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A new symmetry principle in particle physics trio invariance
PHYSICS
Volume 9, number 3
A NEW
SYMMETRY
15 April 1964
LETTERS
PRINCIPLE IN PARTICLE TRIO INVARIANCE
PHYSICS
Y. YAMAGUCHI Institute for Nuclear Study, University of Tokyo Tanashi-machi, Kitatama-gun, Tokyo, Japan Received 12 March 1964
In this letter we shall present a systematic formulation of the internal symmetry properties among the strongly interacting particles based on the simplest type of finite groups : the permutation group of order 3. The existence of three attributes (baryon number N, strangeness S and charge Q ) naturally leads to the existence.of three fundamental objects (fields)(which are Dirac spinors in the ordinary space-time) I). If we attribute them N, Q, S appropriately, we have the Sakata model 2). However, once we give different attributes to three fields, we would wonder why three fields must be distinct and should dig into a subatomism or internal structures of these fields. Thus various strata of atomisms would follow again and again, apparently no end of a story and sharply contradict with the atomism 3). In this respect the Sakata model may be regarded as a stage of chemistry from the atomistic point of view. Towards the ultimate atomistic theory, we remark that the internal symmetries exhibited in the particle world must be discussed solely in terms sf discrete operations, namely by finite groups rather than contineous Lie-groups, in which infinitesimal operations play an essential role. All finite groups are equivalent to symmetry (or permutation) groups (symmetry group of order n will be quoted as Sa) or their subgroups** 6). Another guiding principle is of course simplicity. SI and S2 are too simple, whereas S3 is rich enough as will be seen below. Therefore, we propose to construct our atomistic theory in particle physics upon S3-symmetry: We accept existence of three &+imitive objects, * In this connection see an earlier attempt by Case et
al. 4), who, however, not discussed S3. ** It is the author’s opinion that justification of the use of some contineous groups has tc be made a posteriori in the theory where only discrete operations are used in the premise. As will be seen later, the SU2 for the isospin is of this category.
the chaos field x, y, z, but would not admit any a priori distinction among them. This principle shall be referred to the trioinvariance (invariance with respect to S3). In other words, our theory must be invariant under all (6) permutations among chaos fields t. Notice the marked difference between our and the conventionally accepted philosophies. We postulate that each of three chaos-fields carries baryon number N = 1 (or 5 depending on further development), and the theory should be N-conserving. As is well-known chaos-fields are reducible with respect to the permutation group. Irreducible sets for chaos-fields can easily be found: we have the singlet A and the doublet p, n which span the bases of 1 and 2 dimensional irreducible representations of the permutation group S3: A
=$
(x+y+z);
P =$(-y+z)
n =-&(2x-y-z); 2
(1)
A, n and p may be called the basic Sskata fields. Our postulate of baryon number conservation guarantees that the sum of A-, p- and n-numbers must be conserved in any reactions. The antiSakata-fields n, fr, ii behave like A, p, n under per mutations. We now discuss a two-body system, i.e., the reduction of the product representations (1+ 2) X (1 + 2). The trio-symmetry leads to the following irreducible bases for a two body system of Sakata-and anti-Sakata particles : 1. dimensional representations
XII,
pp +iin *j2’
pn-iip -JZ--*
(2.1)
t In this connection see ref. 2, in which the author constructed SU3 as a product of SU2 for isospina and the permutation n-h. In this paper the permutational invariance is Nly extended for all (three) primitive chaos fields.
281
Volume 9, number
PHYSICS LETTERS -
3
2. dimensional
The trio-invariant given by
Hamiltonian of Fermi type is
H =~~(AA)~+b(AA))(jip+fin)+[b’(AA)@n-iip)+h.c.]
+ c~p+lin)2+c’($p+iin)~n-iip)+&in-iip)2 +f[@p)@A) + [f’{(&)2
+ (Mfill)] (3)
+ (%n)2)+ h.c.]
+ g[ (pp - nn)2 + (fin - iip)2] + [h{(&)@p-lin)+fip)@n+tlp)}+h.c.]
