P H Y S I C A L R E V I E W D V O L U h ' I E 1 0 , N U M B E R 4 1 5 A U G U S T 1974

Pseudo-Goldstone pion masses in a gauge model*

T. C. Yang Center for Theoretical Physics, Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20742

(Received 8 April 1974)

A SU,(2) @J U(1) t3 U(1) gauge model of electromagnetic and weak interactions is presented, which has three pseudo-Goldstone bosons identified as pions. We can include strong interactions in the pion masses without generating mass counterterms, if the strong gauge symmetry is U(l) or "color" SU(3). At the one-loop level (without strong interactions) the PO remains massless, while the a* pick up masses. From our analysis, the one-loop calculation should be interpreted as the electromagnetic and weak contribution to the pion mass difference. Our result is not too large in magnitude as compared with the results found in other models. The pion mass difference including strong interactions has the same form as in the Weinberg model where pions are not pseudo-Goldstone bosons. We also exhibit formulas for the pion masses including strong interactions, which however we can not evaluate.

I . INTRODUCTION

The idea of pions a s pseudo-Goldstone b o s o n ~ , ' ~ ~ perhaps a unique feature of gauge theory, has r e - ceived much attention. Most notably, the pion masses a r e ~ a l c u l a b l e . ~ - * Models with pseudo- Goldstone pions have many distinctive features, some of which we shall discuss below. But in the models3 studied so far, the contribution of weak and electromagnetic interactions to the pseudo- Goldstone boson masses come out to be too large (of the order e 2 m ,,,') to be the pion masses (mass difference). In this note we present a model which contains three pseudo-Goldstone bosons whose masses can be of the order of the pion masses (mass difference). Our model differs from previ- ous ones in that the pseudo-Goldstone boson mass- e s a r e roughly related to the mass differences of the vector mesons. We shall henceforth refer to the pseudo-Goldstone bosons a s pions.

It i s well known that gauge theories in which the symmetry i s spontaneously broken a r e renormal- izable. Because of this property, some physical quantities a r e in principle calculable, among which a r e the masses of the pseudo-Goldstone bo- sons. If the scalar potential of a Lagrangian has a la rger symmetry, called pseudosymmetry, than the gauge symmetry, then certain physical fields remain massless after spontaneous symmetry breaking. These a r e the pseudo-Goldstone bosons, which pick up finite and calculable masses in high- e r order. Such a theory offers the following unique and desirable features.

(a) The role of the pions. The Higgs mechanism in gauge theories renders the ordinary explanation of pions a s Goldstone bosons (for example, in the o model) invalid. Ordinary Goldstone bosons in gauge theory cannot be pions. But if pions a r e pseudo-Goldstone bosons, they pick up masses in

higher-order corrections. The origin of the pion mass i s due to the correction to the pseudosymme- try, in contrast to explicif symmetry-breaking t e rms in nongauge theories (as in the u model).

(b) Hadronic nzass sca le . Although many hadron- ic mass relations have been derived from strong- interaction symmetries, none of the existing the- ories of hadrons relates the scale of the hadronic masses to that of other masses. In renormalizable theories, if the mass i s a calculable quantity, then the square of the ratio of the scalar-meson mass to the vector-meson mass can be calculated in perturbation ser ies . An example i s the model of Coleman and Weinberg."he mass of the pseudo- Goldstone boson i s calculable because of the ab- sence of a mass counterterm; the mass ratio m,/ 112 can be expressed a s a perturbation ser ies .

(c) Zeroth-order s y m m e l r y . An important fea- ture of some gauge theories i s the zeroth-order symmetry. Corrections to such a symmetry a r e of higher order, and a r e finite and in general small, and this might account for approximate symmetry relations observed in nature. In order to have pseudo-Goldstone bosons, the scalar po- tential must have a symmetry la rger than the gauge symmetry. Such a symmetry (the pseudo- symmetry) i s a zeroth-order symmetry. The pseudo-Goldstone pions a r e the consequence of the pseudosymmetry.

