2846 R . K U M A R AND B . K . A G A R W A L

'D. D. Reeder and K. V. L. S a r m a , Phys. Rev. 172, 1566 (1968).

2 ~ . A. Collins, B. J. Hart ley, R. W. Moore, and K. J. Moriarty, Nucl. Phys. E, 381 (1970).

3 ~ . Meyers , Y. Noirat, M. Rimpault, and Ph. Salin, Nucl. Phys. E, 99 (1970).

4 ~ o s h i a k i Karasuno and Shigeo Minami, Phys. Rev. D 2, 1320 (1970).

' ~ a j Kumar , B. K. Agarwal, and C. P . Singh, Phys. Rev. D 4, 2849 (1971).

%. W. Coulter , E. S. Ma, and G. L. Shaw, Phys. Rev. Le t te rs 2 , 1 0 6 (1969).

'A. D. Mart in and C . Michael, Phys. Le t te rs =, 297 (1970).

8 ~ . Imachi, K. Kinoshita, Kunio Shiga, F. Toyoda, and M. Uehara, P r o g r . Theoret. Phys. (Kyoto) 43, 444 (1970).

' pa r t ic le Data Group, Rev. Mod. Phys. 43, S1 (1971). 'OR. Barloutaud, Doung Nhu Hoa, J. Grisei in, D. W.

Merr i l l , J. C. Scheuer, W. Hoogland, J. C. Kluyver, A. Minguzzi-Rauzi, A. M. Ross i , B. Baher , E. Hirsch, J. Goldberg, and M. Laloum, Nucl. Phys. E, 493 (1969). avid J. Crennell , U r i Karshon, Kwan Wu Lai , John

S. O'Neall, J. Michael S c a r r , Rober t M. Lea , Thomas

G. Schumann, and Ernes t M. Urva te r , Phys. Rev. L e t t e r s 23, 1347 (1969). "w. L. Yen, A. C. Ammann, D. D. Carmony, R. L.

E i s n e r , A. F. Garfinkel, L. J. Gutay, R. V. Lakshmi, D. H. Miller , and G. W. Tautfest, Phys. Rev. Le t te rs 22 963 (1969). %. Hodge (see Ref. 1). "0. I. Dahl, Lyndon M. Hardy, Richard J. Hess , Janos

Ki rz , Donald H. Miller , and Joseph A. Schwartz, Phys. Rev. 163, 1430 (1967).

I'M. Abramovich, H. Blumenfeld, V. Chaloupka, S. 0. Chung, J. Diaz, L. Montanot, J. Pernegr , B. Reucroft, J. Rubio, and B. Sadoulet, Nucl. Phys. =7,477 (1971).

1 6 ~ . S. LOOS, U. E. Kruse , and E. L. Goldwasser, Phys. Rev. 173, 1330 (1968).

'IS. M. P r u s s , C. W. Akerlof, D. J. Meyer, S. P. Ying, J. Lales, R. A. Lundy, D. R. Rust , C. E. W. Ward, and D. D. Yovanovitch, Phys. Rev. L e t t e r s 23, 189 (1969). "w. Cooper, W. Manner, B. Musgrave, and L. Voyvo-

dic, Phys. Rev. Le t te rs 20, 472 (1968). "A. Bashian, G. Finocchiaro, M. L. Good, P. D.

Grannis, 0. Guisan, J. Ki rz , Y. Lee , R . Pi t tman, G. C. F i s c h e r , and D. D. Reeder , unpublished repor t , presented a t the Fifth International Conference on Elementary Par t ic les , Lund, Sweden, 1969.

P H Y S I C A L R E V I E W D V O L U M E 5 , N U M B E R 11 1 J U N E 1 9 7 2

Possible Origin for Symmetry Breaking*

J. Schechter and Y. Ueda Physics Department, Syracuse Universi ty , Syracuse, New York 13210

(Received 9 November 1971)

We conjecture that the total Lagrangian of s trong, electromagnetic, and weak interact ions i s invariant under the "left-handed SU(2)" of the weak cur ren ts , which i s however spontane- ously broken. We d iscuss a formula for the "tadpole" p a r t of the K-meson electromagnetic m a s s splitting, which may be related to the presen t scheme.

