P I-I Y S I C A L R E V I E Vjj' VOLUM E 184, NUM 8 ER 2S AUGUST 1969

Nonlinear Realizations. I. The Role of Goldstone Bosons

AHDUS SALAM* AND J. STRATHDEE

International Atomic L&'nergy Agency, International Centre for Theoretical Physics, Miramare, Trieste, Italy(Received 26 December 1968)

A method, based upon analogies with the Wigner boost technique, is presented for setting up the nonlinearrealizations of any continuous symmetry group. It is argued that such realizations are relevant to and usefulin the treatment of cases of spontaneous symmetry breakdown.

1. INTRODUCTION

'HE equations of physics are usually expressed ina form showing manifest covariance under the

transformations of the space-time and internal-sym-metry groups. Recently, however, some attention hasbeen paid to the possibility of expressing internalsymmetries of the chiral type in a fashion which is notmanifestly covariant, some of the transformations ofthese groups being realized nonlinearly. ' In the presentpaper the nonlinear method is shown to provide aneconomical framework within which to treat theproblems of spontaneously broken symmetries, ' i.e.,symmetries which are present in the Lagrangian butnot in the ground state. Beyond this, the view will bepresented that the nonlinear method, as a methodembodying dynamics rather than pure group theory,is applicable only to situations in which the symmetryis broken spontaneously. Much of the following dis-cussion will be in the nature of a review. However, onepurpose of this is to present the notational develop-ments that emerge through a consistent use of thelanguage of Wigner boosts which, it seems to us,greatly clarifies the subject of nonlinear realizations.We shall need this development in particular in PaperII, where we consider nonlinear realizations of theconformal group in space-time.

The discussions in Secs. j. and 2 concern the generalproperties of nonlinear realizations. These have beendealt with in great generality elsewhere. ' However, inorder to make clear the notation adopted in this paperand in Paper II, it will be advantageous to go over thesematters in some detail. Moreover, this review will serveto introduce our main idea, which is that the nonlinearrealizations are intimately involved in situations wherethe symmetry breaking is spontaneous. This idea willbe further developed in Sec. 3. Finally, to illustrate the

* On leave of absence from Imperial College, London, England.For a recent compilation of the very numerous references to

nonlinear realizations see the review by S. Weinberg, in Pro-ceedings of the Folrteenth International Conference on High-LnergyPhysics, Vienna, 196h' (CERN, Geneva, 1968), p. 253.' J. Goldstone, Xuovo Cimento 19, 154 (1961); J. Goldstone,Abdus Salam, and S. Weinberg, Phys. Rev. 127, 965 (1962);T. W. B. Kibble, in Proceedings of the Conference on Particles andFields, Rochester, 1067 (Wiley-Interscience, Inc. , New York,1967).

3 S. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177, 2239(1969); C. G. Callan, S. Coleman, J. Wess, and B. Zumino, i'.177, 2247 (1969); C. J. Isham, Imperial College, London, ReportÃo. ICTP/67/25 (unpublished}.

method, Sec. 4 is devoted to the nonlinear realizationsof SU(3), which become linear with respect to thesubgroup SU(2))& U(1). Thus, the material of Secs. 1

and 2 consists of known results expressed in a newnotation, while that of Secs. 3 and 4 is essentiallynew.

Central to the method of nonlinear realizations isthe notion of a preferred field that provides a bridge torepresentations which are linear but constrained. Thereis in fact a complete duality between sets of fields, onthe one hand, which transform linearly while beingsubject to certain nonlinear constraints and, on theother hand, equivalent sets of unconstrained fieldswhich transform according to nonlinear realizations.The preferred field is used in the formation of covariantequations of constraint upon the set of fields whichtransforms linearly. If these constraints are then usedto eliminate all dependent components, there resultswhat is commonly called a nonlinear realization. Morespecifically, the realization will be linear with respectto all but the preferred field itself, which generallyenters nonlinearly. The structure of the realizationsarrived at in this way must, of course, depend criticallyupon the particular set of fields which are chosen toplay the preferred role. This choice depends in turnupon the nature of the vacuum symmetry since, aswill be seen in the following, the preferred fields areneither more nor less than the field of the Goldstonebosons. 4

A rather trivial and very familiar example of theprocedure just outlined is provided in the case of chiralSU(2)XSU(2).4 Let the chiral four-vector w play therole of preferred field and let it be employed in theformulation of a set of algebraic constraints upon thefieMs P and I" p, a chiral four-vector and six-tensor,respectively. For the equations of constraint one might.take, for example,

~a4'p ~+a f~aii y

7i F p fif p=O, —

where f denotes a numerical constant. The questionof how such constraints could arise within the contextof a dynamical model is not, for the present, at issue.

We wish to emphasize that this paper is not about chiral sym-metries, which are discussed in this section as an illustration only.Our interest is with the general method of nonlinear realizations,of which the chiral realizations are one particular example.

NONLI NEAR REALI ZAT IONS. I 1751

lt follows from (1.1) that, in particular,

~.y.=O,(1 2)

so that the 1.6 various components can be expressed interms of six independent ones, say, n and $. The lineartransformation laws appropriate to the chiral four-vectors x and P are exemplified by

~1,2 ~1,2 )

m3 ~ vr3 COm —x4 Sin&a,

m4 —& +3 Sinco+m4 COSco,

(1.3)

corresponding to a purely chiral transformation. Itremains only to eliminate the dependent componentsn4 and $4 by means of the constraints (1.2). Theindependent components n and $ then transformaccording to nonlinear realizations which are exempli/edby

and

~1q2 ~1q2 )

x,,~ s 3 cosco —(f' —n')"' sina,(1.4)

and, for Q;,

cosa'+ (z 3/f) since

Z 3 cosa —f sin&a

(P3 ~cosco+ (q 3/f) sinu&

(1.7)

4i,2~ 4i, 2,

$3~ gz coma+ (1/f) $ rp sin~, .(1.8)

which replace (1.4) and (1.5), respectively. The real-ization (1.6) was adopted by Schwinger. ' We havereproduced it here because it lends itself readily to"proving" the noninvariance of the vacuum. Thus,for infinitesimal co the transformation (1.7) gives, inparticular,

EQ, a'3j= b~3=~f(1+ ~3'/f')— (1 9)the vacuum expectation value of which cannot possibly

' J. Schwinger, Phys. Letters 248) 473 (1967).