(d) Pa,vliaE conserzlalion of axial-rteclor crcrrertt (PCAC) in gauge theory. The pseudo-Goldstore pions a r e massless in zeroth order. Thus, natu- rally, to f irst order the relevant axial-vector cur - rent has a massless pion pole. Picking up the pion pole we obtain the soft-pion result. It should be emphasized that the soft-pion result i s obtained not because the chiral SU(2)@3SU(2) symmetry i s a good symmetry a s ordinarily interpreted for the soft-pion calculations using PCAC, but simply due

to the fact that the pion mass vanishes to f irst o r - der in perturbation theory. The success of the soft-pion result could be taken to mean that the first-order approximation works so well that high- er-order corrections a r e in fact small. Note that if the pion i s not a pseudo-Goldstone boson, it need not be massless in zeroth order, and in this case something other than the perturbation argu- ment is needed to justify the use of PCAC o r the smallness of the pion mass. Thus, we note that the success of the soft-pion mass differences of Das et al.%an be easily understood if pions a r e pseudo-Goldstone bosons. Furthermore, higher- order contributions to pion masses (on mass shell) a r e finite, thus resolving the old problem of the divergence of on-mass-shell pion mass differ- ences. This last point in (d) has not generally been noticed.

The above four points a r e our motivation for in- vestigating the pseudo-Goldstone boson problem. The outline of the paper i s a s follows: In Sec. 11, we give the formulas for the pion masses and mass differences. In Sec. 111, the model i s presented and the pion masses calculated. In Sec. IV, we give a concluding discussion. Some mathematical details a r e given in the Appendix.

11. THE PION MASS FORMULA

While previous attention has been directed to pseudo-Goldstone pions in gauge theories of weak and electromagnetic interactions o r of strong in- teractions separately, it i s essential to take all three basic interactions into account. In other words, if one has a gauge model of weak and elec- tromagnetic interactions which contains pseudo- Goldstone pions, one must ensure that incorporat- ing strong interaction does not spoil the calculabil- ity of the pion mass. The problem ar ises from the fact that pions a r e strongly interacting particles. Thus the strong interaction, if not properly incor- porated, may require, mass counterterms in order to preserve renormalizability, which would make the pion mass uncalculable. To avoid such coun- te r terms the potential with pseudosymmetry should be the most general one, with respect not only to the weak and electromagnetic interaktions but the strong interaction a s well. From parity consider- ations, it was previously noted7 that the strong gauge symmetry should be neutral to the weak and electromagnetic gauge symmetries, which suggests that it should be U(1) o r "color" SU(3). If the (most general) potential has a pseudosymmetry with respect to the weak and electromagnetic gauge group, and is also color-invariant, then we see that the pseudo-Goldstone boson mass i s finite and

calculable. Let us recall the origin of the pseudo-Goldstone

pion masses. We have a Lagrangian in which the scalar potential has a larger symmetry G than the gauge symmetry G. When G is spontaneously bro- ken to a subgroup S, G i s correspondingly broken to S. With each generator in G not in S, there is associated the unphysical Goldstone boson which i s transformed away. Corresponding to each genera- tor in c, not in S, and also not in G - S, there i s associated the pseudo-Goldstone boson. Since G i s a zeroth-order symmetry, the pseudo-Gold- stone boson remains massless in zeroth order. Higher-order corrections due to pure scalar-me- son exchanges do not contribute to the pion mass since the scalar potential i s C-invariant to all or - ders. With respect to the G-invariant potential the pions a r e like the ordinary Goldstone boson, which remains massless to all orders. To phrase it dif- ferently, if G were the symmetry of the total La- grangian, then after the spontaneous symmetry breakdown the pions remain massless to all o r - ders. Thus the pion masses come from interac- tions which do not respect G symmetry (higher- order corrections). The strong interaction alone being neutral to G, i s -6-invariant to all orders. But the interactions of the weak and electromagnet- ic gauge mesons and the Yukawa interaction need not be C-invariant, and in general they will con- tribute to the pseudo-Goldstone pion mass. For most interesting cases, the Yukawa interaction i s also G-invariant, it then does not contribute to the pion mass (at the one-loop level). We shall not discuss it in this note.

