P H Y S I C A L R E V I E W V O L U M E 1 4 3 , N U M B E R 4 M A R C H 19 66

Goldstone Behavior in Some Model Theories* W- S. HELLMAN AND P. ROMAN

Department of Physics, Boston University, Boston, Massachusetts (Received 28 October 1965)

Simple model theories characterized by bilinear interaction structures are studied. The interrelationship between honvanishing vacuum expectation values of the fields, broken vacuum-state invariance, and con­stant-field translation invariance is pursued, in relation to the appearance of zero-mass particle poles in the propagators. The peculiar role of constant-field translation invariance is commented upon.

I. INTRODUCTION

THERE has been some interest recently in the ap­pearance of constant-field translation invariance1

in theories exhibiting Goldstone behavior.2"4 In a previ­ous paper6 we have noted the connection between the absence of mass renormalization of an w= 0 mass boson gradiently coupled to a spin-J fermion on the one hand, and the invariance of the system with respect to con­stant-field translations, on the other. In this paper we shall study a class of simple solvable field theories charac­terized by interaction structures bilinear in the fields, and examine the occurrences of (a) nonvanishing vac­uum expectation values of field operators, (b) broken symmetries, (c) constant-field translation invariance. The systems to be considered are rather restricted in the sense that they do not give rise to nontrivial scattering. Nevertheless, we feel that the behavior of these models with respect to points (a), (b), and (c) above might provide valuable insight into more realistic situations.

In Sec. II we show that, for a suitably restricted model of our class, a breakdown of a symmetry under a group of homogeneous linear transformations can occur and a massless quantum appears, in accordance with the Goldstone theorem. However, an independent aspect of the system, \iz.j the realization of an invariance under, constant-field translations, implies the same algebraic constraint which, in the symmetry-breakdown argu­ment, is directly responsible for the occurrence of the massless particle. While the conserved current associ­ated with constant-field translation invariance implies a zero-mass state, it does riot require a nonvanishing vacuum expectation value of the field operators for the realization of the state with a nonzero weight.4 (The proof of this last statement will be given in the Ap­pendix.) The realization is guaranteed by the demand of constant-field translation invariance.

* Research sponsored by the U. S. Air Force under Grant No. AF-AFOSR-385-65.

1 We call transformations of "the type, ф —> ф-\-а (where a is a constant) a constant-field translation. In the current literature the misleading term "constant gauge transformation'' is frequently used.

2 H. Umezawa, Nuovo Cimento 38, 1416 (1965). 3 G. S. Guralnik and C. R. Hagen, Imperial College Report No.

ICTP/64/75 (unpublished). This work approaches the problem along similar lines to the present paper.

4 J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127, 965 (1962).

6 W. S. Hellman and P. Roman, Nuovo Cimento 37, 779 (1965).

143

The spin-J version of the system is then discussed, because the symmetry breakdown appears to have a strong bearing on a lepton model described by Feinberg et aU

In Sec. I l l we generalize the system. In these models the separation of the symmetry breakdown from the constant-field translation invariance requirement is made apparent, because of the absence of symmetries under homogeneous linear transformations. We find that the pole in the propagators at zero mass may be throught of as being due to the condition of constant-field translation invariance.

II. TWO-FIELD SYSTEMS WITH EQUAL MASS Let us consider the system of two coupled charged

spin-zero bosons, described by the Lagrangian density

£ = - (dtBfdrPi+tfBfB,) - (д^В2^В2+^В2^В2) -giBfBt+BfBJ. (1)

The field equations are

(n2-fx2)Blt2=gB2tl, (2a)

( • 2 - M 2 ) £ i / = g £ M * . (2b)

The Lagrangian is invariant under the group of con­tinuous linear transformations

B\ —> Bi cosa+iB2 sina,

B2 —> iB\ sma-r-£2 cosa,

where a is a real constant parameter. The invariance with respect to (3) is made transparent if we introduce the two-component quantity

B=( y , tf-iB^B,*). (4)

The Lagrangian can then be written in the form

£~-(d»m„B+№B)-gBtTiB, (5)

where n is the Pauli matrix. The invariance of (5) under the subgroup of SU2 given by

B-*eiar*B (6)

is equivalent to the invariance under (3). We note that if g=0, the system possesses complete SU2 invariance.