')i., 2 ~ 4i, 2,1.5

P3 ~ $3 cosa)+ f$ m/(f' n2)'~2) sin—a,and one sees that only the preferred field ~ entersnonlinearly. As is well known, the form of this resultdepends very much on the manner in which m isparametrized. An alternative scheme would present

in the form

sr, = p,/(1+op'/f')" i=1, 2, 3= f/(1+z'/f')"' (1.6)

where the three-vector q; is independent. Corre-sponding to the purely chiral transformations (1.3),one finds for q; the transformation law

vanish' since the right-hand side is positive-definite.The realization (1.6) is possible only if the zacggm isdegenerate. Notice that this argument does not dependon the existence or nonexistence of a Lagrangian.

For alternative parametrizations such as (1.4), it isnot possible to make such a categorical statement.However, for practical purposes where the nonlinearitiesare always interpreted by power-series expansions inn/f, the implication is the same. Nonlinear constraintscan be dealt with by power-series methods only intheories with degenerate vacuums. The particles asso-ciated with the preferred fields are, in fact, the Gold-stone bosons. One way to see this is to remark thatnowhere in a chiral-invariant Lagrangian does the field~ appear without being accompanied by B„~ as well.It follows that the fields ~ must describe massless(spin-zero) particles. Thus a vacuum state is indis-tinguishable from a state with two zero-frequencypions, four zero-frequency pions, etc. , provided thepions together form an I=O multiplet of the subgroupSU(2). Now, since the pions form an incompletemultiplet of SU(2)XSU(2), it is clear that thesephysically degenerate and indistinguishable states oflowest energy will lead to a (spontaneous) breakdownof chiral symmetry.

So far we have said that if nonlinear realizations areintroduced by considering linear realizations of thepreferred fields together with a constraint, the con-straint implies that the vacuum state in the theorymust be a noninvariant state, and the symmetry aspontaneously broken one, with the independent onesamong the preferred fields playing the role of Goldstonebosons. Consider now the converse problem: Given atheory with a chiral-invariant Lagrangian it mayhappen that the ground state is not chiral-invariant.In this case there must appear Goldstone bosons corre-sponding to the components of the symmetry which areabsent from the vacuum, One may introduce into thetheory a set of spin-zero fields describing these bosons.The problem one is presented with is how to couplesuch mesons with other particles so that the conse-quences of the vacuum asymmetry are made explicit.These sects must include, for example, the guaranteethat these particles remain massless even after inter-acting with other particles. The formalism must alsogive a correct account of the perturbations of massesand coupling constants on account of symmetrybreaking. Olr solltion to the problem is to suggest thatthe appropriate formalism is the one where a nonlinearrealization is employed with these spin zero fields playin-gthe role of the preferredPejds

The problem of making explicit the Goldstone bosonscan be solved in various ways. Suppose, for simplicity,

'S. Gasiorowicz and D. GeBen in their excellent review ofchiral symmetries tArgonne National Laboratory Report No.ANL/HEP 6809 (unpublished) j make this point though they donot stress how vital it is for the dynamical relevance of the wholenonlinear method.

7 L O'. S.Kibble, Phys. Rev. 155, 1554 (1967).

3.752 A. SALAM AND J. STRATH DEE

and belongs to the nonlinear realization (1.5). Thesecond term in (1.10) contains the term ', (8„-+)'together with an infinite number of interaction termswhich arise from the expansion of 1/x in powers, i.e.,

X' X'2

x (x)+x' (x) (x)' (x)'~ ~ ~

)

which is meaningful provided (x)WO. The nonvanishingof (x) must be thought of as a consequence of thesupposed vacuum asymmetry. '

The particle which is characterized by the isoscalarfield X' is of no particular importance in the theory. Itarose as a by-product of the e6'ort to set up an effectiveLagrangian with Goldstone bosons. Having obtainedthe e6'ective Lagrangian, one is at liberty to set X'=0.Only the numerical part (x) must be kept.

An important exception to the theorem which requiresthe presence of massless bosons in situations where asymmetry is broken intrinsically occurs when long-range vector fields are also present. ' If the currents ofthe spontaneously broken symmetries are coupled to agauge field of the Yang-Mills type, then the symmetrybreaking manifests itself not through the appearance ofGoldstone bosons but rather in the acquiring of massby some of the components of the gauge field. Thisphenomenon, which was discovered by Anderson anddeveloped by Higgs and Kibble, will be presented in thenonlinear notation in Sec. 2 where it will be shown thatthe preferred field disappears from the Lagrangian ifthe symmetry group is gauged. In addition —and thisis where we improve on Higgs and Kibbl- the residualquantized objects such as X', whose existence is requiredin their models, can be set equal to zero without doingany violence to the elegant formulation aAorded bythe nonlinear formalism.

Turning back to the formulation of the nonlinearmethod, it will be remarked that, in Eqs. (1.1), thefield x plays a role which is formally similar to that

In a given model it might be possible to compute this quantityby a self-consistent method. That is, substitute x=(x}+p' inthe Lagrangian, and set equal to zero the perturbation corrections«(x').

