P H Y S I C A L R E V I E W D V O L U M E 1 3 , N U M B E R 5 1 M A R C H 1 9 7 6

Nonieptonic decays of hyperons and fl- in SU(4)

Mohinder P. Khanna Physics Department, Panjab University, Chandigarh-160014, India

(Received 24 July 1975)

The nonleptonic decays of octet baryons and are studied in the framework of current algebra assuming SU(4) symmetry andz' dominance of the weak Hamiltonian. For the parity-conserving transitions of hyperon decays, the predictions are in good agreement with experiment. But the results for parity-violating hyperon decays and R- decays are not satisfactory.

It has recentlv been pointed out1 that in the where Glashow -Iliopoulos -Maiani (GIM) s ~ h e m e , ~ the effective weak interaction for the nonleptonic de - cays may t rans form predominantly a s a member of 20"-plet3 representat ion of SU(4). Actually severa l au thors IB4 have already studied the non- leptonic decays of hyperons and of charmed had- rons assuming Su(4) symmetry and 20" domi - nance5 for the weak interaction. In this note we wish to analyze the weak pionic decays of hy - perons and Q - in the framework of cur ren t a lge- b ra , employing Su(4) symmetry and 20" domi- - nance.

We s t a r t with the famil iar cur ren t -current f o r m for the weak Hamiltonian

H, = + ( J J + +J+J) , (1)

where J i s the hadronic weak cur ren t . Let u s f i r s t t r y to understand the SU(4) t ransformation proper t i es of H,. Because of the symmetr ic n a - tu re of H,, in general we should expect

H , = I @ 15,& 20" $ 8 4 . - - - - (2)

Now it i s a highly specific property of the weak cur ren t suggested in the GIM model that i t s b i - l inear contains no 15 at all . F o r the s t rangeness - changing t r a n s i t i o n s t h e singlet a l so does not enter , so f o r these

H, = 20"3 8 4 . - - (3)

It has been shownB that, in an asymptotically f r e e SU(3) -invariant gauge theory of s t rong interac - tions, the gluons enhance the / A I 1 = + par t of H , relat ive to the / A I 1 = $ t e r m s . If these arguments a r e applied to an SU(4) -invariant theory, we would find that 20" is enhanced relat ive to 84.

Let u s now investigate the nonleptonic hyperon and 5 2 - decays in m o r e detail . F o r a weak process

the t ransi t ion amplitude in the s tandard cur ren t - algebra technique i s given by7

Here Bi and Bf a r e , respectively, the initial and the final baryons. A ; ( X ) i s the axial-vector c u r - rent and a, i s the space integral of the t ime com- ponent of axial -vector current , i, j, f , being SU(3) indices. F o r detai ls , reference may be made to the book by Marshalr, Riazuddin, and Ryan.7 Now it is well known that fo r the pari ty -violating (pv) hyperon decays, only the f i r s t t e r m (called the equal-time commutator o r ETC t e r m ) on the right -hand s ide of Eq. (5) contributes, while the contribution to the parity-conserving (pc) t r a n s i - tion comes only f rom the Born t e r m s in the s e c - ond t e r m of Eq. (5). Because of the commutation relat ions of the genera tors of SU(3) xSU(3) with H,, the decay amplitudes a r e then expressible in t e r m s of the mat r ix element (R,IH~IB,). In SU(3) there a r e two reduced mat r ix elements , the d and f type. T h i s is because in the product r e p r e s e n - tation 8 x 8, the representat ion 8 occurs twice. In S U ( ~ ) and in the GIM model, There i s only one reduced mat r ix element. Hence the rat io d / f is fixed and indeed we find it to be -I. The der iva- tion for d j f = -1 i s a s follows: The weak-inter - action Hamiltonian represent ing baryon-baryon weak t ransi t ion for the pari ty -conserving mode can be written a s

where

T I : ; ] sat isf ies T [ ; : :] = - T I : : . We immediately obtain

f r o m which the resu l t d / f = -1 i s straightforward. T h i s value i s remarkably close to the value

d / f = -0.8 needed to fit8 the pc amplitude with only baryon poles.

With the value of (B,(H~;;'~B,) thus determined, it i s known8 that the pv amplitudes a r e about twice

13 N O N L E P T O N I C D E C A Y S O F H Y P E R O N S AND W - I N S U ( 4 )

the experimental values. One could conceivably include the contribution from the K * poleg to ob- tain a reasonable fit.

Let us now discuss the W - decays. Because the products 20 x 20" and 20' x 20" do not contain the - - representation - 20, w e h a v e i n S U ( ~ ) symmetry

(D\H,ID)=O (10)

and

(DIH,IB)=o, (11)

where B and D represent, respectively, the baryon multiplet and the multiplet containing 0 - . Therefore, for the 51- decay only the Zu pole1' contributes. Obviously then the decays

0 - - Z T (12)

and

w - - Z*n (13)

do not occur. Neither does the pv transition of W - - ,IK-. W - thus decays only through the pc transition of

a- -AKW. (14)

Using the value of (B,IH,IB,) obtained from the

hyperon decays, we get

to be compared with experimental value total r = 8 0 ~ 10' sec-'. Thus, current algebra along withSu(4) symmetry and 20" dominance almost forbids 0 - decays. But s i z e SU(4) symmetry i s expected to be very badly broken, W - decays may occur through the breaking of SU(4). Further - more, because of the higher spin of S 2 - , we may not be justified in using the straightforward cur - rent -algebra approach. There could also be a major contribution by the K -meson pole in the t channel to the pc 0 - transitions. Finally 84 rep - resentation of the weak Hamiltonian may b e i m - portant in W- decays. These points will be taken up elsewhere.

Helpful discussions with Professor J. Schechter, Professor V. S. Mathur, and Professor S. Okubo a r e gratefully acknowledged. Thanks a r e also due to Dr. H. F. Jones for reading the manuscript. T wish to express my gratitude to Professor T . W. B. Kibble for the hospitality at Imperial College where this work was done.

'G. Altarelli, N. Cabibbo, and L. Maiani, Nucl. Phys. BEE, 285 (1975); A. Pais and V . Rittenburg, Phys. Rev. - Lett. 34, 707 (1975); M. B. Einhorn and C. Quigg, ~ h y s . y e v . D 2, 2015 (1975).

's. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2, 1285 (1970).

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5Nonleptonic decays in SU(3) demand an effective coupling dominated by the adjoint representation (octet domi- nance). Analogously, in SU (4) nonleptonic decays would go via pentadecuplet dominance. H , in GIM scheme does not contain 15, but only 20" and 84. Since for Ac = 0 transitions 84 3 1@8@27 and 2 0 3 8, the

generalization of the SU (3) octet dominance i s the SU (4) 20" -plet dominance.

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'R. E . Marshak, Rizauddin, and C. R. Ryan, Theory of Weak Interactions in Particle Physics (Wiley, New York, 1969) and references therein.

8 ~ . S. Brown and C. M. Sommerfeld, Phys. Rev. Lett. 16, 75 (1966); M. D. Scadron and L. R. Thebaud, Phys. - Rev. D 5, 2306 (1972).

'B. W. Lee and A. Swift, Phys. Rev. 136, B228 (1964); J. Schwinger, Phys. Rev. Lett. 12, 630 (1964); M. Gro- nau, ibzd. 3, 188 (1972).

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