0 EC 1985
^?,
IC/65/67
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL
PHYSICS
SU9 "COMPOSITE" SYMMETRY
AND WEAK INTERACTIONS
(THE gA/gv RATIO AND SU9 SYMMETRY)
G. COCHO
1965
PIAZZA OBERDAN
TRIESTE
IC/65/67
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
SU "COMPOSITE" SYMMETRY AND WEAK INTERACTIONS*y
G. COCHO*
TRIESTE
October 1965
v Submitted to Nuovo Cimento
** On leave of absence from the lastituto de Fisica, Universidad de Mexico and theComision Nacional de Energia Nuclear de Mexico.
SUMMARY
An SU "composite" symmetry model is considered. A U(9) x U(9)y
group and a relativistic boosting are also discussed. Within suchSUQ composite model, assuming that in the non-leptonic decays the
y
weak hamiitonian transforms as the central component of the
SUjt, B) octet of the {9 , "9) representation of U(9) BJ U(9)' > SU (S^B)
& SUQ(T, Y) the scalar-pseudoscalar mixing is fixed to be s / I S - ^ P .
One obtains with essentially one parameter a reasonable fitting for
both non-leptonic decays of hyperons and K., -> 7T + TC decay. If one
assumes that in the leptonic decays such scalar-pseudoscalar mixing :
is reflected in the vector-axial vector ratio, it also follows that
'i U
SU "COMPOSITE" SYMMETRY AND WEAK INTERACTIONS
I INTRODUCTION
In recent papers some authors'1 ' have streaaed the point of
the permutative origin of unitary symmetries. In such a model,
the central idea is the generation of unitary symmetries from one
conservation law plus the perfect interchangeability of the building
blocks of the hadrons.
We will consider the case when some of these building blocks
are not elementary but composite. (A particular case of composite
states playing the role of fundamental entities is the Wigner super-
multiplet model, where the fundamental entities are the nuclecms
which may be considered as composed of three SU,, quarks).b
In particular, we will consider the case where the building
blocks are not only the SUR spin quarks <LL : f 0( — 1. 2 ->
is a unitary spin index and [ z. i j^ labels the ordinary spin) but.
also_the composite two-antiquark state YoiA ^ ""F LTRV^- 'TYZ^
~ Y/iiWYjC^I) j 6o (A / . If every quark n^* carries
baryonic number ft-I > the S - 0 <b. ^ "composite" quark will
carry Q— —*Z- , We see that this S — O 6 "~"%" quark
has the quantum numbers of the spinless quark in a hypothetical uni-
tary < U^C^ .B) group where the ordinary spin S replaces the
isotopic spin T and the baryonic number B replaces the" hyper-
charge Y . (This SU (S, B) group was first considered by
LIPKIN ). If one considers as basis the quarks ^13 where06 = 1 , 2 , 3 labels the SU, (T, Y) spin and 3 = 1 , 2 , 3 labels
- ? 6
the SU (S , B) spin, we may talk about an SU group which has been
considered by the author and by MIYAZAWA and SUGAWARAV \
We expect this SU symmetry to be broken due to two facts:ya) Due to the composite character of the <3 spinless quark states
•" a n d ^ V i ^ i ' l W • • " a r e c o m p o s e d o f t h e s a m e
elementary entities; although they belong to different representations
- 1 -
of g i
b) As we expect the spin and spinless quark to obey different
statistics, the symmetry will be broken by the statistics. We
assume that the prescription is to project from the matrix elements
the part consistent with the particles being in SU representationsfa
and having the correct symmetry or anti-symmetry properties
(see Section II),
Therefore, the final symmetry will be SU with "some restrictions",-^ b
One might consider also the SUQ (T •, Y) symmetry as a com-
posite broken symmetry if one assumes that the A, quark is compo-
site of two SU0 (T) antiquarks. This would suggest a higher mass
for the ,^quark, which is consistent with the experimental spectrum
of the elementary particles. This case will not be treated in this
paper and therefore we will consider SU and SU as "exact" sym-
metries.
In Section II we study the SU composite group.y
In Section III we consider a relativistic boosting of this, so far,
static symmetry. Assuming that the spin quarks obey a 4-compo-
nent Dirac equation and the spinless quarks a 5-component Kemmer-
Duffin equation.we postulate a U{9)=U(5> 4) non-compact symmetry,
which goes to U(15 , 12) when we include the unitary spin SU,, (T, Y).
This symmetry is broken by the kinetic energy term, at rest redu-
ces to a non-chiral U(9) EJ U (9) and therefore the U(15 , 12) broken
symmetry is a relativistic boosting of such static symmetry.
In Sections IV and V we look to the 3-and 4-point functions.
