Vacuum instability in massless λΦ4 theory

Volume 131B, number 4,5,6

PHYSICS LETTERS

17 November 1983

V A C U U M INSTABILITY IN MASSLESS A~[~ T H E O R Y P. C A S T O R I N A

and M. C O N S O L I

Istituto di Fisica, Universith di Catania, Catania, Italy and INFN-Sezione di Catania, Catania, Italy Received 15 June 1983 The possibility of vacuum instability in massless .h(I~4 theory is discussed by using both functional techniques and canonical quantization. A new symmetric ((~h)= 0) phase exists with energy below the perturbative vacuum for each value of A. The excitations above the new ground state can be described in terms of a Bogoliubov transformation. Massless A qb4 t h e o r y is k n o w n to be unstable due to the infrared divergences of the renormalized theory. In their f u n d a m e n t a l paper [1] C o l e m a n and W e i n b e r g s h o w e d that the r e q u i r e m e n t of vanishing renormalized mass generates a m i n i m u m for the effective potential V(qb) far from the origin of the field space. This effect h o w e v e r is due to the balance of terms of o r d e r A and A 2 so that the value of (~) which arises is roughly

T h e best formalism to a p p r o a c h the p r o b l e m is the calculation of the effective action allowing for composite operators [2,3]. W e shall closely follow the work of Cornwall et al. [3]. Let us consider the massless A ~ 4 lagrangian /~ = ~l(0~,I,)2- (A/4!)~ 4 .

(1)

T h e generating functional for G r e e n ' s functions with sources J ( x ) and K(x, y) is Z[J, K] = exp{(i/h)W[J, K]}

((]~)2 ~ A 2 exp(_32vr2/3A ) (A being the cutoff of the massless theory). For small A (where one expects the o n e - l o o p approximation to be reliable) the m e a n i n g of the solution is not clear. F o r this reason C o l e m a n and W e i n b e r g concluded that the really interesting p h e n o m e n o n occurs in massless scalar electrodynamies, w h e r e for small h (~Of 2) the m i n i m u m in the effective potential is f o u n d in a physically meaningful region. In this p a p e r we shall explore the possibility that a different type of instability arises in massless scalar A qb4 theory. A new symmetric ((4') = 0) phase exists with energy below the trivial perturbative vacuum. This phase is an analogous of the superfluid H e 4, just as the usual s p o n t a n e o u s s y m m e t r y breaking can be c o n n e c t e d to a situation of a pure Bose condensation. T h e excitations built a b o v e the new v a c u u m are the same as in the c o r r e s p o n d i n g massive A ~ 4 theory without tadpole terms.

= f [d~] exp{(i/h){I[~] + Jqb + 2h:bK~}}, 3

(2)

where I [ ~ ] is the classical action

I[qt,] = f d4x L ( x ) ,

(3)

and J ~ = f d4x

J(x)dP(x),

~K,I, = ~f d4x d4y ~,(x)K(x, y)~(y).

(4)

F r o m the above expressions we can obtain F[~b, G], the double L e g e n d r e transform of w [ J , K]:

r[4,, G] :

W[J, K] - J¢~ - )¢~K~ - ~hGK,

(5)

where 6W[J, K]/6J(x) : ~b(x),

0 031-9163/83/0000-0000/$03.00 © 1983 N o r t h - H o l l a n d

(6) 351

Volume 131B, number 4,5,6

PHYSICS LETTERS

6W[J, K]/6K(x, y ) = l[O(x)
holds for E[~b, G] in the Hartree-Fock approximation (h = 1)

Eta, < = f d3x[½(VO)2 + (A/4!)4~4 + ½A~52G(x,x)

~ d4y K(x, y)qS(y), (8)

6F[&, O]/6O(x, y) = - ½hK(x, y).

17 November 1983

(9)

- ~A,,G(x, Y)I,,=y+ ~G '(x, x) + -~aG(x, x)G(x, x)].

(16)

Stationarity requires

It is known [3] that F[qS, G] has the diagrammatic expansion

A&(x) = ~AO3(x) + ½A&(x)G(x, x),

r[,;b, O] = l[&l + ½ih tr ln(DG-~) + ~ih tr(2x-l(&)G)

and

+ Pz[~b, G] - ½ih tr 1,

(10)

(17)

~C-2(x, y) = [- ~Xx+ ~A62(x) + ~G(x, x)]

x 63(x - y).

where iD-l(x, y) = _ IZ]x84(x - y ) , iA-l(~, x, y ) = iD-X(x, y)-Ia(/)2(X)(~a(X

-

-

y).

(11)

F214), G] is given by the sum of all two-particle irreducible vacuum graphs in a theory with vertices determined by the shifted interaction lagrangian (cubic and quartic terms) and propagators G(x, y). At the two-loop level two graphs contribute to F[&, G], they are depicted in fig. 1. Diagram (a) corresponds to the Hartree-Fock approximation and it is exact at the two-loop level in an unshifted theory. Looking for stationary solutions one gets the relation F[&, G]lst= - f dtE[&, G ] ,

(12)

where E[&, G] is the energy of a normalized state ]~b) for which

(18)

Let us look for symmetric (& = 0) solutions allowing only for a translationally invariant G(x, y), i.e.

