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Directional correlation of mesons in a cascade decay of baryons resulting from a baryon-baryon reaction
Nztclear Physics 21 (1960) 237--244; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
D I R E C T I O N A L C O R R E L A T I O N OF M E S O N S IN A C A S C A D E D E C A Y OF B A R Y O N S R E S U L T I N G F R O M A B A R Y O N - B A R Y O N REACTION A. D E L O F F and J. W R Z E C I O N K O
Institute /or Nuclear Research, Warsaw Received 7 J u l y 1960 This is a c o n t i n u a t i o n of a preceding w o r k b y the s a m e a u t h o r s 1). We obtained general expressions of the m o m e n t u m directional correlation f u n c t i o n of m e s o n s resulting from the decay of b a r y o n s being p r o d u c t s of a b a r y o n - b a r y o n reaction. As a particular case we examined a reaction in which b o t h particles before reaction are unpolarized. Calculations were done a s s u m i n g t h a t a) particles before the reaction a r e i n the S state only (low energy zero threshold collision), or b) final s t a t e b a r y o n s are in the S-state only (high threshold reaction). Several e x p e r i m e n t a l possibilities for a d e t e r m i n a t i o n of the relative intrinsic p a r i t y of b a r y o n s are discussed.
Abstract:
1. I n t r o d u c t i o n In a preceding paper 1) which will be hereafter referred to as (I) we considered reactions of the type a + b - + c + d , where all particles involved are ½ spin baryons. The relative intrinsic parity of these particles was assumed to be even ( I a l b = Icld), or odd ( I a l b = -Ida), both cases being taken into account. We used the phenomenological S-matrix approach and found the corresponding cross sections and polarizations, expressed in terms of the scattering angle 0 and some phenomenological parameters, for the two cases of the relative parity. We found further, that, even without assuming a particular interaction there are some possibilities of distinguishing between the relative parities. Assuming that transitions go only from the S state we got different angular dependences of the polarizations for the two values of the relative parity. If the particles in the initial state are unpolarized, the final state particles are a) unpolarized if the relative parity is even, or b) polarized normal to the scattering plane, with the degree of polarization proportional to sin 20, if the relative parity is odd. These results can serve as a test for the determination of the relative parity of baryons. In this paper we discuss some other possibilities of the determination of the relative parity. We will consider the decay of the final state particles (c and d) assuming that they decay by one mode into a baryon and a meson. The shape of the momentum correlation function for mesons emerging in this process depends on the relative parity of the particles occurring in the reaction. We give a 237
2~8
A. D E L O F F AND J . WRZECIONKO
general expression for these correlation functions in b o t h cases of relative parity. As an example we calculate the correlation for transitions in the S-state approxi m a t i o n in which either the a and b particles are in the S-state, or the c and d particles are p r o d u c e d in the S-state only. T h e m e a s u r e m e n t of the correlation can give some i n f o r m a t i o n a b o u t relative parities.
2. T h e C o r r e l a t i o n
Function,
General
Theory
T h e transition T m a t r i x for the reaction a + b --~ c + d , where a, b, c, d particles are ½ spin b a r y o n s was discussed briefly in (I). Using the phenomenological S - m a t r i x t e c h n i q u e we give there the general form of T in the two cases of the relative intrinsic parity. Denoting b y T ~÷~ and T ~-~ the transition m a t r i x in the even ( I a l b = Ielo) and odd ( I a l b = --Ido) relative p a r i t y case respectively we h a v e according to (I) T~+~ = A + B o
1 • n+Co 2 • n+Do
1. n o . , . n + E o
1 • mo,.
m+FOl"
1. m o 2 " 1,
+ G o 1. l o , . m + H o T ~-) =
A'o 1 •
I+B'o 2 • I+C'o
I •
102 • n + D ' o
1
102 • 1
• n o , . I + E ' o t • m+F'o~,
+G' o t . mo,
. n+H'
Ol .
noa. m,
(1) m
(2)
where the angular d e p e n d e n c e of the scalar functions A , . . . , H and A', . . . . H ' was given in (I), o 1 a n d o3 are Pauli spin vectors, and three unit vectors n, m , 1 are defined as follows n--
Pl ^ Pt sin 0 '
m--
Pt--Pt 2 sin ½0'
l--
Pt+Pt 2 cos ½0'
where Pt and pt are unit vectors in the direction of the initial and final c.m. m o m e n t u m respectively. W e will now consider the d e c a y of the final state b a r y o n s assuming t h a t t h e y d e c a y only b y one m o d e into a b a r y o n and a meson. T h e transition m a t r i x describing the d e c a y of the c-particle has the form a l + b l a 1 • k t , where the k 1 v e c t o r is a unit v e c t o r in the direction of m o t i o n of the meson in the rest system of c. Similarly the d e c a y of d is described b y a m a t r i x a 2 + b ~ a 2 • k 2, where k 2 is a unit v e c t o r determining the m o m e n t u m of the second meson in the rest s y s t e m of d. H a v i n g these transition matrices together with expressions (1) and (2) in m i n d we can write functions W ( k 1 , k2) describing the correlation of the mesons m o m e n t a k 1 and k2, W ( k l , k2) _-- ~1~2 T r { T p T t ( l + ~ h o
~•
~t)(1-t-~202.