~ P. %'. Anderson, Phys. Rev. 130, 439 (1963); P. W. Higgs,Phys. Letters 12, 132 (1964); Phys. Rev. 145, 1156 (1966); G. S.Gura»i&, C. R. Hagen, and T. K. B. Kibble, Phys. Rev. Letters13, 585 {1964);T. %.B.Kibble, Ref. 7.

that the Lagrangian contains a zero-spin chiral four-vector C; then it is possible to eliminate C4 in favorof x= (C C )'I'. The kinetic energy then takes the form

—,'(B„c B„c —m'4, 4 )=-', (a„xB„x—m'x')

+~sD„+ D„+, (1.10)

where D„C denotes the so-called covariant derivativeof +. It is given by

taken by the four-momentum p in the formulation ofrelativistically covariant free-field equations. Theanalogy can be extended to interaction terms as well.The involvement of orbital angular momentum in therelativistic coupling of particles with spin is accountedfor by the presence of terms such as y 8/Bx in theLagrangian. Likewise, the involvement of soft pionsin the chiral-invariant coupling of particles with isospinis brought about through terms such as F 7r =m4+pg'0 ' 'z.

We are not advocating the exploitation of thisanalogy as a practical way to make chiral-invariantLagrangians. The existing method which uses nonlinearrealizations directly is a simpler one to apply. However,there is another aspect of the analogy between x and

p which leads to a formal development of some power.This lies in the notion of the boost. It was Wigner'sdiscovery" that the momentum four-vector p„couldwith great advantage be represented in the form

p.= (L,).4m, (1.12)

A ~ E(p,A) = Lp „'AL„, (1 14)

where E(p,h) is, in effect, an ordinary space rotation.This realization is, of course, essentially the same as thenonlinear realization (1.5) obtained in this instance bysetting p4

——(p'+m')"'. It is perhaps worth noticingthat, insofar as finite-dimensional realizations areinvolved, the distinction between the compact groupSU(2)XSU(2) and its noncompact relative SL(2,C)is a minor one.

We propose to adopt the method of Wigner for dealingwith nonlinear realizations. That is, we shall express thepreferred field in the form

s =(L.).4f, (1.15)

where (L ) s denotes a 4X4 matrix whose componentsare dynamical variables. These variables are not allindependent. They are subject to the constraints

(1.16)

or, in other words, L belongs to So(4). Included

10 E. P. signer, Ann. Math. 40, 149 {1939).

where m denotes the rest mass Qp', and (L„) s de-notes a 4&&4 matrix belonging to the Lorentz group.The mass-shell constraint p p =m' is accounted forautomatically in the representation (1.12) by the(pseudo) orthogonality of the matrix L„.The power ofthis representation lies in the fact that it leads to therealization of the Lorentz group in terms of 3)&3orthogonal matrices (or 2X2 unitary and unimodularmatrices). Thus, corresponding to the I orentz trans-formation

pa~ pa =+eppp&

one has the realization

184 NONLI NEAR REAL IZATIONS. I 1753

among the constraints (1.16) is of course the principalone, v.,v. =f .

The main advantage to be gained by replacing thepreferred 6eld v. with the matrix (L ) v is the easewith which such a matrix can be used to eGect a passagebetween linear and nonlinear realizations. Moreover,it enables one to discover the general features of a classof nonlinear realizations without the complication ofhaving to commit oneself to a particular paramet-rization. It is this formal power which makes the boostapproach useful for generalizing beyond the chira1groups. However, to avoid semantic confusion we shallinvent the new name reducing matrix for the matrixI. and its generalizations, since the word "boost" hasalready a rather precise meaning within the context ofthe inhomogeneous Lorentz group and itsrepresentations.

The realizations of a continuous group G, whichbecome linear when restricted to some specified sub-group H, are treated by means of the reducing matrixin Sec. 2. These realizations are then gauged in theYang-Mills manner. Section 3 contains some generalremarks about symmetry breaking, both spontaneousand explicit. The formal techniques are illustrated inSec. 4 on a model which could have practical interest,namely, the nonlinear realizations of SU(3) whichbecome linear with respect to SU(2)rX U(1)r.

D(g)+, a&G (2.1)

and their decomposition into linear irreducible repre-sentations of H are known. It will prove convenient toassume that the basis has been chosen so as to renderthe matrices D(h), where h&H, block-diagonal in form.

The first stage in solving the realization problem isthe de6nition of a matrix (Lz) v, the elements of whichare field variables. This matrix, which we shall call thereducing matrix, mill be subject to a number of algebraicconstraints and will be endowed with a peculiartransformation behavior under the operations of thegroup G. The basic requirements are the following:

(a) The matrix Lq is constrained to belong to thegroup G. i.e., to its self-representation. This is in orderthat, for any finite-dimensional representationg ~ D(g), the functional D(L~) shall be well defined.The number of independent fields, p, needed to pa-rametrize I& is therefore equal to or less than thedimensionality of G.

(b) Under the operations of the group G the Qeldswhich make up the reducing matrix transform according

2. NONLINEAR REALIZATIONS

Consider the problem of constructing nonlinearrealizations of a continuous group G which becomeboth linear and irreducible under some specified sub-group II. One may suppose that the linear irreduciblerepresentations of G,

toLp-+ gL~h '(»g), (2.2)

where gEG and h(g, g) QH. In other words, the columnsof the reducing matrix must be arranged into setswhich transform among themselves according to somerepresentation of the subgroup H.

