been included in the analysis reported in this paper. In sect. 2, the general Faddeev formalism is recalled. The two-body interactions required for its application are presented in sect. 3. S- and P-waves are considered in both T = 0 and T = 1 K’N channels and the Coulomb interaction is included. 462

J. Satiudo / K’d

elastic scattering

The resulting values for the T = 0 scattering presented in sect. 4 and discussed in sect. 5.

463

lengths (in S- and P-waves) are

2. Notation and general formalism Since the pioneering work of Faddeev 9, there has been a considerable interest in the non-relativistic quantum three-body problem, both from a formal point of view and because of its applicability to three-nucleon and hadron-deuteron systems. Relatively recent summaries of the work done on formal aspects of Faddeev equations and on their implementation can be found in the monographs of Schmid and Ziegelmann lo) and Afnan and Thomas ‘l). Let us consider three non-relativistic particles that, for the moment, we assume are distinguishable. We denote by m, (a = 1,2,3) the mass of the particle labelled (Y,and by A& the sum of the masses of the two particles different from CX.Besides this, the reduced mass of each pair of particles, CL~=

mm3/h+m3),

(2.1)

and analogously for ~~ and cam,will be used. As reference frame three-body c.m. one. The linear momenta of the particles will be Obviously, only two of these momenta are independent. In practice, the three-body momentum states by that of one of the particles, pa, momentum of the two others, qa = (m,pp - mpp,)l(mp + m,) ,

a, p, y = 1,2,3

we choose the denoted by pn. we characterize and the relative

cyclic.

(2.2)

The total c.m. energy (excluding masses) is designated by E. The particles are allowed to have spin, s,, and isospin. By 1, we shall denote the orbital angular momentum (in the total c.m. frame) of particle LYreferred to the center of masses of the two others. In order to make the Faddeev formalism useful, the particles are assumed to interact only via two-body separable potentials of the form (units ti = 1 are used throughout this paper) V(q, 4’; E I=

c

T,J,M,S,L,L’,v

x

0

d3q)~E,s(flq)

~~S(E)12CL)P~~LIL’,S(~nq’)+g~i;S(q’)

,

(2.3)

with form factors g(p) and intensities A(F) that we allow to depend on the energy E of the pair. The symbol PT represents the projection operator into the two-particle isospin subspace. The indices T,J,S,L correspond to the isospin, total angular momentum (of third component M), and the spin and orbital angular momenta of the interacting pair. The label v characterizes the different terms of the potential (in the case of rank larger than 1).

464

J. Saiiudo / K’d

elastic scattering

The cross sections for the different three-body processes are obtained from the three-body amplitudes, commonly denoted by X,,,;,,, (pi, pL*; E). Here, the subscripts (Y,m (p, n) denote the initial (final) three-body state, in which the particles with label different from u(o) are coupled to form a two-body state specified by m(n). The usual version of Faddeev equations in the case of separable two-body interactions, reads 12)

x A,,@ -PC’(ml + m + m3)/2m,W)X,,,;,,,

(P;, pa ; El

,

(2.4) where ZO,, Gn,mis closely related to the form factors of the separable two-body potentials and A,,? represents the propagator of the three-body state in which particles with label different from y are interacting. We use boldface types for X, Z and A to indicate that they correspond to matrices in the case of some of the separable two-body interactions having rank larger than 1. For practical purposes it is most convenient to perform a partial wave decomposition of eq. (2.4). Then, instead of the above three-dimensional integral equations, one has to solve the one-dimensional ones for the rotationally invariant amplitudes 13):

Here, the superscript L) stands for the set of quantum numbers (total angular momentum f, isospin T and parity T) conserved in every three-body process. The subindices N, represent quantum numbers characterizing the various ways of coupling states of two of the particles with those of the third one to form the three-body state of quantum numbers D. Different coupling schemes can be chosen. For instance, if one adopts the coupling scheme of Harms r4), the subindex Nr would be explicitly written as N1 = {Ti, Jt, S1, iI, jl) and would represent a threebody state where particles 2 and 3 are coupled in a two-body state of isospin ?“I, total angular momentum J1 and spin Si, and then coupled to particle 1, of orbital and total angular momenta I1 and jr, to give the quantum numbers D. In the scheme of Sloan and Aarons 15), instead, one has iV1={Ti, J1, SI, X:1, II} which corresponds to having particles 2 and 3 with quantum numbers TI, J1, SI, as in the Harms scheme, then coupling J1 with the spin of particle 1 to give the channel spin Xl, and, finally, coupling zi, to the orbital angular momentum II. The poie~~~~l and propagator terms in eq. (2.5) are explicitly given by (for definiteness, the secondly