At the one-loop level, the calculation has been carried out by Weinbere for any group and repre- sentations and for arbitrary gauge. The result can be most easily illustrated by using the Landau gauge. The propagators of the vector mesons A: and the scalar mesons 6, in this gauge a r e given by

where p a andhl , a r e the masses of A; and O i . There a re four diagrams, shown in Fig. 1, which contribute to the pseudo-Goldstone pion masses. Their contributions a r e a s follows:

where B denotes the quantum numbers of the pseu- do-Goldstone pions, and

10 - P S E U D O - G O L D S T O N E P I O N M A S S E S I N A G A U G E M O D E L

pling constants of the interaction t e r m s $,AgA ,,, FIG. 1. Feynman diagrams for the scalar self-energy a,OB6,A;, OB@BA:A,,, and @ B @ B ~ , , respect ive- (Here dashed lines refer to scalar fields, wavy lines ly. We then find refer to gauge fields. )

We can easily check the above formula with Weinberg's resul t which c a n be written a s

m 2 A B = - ( ; - l n - l ) A B ,

where - pLaa = -(oBA)i(eAh)i,

where A, a r e the vacuum expectation value of the r e a l s c a l a r fields, 0, a r e ant isymmetr ic and Her - mit ian mat r ices of the genera tors in the (real) s c a l a r space. Capital A , B denotes the axes which project out the pseudo-Goldstone boson subspace and repeated indices a r e summed over. In the rep- resentation of physical par t ic les (with diagonalized m a s s matrix), one finds that Eqs. (1) and (2) a r e the same, with the aid of the following relations:

g ~ a t - k ( ~ B ~ ) ~ ( ( ~ . a , @ , } A ) l / ~ F B ' , (5a)

~ B B ~ ~ , - - ( Q B * ) ~ ( @ . ' P ~ ~ ) ~ / i i B J, (5b)

C f s ~ ~ g ~ ~ ~ L ~ l ~ ~ = - ( ~ g ~ h ) ~ ( ~ ' ~ ) ~ / l Ila 1 ' ) ( 5 ~ ) 0 1

where I zB / = I -(Q,A),(B,A), and a, P correspond to indices of the diagonalized vector mesons. Eq. (5) can be easily checked from Ref. 2.

As we discussed before, if s t rong gauge symme- t r y i s U(1) o r color SU(3), whereas the weak and

electromagnetic gauge symmetry i s color-singlet, then the s c a l a r potential with the pseudosymmetry i s the most general gauge-invariant one within the pseudo-Goldstone pion sec tor . The pion m a s s to f i r s t -o rder perturbat ion in the weak and electro- magnetic coupling constants i s again given in Fig. 1. Therefore to a l l o r d e r s of s t rong interaction, we can use cur ren t a lgebra and write

where g , i s the weak and electromagnetic coupling constant defined by

C,", - C gaA;1;, a

and 1, i s the hadronic cur ren t coupled to A,. The

YANG 10 -

essential element in obtaining Eq. (6) i s that the strong interaction does not generate infinite cor- responding to the mass counterterm of the pseudo- Goldstone pions, thus the pion mass including al l orders of strong interaction should be finite.

111. THE MODEL

We shall consider a SU, (2) Z3 U(1 ) s U(1) gauge model for weak and electromagnetic interactions. We have an extra U(l) gauge symmetry a s com- pared with the Weinberg SU,(2)8 U(l) model.' As expected, many properties will be the same a s in the Weinberg model, but there a r e two differences: (1) Because of the extra U(1) gauge gluon, the neu- t r a l currents a r e different in this model than in Weinberg's case, since the fermions can couple to the two neutral vector gluons arbitrarily. The neu- t ra l current induced process will have a different amplitude, which i s more flexible than in the Weinberg model. (2) In the SU, ( 2 ) s U(1) model, one cannot incorporate the pseudo-Goldstone pi- o n ~ . ~ But in the SU, (2) 8 U ( l ) @ U(l ) model, the (doublet) representations can be unlocked, by re- quiring a discrete symmetry, namely, charge con- jugation invariance. As a result, the scalar po- tential has a pseudosymmetry, which gives r i s e to three pseudo-Goldstone bosons.

