Three-particle associated production in pion-nucleon collisions

~.B

]

Nuclear Physics 37 (1962) 438--448; ~ ) North-Holland Publishing Co., Amsterdam :Not to be reproduced by photoprint or microfilm w ith o u t w r i t t e n permission from the publisher

THREE-PARTICLE ASSOCIATED P R O D U C T I O N IN P I O N - N U C L E O N C O L L I S I O N S K. R A M A N t

The Institute of Mathematical Sciences, Madras, India Received 27 March 1962 Abstract: The production of Y K ~ a n d N K K in pion-nucleon collisions is studied in the single-particle exchange model. The total cross-sections for the reactions and the angular distributions and energy spectra of the recoil hyperons are calculated. The reactions are compared with each other and with Nnn production. There is a significant difference between the _Y'Kz~and the A K ~ reactions if the Z'A parity is odd.

1. Introduction

The single-particle exchange or peripheral interaction mechanism has been found to be useful in explaining quasi-elastic pion-nucleon and nucleon-nucleon scattering 1'6) and three-particle associated production in nucleon-nucleon collisions 2). In this paper we study the contribution of such a mechanism to the reactions 7c+N ~ Y + K + T r ~+N ~ N+K+K as in the diagrams shown below, where N stands for a nucleon and Y for a Z or A hyperon. _~

.fJ-

J

f

KLI

1

E

I

J

Fig. 1. The reactions ~z+N-+Y+K-+-~ and ~ ÷ N - + N + K ÷ K .

The distribution of the transverse momentum of the recoil A ° in the reaction n + N ~ A ° + K + n has been studied by Blokhintsev et al 3). For the same reaction, G. Costa and L. Tenaglia 4) found in their study of K K n n couplings that couplings of the two forms assumed by them gave a total cross-section that was negligible t A preliminary report of this work was presented at the Annual Symposium on Cosmic Rays, Madras, December 1961. 438

THREE-PARTICLE ASSOCIATED PRODUCTION

~39

compared to that estimated from experiment (which was of the order of magnitude of 0.5 rob); they concluded that the peripheral interaction did not make an important contribution to the reaction considered. However, KKzcrc couplings of different forms can lead to total cross-sections of different orders of magnitude. And experimentally, as noted by S. Barshay 5), the marked backward peaking of the Z hyperons produced in re--nucleus collisions at very high energies suggest that these hyperons are produced by a stripping mechanism. We here assume that the K s --* KTr and ~ ~ K K couplings at the upper vertex of the single-meson exchange diagrams (see fig. 1) are dominated by vector resonances in the Kzc and nn systems. Calculations are restricted to collisions in which A z, the square of the four-momentum of the exchanged particle, has a magnitude less than 10p 2 (where # is the pion mass), as the model can be expected to be valid only for small values of IA2[. The cross-sections thus obtained are considerably larger than those obtained in ref. 4), although they are still small compared to the value of 0.5 mb estimated front experiment. But it may be noted that the cross-sections calculated here refer only to a part of the total number of collisions, viz., those in which the laboratory kinetic energy of the recoil baryon has a given upper limit (fixed by the condition A 2 > -10#2). If such collisions are not a negligible fraction of the total number of collisions, it is likely that the mechanism considered makes a significant contribution. In sect. 2 the limiting curves for the kinematical variables chosen are given. In sect. 3, the total cross-sections for the reactions with z--mesons on hydrogen are given and compared with Nnn production. A crude impulse approximation estimate of the production of YK~ and N K K on deuterium has been made; the relative production of charged and neutral hyperons on hydrogen and deuterium is sensitive to the isobaric spin of the Kzc resonance. In sect. 4 the angular distribution and energy spectrum of the recoil hyperon are considered assuming odd Z A parity. This angular distribution and the total cross-section are sensitive to the S A parity. 2. Kinematics The notation used is as follows: the quantities Pl, Pf, qx etc. are four-momenta; the metric used is such that ( A B ) = A o B o - A • B.

%-toolA

Pj

:,M i

I ~.

~ q3

Mf

Fig. 2. Kinematics o f the reactions.