,
where the baryon number conservation (or, in brief, N-conservation) holds. At this stage we shall make the following assignment : p has unit electric charge Qp = +l while A and n are electrically neutral &A = Qn = 0. We impose that the number of p’s should be conserved *, charge conservation, or, in brief, Q-conservation. Notice that to count integers is a perfectly sensible operation in our atomism. Then we have to set in the interaction (3) : b’ =c’=f
=h ~0
d = -g
1
(4)
and obtain the following interaction : H = a(AA)2 + b(AA)(pp+iin) + ~@p+iin)~
+A(AP)GA)
+ &r)(nA))
(5)
+ g{ @p - iin) + 2(ETn) (iip) + 2(1ip)tin)} . Now we associate A, p and n fields with particles indicated by these letters. If (5) is the basic in-
* It is clear that the n-number conservation is equally good, showingthe symmetry betweenp and n.
15 April 1964
teraction, we can clearly assign the strangeness S and the isospin for A, p, n in usual way. The interaction (5) is precisely the basic interaction of Fermi type for the Sakata model, which is charge independent, S - and N-conserving, but not unitary symmetric. We now reached the theorem in our trio-formalism, namely in the N-conserving trio-invariant theory, the Q-conservation leads to the charge-independent (rather than unitary) and S conserving Sakata theory provided that basic interactions are of Fermi type. This sort of situation is found in any finite group: If multiplets of particles are assigned to irreducible representations of a finite group under which interactions are invariant, the invariance requirement for this group and the charge conservation law are not always compatible. The compatibility means that in effect the symmetry is enlarged with a corresponding enlargement of the irreducible representations. In this way the charge independence (or chargesymmetry, etc., depending on a group choice) 4) is restored. Jn conventional approaches, one tries to obtain a weakly broken (i.e., charge-independent and S-conserving) world of particles, starting from the unitary symmetric Sakata theory or its octet version. Thereby a necessary asymmetry is either introduced as a small perturbation (socalled T&type) in an ad hoc way, or expected to be derived in a self-consistent bootstrap mechanism. Attempts along the second line have been made often, and lead to, e.g., the symmetric as well as asymmetric worlds as possible solutions. However, there seem not to exist so far any physical raison-d’etre why Nature prefers an asymmetric (charge independence) rather than symmetric world (unitary symmetry). On the contrary, in our scheme it is quite natural that the resulting particle world is in general asymmetric though the primitive chaotic world is quite symmetric. The so-called broken symmetry problem has been solved in principle.
Volume 9, number 3
PHYSICS
LETTERS
Thus we gave the basis of strong interactions, from which the original Sakata theory or its octet version a la Gell-Mann 8) can be developed. We briefly mention other interactions. We have two invariance principles : The trio-invariance and Q-conservation (besides the N-conservation which we shall not give up 2,7)). Strong interactions obey these symmetries. So it is natural to suppose that weaker interactions would violate either one or both of two symmetries. First, violate the trio-invariance, while keeping the Q-Conservation. Then we shall have a Qconserving usual weak interactions. It is interesting to note that weak interactions violate simultaneously symmetries in the internal space (trioinvariance) and in the Minkowsky s ace (parity). We notice that the Cabibbo theory ERof leptonic weak interactions is naturally incorporated in our scheme. Another kind of possibility is offered by keeping the trio-invariance but relaxing the Q-conservation law. Some of such possibilities (super weak interactions) have already been remarked 2) (also see ref. 7). Finally, we simply make an announcement on further development already achieved. The triotheory can be built up on the two-valued ((‘spinor”) representation ‘4 5). Then the ‘baryon-lepton sym-
15 April 1964
metry” 2, g, and an approximate unitary symmetric world of particles can naturally be formulated. Weak interactions can be constructed based on a larger finite group (connected with symmetrics possessed by the tetrahedron). Full account will be published in the Progress of Theoretical Physics. The author is grateful to Dr. A. Fujii for his kind advices how to present this work.