In Section VI we apply the composite synametry to the non-
leptonic decays of hyperons and to * i ~^ T[+~TT , We consider
also the leptonic decays of baryons and we obtain | o ^ y I': ' *^
in excellent agreement with the experimental value $-
- 2 -
II SUQ "COMPOSITE" SYMMETRY
In the SU model, the basic elements, "quarks", are supposed6 1
to carry baryonic number B = -r , the pseudoscalar and vector
mesons belong to the regular representation of SU and the baryons
and decuplet —• resonances to the 56-dimensional fully-symmetric
representation. The fact that the elementary entities carry bary-
onic number B = — arises naturally if we assume that the quark is
not only an SUQ (T> Y) spinor but is also a spinor in a 3-dimension-
ai unitary space in which the ordinary spin S replaces the isospin
T and the baryonic number B replaces the hyper charge Y. In
such a case, we may consider the basic entities to be spinors in a
njne-dimensional unitary space and,therefore, SU to be the basic
intrinsic group.
The dimensions of some of the representations of this group
are:
[1]
[2,0]
= (scalar)
= 9 (quark)
= 45
[3]
[2,1]
[I3]
= 165
= 240
= 84
[2,17] = 80 (regular)
We obtain for the decomposition of some of the direct products:
80 38 80 = 1 + 80 + 80 + 1215 + 1540 + 1540*+ 1944 (2-1)
165 ® 165* = 1 + 80 + 1944 + 25200 (2-2)
Under
SU3 ( S , B ) S S U 3 (T, Y)
30 = (8, 8) + ( 1 , 8) + (8, 1) . (2-3)
165 = (10,10) + (8, 8) + ( 1 , 1) (2-4)
- 3 -
When we decompose the 80 + 1 representations [ (9, 9) of a
U(9) Sf U(9) group which will be considered in section III] we
obtain two SU (T, Y) nonetai of spinlesg particles. They are6
linear combinations of fermion-antifermion and boson-anti-
boson quarks and as the "relative parity" in the two cases is
different ( - and + ), those states have not defined parity. As
we want states with defined parity, the physical particles are
not going to be eigeristates of SU (If, B) & SU (T f Y) but of6 6SU,, ( S , T . , Y . , . ) H SUQ (X) K U (B) where SU is the group generated
by the spin-carrying quarks and SU (X) is generated by the
spinless quarks.
Under this chain the 80 , 165-representations contain:
TABLE'I
dim. of
80
B
o
1
- 1
dim. of SU_b
35
6
dimS U 3
1
"3
3
.o f(X)
(dim. of
(1+8)1™,
1 I T O J 3
< 1 + 8 ) | +
SUJJP
6 .
(8)0"
]
Name
(A)
(B)
(C)
(1+8) 0 (D)
165
-2
)i+ , 3/2+(l0)
(8 + 10) i +
(10) 0+
(E)
(8+10)1^ (1+8) 0"1! (F)
(G)
(H)
One may assume as usual that the ordinary pseudoscalar
and vector mesons belong to the 35-representation (A) of SU-
and the baryon and the decuplet -^ + resonances to the 56«repre-
sentation (E). One might also accommodate the baryons in the
80-dimensional representation, in such a case, lye ^ 4-
(see below) and the - - decuplet must be accommodated in a
different representation.
On the other hand, if we assume that the mass operator
transforms as a tensor of the regular representation of SUQ, we(4)obtain a sum rule for the baryon-like particles of the regular
representation which is well obeyed by the physical baryons:
~j ~~ ~ K "*"•'" (2-5)- Experimentally
124 MeV = 128 + 5 MeV
One also obtains:
(2-6) i, v\ - -^ ri
Where TT and l< are scalar particles.(5)
This implies
One possible way out of this paradox is to assume that the
baryons are both in the 80 and the 165 representation (Of course,
in assuming that the baryon part of such representations is identi-
cal we have to break SUq symmetry). In order to allow this we
will assume that the spinless quark fy$ is not an elementary
entity but is a composite entity formed by two spin antiquarks:
4 , = £.ji fr/5 fcr £*W w h e r e ^ = 1 ^ 8 j s a SU3{T, Y)j fr/ fcr W
index and i , j take the values 1 , 2 and therefore tpj f', are spin
antiquarks. In doing this SU will be degraded to SU although
some of the restrictions will remain. This kind of broken
- 5 -
symmetries are present when we consider as "building blocks" of
the matter not only the "elementary entities" but also some of
their composite states. (In our case the spinless quark. One
might choose another SU - "broken" group when considering as
building blocks other composite states). If we consider only the
spacial StL :: SU (S,-B) what we have done essentially is to
"transform" the Uo = SUO (S) IS U- (B) group into an StL (S. B)
group. This may be considered as an example of a more general
case.
Let us consider a system where the elementary entities are
U spinors (quarks). Then, in addition to the quantum numbers
of SU , every quark ty : ( fit 1 m2 , .. n) may be considered as
carrying —~r units of a quantum number N and every antiquark
§>[ as carrying ——r units of N. We define
Trf tl — ^ I t i ^ y i•' • T h i i s i s ' invariant under SU but
has the same quantum numbers as the y * spinor of SU ..
