- Email: [email protected]
Pole approximation model in hyperon decays and unitary symmetry
Volume 6. number 3
PHYSICS
2 N+l ___ 1 +N+2E2-%
LETTERS
15 September 1963
Eq. (13) and hence also (12) are manifestly energy translation invariant and can be used in conjunction with experimental data to obtain experimental esHw2 =N+<-2+s ET1. (8) timates of CDD providing care is taken to make finite sample corrections. If, as a check on our work, we consider the joint The lack of correlation in an invariant distribueigenvalue distribution which arises from an indetion between pairs of off-diagonal elements and bependent matrix distribution tween off-diagonal elements and diagonal elements probably does not persist for squares (and higher PN(EI,E~.. . , EN) =K[ n 1Ek-EvI 1 exP(-$ c Ex2), Powers) of the matrix elements. Calculation of the correlation coefficients for higher powers of the P
H,,+&
Hcyr2=&2
=O,
9
>
(7)
(11)
indicating the correctness of (7) and (8). From (7) and (8) and the relation HIP = q which holds if the mean eigenvalue is not taken to be zero, we have for the diagonal-diagonal matrix element correlation coefficient
l+N ‘DD with ‘6E1,6E2
%E1,bE2
= 3 + (N-l)
C6EI, 6E2 ’
(14)
defined by (13).
I + (N+I) C6EI, 6~~ ‘DD
= 3 + (N-l)
(12)
C6El, 6E2 ’
where 6E1 6E2 C6E1’
6E2 = [@El)2
@$+
*
(13)
References 1) C. E. Porter and N. Rosenzweig, Suomalaisen Tiedeakatemian Toimituksia A VI, no. 44 (1960). 2) A. J. W. Sommerfeld, Partial differential equations in physics, (Academic Press, New York, 1949) p.22’7. 3) M. L. Mehta, Nuclear Phys. 18 (1960) 395.
*****
IN
POLE APPROXIMATION MODEL HYPERON DECAYS AND UNITARY SYMMETRY E. EBERLE * and S. IWAO Istituto
Istituto di Fisica dell’ UniversitB, Geneva Nazionale di Fisica Nucleare, Se&one di Genova Received 8 August 1963
We have investigated the pole approximation model of Feldman et al. l) on the “hadronic decay” of hyperons joined with the unitary symmetry model of Gell-Mann 2) and Ne’eman 3). In this brief communication we give the main results we have obtained. We have studied (i) the parameters appearing in the theory as to be consistent with the experimental information in the C- and A- hadronic decays (except the A-decay rate), (ii) the effect of the kaon pole relative to the baryon pole contribution by making use of the K-n direct weak coupling constant calculated by us 4), (iii) the sign of the asymmetry parameter of the 8- -decay and (iv) the decay probability of the A-hyperon by making use of the parameters preceedently determined. We find that only one solution of the model of Feldman et al. in the unitary symmetry n+ r+ and scheme gives a correct result, i.e., the pure p-wave and the pure s-wave decay for the xiC _ n+ 1; processes respectively. The procedure of our approach is as follows: in the C- and A- hadronic decays there are six unknowns * On leave of absence from INFN-Sezione Siciliana-Catania. 302
Volume 6, number 3
PHYSICS
LETTERS
15 September 1963 -.
by using the unitary symmetry, are reduced to five. We choose five equations furnished by the experimental information, to calculate these parameters. With the values of the parameters, thus obtained, we derive the sign of the a- -asymmetry parameter and the A-decay probability. As usual 5) we c$n write two sets of equations corresponding to: (i) the pure s-wave and the pure (ii) the pure p-wave and the pure s-wave decay for C+ _ n +II+ p-.azv~ decay for C 4 n+ai and C- * n+r-, - n +r- respectively. Including the K-pole contribution we get: which,
(I) 1st set
hc G2K (2/2 G - G2)F
= Gl?