We shall not go into the fermion sector, since the couplings of the fermions with the physical gluons follow easily from the spontaneous symme- t ry breakdown discussed below. For the purpose of calculating the pseudo-Goldstone boson masses, we shall only discuss the scalar Lagrangian. Let g be the SUL(2) gauge coupling constant, and for simplicity, we have the same coupling constant g' for both of the U(1) groups.1° We have three dou- blets of spinless mesons,

with the following gauge-invariant Lagrangian:

where A;, a = l ,2 ,3 , and B,, C, a r e the SU, (2) and U(1)B U(l) gauge vector mesons, and P(6, i, 5) is the scalar potential.' The Lagrangian is like the o model, with the field a +inUT,, C + i l l a ~ , , and

4, +iGar, transforming like ($, $) under chiral SU(2)8 SU(2) symmetry. We have only gauged the left-handed SU(2) symmetry. If we turn off the weak and electromagnetic interaction, the above d: i s chiral SU(2)B SU(2)-invariant with the dis- crete symmetry imposed below." The isospin and parity of the (pion) fields a r e defined a s in the a model. This point is relevant, if pions a r e go- ing to have strong interactions also. The color SU(3) strong-interaction symmetry will be dealt with in Appendix B. But a s far as the masses a r e concerned, we can forget it a t present.

Let us f irst of all see the pseudo-Goldstone bo- sons in the model. We impose a discrete symme- try, namely, a special charge-conjugation invari- ance on the Lagrangian, which plays a crucial role. The Lagrangian invariant under the discrete symmetry i s also renormalizable, if it has the most general (quartic) polynomial which is gauge- invariant but also discrete-symmetry-invariant. In Eq. (7) one notes that the field + couples to B,, whereas ri, and 6 couple to C,, in such a way that the covariant derivatives a r e invariant under the following charge-conjugation transformation: C?I 2 (T,)+*, B - -B ,. In the scalar polynomial, te rms like ($&), ($0, and so on a re not charge- conjugation-invariant. Therefore by imposing the charge-conjugation invariance, the most general gauge-invariant polynomial i s a function of @', @, t 2 , and iJ [ only." As a result, the polynomial has a pseudosymmetry, namely, it is also invari- ant under separate SU(2) transformation on 6, and/ - or on $ and 5 together, i.e., G =SUz(2)'8 SU;.'(~) @ U(1)B U(1). The gauge symmetry G =SUL(2) 3 U(1)a U( l ) i s spontaneously broken to a subgroup S = U(l), i.e., the electromagnetic gauge; corre- spondingly G i s also broken to S = U(1). Thus we have G - S - (G - S) = 3 pseudo-Goldstone bosons, where in this equation i s defined to be the num- ber of generators of the group c, etc.'

Let the vacuum expectation values of the fields be (4 , ) = A , ( a ) =u, and ( C ) = C . The symmetry i s spontaneously broken to the electromagnetic gauge, with the photon

A , = 1

2 x i 2 (g'A3, +gBp + g C , ) ( 2 g 2 + g )

(8)

The charged vector meson has mass

The other two neutral vector mesons a r e also massive. They can be expressed analytically, if we se t h2 = a 2 +C2. In general, the vacuum expec- tation values a r e unrelated, but in order to car ry out the calculation analytically, we shall assume this relation holds without changing the qualitative

10 - P S E U D O - G O L D S T O N E P I O N M A S S E S I N A GAUGE M O D E L 1255

features of our model. We then get mz2 the physical gluons, W:, Z,, X,, and the photon, = (2g2 +gf2)h2, where and in t e rms of the pseudo-Goldstone pion fields,

1 o r be evaluated by using Eq. (5). The latter meth-

Z, = . 2 I,, [JZ g ~ ; + $iBP + c . ) ] , (2g"g od i s simple, once the representation of the gen- era tors in Eq. (5) i s specified. The details will

(9) be presented in Appendix A. (i) Mass 01. the neutral pion. We find by the use and mX2 =g' 'A', where

of (5a) that the coupling of n o with any two vector 1 mesons i s zero; thus the two-vector-meson ex- X u - ( B p - C p ) .

