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Comprehensive sum rules for s-wave amplitudes of nonleptonic hyperon decays
Volume 83B, number 2
PHYSICS LETTERS
7 May 1979
COMPREHENSIVE SUM RULES FOR S-WAVE AMPLITUDES OF NONLEPTONIC HYPERON DECAYS
M. KATUYA Laboratory of Phystcs, Shtzuoka Women's Universtty, Yada 409, Shtzuoka 422, Japan
Received 15 November 1978
Sum rules for the S-wave amplitudes of the nonleptonic hyperon decays are discussed. A new sum rule X/~(Ag + o ) = ( z : - £+) + is proposed as one of the comprehensive sum rules in the SU(4) symmetry theory.
One of the most important discoveries brought by extending the SU(3) symmetry theory to the SU(4) symmetry theory is that of the exact relation [ 1,2] for the parity-violating amplitudes of the nonleptonic hyperon decays * i, A ° . z;:
=
(1)
This exact relation is derived * 2 by assuming the CPmvariant hamlltonian to transform like the 20-dimensional SU(4) representation whmh is expected to be enhanced by the study of the short-distance behavlour of the product of weak currents [5]. Although there ts a rather large (about 40%) discrepancy between the ratios (1) and the experimental ratios [3], A 0 . £ 0 : .~= (1.44 -+ 0.01) : - V ~ × (0.84 -+o.o3): (2.00 -+o.o2), this discrepancy seems not so serious since the SU(4) symmetry is rather strongly broken. It is, therefore, quite reasonable to regard the relation (1) as basic for the S-wave amplitudes of the nonleptonic hyperon decays. It is useful to represent the relation (1) equivalently in terms o f two sum rules as follows: --
+
a °_ - 2 = - - =
(2)
3 A 0- - 3"-- = x/c3Z;.
(3)
The former is the famihar Lee-Sugawara sum rule [6,7] ,1 We use the same conventions for the amphtudes as m ref [3] except for a change of sign in ~o4:2 Relation (1) had already been derived m another context of SU(3) symmetry, see ref. [4]. +
[6,7] which was found in the SU(3) symmetry theory originally by assuming the CP-invariant hamlltonian to transform like an octet, X6. On the other hand, the latter is an unfamiliar sum rule, which does not remain valid in the SU(3) symmetry hmit. In fact, the conslstency of the Lee-Sugawara sum rule with experiment [3] is quite satisfactory, however, that of the latter sum rule (3) is not. For the sake of the subsequent discussion, we rewrite the sum rule (3) as ~ / ~ (A~ + 20) = (2_- - ~ ; ) ,
(4)
by using the A I = 1/2 relations for the A, N and 2 decay amplitudes , t . It is worthwhile to notice that the sum rule (4) is found to be almost consistent with experiment, although the sum rule (3) is found to be violated to approximately 50%. This fact means that there exists a considerable amount of deviation from the A I = 1/2 rule. The purpose of this letter is to insist that that sum rule (4) is indeed theoretically more comprehensive than the Lee-Sugawara sum rule (2) and the unfamiliar sum rule (3) in the SU(4) symmetry theory, in the sense that the sum rule (4) remains valid even in presence of a suppressed 84-dimensional representation ( A I = 1/2 and 3/2, AS = 1) part arising from the product of weak currents. The derivation seems obviously trivial, therefore, we only give an outline of it below. Our discussion is based on the following assumptions" (a) The strong interactions are fully invariant under SU(4). (b) The weak interactions are of the product of the 227
Volume 83B, number 2
PHYSICS LETTERS
weak charged currents [5] and are CP-invariant. Here we take into account the effect of the symmetry breaking perturbationally. That is, the most basic relation is derived in the SU(4) symmetric hmit (m c = rn u = m d = ms) , and a possible modification to it is given by the effect of the SU(4) symmetry breaking, probably by the 1__55interaction with the coefficient proportional to in (mc/mu) [8]. Since the 15 interaction can not be induced due to the property of the weak currents [5] in the SU(4) symmetric limit, the bihnear form o f the weak currents contains only two parts belonging to a 20-dimensional and an 84-dimensional representation of SU(4). One finds that there are two independent couplings for the 84-dimensional spurion as well as two independent couplings for the 20-dimensional spurlon contributing to the S-wave amplitudes of the nonleptonic hyperon decays. Since one has seven independent amplitudes to discuss in this ground, there must still exist three sum rules which should be more comprehensive than the sum rules (2) and (3). These sum rules are exactly the sum rule (4) ,3 and A(A) = A ( - ) ,