Several authorslT4 pointed out that i t i s possible to construct unified theories of the weak and elec- tromagnetic interaction based on the left-handed SU(2) group generated by the leptonic and Cabibbo weak currents . The theory may be arranged s o that the Lagrangian i s exactly invariant under this SU(2) group. This symmetry i s then spontaneously brokenzv3 so that the multiplets of particles in- volved no longer a r e degenerate in mass . Specifi- cally, the e mass splits away from the v, mass (which is zero), the p mass splits away from the v, mass and the intermediate-boson mass spli ts away from the photon mass. Because of the p re s - ence of gauge fields, no Goldstone bosons appear.5 It i s clear that not a l l of these splittings a r e small. Thus, i t i s tempting to speculate that this mecha- nism may in fact be responsible for the apparent

symmetry breaking of the strong Lagrangian which we take to be exactly chiral -SU(3) x SU(3) -invari- ant.

Since the left-handed SU(2) group of the Cabibbo currents i s a subgroup of chiral SU(3) X SU(3) we would then have the situation where G,,, + Gelectromagnetic +Cad i s invariant under this SU(2) group. This is our basic conjecture.

F i r s t we l is t the fields needed to construct al l presently known particles (neglecting gravity):

(i) the leptons e , v, , p , and v, ; (ii) the photon and possibly some intermediate

vector bosons; (iii) the quarks q,, q,, and q,. We next give the transformation properties3 of

the fields with respect to the left-handed SU(2). For convenience we define new quark fields related

5 - P O S S I B L E O R I G I N F O R S Y M M E T R Y B R E A K I N G 2847

to the old ones by a rotation through the Cabibbo angle 8,

Q, = -q, sin8 + q, cos8.

Then the doublets a r e

while the singlets a r e assigned to be

where, for each field 4 we have GL = $(l + y,)$ and +bR = +(I - y5)&. The photon and intermediate bosons may be c o n ~ i d e r e d ~ ' ~ to form a triplet but we shall make no use of this in the present note. A possible way of solving the traditional problems associated with neutral currents in such a scheme has been given in Ref. 3.

To implement the spontaneous breakdown mech- anism we may introduce, a s usual, an auxiliary doublet of scalar fields

which has nonvanishing vacuum expectation value, i.e., ($),= (Ox), where h i s a rea l number. The p - mass te rm in the Lagrangian, for example, comes from the SU(2) -invariant te rm

which simply becomes -mvEF when 4 i s replaced by i t s vacuum expectation value. Analogously we may consider bilinear quark mass t e rms which a r e invariant under the left -handed SU(2) and which may, following Gell-Mann,' be considered respon- sible for a l l breaking of SU(3) x SU(3). The most general SU(2) -invariant bilinear quark te rm in the Lagrangian which conserves electric charge7 is essentially given by

where f , . - . f, a r e some arbitrary constantsS and 7, is the Pauli matrix. If we insist on a CP-con- serving theory we require the f , to be real.

Now Eq. (4) contains chiral SU(3) xSU(3)-symme- try -breaking t e rms belonging to the (3, 3*) + (3*, 3) representation. The usual symmetry -breaking te rm of this type is written

When (4) i s expanded by using (1) and replacing q5 by (@), we see that i t gives r i s e to t e rms of the type given in (5) plus some additional I AS I = 1 t e rms . These / A S I = 1 te rms can be eliminated by the special choice o f f , :

[~ l t e rna t ive ly , the I AS I = 1 t e rms can be "trans - formed" away by SU(3) xSU(3) rotations.] Thus, the usual type of strong-symmetry -breaking t e rms may conceivably result from the mechanism which spontaneously breaks the weak and electromagnetic symmetry. We reemphasize that this same mech- anism is expected to produce the v, -p and pho- ton-intermediate -boson splittings which a r e not small even on the strong-interaction scale.

As i t stands (5) contains three unknown con- stants - the quark "masses" - which a r e not fixed from (4) and (6). A number of interesting modelsg have been made to try to relate these quantities. Here we would like to mention another model which leads to an experimentally interesting relation. The model i s defined by the equation

This ansatz would correspond to setting f , = 0 in (4). The f , te rm i s distinguished from the others by the fact that i t contains the matrix 7,. This is a reflection of the fact that in the group SU(2) the conjugate defining representation i s equivalent to the defining representation itself. Such a circum - stance is accidental to SU(2) and does not hold in la rger groups like SU(3). Perhaps, therefore, the introduction of a higher combined weak-electro- magnetic symmetry might eliminate the f , term.