~2)}/Tr{TpTt},
(3)
DIRECTIONAL
CORRELATION
239
where 2 R e (a~ b~*)
2~=[a~[2q-lb~l 2,
n~ --
la~l,+lb~l 2 ,
(x ---- 1, 2),
a n d the vectors u 1 and u 2 are defined as Pc " k l
~1 :
kl +
X2 :
k2 -~-
1 ' Pc - - - - Pc A k 1 ,
(Ec+mc)mc
me
Pa " k2
1
(Eo+md)md Pal--
--rodPo ^ k2-
Furthermore, p is the d e n s i t y m a t r i x describing the polarizations of a and b particles, T stands here for T ~+~ or T~-L For small m o m e n t a of the particle c and d respectively, where pc/rnc << 1 a n d Pa/mo << 1 the contribution from the motion of these particles can be neglected and we can put ~ ~ k~. The correlation function W can be expressed in terms of the polarizations of the c and d particles a n d the spin correlation tensor. As a m a t t e r of fact we get simply from (3)
W(k~, k2) : 2xAz{1 + ~ P1, Kl,+~/2 Pz, K 2 , + ~ / , C o K~,Kzj},
(4)
where, i, j = x, y, z, the quantities PI~ a n d P2~ are the polarization vectors of the c a n d d particle respectively, and C~ is the spin correlation tensor; the definitions are
PI, --
3. The
Tr{ TpTt Uli }
,
(5)
Tr{ TpTt a~.,} P~'-- Tr{TpTt} '
(6)
Tr{ TpTt alia2~} C° ---- Tr{TpTt}
(7)
S-State
Tr{TpT*}
Approximation
The general form of the W function as given b y (4) is very complicated especially if the particles a a n d b occurring in the initial state of our reaction are polarized. E v e n if one would get the explicit formulae for --I~PI~I,--2,PI+I,C~I expressed b y the scalar functions A . . . . . H or A', . . . . H' it is impossible to draw a n y conclusion about the relative parities. For both values of the relative p a r i t y the expressions for polarizations and C~ are v e r y similar. It is sufficient to know how the ~P~+) depend on the A, • " " , H functions and one li ~ r~c+l ~2i , (-I+l ~tJ can write at once the dependence of the P~.), P ~ ) , C~7) on the A', . . . . H ' functions b y the application of a simple transformation. This is shown in the
~40
A. DSLOFFAND j. WRZZCIONKO
appendix. However, it is possible to distinguish between parities by investigating the angular dependence of the polarizations and the correlation tensor. The angular dependence of these quantities should obviously be much influenced by the types of transitions, the latter being determined by the relative intrinsic parity of the particles involved in the reaction. Using the A,..., H and A', .... H' expressed in terms of the scattering angle 0 as given in (I) one can find the angular dependence of the polarizations and the correlation tensor for both cases of the relative parity. However, it is rather difficult to get general expressions for them preserving all angular momenta. W e will discuss here a simplified case assuming the so called S-state approximation introduced in (1). According to this approximation we assume two special cases: a) the final state particles are only in the S-state and b) the initial state particles are only in the S-statc. Besides that we confine ourselves to reactions in which the initial state particles a and b are unpolarized. Putting p = ~- we obtain the polarizations and the correlation tensor
a c+)P(+~ = n~2[Re(AB*+CD*) +Im(GE*+FH*)],
(8a)
a c+~P(+~ = n t 2 [Re (A C* + BD*) + Im (HE* + FG*) ],
(Sb)
a~+)C~+~ = n, nj2 Re (AD* + BC* +HG*-- EF*) + m~m j2 [Re (A E* -- FD*) + Im (CH* + BG* )] +l,l~2 [Re (A F*-- ED*) + Im (GC* + H B * ) ] + l~mj2[Re(A G* + HD* ) + Im(C F* + E B*) ] +m~l~2[Re(AH* +GD*) + I m ( F * B +C* E)],
(8c)
a~+~ = [A[2+[B]*+[C[2+[D[*+]E[2+IF[2+[G[*+[H[2.
