Unstable Goldstone bosons and the elimination of scalar particles from gauge theories

P H Y S I C A L R E V I E W D V O L U M E 1 0 , N U b I B E R 4 1 5 A U G I J S T 1 9 7 4

Unstable Goldstone bosons and the elimination of scalar particles from gauge theories

C. L. Hammer and B. DeFacio* Ames Laboratory-Un~trd States Atomic Energy Commission, and Department of Physics. Iowa State Unliersity, Ames, Iowa 50010

(Received 13 August 1973; rev~sed manuscript received 20 May 1974)

One of the key ingredients in the gauge field theories which unify electromagnetism and weak interactions is spontaneous symmetry breaking. A source of instability is given for the Goldstone bosons and the details of the procedure for taking a quotient space and eliminating certain negative metric and/or unstable fields are given. 11 is then shown that a suitable symmetry breaking combined with unstable Goldstone bosons can lead to massive vector bosons without any superfluous massless scalar bosons. The gauge transformation properties of the "in" fields are also discussed.

I INTRODUCTION

Weinberg',' has used spontaneous symmetry breaking3-'' to c rea te a Lagrangian model which unifies electromagnetism and weak interactions. A number of questions concerning renormalization and unitarity which were raised by the Weinberg model have been answered. In ordinary (positive- metric) quantum field theory ' t ~ o o f t " ~ " and Lee and c ~ l l a b o r a t o r s ' ~ - ~ ~ have treated these matters thoroughly. ~ a k a n i s h i " - ~ ~ has used his indefinite- metric field theoryL8 to discuss Higgs phenomena" and chiral gauge spontaneous breakdown."

Spontaneous symmetry breaking has several as - pects. The ~iggs-phenomena1-' version i s based on a model Lagrangian and i s characterized by a nonunique physical vacuum. The unwanted mass- l e s s scalar bosons a r e eliminated by coupling the conserved currents in the original Lagrangian to A b e l i a ~ ~ ' . ~ or non-Abelian7 gauge fields. In the ax- io lnat~c approach to quantum fields, the vacuum i s unique, but the commutators between current com- ponents contain the Goldstone bosons if the theory i s manifestly covariant at every stage. Fer rar i" has shown that the current need not be Lorentz- covariant nor need the metric be positive in the Wightman formulation for the Goldstone theorem to remain true. Fe r r a r i ' s theorem allows one to apply the Goldstone theorem of Ref. 3 to the class- e s of nonlocality found both in electromagnetism2' and Nakanishi's indefinite-metric field theory" for the f i r s t time. Further approaches to spontaneous symmetry breaking include Freundlich and L ~ r i 6 , ~ ~ who used the chain approximation to the Bethe-Salpeter equation. This approach allows one to exclude the unwanted scalar bosons from off-shell s tates a s well a s the physical on-shell states. The other approach that we a r e aware of i s due to Aurilia, Takahashi, Papastamatiou, and ~ m e z a w a ~ ~ - ~ ~ in a s e r i e s of papers. These au- thorsZ3 have studied the changes in the dilation generator whenever the dilation current can be ex-

pressed a s the divergence of a local current . This study was then used in Ref. 24 to develop two models of spontaneous symmetry breaking in which all masses a r e dynamical in origin. One of the mod- e ls of Aurilia et 0 1 . ' ~ resembles the present work in the interesting fact that the Goldstone bosons a r e conlposite. Since there seems to be some con- fusion,'' we wish to state specifically our opinion that the study of the dilation operator for a gauge- independent ~ a g r a n g i a n ' ~ i s a very d i f f e ~ e n t model of spontaneous symmetry breaking from the Higgs- type models presented in Ref. 1-6 and 10-20 where gauge fields a r e used. The only overlap between these two approaches i s to be found in the axiomatic studies (see, for example, Refs. 8-10 and their references), and i s not yet completely clear .

In the present work, the Higgs-type gauge field approach will be taken. Good pedagogical discussions of these spontaneous-symmetry-breaking models a r e available including an expository dis- c u ~ s i o n , ' ~ a positive-metric d i ~ c u s s i o n , ' ~ and an indefinite-metric d i s c u ~ s i o n . ' ~ Thus, it i s only necessary to present the results we need in our work.

In Sec. I1 we will present the results needed on unstable-particle field t h e o r i e ~ . ' ~ , ' ~ In Sec. I11 and Sec. IV spontaneous symmetry breaking is discussed and a new kind of Goldstone boson i s ex- hibited. The gauge transformation properties of the "in" fields a r e discussed in Sec. V. These new massless bosons a r e shown to be unstable in the sense of Hammer and Weber," which i s quite different from the usual Goldstone bosons. This formalism suggests a physical reason why the ext??a scalar particles, which clutter conventional gauge theories, do not appear a s "physical" particles.