(5)
2A(L--S)-- A(~) = 3X/~ A(2;),
(6)
where A ( A ) = A 0- + v f f A 0, A(--) - ZZ - V~200, A(~;) - ~_ - ~ Z - X/~E; and A ( L - S ) -= A 0_ - 2 ~ 2 x/~P~. The sum rules (5) and (6) have already been pointed out by Kobayashl et al. [9], explicitly and implicitly, respectively, on the basis of the SU(4) symmetry. In our scheme we can also derive corresponding to (1) the following relation:
7 May 1979
= 1.11 -Vr3- X 0.99 : 2. The agreement of these ratios with experimental ratios is still unsatisfactory although the inconsistency in (7) is faintly milder than that in (1). So far we have discussed the sum rules in the SU(4) symmetric limit, although the real world is far from SU(4) symmetric (m c >> m u = m d ~ ms). Finally, we gave a few comments on the effect of the symmetry breaking. The mass relation m c >>m u induces the 15dimensional interaction and it may give an Important contribution to the hyperon nonleptonic decays [8]. It is easy to see that the sum rule (4) does not remain valid, but the sum rules (5) and (6) do, in the presence of the 15-dimensional interaction in addition to the SU(4) symmetric interactions. It should be noticed that the sum rules (5) and (6) hold even in the SU(3) symmetric limit [10]. In conclusion, we would like to recommend the sum rules (5) and (6) which were pointed out In refs. [9] and [10] as the most comprehensive sum rules, since they remain vahd in the SU(3) symmetry theory, too. And we would like to propose the sum rule (4) which is consistent with experiment as one o f the comprehensive sum rules in the SU(4) symmetry theory. Finally, we would like to comment that the contribution from the 84-dimensional spurion to eq. (1) shows a favourable tendency phenomenologlcally as represented In eq. (7).
-
AO: ~0" Z- = 1 +
2 ~
- X / ~ [1 + A ( Z ) + ~V~-~(Z)] : 2 . L 2,/8_--3
References
-J (7)
If we use A ( A ) = A(~) = --0.15, A ( Z ) = 0.23, ,_----= 2.04 in units of 2.213 X 10 - 7 [3], we get A 0_ : Y~. 2-_
¢3 Instead of the sum rule (4), we may propose a similar sum rule, X/6(AO -- -~-_)= (2;~- ~-) as one of the comprehensive sum rules m the SU(4) symmetry, by using the sum rule (5). In this letter, however, we recomment the sum rule (4) by taking the phenomenology of the S-wave decays into consideration. 228
The author would like to express his sincere thanks to professor Y. Kolde for helpful discussions. Also he wishes to express his gratitude to professor M. Nakagawa for his useful comments, especially, on the eqs. (5), (6) and (7).
[1 ] G. AltareUi, N. Cabibbo and L. Maiam, Phys Lett. 57B (1975) 277. [2] Y. Iwasakl, Phys. Rev. Lett. 22 (1975) 1407. [3] Particle Data Group, Phys. Lett. 75B (1978) [4] S. Ishlda, K. Nakamura and M. Oda, Prog. Theor.Phys. 47 (1972) 304. [5 ] Y. Katayama, K. Matumoto, S Tanaka and E. Yamada, Prog Theor. Phys 28 (1962) 675; Z. Makl, M. Nakagawa and S. Sakata, Prog Theor. Phys. 28 (1962) 870, S.L. Glashow, J. lhopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285 [6] B.W Lee, Phys. Rev. Lett. 12 (1964) 83.
Volume 83B, numbei 2
PHYSICS LETTERS
[7] H. Sugawara, Prog. Theor. Phys. 31 (1964) 213. [8] M.A. Shlfman, A.I. Vamshtein and V.I. Zakharov, Nucl. Phys. B120 (1977) 316; A.I. Valnshtein, V.I Zakharov and M.A. Shffman, Soy. Phys. JETP 45 (1977) 670.
7 May 1979
[9] M. Kobayashl, M. Nakagawa and H Nitto, Prog. Theor. Phys. 47 (1972) 982. [10] M. Suzuki, Phys Rev. B137 (1965) 1602.