Pf

440

K. ~MAN

We define the variables S = ( p i + q l ) 2,

S/', = (q2+q3) 2,

A2 = ( p i - p f ) 2.

(1)

The centre-of-mass frame of the initial pion and nucleon is denoted 6) by a suffix U attached to the momenta, as in Pit;; in this frame, the quantity S is the total energy of the initial pion and nucleon. The centre-of-mass frame of particles 2 and 3 is denoted by a suffix V; the quantity Sex is the total energy of particles 2 and 3 in this frame. The laboratory frame in which the initial nucleon is at rest, is denoted by a suffix L. The quantity A2 is the square of the four-momentum of the exchanged particle. We use the symbols p, rn, MN and My to denote the masses of the ~r-meson, K-meson, nucleon and hyperon, respectively. The limiting values of the different variables may be obtained by using the expression for A2:

/I 2 = Mf +Mp--(S+Mf -/+2)(S+Mp - ~I)/2S

r<++

+2L

~

_++]+.rL(+

4#

_M+]+co+o+.

Here, Ofu is the angle between the momentum vectors Piu and Pfu, i.e. the angle o f the recoil baryon measured from the backward direction in the over-all centre-of-mass frame. The threshold value of S is obtained from the condition IPfu[ = 0. For given S, the quantity Sex is limited by the inequality (rn2+rns) 2 < Sex <(~/S-M~), while the upper and lower limits for A2 are obtained by taking for Sex its minimum value (m2 + m3) 2 and putting cos 0m = + 1 and - 1, respectively. At very high incident energies, i.e. large S, the quantity A2 approaches its maximum possible value (Mi - M r ) z. For given S, the condition A2 > - 1 0 # 2 fixes an upper limit Se~ax on Sei. The i0 6

,..,

0$

,., po3 ....f / "/> ' ~ ~

['J ,o~ ,64

-

,d+

+6$

,6$

, 6 ++

,(~6

,@' "

TrrClab)

Fig. 3. Here, TTris the laboratory kinetic energy of the incident pion, given in units of the pion mass/t.

THREE-PARTICLE ASSOCIATED PRODUCTION

441

variation of ~,~ax with S is shown in fig. 3, for the reaction 7z+N -~ A + K + ~ , S _ MO . compared with that of the limiting value ~-I =

and

o

!2--°-~o---o-.._~_v..~ go

m&x

6fu 6*

Io 5 MeV 3oof-

o

'

o

~o ;i/~? ,;o

....

2.0

Fig. 4. Here, 0~a~x is the angle o f the backward cone in which the A ° can emerge in the c.m. frame when A z ~ _ 10/z 2. The two curves are for incident pion kinetic energies o f 10 4 and 10~ MeV (lab.). 0 o

.~

'?o

,5.°

oi ?° ~ -si//z2

rs

7TO

i

-I 5

t-

Fig. 5. The curves here are for S = 1022/t 2, corresponding to an incident pion kinetic energy o f 104 MeV (lab.).

F o r given S and ~ P I , the condition A 2 > -- 10/z 2 fixes an upper limit 0f%ax on the angle 0re; the variation of 0~Cx with ~ I is shown in fig. 4 for 2 values of S, corresponding to incident pion energies of 10 4 and 10 5 MeV (lab.). For given S and 5al, the quantity A 2 is limited from above by the condition cos 0fu = + 1; this limit becomes more and more negative forincreasing b°i . Emergence

442

K. RAMAN

at larger angles 0rtj (for given 6ai) is possible only for more and more negative values of ~A2. These features are seen in fig. 5, where A2 is given as a function of 6ai for different values of 0fu, for fixed S. 3.