f. SaZudo / R id elastic scattering

465

where we have denoted by C[J~&, S,)] and C[r(Tp, Ta)] the spin and isospin recoupling factors 16) corresponding to the channels LYand & and M(K $3, &, sn, & ; A, 0, L,, L) IB+3S,+L.,~Ju+3S~+Lut2r-c2n_2R

=(-1) x

[(2.&3+ 1)(2&F* + 1)(2X@-t 1)(2& + l)]l’* lp

S$

S,

n

x I LP

43

43 11 R

Lp r

(GT~,J~,s,,~,,~,:T,,J,,s,,I,,L,:;~ (~b, ~a ;

2

I.,

A j ( !a -mL, mi,

s,

s,

2;, 11 L,

n

Gw

lYa J,

-E))vo,vp

c (_#3+“-=“-=-(2& %_p.mLJ%,

=(2fl+l)

x k (

n

‘a

+ I)‘/2

A j 4rr312

rnL_ -m+

I” d (cos ePbPa) -1

Although the system of coupled integral equations (2.5) can, in principle, be solved without restriction on the rank of the potentials and the number of affected angular momentum waves in each two-body subsystem, one is compelled, in practice, to consider separable potentials of the lowest rank compatible with experimental data and effective only in the lowest angular momentum waves. Otherwise, the computational task becomes unmanagable. Moreover, the form factors g(p) are usually taken so as to allow the analytic evaluation of the integrals in the right-hand side of eqs. (2.7) and (2.9). There exists, however, a procedure

466

_T.Sariudo / K*d elastic se&ring

suggested by Sloan I’) for taking into account the effect of those terms of the potential excluded, for the above mentioned reasons, from the Faddeev algorithm. Such secondary terms, handled as a perturbation, give correctians to the Faddeev amplitudes. If we denote by X”’ the unperturbed amplitudes calculated by means of eq. (2.5) with only the principal terms in the potentials, the lowest-order corrections to these ~~Iitudes are given by

~A~,,(E-p~~(rn~+rn~+rn~)/2rn,M,)C~,,;,,,(p~,p,;E),

(2.10)

being

xAg,,(E-pb2tml+m2-tm3)/2mdM,)X~l,,,,(p~,p,;E). (2.11) We use primes on 2” and c’ to indicate that they are matrices whose rows (or co~urnns~ correspond to the principal terms in the interaction whereas the columns (or rows) are in correspondence with the secondary ones. Rows and columns of the square matrix A” correspond to the secondary terms in the interaction between the two particleswith label different from y, The use of eqs. (2.10) and (2.11) will be discussed in sect. 4. 3. Two-body

interactions

For the reasons quoted near the end of the preceding section, we have chosen ah the form factors to be of the ~amaguchi type”):

g(p) = (p2+p2f-’ ’

For the NN subsystem, the (two-body energy-dependent) taken as recently proposed by Garcilazo 19): ~(&~=~~~tanb(~-~~~=~

(3.1) intensity has been (3.21

with ANN= -9.41 fmv3 and EC = 0.816 fm-‘. This rank-one potential, with a range parameter &N = 5.632 fm-I, fits the triplet scattering length (a, = 5.39 fm) and the deuteron binding energy (~a = -2.225 MeV), The small admixture of D-wave in the deuteron wave function and, consistently, in the potential has been neglected. Now, let us discuss the KN potentials. In the T = 1 channel, the S-wave interaction at low energies can be fairly well represented by a rank-one potential suggested in a recent paper 20). Its energy dependence is taken to be of the form ~~~(~~=~~~

exp(ba),

(3.3)