(8d)
The corresponding expressions for the odd relative parity can be obtained using the substitution (A. 1) from the appendix. Now let us consider the two particular TABLg 1 Possible transitions Parity
according
to assumptions
even
a) a n d b ) Parity
odd
transition
amplitude
transition
~ ~
1So -~ 1S o
~
s P o --> 1S o
c¢'
"~ ~
'S1 ~
3S 1
fl
1P1 - + sS 1
fl'
'~ ~
1S o -~ 1S o
a
3S 1 ~ 8P 1
a'
" ~.~ ~ "4~
8S1 "--Y"$SI
b
8S 1 -+ 1P 1
b'
~" ~ ~
3S I -+ 3 D I
c
IS o --~ 8 p o
c'
'~
amplitude
241
D I R E C T I O N A L CORRELATION
cases q u o t e d as a) and b) above. If the initial state particles are assumed to be in the S-state we h a v e only three transition amplitudes different from zero, a similar situation results for the final state particles in the S-state. T h e corresponding transitions are w r i t t e n in table 1. Using the definitions of A . . . . . H and A ' , . . . . H ' given in (I) we express t h e m b y the corresponding transition amplitudes a n d the scattering angle 0. The obtained A,..., H and A ' , . . . . H ' are n e x t inserted to (8). After a lot of algebra we get finally W(±~ (kl, ks) = ~1 Is{ 1 +~1 P t W " k i + ~ s Ps (±) " k2 -~-'~'/1'/2l.a° FC(±) nnn
•
k i n • k~.+ Cmmm (±) • k ~ m • k s + C~)I • k l l • k s
(9)
r(±) ! . k l m . ks+C(m~) m . k l 1 . ks]} ' vtm
where for the final state particles in the S-state we h a v e
P1 (±) -----P2 (4-) : O. In t h a t case, we h a v e for p a r i t y even C(+) --= C.!+)n.n. = [lfllS+21ysl+2 Re (y/~*)]/16psa (+', C~+) =-- ~ ,
~ , ---- fl-- ~ Y
I~[s-~-c°s0{ly[s-]-2v/~Re (fly*))
16psa (+),
Ci i(+) " " i r r ~ J --ram(7"(+) ~--~ ~ m
--
~
1
s
Re (fly,))] /
16psa(+),
(lo)
C(+)l-,-.-~ m ~r(+) - - ~'~; = a sin 0 [ l y l s + 2 ~ / ~ Re ( f l y * ) ] / 3 2 p s a (+), C(m+~) ~
C{+)m.l. = Ctm
where e (+) is the cross section ~+) = D<2+31312-t-alYlS3/16~ z. F o r p a r i t y odd we h a v e C(,~) :~ Cl~-)n,n; = [3lfl'[s--]oc'12]/16ps(~(-), C[7 /
C~7) l J j = [ f I3y , 12-[:¢'lS+cos O ( 3 [ y ' i s - 3 [ f l ' l s ) ] / 1 6 p 2 a (-~
C,~m (-) ~ C l T ) m i r n J = r-al-'r2--l£t2--cos L2,~ 0(}ly'lS--al/~'la)J/16ped -), CI; t ~
C~7)l~mj = 3 sin O[j),']2-21fl'12]/32p2a(-),
c;J
cl;
= <2,
where ~(-) =
([~'12-}-3[fl'12+3[y'12).
{11)
2~2
A.
DELOFF
AND
J.
WRZECIONKO
The corresponding expressions for the case when the S-state is assumed as the initial state are for parity even P 1 (+) =
P~+) = 0,
= [Ib12--1alZ--21c12+2v/-2 Re (bc*)]/16p2a ¢+~, (12) C{+) -= [Ibl s - lal2+½1cl ~ - v/2 Re(bc*) --cos 0(3V'2 Re(bc*)+{Icl2)]/16p22 (+), H C(+) ~n
CI+) q?if't$ = [Ibl ~ - lal~+½1cl m- v/~ Re(bc*)+cos 0(3X/~ Re(bc*)+~-Icl2)3/16p2~'+L CI+) = C ~ ) = sin 013V/~ Re(bc*)+~lc12]/16p22 ~+~, Ira where
~(+~ = (lal2 + 31b1'--1-31cl2) /162#2, and for parity odd p l ( - ) ~- --P2 ¢-~ = n 3 v ' 2 sin
C#2 =
20 Im(a'b'*)/16p22 (-~,
[31b']2 - Ic'12]/16p~ a~-~,
1
= [3(la'12--21b'12)--(312'12+21c'12) cos O+6la'l z cos 0 cos 20]/32p22 ¢-),
(-)
[3([a'12-21b'12)+ (312'12+21c'1 z) cos 0-6[a'1 ~ cos 0 cos 20]/32p22 '-~,
c z2
=
=
(13)
sin 0[--(312'[2--21c'[ 2) + 6 V / 2 Re (a' b'*)cos 0+61£12 cos20]/32p22 ¢-~, sin 0[--(3[a'12-2[c'12)-6~¢/2 Re(a'b'*)cos 0+6]a'l 2 cos20]/32p22 c-~,
where
cr(-~ = (l£12+ 31b'lZ-t- 3[c'12) /16p 2. 4. D i s c u s s i o n
First let us discuss processes with the S state taken in the final state. To this category belong reactions with sufficiently high threshold where the kinetic energy of the initial state particles is mostly transformed into the big masses of the final state particles. We can therefore assume that the final state particles are very slow, and are produced really in the S state predominantly. This assumption, at least, is to be checked experimentally if we choose only reactions with isotropic cross section. We will consider here the following examples examined already b y Barshay 3). A+Z-,
X+Z,
-+ A + A -