11. UNSTABLE PARTICLES

The fields in Hammer-Tucker (HT) t h e ~ r y ~ ' . ~ ~ a r e assumed to satisfy linear partial differential

1226 C . L . H A M M E R A N D B . D e F A C I O

equations with external s o u r c e s J(x) and configura- conserved cur ren t j, according to tion-space c-number differential opera tors D(a), given by

and

%(x)E(-a ) =5(x),

where

T h e symbol t means to take the Hermit ian adjoint of the object whether c number o r q number, the a r r o w in 5 indicates that the D operator operates t o the left, and y , i s the 2 ( 2 s + l ) x 2 ( 2 s + l ) generalization of the Dirac mat r ix y,. A Lorentz four- vector A, is defined a s

A , = ( ; i , i A , ) = ( A , , i A , ) , (2.4)

where p = 1 , 2 , 3 , 4 , j = l , 2 , 3 , a n d A , , A , a r e p u r e real . A Lagrangian density d: is assumed t o exis t such that a conserved Hermit ian cur ren t J , i s given by

with

where S ( A , , ) i s a f r e e Lagrangian density for the vector field A,, s ( ~ + , cp) i s a f r e e Lagrangian den- s i ty fo r the s c a l a r fields q and cpt, and s,, is a n interaction Lagrangian. The 6d:/6AP denotes the functional d e r i v a t i ~ e ~ ~ , ~ ' of 6: with respec t to A ,. The homogeneous configuration space c-number solutions, u, (8, x) and r , (5, x), a r e labeled by dis- c r e t e quantum numbers v, three-momenta c, and depend upon the space-t ime point x. They a r e solutions to the equations

and

The charge-conjugation propert ies of 11, and u , a r e

and

l', (6 , X) =cu:($, Y ) , (2.10)

where * means complex conjugation and C, the c- number charge-conjugation matrix, is chosen such that C C * = l .

The creat ion and destruction opera tors for "in" and "out" s t a t e s for par t ic les at, a and antiparti- c l e s D t , b a r e defined in t e r m s of the free-part ic le

a m ( o u ~ ) , ( h ) = w-l im IA, (P , so)], ro * - * i ' ")

(2.11)

h ( 6 ) = w -1im B, ( p , x,)] , (2.12) X O + - ( f *

where

A, (p , xo) = j d?~, , (@~( P, X I , $(,Y)), (2.13)

and

B ~ ( P , X ~ ) = J ~ ~ J ~ ( T , ( D , x), 4(x)). (2.16)

T h e conserved cur ren t 1 , i s associated with the free-field equation

a,, ~ ~ ( $ 2 , $1)=$2D(a)+l -&5(-d)d1 , (2.17)

where fac tors of L a r e chosen such that

Some nonvanishing commutation relat ions among the fields and the creat ion and destruction opera- t o r s include

1 (I : ($1, 4 In (XI ]= Z, (6 , Y) , (2.19)

Ib;"($), $ln(x)1 = l ' , ( h , A ) , (2.20)

both of which a r e derived from the quantization postulates,

and

where da,(x) i s an integration over a spacelike sur face o,(x) that contains the space-t ime points t . Similar equations apply when "in" is replaced by "out" in Eqs. (2.19-(2.22).

Note that ir,, a,, L.,, and F , need not be a complete or thonormal s e t of functions a s i s usually the case. If the u's and 1,'s a r e a complete ortho- normal s e t then the HT formal i sm reduces to the Yang-Feldman33 formalism. The charge conjugation proper t i es of the fields a r e ~ t a n d a r d . ~ ,

As xo - -m (+m), $(x) - IJ"(O~~)(X) and the operator $(x) is related to the "in" and "out" opera tors

10 - U N S T A B L E G O L D S T O N E BOSONS AND T H E E L I M I N A T I O N O F . . . 1227

according to

where G,, , a r e the retarded and advanced Green's functions given by

D(a)G., , ( ~ - y , r n , 2 ) = 6 ( ~ ' ( x - y ) (2.25)

together with appropriate boundary conditions. If the single-particle configuration-space states a r e defined in te rms of the vacuum state 152 ), P , 1 i2 ) =0, where P, i s the 4-vector momentum operator,

then it follows from Hammer and WeberZ8 that the configuration representation for the single-particle state i s

Similarly, one has

( x - / b ' n ~ O u t t ( $ ) / ~ ) ~ ( x - ~ ~ ( i n , o u t ) - ) ,

= v($, x) . (2.28)

The norm / / ) / of a generic field operator A(.Y,) is defined a s

I I A / ! ~ lim ( i 2 I [ ~ ( x ~ ) , ~ ~ ( y ~ ) l ! ~ ) , (2.29) 1X0 I - -

t Y O ' * "

where x, and yo independently approach either +m

or -m. AS a consequence of this definition and Eqs. (2.11)-(2.16), Eqs. (2.21) and (2.22) can be written a s

Using the argument in the proof of Theorem I of HT, one can combine Eqs. (2.13)-(2.16) with Eq. (2.30) to show that

where G(x - y, m:) i s the single-particle homogeneous Green's function

G =G, -GR.