4 7 ( lo ) change diagram (Fig. 1) does not contribute. We

Note that our definition of g and g' [in Eq. (7)] i s a factor of 2 smaller than the ordinary gauge cou- pling constants. The electric charge is given by e = 2gg'/(2g2 +g ' 2)"2 and Q = 3 1 ~ ~ + Y, where Y i s the sum of hypercharges corresponding to the two U(1) groups.

It i s straightforward to find that the four Gold- stone bosons a r e

1 ~ ( ~ @ ' + { o ) n ' + ( 2 ) n i ) , i = l i i 2 , 3 (11)

and

1 - ( - x g 4 + ( u ) o + ( c ) c ) , Jz

and the three pseudo-Goldstone pions a re

where X2 = ( a )' +( C )'. One recalls that pseudo- Goldstone bosons correspond to generators of the pseudosymmetry group (but not the gauge group). This is evidenced by the different sign in Eq. (12), which corresponds to SU; (~)@ S U ~ *"2) transfor- mation a s compared with gauge transformation, which acts on all fields simultaneously [as in Eq. (1 I)]. This point will be clearly seen in Appendix A, which i s patterned after Ref. 2.

To calculate the pseudo-Goldstone pion masses at the one-loop level, we use Eq. (1) [or (2)]. (Here the strong interaction i s turned off, setting the strong coupling constant, which we call f , equal to zero.) The coupling constants in Eq. (1) can either be calculated directly by expressing the interaction Lagrangian [ ~ q . (7)] in te rms of

have checked this by direct calculation from the Lagrangian (7). The four-point vertex (5b) i s non- vanishing for (n Wt W;, (r i 0 ) 2 ~ , ~ , , and (n o ) 2 ~ , ~ , with coupling constants g,o,oW + ,- =g2, g ,o ,oZz =g2 +igt2, and g,onoXx = ig", which one can also easily check from the Lagrangian (7). But i ts contribution (i.e., m2( A ' ) i s exactly can- celed by the tadpole te rm (m2'*'). One can see this point from Eqs. (5b) and (5c), where the two equations a r e equal but opposite in sign i f Oa2 com- mutes with 8, (here B =n O, note that 0 i s antisym- metric). Thus, we find m .02 = O at the one-loop level (and f = 0).

(ii) &lass of the charged pion. The only nonvan- ishing contribution of m .+ 2'AA' comes from W- and X-exchange diagram with

thus

m n t 2 ' A ' has a contribution from Z exchange, X ex- change, and photon exchange, with coupling of the four-point vertices a s follows:

(2 - 2)2 .. nzz = 2(;g2 +gg2) 9

We have

Equations (13) and (14) can be checked directly f rom Lagrangian (7). Similarly, the tadpole te rm has non- vanishing contribution from Z and X. We find [Eq. (5c)]

12 56 T . C . Y A N G

Summing up the above contributions, we have

We see that the quadratic divergence and logarith- mic divergence cancel; we have a finite integral,

The last te rm in the parentheses makes this model different from previous ones. Without it, the mass will be too big to be interpreted a s the pion mass o r mass difference.

Higher-order corrections to the pion masses, including the no, will presumably be nonzero. Un- less n o i s associated with an additional global symmetry which requires it to be massless to all orders (which we do not find here), n o will pick up mass by the very nature of pseudo-Goldstone bo- sons. It should be noted that a next-order correc- tion of form g 4 m w2 will be of the right order of magnitude for the pion mass. Next, we shall in- clude the effects of strong interactions in the pion mass formulas, and from the analysis below, it will be seen that the isospin breaking in this model i s of the order e2. In view of this, (16') should be interpreted a s the lowest-order contribution of electromagnetic and weak interactions to the pion mass differences. Our result i s satisfactory in that this contribution i s not too large a s i s the case in other model^.^ Since A m .'= ie2(3/2n)m ,', we have

where the mixing angle i s defined by g1/2,g= tan@. Expressing the masses in t e rms of the mixing an- gle, one has

We note that if the mixing angle is (near) 0 or i n the pion mass difference i s (near) zero. It should be noted that the mixing angle i s different from the Weinberg angle and the neutral current in this model (which measures the angle) i s different from that in the Weinberg model.