6 G. Feinberg, P. Kabir, and S. Weinberg, Phys. Rev. Letters 3, 527 (1959).

1247

1248 W. S. H E L L MAN AND P . ROMAN 143

The transformation (6) is represented in the Hilbert space by the unitary operator

where U(a) = eie<">,

G(a) = ia ЫЧфЧхВ-В^пВ).

(7a)

(7b)

If the vacuum state is invariant under the operation U(a), then we obtain

<0|J3|0)=e8''"'i<0|.B|0>. (8) This implies {0|2?|0)=0, since a is arbitrary. Thus, we see that a necessary condition for a nonvanishing vacuum expectation value of В is the noninvariance of the vac­uum state under the continuous group represented by U(a), as expected. However, there exists a second necessary condition as well. From the field equations (2) we obtain, assuming the usual space-time translation invariance of the vacuum, the constraint

»*(0\B\0)=g*(0\B\0). (9) Thus, in order to have (01В |0)^0, we must satisfy also the condition

M4=f (10) If the system is constrained to have a nonvanishing

vacuum expectation value for the fields, we expect, in accordance with Goldstone's theorem, the occurrence of a pole at zero mass in the propagators. We now verify this explicitly.

The propagators are obtained by the external-source technique. We add to £ the terms —J?Bi—Jj?B% ~Bi*Ji—B2*J2 and obtain the equations of motion

( D 2 - M 2 ) £ l , 2 = g # 2 , l + / l , 2 ,

( П 2 - М 2 ) ^ 1 , 2 * = ^ 2 Д * + Л , 2 * .

We define the propagators by

t(BM)\

where

Al,2+(*,*') =

<*м>-<0|Bi.*|0>

/1,2=0

(11)

(12a)

(12b) <0|0>

Now, from Eq. (11), we obtain

Вг,2(х) = Jd'y Ao+(x-y)l-gB2tl(y)+Jlt2(y)2,

where Ao+ satisfies (n2-fx2)A0+(x~-y) = b\x-y).

Equations (12a), (13a), (13b) easily yield the relation

0 - Л Д М + ( * - ^ ) = ^ / ^ Д О + ( * - У ) А 1 . « + ( У - ^ )

+5A(x-x'). (14)

(13a)

(13b)

By taking the Fourier transform, we finally obtain

Ai,2+W = [-*2-iU2+g2/№2+M2)]-1. (IS) There are two poles (in each propagator), occurring at

&=-lfi±g. (16) If now the condition (10) is satisfied, we have

Ai..+(*) I , W = [ - ^ - ^ H V / ^ + M 2 ) ] - 1 . (17) This expression has poles at k2= — (V2/*)2 and at &2=0, i.e., a Goldstone pole appears.7 We note that the ap­pearance of this pole is a consequence of satisfying only one of the necessary conditions for the nonvanishing of vacuum expectation values, and the k2=0 pole can occur even if the vacuum state is invariant under U(a). It is therefore reasonable to ask as to whether the condi­tion g2=v* follows from some other constraint that may be imposed upon the system. We now turn to this question.

Let us consider the transformation group1

(18) £1,2—>Bi t 2+a }

В1Л*->В1Л*+а*, where a is an arbitrary constant scalar. This group is generated by the canonical momenta of the system. The change in the Lagrangian density, arising from (18), is 8£= - №+g)lBx4+B24+a*Bi+a*B2+2a4^. (19) For arbitrary a we can have invariance only if

g=-M 2 . (20) This condition for invariance under the constant-field translation implies Eq. (10), which was one of the neces­sary conditions for (0|2?|0)^0. However, condition (20) is now a necessary and sufficient condition for invariance of the system under constant-field transla­tions. Since the appearance of the zero mass pole hinged only on condition (10), the present result gives some in­sight into the cause of the existence of a k2=0 pole when the vacuum-state invariance is not broken.

