VOLUME 14,NUMBER26 P H Y S I C A L R E V I E W L E T T E R S 28JUNE 1965

ANTIPROTON-PROTON ANNIHILATION AT REST INTO TWO PSEUDOSCALAR MESONS AND SU(6) SYMMETRY

Michiji Konuma*

Scuola Normale Superiore, Pisa, Italy, and Centro Ricerche Fisica e Matematica, Pisa, Italy

and

Et tore Remiddi

Scuola Normale Superiore, Pisa, Italy (Received 19 May 1965)

In this note we d i scus s , in the SU(6) s y m m e ­try model , 1" 3 the proton-ant iproton annihilation at r e s t into two pseudosca la r m e s o n s :

p+p-*Mx+M2. (1)

This problem has been recent ly considered by Dyson and Xuong,4 including in the final s ta te al l the mesons of the 35-plet . As our t rea tment and r e s u l t s a r e different from the i r s , however, we think it would be worthwhile to r epo r t them.

For annihilation at r e s t , we take the init ial and final s t a t e s in Reaction (1) a s 3SX and P wave, respec t ive ly . 5 Considering that the final two-meson s ta te must be s y m m e t r i c a l under the exchange of the two pa r t i c l e s , we see that in the odd angular momentum case it mus t be an t i symmet r i ca l in the SU(6) par t , which t h e r e ­fore mus t , in genera l , be of the form

M P(1)M 6 (2 ) -M 6(1)M P(2). (2) a y y a

The ma t r ix element A for (1) can be wri t ten as

A = X . i j

(3)

where x^> x '̂ a r e the antiproton and proton two-component sp ino r s , q is the re la t ive momen­tum of the mesons , and a; is a suitable e x p r e s ­sion involving in terna l SU(3) quant i t ies . We introduce the hypothesis of minimal SU(6) vio­lation by requi r ing that Xi^X^ behaves like the U,_3) component of the 35-plet .

As in re fe rence 4, we complete (5q) to a spur ion Qv^ of the s t r u c t u r e of the 35-plet of SU(6) by defining it a s

Q M = ( 5 q ) m 6 M

N

[with the usual cor respondence (i = (m,M) be ­tween SU(6) and SU(2),SU(3) indices] , so that A becomes formal ly SU(6) invariant . C inva r i -

ance can be taken into account for the initial S3 = 0 s ta te by imposing

A=-AT, (4)

where A? means t r anspose of A in SU(6) indices . The re a r e eighteen 35-plets which can be con­

s t ruc ted out of 56®56®35®35, corresponding to the product of Ba(3y, BaPr, MaP{l), and MQP{2) mul t ip le ts . Requiring the conditions (2) and (4) to hold, however, we a r e left with just four t e r m s 6 :

/ . "=B R BayV[Mhl)M \2)-M 5 ( 1 ) M / ( 2 ) J , IJLL ajSpi L 6 y y 6 J

l " = k{B . Ba^[M 6 ( l ) M / ( 2 ) 2/i apti y 6

- M / ( 1 ) M 6 ( 2 ) ] - B Q Baf*V[M?{l)M 6(2) 6 y J afiy L 6 /i

-M^W^U)]} ,

and

- M / ( l ) M / ( 2 ) ] - 5 a / 3 /a 5 > / ( l ) M / ( 2 )

-Mj(l)MQP(2)\\,

so that , in genera l ,

A*h*tji*<- (5)

i = i

Explicit calculation shows that 74 does not contribute to final two-pseudosca la r -meson s ta te , while I2 and I3 contributions a r e p ropor ­tional. The final express ion for the ma t r ix

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VOLUME 14, NUMBER 26 P H Y S I C A L R E V I E W L E T T E R S 28 JUNE 1965

elements is

A(K+K-) = -gi + 4(g2+g3),

A(K°K°) = -2g1-(g2+g3) (6)

(apart from an irrelevant factor), where A (ir+Tt~~) stands for the matrix element between the ini­tial pp and the final 7r+(l)7r~(2)-7r"(l)7r+(2) state, etc. [see Eq. (2)]. Of course, A(n%°) =A(r\r\) -A (Tf°r]) = 0, as it should be.