References 1) W.Thirring, Nuclear Phys. 10 (1959) 9’7. 2) Y . Yamaguchi, Suppl. Progr . Theoret.Phys. No. 11 (1959) 1 and 37. 3) P.A. MDirac, The Principles of Quantum Mechanics, (Cxford University Press, Third edition, 1947) Ch. I, in particular p. 34. 4) K.M.Case, R.Karplus and C.N.Yang, Phys.Rev. 101 (1956) 874. 5) L.D 1Landau and E. M. Lifshitz, Quantum Mechanics, translated by J. B. Sykes and J. S. Bell (Pergamon Press, London-Paris, 1958) Ch. KII. 6) M. Gell-Mann, Physics Letters, to be published. 7) Y. Yamaguchi, Prog. Theoret. Phys. 22 (1959) 373. 8) N. Cabibbo, Phys.Rev. Letters 10 (1963) 531. 9) A. Gamba, R. E. Marshak and S. Okubo, Proc. Nat. Acad. Sci. 45 (1959) 881.
*****
ON THE ANALYTIC PROPERTIES AMPLITUDE IN THE COMPLEX
OF THE PION PION SCATTERING ANGULAR MOMENTUM PLANE
J. XWIECINSXI * and P. SURANYI ** Joint Institute for Nuclear Research Laboratory of Theoretical Physics Received 2 March 1964
In the last years one of the central problems of high energy physics has been the existence of Regge poles in field theories. This problem was investigated using the Bethe-Salpeter equation with simple kernels. In the case of the scattering of two scalar particles it was shown 1) in the so called ladder approximation, that the high energy * On leave from the Institute of Nuclear Physics, Krakow, Poland. ** On leave from the Central Institute for Physics of the Hungarian Academy of Sciences, Budapest.
behaviour of the amplitude in the crossed channel is determined by one Regge pole. On the other hand for the scattering of pseudoscalar particles the bubble exchange approximation gives for the rightmost singularity in the angular momentum plane an energy independent branch point 2). The appearance of the cut is closely connected with the behaviour of the kernel of the integral equation for high momentum transfers. For the pseudoscalar theory the bubble exchange diagram does not give a Fredholm type kernel in contrast to the exchange of a simple pole. However, in principle, 283
Volume 9, number 3
A NEW
SYMMETRY
15 April 1964
LETTERS
PRINCIPLE IN PARTICLE TRIO INVARIANCE
PHYSICS
Y. YAMAGUCHI Institute for Nuclear Study, University of Tokyo Tanashi-machi, Kitatama-gun, Tokyo, Japan Received 12 March 1964
In this letter we shall present a systematic formulation of the internal symmetry properties among the strongly interacting particles based on the simplest type of finite groups : the permutation group of order 3. The existence of three attributes (baryon number N, strangeness S and charge Q ) naturally leads to the existence.of three fundamental objects (fields)(which are Dirac spinors in the ordinary space-time) I). If we attribute them N, Q, S appropriately, we have the Sakata model 2). However, once we give different attributes to three fields, we would wonder why three fields must be distinct and should dig into a subatomism or internal structures of these fields. Thus various strata of atomisms would follow again and again, apparently no end of a story and sharply contradict with the atomism 3). In this respect the Sakata model may be regarded as a stage of chemistry from the atomistic point of view. Towards the ultimate atomistic theory, we remark that the internal symmetries exhibited in the particle world must be discussed solely in terms sf discrete operations, namely by finite groups rather than contineous Lie-groups, in which infinitesimal operations play an essential role. All finite groups are equivalent to symmetry (or permutation) groups (symmetry group of order n will be quoted as Sa) or their subgroups** 6). Another guiding principle is of course simplicity. SI and S2 are too simple, whereas S3 is rich enough as will be seen below. Therefore, we propose to construct our atomistic theory in particle physics upon S3-symmetry: We accept existence of three &+imitive objects, * In this connection see an earlier attempt by Case et
al. 4), who, however, not discussed S3. ** It is the author’s opinion that justification of the use of some contineous groups has tc be made a posteriori in the theory where only discrete operations are used in the premise. As will be seen later, the SU2 for the isospin is of this category.