(The equivalent of this model in the'old Wigner supermultipletf
theory might be considered as being constituted of nucleifwhei>e
the elementary components are proton, neutron and anti-alpha par-
ticles generating a SU group). Even if the original U , quarkso n
have the same "mass", we expect, in general, the mass of the
composite quark (j) to be different, and therefore that SU.
symmetry will be broken. As the states b - 6. and y y , . .
0 0 . are composed of the same entities, although they
belong to different representations of SU - , one expects the physi-
cal particles to be, in many cases, superpositions of irreducible
representations of SU ... Now let us consider some specific
cases:
a) SU (S, B). If we consider a U group generated by the
ordinary spin £•> and the baryonic number B, then for J 1 s Kf'
we have s'= ~ B = ~ while for f o = W d e t J t- Cj)l S - 02 I 6 6 JZ i
B = - — . This is our case.
- 6 -
b) If we begin with the T = — non-strange quarks f- j f _1 J 1
all having Y = - , then f = -~- det i (j)| is a T = 0
Y = - - entity. It is worth while to note that in an intuitive way-
one would expect m x „ ^ m -i . This agrees with the spectrum
of the elementary particles which seems to be consistent with a
heavier A quark Cf~ quark).
c) One might consider as elementary entities the product
(j . y . i , j = 1 , 2. This defines a U, group that might be more
or less exact. From this we may obtain an StL broken symmetry
containing non-strange particles.
d) Another possibility is the group in which the basic entities
are <J^^ ^ , 6 = 1 , 2 , 3 with fa [ ^ p l belonging to the
SU {T , Y) ['SU,, (sT, B)] broken symmetry group. This defines
an SUQ broken-symmetry group. This very interesting case will
not be treated in this paper. We will consider SUQ (T , Y) to be
more or less exact, SU^ ( S , Y , T ) to be an "exact" symmetryb
and therefore our final symmetry to be SU^ with "some restrictions".o
Until now we have not considered the fact that the spinless and
spin quarks might {and it is a reasonable expectation) have different
statistics. Therefore, we must expect this SUg composite symmetry
to be broken not only due to the composite character of the spinless quark
but also due to the presence of statistics. Although it is unclear
which are the statistics of the quarks, we know that the baryons are
lermions and the mesons are bosons; we will consider the statistics
as something external to SUq symmetry and we will assume that the
prescription is to project from the matrix elements the part consist-
ent with the correct symmetry or anti-symmetry properties
(depending on whether we have mesons or baryons in the ingoing or
outgoing states). In the,same way, we will consider char'ge conju-
gations and parity'as something external and we will assume that
such operation projects from the matrix elements the corresponding
Invariant part.- 7 -
Regular x regular x regular coupling
In general there will be two couplings
where i , 3 , k = 1J^3 °d(3 f = I^3 . ' t he X are the usual Gell-Mann
unitary spin matrices and the T are the corresponding matrices in-* s
SUQ (S , B) space.Explicitly, (2-8) may be written as
+ (g.'+ gn) (mw + vw + mmm + sssL
+ (g. - g_) (mw + vw + mmm+ sss) (2-9)
We see that the D/F ratios are not the ones predicted by
« symmetry and which seem to be. present in the experiments,b
We will assume that the effect of "compositeness" is to project from
these D/F ratios the part compatible with B being identified with
the baryons in the (56, 1) representations of the non-chiral
U(6) !& U(6) *6 ' and S with the scalar bosons in the 3 5+ + 1 part
of the (21, 21) representation of U(6) S U(6) (405+ + 35+ + 1+ of
SU(6)). Then (2-9) will read
^ (g1 + g9) (mw + vw + mmm + sss)
-i g<J ( m w + vw + mmm + sss)_
- 8 -
If g1 = 4-g we conserve in the 3 mesons vertex only the D
part, which is consistent with charge conjugation, We prefer to
assume that charge conjugation projects from the 3-m©eon matrix
elements the D part; in such a case g and g might take arbi-J. Li
trary values. We discuss this in more detail in the relativlstic
version.
HI U(9) (S>U(9) and U(15, 12) BROKEN SYMMETRY
In order to obtain a relativistic boosting we define the algebra
I \ A where ^ i = 1, „. .9 are the 9 generators of
Un (T, Y) and £>. will contain the generators of the relativistic—^ - f\ i 1 r
version of SU (S, B). As the \ part is trivial we will con-centrate on the A part, in the relativistic boosting of SU (S , B).