In the meson correlation function obtained b y Barshay there was only one direction connected with the reaction, and this was the direction of the ingoing beam. To get more information about the correlation he assumed that the particles in the initial state are polarized. However, if the full kinematics of the reaction is taken into account, i.e. if the directions of the final state particles momenta are included in the correlation, we obtain more information even when
DIRECTIONAL CORRELATION
¢243
the initial state particles are unpolarized. The dependence of the components of C ~ I on the scattering angle 0 does not give any possibility of distinguishing between parities. From the other side we can here think about a full experiment aiming at the determination of the two sets of the scattering amplitudes, defined b y the upper part of table 1. We have 5 independent real parameters and 5 conditions for them given b y the measurement of the cross section and the four components of C~#~. This could be very important in the determination of the parity. Consider now the second class of reactions without a threshold. Here the energy is released due to the total rest mass of the particles, bigger before then after the reaction. For low energy collisions we can assume the S-wave in the initial state. In this case the correlation function shows quite interesting features. First of all in the odd parity case the polarization vectors are different from zero. The other important property is the angular dependence of the C ~ I tensor different in each relative parity case. However, we know only two reactions available for the experimental analysis. These are reactions examined by Pais and Treiman 8) and Treiman 4): 1-+p-+
A+n,
(i)
N - + p -+ A + A .
(ii)
In the final state of the processes (ii) we have like particles so the results (12), (13) should be corrected b y taking Pauli principle into account. Due to this principle the a, a' and c' amplitudes are absent. The components of cl+l are not much altered b y putting a = 0. It is not the same for the odd parity case, where because of the vanishing of two amplitudes a' and c' we have simply P1 ~-~ =
P2 ~-~ =
0,
C~: ~ =
+1,
Ci~-I =
c,,,,,, c-I =
-i,
c~-;I
c17,, ~ =
o.
The above results are strongly dependent on the relative intrinsic parity of the particles involved in the reaction. The measurement of the correlation of the pion momenta could be a sensitive test for distinguishing between parities. We notice also that in this measurement it is not necessary to have the initial particles polarized. The authors are much indebted to Professor J. Werle for reading the manuscript and helpful remarks. Appendix T h e T (-) m a t r i x w h i c h is a p s e u d o s c a l a r c a n b e w r i t t e n as a s c a l a r t i m e s a p s e u d o s c a l a r . E x c l u d i n g f r o m T (-~ t h e o 2 • m m a t r i x w e c a n w r i t e T ~-~ ---- ( A + B a l -~/~a
• n+0a2
I " IG 2 " I-~-OG
• n+Dal 1 " IG 2 "
• nG 2
m+Ha
1
'
n+/~al
• mG
• me2"
2 • I)G 2 •
m :
m ~2
" m,
244
A. DEI~OFF AND J. WRZECIONKO
where the new scalar functions are simply related to the primed quantities
,ff = F',
I~=H',
O=--iB',
D=--iD',
~=E',
P=iC',
O-~A',
B=iG'.
(A.1)
The ~ m a t r i x introduced above is of the same form as To+); this is very useful in m a n y calculations. Consider the T~-~pT ~-~ * matrix, where p is given b y P =
1(1-[-O"1" gl)(l+g2
" g2);
here, ~1 and g~ are the polarization vectors of the particles a and b respectively. Introducing ~ we have
TC-) pT<-) t = ~p' ~ t, where p' is of the same form as p with g2 replaced b y g'2 = - - g , + 2 m ' . g2m.
(A.2)
Due to the same form of £P and T (+) a n y result obtained for the even p a r i t y case can be written at once for the odd p a r i t y case. One should only apply the transformation A -+ A, . .., H -+/-), g2 -+ g'2 and next make use of (A.1) and (A.2).