the chaos field x, y, z, but would not admit any a priori distinction among them. This principle shall be referred to the trioinvariance (invariance with respect to S3). In other words, our theory must be invariant under all (6) permutations among chaos fields t. Notice the marked difference between our and the conventionally accepted philosophies. We postulate that each of three chaos-fields carries baryon number N = 1 (or 5 depending on further development), and the theory should be N-conserving. As is well-known chaos-fields are reducible with respect to the permutation group. Irreducible sets for chaos-fields can easily be found: we have the singlet A and the doublet p, n which span the bases of 1 and 2 dimensional irreducible representations of the permutation group S3: A
=$
(x+y+z);
P =$(-y+z)
n =-&(2x-y-z); 2
(1)
A, n and p may be called the basic Sskata fields. Our postulate of baryon number conservation guarantees that the sum of A-, p- and n-numbers must be conserved in any reactions. The antiSakata-fields n, fr, ii behave like A, p, n under per mutations. We now discuss a two-body system, i.e., the reduction of the product representations (1+ 2) X (1 + 2). The trio-symmetry leads to the following irreducible bases for a two body system of Sakata-and anti-Sakata particles : 1. dimensional representations
XII,
pp +iin *j2’
pn-iip -JZ--*
(2.1)
t In this connection see ref. 2, in which the author constructed SU3 as a product of SU2 for isospina and the permutation n-h. In this paper the permutational invariance is Nly extended for all (three) primitive chaos fields.
281
Volume 9, number
PHYSICS LETTERS -
3
2. dimensional
The trio-invariant given by
Hamiltonian of Fermi type is
H =~~(AA)~+b(AA))(jip+fin)+[b’(AA)@n-iip)+h.c.]
+ c~p+lin)2+c’($p+iin)~n-iip)+&in-iip)2 +f[@p)@A) + [f’{(&)2
+ (Mfill)] (3)
+ (%n)2)+ h.c.]
+ g[ (pp - nn)2 + (fin - iip)2] + [h{(&)@p-lin)+fip)@n+tlp)}+h.c.]
,
where the baryon number conservation (or, in brief, N-conservation) holds. At this stage we shall make the following assignment : p has unit electric charge Qp = +l while A and n are electrically neutral &A = Qn = 0. We impose that the number of p’s should be conserved *, charge conservation, or, in brief, Q-conservation. Notice that to count integers is a perfectly sensible operation in our atomism. Then we have to set in the interaction (3) : b’ =c’=f
=h ~0
d = -g
1
(4)
and obtain the following interaction : H = a(AA)2 + b(AA)(pp+iin) + ~@p+iin)~
+A(AP)GA)
+ &r)(nA))
(5)
+ g{ @p - iin) + 2(ETn) (iip) + 2(1ip)tin)} . Now we associate A, p and n fields with particles indicated by these letters. If (5) is the basic in-
* It is clear that the n-number conservation is equally good, showingthe symmetry betweenp and n.
15 April 1964
teraction, we can clearly assign the strangeness S and the isospin for A, p, n in usual way. The interaction (5) is precisely the basic interaction of Fermi type for the Sakata model, which is charge independent, S - and N-conserving, but not unitary symmetric. We now reached the theorem in our trio-formalism, namely in the N-conserving trio-invariant theory, the Q-conservation leads to the charge-independent (rather than unitary) and S conserving Sakata theory provided that basic interactions are of Fermi type. This sort of situation is found in any finite group: If multiplets of particles are assigned to irreducible representations of a finite group under which interactions are invariant, the invariance requirement for this group and the charge conservation law are not always compatible. The compatibility means that in effect the symmetry is enlarged with a corresponding enlargement of the irreducible representations. In this way the charge independence (or chargesymmetry, etc., depending on a group choice) 4) is restored. Jn conventional approaches, one tries to obtain a weakly broken (i.e., charge-independent and S-conserving) world of particles, starting from the unitary symmetric Sakata theory or its octet version. Thereby a necessary asymmetry is either introduced as a small perturbation (socalled T&type) in an ad hoc way, or expected to be derived in a self-consistent bootstrap mechanism. Attempts along the second line have been made often, and lead to, e.g., the symmetric as well as asymmetric worlds as possible solutions. However, there seem not to exist so far any physical raison-d’etre why Nature prefers an asymmetric (charge independence) rather than symmetric world (unitary symmetry). On the contrary, in our scheme it is quite natural that the resulting particle world is in general asymmetric though the primitive chaotic world is quite symmetric. The so-called broken symmetry problem has been solved in principle.