Cross-Sections

Summing over spins and equating to unity the unknown form factors at the vertices and in the propagator gives the following result 6). (Final state interactions between particles emerging from different vertices are neglected; the KNA vertex has been assumed to be pseudoscalar and the KN2; vertex scalar).

t~2ff/t~A 2OoqPl

=

M iMf [Vuia2 1 2(2z)~qLM 2 ('12_ mo2)2 qv ~ i ai(~e~, '12),

= G~" [8(2702q2LM2]-lq~(A2)qv~a,(Se,, A2),

(3)

where VII and GH are the vertex function and coupling constant, respectively, for vertex II, qv is the final momentum of particles 2 and 3 (emerging from vertex I)in their c.m.s., and O(A 2) = ~a(A 2) = (MN-Ma) 2-A2 (A2_rn2)2 ' =

2) =

for n + N ~

(MN+M,)2-A2

('1~_m2) 2 _'12 = ~N('12) (,t 2-/z2) 2'

A+K+n,

for n + N ~ Z + K + n ,

'

(4)

for n + N ~ N + K + K .

The variation of these with A2 is shown in fig. 6.

-2

o

4

a

Fig. 6. F o r convenience in the choice o f scale, the functions plotted are

~z(A 2)× 0.1.

12

-- Li~/~2

~A(A ~) ×

10, ~N(A 2) and

THREE-PARTICLE ASSOCIATED PRODUCTION

443

For the K N Z vertex, Sakurai's estimate 7) of 1/4re G~2~ ~ 0.6 is used; for KNA, the value 1/47: Gr~a ~ 10, estimated in a previous calculation 8) on K + N scattering (assuming odd ZA parity) is taken. 1/&r G2N~ is taken to be 15. In eq. (3), ai(S~i, A 2) is the off-shell scattering cross-section at the vertex I. For r~rc scattering, Bowcock, Cottingham and Lurie 9) have adopted a P-wave resonant scattering amplitude of the form T - 8~x/~9° 3 cos 0 YkS k S ~ - W~ - i~k 3 '

(5)

where T is related to the differential cross-section by ITI2 = (8zx/6P)2 Oa/df2, and 9~ may be related to the energy Wr and width Fr o f the bipion resonance and the z.n_ bipion coupling constant G by lo) 2Wr G2 Y = S T Fr " k~ 6zWr

(6)

In eq. (5), k is the centre-of-mass m o m e n t u m and 6 a is the c.m. energy. A scattering amplitude of similar form is adopted here for the Kzc scattering at vertex I for n + N ~ Y + K + T c , and for the reaction zc+rc ~ K + / ~ for r c + N ~ N + K + K . , the resultant factor k 2 in (5) being replaced by kvqv, where kv, qv are the initial and final c.m. momenta, respectively, for the off-shell scattering at vertex I. The energies o f the bipion and Kn resonance are taken as 765 MeV and 885 MeV, respectively, and the half-widths as 80 MeV and 16 MeV, respectively; GpK~,is taken lo) to be 3.9. These give ? = 0.175 # - 1 for K + r c --, K+~z, = 0.235 # - 1 for n+rc --> K + K , = 0.326/~-1 for ~ + n ~ rc+~. Fig. 7 shows qvda~ ai(oo'i, A 2) at a function of ~9°i, A2 being replaced in it by an average value. Total cross-sections for collisions with A 2 >__ - 1 0 # 2 are obtained by integrating d2a/dA20~g°I graphically, with respect to A 2 between the lower limit A 2 = - 1 0 # 2 and the upper limit obtained from eq. (2) (as indicated in sect. 2), and with respect to ._9°1between (rn + / 0 2 and the value necessary to give Amin 2 = - 10# 2. Taking into account isobaric spin, total cross-sections for ~ - incident on hydrogen are given in table 1 and are compared with N~n production calculated in the same approximation. Thus, in the range of A 2 considered, the ratio O'tot(Z~)/o'tot(A ) is seen to be 2 to 4 when the ZA parity is assumed to be odd. For even ZA parity we can expect that 2

These results can only be rough estimates mainly because of the uncertainty in the coupling constants.

444

K. RAMAN

The effect o f the I A parity would be accentuated if events with m u c h smaller A 2 only were compared; for instance, taking A 2 > 0 only, trtot(I)/trtot(A ) at 107 to 108 MeV is f o u n d to be 8 to 10. But such events would be only a small fraction o f the total n u m b e r o f collisions.

