Yukawa terms in the unitary gauge theory

Nuclear Physics 26 (1961) 230---232; ~ ) North-Hollat~ -Oublishing Co., Amsterdam Not to be reproduced by photoprint or .mtca~filmwithout written permission from the publisher

YUKAWA T E R M S IN THE U N I T A R Y GAUGE T H E O R Y Y. N E ' E M A N

Department o/ Physics, Imperial College London Received 24 March 1961 Considering higher order graphs, Yukawa terms are shown to appear in the unitary gauge theory under certain CP assumptions.

Abstract:

1. Introduction Three recent papers 1, 2,8) have derived the strong interactions from a 3-dimensional u n i t a r y gauge. Though the symmetries implied seem to fit most experimental facts, all three versions lack Y u k a w a terms, i.e. no single pions or kaons are e m i t t e d or absorbed. In a former theory, Salam a n d W a r d *) have shown t h a t such terms appear in a gauge-theory if it contains a neutral meson with CP=

1.

(1)

Such an x ° meson can t h e n have

o 4= o,

(2)

so t h a t terms like

#01,rcB~,

~r/,fl~oB~/"

(3)

give a Y u k a w a vector interaction mediated b y the vector boson B ~ (here fl~ is the group operator).

2. The ~o Meson Looking for an z ° in the representations of the 3-unitary group, we first notice t h a t there exists a one-dimensional representation given b y the trace of the set. I t could accommodate an z o, b u t its scalar n a t u r e with respect to the group does not allow it to contribute to the interaction Lagrangian. Our answer therefore points to the isosinglet within the 8-representation, which we denoted in ref. s) b y ~ ' , otherwise known as p0. For this meson to fulfil eq. (1) there are 3 possible choices. (A) The first choice is C = 1 (as for the pions), P = 1, i.e. the mesons' 8-vector does not have one p a r i t y for all its components. Such was also the case in refs. ¢ ~). In b o t h studies, this was based on a group 230

Y U K A W X T]~RMS

231

structure allowing such a difference between the hypothetical sigma meson and the pions. In our case, the p3 is also an isolated singlet. In the formalism of ref. 3), and with ((u~)) denoting the linear envelope of vectors u ~, namely subspaces of the algebra's 8-space P, we have

po ((uS)), [uS, u

(4)

= (CuS, uS, u ' ) ) ,

C5)

= 0,

(6)

u" e

On the other hand, we can form two other 3-rotation subspaces similar to U s with the kaons. Defining vectors

b s = --½(u 7 - ~/3uS),

cO =

(7)

K +, K - e U o = ((u 1, u', c°)),

(8)

we find

K °, K ° e U~ = ((u', u 3, b°)),

p0 e ((uS)) = ( ( b ° - - c ° ) ) .

(9)

This defines a certain limited connection with the kaons; b u t assumption (A) is certainly allowed as long as the parity of the latter is undetermined. Still, we should note that the analogy with refs. • 6) m a y be too stretched, as these deal with 4-dimensional rotations (a semisimple algebra splitting into two ideals) whereas our gauge is given b y a simple algebra with no ideals. (B) The second choice one can make is the following: t h e po is a complex field po :_ ½~/~(plo+ip o) like the kaons, and pl ° is an z °. This would induce some changes in the formalism of ref. 3) through putting ~ ~

(10)

in the P 8-space. Still, as this is the case with the fermions, it could also happen here. (C) The third possible choice is P = --1 C = --1

(as for the pions), (in opposition to the eigenvalue of the pions).

3. D i s c u s s i o n If eq. (1) holds for p0 s ( (u 8) ) while ~ e U Ghas C P = -- 1 we should get opposite relative CP eigenvalues for the corresponding baryon bilinears. As these have C = I both g and P, this means opposite parities. (S and P are conventional scalar and pseudoscalar Fermion bilinears). Thus the assumption that p0 is the x ° of eq. (2) and (3) gives us a prediction for an odd relative A--27 parity. It is also edifying to note the difference in the order of the diagrams leading

232

:

Y.

N]~'~MAN

to Yukawa terms for the kaons and the pions respectively. Single kaons can be emitted as in eq. (3), mediated by Z~ (the isospinor t h a t m a y be identical with the observed K* according to Chan's calculation e) as in fig. 1. /S

/

N

I I

z:,AK °

^ / I .

.

.

.

.

.

P

P Fig. 1. Emission of ~ K+ meson (Z~+ is a K+-like vector boson).

<

Fig. 2. Emission of a ~+ meson (V~+, Z~ +, Z~ ° are vector bosons behaving like ~+, K +, K ° in isobaric sI~ace ).

Because of eq. (6), there are no terms mixing go and ~ mesons in the 8representation of the unitary group (as it is constructed from the Lie coefficients of the u ~ basis). Thus, pion Yukawa terms appear only in higher order graphs, like the example in fig. 2. The possible effect on the ratio of the phenomenological couplings is interesting; g, > gK would preclude the attribution of the strength itself to eq. (2) as was assumed in ref. 4) namely terms with no po would still be strongly coupled. I would like to t h a n k Prof. A. Salam for guidance, criticism and discussions of this problem. References 1) 2) 3) 4) 5)

A. Salam and J. C. Ward, Nuovo Cim., to be published M. Cell Mann, preprint Y. Ne'eman, Nuclear Physics 26 (1961) 222 A. Salam a n d J. C. Ward, Nuovo Cim. 19 (1961) 167 J. Schwinger, Ann. Phys. 2 (1957) 407; M. Cell Mann and Levy, Nuovo Cim. l b (1960) 4, 705 6) C. H. Chan, Phys. Rev. Letters, to be published