(c) Under the operations of the subgroup P thereducing matrix transforms in the ordinary way,

l.~ —+ hl.ph '.It follows from (2.2) that the functionals D(L~),

which are defined for any finite-dimensional repre-sentation, transform according to

D(4) ~ D(gL~h ') =D(g)D(L~)D(h ') (2 4)

It is this property which enables one to project non-linear realizations out of linear ones such as (2.1) bythe operation

4=D(L~ ')+. (2.5)

A comparison of (2.1) and (2.4) yields for f the trans-formation law

4 ~D(h)4, (2.6)

where h=h(g, g) is, in general, a nonlinear structurewhich depends upon the preferred fields P, whichparametrize Lz.

The detailed form of the matrix h(g, g) is dependentupon the parametrization scheme, i.e., it depends uponwhich combinations of the components (L&) s aretaken as independent variables. Perhaps the simplestscheme is the one adopted by Coleman, Ress, andZumino' and, earlier, by Kibble~:

where A, denotes the set of infinitesimal generators ofG that are not contained in the algebra of II. %hateverthe scheme chosen, one can discover the matrix h(»g)by referring Eq. (2.4) to a representation of G whichcontains a singlet of H. Such a representation includesat least one column, X, for which (2.4) takes the form

D(Lq)x ~ D(Lg.)x.=D(g)D(Lq)x, (2.I)i.e., for which D(h) is represented by the identity. Ifthe chosen parameters p, are expressed in terms of thecomponents of the column which transforms accordingto (2.7), then it is straightforward to compute thetransformation law of these parameters. Having donethis, one can compute the matrix h by comparison with(2.2), i.e.,

h(»g)=L~ ' gL~. (2 g)

The method will be illustrated in Paper II for the caseof the conformal group "

Consider now the problem of defining a coeariarltderivative operator for the nonlinear realization (2.6).

~1 A. Salam and J. Strathdee, following paper, Phys. Rev. 184,1760 (1969).

A. SALAMI A X D J. STRATH BEE 184

A„iP=D(Lp ')B„V, (2 9)

which is clearly covariant. However, one should notadopt A„as the desired covariant derivative since itdepends upon the imbedding representation D(g). Inorder to remove this dependence it is necessary toanalyze (2.9) more closely. Write

~u0=~A+D(Ld ')~ID(L~)4 (2 1o)

The matrix D B„D can be simplified if use is made ofthe constraint L~gG for all x. In particular, it follows

that the matrix

L~ '(x)Lq(x+b—x) =1+bx„Lq '&„Lp+

is an infinitesimal transformation of G. In other words,the matrices I& '8„1.~ belong to the infinitesimal

algebra of G. They can therefore be expanded in theform"

L~ 'B„L~=i(Lq 'B„Lq);s;, (2.11)

where the matrices s; constitute a basis of the algebraand the coefFicients of the expansion are denoted by(Lz 'B„L&);. In the representation D(g), where theinfinitesimal generators s; are represented by 5;, theexpansion (2.11) takes the form

D(Lq ')B„D(L~)=i(L~ —'B„L~);S;. (2.—12)

The transformation behavior of the coefFicients in(2.11) is complicated by the presence of the derivativeoperator. From (2.2) one finds that

L~ 'B„Lp-+ It(Lp 'B„Lp)h '+hit„h '. (2.13)

That is, there is present, in general, an inhomogeneousterm in the transformation law. Clearly, however, theinhomogeneity belongs to the algebra of IJ. This pointis of crucial importance because it means that the fields

(L& 'B„Lz); can be divided into two sets, one of whichtransforms covariantly while the other contains theinhomogeneity.

Let us suppose that the algebraic basis s; has beenchosen in such a way that it can split into two com-ponents m and n, I2 which transform independently

'~ A summation over the repeated subscript i is implied. Onoccasion it will be useful to divide the sum into two parts, oneover the subalgebra H indicated by repeated greek subscriptso., p, ., and the remainder indicated by repeated latin subscriptsfrom the beginning of the alphabet, a, b,

It is evident that the ordinary derivative is not co-

variant:tt„$ D(h) it„$+B„D(h)iP

In order to be able to make covariant field equationsit is essential that one define a covariant operatorresembling the derivative. This can be done in thefollowing way.

Let us imbed ip in some linear representation D(g):

iP=D(Lq ')0';

and let us define, relative to it, the operator 6„:

ct„%'(x) ~ D(g)8„%'(x)+B„D(g)%(x). (2.19)

With the basis s; defined above, one can write

(I/t)g~. g '= (g~.g ')'s* (2 2o)

since the matrices g '8„g belong to the algebra of G.Therefore, (2.19) can be expressed in the form

~.+(x) ~ D(g)L~.+i(g~.g ')*~*X(x) (2 21)

In order to replace this with a covariant formula, onemust introduce a set of gauge fields

A„=A„;s;, (2.22)

under H. Moreover, suppose that the m constitute abasis for the subalgebra H. In this basis the expansion(2.11) takes the form

(1/i)Lg 'B„Lg=I'„m +AD„g,n, , (2.14)

which is to be looked upon as the definition of the field

quantities I'„and D„P . The inhomogeneous term inthe transformation (2.13) affects only the I'„; thefields D„@, therefore belong to a bona fide nonlinearrealization of the group G. They are to be interpreted asthe corariant derivatii, es of the preferred fields p, iN terms

of which the reducing matrix is parametrized. The realparameter A. will be fixed by normalizing the kinetic-energy term associated with g, .

