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Baryons and leptons and group SU4
I ': 8.B
I I
Nuclear Physics 52 (1964) 342--344; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photopr!nt or microfilm without written permission from the publishe
BARYONS AND LEPTONS AND GROUP SU~ W. K R O L I K O W S K I
Institute for Nuclear Research, Polish Academy of Sciences, Warsaw and Institute of Theoretical Physics, University of Warsaw Received 7 October 1963 Abstract: T h e regular representation o f t h e u n i t a r y unirnodular group in four d i m e n s i o n s SU~ is pointed o u t as a possible f r a m e for a c o m m o n classification o f b a r y o n s a n d leptons.
As is well known, the charge independence of strong interactions can be described by invariance under the unitary unimodular group in two dimensions SU2. Its generators constitute a three-dimensional vector T called isospin. The question whether there is a higher symmetry of strong interactions is essentially equivalent to the question, how the strong interactions depend on the hypercharge Y. Since the group SU2 is of rank one and one extra observableis furnished here by the hypercharge, the group of possible higher symmetry of strong interactions must have rank two. Indeed, it appears likely to many people that in strong interactions there exists a broken invariance under the unitary unimodular group in three dimensions SU3, generated by eight operators, three of them being identified with the isospin and one with the hypercharge 1, 2). It happens that those four of the eight generators are strictly conserved in strong interactions. Among the representations of SU3, the regular representation seems to play a pronounced role. It has eight dimensions and the following isospin structure: D(1, 1) =
D(0), D(1)/.
(1)
D*(½) / This structure enables us to identify the octet of the metastable baryons with a regular representation of SU3, if mixing of elements between different representations of SU 3 , due to the experimentally observed violation of the SU3 symmetry, is negligible. (E.g. D(0) ~D(1, 1) can a priori mix with D(0) -- D(0, 0) and D*(½) ~ D(1, 1) with D*(½) e D(3, 0). It would mean that there is a mixing of "bare" A with "bare" Y~ (1405) and "bare" N with "bare" ~*(1530); it may be prohibited however by conservation of spin and parity.) Mixing of elements inside representations of SU3 caused by strong interactions is forbidden by conservation of isospin and hypercharge. The aim of the present note is to point out that there is a possibility of common classification of baryons and leptons by means of a regular representation of the unitary unimodular group in four dimensions SU4. 342
343
BARYONS AND LEPTONS
First of all let us notice that to distinguish baryons and leptons from each other it is enough to consider one extra observable, e.g. the baryonic number or the leptonic number. Hence a group of rank three is sufficient to classify baryons and leptons, if it happens that all interactions of baryons and leptons do not mix too strongly between and inside representations of this group. Such a group cannot be, of course, a group of symmetry of interactions, as strong interactions display a strong asymmetry between baryons and leptons. Let us consider the group SU4. It has fifteen generators. Eight of them we shall identify with generators of SU3 and one with an observable which takes the values _+ 1 for leptons and 0 for baryons. Let us call this observable the leptonic charge Z. Six other generators of S.U4 change the leptonic charge by + 1 and the hypercharge by _+ 1 or 0. The regular representation of SU4 has fifteen dimensions and the following structure in terms of multiplets of SU3 : [ D(1, 0) n(1, 0, 1) = | D ( 0 , 0), n(1, 1 ) ) , \ D(0, 1)
(2)
where D(1, 0) and D(0, 1) are the triplets Z = + 1 and Z = - l, D(0, 0) is a singlet Z = 0 and D(1, l) an octet Z = 0 with respect to SU3. Here D(1, 1) would represent the octet of metastabl¢ baryons N, A, ,~, N and D(0, 0) might be the isobar Y~(1405) 3), whereas D(1, 0) and D(0, l) would denote two lepton triplets. Indeed, in the experiment we can distinguish six leptons, say, e-,
½(1 ---Vs)vc,
#+, ½(l___ys)v.
(3)
(and their antiparticles), which can be tentatively grouped into two triplets, D(1, 0) (Z = + 1) and O(0, 1)(Z = - 1). It is interesting to remember that in the representations D(1, 0) and D(0, 1) the hypercharge Y(defined by one of the generators of SU3 in the same way as for D(1, 1)) is not given by integers; namely, for D(1, 0) we have 1) Y
=
(!i :)
and T 3
=
-~
(! 00) -½0
.