The above considerations are easily extended for the case of a two-fermion system. Let

£ = - ^ ( - i 7 ^ M + m ) i A + ^ r ^ , (21a) where

^ 0 ' ftW. (21b)

The equations of motion are

(-*у>д,+т)ф1Л=&%Л9 (22) and the corresponding adjoint equations. The system is invariant under , . . ,n~,

ф—>егапф. (23) 7 The simple pole structure of two-field models with bilinear

interactions was first studied, from another point of view, by В. В. Deo, Nucl. Phys. 28, 135 (1961). In connection with Goldstone's theorem for free fields, cf. G. S. Guralnik, Phys. Rev. Letters 13, 295 (1964).

143 G O L D S T O N E B E H A V I O R IN SOME MODEL T H E O R I E S 1249

The nonvanishing of (0|^i,2|0) requires, as before that the vacuum be noninvariant under U(a). However, the vacuum must also be noninvariant under the homogene­ous Lorentz transformations including spatial reflection. Under this group, the spinor transforms as

H%'&) = l / ( I ) ^ 4 ( £ ) s ^ w W • (24)

Thus, if we had U(L)\0)= |0), we would obtain

( 0 | * M | 0 > = 5 L < 0 | * I . , | 0 > ,

so that (0|^i,2|0) would vanish, since SL is arbitrary. From Eq. (22) we get the analog of Eq. (9),

^(0|#1,,|0)=^<р|*1,,|0>. (25)

Thus, for (0\ф\0)7*0, the further necessary condition

m>=g2 (26)

must also be satisfied. For the propagators we obtain, in the general case,

Gi*+(p) = Zyp+m-?/(yp+m)J*. (27)

Each propagator has two poles; they occur at £2=--(w-|-g)* and at p2=-~(ni-g)\ If the condition (26) is satisfied, then

GI,*4P) I , w « Zyp+f»-nfi/(yp+m)JrK (28)

As in the case of the boson system, we now still have two poles, viz.,

p2=0 and £2=-(2m)2 , (29)

one of them being at zero mass. We note here the in­teresting possibility of explaining the muon-electron mass difference by a broken-vacuum mechanism. If we start with a mass-degenerate doublet and a n-sym-metric Lagrangian and then impose the condition (0\ф\0)?*0, then one of the necessary constraints causes a mass splitting while maintaining the r\ symmetry. The zero-mass excitation could be the electron (in the absence of electromagnetic and weak interactions), and the other excitation at (—2m)2 would be a candidate for the muon.8

The condition (26) is again sufficient to render the system invariant under constant-field translations. This is most easily seen by requiring the equations of motion to be invariant under the transformation

T T f - V ^ /

where a is a constant spinor. Equation (22) yields the condition for invariance,

m=g. (31)

This then again produces the two poles (29). 8 A more detailed lepton theory, based on this idea, will be

discussed elsewhere.

III. MORE GENERAL SYSTEMS In the previous section we have discussed models

possessing an invariance under a continuous symmetry group and having the property that the invariance of the vacuum state under the group would prevent the vacuum expectation values of the field operators from being different from zero. We now consider systems where no such symmetry group is present but which, nevertheless, exhibit a massless particle pole in the propagator when the conditions for a nonvanishing vacuum expectation value of the fields are satisfied.

Consider first the boson model of Sec. I l l , but with different masses. The equations of motion are

-(n*-intf)Bi,*=gBttl (32) and their Hermitian conjugates. Since MIT JU2, the system no longer possesses a symmetry under a continuous group of linear homogeneous transformations. If Б is a proper scalar, then there is no symmetry group at all under which the vacuum must be noninvariant to permit a nonvanishing vacuum expectation value (0|Б|0)?^0. But if Bit2 is pseudoscalar,9 then it is necessary that the vacuum be noninvariant under the discrete parity group, i.e., that P(0)=^|0), so as to allow for (0| J510)^0. A further necessary condition follows from the field equations (32) and we get

g2=MiV. (33)

The propagators are obtained in the same manner as in Sec. II and are

Ai.i+(*) = C - ^ - M M M ^ V ^ M W ) ] - 1 . (34)

For both propagators the poles occur at

£2= W+tfteiLW-rty+WJ». (35) The condition (33) then produces, besides the finite mass (JUI2+#22)J a pole to occur at k2=0.