PHYSICS LETTERS
7 May 1979
COMPREHENSIVE SUM RULES FOR S-WAVE AMPLITUDES OF NONLEPTONIC HYPERON DECAYS
M. KATUYA Laboratory of Phystcs, Shtzuoka Women's Universtty, Yada 409, Shtzuoka 422, Japan
Received 15 November 1978
Sum rules for the S-wave amplitudes of the nonleptonic hyperon decays are discussed. A new sum rule X/~(Ag + o ) = ( z : - £+) + is proposed as one of the comprehensive sum rules in the SU(4) symmetry theory.
One of the most important discoveries brought by extending the SU(3) symmetry theory to the SU(4) symmetry theory is that of the exact relation [ 1,2] for the parity-violating amplitudes of the nonleptonic hyperon decays * i, A ° . z;:
=
(1)
This exact relation is derived * 2 by assuming the CPmvariant hamlltonian to transform like the 20-dimensional SU(4) representation whmh is expected to be enhanced by the study of the short-distance behavlour of the product of weak currents [5]. Although there ts a rather large (about 40%) discrepancy between the ratios (1) and the experimental ratios [3], A 0 . £ 0 : .~= (1.44 -+ 0.01) : - V ~ × (0.84 -+o.o3): (2.00 -+o.o2), this discrepancy seems not so serious since the SU(4) symmetry is rather strongly broken. It is, therefore, quite reasonable to regard the relation (1) as basic for the S-wave amplitudes of the nonleptonic hyperon decays. It is useful to represent the relation (1) equivalently in terms o f two sum rules as follows: --
+
a °_ - 2 = - - =
(2)
3 A 0- - 3"-- = x/c3Z;.
(3)
The former is the famihar Lee-Sugawara sum rule [6,7] ,1 We use the same conventions for the amphtudes as m ref [3] except for a change of sign in ~o4:2 Relation (1) had already been derived m another context of SU(3) symmetry, see ref. [4]. +
[6,7] which was found in the SU(3) symmetry theory originally by assuming the CP-invariant hamlltonian to transform like an octet, X6. On the other hand, the latter is an unfamiliar sum rule, which does not remain valid in the SU(3) symmetry hmit. In fact, the conslstency of the Lee-Sugawara sum rule with experiment [3] is quite satisfactory, however, that of the latter sum rule (3) is not. For the sake of the subsequent discussion, we rewrite the sum rule (3) as ~ / ~ (A~ + 20) = (2_- - ~ ; ) ,
(4)
by using the A I = 1/2 relations for the A, N and 2 decay amplitudes , t . It is worthwhile to notice that the sum rule (4) is found to be almost consistent with experiment, although the sum rule (3) is found to be violated to approximately 50%. This fact means that there exists a considerable amount of deviation from the A I = 1/2 rule. The purpose of this letter is to insist that that sum rule (4) is indeed theoretically more comprehensive than the Lee-Sugawara sum rule (2) and the unfamiliar sum rule (3) in the SU(4) symmetry theory, in the sense that the sum rule (4) remains valid even in presence of a suppressed 84-dimensional representation ( A I = 1/2 and 3/2, AS = 1) part arising from the product of weak currents. The derivation seems obviously trivial, therefore, we only give an outline of it below. Our discussion is based on the following assumptions" (a) The strong interactions are fully invariant under SU(4). (b) The weak interactions are of the product of the 227
Volume 83B, number 2
PHYSICS LETTERS
weak charged currents [5] and are CP-invariant. Here we take into account the effect of the symmetry breaking perturbationally. That is, the most basic relation is derived in the SU(4) symmetric hmit (m c = rn u = m d = ms) , and a possible modification to it is given by the effect of the SU(4) symmetry breaking, probably by the 1__55interaction with the coefficient proportional to in (mc/mu) [8]. Since the 15 interaction can not be induced due to the property of the weak currents [5] in the SU(4) symmetric limit, the bihnear form o f the weak currents contains only two parts belonging to a 20-dimensional and an 84-dimensional representation of SU(4). One finds that there are two independent couplings for the 84-dimensional spurion as well as two independent couplings for the 20-dimensional spurlon contributing to the S-wave amplitudes of the nonleptonic hyperon decays. Since one has seven independent amplitudes to discuss in this ground, there must still exist three sum rules which should be more comprehensive than the sum rules (2) and (3). These sum rules are exactly the sum rule (4) ,3 and A(A) = A ( - ) ,