From (6) we obtain the triangular relation7

A(ir + ir-)-A(K+K-)+A(K°Ko) = 0. (7)

From the experimental data of reference 5, if we take the rate for Reaction (1) to be pro­portional to the meson momentum times the squared absolute value of the amplitude, we get, in arbitrary units, the following values:

!A(7T+77-)| = 1.99±0.10,

U(K+K~)I = 1.32±0.19,

\A(K°K°)\ = 0.87 ±0.06.

Equation (7) seems very well satisfied by these quantities.

We conclude that, at least for annihilation at rest into two pseudoscalar mesons, the SU(6)-symmetry model seems to give a rather sat­isfactory agreement with experiments.

The authors wish to thank Professor L. A. Radicati for constant encouragement, and

Dr. R. Fer rar i for useful comments and clar­ifying discussions. One of the authors (M.K.) wishes also to express his gratitude to Profes­sor G. Bernardini and Professor L. A. Radi­cati for their hospitality at the Scuola Normale Superiore di Pisa.

*On leave of absence from the Department of Physics, University of Tokyo, Tokyo, Japan.

1 F. Giirsey and L. A. Radicati, Phys. Rev. Letters 13, 173 (1964).

2A. Pais, Phys. Rev. Letters 13, 175 (1964). 3B. Sakita, Phys. Rev. 136, B1756 (1964). 4F. J. Dyson and N. Xuong, Physc Rev. Letters .14,

654 (1965). 5R0 Armenteros, L. Montanet, D. R. O. Morrison,

S. Nilsson, Ao Shapira, J. Vandermeulen, Ch. d'Andlau, A. Astier, J. Ballam, C. Ghesquiere, B. P. Gregory, D0 Rahm, P. Rivet, and F. Solmitz, Proceedings of the International Conference on High-Energy Nuclear Physics, Geneva, 1962, edited by J. Prentki (CERN Scientific Information Service, Geneva, Switzerland, 1962), p. 351D

6The terms I2 and /4, apart from exchange antisym-metrization, correspond to the two independent terms of Eq. (4) in reference 4.

7It should be noted that Eq. (5) is fully symmetrical as far as the SU(3) part is concerned. In SU(3) sym­metry alone, however, there is no relation for charged meson production, so Eq. (7) is a pure SU(6) prediction. We recall that the only prediction in SU(3) involves apparently S -wave neutral mesons, and gives no con­tribution in the low-energy region. See M. Konuma and Y. Tomozawa, Phys. Rev. Letters 12, 425 (1964).

EQUIVALENCE OF THE CURRENT ALGEBRA AND GROUP-THEORETIC FORMULATIONS IN PARTICLE PHYSICS*

E. C. G. Sudarshan

Physics Department, Syracuse University, Syracuse, New York (Received 1 June 1965)

Recent developments in particle physics have led to a phenomenological but quantitatively successful correlation of a variety of proper­ties of strongly interacting particles (including the low-lying resonances). The success of this approach raises fresh problems,1 since the symmetry groups underlying these systemiza-tions were by no means symmetry groups of the Hamiltonian of the system2; the apparent violations of the symmetry were sufficiently large to invalidate any traditional method of computing the effects of symmetry violation. And yet the simpler predictions of these (uni­

tary) symmetry groups have had significant success. With a view to avoiding this embar­rassment, several people have attempted to depart from a conventional approach to sym­metry and to try to base it on the algebra of certain ("current") operators.3 The formula­tion involving the (space)-integrated currents has been used recently4 to rederive several results of the SU(6) and SU(4) theories in a more direct fashion. It is the purpose of this note to show that insofar as one uses only the integrated currents, the formulation in terms of the algebra of currents is completely equi-

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