A
2
= Gl
2 hc (d2 G+ G2)h+G1$=-~v~2(~1-~2h+JZ G2”-
h
A
x
Gl -,/2
-=-_n h/I
m&
hc
m,Fc -) hA
A
from A- = 0 ,
(1)
fromB+=O,
(2)
fromxo
1
= -(&Rxi $ A ,
(3)
from B-
h,
(4)
from x* = (!!$“$
,
(5)
A
AA =-mT
(&G-+
hA
BA = m,, (.@ Gh, - Glhc
Cl:),
x2
- mnFA) ,
(6)
0
where G, G1 and G22are NN?r, ACn an; CCa coupling constants. K’ = (&/y$ (l/x0), K = (j3h/r~$ (~/xA), fly = (my-mN)2-mT , YY = (my+mN) -m712 (Y = C , A only) and Sa = B-/ m, . Here x0 and XA are related parameters o. of C+ - p + no and 01~of A - p + 1~~by cue = - 2 Re x0/ (l+ 1x0 ( 2, and to the decay asymmetr Other notations are completely equivalent to those of Feldman et 0’~ = - 2Re Q/(l+].XA 1I ) respectively. al. except for J2 instead of 2 for the coefficients of the charged pion and the charged kaon coupling constants to the nucleons and to CN. Combining eqs. (1) to (5) and G1 = t (1-y)G and G2 = 2yG, as prescribed by the unitary symmetry 6, (y is the ratio between the coupling constant in F-coupling and the pion-nucleon coupling constant with an appropriate coefficient), we obtain *(&+$(l-y)2)*n
=&!-+! . .
K/K’
JZ+~Y
($(1-~)2+fly-
l)(*n+flqF&+&(l-y)(4y-&)m,pA
0)
,
where 6: = (l?:)i(&? fl rnc/qs /I$ mn) and r: is the C- _ n+a- decay width. From eq. (7) we can derive the value of the parameter y. The alternative set of equations is: m
2nd set h, (/2G+G2)G+Glx2=0
G1 - Jz-=0
G,"-
m&x
h/l
A
G22
*2
- Gl$=
h,
(1/2G- G2)h
A
-
-K’a2
G1+
[fl
G - G2):
from A+ = 0,
(8)
from B- = 0 ,
(9)
from x0 = -($tg
- Gl]
,(lO)
(11)
0’
A
where t7 is connected with l7+ in the same way as 0 is connected with I’:. The A-decay is specified by (5) and (6). From (8), (9), (lh), (11) and (5) with G1 and G2 always given by unitary symmetry we get K/K
=s._ +
($(l -y)2 23(l-Y)2-flY+&-Y)U2-4Y)
- J2 y - 1) *ra,_[hrz_~);yflm
(12) F * 71 z 303
vTolume6, number 3
PHYSICS
15 September 1963
LETTERS
In solving the two sets of equations numerically, we use x0 = - 1 (consistent with )AI / = a rule) 7, and cl\= 0.36. We have solved the equations putting F* = Fz = 0. Although the exclusion of the terms conaining FA and FE does not change the final conclusions, these contributions are not generally negligible. We shall discuss this point, in more detail, in a forthcoming paper. A rough idea of the effect of the I-pole is given by the modification caused to the ratio K/K’ which is in case (I) of - 15% and in case (II) )f - 300/O. There is one solution for each of the eqs. (7) and (12) which we took as the good one, since the other mes are not reasonable; it is the following one; case (I) y = 0.52 and case (II) y x 0.29. We calculate now the sign of the E- -decay asymmetry parameter. For this purpose we define the p/s ratio in the Z--decay by : 2G3h,
vhereK”
= (&/y&~(l/x~),
/x2 + Glhi /<2 =K"(,,'z
BE = (mE-mA)2-m,2,
G3hE - Glh;)
ye = (mE+m&mlr2,
,
(13)
hs =_faz/(mn-mn),
hh= x2 =aa(mz+mA)/