For unstable particles (U) there a r e no single- particle, homogeneous solutions to the Klein-Gor- don equation. To see this one notes that because of the definitions Eqs. (2.11)-(2.16), 11 A&,) 11 and

/IBu(xo) / / a r e proportional to 11 $,(x) '1, which will later be shown to be zero for unstable particles by the arguments leading to Eq. (2.43). Thus

and by Eq. (2.30), u(5, x) = v(5, .t.) = 0. In fact, the techniques used in Ref. 28 can be used to show that the left-hand sides of Eq. (2.28) vanish for unstable particles thereby directly establishing the result u(5, X) = ~ ' (5 , x) = 0. This is a mathematical state- ment of the physical fact that an unstable particle does not exist for infinite time intervals. There a r e no "in" o r "out" states that contain an unstable particle.

However, this does not necessarily mean that

for an unstable particle since Eqs. (2.13)-(2.16) cannot be inverted in general. To study this question and the question of constructing the operator norm for an unstable particle, consider the following arguments. For both stable and unstable particle fields,

where P , = (5, iH ). The configuration states satisfy

If I +,) i s an arbitrary Heisenberg state, then one obtains the Schrtidinger equation

For a nontrivial state ( x * i , there will be c-number states ( x -t 1 +,) # 0 which a r e the exact continuum solutions of Eq. (2.35).

If / $,) is chosen a s an eigenstate of P,,

and one combines this condition with Eq. (2.33), then

where L = a ,a,. For a stable particle, when 4,) = 15, in, * ), M = m ,, the rest mass of the particle, and ( x i /5, in, +) i s the c-number single-particle Klein-Gordon solution. For an unstable particle, ( x i 1 $,) i s a multiparticle c-number solution to the Klein-Gordon equation corresponding to the decay-products state and M is a spectrum which comprises the mass spectrum of the unstable particle.

It i s clear from the foregoing arguments that

(x , in (out), * ) = w-lim 1 / x i ) ] (2.37) x(, + - m f , *)

1228 C . L . H A M M E R A N D B . D e F A C I O

corresponds in the unstable-particle case to a noninteracting multiparticle state of the decay products. It i s only in this sense that a +h.OU1(x) field exists for an unstable particle. One now sees that it i s essential to define the unstable (1 insout , bin.out a s functionals of +bin.out(x), according to Eqs. (2.13)-(2.16) rather than to use the non- existent inverted counterparts. This is because 115, in (out)) and the (I in,oUt commutators vanish for an unstable particle, independently of +m.oU'(x), because urn($, x) and v,( $, x ) vanish.

The equations corresponding to Eqs. (2.23) and (2.24) for an unstable particle become

and

where f and g a r e functionals only of the stable- particle fields $, which correspond to the open decay channels for the unstable particle.

It follows from Eq. (2.36) that an unstable particle has a spectral representation

(a I 1 +u(x), 4;(~1)1 la)

where G ( x - y , ~ ' ) i s a homogeneous Green's function of the Klein-Gordon equation. For a general spin-s field, Hammer and WeberZ8 have shown that for both stable and unstable particles, Eq. (2.40) can be written a s

where ~3 i s a contour in the negative direction which encircles al l of the singularities on the physical sheet a s shown in Fig. 1. For a stable particle this reduces to

114, / l = ~ ( m , , m ~ ) ~ ( x - ~ , m , 2 ) , (2.42)

where Z(m ,, m,) i s the propagator renormalization constant. In this case, only the pole te rm shown in Fig. l (b) contributes since the cut contribution can be shown to vanish like / x, -yo / -' where y > 1. Thus, Eqs. (2.31) and (2.40) agree provided renormalized fields +, = Z-112~s a r e used. For unstable particles, the pole te rms damp exponentially a s / x, -yo I - m and the cut contribution vanishes just a s in the stable case. Therefore, the norm of an unstable particle vanishes:

m' PLANE

I ( 0 1

FIG. 1. The m' plane showing singularities corresponding to the stable- and unstable-particle states. The poles a t mi a r e in the unphysical sheet.

It i s of interest to note that, in general, an unstable-particle field can be used a s a projection operator which destroys any Heisenberg state j +b,, c c ) which i s a closed channel (cc) of the unstable particle. That is, one can say that for the positive-frequency part of $,," written a s $::',

o r equivalently,

Consequently, if cp, is a field operator corresponding to a stable particle A and if for a l l / $,,, cc ) one has

thenA cannot be contained in the spectrum of li. Since an unstable particle cannot decay into a single particle (energy-momentum conservation), a more convenient but somewhat weaker form of Eq. (2.46) i s

Y o - *

Thus, the development from Eqs. (2.44)-(2.47) provides a nontrivial example of taking a quotient space of / $,)'s by removing the subset of vectors, { I+,, CC)) , a s i s required for any indefinite-met- r i c theory.

In this regard, an especially interesting case occurs when

10 - U N S T A B L E G O L D S T O N E B O S O N S A N D T H E E L I M I N A T I O N O F . . .

fo r a l l x, and 1,. This requ i res that p(iMZ) = 0 (ev- erywhere), a s can be seen from Eqs. (2.40) o r (2.41). This means that t h e r e a r e no singularities, not even continuum cuts, enclosed by the contour e of Eq. (2.41) and Fig. 1. Since Eq. (2.40) de- pends upon the vacuum expectation value of a corn - ri2zctator, the re i s no unstable-particle vacuum contribution to p ( ! d 2 ) even if (0 1 $,(x) ) i 0. Thus, Eq. (2.48) implies that either

i.e., the vacuum 5 2 ) i s the only open physical channel for U. The field $, is therefore totally unobservable and can only c r e a t e o r destroy mathematical s ta tes Ia(.u)), for which (52(x) P , lSl(x)) # 0, which in a renormalized theory correspond to the physical vacuum. In th i s case, the l inear span of 1 $,, c c ) ' s contains a l l possible physical s ta tes , except the vacuum state , i.e., ' 5 2 ) and { / $,, c c ) ) span the physical Hilbert space.

