K+-meson—nucleon scattering and K+-Y relative parity

Nuclear Physics Not

12 (1959) 521-526;

to be reproduced

by

K+-MESON-NUCLEON

photoprint

@

North-Holland

or microfilm

without

SCATTERING

mitten

AND

Publishing permission

K+-Y

Co., Amsterdam from

the

publisher

RELATIVE

PARITY L. K. PANDIT Tata Institute

and S. N. BISWAS

of Fkkndamental Received

2 April

Research,

Bombay

1959

Abstract: A covariant integral equation in the ladder approximation has been set up for the Feynman amplitude describing the K+-nucleon scattering through a strong direct K-Y-N interaction. The two possibilities of the relative K-Y parities have been considered, it being assumed that both K” and K+ have identical parity. The equation has been solved by the Fredholm method employing the prescription of McCarthy and Green to obtain a divergence-free solution. Our calculations favour even relative K-Y parity. For best fit with (K+-p) scattering experiments, the coupling constants required are GNK,I = ~/~GNIQ_‘, G’NK&Z = 0.5.

1.

Introduction

In recent times the moderately strong interactions between the strange particles have become an important field of research. One of the most direct ways of investigating these is to study the scattering of the particles concerned. With this in view, we are interested in the present work in calculating covariantly the K+-nucleon scattering through a direct K-Y-N interaction. Calculations l) using non-relativistic models have been performed by Stapp, Vitale and Amati, and Ceolin and Taffara. In formulating the interaction 2), conservation of the total isobaric spin T and the strangeness number S is taken into account, the T and S assignments being according to the Gell-Mann-Nishijima scheme. One also needs, of course, the knowledge of the spins and parities of the particles. Regarding these, one could make assumptions which may then be directly confronted with the experiments. We shall assume that the spins of Y (LI, Z) are $ and the spin of K is 0. We shall further consider both possibilities of relative K+-Y parity, it being assumed that n and Z have the same parity relative to the nucleons. If one uses the convention that the parity of the hyperons is the same as that of the nucleons, our parity nomenclature (i.e. relative K+-Y parity even or odd) refers directly to the parity of the K+-meson. Our calculation is, of course, quite general, since, as seen below, only the relative parity of K-Y enters the discussion on account of “associated production”. 521

522

L. K.

PANDIT

AND

2. Interaction

S.

N.

BISWAS

Hamiltonian

In the isobaric spin notation, the interaction between the hyperons, nucleons and K-mesons can be written as H, = GNI
Diagrams

In the Nishijima-Gell-Mann scheme the K+*O-mesonhas isobaric spin $ with strangeness +l and the nucleon has isobaric spin i with strangeness zero. Therefore, there are two isobaric spin states, namely the triplet (T = 1) and the singlet (T = 0), of the K+-meson-nucleon system. If a* denotes the amplitude for the scattering of K+-mesons by nucleons in the isobaric spin state T, the K++p + K+.+p scattering amplitude is a l; while the K++n + -+ K++n amplitude is $(ul+ao) and the charge-exchange K++n --f K”+p amplitude is +(Lz~-~0). The lowest-order Feynman diagram in accordance with the above interaction is drawn in fig. 1. In view of strangeness-conservation, only the

Fig.

1

diagram with crossed K-meson lines can occur. Besides this diagram, we include a whole series of higher order diagrams of the type shown in fig. 2. l. \.

,x_ --

__---__

,-..: -

I’

-----.__,.

-

:=.

Fig. 2

This means that we shall be working in the “ladder-approximation.” The ladder approximation scattering amplitude in the total isobaric spin state T(= 1, 0) is given by M,

= C,fi,@,(Z)u,

where @r(Z) satisfies the integral equation @Q(Z) = F,(Z)+&. s K(Z, Z’)@,(Z’)d4Z’.

(2)

K+-MESON-NUCLEON

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523

Here zc2and 1crare the Dirac-spinors describing the final and initial nucleon, and we have (&=c= 1)t

’

CT =-

Gac

Fw

m

=22/E,

64h-Pf)

E,o,o,

(34

where pi, pf denote the initial and final four-momenta of the system, E,, E, being the initial and final energy of the nucleon, ol, oa those of the K-meson; F(Z) = F{y. (p--k-z)-iq-lr, K&Z’) = T{y * (p-Z’-Z)-M)-‘I+

iz, = -

. (P-Z’)--WZ}-~(Z’~-,U~)-~,

(3b)

k4G2cT.

Here we have taken GNU = aG,,,

= aG,

in which case “=

l+a2

for

T = 1,

13-a2

for

T = 0.

As stated earlier we shall put I’ = 1 or ys according as the K+-Y relative parity is even or odd. M is the mass of d, .X (neglecting their mass difference), m the nucleon mass and ,V the K-meson mass. The integral equation (2) is of the Fredholm type and hence the solution may be obtained by means of the Fredholm’s resolvent kernel. A modification of this solution, developed by Green and McCarthy s), will be used here. 4. Fredholm

Solution

Writing equation (2) in the operator form @ = F+LK@ the Fredholm solution is F+J(K-trK)F+ @ = l-1

...

trK+$12{(trK)2-trK2}

. . .’

(4

where tr K = trK2 =

I

K,(t, t)d4t, K,,(t, Z)K,(Z, t)d4Zd4t,

(5)

KF = j-K,,(Z, t)F,(t)d4t. Actually, tr K diverges logarithmically. This divergence is eliminated here by using the prescription of McCarthy and Green S), which results in the t The notations used here for Dirac operators, the space-time metric etc. are the same as used in Schweber, Bethe and de Hoffmann, Mesons and Fields, Vol. 1 (New York, 1966).