Corresponding to the expansion (2.14) one has, inthe representation D(g},

(1/i)D(Iz ')B„D(L&)=1'„M +AD„g,iY, , (2.15)

which can be substituted into the expression (2.10)for 6„:

h„ip= B„i'd+i I'„7tl i'd+i AD„Q,X f. (2.16)

Since the left-hand side of (2.16) transforms covariantly,as does the third term on the right, therefore the sumof the remaining two terms must also be covariant.The latter part, denoted D„|P, has in addition the re-quired property of being independent of the imbeddingrepresentation. Thus, the covariant derivative of |P is

D„P= a„P+ r„.3d.y. (2. 7)

Finally, consider the problems which arise when thetransformations of the group G are made space-time-dependent; i.e.„when G is turned into a gauge groupof the Yang-Mills type, we have

+(x) D(g)+(x), g=g(x)EG. (2.»)There is no need to alter the prescription (2.5) forextracting the nonlinear realizations from 0 . Indeed,the nonlinear transformation law (2.6) is formallyunchanged since the matrix h(gg) is defined forarbitrary g(x)CG. The modifications are needed, ofcourse, in the definition of covariant derivatives.

Now, it is well known that the ordinary derivativeis not covariant under space-time —dependent trans-formations:

NONLI NEAR REALIZATIONS. I 1755

which transform, according to the following law:

A„+g—A„g '+(1/if)gB„g ' . (2.23)

The covariant derivative for linear representations isthen defined by

n„+= (8„+ifA„;$,)% . (2.24)

In the usual fashion, the covariant derivative of thegauge field itself is contained in the antisymmetrictensor

F„„=B„A„B„—A„+if/A„,A„1 . (2.25)

From the expressions (2.24) and (2.25), which arecovariant in the linear sense, one can project out thegeneralized nonlinear covariant derivatives. First, if fis defined by (2.5) its covariant derivative must becontained in the operator

h„P=D(Lg ')(8„+ifA„;S;)4,which can be simplified to the form

h„P= (8„+ifB„;S;)f, (2.26)

where B„;is a modified gauge field defined by

B =L~ iA Lq+(1/if)Lq i8 Lq. (2.27)

It transforms under a general gauge transformationg(x) according to

B -+hB li '+(1/if)hr7 li '. (2.28)

The most important feature of this transformation lawis the fact that the inhomogeneous term belongs tothe algebra of H. This means that the operator h„of(2.26) can be separated covariantly into two pieces,

h„f=D„/+A(D„p, )X,f,which defines the generalized covariant derivatives

D„f= (8„+ifB„M)P (2.29)

covariant form D„f as given in (2.17) and to takeaccount of the new zero-spin boson field P„whoseexistence this implies, by adding a covariant kinetic-energy term

and, possibly, adding other derivative coupling termsusing D„p . It is not possible to construct any covariantobject which contains a term like qb, p, and it thereforefollows that the new bosons must be without mass.

The invariance of the Lagrangian can be furtherenlarged to include the space-time —dependent trans-formations of G by introducing a gauge field B„whichtransforms according to the reducible nonlinear real-ization (2.28). The covariant derivatives D„f andD„P, are given the new forms (2.29) and (2.30),respectively, while for the gauge field it is necessaryto adjoin the kinetic-energy term

where B„,is given by (2.31).The upshot of these 6nalmodifications is that the preferred field P and itsmassless quanta have disappeared from the Lagrangian.They have been absorbed by a redefinition (2.27) ofthe gauge field. Not only has the multiplet of masslessbosons P disappeared, but part of the gauge field,B„„has acquired a well-defined mass f/X. The otherpart, B„, which enters the covariant derivatives,remains without mass.

This phenomenon, whereby the introduction of agauge multiplet of vector particles causes the dis-appearance of the massless zero-spin particles, has beendiscussed by a number of authors' in the context ofspontaneous symmetry breaking. In Sec. 3 we showthat this is precisely the context in which nonlinearrealizations have meaning. The massless zero-spinparticles are indeed the Goldstone bosons.

kD„&,= fB„,.In this section we demonstrate that the formalism of

nonlinear realizations and effective Lagrangians pro-vides a natural framework for treating intrinsicallybroken symmetries. The arguments given here parallelthose of Kibble. 7

To discuss a system with spontaneous symmetrybreaking one must have in mind a Lagrangian whichis invariant with respect to the transformations ofsome continuous group G. Secondly, one must assumethat the ground state or vacuum is not an invariantof G but only of some subgroup H. This property ofthe ground state is signalled by the nonvanishingexpectation values of fields or combinations of fieldswhich belong to nontrivial representations of G. Itsconsequences include symmetry-breaking perturbationsof the masses and couplings of physical particles and,in particular, the appearance of spin-zero masslessbosons.

These expressions are covariant against gauge trans-formations of the second kind. It remains only to findthe covariant derivatives of the fields B„and B„,.These are contained in the antisymmetric tensor

B„„=Lp 'F„„I.~,

which, in view of the delinition (2.27), takes the form

B„„=a„B, a„B„+if)a„,—B„j (2.31).The realizations discussed in this section can be used

in the construction of Lagrangians that are invariantwith respect to the transformations of G. First, aLagrangian which is manifestly invariant with respectto the subgroup H can be modified so as to becomeinvariant with respect to the space-time —independenttransformations of the larger group G. It is necessaryonly to replace the ordinary derivatives 8„$ by their

(2 30) 3. SPONTANEOUS SYMMETRY BREAKING

A. SAr AM AND J. STRATHDEE

Consider a system of fields, fermions and bosons,denoted collectively by 0' and, in addition, a spin-zero

multiplet M (of which some components M will

correspond to Goldstone particles), which transform

according to the reducible linear representations

4' —+ D(g)%', M +D—(g)M, (3.1)

where g denotes an element of G, and D and D are thematrices appropriate to the representations concerned.The Lagrangian of this system is supposed to be in-

variant under these transformations:

L(%',8„+,M, B„M)=L(1PP,DB„%',DM,DB„M) . (3.2)

This means that the system is classified into completemultiplets of G with the various couplings that areallowed by this syrrnnetry. It m.ay therefore be quiteunlike the physical reality which reflects the ground-

state asymmetry. The mass splittings of the physicalmultiplets can be large and, in fact, so large that someof the multiplets may be regarded as incomplete. Thesame is true for the couplings.