It follows fur ther f rom Eq. (2.50) that M=O i n Eq. (2.36) since I$,) = 152) if , ( x i l ~ , ) # O . This in tu rn implies

s o that &(x) represen ts a m a s s l e s s field. Any field operator q, which does not satisfy

Eq. (2.46) can be written a s a superposition of fields

@ a = W s + b W u , (2.51)

where a, b a r e complex numbers, qs, generate s ta tes in { / $,, cc )), and q,, q ~ : generate s ta tes in the domain of U. If 4, sat isf ies Eqs. (2.48)-(2.50) a l l of the fields in the domain of i t s decay products must a l so satisfy these constraints. Thus

( ~ 2 ~ ~ C P , ( X ) , U , : ( Y ) ~ ~ ~ ? ~ O , (2.52)

and q,(x) i s a n unstable field, totally unobservable, in the s a m e sense a s i,.

These equations can be used to demonstrate the nonuniqueness of the vacuum and one form of the Higgs transformation. Since q , generates s ta tes in { 1 $ H , cc 1,

:a I@, 52) = o . (2.53)

Fur thermore , because of Eq. (,2.52), if q,(x) i s nontrivial,

( n / ~ , a ) i - o ,

C q c = 0 .

It therefore follows from Eq. (2.51) that

where b,u3 a r e c numbers. The physical field cor - responding to part ic le A can be written with the help of Eq. (2.51) a s

This i s the Higgs t ransformation i f one identifies UJ, with a m a s s l e s s vector field, c p , a s the gradient of a m a s s l e s s s c a l a r field (the unstable Gold- stone boson), and Q~ with a mass ive vector field in the Higgs-model Hamiltonian.

There is a t least one generalization of these ideas from the special c a s e of pure vacuum fluctu- at ions. Suppose that instead of Eq. (2.48) only the instability condition / j 4, ~' = 0 is required and that 152 j i s in the domain of d,. (This is possible s ince the norm is defined in t e r m s of the vacuum expectation value of an operator commutator . ) Fur ther a s s u m e that !I@, / I vanishes and a l so that /R) is in the domain of q,. Then Eqs. (2.53) and (2.54) s t i l l apply s ince (a 1 $;sou' 52 ; and ( a i~,:,""' i a r e nonzero c numbers . In general I!@, I / T 0 s o that the unwanted vacuum expectation values can be made to cancel just a s i n Eq. (2.55).

Furthermore, following the arguments leading to Eq. (2.51),

F o r this equation to be consistent with the field equation of q , , say Eq. (2.1) with d replaced by q,., requ i res (52 I J".""' / a ) =O. In addition,

-. ouf = 0 (2.57)

i f J'"~oU' vanishes. Thus cp;,""' represen ts the decay products of an

unstable field of ze ro mass . One expects such a field to give r i s e to a spec t ra l function p(nzZ) with poles in the unphysical sheet shown in Fig. 1, which l ie on the imaginary axis, in general well below the threshold of i t s possible decay products. T h i s field i s therefore essentially unobservable.

In the next section these ideas will be examined in detail using a model due to Nakanishi.'' The field q, will be identified with a massive, s table vector field, and q, a s a m a s s l e s s vector field which contains co, a s well a s q,, the gradient of the unstable, Goldstone-boson field.

111. SPONTANEOUS SYMMETRY BREAKING

The Lagrangian density proposed by Nakanishi'' i s a par t icular ly useful model for exploring the analysis presented in the previous section. This Lagrangian density is written a s

1230 C . L . H A M M E R AND B . D e F A C I O

where

where ~ ( c p 'cp) i s a quadratic, rea l polynomial and B(x) i s a Lagrange multiplier field introduced so that the Lorentz gauge condition is a field equation. We use the 4-vector definitionA , - (s, LA,) throughout.

The field equations for U, obtained from d: a r e

a ,u , (x )=o , (3.4)

(-n+m,")u, - a , B = j , , (3.5)

where

j,, =-ig[qt(a,cp)-(a,pf)cpl-2g2cp+cpii,. (3.6)

It can be verified directly (using the field equations for Q ) that

a , j , - 0 . (3.7)

For Eqs. (3.4), (3.5), and (3.7) to be consistent requires the field equation for B ( x ) to be

3 B = 0 . (3.8)

The canonical conjugates of Ui, U,, cp, and c p t a r e

T + =$ -igcpu0.

These equations lead to the equal-time commutation rules

[ LTk (x), i{ ( y ) ~ x o = r o = i 6 k 1 6(% -q ) ,

1 u,(x), B(Y)I,,=,, =iG(K -9,

Here the usual choice has, been made that al l other combinations of U,, U,, U,, B, (o, 4, (o ', and it have vanishing commutators.