We consider now the group

Bi-+Bx+a, B2^B2+b;

Bx*-*Bi*+o*, B2*~~>B2*+b*. ( 6 )

From Eq. (32) we obtain as conditions for invariance

From these equations follows>i2ju22:==g2, i.e., the previous

necessary condition of broken vacuum, and also10

b=±(»1/m)a. (38)

Thus, the system is invariant under the constant-field 9 This comment holds true, of course, even for the model of

Sec. I. 10 The appropriate sign in Eq. (38) is the one which is compatible

with the sign of the root in g = ± ^ 2 .

1250 W. S . H E L L M A N A N D P . R O M A N 143

translation group

Bi-*Bi+a, B2 —> B2±(jjn/»2)a;

Bi* -> £ i * + a * , B2* -> Б 2 * ± (Mi/A*2)a*, ( 3 9 )

provided the condition (33) is satisfied. Then the Goldstone pole will again occur.

Our final generalization of systems with bilinear interaction is the Thirring version of the Zachariasen model.11 The Lagrangian density is

£ = -i(d»BdtlP+it&) / (d^Z(<r)dfiZ((r)+Z2((T))do-

-gjf(cr)BZ(a)d<T-J1B- fz(<r)J2(cr)d<T, (40)

where / i , J2 are external sources. The equations of motion are

(П2- tfB^gJfMZtoda+Jx, (41a)

0-^W=g/W5+/,W. (41b) If the external sources are off, the necessary condition for the nonvanishing of the field vacuum expectation values (0\B\0) and <0|Z(<r)0|) is

M2 = W AT, (42) J <r

as follows from Eq. (41). The propagator for the В field is easily obtained and we find

Г Г f*(?) T 1

A*+(*) = l -V-S+£ I -^—dv\ . (43)

If the condition (42) is fulfilled, this propagator has only one pole which occurs at k2=0.

Let us consider the generalized constant-field trans­lations

B-^B+a, (44)

Z(a)-*Z(*)+b(c). V ;

The system will be invariant under (44) provided the conditions

-fx2a=g f(*)b(<r)da,

-<rb(<r) = gf(p)a

are fulfilled. From (45a) and (45b) we obtain

№.

(45a)

(45b)

M2=g2/ dc, J <7

(46)

11 W. Thirring, Phys. Rev. 126, 1209 (1962). See also C. R. Hagen, Ann. Phys. (N. Y.) 31, 185 (1965).

Thus, the conditions which guarantee that Eq. (44) is an invariance group will again imply the pole at zero mass.

A realistic theory which exhibits the type of behavior of the models discussed, is the gradient-coupled-pseudo-scalar theory. Here

<£= -ф(-гу»д»+м)ф-%(д>>Вд»В+^В2) +г/Фу"Уьфд,В, (47)

so that the field equation for В is

(n*-/t)B=ifd,tfy»Yrf). (48)

Because of space-time translation invariance, дм(0|^7"7б1Д|0) vanishes. We then obtain from Eq. (48)

M 2 ( o | £ | o ) = 0 . (49)

Thus, for nonvanishing bare boson mass the vacuum expectation value of В must vanish. In the limit of zero bare mass, however, we cannot conclude to (01В \ 0 ) = 0 . In this case there appears a physical В particle with zero mass, since the mass renormalization of the В field vanishes in the same limit.6 Furthermore, the invari­ance of the theory under B~>Б+а in the zero-mass limit is apparent from Eq. (48). Thus the properties of our solvable models appear to have some basis for generalization.12

In this section we have examined models in which the conditions for nonvanishing vacuum expectation values of the fields are compatible with the constant-field trans­lation invariance requirements. Massless particles ap­pear when these conditions are satisfied, but the theories do not support broken-vacuum invariance conditions since invariance groups of continuous linear homogene­ous transformations are not present.

IV. DISCUSSION

We have considered in some detail the roles played by broken-vacuum invariances and constant-field transla­tion invariance in the appearance of zero-mass particles for several field-theory models. The analysis exhibited a strong correlation between conditions required for constant-field translation invariance and the realization of a nonvanishing vacuum expectation value for the fields. The conditions were compatible with those arising from the broken symmetry arguments and produced massless quanta. However, as exemplified by the models of Sec. I l l , including the gradient coupled theory, the requirements of constant-field translation invariance can be compatible with the broken-vacuum condition even when continuous internal symmetry groups of homogeneous linear transformations are not present.