References 1) 2) 3) 4)
A. S. A. S.
Deloff and J. Wrzecionko, Nuclear Physics (1960) Barshay, Phys. Rev. l l 3 (1959) 349 Pais and S. B. Treiman, Phys. Rev. 109 (1958) 1759 B. Treiman, Phys. Rev. l l 3 (1959) 355
D I R E C T I O N A L C O R R E L A T I O N OF M E S O N S IN A C A S C A D E D E C A Y OF B A R Y O N S R E S U L T I N G F R O M A B A R Y O N - B A R Y O N REACTION A. D E L O F F and J. W R Z E C I O N K O
Institute /or Nuclear Research, Warsaw Received 7 J u l y 1960 This is a c o n t i n u a t i o n of a preceding w o r k b y the s a m e a u t h o r s 1). We obtained general expressions of the m o m e n t u m directional correlation f u n c t i o n of m e s o n s resulting from the decay of b a r y o n s being p r o d u c t s of a b a r y o n - b a r y o n reaction. As a particular case we examined a reaction in which b o t h particles before reaction are unpolarized. Calculations were done a s s u m i n g t h a t a) particles before the reaction a r e i n the S state only (low energy zero threshold collision), or b) final s t a t e b a r y o n s are in the S-state only (high threshold reaction). Several e x p e r i m e n t a l possibilities for a d e t e r m i n a t i o n of the relative intrinsic p a r i t y of b a r y o n s are discussed.
Abstract:
1. I n t r o d u c t i o n In a preceding paper 1) which will be hereafter referred to as (I) we considered reactions of the type a + b - + c + d , where all particles involved are ½ spin baryons. The relative intrinsic parity of these particles was assumed to be even ( I a l b = Icld), or odd ( I a l b = -Ida), both cases being taken into account. We used the phenomenological S-matrix approach and found the corresponding cross sections and polarizations, expressed in terms of the scattering angle 0 and some phenomenological parameters, for the two cases of the relative parity. We found further, that, even without assuming a particular interaction there are some possibilities of distinguishing between the relative parities. Assuming that transitions go only from the S state we got different angular dependences of the polarizations for the two values of the relative parity. If the particles in the initial state are unpolarized, the final state particles are a) unpolarized if the relative parity is even, or b) polarized normal to the scattering plane, with the degree of polarization proportional to sin 20, if the relative parity is odd. These results can serve as a test for the determination of the relative parity of baryons. In this paper we discuss some other possibilities of the determination of the relative parity. We will consider the decay of the final state particles (c and d) assuming that they decay by one mode into a baryon and a meson. The shape of the momentum correlation function for mesons emerging in this process depends on the relative parity of the particles occurring in the reaction. We give a 237
2~8
A. D E L O F F AND J . WRZECIONKO
general expression for these correlation functions in b o t h cases of relative parity. As an example we calculate the correlation for transitions in the S-state approxi m a t i o n in which either the a and b particles are in the S-state, or the c and d particles are p r o d u c e d in the S-state only. T h e m e a s u r e m e n t of the correlation can give some i n f o r m a t i o n a b o u t relative parities.
2. T h e C o r r e l a t i o n
Function,
General
Theory
T h e transition T m a t r i x for the reaction a + b --~ c + d , where a, b, c, d particles are ½ spin b a r y o n s was discussed briefly in (I). Using the phenomenological S - m a t r i x t e c h n i q u e we give there the general form of T in the two cases of the relative intrinsic parity. Denoting b y T ~÷~ and T ~-~ the transition m a t r i x in the even ( I a l b = Ielo) and odd ( I a l b = --Ido) relative p a r i t y case respectively we h a v e according to (I) T~+~ = A + B o
1 • n+Co 2 • n+Do
1. n o . , . n + E o
1 • mo,.
m+FOl"
1. m o 2 " 1,
+ G o 1. l o , . m + H o T ~-) =
A'o 1 •
I+B'o 2 • I+C'o
I •
102 • n + D ' o
1
102 • 1
• n o , . I + E ' o t • m+F'o~,
+G' o t . mo,
. n+H'
Ol .
noa. m,
(1) m
(2)
where the angular d e p e n d e n c e of the scalar functions A , . . . , H and A', . . . . H ' was given in (I), o 1 a n d o3 are Pauli spin vectors, and three unit vectors n, m , 1 are defined as follows n--
Pl ^ Pt sin 0 '
m--
Pt--Pt 2 sin ½0'
l--
Pt+Pt 2 cos ½0'
where Pt and pt are unit vectors in the direction of the initial and final c.m. m o m e n t u m respectively. W e will now consider the d e c a y of the final state b a r y o n s assuming t h a t t h e y d e c a y only b y one m o d e into a b a r y o n and a meson. T h e transition m a t r i x describing the d e c a y of the c-particle has the form a l + b l a 1 • k t , where the k 1 v e c t o r is a unit v e c t o r in the direction of m o t i o n of the meson in the rest system of c. Similarly the d e c a y of d is described b y a m a t r i x a 2 + b ~ a 2 • k 2, where k 2 is a unit v e c t o r determining the m o m e n t u m of the second meson in the rest s y s t e m of d. H a v i n g these transition matrices together with expressions (1) and (2) in m i n d we can write functions W ( k 1 , k2) describing the correlation of the mesons m o m e n t a k 1 and k2, W ( k l , k2) _-- ~1~2 T r { T p T t ( l + ~ h o
~•
~t)(1-t-~202.
