Volume 157B, number 1 PHYSICS LETTERS 4 July 1985

S P O N T A N E O U S S Y M M E T R Y B R E A K I N G IN AN 0(2 ) I N V A R I A N T S C A L A R T H E O R Y

Yves B R I H A Y E

University of Mons, Mons, Belgium

and

Maurizio C O N S O L I

CERN, 1211 Geneva 23, Switzerland

Received 15 March 1985

A simple variational principle for an 0(2) invariant scalar theory, within the framework of gaussian wave functionals, is presented. The conditions for the occurrence of spontaneous symmetry breaking are discussed. The limits of validity of the gaussian approximation are also analyzed in detail.

Self-interacting scalar theories play an important role in the framework of the presently accepted theory of electroweak interactions. The vacuum ex- pectation o f a scalar field is a convenient device to gen- erate the mass o f the vector bosons without destroy- ing the renormalizability properties of the theory.

At the same time there are some problems con- cerning the internal consistency of these scalar theo- ries independently of their embedding in more com- plicated structures. Indeed even the analysis of the simplest possible self-interacting theory (;k~b 4 in (3 + 1) dimensions) is not free o f ambiguities * l . An inter- esting recent analysis [ 1 ] suggests, for example, the existence o f two very different ?re 4 theories. The first case corresponds to the regularized version (i.e. cutoff dependent) which exhibits spontaneous symmetry breaking [2] within a class o f well defined quantum states (gaussian wave functionals). The second situa- tion on the other hand, in which the limit of the con- tinuum (cutoff to infinity) is taken before drawing any conclusion about the physics of the theory,

,1 For a complete set of references on theory and gaussian quantization see ref. [ 1 ].

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shows very different features (i.e. no spontaneous symmetry breaking). Moreover, the stability analysis of the two regimes requires opposite signs of the bare coupling constant ~k appearing in the hamiltonian. In- deed the regularized version requires ;k to be positive, the continuum theory of ref. [ 1 ], has a meaning only with a negative (and vanishing small) coupling con- stant. We believe that, besides the boundedness of the hamiltonian in any defined subspace one should also consider the stability properties o f the Green's func- tions generating functional. Indeed euclidean pro- longation is defined only if ;k > 0, in order for the large field components to be damped in the functional in- tegration. Without such an assumption the meaning of the resulting theory is very difficult to understand.

At the same time we agree with ref. [1] that the regularized version has no consistent continuum limit, since the relevant physics associated to the sponta- neously broken phase drifts to infinity with the cut- off. This trouble, however, is shared by any non- asymptotically free theory and is not peculiar of our simple model field theories, since we are not able to discover non-trivial ultraviolet fixed points by means other than perturbation theory. Therefore, there is

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Volume 157B, number 1 PHYSICS LETTERS 4 July 1985

the common belief that only asymptotically free theories may (possibly) generate self-consistently a scale which can be held fixed, in the limit of infinite cutoff, by suitably rescaling the coupling constant at the cutoff according to the renormalization group. In other words, a meaningful continuum limit, whenever possible, requires the understanding of the infinities "in fieri" rather than the explicit introduction of in- finities "in atto". This is, after all, the underlying ra- tionale of lattice gauge theory calculations.

In our case, if we want to give any meaning to the scalar self-interacting theories, we have to assume some mechanism which provides an external cutoff. As shown in ref. [2] the possibility ofdecoupling the spontaneously broken phase from the cutoff is relat- ed to the addition hypothesis that the interaction at the cutoff scale (A) is perturbatively small.

One may object that the use of a "'large"A and a "small" X is an obvious contradiction since Xq~ 4 theo- ries are not asymptotically free. This is not a good ob- jection. In fact we cannot take the limit A ~ oo in our analytical expressions but we can make A larger than any meaningful energy scale. Let us consider for a moment the situation of massless scalar electrodyna- mics. In this theory [3], spontaneous symmetry breaking can be discovered at the one-loop level. The phenomenon takes place at a scale M at which the scalar running coupling constant X(M 2) satisfies the relation

X(M2)/16rr2 ~ a 2 (a ~ 1/137).

Now let us assume this relation for our analysis of R¢4 theory. If we "limit" ourselves to analyse regions as large as (M -= (¢))

A 2 ~ M 2 exp [(1 - a)/a 2 ] ,

we still remain in a perturbative regime (by using the leading log evolution of the running coupling con- stant) since

X(A2)/167r 2 ~ a .

By adopting this point of view we shall extend the analysis of ref. [2] to an 0(2) invariant scalar theory in order to understand the reliability of a gaussian va- riational principle in the case o f a continuous sym- metry group. In this case, in fact, there are exact field theoretical results (Goldstone theorem) which can shade some light on the virtues and the limits o f our

approximation. Our theory is described by the lagrangian density

1 2 2 = ~ ( ~ , ) - v ( ~ , . , ) , (1)

with

~= ( ¢ 1 , ¢ 2 ) ,

and

1 2 V(~'~b) = ~mB(~b- ~) + [~(4[ • 2)] (~b'~b) 2 (2)

We assume here X > 0 while m 2 is an arbitrary param- eter. Gaussian quantization is performed by introduc- ing the class of wave functionals

,I, [¢] =exp ( _ 1 fd3xd3y[¢ ' - (¢P](x)

\ ci7 i (x ,y) [¢: - %>I~) ) , X (3)

where <¢i>(x) and Gi/(x , y) are variational parameters. We will limit ourselves to the situation (¢i>(x) = F i = const.