The fact that B satisfies the free, massless field equation, Eq. (3.8), and has the above stated commutation rules with U,, leads Nakanishi to

asser t

and

[ ~ ( x ) , B(y)i =-irn,2D(x-y), (3.12)

where D(x- y) i s the homogeneous Green's function that satisfies

The commutator of Eq. (3.12) leads to the norm

I \B(x ) / I= - im,"~(x -y ) , (3.14)

with corresponding commutation rules

l a m ( 6 ) , a " ( 9 ) l < o ,

for the in operators. To avoid difficulties with this "negative" norm, m, is set to zero. All field equations and commutators discussed above r e - main the same except

1 ~ ( x ) , B(Y)] = o , (3.15)

-mu, - a ,B = j,, (3.16)

c ( c , ) = - + ( ~ , L L - a , . L',)'. (3.17)

To avoid cluttering up the field theory with massless fields, Nakanishi postulates the constraint on the physical states

where ( + ) indicates the positive-frequency part of B(x).

From the point of view expressed in the previous section, the commutator Eq. (3.15) means llB(x)lj i s trivially equal to zero regardless of which vacuum state, physical o r mathematical, i s used to define the norm. Since B(.Y) + 0 for all x, a s i s shown by i ts various commutator rules,

This means the vacuum state i s degenerate. The field equations, Eq. (3.16), further imply that in general

( ~ l j ~ l ~ ~ + O ,

(n lu : i n ) + o ,

so that no operation destroys the vacuum. To exhibit symmetry breaking Nakanishi per-

forms the Higgs transformation6

where 1 = i+bt, x = Xt, and ZJ = I ) * . It is assumed that


( 51 /$ ( x ) /O )= (O /x ( x ) i 51 j=O . (3.21)

Under this transformation C(V ', q ) becomes

c((P+, (P)=-~M~u, ,u , , - + ( a , # ) ( a , $ ) - ~ ( a , x ) ( a , x ) + ~ ~ , a , x -+g2Uiibr,($' + x 2 ) - g ~ ~ , ~ 7 ~ $ - g ~ r , ( ~ a , ~ - ~ , x ) - ~ ( ~ ( ~ + $ ) 2 +4x2), (3.22)

where

M = g v .

The field equations, Eqs. (3.4) and (3.8), remain

but now the field U , picks up the mass M a s is indicated by

where

In the limit g - 0, v- m with M finite, J, vanishes leaving the free-field equations

Nakanishi therefore identifies x a s the Goldstone- boson field. The corresponding interacting field equation for x i s

The gauge symmetry i s broken because a ,J, # 0. From the free-field equations and the free-field

counterparts of Eq. (3.10), Nakanishi infers the following:

[ ~ ' " ( x ) , Xm(y)l =-zMD(x- t ) , (3.30)

[ Uy(x), xm (>) I = 0,

[ xm(x), xm(y)l =ZYD(X - Y ) ,

where y is a rea l nonzero, dimensionless constant. It has been assumed here that the "in" fields satisfy the free-field equations and commutation rules, that is, those which a r e obtained in the limit g- 0 with M fixed. It is apparent f rom Eqs. (3.30) that both U ; and xm operate upon states in the domain of Em. Since the constraint remains

these fields a r e not physical. However, the vector field defined a s

commutes with Bin at all space-time points. For Nakanishi the physical subspace i s therefore generated by V:, $", B'" and their Hermitian conjugates. The boson field xin is therefore eliminated because it is unphysical. Note that the state B1"(x)jO) i s a physical state by this definition al- though it i s unobservable.

However, it i s clear f rom Eq. (3.30) that

so that there remains some overlap between the physical and unphysical states. This overlap must be eliminated since otherwise part of ~ ( x ) must be contained in the physical subspace. The overlap can be eliminated by a subtraction a s shown by the development leading to Eq. (2.55). This requires

and, if Eq. (3.32) remains valid,

Thus / / x / I = 0 and x becomes an unstable Goldstone- boson field. In this respect the analysis given below is quite different from Nakanishi's.

Equation (3.35) implies that the equal-time commutator [ x '(x), x O(y)]Xo,yo must vanish. The equal- time commutators which replace Eq. (3.10b), a s obtained from the transformed Lagrangian, Eq. (3.22), a r e

and

A consistent set of equal-time commutators for the f ree fields (g = 0, M finite) i s

1232 C . L . H A M M E R AND B . D e F A C I O - 10

and Eq. (3.10a) with Cru and B replaced by their free-field counterparts. The difference between these commutators and those of Nakanishi i s s imply that we forced xO(x), X o ~ ) ] r o = y , to vanish instead of

L x"c),

In contrast to Eq. (3.30), these commutators and the free-field equations lead to the unequal time commutators,

[ x"x1, xOb 11 = 0 ,

BO(x), Bo0')l = 0 ,

[ B '(XI, xO(y)l = -iMD(x - Y 1, a (3.38)

[B0(x), ~ i ; (y) ] = i -D(x-g), ax,

All other commutators between $(y) and the fields ~ ( x ) , B(x) , or U,(x) vanish.