~2)}/Tr{TpTt},
(3)
DIRECTIONAL
CORRELATION
239
where 2 R e (a~ b~*)
2~=[a~[2q-lb~l 2,
n~ --
la~l,+lb~l 2 ,
(x ---- 1, 2),
a n d the vectors u 1 and u 2 are defined as Pc " k l
~1 :
kl +
X2 :
k2 -~-
1 ' Pc - - - - Pc A k 1 ,
(Ec+mc)mc
me
Pa " k2
1
(Eo+md)md Pal--
--rodPo ^ k2-
Furthermore, p is the d e n s i t y m a t r i x describing the polarizations of a and b particles, T stands here for T ~+~ or T~-L For small m o m e n t a of the particle c and d respectively, where pc/rnc << 1 a n d Pa/mo << 1 the contribution from the motion of these particles can be neglected and we can put ~ ~ k~. The correlation function W can be expressed in terms of the polarizations of the c and d particles a n d the spin correlation tensor. As a m a t t e r of fact we get simply from (3)
W(k~, k2) : 2xAz{1 + ~ P1, Kl,+~/2 Pz, K 2 , + ~ / , C o K~,Kzj},
(4)
where, i, j = x, y, z, the quantities PI~ a n d P2~ are the polarization vectors of the c a n d d particle respectively, and C~ is the spin correlation tensor; the definitions are
PI, --
3. The
Tr{ TpTt Uli }
,
(5)
Tr{ TpTt a~.,} P~'-- Tr{TpTt} '
(6)
Tr{ TpTt alia2~} C° ---- Tr{TpTt}
(7)
S-State
Tr{TpT*}
Approximation
The general form of the W function as given b y (4) is very complicated especially if the particles a a n d b occurring in the initial state of our reaction are polarized. E v e n if one would get the explicit formulae for --I~PI~I,--2,PI+I,C~I expressed b y the scalar functions A . . . . . H or A', . . . . H' it is impossible to draw a n y conclusion about the relative parities. For both values of the relative p a r i t y the expressions for polarizations and C~ are v e r y similar. It is sufficient to know how the ~P~+) depend on the A, • " " , H functions and one li ~ r~c+l ~2i , (-I+l ~tJ can write at once the dependence of the P~.), P ~ ) , C~7) on the A', . . . . H ' functions b y the application of a simple transformation. This is shown in the
~40
A. DSLOFFAND j. WRZZCIONKO
appendix. However, it is possible to distinguish between parities by investigating the angular dependence of the polarizations and the correlation tensor. The angular dependence of these quantities should obviously be much influenced by the types of transitions, the latter being determined by the relative intrinsic parity of the particles involved in the reaction. Using the A,..., H and A', .... H' expressed in terms of the scattering angle 0 as given in (I) one can find the angular dependence of the polarizations and the correlation tensor for both cases of the relative parity. However, it is rather difficult to get general expressions for them preserving all angular momenta. W e will discuss here a simplified case assuming the so called S-state approximation introduced in (1). According to this approximation we assume two special cases: a) the final state particles are only in the S-state and b) the initial state particles are only in the S-statc. Besides that we confine ourselves to reactions in which the initial state particles a and b are unpolarized. Putting p = ~- we obtain the polarizations and the correlation tensor
a c+)P(+~ = n~2[Re(AB*+CD*) +Im(GE*+FH*)],
(8a)
a c+~P(+~ = n t 2 [Re (A C* + BD*) + Im (HE* + FG*) ],
(Sb)
a~+)C~+~ = n, nj2 Re (AD* + BC* +HG*-- EF*) + m~m j2 [Re (A E* -- FD*) + Im (CH* + BG* )] +l,l~2 [Re (A F*-- ED*) + Im (GC* + H B * ) ] + l~mj2[Re(A G* + HD* ) + Im(C F* + E B*) ] +m~l~2[Re(AH* +GD*) + I m ( F * B +C* E)],
(8c)
a~+~ = [A[2+[B]*+[C[2+[D[*+]E[2+IF[2+[G[*+[H[2.