(7)Similarly to U(6 , 6) we assume:
(1) The spin quark will obey the 4-component equation
X'7i\) T '~^" w n e r e Y- i = 1 # 2 J 3', 4 are the 4 Dirac matriceswith y \ hermitian ^ antihermitian and the metric (-1 , - 1 , - 1 , 1 )
(2) The spinless quark will obey a 5-component Kemmer-
Duffin equation. This is consistent with our hypothesis (see
Section n) of considering the spinless quark as composed of 2 spin
antiquarks. Another way of expressing this is to assume that the
spinless quarks are defined by Y '' ) i , j = 1., 2 , 3 , 4 obeying
the Bargmann-Wigner equation
This equation already shows the "composite" character of the
spinless quark. However, we prefer to use the Kemmer-Duffin
equation and therefore the spinless quark ^ °C = £, . . . , 9 will
obey the equation, / /O , £ . ) ty,t — P
where X. is the 4-velocity. The V.
n l
- 9 —
a r e the 4 x 4 Di rac m a t r i c e s obeying V", Y- + ^• V . = 2 g..
and / ^ a r e the 5 x 5 Kemmer-Duffin m a t r i c e s obeying
< ft ft ftp =In an explicit representation we have:
(3-2)
(3-3)
If
f 0 0 0 0 1 \0 0 0 0 00 0 0 0 00 0 0 0 0
a o o o o
k--
/0 0 0 00 0 0 0 1
= i 0 0 0 0 00 0 0 0 0]
VD 1 0 0 V
r
o o o o o1
0 0 0 0 00 0 0 0 00 0 0 0 10 0 0 1 0
then we may write
a'
The antiquarks will obey the equation
/o o o o o\'o o o o o N
o o^o o i0 0 0 0 0
\o o i o o/
(3-4)
H r'=-(3-5)
(3-6)
Therefore we may consider a formal in variance broken by
the kinetic energy with 9-component spinors as the fundamental
entities, transforming under the non-compact symmetry group
U(5 , 4) which leaves invariant the form
( 3 . 7 )
- 1 0 -
and where the generators may be represented by L \ ^ vt- >iV"«J
In general we will assume that a multispinor 0 . ' ' ' obeys the7 A B C I • *
restrictions:
(3-8)
(3-9)
At reet^eqs. (3-8), (3-9) read
D-
The set of multispinors satisfying (3-10), (3-11) is not invari
ant under the full group U(5 » 4) but under the subgroup containing
such unitary matrices Spe such that
This group is the U(3) BJ U(3) group generated by ( I + £>0 ) T
We have factored out the submatrices acting on the 5,, 6 , 7 indices
as at rest those components are identically zero.
-11-
To consider the SU (T", Y) part, substitute for U(5 > 4),
U(15, 12} and for U(3) K U(3) substitute U(9) x U{9) j 8 ' There-
fore eq. (3-10), (3-11) break the reducible spaces of the non-
compact U(15 > 12) in irreducible spaces invariant under the com-
pact U(9) H U(9), eq. (3-8), (3-9) constitute a relativistic boost-
ing of this U(9) 33 U(9) static symmetry and for a particle with
4-moraentum p the invariant subgroup will be U(9) JS U(9)p
defined by
(3-13)
We apply now the restrictions given by eq. (3-8), (3-9) to second
and third rank spinors.
1) Mixed spinor of rank 2
We may write ;
with
with (bfl 3 9 x 9 matrix and P, B, B and C 4 x 4, 4 x 5,
5 x 4 and 5 x 5 matrices respectively.
When we apply the eq, (3-8), (3-9) we obtain
( 3 . l 6 )
& - -
-12-
the conditions (3-16) to (3-19) imply
J (3-20)
and i = 1, ... 4 0C= 1, . . . 5
. obeys the 4-component Dirac equation P • If 1 :
(3-22)
(3-23)
with
(3-24)
( 3 ' 2 6 )
( 3 - 2 6 )
( 3 - 2 7 )
-13 -
2) Full symmetric spinorARC
When we apply the restriction given by eq. (4-1) we obtain
(3-28)
(3-29)
MO)
' ( 3 " 3 0 )
where C obeys:
" ^C (3-32)
The content of the mixed spinor hg~ is given by the (9 , 9)_ repre
sentation of U(9)'x U(9) (or 80 + 1 of SU )and the content of
-14-
A R r is given by the (165 , 1)+ representation of U(9) x U{9).
The subindex + or - gives the eigenvalues of ^ ,
IV REGULAR-REGULAR-REGULAR VERTEX
If we consider the vertex
(1) + (2)+ (3)->0
:PX + P2 + P3 = 0 (4-1)
where p , p_ and p are the 4-momenta of the particles (1), (2)
and (3) wit
couplings:
and (3) with masses m 1 , m and m * in general there will be two
h ; (4-2)
^ = T y l I •*• (4-3)
-15-
Tr[-B;(l)BWR,
( 4 . 4 )
If we substitute eq. (3-20,23) in'(4-4) we obtain
- 1 6 -
_ ]
(4-6)
where(4-7)
( 4"
Tr
+
(4-9)
- 1 7 -
As we did in the static case we will consider statistics and
discrete symmetries as charge conjugation as external and we will
assume that the prescription is to project from the matrix element
the part with the correct symmetry. . On the other hand we will
consider the spinless quark as composed of two spin antiquaries
(see Section II) and therefore:
If as usual we assume that the quarks "like" to be in a sym-
metrical state we may identify the baryonlike particles with the ones
in the (165,, 1) representation of U(9) J3,U(9) containing the
(56, 1) representation of U(6) E> U(6.) and the scalar particles with
the ones in. the (21, 21) repre sentation of U(6) J3 U(6) : k\c\>)
projecting the D/F part consistent with such identification (see
Section II).