A remark: Suppose we treat the second U(l) gauge symmetry a s the strong-interaction gauge group, where C, i s the strong vector gluon. Then we have the Weinberg model for weak and electro- magnetic interaction and an Abelian gluon for the strong interaction. The strong U(1) group should presumably be characterized by a different cou- pling constant than g, but we expect that the quali- tative features will not be changed. From the pre- vious section, we should expect no pure strong in- teraction contribution to the pion mass, and this i s checked from Eq. (16) where diagrams with X- meson exchanges alone do not contribute. If the strong-interaction gauge symmetry i s non-Abelian, [e.g., color S U ( ~ ) ] , then gauge invariance (color conservation) prohibits diagrams with one weak vector-meson and one strong vector-meson ex- change. On the other hand, for Abelian strong gauge symmetry, this i s allowed [i.e., U'-meson and X-meson exchanges in Eq. (16)].

How to incorporate color SU(3) gauge group in the present model will be dealt with in Appendix B. The crucial points are: (1) The pions a r e pseudo- Goldstone bosons even in the presence of strong interaction; (2) the pions interact strongly also.

Let us now turn to the pion masses including al l orders of strong interaction. Using Eq. (6) we find

10 - P S E U D O - G O L D S T O N E P I O N M A S S E S I N A G A U G E M O D E L

where A;, (A = W, 2, X, and photon) is the propa- coupling constants [in (17), (18)l a r e the three- gator of the vector meson A, j * i s the associated point vertices gndA2 (4 the exchanged meson) o r current and so on. The strong-interaction effect the coefficient of the seagull term, i.e., g,,,,, i s represented by T * product of the currents, with which were given before. the lowest-order weak and electromagnetic cou- Since 8(2g2 -g'2)2/(2gz + g 2 ) =$(2g2 + g i 2 ) - eZ, we pling constants given by the Lagrangian (7). The see that the pion mass difference i s given by

which has the same formlZ a s in the Weinberg mod- e l where pions a r e not pseudo-Goldstone bosons. In the soft-pion limit, the f i r s t te rm seems to dominate,12 which i s the result of Das et aL6 One could evaluate the pion mass itself a s well in the soft-pion limit. The evaluation of the spectral function becomes a problem, since low-lying pole dominance seems unwarranted.

From (19), we see that the pion mass difference is characterized by the square of the electromag- netic coupling constant e2. We have neutral pion mass m ,02 = m - A m ,.' a s given by (17). The neutral pion mass need not be too small; the value depends on the evaluation of the spectral functions a s we have commented above. In this model, a s in the Weinberg model, the isospin breaking i s of the order e2, which can also be seen directly from the mixing of the SU(2) and U(1) gauge gluons.12 In comparison, the isospin breaking in some other models3 need not be of order e2, which would pre- sumably spoil the result of Das et al.'

IV. DISCUSSION AND CONCLUSION

We have generalized the Weinberg SU, (2)B U(1) model by introducing an additional U(1) group. By so doing, one notes the following: (1) The model has more flexibility, since two neutral vector me- sons can be exchanged. If experiments should turn out not to agree consistently with the Weinberg model, an additional gluon exchange will be the most simple and natural explanation. (2) The mod- e l has three pseudo-Goldstone bosons i f one im-

poses a charge conjugation invariance a s specified before. (3) The masses of the pseudo-Goldstone bosons a r e given by functions of the masses of the gauge vector mesons which can be made small by properly restricting the mixing angle of the cou- pling constants. The pseudo-Goldstone boson masses in models studied before turn out to be re- lated to the masses of the gauge vector mesons, which will be an order of magnitude too large to be pion masses. The present model does not have such a drawback. We identify the pseudo-Gold- stone bosons a s the pions.

One notes that the pseudo-Goldstone boson mass- e s have no contribution from the scalar-meson ex- change diagrams. Thus at the one-loop level we a r e able to write the masses in t e rms of time-or- dered products of vector o r axial-vector currents (corresponding to vector-meson exchange). On the other hand, if pions a r e bound states of quarks, then there i s no a p i o r i reason why the scalar current contribution (corresponding to scalar-me- son exchange) i s small o r not present. Since one lacks knowledge about commutators involving sca- l a r currents, this presents an obstacle to the eval- uation of the pion mass by the use of current alge- bra.