A vector field V i can now be defined a s

that satisfies the field equations

and the commutation rules

[ B0(x), vO,(y )I = [ x0(x), v O , ~ ) ] = 0 ,

[ V",x), V: b)l = $[ U;(x), LT; (y)] (3.41)

The field equations for V t imply

where

Therefore we deduce the commutator

This result is consistent with the free-field equations for U ; ,

and with Nakanishi's results1g if we identify 2 - " 2 ~ ; to his UF. The subtraction indicated by Eq. (3.39) therefore eliminates the unphysical mass less fields from the vector field V; just a s the subtraction indicated by Eq. (2.55) eliminates the unphysical parts of Q,.

If we now assume the in and out fields satisfy the same equations a s their free-field counterparts, we can summarize our results in te rms of the previous section a s follows:

1. The physical subspace i s generated from 152 j by VLn, Sin and their Hermitian conjugates. The physical vacuum is defined by P , ! 0 ) = 0. All of these states a re observable.

2. The mathematical state or vacuum subspace i s generated by B'", x'.

3. Since [ ~ ( x ) , ~ ( y ) ] = 0 the states created or destroyed by ~ ' " ( x ) a r e all vacuum states. In general, if ~ ' " ( x ) / a ) = IQ(x)>,, (n(x) I P , i a (~ ) ) -f 0.

4. Since x m creates states only in the domain of B'", these states also correspond to vacuum states.

5. The unphysical vacuum states a re unstable and massless.

It should be pointed out that if Nakanishi had defined

then

even though

However, xm(x) crea tes o r destroys only states which a re in the domain of B(.r). These states must still renormalize into the physical vacuum so that

That is, the spectral density function for x(.w) can contain no singularities which correspond to the physical in, out states which span the physical Hilbert space. Our choice for the x0(s ) commutation rule given in Eq. (3.38) simply emphasized this point.

IV. A GENERALIZATION OF THE MODEL

It is possible to extend slightly Nakanishi's mod- e l in a formal o r heuristic way to include the pos- sibility of generating unstable fields B(x) and x(x) that a r e not simply vacuum fluctuations. Suppose we postulate that Eqs. (3.4) and (3.5) apply but require only that


W-lim a, j , (x)=O. ir -" (4.1)

Then we obtain

from which follows

s o that the B" field is again massless. We further assume that after the Higgs transformation i s made, Eq. (3.25) i s obtained again with J, - 0 as g- 0 with M finite, and

J, r j p + M 2 ~ , , - M a , x (4.3)

a s before. We now choose the field x such that

M 3 x = a , j P . (4.4)

This choice then enables

d , J , = O (4.5a)

and

ax" = 0 , (4.5b)

so that the x'" field i s also massless. The transformation for the interacting fields

gives the field equations

Because of Eq. (4.1), the free-field equations and free-field commutators should be unchanged from those in the previous section. We may therefore surmise

i lx / I = llBil=o.

The Goldstone-boson field x and the auxiliary field B a r e therefore unstable. However, in contrast to B(x) of the previous section, now [ B(x), B ( ~ )] + 0 for all space-time points. Consequently B(x) and ~ ( x ) can represent physical resonances rather than vacuum fluctuations.

V. OTHER GAUGES

To exhibit spontaneous symmetry breaking one s tar t s with a gauge-independent theory, such a s Eqs. (3.1)-(3.3) with m , =0, and introduces the Higgs transformation, Eq. (3.19), which destroys the gauge symmetry and introduces a new field, the Goldstone-boson field ~ ( x ) , into the theory. Since the starting point i s independent of the gauge one expects the final result also to be independent of the original gauge. This i s not possible unless

the Goldstone field can be used to "soak up" the loss in gauge invariance of the massive vector field. In fact if the Goldstone field cannot be used in this manner then a s shown by Aurilia, Taka- hashi, and U m e z a ~ a ~ ~ . ~ ~ the Heisenberg fields do not asymptotically approach the free-field limit.

To investigate the general gauge, the quantity BaB2 can be added1' to Eq. (3.1) (with m o =O). The field equations, Eqs. (3.24)-(3.29), become

CB(x)=O, (5.4)

where J,(x) i s again given by Eq. (3.26). The noninteracting fields now satisfy the field equations

aXo(x) =a MB'(X), (5.5)

so that only BO(x) now satisfies a free-field equation. The Heisenberg fields U,(.Y) and ~ ( x ) therefore do not have free-field limits.

The unequal time commutators a r e unaltered from Eqs. (3.38) except

a + i c u M - E ( X - Y ) ,

a r u where"

a --- - aMZ G ( x - s , x 2 ) , M = O

and

Thus in contrast to the a = 0 case, / k ( 1 0 and the two-point function ( a / [ X ( x ) , U,(y)] 152 i does not satisfy any free-field equation.

Nevertheless Nakanishi showslg that Eqs. (5.1)-

1234 C . L . H A M M E R AND B . D e F A C I O 10 - (5.4) a r e independent of the initial gauge by using a gauge transformation of the second kind to transform both U , and X,

The gauge field A(x) can be chosen so that

a , G,(,c) = 0 .