Now, we may write expressions for the different "currents".
We write explicitly the result for the vector and pseudoscalar ones:
-18-
, + meson terms|*F
(4-11)
where
P= f+?1 (4-12)
Q= (>-? (4-13)
(4-14)
and Rrfl and P fn1 are the 4-momentum and mass of the ingoing
and outgoing particle and JA. is the mass of the vector and pseudo-
scalar mesons, '•
In (4-10) we note that in order to obtain the coupling to con-
served vector currents we need g« = 0. This would imply no coup-
ling of the scalar bosons to the baryons which might help explain
why such particles are hard to detect. {
Until now reliable candidates for scalar particles have not
been found around 1200 MeV where we . expect them. Although one
must expect new experimental data a different possibility is to
assume that P, B, B and C represent not particles but densities
("generalised currents") not neccessarily dominated by the onei-
particle state. In the case of .the scalar current C the contribu-
tion of two-particle states might'be more important than the hypo-
thetical one-scalar-particle state of higher mass. In such a case
BBC might represent mainly a BBmm contribution .
In order to compare with the physical decays and coupling
constants we must know what values we must substitute for M ^ '
and /U» . We will assume that the answer must be given in terms(9)
of the operator A (which eigenvalues will be the number of- l
-19-
quarks + the number of antiquarks) and in terms of 4-velocities.
The operator/! will give m = m' =/JL= 2 or m = m ' =3 = 2
depending on whether we consider the composite spinless quark
as fundamental or not. (These values one might obtain considering
mean masses but we believe that the number of quarks + antiquarks
is something more clear).
When we have a 4-velocity, i. e. -*—• we will retain the•p m
physical masses as in this way —s— = 1 at rest. If energies
or momenta are present we will transform them into 4-velocities.
Now, we consider the decay of a vector meson v in two
pseudoscalar mesons m and m' with momenta <a p and p'
respectively. The relevant quantity is
(4-15)
For the factor f i+ ~rr;,i ) we use "central masses"
(eigenvalues of A ) and therefore it takes the value — . For the
other factors we will leave the physical masses (as they are compo
nents of 4-velocities). If we compute in thisway the widths for
the p-*TUT( j K V - * K T H ^ ^ - ^ X ^ K decays we
obtain the values given in Table II i
- 2 0 -
TABLE II
p
Theoretical
112 MeV
52 MeV
2. 7 MeV
Experimental
112 + 4*
50±2
2.6 + 0.6**
i
Theoretical(Usual way)
112*
32
2 ;
* Normalized with the experimental data.
** Assuming a q)»-UJ mixing angle of 38 (consistent with the
observed masses and taking : '•
R- r*(10)
Prom table II we see that in this way we obtain good agree-
ment with the experimental data.
N N m and y m m
The ratio between the gf^
constant is given by
a n d effective coupling
• Tin
(4-16)
If we take m = - M- , we obtain R = 4. 4 in reasonable
agreement with the experimental value R— 5. (If we take m = H
- 2 1 -
we obtain R = 3. 3 which does not agree with the experiment.)
Electromagnetic vertex
If one assumes the electromagnetic interactions proceed
through vecton intermediary states, we obtain the same U(6, 6)
result for the Sachs form factors."
If due to the pole dominance at Q =/A one replaces the
factor in [ ] in (4-17) (4-18) by their residues at u?~ one obtains
F-type
! * U ( D + | F ) . ^
which gives us
As experimentally /G< " " ^ * this value is better
with m = — U- than with m = A>- .
- 2 2 -
T
V FOUR POINTS FUNCTIONS
We will write U(15,12) invariant amplitudes for such ampli-
tudes. Although in doing this we will find all the usual "unitarity
troubles" •" of U(6 , 6), we expect the expression to have sense
for the static and forward scattering limit.