We know that in such a system Goldstone bosonsmust be present. Being massless, these particles causea readjustment of the stable states of the system. Inparticular, the physical vacuum should contain anadmixture of zero-energy Goldstone particles (whichis to say that the vacuum degenerate). This propertyof the Goldstone particles can be formulated as eGectinga redefinition of the "bare" masses and couplingconstants that takes account of the ground-stateasymmetry. We wish to introduce a set of fields p, (x)to represent the Goldstone particles and to put theLagrangian (3.2) into a form which, lhough still invariant under the trausformatious of G, shows explicitly,in its bare masses and coupling constants, the effectsof the underlying asymmetry. As stated in the Intro-duction, the method of nonlinear realizations, with thepreferred fields p, (x), appears to be just the rightconstruct to solve this problem.

The subset M of M referred to earlier transformsunder the subgroup H, like the set of those gen-erators n, Lsee (2.14)j that correspond to the spon-taneously broken symmetries of G. The remainingcomponents of M, which are not included among theset M„will be labelled as Mg.

In such a case it is possible to invent a transformationLz(x)QG which transforms away the components M,in the sense that one can represent M in the form

The components of the matrix I.&—obtained as non-linear functions of M by solving (3.4)—must satisfyvarious constraint conditions in order that I.~ belongto G, but they can be expressed in terms of a set ofsuitably chosen independent parameters p, equal innumber to the m, of (3.4). In principle, therefore, onecan solve for these parameters p, (the preferred fields)in terms of the original set of fields M. (The numberof m 's being set equal to zero equals the number ofP, 's introduced. ) Since the latter fields transformaccording to the given linear rule (3.1), one can deter-mine the transformation law of the preferred set @,.This law is quite generally nonlinear and, if the non-linearities are expanded in powers, inhornogeneous. Ittherefore follows, as has been emphasized in Sec. 1,that the representation (3.3) can be used only intheories with noninvariant vacuums. "

It may be that some of the fields 0, and in particularMz in (3.2), do not represent physical particles —inother words, they represent particles which are so farremoved in mass from their partners in the set M asto be irrelevant dynamically Lthe analogy of M, iswith ~ (a=1, 2, 3) and of M& with 0 in the chiralmodel). The chief problem, therefore, is to exhibit theformalism in such a way that these fields can be re-moved from consideration. Ke do this by following thestandard nonlinear prescription of imposing constraints;the details of the method are as follows.

In the Lagrangian (3.2) substitute the expression(3.3) for the fields M and its analog for 4' to give

L(4,8„%,M, B„M)=L(D(L&)4,&„D(L,)P,D(L,)m, ag(L, )m)=L(&»(Lr') ~.D(L~)4,m»(Lp ')

X &P(L~)m), (3.5)

where the validity of the second step depends upon theinvariance of this Lagrangian under the transformationsof G. The derivative terms in (3.5) involve the operatorh„defined in Sec. 2,

D(Lp ')8+(Lq)P=D„P=D„f+iU)„&,N,Q, (3.6)

where D„P and D„P, denote the covariant derivativesdefined by (2.14) and (2.17). Similar relations hold form. The fields P and m defined by (3.3) and (3.4) belongto a reducible nonlinear realization of G which becomeslinear with respect to the subgroup II. The number ofcomponents f, p„and mz is equal to the number oforiginal field components 0' and M. However, it isclear that any subset of the fields and mz which spansa (linear) representation of H can be set equal to zerowithout doing violence to the invariance of I.. Such adisappearance of some of the components of f is

M=D(Lg)m,

with L&(x) so determined that

m =(D(Lp ')M)o=0.

(3.3)

(3 4)

"We do not have a general method which defines the trans-formation I.~ in cases where the Lagrangian contains explicitlyno Goldstone boson 6elds with the requisite quantum numbers.For the above method to work, it is necessary to introduce extrafIelds with the requisite quantum numbers, perhaps by some self-consistent method which we have not investigated.

NONLI NEAR REAL IZATIONS. I

balanced by the appearance of algebraic constraintson O'. This can be seen by inverting the formulas (3.3)and (3.4) and expressing the components P and mgas nonlinear functions" of 0' and M.

Thus, one can express the Lagrangian (3.2) in termsof the nonlinear variables

L =L(4, ~A+~f&.*~4)+ (3.9)'4 To illustrate, take the case of M =m„a=1, 2, 3, and &~=a.

One can introduce (following Higgs and Kibble) a field x(g)=g{m'+0'). For the nonlinear method to apply, it is essentialthat X11-—(x(x)) should not vanish. Writing g=yo+y', where g'is the quantized field, what we are saying in the text is that g' canbe set equal to zero.

» We have of course set the components m =0 by way ofdefining the preferred set p . This does not reduce the number ofindependent variables, nor does it imply any constraints upon the3f. When constraints are introduced, then mg can also be setequal to zero so that all m's are out of (3.7). The Goldstonebosons„however, remain, being described by the p 's whichappear in the definitions of 6„.The formula (3.7) is analogous toKibble's formula (10}(Ref. 7).Our formulation, being completelygeneral, is valid for any symmetry group and any representa, tion.

L(%',8„4,3f,B„M)=L(g,h„f,mg, h„mg), (3.7)

and feed in the realistic bare masses and couplingconstants. Unwanted components of P and mg cannow be set equal to zero"; the only rule to be observedis the manifest invariance of the right-hand side of(3.7) under the transformations of the subgroup H.