O(x, y)=

i

f

d3k

+~c

× ( &o exp{j[k • (x - y) - o)(xoW2-- t.o2(k) + ie J Then

G(x, y)= (~)3 f d3k 2 w ~ exp[ik " (x - y)} , (20) and w(k) has to be determined by minimization of the energy. By defining E through (21)

(4~l,i,(x, o)140 = 4,(x),

(13)

we get

<4,1+(x, 0)+(y, 0)10) = 4)(x)&(y) + hO(x, y),

(14)

1 1 f d3k e --- ~ (2rr)----3 J ~ [k 2 + w2(k)]

E l 6 , G] = (4,1DI0) •

(15)

In ref. [3] it is shown that the following form

(19)

-zc

E = f d3x e ,

i.e.

yo)]}

+ 1 g

d3k ' f 2oJ(k')"

(22)

w(k) = k[1 + a(k)]/[1 - a ( k ) l ,

(23)

A

f J

d3k

We put

with {a)

{b)

Fig. 1. Graphs contributing to F2[&, G] at the two-loop level. Diagram (b) arises in the shifted theory.

352

0 ~ a (k) ~ 1.

(24)

For u = 0 we get the massless perturbative vacuum with ~o(k)= k. By substituting eq. (23)

V o l u m e 131B, n u m b e r 4,5,6

PHYSICS LETI'ERS

into eq. (22) we obtain (x = k 2) 1

17 N o v e m b e r 1983

formation, i.e. /i(k) = c { a ( k ) - [O(k)/[q4k)l] tghlg,(k)lfit(-k)}

dx 4x(1 + a 2)

(32) + a~(fdxl-~h2] 1672 ,

(25)

1-~d~} l"

The reason for the choice (23)-(24) will be explained in the following. Since the Hartree-Fock approximation is equivalent to the Rayleigh-Ritz variational principle we can look for a trial state (different from the trivial vacuum) for which a canonical quantization of the hamiltonian reproduces the results obtained using functional methods. Let us define the hamiltonian density operatot 3c-/f(x)

and the constant c has to be determined from the requirement that [~/t(k),/~J(k')]

= (27r)32a)(k)•3(k

- k'),

(33)

w(k) being fixed from the diagonalization of the quadratic part of the normal ordered hamiltonian in the state IO) [51. We will transform to the ft operators, choosing ,1 O = -10] and defining a(k)= tghlO(k)[. If we assume (()l~,(x, 0)+(y, 0)10) = G(x, y),

(34)

then

f

@ ( X ) = ½[l~[2(X) + (V(I)) 2] + ( A / 4 ! ) ( I I 4 ( x ) ,

(26)

iNN)

and

with C~(x, 0)=

1

X(Ol(v,)21O) = -

fd3k 2k

[l(x,O)=

l

'~AxG(X,Y)lx-,,

(36)

provided that

x [d(k) exp(ik, x) + fi?(k) exp(- ik. x)],

(27)

fd3k

,o(k) : k[1 + o~(k)]/[1 - o~(k)]. Therefore

2

x [fi+(k) exp(- ik. x) - fi(k) exp(ik, x)],

(28)

and (we follow the convention of ref. [4])

[a(k), a-~(k')l = (27r)32k63(k - k').

(29)

Let us introduce a zero-momentum, symmetric ((~) = 0) normalized trial state

('f d/~[g,(k)at(k)a+(-k)

-- 4,*(t,)a (k)a(- k)l) ~0,

(61 f---f Io) = e [ a ] ,

(37)

with e[a] being defined in eq. (25). We can now investigate the possibility that the state [13)has a lower energy than the perturbative vacuum 10). By variation we obtain

(x = k 2)

IO) = exp ~

& D ] / a - (x) = (1/167r 2) x [8xa/(1 - 22) z - (A/8~-2)(1 + a)-ZN] = 0 (3o)

(38)

with A2

with dk

~<6lt}2(x)lo) = '~a-~(x, x),

= (277") -3

a N = f dx 1 -+--~'

d3k/2k,

(39)

0

and 10) is the vacuum of the h operators. Assuming ~O(k)--- t/,([k[) one gets

a(k)]O) =

a+(-k)lO)[O(k)ll,l,(k)[]

tghlO(k)[.

and we cut the divergent integral at k 2= A2. (31)

The operators which annihilate ]0) can be obtained through a Bogoliubov-Valatin trans-

,1 T h e i m a g i n a r y part of g, does not a p p e a r in the expression for the e n e r g y which is e x t r e m i z e d by the choice g~ = - 14,1. T h i s condition a m o u n t s to a s s u m e that q, m u s t h a v e the s a m e sign for all values of k.