P=‘n coupling constant which is specified by G3 = -(l-2y)G, c and G% 5 ~~~~rn~+rn~)/b~(rn~-rn~) - a > b and a’ , b’ specify the S-Aand E-C weak vertices simi.arly to the A-N and C-N vertices. In order to have the experimental sign of 01~we have to havexs < 0. Under an assumption discussed below, it turns out that the solution of the 1st set of equations gives the wrong sign for ok, while the solution of the 2nd set leads to the correct sign. It is a lucky opportunity that this result does not depend strongly on the value of hE, h’z, x2, t2. We iave no way of calculating these parameters and we have assumed, physically enough in our opinion, that no other assumption is needed to derive the hey have the same sign than h,, hx, 12 and 02 respectively; :onclusion obtained above. As a consequence we are led to exclude the 1st set of equations. It is worthvhile to notice, at this point, that the value y ET0.29, corresponding to the 2nd set of equations, is very iear to that obtained from different sources 3). We then calculate the A-decay probability by making use of the five parameters obtained from the 2nd Set of equations. We get r(h - p+n-) = 0.27 X lo-l4 GeV which is not very much different from the exper.mental value r(h + p+ a-) = 0.17 X lo-l4 GeV. As we said before we are not considering here the K-pole; iowever, we can say that its contribution makes the agreement with the experimental width better. To conclude it seems to us that, from the above results, the unitary symmetry model can be a nice ,001 to study the non-leptonic decay of hyperons, and that it would be interesting to have more experinental information in order to test this possibility to a wider extent. ‘ak’(m’-md’ i:(ms-mA)
References
G.Feldman, P.T. Matthews and A.Salam, Phys.Rev. 121 (1961) 302. M. Cell-Mann, The eightfold way : A theory of strong interaction symmetry, CTSL-20 (1961). Y.Ne’eman, Nuclear Phys.26 (1961) 222. E.Eberle and S.Iwao, Physics Letters 6 (1963) 238. E.g., J.C. Pati, Phys.Rev. 130 (1963) 2097. J. J.de Swart, The octet model and its Clebsch-Cordan coefficients, CERN, preprint (1963). 7) F. S. Crawford, Proc. of the 1962 Int. Conf. on High-Energy Physics at CERN, p. 827. 8) J. J.de Swart and C.K.Iddings, Phys.Rev. 130 (1963) to be published.
1) 2) 3) 4) 5) 6)
*****
104
PHYSICS
2 N+l ___ 1 +N+2E2-%
LETTERS
15 September 1963
Eq. (13) and hence also (12) are manifestly energy translation invariant and can be used in conjunction with experimental data to obtain experimental esHw2 =N+<-2+s ET1. (8) timates of CDD providing care is taken to make finite sample corrections. If, as a check on our work, we consider the joint The lack of correlation in an invariant distribueigenvalue distribution which arises from an indetion between pairs of off-diagonal elements and bependent matrix distribution tween off-diagonal elements and diagonal elements probably does not persist for squares (and higher PN(EI,E~.. . , EN) =K[ n 1Ek-EvI 1 exP(-$ c Ex2), Powers) of the matrix elements. Calculation of the correlation coefficients for higher powers of the P
H,,+&
Hcyr2=&2
=O,
9
>
(7)
(11)
indicating the correctness of (7) and (8). From (7) and (8) and the relation HIP = q which holds if the mean eigenvalue is not taken to be zero, we have for the diagonal-diagonal matrix element correlation coefficient
l+N ‘DD with ‘6E1,6E2
%E1,bE2
= 3 + (N-l)
C6EI, 6E2 ’
(14)
defined by (13).
I + (N+I) C6EI, 6~~ ‘DD
= 3 + (N-l)
(12)
C6El, 6E2 ’
where 6E1 6E2 C6E1’
6E2 = [@El)2
@$+
*
(13)
References 1) C. E. Porter and N. Rosenzweig, Suomalaisen Tiedeakatemian Toimituksia A VI, no. 44 (1960). 2) A. J. W. Sommerfeld, Partial differential equations in physics, (Academic Press, New York, 1949) p.22’7. 3) M. L. Mehta, Nuclear Phys. 18 (1960) 395.