J. Satiudo / K’d

elastic scattering

467

with A$ = 371.1 fmh3 and b = 0.4982 fm-i. For the range we take p& = 4.74 fm-‘. The resulting potential fits the scattering length a gh = -0.309 fm and the effective range r. = 0.32 fm [ref. 5)]. The P-wave interaction is not negligible, but a comparison of the scattering lengths 5*6)shows that it is considerably less important than the S-wave one. For this reason we treat it as a secondary term. Although there is not a conclusive determination of the scattering lengths for the T = 0 channel, it is well known *l***)that the intensity of the interaction is much smaller than in the T = 1 case. In consequence, we include the T = 0 interactions, in both S- and P-waves, among the secondary terms, that are treated perturbatively. 4. Results The potentials presented in sect. 3 have been introduced in the Faddeev algorithm. The Gauss-Seidel iterative method has been used to solve the linear integral equations (2.5) with only the principal terms of these potentials, i.e. T = 0, L = 0, J = 1 for NN and T = 1, L = 0, J = i for K+N. The integration path has been rotated, as usual, to avoid the singularities on the real axis. The maximum rotation angle, CD,allowed by the singularities occurring in the complex plane is given by 23) mN being the nucleon mass and p the K’d c.m. @max= arctan (2(m,~,)~‘*/p), momentum. Usually, a contour rotation angle of about @,,, is chosen. Following this common practice, we have taken @ = 10” and @ = 7.5”, respectively, in the cases of 342 MeV/c and 470 MeV/c incident K’ laboratory momentum.

TABLE

1

Elastic partial wave amplitudes, for total angular momentum A = 0 and 1, obtained for different choices of the contour rotation angle @ and the number N of Gauss-Legendre quadrature points in the cases of (a) 342 MeV/c and (b) 470 MeV/c incident K’ laboratory momentum

@Wed (a) A=0

A=1

(b) A=0

A=1

N=6

N=lS

N=32

N=48

N=64

-2 -10 -18

0.849-i0.833 0.746-i0.357

1.318-i0.761 1.197-i0.464

1.706-iO.720 1.426-iO.780 1.255-i0.785

1..595-i0.785 1.426-i0.779 1.511-iO.720

1.521 -iO.802 1.426-i0.779 1.382-iO.805

-2 -10 -18

0.342-iO.086 0.228-iO.002 0.107-iO.088

0.384-i0.212 0.393-iO.085 0.274-iO.067

0.434-iO.097 0.419-i0.118 0.303 -iO.126

0.433-iO.109 0.421-i0.117 0.481-iO.071

0.429-i0.115 0.421-i0.117 0.3899i0.138

-2 -7.5 -13

0.306 - i0.597 0.335 - iO.403

0.727 - iO.640 0.688 - iO.504

1.083 - i0.694 0.893 - i0.679 0.802 - i0.736

0.993 - iO.711 0.889 - i0.669 0.891- i0.623

0.942 - iO.705 0.889 - i0.669 0.898 - i0.691

-2 -7.5 -13

0.239-i0.923 0.171-i0.423 O.lOO-i0.453

0.291 -iO.229 0.320-i0.125 0.237-iO.460

0.384-i0.129 0.365-iO1152 0.305-iO.202

0.381 -iO.142 0.366-i0.148 0.366-i0.114

0.376-i0.147 0.366-i0.148 0.375-i0.166

The entries in the table are in units 27rZ10-s MeV2. For @ = -2” and N = 6 or 15 the Gauss-Seidel method, used to solve the linear equations (2.5), does not converge in the case A = 0.

468

J. Satiudo / KCd elastic scattering

In this way, a mesh of 48 Gauss-Legendre points in each integration is enough to reach an accuracy better than 1%. As shown in table 1, a choice of Cp either small or near @,,, would require a finer mesh. For higher total orbital angular momentum, A, when multiple scattering terms become less important, rotation angles smaller than $@,a, could equally well be adopted, since the dominant single scattering term does not present singularities on the real axis. The amplitudes obtained in this way ought to be corrected by the effect of the secondary terms in the potentials namely T = 1, L = 1, J = $, $; T = 0, L = 0, J = 4 and T = 0, L = 1, J = $, $ for KN, and by the Coulomb interaction. For the secondary terms, the perturbational method sketched in sect. 2 has been used, with additional approximations that we are going to explain. Firstly, we have neglected the second term in the right-hand side of eq. (2.11), that is, we have taken c:,Wn(P;~