This gives

There i s no loss in generality in ignoring the homogeneous solution of Eq. (5.12a) since and 6, can always be redefined to include this contribution.

In te rms of i ( x ) and G,(x), Eqs. (5.1)-(5.4) become

The theory then reduces in form to the a = O case regardless of the magnitude of a, since Eq. (5.12b) can be used to show directly that the commutation rules Eqs. (3.38) also remain unaltered in form with y ( s ) and fi,(x) replaced by ~ ( x ) and L r , ( x ) :

1 XO(.x), iO(y)l = 0 ,

Thus it i s apparent that the free fields a r e the asymptotic limits of the Heisenberg fields ; and I;', rather than x and U , and that 1) ( 1 = 0. The instability of the Goldstone boson is independent of the choice of a,.

Other useful commutators which follow from Eq. (5.12b) a r e

These commutators agree with those found by Nakanishi. l g

The discussion of gauge invariance i s further il- luminated by an argument that parallels a recent one also due to Nakanishi." He notes that the original theory expressed by Eqs. (3.1)-(3.14), with rn , = 0, i s invariant under the gauge transformation

fi;(y) = e - ~ P ( r ) f i , ( x ) e ~ ~ l x :

where the gauge filnction ['(x) i s any rea l c-number function that satisfies

Since j e (x) i s conserved [see Eq. (3.7)] a time-independent charge operator can be defined a s

= J do, (x)j; ( X I , (5.18)

where in the last equation the asymptotic limit ?i0 - -m has been chosen. In addition to the charge operator which usually

generates the gauge transformation of the f irst kind, a conserved current, which generates the gauge transformation of the second kind, can be constructed from B(x) and a gauge function [(x) a s

where

~ ( ~ 1 5 , ~ ( x ) = ~ ( x ) a , ~ ( x ) - [ a , B(.x), (5.20)

and again the limit xo- -m is used to evaluate I. Now assume!hat the in limits of the Heisenberg

fields 4, IT,, C',, i , and B approach their free- field values,

A . 4" = $0, Vv"= v", fi,h = f i u U 9 X l n = X O , and, because of Eq. (5.13),

B =BO=B'"

These limits ~ m p l y J:(x) = 0, so that with the help of Eqs. (3.26), (3.39), and (5.10), Eq. (5.18) becomes

Thus [ Q, I ] = 0 and the gauge generator Z(x) can in


general be written a s

where a and b a r e real constants. While the Higgs transformation destroyed the

gauge invariance it did not affect the field Nuation for B ( x ) . Thus, I remains a conserved quantity in the final theory. By using Eqs. ( 5 . 1 4 ) and the theorem''

for any W ( g ) such that

it is easy to show

and because B ( x ) commutes with the physical fields

It therefore follows that under this gauge transformation the in fields transform a s

These transformations a r e consistent with Eq. ( 3 . 3 9 ) which can be rewritten a s

Thus once again ;ln ( x ) field "soaks up" the gauge function produced from f i m ( x ) .

The form of Eq. ( 5 . 2 7 ) and the invariance under the I -gauge transformations suggest that in gen- e ra l

where @ is some unknown functional of the fields a s shown. This i s also consistent with the full gauge transformation generated by Z(x) and therefore with Eq. ( 5 . 1 6 a ) a s can be seen from the following argument.

The commutation rules and Q , a s defined in Eq. ( 5 . 2 1 ) , can be used to give (see also Ref. 2 0 )

Then from Eqs. ( 5 . 2 4 ) and ( 5 . 2 5 ) the corresponding commutators for E ( x ) a r e

[ 5 ( x ) , ~ l n ( s ) ] = - ~ ( a + b ) M < ( x ) .

Now Eq. ( 5 . 2 8 ) can be used to show

e - ' x i ' u ( x ) e ' z = @ ( V , " , $In, B " ) + M - ' a , i l n ( x )

- ( a + b ) a , t ( x )

= +a, ('(x), ( 5 . 3 1 )

in agreement with Eq. ( 5 . 1 6 a ) . No inconsistency is incurred by the requirement that 3: = ir: be the asymptotic limit of the Heisenberg field i ' , ( x ) . This in turn implies, because of Eq. ( 3 . 2 8 ) , it is also consistent to require J r ( x ) = O from which it immediately follows that $In = 4O, X I n = X O , and VF = V t . Note also that the physical fields V r and $In a r e invariant under the full Z-gauge transformation. Thus no observations can be affected by the original choice of gauge.

ACKNOWLEDGMENTS

The authors wish to acknowledge many helpful discussions with Professor Bing-lin Young of Iowa State University.

*Present address : Department of Physics, University of Missouri-Columbia, Columbia, Missouri 65201.

's. Weinberg, Phys. Rev. Lett. 19, 1264 (1967). 's. Weinberg, Phys. Rev. Lett. 27, 1688 (1971). 3 ~ . Goldstone, Nuovo Cimento 19, 154 (1961). *J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev.