12 A related discussion of the gradient-coupled theory in relation to the Goldstone theorem appears also in Ref. 3. I t should also be mentioned that a discussion of the gradient-coupled theory with respect to constant-field translations has been discussed with refer­ence to partially conserved currents by M. Gell-Mann and M. Levy, Nuovo Cimento 16, 705 (1960).

143 G O L D S T O N E B E H A V I O R IN SOME MODEL T H E O R I E S 1251

The behavior of the models with respect to the con­stant-field translation groups is not contingent on the noninvariance of the vacuum. This provides an example of theories invariant under a continuous symmetry group which necessarily produces particles of zero mass. The appearance of massless particles arising from con­served currents is thus seen to be separable (at least in the cases studied) from the broken symmetries arising from linear internal transformation groups which require mass degeneracy in the Lagrangian structure for invariance.

The significance of the constant-field translation group is obscure. It is neither an internal symmetry transfor­mation nor a space-time invariance. Furthermore, unlike the familiar gauge transformations of second kind which it resembles,13 there appears to be no raison d'etre for its existence.14 We are tempted to believe that when such a symmetry exists, it is a consequence of some deeper structural feature rather than an a priori symmetry. In particular, if the vacuum is broken so that certain necessary conditions on the parameters of the theory are satisfied, the constant-field translation symmetry may arise as a consequence. Furthermore, even if there are no algebraic constraints for realizing a nonvanishing vacuum expectation value, a constant-field translation invariance may arise as a consequence of demanding (0|ф|0)^0, as was exemplified by the gradient-coupled theory.

We note that any constant-field translation is indeed a "broken symmetry," since invariance of the vacuum under such a transformation implies a=0, whether or not the field has zero vacuum expectation value. Con­versely, if the vacuum is broken, the field vacuum expectation values themselves may serve as.constant fields. For example, in the case of the boson model of Sec. I, assuming that the vacuum is broken, and taking in particular the solution

g=+M2 (50)

of Eq. (10), we find from the field equation (2a) that

<0|5i |0>=-<0|58 |0) .

Then, under the transformation

£ i - + £ i + ( 0 | £ x | 0 ) , B2-+B2-(0\B1\0) (51)

the system is invariant. This would be a modified constant-field translation invariance. If we take the solution

«=-M2 (52) 13 Note, however, that gauge transformations are not Lie groups

in the ordinary sense, whereas the constant-field translation is a one-parameter Lie group.

14 It may also be interesting to observe that the constant-field translation group is noncompact.

of Eq. (10), we still have invariance under

B1-^B1+(0\B1\0)7 B2^B2+(0\B1\0). (53)

The reason is that when (52) holds, Eq. (2a) gives

<0|£i|0>=<0|ft|0). (54)

Finally, we note that if we define, in either case,

Вг'^Вг-ЩВгЩ, B2f^B2- (0 |£2 |0) , (55)

i.e., if we subtract the respective constant fields, we clearly have

(0 |Б / |0 )=(0 |Б 2, | 0 )=0 . (56)

These comments may serve to put the question of constant-field translations into proper perspective.

APPENDIX We demonstrate that the zero-mass state arising from

constant-field translation invariance has nonzero weight. Assume the theory is invariant under

0t->0d-0». (Al) Then we have a conserved current,

V = 0 . (A2)

On grounds of Lorentz invariance, we have the spectral decomposition

<0| [/*(*), Ф < Ш 0 ) = (2T)-*fd*k е*<*-">

XWfi*(»)+eW)k*f**(#)). (A3) From (A2) and (A3) we obtain

Л Л ^ Ь ^ Э Д . (А4) We have to show that at least one of the two coef­ficients, cit2\ is not zero.

By definition of the generator,

a{= J d*x(0\LJ0(x),<t>i№\0)\*°-v°- (А5)

Using (A3) and (A4), this equation becomes

a . = (27Г)-4 \d*xd*k e^-^k^UW+cfidik?). (A6)

Performing the integrations in turn,

a{= (2тг)-1с2г* jdk°e(k0)8(k2) = ( 2 T ) ~ W . (A7)

Hence c2{= (2т)а^0. Q.E.D.