(4)
oo
Hence the Gell-Mann-Nishijima charge formula must be reformulated in order to apply both for baryons and leptons. E.g. the formula O = r3+½Y+~z
(5)
reduces to the mentioned formula for D(1, 1)(Z = 0) and gives
[lo0
Q=/ooo! \0 0 0/
(6)
344
W. KR6LIKOWSKI
for D(I, 0)(Z = + 1). In this case leptons with charges + 1, 0, 0 would form the triplet D(1, 0). Then the triplet D(0, 1) would consist of leptons of charges 0, - 1, 0. In order to have the experimentally observed conservation of baryonic number we must assume the conservation of Z and conservation of all fermions, the latter provided by overall gauge invariance for spinor particles. The product of the group SU 3 and the gauge group forms the full group of unitary transformations in four dimensions U#. The conservation of Z forbids the mixing between D(1, 0) and D(0, 1) as well as between both of them from one side and D(0, 0) and D(1, 1)(and further baryon representations of SU3) qfrom the other side. Mixing inside D(1, 0) and D(0, 1) should be excluded by proper ordering of six leptons into two triplets, D(1, 0) and D(0, 1), and by a proper form of lepton interaction. Then we could say that mixing between and inside representations of SU# is negligible, if only (i) mixing effects for baryon representations of SU 3 are negligible and (ii) all existing leptons belong to the representation D(1, 0, 1). It is known that the Lie algebra of the group SU 4 can be represented by fifteen traceless Dirac matrices 4). Since they are 4 x 4 matrices, they give a representation of this algebra generating the fundamental representations D(1, 0, 0) and D(0, 0, 1) of SU 4. References 1) 2) 3) 4)
M. Gell-Mann, Phys. Rev. 125 (1962) 1067 Y. Ne'eman, Nuclear Physics 26 (1961) 222 S. L. Glashow and A. H. Rosenfeld, Phys. Rev. Lett. 10 (1963) 192 R. E. Behrends, J. Dreitlein, C. Fronsdal and B. W. Lee, Revs. Mod. Phys. 34 (1962) 1
I I
Nuclear Physics 52 (1964) 342--344; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photopr!nt or microfilm without written permission from the publishe
BARYONS AND LEPTONS AND GROUP SU~ W. K R O L I K O W S K I
Institute for Nuclear Research, Polish Academy of Sciences, Warsaw and Institute of Theoretical Physics, University of Warsaw Received 7 October 1963 Abstract: T h e regular representation o f t h e u n i t a r y unirnodular group in four d i m e n s i o n s SU~ is pointed o u t as a possible f r a m e for a c o m m o n classification o f b a r y o n s a n d leptons.
As is well known, the charge independence of strong interactions can be described by invariance under the unitary unimodular group in two dimensions SU2. Its generators constitute a three-dimensional vector T called isospin. The question whether there is a higher symmetry of strong interactions is essentially equivalent to the question, how the strong interactions depend on the hypercharge Y. Since the group SU2 is of rank one and one extra observableis furnished here by the hypercharge, the group of possible higher symmetry of strong interactions must have rank two. Indeed, it appears likely to many people that in strong interactions there exists a broken invariance under the unitary unimodular group in three dimensions SU3, generated by eight operators, three of them being identified with the isospin and one with the hypercharge 1, 2). It happens that those four of the eight generators are strictly conserved in strong interactions. Among the representations of SU3, the regular representation seems to play a pronounced role. It has eight dimensions and the following isospin structure: D(1, 1) =
D(0), D(1)/.