*****
IN
POLE APPROXIMATION MODEL HYPERON DECAYS AND UNITARY SYMMETRY E. EBERLE * and S. IWAO Istituto
Istituto di Fisica dell’ UniversitB, Geneva Nazionale di Fisica Nucleare, Se&one di Genova Received 8 August 1963
We have investigated the pole approximation model of Feldman et al. l) on the “hadronic decay” of hyperons joined with the unitary symmetry model of Gell-Mann 2) and Ne’eman 3). In this brief communication we give the main results we have obtained. We have studied (i) the parameters appearing in the theory as to be consistent with the experimental information in the C- and A- hadronic decays (except the A-decay rate), (ii) the effect of the kaon pole relative to the baryon pole contribution by making use of the K-n direct weak coupling constant calculated by us 4), (iii) the sign of the asymmetry parameter of the 8- -decay and (iv) the decay probability of the A-hyperon by making use of the parameters preceedently determined. We find that only one solution of the model of Feldman et al. in the unitary symmetry n+ r+ and scheme gives a correct result, i.e., the pure p-wave and the pure s-wave decay for the xiC _ n+ 1; processes respectively. The procedure of our approach is as follows: in the C- and A- hadronic decays there are six unknowns * On leave of absence from INFN-Sezione Siciliana-Catania. 302
Volume 6, number 3
PHYSICS
LETTERS
15 September 1963 -.
by using the unitary symmetry, are reduced to five. We choose five equations furnished by the experimental information, to calculate these parameters. With the values of the parameters, thus obtained, we derive the sign of the a- -asymmetry parameter and the A-decay probability. As usual 5) we c$n write two sets of equations corresponding to: (i) the pure s-wave and the pure (ii) the pure p-wave and the pure s-wave decay for C+ _ n +II+ p-.azv~ decay for C 4 n+ai and C- * n+r-, - n +r- respectively. Including the K-pole contribution we get: which,
(I) 1st set
hc G2K (2/2 G - G2)F
= Gl?
A
2
= Gl
2 hc (d2 G+ G2)h+G1$=-~v~2(~1-~2h+JZ G2”-
h
A
x
Gl -,/2
-=-_n h/I
m&
hc
m,Fc -) hA
A
from A- = 0 ,
(1)
fromB+=O,
(2)
fromxo
1
= -(&Rxi $ A ,
(3)
from B-
h,
(4)
from x* = (!!$“$
,
(5)
A
AA =-mT
(&G-+
hA
BA = m,, (.@ Gh, - Glhc
Cl:),
x2
- mnFA) ,
(6)
0
where G, G1 and G22are NN?r, ACn an; CCa coupling constants. K’ = (&/y$ (l/x0), K = (j3h/r~$ (~/xA), fly = (my-mN)2-mT , YY = (my+mN) -m712 (Y = C , A only) and Sa = B-/ m, . Here x0 and XA are related parameters o. of C+ - p + no and 01~of A - p + 1~~by cue = - 2 Re x0/ (l+ 1x0 ( 2, and to the decay asymmetr Other notations are completely equivalent to those of Feldman et 0’~ = - 2Re Q/(l+].XA 1I ) respectively. al. except for J2 instead of 2 for the coefficients of the charged pion and the charged kaon coupling constants to the nucleons and to CN. Combining eqs. (1) to (5) and G1 = t (1-y)G and G2 = 2yG, as prescribed by the unitary symmetry 6, (y is the ratio between the coupling constant in F-coupling and the pion-nucleon coupling constant with an appropriate coefficient), we obtain *(&+$(l-y)2)*n
=&!-+! . .