(8d)
The corresponding expressions for the odd relative parity can be obtained using the substitution (A. 1) from the appendix. Now let us consider the two particular TABLg 1 Possible transitions Parity
according
to assumptions
even
a) a n d b ) Parity
odd
transition
amplitude
transition
~ ~
1So -~ 1S o
~
s P o --> 1S o
c¢'
"~ ~
'S1 ~
3S 1
fl
1P1 - + sS 1
fl'
'~ ~
1S o -~ 1S o
a
3S 1 ~ 8P 1
a'
" ~.~ ~ "4~
8S1 "--Y"$SI
b
8S 1 -+ 1P 1
b'
~" ~ ~
3S I -+ 3 D I
c
IS o --~ 8 p o
c'
'~
amplitude
241
D I R E C T I O N A L CORRELATION
cases q u o t e d as a) and b) above. If the initial state particles are assumed to be in the S-state we h a v e only three transition amplitudes different from zero, a similar situation results for the final state particles in the S-state. T h e corresponding transitions are w r i t t e n in table 1. Using the definitions of A . . . . . H and A ' , . . . . H ' given in (I) we express t h e m b y the corresponding transition amplitudes a n d the scattering angle 0. The obtained A,..., H and A ' , . . . . H ' are n e x t inserted to (8). After a lot of algebra we get finally W(±~ (kl, ks) = ~1 Is{ 1 +~1 P t W " k i + ~ s Ps (±) " k2 -~-'~'/1'/2l.a° FC(±) nnn
•
k i n • k~.+ Cmmm (±) • k ~ m • k s + C~)I • k l l • k s
(9)
r(±) ! . k l m . ks+C(m~) m . k l 1 . ks]} ' vtm
where for the final state particles in the S-state we h a v e
P1 (±) -----P2 (4-) : O. In t h a t case, we h a v e for p a r i t y even C(+) --= C.!+)n.n. = [lfllS+21ysl+2 Re (y/~*)]/16psa (+', C~+) =-- ~ ,
~ , ---- fl-- ~ Y
I~[s-~-c°s0{ly[s-]-2v/~Re (fly*))
16psa (+),
Ci i(+) " " i r r ~ J --ram(7"(+) ~--~ ~ m
--
~
1
s
Re (fly,))] /
16psa(+),
(lo)
C(+)l-,-.-~ m ~r(+) - - ~'~; = a sin 0 [ l y l s + 2 ~ / ~ Re ( f l y * ) ] / 3 2 p s a (+), C(m+~) ~
C{+)m.l. = Ctm
where e (+) is the cross section ~+) = D<2+31312-t-alYlS3/16~ z. F o r p a r i t y odd we h a v e C(,~) :~ Cl~-)n,n; = [3lfl'[s--]oc'12]/16ps(~(-), C[7 /
C~7) l J j = [ f I3y , 12-[:¢'lS+cos O ( 3 [ y ' i s - 3 [ f l ' l s ) ] / 1 6 p 2 a (-~
C,~m (-) ~ C l T ) m i r n J = r-al-'r2--l£t2--cos L2,~ 0(}ly'lS--al/~'la)J/16ped -), CI; t ~
C~7)l~mj = 3 sin O[j),']2-21fl'12]/32p2a(-),
c;J
cl;
= <2,
where ~(-) =
([~'12-}-3[fl'12+3[y'12).
{11)
2~2
A.
DELOFF
AND
J.
WRZECIONKO
The corresponding expressions for the case when the S-state is assumed as the initial state are for parity even P 1 (+) =
P~+) = 0,
= [Ib12--1alZ--21c12+2v/-2 Re (bc*)]/16p2a ¢+~, (12) C{+) -= [Ibl s - lal2+½1cl ~ - v/2 Re(bc*) --cos 0(3V'2 Re(bc*)+{Icl2)]/16p22 (+), H C(+) ~n
CI+) q?if't$ = [Ibl ~ - lal~+½1cl m- v/~ Re(bc*)+cos 0(3X/~ Re(bc*)+~-Icl2)3/16p2~'+L CI+) = C ~ ) = sin 013V/~ Re(bc*)+~lc12]/16p22 ~+~, Ira where
~(+~ = (lal2 + 31b1'--1-31cl2) /162#2, and for parity odd p l ( - ) ~- --P2 ¢-~ = n 3 v ' 2 sin
C#2 =
20 Im(a'b'*)/16p22 (-~,
[31b']2 - Ic'12]/16p~ a~-~,
1
= [3(la'12--21b'12)--(312'12+21c'12) cos O+6la'l z cos 0 cos 20]/32p22 ¢-),
(-)
[3([a'12-21b'12)+ (312'12+21c'1 z) cos 0-6[a'1 ~ cos 0 cos 20]/32p22 '-~,
c z2
=
=
(13)
sin 0[--(312'[2--21c'[ 2) + 6 V / 2 Re (a' b'*)cos 0+61£12 cos20]/32p22 ¢-~, sin 0[--(3[a'12-2[c'12)-6~¢/2 Re(a'b'*)cos 0+6]a'l 2 cos20]/32p22 c-~,
where
cr(-~ = (l£12+ 31b'lZ-t- 3[c'12) /16p 2. 4. D i s c u s s i o n
First let us discuss processes with the S state taken in the final state. To this category belong reactions with sufficiently high threshold where the kinetic energy of the initial state particles is mostly transformed into the big masses of the final state particles. We can therefore assume that the final state particles are very slow, and are produced really in the S state predominantly. This assumption, at least, is to be checked experimentally if we choose only reactions with isotropic cross section. We will consider here the following examples examined already b y Barshay 3). A+Z-,
X+Z,
-+ A + A -