In order to diagonalize the quadratic part of the hamiltonian it is convenient to define shifted fields and 7? as follows

¢I = F I + ~cos0 - ~?sin0 ,

¢2 = F2 + ~ sin 0 + r/cos 0 , (4)

with

cosO=F1/p, sinO=F2/P , p=(F2 + F2) 1/2.

The shifted fields satisfy (in the gaussian states)

(~)~, = (n>v, = 0 , (5)

we will restrict to functions Gi/(x,y ) such that

(~2),i,=Io(m~), (n2),i,=Io(mn), (~r/),i ,=0, (6)

with

- f d 3 k ½(k2+m2) -1/2 (7) Io(m) (2rr)3

transferring all the remaining variational arbitrariness on the ~ and r/masses only.

The momentum integration in (7) is extended up to a maximum value Ik[ = A. As shown in ref. [2] this restriction does not violate Lorentz invariance at the extrema of the variational procedure. Indeed the

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Volume 157B, number 1 PHYSICS LETTERS 4 July 1985

gaussian states can be constructed explicitly [2] by using creation and annihilation operators and a Lorentz covariant spectrum of excitations is discover- ed at those states i~) for which

~i(('¢' IH i'-I')/('.I'l'-I' )) = 0 . (8)

In other words Lorentz invariance is generally violated by introducing a cutoff in momentum space but is preserved at the extrema of the energy. This can be understood by observing that the hamiltonian com- mutes with the generators o f the Lorentz group and with a variational procedure (even in a restricted sub- space) we can get very close to some energy eigen- state.

By defining the hamiltonian operator

H -fd3x ~¢(x): fd3x~ (.~ +~r 2) + 1 (v¢1)2

1 + 5( V~b2)2 + V(d~'~)] , (9)

and transforming to the g and r/fields, we obtain the energy density of our gaussian states

( ~ i ~ i ~ ) / ( ~ I~I ,) = E(m~ , m n , p) , (10)

with

E(m~ , m n, p) = I i ( m t ) + Ii (m n)

-½ [m~lo(m ~) + m2Io(mn)]

1 2 + ~ mB [p2 + io(m~) + io(mn) ]

+ [2,/(4!" 2)] [p4 +p2 [610(rn~ ) + 210(mn)]

+ 3102(m~) + 3120(mn) + 2Io(m~)lo(mn)], (11)

where

d3k I I ( m ) = f ( 2 - ~ ½(k2 + m2)1/2 " (12)

By extremizing the energy density (11) with respect to m~ and m n we get the gap equations

rn~(p) = m2B + 0, /4!)[6P 2 + 610(m~) + 2Io(mn)], (13)

2 - 2 mn(p) - m B + 0, /4!) [2p 2 + 2Io(m ~) + 6Io(mn)] •

The above equations can be solved in order to define the gaussian effective potential

V(p) = E (m~(p), ran(p), p) . (14)

Absolute extrema occur at those values of/9 for which

V'(p) = p{m 2 +IX/(4! • 2)] [4p2+ 1210(m ~) +410(rnn)]}

= 0 , (15)

therefore we get the origin in field space (p = 0) and those values o f F I and F 2 for which

m 2 + [X/(4! • 2)] [4p 2 + 1210(m~) + 410(mn) ] = 0. (16)

We note that, if we keep in eqs. (13) and (16) only those terms which do not involve the f luc tua t ion- fluctuation interaction (the linearized approximation), eq. (16) is equivalent to the condition

m2(p) = 0 . (17)

In other words the Goldstone theorem is easily dis- covered at the linearized level but is violated in the gaussian approximation. This is understood by notic- ing that the gaussian approximation becomes exact (at the two-loop level) in the case of an O(N) symme- try only when N ~ oo [4]. The linearized approxima- tion, however, does not correspond to any variational procedure and only by exploring a defined subspace with the whole hamiltonian we can get to some con- clusions. Therefore we will continue our analysis by keeping in mind that no "positive" statements (as in any limited variational principle) can be obtained from our final results.

For simplicity we shall fix m 2 from the condition that

V"(p) Its= 0 = m~(p)lp=o = m2(p)lp=o = O. (18)

This is not a restriction at all as shown in ref. [2] and it can be also understood by using the following ar- gument. Let us assume for a moment a different mass renormalization condition, say

" rn 2 g (P)lp=0 = R , (19)

then we can define the renormalized coupling con- stant at zero momentum ,2

' XR(0) -= V""(p) b=0 1

1 -- g X/_I (mR) -- ~ X212_I(mR) = x (20)

1 2 + ~ X/l(m R) + ~ ~,212_1(m R)

1. .2 The factor ~ m due to our choice of the in teract ion term in

eq. (2).