Pa; m -~:,r;%m(P;,

pa; -w *

(4.1)

This is a kind of impulse approximation (we mean here those calculations which neglect the multiple scattering) for the evaluation of the corrections and is justified by the fact that these corrections are small. For the elastic kaon-deuteron process (denoting by 2 the kaon and by 1 and 3 the nucleons, indistinguishable in the isospin formalism) it results in, after a few manipulations, X:%,,

(6

PZ;~3 = .s

3 K&O,

W1)2

T,=d,l

Here 4d denotes the unnormalized

deuteron wave function r5)

dd(p) = -miJ((4rr)1’2(p2+~?i)(p2+P&N)),

(4.3)

with ffd = 0.23 fm-‘, and tbTYrepresents the two-body matrix for the secondary terms in the KN interaction. The matrix element in the integrand of eq. (4.2) has a slight variation in the range of integration where 4d is significant. It is, therefore, reasonable to take this matrix element as independent of pl: and equal to its on-shell value for relative KN linear momenta, k:, k,, corresponding to the nucleons being frozen in the deuteron. This is a second approximation [fixed scatterer assumption 24)] that, together with the first one, corresponds to what may be called single-scattering impulse-approximation 25*26)(SSIA). The corrections to the elastic kaon-deuteron amplitudes become finally X?,:2,m(~L~2;E)=

C y=1,3 T,=O,l

(CC%&

T,)1)2(kL; nlt\Ty(E)ik,; m)Sh-pi)],

(4.4)

J. Sa~udo / IT +d elastic scfftfef~ng

469

where S&p;! -pL)] represents the Jnnormalized deuteron form factor 27). To be normalized, the elastic amplitude X 2,,,,+, must be multiplied by N2, being N2= 4c&N&d+PNN)31rrmL The advantage of using SSIA lies in the fact that the corrections can then be directly related to the respective scattering lengths. In this way, the interactions are described by the low-energy parameters and the search of a suitable potential can be avoided. Moreover, since some of such low-energy parameters are to be determined so as to fit the experimental cross sections, the necessary minimization procedure becomes, when SSIA is used, considerably quicker and safer. On the other hand, numerical tests have shown that the errors due to this approximation are not larger than 10% in the corrections and consequently, about 1% in the total amplitudes. In our calculations we have taken (cz~,~) a :,,,, = -0.021 fm3

and

a&

= 0.013 fm3

(4.5)

for the KN T = 1 P-wave scattering lengths 5), whereas the T = 0 ones, u~,~,~, ~?,i,~, 0 have been left as parameters to be determined. a 1,3/2? The Coulomb corrections have been taken into account by adding the Coulomb scattering amplitude to the strong one, following the method used by Thomas 28) in the treatment of the rd system. We have tried to fit the K’d elastic differential cross section measured by Glasser ef al. 26) at 342 MeV/c K’ incident laboratory momentum. Relativistic kinematics has been used to obtain the cross section from the amplitude and to transform quantities (momenta and cross section) from the c.m. to the lab frames. Using the CERN Minuit code, the optimum values of the KN T = 0 scattering lengths turn out to be aZ,ij2 = 0.002 * 0.02 fm3 ,

a:112 =0.08*0.1

fm3,

u&,2

= 0.008 f 0.05 fm3 ,

(4.6) with a x2 per degree of freedom of 0.79. The experimental data and the theoretical cross section calculated with these values of the parameters are shown in fig. 1. 5. Discussion

The importance of the succesive terms in the multiple scattering series can be seen in table 2, where we have shown in the elastic partial wave amplitudes evaluated by retaining only up to single scattering (SS), double scattering (DS), or triple scattering (TS) terms, respectively. To illustrate, the Faddeev amplitudes are also shown. For comparison with the T = 0 KN scattering lengths determined by us, we list in table 3 the values obtained previously by other authors with different methods. The large errors, induced by the experimental uncertainties, in the various determinations of a&i2 make all them compatible. However, in view of the relevance at low energies of multiple scattering terms ‘**), we find our determirlation more