127, 965 (1962). -

'P. W. Higgs, P h y s Lett. 2, 132 (1964). 6 ~ . W. Higgs, Phys. Rev. 145, 1156 (1966). 7 ~ . W. B. Kibble, Phys. Rev. 155, 1554 (1967). 'R. F. Streater, Proc. R. Soc. E, 510 (1965). 'H. Ezawa and J. A . Swieca, Commun. Math. Phys. 5,

330 (1967). 'OR. F e r r a r i , Nuovo Cimento H, 386 (1973).

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llG.'t Hooft, Nucl. Phys. E, 173 (1971). "G. 't Hooft, Nucl. Phys. z, 167 (1971). I3B. W. Lee , Phys. Rev. D 5, 823 (1972). "B. W. Lee and J. Zinn-Justin, Phys. Rev. D 5, 3121

(1972). I5B. W. Lee and J. Zinn-Justin, Phys. Rev. D 5, 3137

(1972); 8, 4654(E) (1973). I6B. W. Lee and J. Zinn-Justin, Phys. Rev. D 5 , 3155

(1972). "K. Fujikawa, B. W. Lee , and A. I. Sanda, Phys. Rev.

D 6, 2923 (1972) "N. Nakanishi, Prog. Theor. Phys. Suppl. 51, 1 (1973). *$N. Nakanishi, Prog . Theor. Phys. 49, 640 (1973). '%. Nakanishi, Prog. Theor. Phys. 50, 1388 (1973). "c. L. Hammer and R. H. Good, Jr . , Ann. Phys. (N.Y.)

1 2 , 463 (l961). 22r Freundlich and D. ~ u r i 6 , Nucl. Phys. E, 557 (1970).

2 3 ~ . Aurilia, Y. Takahashi, and H. Umezawa, Phys. Rev. D 5, 851 (1972).

"A. Aurilia, Y. Takahashi, N. J. Papastamatiou, and H. Umezawa, Phys. Rev. D 5, 3066 (1972).

"A. Auri l ia , Y. Takahashi, and H. Umezawa, Prog.

AND B. D e F A C I O 10 -

Theor . Phys. 48, 290 (1972). 2 6 ~ . Takahashi and R. P a l m e r , Phys. Rev. D l, 2974

(1970); R. P a l m e r and Y. Takahashi , ibid. 2, 3086 (1970).

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2 8 ~ . L , Hammer and T. A. Weber, Phys. Rev. D 5, 3087 (1972).

'$c. L. Hammer and R. H. T u c k e r , J . Math. Phys. 12, 1327 (1971).

3 0 ~ . H. Tucker and C. L . Hammer , Phys. Rev. D 3, 2448 (1971).

3 1 ~ . Nishijima, Fields and Particles (Benjamin, New York, 1969), especiaIly Chaps. 1-3, 7, and 8.

3 2 ~ . M. F r i e d , Functional Methods and Models in Quantum Field Theory W I T , Cambridge, Massachu- s e t t s , 1972).

3 3 ~ . N, Yang and D. Feldman, Phys. Rev. 79, 972 (1950). 3 4 ~ . J . Nelson and R. H. Good, J r . , Rev. Mod. Phys. 40,

508 (1968). 3 5 ~ . Umezawa, Nuovo Cimento 40, 450 (1965).

P H Y S I C A L R E V I E W D V O L U M E 1 0 , N U M B E R 4 1 5 A U G U S T 1 9 7 4

Analytic continuation of the two-Reggeon-cut discontinuity formula

A. R. White National Accelerator Labomtory, Batavra, Illinois 60510'

(Received 28 February 1974)

We give a detailed treatment of the analytic continuation of the two-Reggeon-cut discontinuity formula from above the four-particle threshold in the r channel down to negative t . We confirm the negative sign of the two-Pomeron-cut contribution to the total cross section. We show how the Mandelstam graphs can 5e used as a check on this result. We trace the negative sign to signature factors and use this to argue that multi-Pomeron cuts should contribute to the total cross section with alternating signs.

I . INTRODUCTION

In this paper we give a detailed treatment of the analytic continuation of the discontinuity formula'-3 for the two-Reggeon cut from t > 16 m2, where it is initially derived, to the scattering region t < 0. This continuation i s not treated in sufficient detail in Ref. 2, and it has been suggested4 that a more detailed treatment could lead to a reversa l of the claimed negative sign for the contribution of the two-Pomeron cut to the total c ros s section. How- ever, the analysis we present in this paper con- f i rms that the sign i s indeed negative. We empha- s ize that this result is based only on the combina- tion of t -channel unitarity with standard analyticity assumptions for both multiparticle and Pomer- on scattering amplitudes.

As we discuss in Sec. VII, any treatment of the two-Reggeon cut based on the analytic continuation of multiparticle t -channel unitarity to complex angular momentum must be applicable to the fa- miliar Mandelstam Feynman graphs. It then follows from the real-analyticity property of Pomer- on scattering amplitudes that the sign of the two- Pomeron cut in the full amplitude must be the same a s that of the two-Reggeon cut given by the Mandelstam graphs, since the signature and other kinematic factors a r e the same in the two cases. This argument in one sense confirms our result that the sign of the two-Pomeron cut i s negative. Alternatively we could say that the argument ac- tually requires u s to prove that we obtain the negative sign in order to check that the precise form of our complex angular momentum analytic con-

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