We will consider the process:
(1) + (2)+ (3) + (4)->0(5-1)
Pi + P2 + P3 + P4 = °
where the particles (1), (2), (3) and (4) belong to the regular
representation of U(l 5,12) and P , P , ? and P4 are their
4-momentum vectors, We may write for the relevant matrix,
element:
r
+other4non-
cyclical permutation terms + p , 2 | | c 0 §*(I) | ^ j > | ^ )
"" *" f5-2)+ other 2 terms; where
with YO^ = T r r ) ^ k r (5-4)
and the 0^ and n are invariant amplitudes. When we substitute
the explicit structure of <£fl we obtain
-23-
6
r«) e3
) s & (
-vTrC e;(l^ C(0 £B> «'« -v C ^ ^ A ^
(5-5)
•r-Tr l\\)%\l)
(5-6)
-24-
As in the case of the vertex function we need to take into
account: 1) The composite character of the spinless quarks in
B and C. 2) The presence of statistics. (In doing this we
break the symmetry). We will assume that the prescription is
to project from the matrix element linear combinations compat-
ible with B and C belonging to f &X3n and Y i ^ J of
U(6 , 6) and which obey the correct symmetry (antisymmetry)'
properties in the boson (baryon) case.
From.. (5-2) to (5-6) we obtain:
) ^ with ' "
[pto rV) r.Wfo+C\\\C\D c
-t-*U\34Tf [ r^rVn n^vV) 4. cln) ch\)c\i)
TY
-25-
+|3B [Tr r lt0 r
Jo) r
a [Tr B l(o eff
Tr C l
B [u) & dm
8 me
(5-8)
-26-
As " and C are bosons we have as restrictions:
) (5-9)
(5-10)
The 3^ are symmetrical with respect to interchange of the last two
variables. The fermion character of the baryons imposes
restrictions:
We see that conditions (5-9), (5-10) and (5-11), (5-12) are incompat-
ible, which reflects the fact that the statistics break the symmetry.
We assume that the prescription is to project from (5-7) the sym-
metrical or antisymmetrical part in either case.
Then we obtain
+•
~ 97 -
+ other four terms +
^
BJW 8
- 28 -
#*
TV 6
~
T,
,^3^a) I
ra TV.C ta
TrBVnB
(5-13)
- 2 9 -
From (5-13) we must remark:
1) For the ^ ^ I1 4-point function the answer is the same
as given by U{6, 6). However, the unphysical character of this
amplitude makes comparison with the experimental data difficult.
2) The BBTP amplitude is the same as in U{6 , 6) but— A P?C A* it'
the amplitude B B .i j _ , Pf\ Pg is absent.
3} As in the B B B B amplitude, the experimental data (even
in the static limit) are in contradiction not only with U(6j> 6) but
also with SLL , and SUO is unclear how to compare with
the data.
4) The predictions given b y B P B C and B B C C need the
presence of the scalar particles C, which should have mean masses
around 1200 MeV (as no reliable candidates have been found ).
If C represents not a scalar particle but a scalar density (see
Section IV) which might not be dominated by one-particle states,
but by 2-particle states, then B P B C might be dominated by a 5-
particle amplitude (i. e .BBmmm) and jBBCC by a 6-particle
amplitude (i .e. B B m m m m ).
5) If we neglect the large N-N scattering length (very sens-
itive to the presence of the deuteron and the triplet virtual state,
the scattering lengths for the process m + m -^ m + m, (
the scattering length goes from 0. 2 to 2) B + B -» B + B
( cl^f - ^ 0 - 5 3i7 ~ (f). 5 ) a r e ° ^n e s a m e order of mag-nitude, so perhaps in zero order approximation one might take all
the dLs as equal in absolute value and all the (3- also equal.in
The unitarity troubles "inherent"^11(15 , 12) invariant amplitudes
and the preceding remarks restrict one to non-leptonic decays*
where non-unitarity problems are not as important due to the col-
linearity of the process.
-30-
VI WEAK INTERACTIONS
1) Non-legtonic decays. We may assume that the weak inter-
action hamiltonian transforms as the (9 , !§) representation of
U{9) x U(9) ( Regular of U(l 5 ,12)) . In particular we will assume
that the weak hamiltonian belongs to the central component of the
SU- (S, B) octet in SUO o SUQ (S, B} Ei SUQ (T, Y). This gives us
for the pseudoscalar-scalar mixing the ratio:
Let vis consider the non-leptonic decay of a baryon. B into a
baryon B and a meson m. If we use the spurion analysis, we
have to consider the process
B •-* B' + m + S (P)
p -?• p1 + k
where S(P) is the scalar (pseudoscalar) spurion which will enter in
the parity-conserving (parity-violating) part of the decay.
From eq. (5-13) the relevant part of the matrix element is
where
5 (6-3)
S = I (6-4)
-31-
and m, m1 andyUare the masses of the B, B* and m particles,
(Note that we have included the mass fx in the factor / J 'ffl p\l ft pJ W
in order to express it in terms of 4-velocities (see Section IV).
When we consider the spinless quark as composite and there-
fore we reduce the symmetry from U(15 , 12) to U(6 , 6) (see Section
IV), the net effect is to project the F part of the parity-violating
part of the interaction. Then (6-2) becomes
^
(6-5)
where A is the spurion unitary spin matrix and v* , v are
Dirac spinors. Note that no restrictions have been imposed con-
cerning PC -invariance or the current x current form of the inter-
action (If P C-invariance then a =~a1)
From (6-5) we obtain for the observed decays:
-32-
NUMERICAL VALUES FOR THE
OBSERVED NON-LEPTONIC DECAYS (14)
Theor*
Exp
Theor
Exp
A .