K/K’
JZ+~Y
($(1-~)2+fly-
l)(*n+flqF&+&(l-y)(4y-&)m,pA
0)
,
where 6: = (l?:)i(&? fl rnc/qs /I$ mn) and r: is the C- _ n+a- decay width. From eq. (7) we can derive the value of the parameter y. The alternative set of equations is: m
2nd set h, (/2G+G2)G+Glx2=0
G1 - Jz-=0
G,"-
m&x
h/l
A
G22
*2
- Gl$=
h,
(1/2G- G2)h
A
-
-K’a2
G1+
[fl
G - G2):
from A+ = 0,
(8)
from B- = 0 ,
(9)
from x0 = -($tg
- Gl]
,(lO)
(11)
0’
A
where t7 is connected with l7+ in the same way as 0 is connected with I’:. The A-decay is specified by (5) and (6). From (8), (9), (lh), (11) and (5) with G1 and G2 always given by unitary symmetry we get K/K
=s._ +
($(l -y)2 23(l-Y)2-flY+&-Y)U2-4Y)
- J2 y - 1) *ra,_[hrz_~);yflm
(12) F * 71 z 303
vTolume6, number 3
PHYSICS
15 September 1963
LETTERS
In solving the two sets of equations numerically, we use x0 = - 1 (consistent with )AI / = a rule) 7, and cl\= 0.36. We have solved the equations putting F* = Fz = 0. Although the exclusion of the terms conaining FA and FE does not change the final conclusions, these contributions are not generally negligible. We shall discuss this point, in more detail, in a forthcoming paper. A rough idea of the effect of the I-pole is given by the modification caused to the ratio K/K’ which is in case (I) of - 15% and in case (II) )f - 300/O. There is one solution for each of the eqs. (7) and (12) which we took as the good one, since the other mes are not reasonable; it is the following one; case (I) y = 0.52 and case (II) y x 0.29. We calculate now the sign of the E- -decay asymmetry parameter. For this purpose we define the p/s ratio in the Z--decay by : 2G3h,
vhereK”
= (&/y&~(l/x~),
/x2 + Glhi /<2 =K"(,,'z
BE = (mE-mA)2-m,2,
G3hE - Glh;)
ye = (mE+m&mlr2,
,
(13)
hs =_faz/(mn-mn),
hh= x2 =aa(mz+mA)/
P=‘n coupling constant which is specified by G3 = -(l-2y)G, c and G% 5 ~~~~rn~+rn~)/b~(rn~-rn~) - a > b and a’ , b’ specify the S-Aand E-C weak vertices simi.arly to the A-N and C-N vertices. In order to have the experimental sign of 01~we have to havexs < 0. Under an assumption discussed below, it turns out that the solution of the 1st set of equations gives the wrong sign for ok, while the solution of the 2nd set leads to the correct sign. It is a lucky opportunity that this result does not depend strongly on the value of hE, h’z, x2, t2. We iave no way of calculating these parameters and we have assumed, physically enough in our opinion, that no other assumption is needed to derive the hey have the same sign than h,, hx, 12 and 02 respectively; :onclusion obtained above. As a consequence we are led to exclude the 1st set of equations. It is worthvhile to notice, at this point, that the value y ET0.29, corresponding to the 2nd set of equations, is very iear to that obtained from different sources 3). We then calculate the A-decay probability by making use of the five parameters obtained from the 2nd Set of equations. We get r(h - p+n-) = 0.27 X lo-l4 GeV which is not very much different from the exper.mental value r(h + p+ a-) = 0.17 X lo-l4 GeV. As we said before we are not considering here the K-pole; iowever, we can say that its contribution makes the agreement with the experimental width better. To conclude it seems to us that, from the above results, the unitary symmetry model can be a nice ,001 to study the non-leptonic decay of hyperons, and that it would be interesting to have more experinental information in order to test this possibility to a wider extent. ‘ak’(m’-md’ i:(ms-mA)
References
G.Feldman, P.T. Matthews and A.Salam, Phys.Rev. 121 (1961) 302. M. Cell-Mann, The eightfold way : A theory of strong interaction symmetry, CTSL-20 (1961). Y.Ne’eman, Nuclear Phys.26 (1961) 222. E.Eberle and S.Iwao, Physics Letters 6 (1963) 238. E.g., J.C. Pati, Phys.Rev. 130 (1963) 2097. J. J.de Swart, The octet model and its Clebsch-Cordan coefficients, CERN, preprint (1963). 7) F. S. Crawford, Proc. of the 1962 Int. Conf. on High-Energy Physics at CERN, p. 827. 8) J. J.de Swart and C.K.Iddings, Phys.Rev. 130 (1963) to be published.
1) 2) 3) 4) 5) 6)
*****
104