Contained in the Lagrangian (3.7) there will ingeneral be the term

~D„Q D„Q *=28„$,8„@*+interaction terms.

The first term on the right-hand side of this is to beinterpreted as the kinetic energy of the Goldstonebosons. These are massless bosons because, althoughP, appears elsewhere in the Lagrangian (in the co-variant derivatives of f), it is always accompanied by8„&,. There is no mass term. That they are the Gold-stone bosons is clear since they appear only when thesymmetry is broken spontaneously by the impositionof constraints. It is clear from the above discussionthat these particles could not even be defined if thevacuum were symmetric. On the other hand, if theLagrangian were not symmetric they would acquire amass since the matrix D(L~) would, in such a case,fail to be eliminated completely out of the right-handside in (3.5).

It is of interest to see what happens if a Iong-rangegauge field is present. Let us therefore replace theLagrangian (3.2) by one which is invariant undertransformations of the second kind. This means intro-ducing a set of gauge fields, i.e.,

L(@,B„+)~L,=L(%', 8„%+ifA„;$,%')

,'F„„P„„,, (3.8)-in the notation of Sec. 2. If the expression (3.3) for 4in terms of the nonlinear rea1ization f is substitutedin (3.8), one 6nds that

where 8„is defined in terms of the gauge 6elds A„andthe reducing matrix Lq as in (2.27). 8„ transformsaccording to the nonlinear rule (2.28). The covariantderivative of 8„ is contained in the expression (2.31)for 8„„.The Goldstone particles, represented by J~,have been absorbed in the rede6ned gauge qelds 8„.They no longer exist as independent particles. The6elds p, still appear itnplicitly in the "nonlinear"transformation laws of P and 8„, but they no longerhave any dynamical signi6cance. Moreover, theLagrangian I.~, being independent of the P, is notof the nonlinear variety.

Since the gauge 6elds 8„ transform inhomogeneouslyLEq. (2.28)j, it is essential for the preservation ofgauge invariance of the second kind that there shouldbe no mass term tn'8„'. However, it is possible tomaintain gauge invariance of the 6rst kind in thepresence of a term such as

l 'L&.-—(I/f)(L '~.L )-j', (3.10)

or, in other words, if the Goldstone particles are revived.This means that intrinsic symmetry breaking in thepresence of a gauge Geld of finit range requires thepresence of Goldstone particles.

In conclusion, the Goldstone 6elds P, will becomemassive if and only if there is introduced in (3.2) anexplicit symmetry breaker. If that is done, it is clearthat the reducing matrix I& must appear explicitly inthe transformed Lagrangian (3.5) and not merely inthe covariant derivatives. This means that P is nolonger everywhere accompanied by 8„&„ thus, byexpanding I.& in powers of p, one can always 6nd a termproportional to P. For example, one could add to theLagrangian (3.5) an explicit symmetry breaker of theform

~'D, ~(L~), (3.11)

where D(g) denotes some chosen (self-conjugate) ir-reducible representation of G and Du(g) indicates amatrix element of D(g) between states which aresinglets of II, i.e.,

Bu(hg) =Du(gh) =Du(g) . (3.12)

The presence of a term like (3.11) in the Lagrangiantherefore does not violate the symmetry under H.

This is the approach advocated by Vfeinberg. " Astill more satisfactory Lagrangian, fully invariantunder G but still producing a mass for the p particles,could be

L= (L—-', m'y')+-', nPP,

where m2 is computed self-consistently by setting upan interaction representation and computing the self-mass of the p particle which is then put equal to thephysical mass, i.e., its bare mass is zero. Whether this

"Weinberg actually writes out the expression DgI(Ly) as apower series, and in his way of setting it out it is hard to recognizethat it can be so compactly expressed.

1758 A. SALAM AN D J. STRATH BEE

K and E. First, under the subgroup SU(2)XU(1),since h(K,g)=g, it is clear that K and E transformlike I= 2 fields with V=+1 and V= —1, respectively.To discover their behavior under the hyperchargechanging transformations it is necessary to use thefact that h '(K,g) multiplies the third column of L»by a phase factor but does not mix into it the first twocolumns. I.et us consider the infinitesima1 hyperchargechanging transformation

self-consistency procedure will introduce other Gold-

stone particles into the theory is an open question.

4. A SIMPLE EXAMPLE

In order to illustrate the techniques presented inSec. 2 we consider here the nonlinear realizations ofSU(3) which become linear with respect to the sub-

group SU(2)rXU(1)r. This example is sufficientlycomplicated to exhibit the main features of the non-linear formalism and, moreover, it has some physicalrelevance in that the breaking of SU(3) symmetrymay well be, at least in part, intrinsic. In addition, ascan easily be seen, only very little eA'ort will be neededto extend the formulas given here to the case of chiralSU(3) X

SU�(3)

broken spontaneously to chiralSU(2) XSU(2).