In the meson correlation function obtained b y Barshay there was only one direction connected with the reaction, and this was the direction of the ingoing beam. To get more information about the correlation he assumed that the particles in the initial state are polarized. However, if the full kinematics of the reaction is taken into account, i.e. if the directions of the final state particles momenta are included in the correlation, we obtain more information even when
DIRECTIONAL CORRELATION
¢243
the initial state particles are unpolarized. The dependence of the components of C ~ I on the scattering angle 0 does not give any possibility of distinguishing between parities. From the other side we can here think about a full experiment aiming at the determination of the two sets of the scattering amplitudes, defined b y the upper part of table 1. We have 5 independent real parameters and 5 conditions for them given b y the measurement of the cross section and the four components of C~#~. This could be very important in the determination of the parity. Consider now the second class of reactions without a threshold. Here the energy is released due to the total rest mass of the particles, bigger before then after the reaction. For low energy collisions we can assume the S-wave in the initial state. In this case the correlation function shows quite interesting features. First of all in the odd parity case the polarization vectors are different from zero. The other important property is the angular dependence of the C ~ I tensor different in each relative parity case. However, we know only two reactions available for the experimental analysis. These are reactions examined by Pais and Treiman 8) and Treiman 4): 1-+p-+
A+n,
(i)
N - + p -+ A + A .
(ii)
In the final state of the processes (ii) we have like particles so the results (12), (13) should be corrected b y taking Pauli principle into account. Due to this principle the a, a' and c' amplitudes are absent. The components of cl+l are not much altered b y putting a = 0. It is not the same for the odd parity case, where because of the vanishing of two amplitudes a' and c' we have simply P1 ~-~ =
P2 ~-~ =
0,
C~: ~ =
+1,
Ci~-I =
c,,,,,, c-I =
-i,
c~-;I
c17,, ~ =
o.
The above results are strongly dependent on the relative intrinsic parity of the particles involved in the reaction. The measurement of the correlation of the pion momenta could be a sensitive test for distinguishing between parities. We notice also that in this measurement it is not necessary to have the initial particles polarized. The authors are much indebted to Professor J. Werle for reading the manuscript and helpful remarks. Appendix T h e T (-) m a t r i x w h i c h is a p s e u d o s c a l a r c a n b e w r i t t e n as a s c a l a r t i m e s a p s e u d o s c a l a r . E x c l u d i n g f r o m T (-~ t h e o 2 • m m a t r i x w e c a n w r i t e T ~-~ ---- ( A + B a l -~/~a
• n+0a2
I " IG 2 " I-~-OG
• n+Dal 1 " IG 2 "
• nG 2
m+Ha
1
'
n+/~al
• mG
• me2"
2 • I)G 2 •
m :
m ~2
" m,
244
A. DEI~OFF AND J. WRZECIONKO
where the new scalar functions are simply related to the primed quantities
,ff = F',
I~=H',
O=--iB',
D=--iD',
~=E',
P=iC',
O-~A',
B=iG'.
(A.1)
The ~ m a t r i x introduced above is of the same form as To+); this is very useful in m a n y calculations. Consider the T~-~pT ~-~ * matrix, where p is given b y P =
1(1-[-O"1" gl)(l+g2
" g2);
here, ~1 and g~ are the polarization vectors of the particles a and b respectively. Introducing ~ we have
TC-) pT<-) t = ~p' ~ t, where p' is of the same form as p with g2 replaced b y g'2 = - - g , + 2 m ' . g2m.
(A.2)
Due to the same form of £P and T (+) a n y result obtained for the even p a r i t y case can be written at once for the odd p a r i t y case. One should only apply the transformation A -+ A, . .., H -+/-), g2 -+ g'2 and next make use of (A.1) and (A.2).
References 1) 2) 3) 4)
A. S. A. S.
Deloff and J. Wrzecionko, Nuclear Physics (1960) Barshay, Phys. Rev. l l 3 (1959) 349 Pais and S. B. Treiman, Phys. Rev. 109 (1958) 1759 B. Treiman, Phys. Rev. l l 3 (1959) 355