L. X. PANDIT

524

replacement

of tr

AND

S. N. BISWAS

K, tr K2 etc. by tr’ K, tr’ KS,where tr’ K, for example, is

obtained by substracting from tr K the value it assumes when fi = 0 (fi = = resultant 4-vector energy-momentum). The solution of (2), in the first Fredholm approximation, is then given by @ = F/(1-Au’,) where zl’r = tr’ K, %

=

tr

K =

I

tr[r{r

- (p-21’)

-jWj-rr{y

. (p-4’)

-m}-l]

(Z’2-p2)-1d4Z’.

5. Cross Sections Using (6), the scattering cross section is obtained in the usual manner. The differential cross section in the centre of mass system is given by (g2 = G2/4n) do,+ -= dL’

i(g2cA2

and the total cross section

1 1 A*+B* cos 8 ~ W2 [a+b cos e]2 (l+gscr

6’)”

’

(7)

by

The + sign is for relative K+-Y parity even and the - sign for the relative parity odd. The various quantities occurring in (7) and (8) are listed in the appendix. In the above cross section formulae, if we drop the last factor (containing t*) we get the result one obtains from the first Born-approximation. the scattering amplitudes for the If we take cc = l(G,,,, = G,,,), T = 1 and T = 0 states become equal (cr = c,, = 2), and hence we get no charge-exchange scattering. Experiments give a value of 0.2 to 0.3 for the ratio of charge-exchange to no-charge-exchange scattering *) . We have therefore taken the value a = 43, (G,,, = 1/3G,,), so that cr = 4 and c0 = 0, and hence the scattering takes place only through the T = 1 state. This leads to a value of 0.2 for the ratio of charge-exchange to no-chargeexchange scattering. In fig. 3 the total cross section for the scattering of K+-mesons from protons is plotted as a function of the kinetic energy E of the incident K+-mesons in the laboratory system. The value g2 = 0.5 has been used along with u = 1/3. It is seen that the Fredholm curve with relative K+-Y parity even gives the experimentally found increase with energy, and also the magnitude of the cross section in the region E = 50 to 300 MeV. Thus, from this calculation it appears that, within the framework of the interaction scheme used, relative K+-Y parity even is preferred.

K+-MESON-NUCLEON

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525

Regarding the angular distribution, both experiments and our calculations at 50 MeV give a more or less isotropic distribution in the centre of mass system. The angular dependence alone does not provide any information on the relative parity of K+ and Y. Experiments “) seem also to point to a repulsive potential between K+ and proton. From the cross sections alone, one cannot make any statement regarding the nature of the potential. A separate investigation, using, for example, dispersion relations, is needed to determine unambiguously the sign of the potential. We hope to present the results of such an investigation in the near future.

QO80 -

70-

60

t

250

z

b

i

d-j- BORN 0

50

100

150

200

300

Fig. 3. Total cross section IJ for (K+-p) scattering plotted as a function of E, the kinetic energy of the incident K+-meson in the laboratory system. The + or - signs refer to relative K+-Y parity even or odd, respectively.

Recently, Pais “) has put forward the conjecture that K+ and KO might have opposite parities, making it possible to formulate a direct (K+ROn’) interaction, which may contribute substantially to the K+-N scattering. Work is now being done also to take this possibility into account. We should like to express our thanks to Dr. Yash Pal and Dr. M. S. Swamy for useful discussions on various experimental points.

I..

526

K.

PANDIT

AND

S.

N.

BISWAS

Appendix W2 = (,~+m)~+2mE,

d* = m&M.

A* = -~2(W2-W2-2dfrn--m2)+20W(oW--o2-22d*:m) +2Ll*mw2+ (Ll’)“{(W-t0)2+m2}, B* = K2[-~2+2wW-2d*‘m-

(4*)2].

(m2-p2)2

-2(m2+#4

w2

[4+9,+9,-(2m;2M)2 (.&+9-J)

t* = ; a =

)

b = -2k2.

$+m2-M2-2&k2+m2, 1

Y1 =

1.

s 0

dv log

v2-v(l+a-p)+a v(B-a)+a

1

dv log

92=; s 0

'

v2-v(l--y+4a)+4a ’

v(y-4a)+4a

1 9,=

X dx

s 0

Idz s 0

1 Y4=

'

X da?

s 0

a(l-X)+/?X2+~yX(l-.Z)-~X[l+3Z-X(l+Z)2]

Id2 I 0

a(l-x)+/3x2+&(1--.z)

’

where

In the computations given here a has been neglected in 9’3 and Y4. References 1) H. Stapp, Phys. Rev. 106 (1957) 136; D. Amati and B. Vitale, Nuovo Cimento 5 (1967) 1533; C. Ceolin and L. Taffara, Nuovo Cimento 6 (1967) 425 S. Barshay, Phys. Rev. 110 (1958)743 2) M. Gell-Mann, Phys. Rev. 106 (1957) 1297 J. Schwinger, Ann. Phys. 2 (1957) 407 A. Pais, Phys. Rev. 110 (1958)574 3) I. E. McCarthy and H. S. Green, Proc. Phys. Sot. A 67 (1954) 719 4) M. F. Kaplon, Cem Conference Report (Geneva, 1968) 6) A. Pais, Phys. Rev. 112 (1958) 624