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Volume 157B, number 1 PHYSICS LETTERS 4 July 1985

with

d3k I - l ( m ) =f + m2)-3 /2 . (21)

From eq. (20) we see that we can "force" the fourth derivative at the origin to be negative either by taking the limit m R -+ 0 or A ~ oo (and m R fixed). In other words the absolute minima of our function V(O) de- pend on the relative magnitude of the "ultraviolet" and "infrared" physics. Again, let us assume m R = 0 but let us enclose our system in a finite volume charac- terized by a maximum size L and a minimum lattice spacing d. Then XR(0 ) will become negative when

In ( L 2 M 2) >> k -1 . (22)

I f we assume a perturbative regime at the lattice level, the instability will show up in exponentially large regions. These are the predictions of the gaussian quan- tization. From now on we will assume the value m R =0 and the asymptotic relation

in (L/d) -+ o o . (23)

By using the identities of ref. [5] among the I func- tions we can then compute the difference AE = V(~) - V(0) (~ being defined as the solution of eq. (16)). We fmd

AE = (--1/128rr 2) ( r ~ + m 4) (24)

( r~ and rfi n being defined as the value of m~(0) and mn(p ) at O = 0).

Eq. (24) means that the gaussian state at the origin (the perturbative vacuum) is not stable when we choose extreme situations (very large volume and massless particles) described above.

Which conclusions can be extracted from the abso- lute minima (i.e. the whole set of states (3) with F 2 + F 2 = ~2)? The spectrum o f those states is fixed by the following equations:

1 - - 2 - -

1 = ~y(~n~) + g y ( m n ) [ ~ y ( m ~ ) - 1] ,

- 2 = fftff []y(rn~ - 1)] ~2 = (6/~.)rn~ (25) m n , ,

and

y ( m ) - (h /16rr2)[ ln (4A 2 /m2) - 1].

For small 3,, the following relations hold:

2 ..~ [0.23 + O(k /16rrZ)]m~, (26a) m n

rn~ = O(e-1 /h) A 2 , (26b)

the first of which represents the Goldstone theorem* a, in the language of ganssian quantization. At the same time (26b) expresses the decoupling of the cutoff from the scale associated to the spontaneously broken phase.

To conclude our analysis we want to do some con- siderations about the convexity properties of the " t rue" effective potential. Our gaussian potential is not convex downward under the conditions described above, convexity being recovered in the situation XR(0) > 0. One can adopt the point o f view that gaussian quantization is reliable as far as the potential maintains the good convexity properties (only one minimum at the origin). In this case we should con- clude that the whole approach to detect spontaneous symmetry breaking through gaussian states is wrong. Another possibility, however, is that the true effec- tive potential is "f la t" in the region enclosed by the gaussian minima. The "gaussian boundary" would then define the class of values of (F1, F2)which can be meaningfully described by means of gaussian states, all the others being analyzable only by means of more complicated wave functionals. This can be understood from the following argument. Let us assume for a mo- ment the simple Xq~ 4 case (i.e. no continuous symme- try). From the analysis of ref. [2] we deduce the ex- istence of two (degenerate) absolute gaussian minima at (~b} = +/~, ~t'p [~b] and ~_ff. [~]. In the infinite-vo- lume limit the overlapping of these two states is zero,

f [ d ~ ] ¢ ~ [ ~ ] , I , _pD] - e x p ( - l F l 3 V ) (27)

(V being the quantization volume). Now any normal- ized state of the form

~ ' [ ¢ ] = a ~ [ ¢ ] + / ~ _ ; [ ~ b ] (28)

(with ot 2 + f12 = 1), has the same energy as q'p[~b] and ~I,_p [q~] (in the limit V-+ oo) but expectation value

In other words we have an infinite set of states with the same energy as the gaussian minima (fixing a

~a The theorem is only roughly verified but the situation can be improved when going to larger NO(N) scalar theories.

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boundary in the one-dimensional space of constant fields). In this case the convexity properties o f the ef- fective potential are trivially discovered. It is not too difficult to extend the above arguments to our 0(2) case (with the obvious generalization that the gaus- sian boundary is fixed on a circle).

From the above considerations we conclude that gaussian quantization may be a useful device to ob- tain interesting information about the effective po- tential even if the gaussian states exhibit those general properties (existence o f massless particles) only ap- proximately, required by exact field theoretical re- suits.

The authors are deeply indebted to Professor D. Amati, Professor A. Neveu, Professor G. Preparata and Professor L. Van Hove for useful discussions and for reading the manuscript.

References

[ 1 ] P.M. Stevenson,Wisconsin pmprint MAD-TH-226 (1984). [2] M. Consoli and A. Ciancitto, CERN preprint TH-3951

(1984). [3] S. Coleman and E.Weinberg, Phys. Rev. D7 (1973) 1888. [4] J.M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev.

D105 (1974) 2428. [5] P.M. Stevenson, Z. Phys. C24 (1984) 87.

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