(5)
2A(L--S)-- A(~) = 3X/~ A(2;),
(6)
where A ( A ) = A 0- + v f f A 0, A(--) - ZZ - V~200, A(~;) - ~_ - ~ Z - X/~E; and A ( L - S ) -= A 0_ - 2 ~ 2 x/~P~. The sum rules (5) and (6) have already been pointed out by Kobayashl et al. [9], explicitly and implicitly, respectively, on the basis of the SU(4) symmetry. In our scheme we can also derive corresponding to (1) the following relation:
7 May 1979
= 1.11 -Vr3- X 0.99 : 2. The agreement of these ratios with experimental ratios is still unsatisfactory although the inconsistency in (7) is faintly milder than that in (1). So far we have discussed the sum rules in the SU(4) symmetric limit, although the real world is far from SU(4) symmetric (m c >> m u = m d ~ ms). Finally, we gave a few comments on the effect of the symmetry breaking. The mass relation m c >>m u induces the 15dimensional interaction and it may give an Important contribution to the hyperon nonleptonic decays [8]. It is easy to see that the sum rule (4) does not remain valid, but the sum rules (5) and (6) do, in the presence of the 15-dimensional interaction in addition to the SU(4) symmetric interactions. It should be noticed that the sum rules (5) and (6) hold even in the SU(3) symmetric limit [10]. In conclusion, we would like to recommend the sum rules (5) and (6) which were pointed out In refs. [9] and [10] as the most comprehensive sum rules, since they remain vahd in the SU(3) symmetry theory, too. And we would like to propose the sum rule (4) which is consistent with experiment as one o f the comprehensive sum rules in the SU(4) symmetry theory. Finally, we would like to comment that the contribution from the 84-dimensional spurion to eq. (1) shows a favourable tendency phenomenologlcally as represented In eq. (7).
-
AO: ~0" Z- = 1 +
2 ~
- X / ~ [1 + A ( Z ) + ~V~-~(Z)] : 2 . L 2,/8_--3
References
-J (7)
If we use A ( A ) = A(~) = --0.15, A ( Z ) = 0.23, ,_----= 2.04 in units of 2.213 X 10 - 7 [3], we get A 0_ : Y~. 2-_
¢3 Instead of the sum rule (4), we may propose a similar sum rule, X/6(AO -- -~-_)= (2;~- ~-) as one of the comprehensive sum rules m the SU(4) symmetry, by using the sum rule (5). In this letter, however, we recomment the sum rule (4) by taking the phenomenology of the S-wave decays into consideration. 228
The author would like to express his sincere thanks to professor Y. Kolde for helpful discussions. Also he wishes to express his gratitude to professor M. Nakagawa for his useful comments, especially, on the eqs. (5), (6) and (7).
[1 ] G. AltareUi, N. Cabibbo and L. Maiam, Phys Lett. 57B (1975) 277. [2] Y. Iwasakl, Phys. Rev. Lett. 22 (1975) 1407. [3] Particle Data Group, Phys. Lett. 75B (1978) [4] S. Ishlda, K. Nakamura and M. Oda, Prog. Theor.Phys. 47 (1972) 304. [5 ] Y. Katayama, K. Matumoto, S Tanaka and E. Yamada, Prog Theor. Phys 28 (1962) 675; Z. Makl, M. Nakagawa and S. Sakata, Prog Theor. Phys. 28 (1962) 870, S.L. Glashow, J. lhopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285 [6] B.W Lee, Phys. Rev. Lett. 12 (1964) 83.
Volume 83B, numbei 2
PHYSICS LETTERS
[7] H. Sugawara, Prog. Theor. Phys. 31 (1964) 213. [8] M.A. Shlfman, A.I. Vamshtein and V.I. Zakharov, Nucl. Phys. B120 (1977) 316; A.I. Valnshtein, V.I Zakharov and M.A. Shffman, Soy. Phys. JETP 45 (1977) 670.
7 May 1979
[9] M. Kobayashl, M. Nakagawa and H Nitto, Prog. Theor. Phys. 47 (1972) 982. [10] M. Suzuki, Phys Rev. B137 (1965) 1602.