0.4
0.31±0.01
1.6
2.0*0.25
S:0.4
0.41± 0.02
0
-1.4±0.12
-0.23
i)-0.17±0.02
ii) -0.36iO.035
2.8
i)3.6i,0.35
ii)1.7± 0.2
£X0
-0.012± 0.014
1.08 ±0.14
^ -
0.33
0.39±0.015
0
0.34±0.6
• * *
where B. labels the amplitude for the clecay B-^m. + B.i " J I •
We see that the agreement is good except for B ( 3 J ) = 0 .
To take a, =-b is consistent with the experimental data (this
equality suggests that our guessing in Section V about the possible
equality of the oC amplitudes might not be too wild).
It is curious that the S wave term is RFC- invariant and
that we get a more reasonable value for B{£ " ) (i. e. B(3!< ) =
-B ( A - )) if we project from B the RPC-invariant part. How-
ever, in such a case all the parity-conserving terms are scaled
down by a factor -^ . , (R is the inversion in the origin of the
SU plane defined by Gell-Mann and P and C are the parity and
charge conjugation operations).
* with a, =»b and m = m' = - u • If we take m = M. all the values of A must be multiplied
by .3 •
** Normalized with the experimental data.-33-
We may also compute the amplitude for the decay K -> 27T.
If one assumes the current x current forms of the interaction, octet
dominance and C P—invariance, then this implies ^ = -1
C P = 1 for the P spurions and therefore it transforms as the sixth
component A ( CT = + C for the 1, 3, 4, 6 and 8 component
and QJ = - C for the 2, 5 and 7 component) . As K| t rans-
forms as A with C ~ 1 C P = 1, then this implies that SU
symmetry forbids this decay. However, experimentally the K,
life is of the same order of magnitude as the hyperon lives ( TK ~
As in our model we have not assumed the current x current
form of the interaction.we may consider the case when the spurion P
has Cr = 1 C P - 1 ( as is the case for the 0~ mesons ) and there-
fore the K| ->2 7T decay is allowed.
If p is the 4-momentum of K, and fe; L^-TJ the 4-momentum
of TT-t- CTT,J then the relevant quantity is
-34 -
In the last step we have taken KYl - yyU in the trace part
(not in the square root as we want this factor equal to one if the par-
ticles are at rest. See Section IV),
From (6-5) and (6-6) we obtain
d i
As experimentally £ ~-—• — *****
{ 6 . 7 )
we need
j \ «/- nfc / i 4.' C \*- _ n l which is consistent with \£fci
and /£ small. (&Pid|jis consistent with the possible "equality"
of the CL amplitudes in the 4-particles matrix element) .
2) Leptonic decays. If one tries to extend the theory to the lep-
tonic decays, the obvious way to generalize itr is to look in the/
(9 , 9) representation of U(9) IS U(9) for tensors transforming as
vector and axial vector densities. However, although vector par-
ticles are contained in such representation, 1 particles are mis-
sing. Although one might argue that perhaps the axial vector
current transforms as a tensor of a different representation we pre-
fer to assume that in some sense the pseudoscalar and scalar densi-
ties are more fundamental than the vector and axial vector ones and
that the S«A ratio "reflects" the ^*S/<1, ratio.
Let us consider the leptonic decay of a baryon B into -a
baryon B* plus lepton and antilepton.
(6-8)
-35-
where f L ' J ig the 4-momentum vector of the baryon B [ B' ]
and Q is the 4-momentum vector of the lepton-antilepton pair.
If we accept the usual current x current form for these leptort-
ic decays, the relevant matrix element may be written as
- ^ i ^ <e'w/+A/ie>
where Vj< is the vector current and A* is the axial-vector
current. The upper index i indicates the SU« (T, Y) properties
of the currents. If Lorentz invariance is assumed and only first
kind currents are allowed then the most general form for this
matrix element is:
(6-10)
:
where Q ~ ?'- p P" f * + f (6-11)
(6-12)
-36-
If m = m' (exact symmetry) we observe that
(6-15)
-fs (6-16)
fy Off) ty urp) - (6-17)
The right-hand sides of (6-14) and ,{6 -17) transform as the
expectation values of scalar and pseudoscalar densities respectively.
Our model will be to assume that the relative weight of such scalar
and pseudoscalar densities is given by (6-1), which is the same ratio
that enters in the non-Leptonic decays. When we "degrade" SU_ toy
SU symmetry, the final result is that the' "fy. term of the vectorcurrent is multiplied by an extra factor ^Tz, which does not appear
in the usual SU or U(6,6) calcula
momentum transfer (Q H = 0 ) we have
in the usual SU or U(6,6) calculations. Therefore in zero
D4F
-37-
\ V^j -=\&! \ -- \ % >^\ -_ 1.18This gives \ V ^ j \
in excellent agreement with the experimental value
= - 1-18 3: 0.02 (W)
It is worth noting the symmetrical aspect of equations (6-14,15)
and (6-16,17). They may be written as
(6-14')
= 0 (6-l5")
^ O (6-16')
where J L 0$ J is a scalar ( pseudoscalar ) density. In momentum
space
(6-19) K^fi^f^,^ (6-20)
with Pu the 4-momentum operator.