In the study of PCAC and symmetry breaking of strong interactions, one recalls that the u model with Goldstone pions has proven to be very useful. We hope that the study of the idea of pseudo-Gold- stone pions will shed some light on the origin of the pion mass, and may be helpful in the bound- state model of pions.

T . C . Y A N G 10

ACKNOWLEDGMENT

The author would like to thank Professor J. Pati, Dr. R. N. Mohapatra for useful discussions, and Dr. D. Kershaw and Dr. G. S te rman f o r reading the manuscript.

APPENDIX A

In this appendix, we evaluate the coupling con- s tants by the use of Eq. (5). We note that Eq. (5) is defined ( see Ref. 2 ) by grouping a l l the r e a l s c a - l a r fields a s a big column matr ix, and 6 1 , a r e the rea l representat ions of the genera tors t imes the coupling constants acting on the big column matrix; h i s the vacuum expectation value of the column matr ix. One of the (equivalent) r e a l represen ta - tions of the T mat r ices can be wri t ten a s

O i 0

0, = 0 0 0 - i

O O i 0 0 - 1

which sat isfy the algebra Big, = i2ci jk O k , i , j , k = 1,2, 3, and [ @,, B,] = 0 , The corresponding s c a l a r fields can be represented by a T = (-bl, a2, a3 , 44), $T = (-n I , n 2 , 7r3,u), and f ' = ( - I I ' , ~ I ~ , I I ~ , 2 ) .

Writing al l s c a l a r fields a s a big column matr ix J ( ,

where 9T = (bT, bT, 5 *), the Lagrangian (7 ) can be written a s

where

In o r d e r to use Eq. (5), we need to find out the representat ion BA corresponding to the physical vector mesons A, where A = W1, 1Y2, 2, X, and the photon A . The 0,'s a r e defined by

By the use of Eqs. (8)-(lo), we find

10 - P S E U D O - G O L D S T O N E P I O N M A S S E S I N A G A U G E M O D E L 12 59

Since ( q T ) =(o, O,0, h; 0 ,0,0,u; 0 ,0,0, z), we im- of the minus sign. F r o m Eq. ( l l ) , one c a n easily mediately find that the m a s s mat r ix is diagonal check that the r e a l Goldstone boson projection op- with m a s s e s given before, namely,

mW2 = -(Bw ( 9 )Ii(&(* )Ii

e r a t o r s belong to the gauge group. Using (A1)-(A3), Eq. (5) can be straightforward-

ly calculated. We find that the only nonvanishing =g2(X2 + u +C2) coupling constant of (5a) i s the n i W Z vertex, a s

given by (13). The nonvanishing four-point v e r - t i ces (5b) and the tadpole contribution (5c) a r e giv-

m , 2 = - ( e z ( * i ) i ( ~ Z ( 9 ) ) i en by (14) and (15), respectively.

= (2g2 + g ' 2 ) ~ 2 APPENDIX B

mx"-(8, ( a ) I i (& (*)I, In th i s appendix, we shal l give a n example in = g ' 2 ~ 2 which the pions interact strongly with the color

vector gluons, but remain a s pseudo-Goldstone m A 2 = 0 ,

bosons. T h e model cons i s t s of two doublets $I and where we use u + C2 = h2. 4 which have no s trong interaction a s given before.