The first stage in establishing the nonlinear real-izations is the construction of a reducing matrix(L») &+SU(3). This matrix must transform according

L»~ gL»h '(K,g), (4.1)

(4 3)

where ~ denotes a two-component infinitesimal quantityand e its Hermitian adjoint. The components of thecolumn (L») ' are transformed according to

2'' 1 —X'EK 2XK=26 +ibp-

&1+X2EK I+X'EK 1yz&EK'

(1 X'KK —2X«K 1 X'EK—=i +ibp

&1+X2EK 1+&«EK 1+x&RK

where bq =by(K, «) is a nonlinear effect coming fromh '. It can be eliminated from these formulas, whichthen yield the transformation law

(2lIK 2XK 2X(i«K)

b = i« — . (4.4)1 X'EK —1 X'EK 1—X'EK—

From this formula and its Hermitian adjoint it is asimple matter to extract the resultKE 1 X'EK KE —2kfC

+EK 1+)PEKEK 1+)PEK '

(4.2)1 )PEK—

(1/«)bK=D2A) '(1 X'E K—)« Xb K—Kj1+X'EK

(«K+R «)K , (4.5)'E1+X'EK.1+X'EK

where g+SU(3) and h+SU(2)XU(1). Since SU(2)XU(1) is a four-parameter group, while SU(3) haseight parameters, one expects that there should be aset of four preferred fields, K and E, with which toparametrize I.~. These fields correspond to the hyper-charge changing transformations of SU(3). Out of allthe possible parametrizations we shall pick one thatdoes not involve square roots Pand is therefore thenearest in spirit to %einberg's treatment of chiralSU(2) XSU(2)j. It is given by

where X denotes a real parameter to be fixed later. Thefield K is a two-component column vector and Edenotes its Hermitian adjoint, a two-component rowvector.

It is necessary to demonstrate that the paramet-rization (4.2) is consistent with the transformationrequirements (4.1). This can be done by setting upexplicit transformation rules for the preferred fields

1+bh= (L» '+bL» ') (1+bg)L», —(4.6)

where bL» is obtained by using (4.5) in conjunctionwith the form (4.2). The result is

which is, therefore, the transformation law implied by(4.1) in the parametrization (4.2).

The inanitesimal form of the transformation h(K, g)is determined by the method of Sec. 2,

(1ji)bh=

K(E «+« K)E.—X(K«+ «E) —X'

E «+«K

1—X2 E.(4.7)

which clearly belongs to SU(2) X U(1). This matrix controls all of the nonlinear realizations with the exception ofthe preferred realization (4.5). Suppose that g transforms under SU(2)X U(1) according to a linear irreduciblerepresentation which is generated by the isospin and hypercharge matrices I and Y. These are defined, in the

NO N I. I i%EAR REAL I ZAT IO NS. I 1759

context of SU(3), bybf =bh, ~Fsg = [bh, '(Is'+ ', bs'-Y) bh—ss YQ

= [bh, 'Is'+ ,'(bh-ii+bhss 25—hss) YQ= [tr(bh~) I——ssbhssYQ,

where the Ii ~ are defined on the SU(3) quark by Ii Q~= b~Q ', b—P—P~ Th. en, corresponding to the hyperchargechanging transformation (4.3),one finds, using (4.7) in (2.6),

1 sK+E s(ssK+Kss) I+ (XsEsK I+ssY)

1 —~sEK(4.8)

where ~ denotes the Pauli matrices.The covariant derivatives are determined in the parametrization (4.2) by the matrix

2P.'(KK„Kg)—2X4K(K„K KK„—)K1+AsKK (1+X'KK)'

26K„K„K—K E„E1+PPKK (1+X'KK)'

2ltK„E„K EK„'—+2X'E

1+X'EK (1+XsEK)s

E K„E„K-2A.'

(1+XsEK)'

(4.9)

where K„=B„K and K„=B„E. This matrix, beinganti-Hermitian and traceless, belongs to the algebraof SU(3). The part which belongs to the algebra ofSU(2)XU(1) can be separated out and used in theconstruction of the covariant derivative of f,

-E„~E—K~K„D„P=B„P+2X' I

1+X'EK

E„K KK„—(X'E~K I+-', Y) f, (4.10)

(1+PPEK)'

and the remainder is used to define the covariantderivatives of K and E:

E„K EK„—D~ = +X'K

1+~sEK (1+XsEK)s'

K„K~ KK„—Dg= —Xs'

1+X'EK (1+X'EK)s

Any Lagrangian made out of P and P and theircovariant derivatives together with D„K and Dgwill be SU(3)-invariant if it conserves isospin andhypercharge. The vacuum state corresponding to sucha Lagrangian will not be invariant under the hyper-charge-changing transformations, and the resultingGoldstone bosons are characterized by the fields Eand E.

The interaction of, for example, nucleons cV andhyerpons A with the Goldstone fields K and E couldbe described by the SU(3)-invariant Lagrangian

L=N(iy„D„m~)cV+It (iy„D„ms)—it+D„ED„K-+gsx~Ny„IUD„K+, (4.12)

=M'/6X' M'EK+2M—'X'(KK)'+ (4.13)

where the normalization has been adapted to give thekaons mass M. There is no necessity to stop at octetbreaking. It would be a fairly straightforward calcu-lation to derive the explicit form of a symmetry breakerbelonging to any higher representation. In such calcu-lations it is useful to define the field

C =LKkgL~ ',which transforms like an ordinary (linear) octet,

C —+ gCg ', g&SU(3)

but satisfies the algebraic constraints

(4.14)

4'(4+1)= 2. (4.15)

The I= I'=0 component of C is proportional to theoctet symmetry breaker (4.13). The generalized sym-metry breakers, belonging to higher representations ofSU(3), could be expressed in terms of the I= Y=Ocomponents of the appropriate irreducible tensorpolynomials in C.

where D„A=8„A and D„N is given by (4.10) withI= ~/2 and Y=1. To the invariant part (4.12) onemight add an explicit symmetry-breaking term inorder to give mass to E and K. This term could takethe form of a matrix element between SU(2)XU(1)singlets of the transformation D(Lrr) in the manneroutlined in Sec. 3. The simplest representation Dwhich can serve in this role is of course the octet. Theoctet symmetry breaking is given by

M' M' 1 4X'EK+X4(—KK)s

tr(LzhsLx 'Xs) =12&2 6~»+2~'RE+&4(EK)s