If in unitary space K transforms as ^ ., ., ,., antisym-13k
metrical in upper and lower indices, we are sure of having pure F
vector current in spite of having BSB vertex.The symmetrical aspect of equation (6-14 , 17 ) suggests that we
may talk about "conservation" of A, with % and K ^ switching
roles.
- 3 8 -
VII CONCLUDING REMARKS
We close this paper with a summary of the results and with
a note about some points to be investigated.
1) For the 3-point function in strong and E-M interactions, in
addition to the results of U(6 , 6) one obtains that the conserved
vector coupling implies zero coupling for the BBS vertex, which
might help explain why the scalar mesons have not been clearly
observed.
2) For the 4-point amplitude if one writes U(15,12) invariant
amplitudes (with all the inherent "unitarity troubles") one may-
correlate the B + B -*B + B m + m»* m + m and m + Bs»- m + B
processes. However, as not only U(6., 6) but also SUC and SU
do not agree well with the experiments in the reaction domain it is
not clear how to make this comparison. The fact that, if we ne-
glect the large N-N scattering length, the m + m ^ m + m
m + B-*m + B, B + B-^B + B are of the same order of magnitude,
might imply that the symmetry has sense in zero order approxi-
mation,
3) If we assume that in the non-leptonic decays the weak inter-
action hamiltonian transforms as the central component of the
SU ( s \ B) octet of the (9 , "9) representation of U(9) ® U(9), one
obtains a reasonable fit for the observed non-leptonic decay of
hyperonsexcept by B( c»- K The same parameter gives for the
K , ->> "TT-t-TT partial width a value within 20% of the experimental
one.
4) If one assumes that the same pseudoscalar-scalar ratio
{\ ^ ? — J;§S ) is "reflected" in the axial vector and vector
current in the leptonic decays of baryons one obtains \ fyVj I ~\»\8in excellent agreement with the experimental value $^L -_ i |vi A [
° J V '
-39-
Some points to be investigated.
a) The introduction of statistics as something external is not
very attractive. A different possible model is to consider a "mixed
algebra" where the product for the fermionic densities is given
by anticommutators instead of commutator (although this is not an
algebra in the usual sense of the word.)
b) One might consider the ordinary S£L (T, Y) as a "com-
posite" broken symmetry.
c) Another possible generalization is to consider as composite
building blocks not only ( ^ r £ 0 ^ ^ £p, ^ \2(l) - fyjitt) <Sy2(D*Jbut all thetwoSU,, quark symmetric states.
This would give a U(27) S3 U(27) symmetry containing in its (27, 27)
representation all the particles of the (6 , 6), (56 , 1), ( 1 , 56) and
(21, 2l) representations of U(6) H U(6).'
d) One might also consider a U(27) compact version which
would include as a subgroup the U(12) of GELL-MANN
ACKNOWLEDGMENTS
The author is grateful to Professor Abdus Salam, Professor
Paolo Budini and the IAEA for the hospitality extended to him at the
International Centre for Theoretical PhysicSj Trieste. He would
like to thank Dr. R. White for reading the manuscript.
-40-
REFERENCES AND NOTES
(1) Y. YAMAGUCHI, Phy®. Letters B, 281 (1964).
J. SCHECTER, Y. UEDA and S. OKUBO, preprint, Rochester
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P. CERULUS and J. WEYERS, preprint, University de
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(2) H. J. LIPKIN, Phys. Letters £, 203 (1964).
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H. MIYAZAWA and H. SUGAWARA, Progr. Theoret. Phys. 33!,
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(4) G. COCHO, ICTP preprint, iIC/65/25, Trieste.
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(1965) might be a candidate if its spin-parity results to be 0 .
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34, 263 (1965), have considered a I?(18) as a relativistic general-ization of SU .
y
-41-
(9) H . J . LIPKIN and S. MESHKOV, Phys. Rev. Letters U,
845 (1965).
(10) Data on particles and resonant states, A. H. ROSENFELD et
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(13) J. HAMILTON, P. MENOTTL G. G. OADESandL.J . VICK,
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(16) For the actual calculations one must weigh the AS = 0 relative
to the AS j- 0 transitions in terms of the Cabibbo angle 0 .
2 2(17) Assuming Go . (Q ) does not have a pole in Q . i
(18) C.S.W (unpublished). Quoted by M. GOURDIN, Phys. Rev.
Letters 1JJ, 82 (1965).
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-42-