To find a consequence of (7) , we note that a great deal of effort has been expended by many workers on an SU(3) xSU(3)-invariant strong interaction with the symmetry-breaking t e rm of (5). One can proceed by using the so-called "current-algebra" approach or , equivalently, to the approximation in which people work, by using a linear SU(3) o mod- el . We have the problem by the lat ter method so i t i s convenient for u s to car ry over the previous results and see what happens when (7) i s imposed. We consider three quantities A, a m i and three quantities a , a ( 3, q, )o. The weak decay constants and some of the spin-zero particles' masses a r e given in t e rms of these. Specifically the squared masses of the pion and the scalar i so- vector part icle [6(960) presumably] a r e givenlo by

2848 J . S C H E C H T E R A N D Y . U E D A

In addition, that par t ("tadpole" part) of the K + - K O m a s s difference which is not due t o the one- photon self-energy diagram is"

where W = a3/a and a = i ( a , + a 2 ) . Imposing A , = 0 in (8) and (9) gives

Since (m ./m ,I2 is s m a l l a , only differs f rom a, by about 2%. Substituting (11) into (10) gives the formula

Numerically (12) yields -0.57m ,02 if W = 1.56 (from the equation1' W = 2 F K / F , - 1) o r - 0 . 5 4 ~ 1 if W = 1.7 (from fitting the 17' mass1'). Equation (12) should be compared with

The est imate of the one-photon contribution given above1' is model dependent s o we may consider the agreement reasonable. The fact that the sign is right is interest ing s ince the one-photon contribu- tion has the wrong sign. Also the presence of vec- t o r and axial mesons was not taken into account in the derivation of (12).

ACKNOWLEDGMENTS

After the f i r s t version of this note was submitted, we received a preprint by Weinberg13 in which the left-handed SU(2) invariance of the total Lagran- gian w a s independently conjectured on the b a s i s of renormalizability. We a l s o received a repor t by ~ a j a s e k a r a n ' ~ which gives a discussion of theor ies of this type. We would like t o thank these authors fo r sending u s copies of the i r work.

*Work supported by the U . S. Atomic Energy Commis- sion.

'J. Schwinger, Ann. Phys. (N.Y.) 2, 407 (1957); S. L. Glashow, Nucl. Phys. 22, 579 (1961); A. Salarn and J. C . Ward, Phys. Letters 2, 168 (1964).

's. Weinberg, Phys. Rev. Letters 19, 1264 (1967). 3 ~ . Schechter and Y. Ueda, Phys. Rev. D 2, 736

(1970). 4 ~ . D. Lee, Phys. Rev. Letters 26, 801 (1971). his mechanism i s discussed by P. W. Anderson,

Phys. Rev. 130, 439 (1963); P. W. Higgs, Phys. Letters 12, 132 (1964); F. Englert and R. Brout, Phys. Rev. - Letters 2, 321 (1964); G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, ibid. 13, 585 (1964).

6 ~ . Gell-Mann, Phys. Rev. 125, 1067 (1962); M. Gell- Mann, R. Oakes, and B. Renner, ibid. 175, 2195 (1968); S. L. Glashow and S. Weinberg, Phys. Rev. Letters 20, 224 (1968). -

'or equivalently we may impose "weak hypercharge" conservation. See Ref. 3.

8 ~ h e s e terms were previously given in Eq. (39) of Ref. 3, where the f, te rm was unfortunately omitted. This implies that the statement made in Ref. 3 that the mass of ql cannot come from this mechanism is in- correct . The existence of this additional te rm has been pointed out by Mukunda (see Ref. 14). k. J. Oakes, Phys. Letters E, 683 (1969); R. Gatto,

G. Sartori , and M. Tonin, ibid. E, 128 (1968); N. Cabibbo and L. Maiani, Phys. Rev. D 1, 707 (1971). The scheme of Oakes can be obtained by imposing f = &Ajl(cot29+ tan2@) a s well a s f = 0 and (6) on the coefficients in (4). Thus, it i s a special ca se of our model.

'OJ. Schechter and Y. Ueda, Phys. Rev. D 3, 2874 (1971). "J. Schechter and Y. Ueda, Phys. Rev. D 4, 733 (1971). "R. H. Socolow, Phys. Rev. 137, B1221 (1965); J. H.

Wojtaszek, R. E. hlarshak, and Riazuddin, ibkd. 136, B1053 (1964). 13S. Weinberg, Phys. Rev. Letters 3, 1688 (1971). I*G. Rajasekaran, Saha Institute report (unpublished).