The l a s t thing we need t o know is the projection In addition we have th ree s c a l a r representat ions operator of the pseudo-Goldstone bosons, defined which t rans form a s doublets with respect to SU, (2),

by and t r iplets with respect to the color SU(3) gauge

where j F, / [-(8, ( Q ) ) ~ ( @ , ( P \ ) ~ ] ' ! ~ . F r o m Eq. (121, jv I = i ( H t ) i , , + , i ( r ~ ' ) , ~ , we find that (C - in3) i1 , (C - (C - in3),,

- 8' i = 1, 2, 3

e B = & ? ( @' @,) 2' ' (A3) where SU(2) indices a r e labeled by column and the color SU(3) indices a r e labeled by row. ,lil'" t r ans - f o r m s a s a triplet, i = 1, 2, 3, with respec t to a

It i s c l e a r that 0, belongs to the group S U * ( ~ ) global color SU'(3) group, where Q, & a r e singlet. @ S U ' ~ " ~ ) , but not the gauge group SU(2), because The gauge-invariant Lagrangian is

where A:, B,, C, a r e the weak and electromagnetic +ni,,) and ( C ) replaced by a, C ) (note that h2 = u gauge mesons, V , the color gluons, with g, g', +C2 i s replaced by h2 = u 2 +3C2). AS discussed be- and f the corresponding coupling constants. We fore, the color-gluon exchange does not contribute s e e a s before the gauge-covariant der ivat ives a r e to the pion m a s s a t the one-loop level. Indeed, invariant under the charge conjugation @ 2 T,+*, one finds that the coupling constants fo r pseudo- B, - -B,. Thus by imposing the above d i sc re te Goldstone pions with the weak gauge mesons a r e symmetry, the potential h a s a pseudosymmetry, the s a m e a s before.

G = S U ? ( ~ ) % SU:*"~(Z)@ ~ ( 1 ) a U ( ~ ) B S U ~ ( ~ )

a s compared with

G =SU,(Z)P U(1)S U(1)% s u C ( 3 ) .

The color SU(3) i s completely broken, by assigning vacuum expectation values ( bi ) =h, ( o ) =u, ( C i j ) = C6ij, with m V 2 = f 'c'. The m a s s e s of the weak gauge mesons a r e given a s before by replacing C2 by 3C2. The pseudo-Goldstone pions a r e again given by (12) with n i replaced by ( l / n ) ( n : , +n:,

We note that the pions a r e not singlet with r e - spec t to the color SU(3) group and thus interact strongly with the color gauge gluons. But the pi- ons a r e singlet with respec t to the color group SU"(3) consisting of the genera tors {pa B 1 + 18 p&), where pa and p& a r e genera tors of the color SU(3) and SU'(3) group. In this model, the classification of the hadrons should go by the SUN(3) group. The pari ty and isospin of the pions a r e classif ied by the ch i ra l SU(2)E SU(2) group, which is a n exact group of the s t rong interactions.

1260 T. C. YANG

*Work supported in pa r t by the National Science Foun- dation under Grant No. GP 32418.

's. Weinberg, Phys. Rev. Lett. 2, 1698 (1972). 2 ~ . Weinberg, Phys. Rev. D 1, 2887 (1973). 3 ~ . Y. Lee, J. M. Rawls, and L.-P. Yu, Nucl. Phys.

B68, 255 (1974); J. Lieberman, Phys. Rev. D 9, 1749 - (1974).

4 ~ . Bars and K. Lane, Phys. Rev. D 8, 1169 (1973) ; 8 , 1257 (1973).

'ST Coleman and E. Weinberg, Phys. Rev. D 1, 1888 (1973).

%. Das, G. S. Guralnik, V. S. Mathur, F. E. Low, and J. E. Young, Phys. Rev. Lett. 18, 759 (1967).

's. Weinberg, Phys. Rev. D 8, 605 (1973); 8, 4482 (1973); R. N. Mohapatra, J. C. Pati, and P. Vinciarelli, i t id .

8 , 3652 (1973). Weinberg, Phys. Rev. D 5, 1412 (1972).

'T. C. Yang, Nucl. Phys. E, 541 (1974). l o w e mean setting the Abelian coupling constants equal

after the renormalization. For the present calculation, this does not cause any complication.

11 P(@,1/',&) = a 1 @ 2 ~ b 1 + 2 a c 1 t 2 +a2(q2)'

We see L i s also invariant under 5- + (we can also impose reflection symmetry q- -$ o r 5 -- k ; then d , = 0) . P is automatically chiral SU(2) 8 SU(2)-invari- ant.

"D. A , Dicus and V. S. Mathur, Phys. Rev. D 2, 525 (1973).