(1)
D*(½) / This structure enables us to identify the octet of the metastable baryons with a regular representation of SU3, if mixing of elements between different representations of SU 3 , due to the experimentally observed violation of the SU3 symmetry, is negligible. (E.g. D(0) ~D(1, 1) can a priori mix with D(0) -- D(0, 0) and D*(½) ~ D(1, 1) with D*(½) e D(3, 0). It would mean that there is a mixing of "bare" A with "bare" Y~ (1405) and "bare" N with "bare" ~*(1530); it may be prohibited however by conservation of spin and parity.) Mixing of elements inside representations of SU3 caused by strong interactions is forbidden by conservation of isospin and hypercharge. The aim of the present note is to point out that there is a possibility of common classification of baryons and leptons by means of a regular representation of the unitary unimodular group in four dimensions SU4. 342
343
BARYONS AND LEPTONS
First of all let us notice that to distinguish baryons and leptons from each other it is enough to consider one extra observable, e.g. the baryonic number or the leptonic number. Hence a group of rank three is sufficient to classify baryons and leptons, if it happens that all interactions of baryons and leptons do not mix too strongly between and inside representations of this group. Such a group cannot be, of course, a group of symmetry of interactions, as strong interactions display a strong asymmetry between baryons and leptons. Let us consider the group SU4. It has fifteen generators. Eight of them we shall identify with generators of SU3 and one with an observable which takes the values _+ 1 for leptons and 0 for baryons. Let us call this observable the leptonic charge Z. Six other generators of S.U4 change the leptonic charge by + 1 and the hypercharge by _+ 1 or 0. The regular representation of SU4 has fifteen dimensions and the following structure in terms of multiplets of SU3 : [ D(1, 0) n(1, 0, 1) = | D ( 0 , 0), n(1, 1 ) ) , \ D(0, 1)
(2)
where D(1, 0) and D(0, 1) are the triplets Z = + 1 and Z = - l, D(0, 0) is a singlet Z = 0 and D(1, l) an octet Z = 0 with respect to SU3. Here D(1, 1) would represent the octet of metastabl¢ baryons N, A, ,~, N and D(0, 0) might be the isobar Y~(1405) 3), whereas D(1, 0) and D(0, l) would denote two lepton triplets. Indeed, in the experiment we can distinguish six leptons, say, e-,
½(1 ---Vs)vc,
#+, ½(l___ys)v.
(3)
(and their antiparticles), which can be tentatively grouped into two triplets, D(1, 0) (Z = + 1) and O(0, 1)(Z = - 1). It is interesting to remember that in the representations D(1, 0) and D(0, 1) the hypercharge Y(defined by one of the generators of SU3 in the same way as for D(1, 1)) is not given by integers; namely, for D(1, 0) we have 1) Y
=
(!i :)
and T 3
=
-~
(! 00) -½0
.
(4)
oo
Hence the Gell-Mann-Nishijima charge formula must be reformulated in order to apply both for baryons and leptons. E.g. the formula O = r3+½Y+~z
(5)
reduces to the mentioned formula for D(1, 1)(Z = 0) and gives
[lo0
Q=/ooo! \0 0 0/
(6)
344
W. KR6LIKOWSKI
for D(I, 0)(Z = + 1). In this case leptons with charges + 1, 0, 0 would form the triplet D(1, 0). Then the triplet D(0, 1) would consist of leptons of charges 0, - 1, 0. In order to have the experimentally observed conservation of baryonic number we must assume the conservation of Z and conservation of all fermions, the latter provided by overall gauge invariance for spinor particles. The product of the group SU 3 and the gauge group forms the full group of unitary transformations in four dimensions U#. The conservation of Z forbids the mixing between D(1, 0) and D(0, 1) as well as between both of them from one side and D(0, 0) and D(1, 1)(and further baryon representations of SU3) qfrom the other side. Mixing inside D(1, 0) and D(0, 1) should be excluded by proper ordering of six leptons into two triplets, D(1, 0) and D(0, 1), and by a proper form of lepton interaction. Then we could say that mixing between and inside representations of SU# is negligible, if only (i) mixing effects for baryon representations of SU 3 are negligible and (ii) all existing leptons belong to the representation D(1, 0, 1). It is known that the Lie algebra of the group SU 4 can be represented by fifteen traceless Dirac matrices 4). Since they are 4 x 4 matrices, they give a representation of this algebra generating the fundamental representations D(1, 0, 0) and D(0, 0, 1) of SU 4. References 1) 2) 3) 4)
M. Gell-Mann, Phys. Rev. 125 (1962) 1067 Y. Ne'eman, Nuclear Physics 26 (1961) 222 S. L. Glashow and A. H. Rosenfeld, Phys. Rev. Lett. 10 (1963) 192 R. E. Behrends, J. Dreitlein, C. Fronsdal and B. W. Lee, Revs. Mod. Phys. 34 (1962) 1