J. Sariudo / K’d

470

elastic scattering

Fig. 1. Differential cross section, in the laboratory frame, for K’d elastic scattering at 342 MeV/c incident K’ momentum. The experimental data are from Glasser et al. 26). The curve represents the theoretical prediction with the formalism explained in the text and parameters given in eq. (4.6).

reliable than that of ref. *‘), where such terms have been ignored, or that of ref. 29), where they have been evaluated approximately by assuming a zero-range KN interaction. Our value of a?,~,~ is also in accordance with those reported in table 3 whereas that of a& is compatible with that of ref. 31), but not with those of refs. 30*22).Nevertheless, the methods used in refs. 22230-34)make use of a theoretical approach quite different from ours, and therefore, it is difficult to draw any conclusion from a comparison of the corresponding values of a&/2 LIZ?,~,~and ~~:,3/2. TABLE

2

Elastic partial wave amplitudes, corresponding to total angular momenta A = 0, 1,2, and 3, evaluated by means of eqs. (2.5) (see the text) in the cases of (a) 342 MeV/c and (b) 470 MeV/c incident Kf laboratory momentum and by either retaining only single scattering terms (SS), or considering also double scattering terms (SS + DS), or including triple scattering terms (SS + DS +TS)

(a) SS SS+DS SS+DS+TS FAD (b)

SS SS+DS SS+DS+TS FAD

A=0

A=1

A=2

A=3

1.677 - i0.479 1.495 - i0.597 1.438 + i0.898

0.422 - i0.126 0.422 - iO.114 0.422-i0.117

0.119 - iO.037 0.120-iO.037 0.120-iO.037

0.038 - iO.012 0.038 - iO.012 0.038-iO.012

1.426 - i0.779

0.422 - i0.117

0.120 - iO.037

0.038 - iO.012

1.095 - i0.452 0.962 - i0.549 0.899 - i0.756

0.369-i0.156 0.367 - i0.143 0.366 - i0.147

0.131- iO.056 0.132 - iO.056 0.132 - iO.056

0.051- iO.022 0.051- iO.022 0.05 1 - iO.022

0.889 - i0.668

0.367-i0.148

0.132-iO.056

0.051-

The Faddeev amplitudes (FAD), equivalent to the complete The entries in the table are in units 2rr210-s MeV’.

multiple

scattering

iO.022

series, are also shown.

J. San’udo / K+d elastic scattering TABLE Previous

determinations

ah,2 (fd

-0.035

of ai,l,2, ay,l,z and aT,3,2

25

1

29)

-0.11::::,” -0.005

3

Ref.

0.04*0.04

0.123 f 0.007

-0.046

i 0.002

0.035 f 0.013 -0.019

0.125rtO.04 0.086

30) 31) 22) 32) 33)

-0.23*0.18 0.02

34)

-0.17

471

Method impulse-approximation analysis of K’d data K+d multiple scattering with zero-range boundary-condition formalism KN phase-shift analysis with dispersion relation constraints KN partial wave dispersion relations KN phase-shift analysis with parametrized inverse amplitudes KN forward dispersion relations KN multichannel analysis with dispersion relation constraints KN effective lagrangian

25 i

Fig. 2. Differential cross section, in the laboratory frame, for K*d elastic scattering at 470 MeV/c incident K+ momentum. The experimental data are from Glasser et al. 26). The theoretical curve has been obtained with the formalism explained in the text and with those parameters giving the optimum fit at 342 MeV/c (see fig. 1).

472

J. Saiiudo / K’d

elastic scattering

Besides the experimental data of Glasser et al. 26) at 342 MeV/c, we could have used in our fit also their results at 470 MeV/c. The same minimization procedure gives, in this case, values for the T = 0 KN scattering lengths rather different from, although compatible with, those of eq. (4.6). Furthermore, the inclusion of P-waves in the T = 0 KN channel does not imply a significant improvement of the x2 value with respect to that obtained when only S-wave is considered, in spite of the fact that two new adjustable parameters are available. Such improvement, instead, is notorious when only the 342 MeV/c data are considered. For this reason we find more reliable the values of the parameters given in eq. (4.6). The differential cross section evaluated with these parameters is compared with the experimental data at 470 MeV/c in fig. 2. The discrepancies could be interpreted as due to an unexpected abnormal importance of the effective range in someone of the waves or (difficult to believe) to an error in some of the data reported by Glasser er al. The effect on the calculated differential cross section of the uncertainty in the input, namely the errors in the current value ‘) of the T = 1 S-wave scattering length, u& = -0.309*0.002 fm, is shown in fig. 3. Clearly, the influence on the cross section of such uncertainty is negligible. The large errors in the scattering lengths, eq. (4.6), determined in this work are entirely due to the K+d scattering data. As explained in sect. 4, the secondary terms in the K’N interaction have not