400 t"

~J rr-

300

Kt-'IT

I:: + xt

~- K + Tr

2o,

L~

I::: -i- I o

_1oo

n + 1T---~ K + K ~.o-c

~' ,a

o-

,~o

so

~o

s-

~,/~

o ~o

Fig. 7. Here, gvSax½a~(Sax,A2) is plotted as a function of Se~ .The natural units used are i~ = c = # = l. TABLE 1 Total cross-sections and ratios Incident pion lab. kinetic energy a(~°K+~-) (MeV) (#b)

Total cross-sections for all aUowed d z, for z~- on hydrogen

Total cross-sections and ratios for A~ --> -- 10#2, for n - on hydrogen a) a(A°K+z~-) a(Z'°K+~t -)

104

29.8

0.4

105 106 107

10.0 18.1 19.4

0.34 0.23 0.23

a(pK°K-) (#b)

a(NKKI) a(YKzt)

a(NKR~) a(l°K+er-) a(A°K+~-) a(Nz~) (mb) a(~'°K+zr -)

124

4.0

0.27

0.51

1.7

82 77 70

8.1 4.6 3.9

0.46 0.52 0.52

0.75 0.99 1.15

2.8 3.6 4.5

a(AOK0z1-0) = ½a(AOK+z¢-);

a(nK0 K °) = a(nK+K -) = ½a(pK0K -)

a) a(NK~), aCYK~) and a(Nnz0 include all charge-states, superposed incoherently; their ratios are the same on deuterium as on hydrogen. All cross-sections here have been computed assuming an isobaric spin of ½ for the Kzl resonance. The c o n d i t i o n A2 => - - 1 0 # 2 implies that the total cross-sections refer to collisions i n which the final A, I a n d n u c l e o n have l a b o r a t o r y kinetic energies less t h a n 120.5 MeV, 138.2 MeV a n d 103.9 MeV, respectively; they m a y be expected to be appreciably smaller t h a n the total 3-particle p r o d u c t i o n cross-sections observed. F o r comparison, the c o n t r i b u t i o n to o'tot(Y ) f r o m the diagrams considered for the whole allowed

THREE-PARTICLE

ASSOCIATED

PRODUCTION

445

range o f A z and SPI is shown for n + N --, Y + K + n in the last two columns of table 1. A crude impulse approximation estimate gives the following features of the total cross-sections for production on deuterium. The isobaric spin o f the K n resonance determines the relative production of charged and neutral hyperons on hydrogen and deuterium in this model. An isobaric spin o f ½ leads to A ° and ~o production by n - on deuterium of the same order as on free protons, ~ - production (suppressed on hydrogen) is enhanced to the same order of magnitude as that of 2~°, while Z + production is still suppressed. An isobaric spin of 3 would mean much greater production of neutral hyperons on deuterium than on hydrogen; X+ production would be o f the same order on hydrogen and deuterium and comparable to that of Z ° on hydrogen; X - production, though appreciable on deuterium (while being suppressed on hydrogen), would be only a b o u t ½ o f ~+ production. The K K pair production by n - incident on deuterium is expected to be about 1.5 times as large as that on hydrogen.

4. Angular Distribution and Energy Spectrum The energy spectrum and angular distribution of the recoil hyperon have been calculated at different incident energies. In the laboratory frame, the hyperon energy spectrum measures the variation of

t ~o- x 103(in nat. units)

-4

) )

•

~3

.......

43-------¸

i~4

O .

.

.

.

.

.

" J

ig

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

a._o

i7

£Iu (in units of if) Fig. 8. The abscissa E~v is the total c.m.s, energy of the hyperon; the peak corresponds to a kinetic energy of 8.37/z. the cross-section with A2; the A ° laboratory energy spectrum has a peak at a recoil kinetic energy of 113.4 MeV, while the I; spectrum~rises at zero kinetic energy. (The

446

K. RAMAN

recoil n u c l e o n s p e c t r u m for ~ r + N ~ N + K + K a n d 7 r + N ~ N + ~ r + ~ z have p e a k s at a kinetic energy o f 10.4 MeV). The energy spectrum o f the recoil b a r y o n in the overall centre-of-mass f r a m e measures essentially the v a r i a t i o n o f the cross-section with ~ i a n d is p e a k e d at a n energy d e t e r m i n e d b y the K n (or 7rTr) r e s o n a n c e in t h e KTz(or Irn) scattering at the TABLE 2

