Unified interactions in the unitary gauge theory

8.B

] I

Nuclear Physics 30 (1962) 347--349; (~) North-Holland Pubhshing Co., Amsterdam N o t to be reproduced by p h o t o p n n t or mierofdm without written permis~aon from the publisher

UNIFIED INTERACTIONS

IN T H E U N I T A R Y G A U G E T H E O R Y Y. N E ' E M A N

Department o/ Physws, Imper)al College, London Received 15 September 1961

Abstract: Postulation of ~inite vacuum expectation values for K1° and Po generates the strong, weak and electromagnetic interactions and produces the mass differences within the baryon and boson multlplets.

1. I n t r o d u c t i o n

The three-dimensional unitary gauge theory has been shown to reproduce the strong interactions--whether in the SU(3)form suggested b y Gell-Mann x) and the author 2) or in the U(3) form of Salam and Ward 3). Yukawa terms appear 4) upon postulation of

<:)o

0,

(1)

where 9o is the T = 0, Y = 0, S = 0 meson accompanying K and ~ in the meson octet. This followed a previous suggestion of (Kx°>0 ~ 0,

(2)

proposed b y Salam and Ward 5) as the materialization of the "spurion" concept for the IATI = ½ rule of weak interactions. We shall now show that the above postulates, when superposed upon the unitary gauge, are sufficient to the production of a mass spectrum and the generation of the electromagnetic interactions, besides engendering the Yukawalike strong and the weak interactions. 2. W e a k and E l e c t r o m a g n e t i c I n t e r a c t i o n s

A bedevilling feature of attempts at local-gauge theories based on the conventional symmetry schemes is the emergence of "doubled" electromagnetic fields. This results from the fact that electromagnetic interactions are known to conserve both T z and Y (or/'+3 and /'-3 in the Salam-Polkinghorne isospace), i.e. they correspond to a 2-- parameter abelian group. We shall now see how the introduction of postulate (2) in the unitary gauge produces only a one-parameter gauge with a single vector meson, the Maxwell field. 347

348

Y. NE'EMAN

The interaction lagrangian in the SU (3) or U (3) theories can be written as 8

L , = ~, idB~,, ,,

(3)

1

where 1'~ is an 8-component vector or axial vector charge-current (we do not deal with the baryon conservation one-parameter abelian invariance of U(3) that does not exist in SU(3)). To the extent implied b y the finite expression (2), the expression (3) ceases to be a scalar with respect to SU(3). All 8 components of i ~ (taken in the U basis of ref. 3)) are disturbed b y this perturbation. On the other hand, if we work in the U' basis of ref. e), we find that we can isolate one of the 8 currents which is undisturbed b y (2) as it does not contain K1°. As was shown in ref. ~), this is the Maxwell current, coupled through the group operator C' s to the A z field. We found for the 3-dimensional representation U'8 ---- ~ v " 3 ( 2 j n - - J , ~ - J 3 3 ) ,

(4)

where J,%~ = 6,~b,

(5)

and got Q

2

-

~v/3 C' s

(6)

in the 8-representation. From the point of view of gauge postulates, this implies an unrelated coupling. The breakdown of the full unitary invariance through K1° absorption leaves us with a one-parameter invariance corresponding to the conservation of electric charge. The weak interaction lagrangian resulting from (2) and (3) is invariant to this new abelian gauge - - and so are the free field lagrangians. We can thus postulate a new gauge invariance generating the electromagnetic interactions. This answers a question we left open in the discussion of ref. 3) and provides a model for the appearance of new couplings in a gauge-theory; it replaces Pais' idea of a hierarchy of symmetries b y a concept involving a pyramid of gauges. We note that the C' 8 operator is diagonal and commutes with all other diagonal operators of U (3), i.e. leaves T, and Y undisturbed. This is a "passive" conservation law, in contrast to the "active" survival of Q conservation.

3. The Mass Spectrum We now follow this model for (1). We find that pO absorption disturbs 4 out of the 8 currents. These are the currents coupled to the K-like quartet Z,; the p0 has neither isospin nor hypercharge, and it has only the kind of charge corresponding to the remaining 4 operators of SU (3); a quartet with no diagonal

UNIFIED INTERACTIONS

349

component (when T, and Y are diagonal) whose role in SU(3) when compared to the Gell-Mann-Nishijima scheme reminds us of the notorious submerged part of an iceberg. Thus postulate (1) breaks down this "submerged" current's conservation, but does not touch the isospin and hypercharge current. Again we see that the interaction lagrangian yields an element respecting only T and Y invariance; and we can postulate a local gauge resembling the B 7 and Ba Y of Sakurai's theory s), with two new unrelated couplings (from the point of view of the group). Such an interaction reproduces the symmetry displayed by the mass spectrum. This seems to be the "mysterious interaction" required by S. L. Glashow in a paper on the anomalous magnetic moments and the unitary gauge ~). When discussing these results with Prof. A. Salam, the author learned of a similar suggestion put forward by him at the La Jolla conference. The author would like to thank Prof. A. Salam for a discussion of the problem. References I) 2) 3) 4) 5) 6) 7)

i%1. GeU-Mann, Report CTSL-20 (1961) Y. Ne'eman, Nuclear Physics 26 (1961) A Salam and J Ward, Nuovo Cim 20 Y. Ne'eman, Nuclear Physics 2b (1961) A. Salam and J. Ward, Nuovo Cim. 19 J. J. Sakurai, Ann. Phys. I I (1960) 1 S. L. Glashow, unpublished

222 (1961) 419 230 (1961) 165