Volume 9, number 3
PHYSICS
LETTERS
Thus we gave the basis of strong interactions, from which the original Sakata theory or its octet version a la Gell-Mann 8) can be developed. We briefly mention other interactions. We have two invariance principles : The trio-invariance and Q-conservation (besides the N-conservation which we shall not give up 2,7)). Strong interactions obey these symmetries. So it is natural to suppose that weaker interactions would violate either one or both of two symmetries. First, violate the trio-invariance, while keeping the Q-Conservation. Then we shall have a Qconserving usual weak interactions. It is interesting to note that weak interactions violate simultaneously symmetries in the internal space (trioinvariance) and in the Minkowsky s ace (parity). We notice that the Cabibbo theory ERof leptonic weak interactions is naturally incorporated in our scheme. Another kind of possibility is offered by keeping the trio-invariance but relaxing the Q-conservation law. Some of such possibilities (super weak interactions) have already been remarked 2) (also see ref. 7). Finally, we simply make an announcement on further development already achieved. The triotheory can be built up on the two-valued ((‘spinor”) representation ‘4 5). Then the ‘baryon-lepton sym-
15 April 1964
metry” 2, g, and an approximate unitary symmetric world of particles can naturally be formulated. Weak interactions can be constructed based on a larger finite group (connected with symmetrics possessed by the tetrahedron). Full account will be published in the Progress of Theoretical Physics. The author is grateful to Dr. A. Fujii for his kind advices how to present this work.
References 1) W.Thirring, Nuclear Phys. 10 (1959) 9’7. 2) Y . Yamaguchi, Suppl. Progr . Theoret.Phys. No. 11 (1959) 1 and 37. 3) P.A. MDirac, The Principles of Quantum Mechanics, (Cxford University Press, Third edition, 1947) Ch. I, in particular p. 34. 4) K.M.Case, R.Karplus and C.N.Yang, Phys.Rev. 101 (1956) 874. 5) L.D 1Landau and E. M. Lifshitz, Quantum Mechanics, translated by J. B. Sykes and J. S. Bell (Pergamon Press, London-Paris, 1958) Ch. KII. 6) M. Gell-Mann, Physics Letters, to be published. 7) Y. Yamaguchi, Prog. Theoret. Phys. 22 (1959) 373. 8) N. Cabibbo, Phys.Rev. Letters 10 (1963) 531. 9) A. Gamba, R. E. Marshak and S. Okubo, Proc. Nat. Acad. Sci. 45 (1959) 881.
*****
ON THE ANALYTIC PROPERTIES AMPLITUDE IN THE COMPLEX
OF THE PION PION SCATTERING ANGULAR MOMENTUM PLANE
J. XWIECINSXI * and P. SURANYI ** Joint Institute for Nuclear Research Laboratory of Theoretical Physics Received 2 March 1964
In the last years one of the central problems of high energy physics has been the existence of Regge poles in field theories. This problem was investigated using the Bethe-Salpeter equation with simple kernels. In the case of the scattering of two scalar particles it was shown 1) in the so called ladder approximation, that the high energy * On leave from the Institute of Nuclear Physics, Krakow, Poland. ** On leave from the Central Institute for Physics of the Hungarian Academy of Sciences, Budapest.
behaviour of the amplitude in the crossed channel is determined by one Regge pole. On the other hand for the scattering of pseudoscalar particles the bubble exchange approximation gives for the rightmost singularity in the angular momentum plane an energy independent branch point 2). The appearance of the cut is closely connected with the behaviour of the kernel of the integral equation for high momentum transfers. For the pseudoscalar theory the bubble exchange diagram does not give a Fredholm type kernel in contrast to the exchange of a simple pole. However, in principle, 283