Fig. 3. Differential cross section, in the laboratory frame, for K+d elastic scattering in the cases (a) 342 MeV/c and (b) 470 MeV/c incident K.’ momentum. The experimental data are the same as in figs. 1 and 2, respectively. The theoretical curves have been obtained in the same manner as those in figs. 1 and 2, the only difference being in the input value of the T = 1 S-wave scattering length, which has been taken

equal to (I) -0.311

fm and (II) -0.307

fm.

J. S&do

/ K”d

elastic scattering

473

(b)

(al

-1

-05

0 cos 6

05

1

-1

-05

0 cos0

05

Fig. 4. Differential cross section, in the laboratory frame, for K’d elastic scattering in the cases (a) 342 MeV/c and (b) 470 MeV/c incident K’ momentum. The experimental data are the same as in figs. 1 and 2. Curves III and IV represent the cross section calculated respectively ignoring and including the secondary terms in the K’N interaction treated in the WA (see the text).

(a)

(b)

COSB Fig. 5. Differential cross section, in the laboratory frame, for (a) 342 MeV/c and (b) 470 MeV/c incident K” momentum. The figs. 1 and 2. The values of the T = 0 scattering lengths used to 0.08 fm3 and a‘& = 0.008 fm3 in both cases, whereas a0o,1,2= 0.022 fm for curve VI.

K’d elastic scattering in the cases experimental data are the same as in obtain curves V and VI are r~:,~,:! = -0.018 fm for curve V and a& =

J.S&udo/ K'd elastic scattering

(b)

(a)

15

I

Fig. 6. Differential cross section, in the laboratory frame, for K’d elastic scattering in the cases (a) 342 MeV/c and (b) 470 MeV/c incident Ki momentum. The experimental data are the same as in figs. 1 and 2. The values of the T = 0 scattering lengths used to obtain curves VII and VIII are 0 ao.l/z = 0.002fm and a&l = 0.008fm3 in both cases, whereas ay,r,a = -0.02 fm’ for curve VII and a$,a = 0.18 fm3 for curve VIII.

(al

Fig. 7. Differential cross section, (a) 342 MeV/c and jb) 470 MeV/c figs. 1 and 2. The values of the T 0.002 fm and a& = 0.08 fm3 in

(b)

in the taboratory frame, for Kfd elastic scattering in the cases incident Kf momentum. The experimental data are the same as in = 0 scattering lengths used to obtain curves IX and X are a& = both cases, whereas ay,a,a = -0.042 fm3 for curve IX and a&,lz = 0.058 fm3 for curve X.

J. Saiiudo / K’d

elastic scattering

475

been included in the Faddeev treatment, but their effect has been taken into account via the SSIA. In fig. 4 we have represented the elastic K+d differential cross section with and without that SSIA correction. It can be seen that such correction is not negligible at all. However, it turns out to be small enough to justify having used SSIA in its evaluation. In order to show how sensitive the differential cross section is to the variation of the extracted T = 0 scattering lengths, we have calculated it by taking the best-fit values for two of those scattering lengths and allowing the third one to take its maximum and minimum values, as given in eq. (4.6). The results are represented in figs. 5 to 7. To conclude, it seems important to stress that the present experimental knowledge of the low-energy kaon-deuteron interaction is very unsatisfactory. More precise data at lower energies are necessary. I wish to thank A. Cruz and J. Sesma for a critical reading of the manuscript. I am also grateful to the referee for suggestions contributing to improve the presentation of this work. This paper has been supported in part by the Comision Asesora de Investigation Cientifica y Tecnica. Financial help of the Instituto de Estudios Nucleares is also acknowledged.

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