~iaifts in peak position with varying incident energy Incident pion lab. kinetic energy TTrL (MeV) 2 × 108 104 105 106

Peak in recoil A 0 c.m. energy spectrum

Try(A) 0.69/~ 8.37 # 41.54/~ 154.6 #

u p p e r vertex. The c.m.s, energy s p e c t r u m atrlt?Efu o f the A ° in n + N ~ A ° + K + n f o r an incident p i o n kinetic energy o f 104 M e V (lab.) is shown in fig. 8. The shift in the p o s i t i o n o f the p e a k (in terms o f the A ° c.m. kinetic energy Tf~)

Fig. 9. The centre-of-mass angular distribution function of the A ° at 2000 MeV and 104 MeV incident pion kinetic energy (lab.). with varying incident energy is shown in table 2. F o r the Z energy spectrum, the p e a k is at a slightly different energy for the same value o f S.

THREE-PARTICLE ASSOCIATED PRODUCTION

447

The angular distribution of the hyperon in the centre-of-mass frame has been calculated. We have the relation

02a/OEfu~(cos 0fu) = [2(2z02qm MN]-I G~,~y kge(cos 0fu, Sa,) " F(Efu, A 2), where F(Efu, A2) is qv 5¢ik(Efu--Ma) 2 2 • aK~(Sal, A 2) with SaI expressed in terms of Efu by the relation Efu = (S+M~-Sal)/2S ~, and ~y(cos 0fu, 5al) is obtained by replacing Az in ~y(A 2) by its expression (2) in terms of cos 0fu. The function 7ty varies slowly with 5al and F(Efu, A 2) slowly with A2; hence in evaluating them, 2T1 in 7ty (cos 0fu, S°l) and A2 in F(Efu, A 2) are replaced by average values for each incident energy. "~ --..3 0 °

\\

°°

~

....

LJ

1"20 °

Fig. 10. The centre-of-mass angular distribution function of the Z' at 2000 MeV and 104 MeV. incident pinn kinetic energy (lab.). The centre of-mass angular distribution functions 7Jy(cos 0fu) have been given on polar diagrams in fig. 9 and fig. 10, respectively. The distributions are peaked backwards, more so for the Z than for the A and more sharply at higher energies. If the EA parity had been taken as even, the angular distributions for E and A would have been alike. The considerations of this paper are being extended to 3-particle associated production by photons on nucleons and to 4-particle associated production. Other types of contributions to such inelastic collisions are also being investigated. The author is grateful to Professor Alladi Ramakrishnan for constant encouragement and to Mr. A. P. Balachandran and Mr. K. Venkatesan for useful discussions. A fellowship from the University o f Madras is gratefully acknowledged.

Note added in proof." Recent evidence indicates that the Kn resonance at 885 MeV is a broad one (and therefore cannot be assumed to be a vector resonance without further evidence about its spin); there is also a narrow Kn resonance (presumably vector at about 730 MeV (ref. 11)). The latter resonance would lead to essen-

448

K. l~tAN

tially the same features for the reactions considered here. The consequences of the superposition o f the effects o f the two resonances i n these a n d other reactions are being studied.

ReferenCes 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)

G. F. Chew and F. E. Low, Phys. Rev. 113 (1959) 1640 E. Ferrari, Phys. Rev. 120 (1960) 988 D. I. Blokhintsev and Wang Yung, Nuclear Physics 22 (1961) 410 G. Costa and L. Tenaglia, Nuovo Cim. 18 (1960) 368 S. Barshay, Nuovo Cim. 21 (1961) 671 F. Salzman and G. Salzman, Phys. Rev. 120 (1960) 599 J. J. Sakurai, Nuovo Cim. 20 (1961) 1212 A. Ramakrishnan, A. P. Balachandran and K. Raman, Nuovo Cim., in press J. Bowcock, W. N. Cottingham and D. Lurie, Nuovo Cim. 16 (1960) 918 Ph. $alin, preprint G. Alexander, G. R. Kalbfleisch, D. H. Miller and G. A. Smith, UCRL-10195