353

Volume 13lB. number 4,5,6

PHYSICS LETTERS

The acceptable solution (0 <~ a ~< 1) is a = 1 + Z-

(Z 2+2Z) m.

(40)

with Z = 4 x / A and A = A N / 8 ~ 2. By defining ,2 A = 4A2/(B - 1),

(41)

and explicitly performing the integral N by using eq. (40) for a, we obtain the non-trivial consistency condition (32rr2/A)A2/(B - 1) = [A2/(B - 1)](sinh Y0- Y0),

(42) with B = cosh Y0. All cutoff dependence disappears and we are left with the transcendent equation 32"n'2/A = sinh y o - y0

(43)

By solving the above equation for each value of A we can get the function a ( Z ) in the range O < ~ Z < < - B - 1.

(44)

It is now possible to calculate the difference e[,~]

-

e[O]

-=

A(a) B-1

1 A4 -16-rr2(B:l)2(

4Za 2

f

d Z - - l _ o ~2

B-1 8,JT2 f 0

B-I f 0

dZ'l~,),

(45)

with & = a ( Z ) l z = s _ , = B - (B 2 - 1) m = e -'° .

(46)

By performing the integrals we obtain in the end h ( d ) = (1/16~2)[A4/(B - 1)2][f~-

(x/32~-2)fzf31, (47)

with fl = 26 - c~2/2 - 3 _ In ~ , f2 = d - 1 - In &, f3 = 1/c7 - 1 + In &,

(51)

with

0,2(0) = 2A2/(B - 1).

(52)

Eq. (51) establishes the Lorentz covariance of the whole procedure. Eq. (52) has a diagramatic representation as in the case of N a m b u and Jona-Lasinio [6]. Indeed from the self-consistent equation of fig. 2 one gets +

A2/m2) 1/2- sinh-l(A/m),

(53)

rn 2= 2 A 2 / ( B - 1 )

and

B=coshy0.

Radiative corrections to the mass can arise only to order A2, therefore the new phase is equivalent to the massive A ~ 4 theory without tadpole terms. Let us try to draw some physical consequences. The meaning of the function a ( k ) is the following. By calculating the density per unit volume in the condensate we find [fi(k)= g~t(k ) a ( k ) / 2 k ]

(48)

V-l(Olri(k)16)=

a2(k)/[1

-

o~2(k)].

(54)

(49,50)

*2B is a finite number depending only on A, since A is quadratically divergent and there is no other dimensional quantity.

354

~o2(k) = k 2 + ~o2(0)

which reduces to eq. (43) with the substitutions

a dZl-i-~

It is r e m a r k a b l e that all the dependence on the cutoff is in front of a quantity only depending on A. It is a simple calculation to see that the quantity A(cT) is negative, even for vanishingly small values of the coupling constant, if Y0 and hence d are determined from eq. (43). Therefore the state [0) is the true ground state of the massless A~ 4 theory in the symmetrical case. The canonical transformation (32) brings H into normal order with the choice (23) for the energy of the "quasiparticles" associated to the operators A ( k ) [provided a ( k ) satisfies eq. (39)]. By using eq. (38) we also obtain

16~r2/A = (A/m)(1

0

A

17 November 1983

)~

+

Q)

=o

Fig. 2. Self-consistent mass equation for Ad~4 theory.

Volume 131B, number 4,5,6

PHYSICS LETTERS

T h e r e f o r e a ( k ) describes a " d e p l e t i o n " , f r o m the pure k = 0 state, of the condensate. I n d e e d a (Z) in eq. (40) has the same f o r m as in the case of a zero t e m p e r a t u r e , self-interacting " h a r d - s p h e r e B o s e gas" [7] of radius d - 1/A and average v o l u m e per particle v ~ d 3 ( B - l). M o r e o v e r by recalling eq. (40) and eq. (43) it is easy to see that for A -~ 0 only the state k = 0 is macroscopically p o p u l a t e d (pure B o s e - E i n s t e i n condensation) while in the case A -~ oc we get a ( k ) = 1 for each value of k. It is interesting to c o m p a r e our results with the case of the self-interacting f e r m i o n t h e o r y of ref. [6]. In that case a critical coupling constant is required for the transition to the chiral symm e t r y breaking phase. In o u r case the t h e o r y is "supercritical" even for vanishingly small values of A and this can be u n d e r s t o o d in terms of Bose statistic. In the case of fermions, due to the Pauli principle, the positive kinetic energy terms can be c o n p e n s a t e d only for a sufficiently large value of the coupling constant :~3 W e have also calculated the case of a complex scalar field. In this case, by defining

(~D = 2-1/211~D1-~- i•2],

(55)

and d(k) = 2-'/2[aa(k) + i&2(k)], /~(k) = 2-'/z[/)l(k) + i/~2(k)], the c o n d e n s a t e state is the neutral one, i.e.

IO'?: exp(f

d/~ 0(k

)

x [a,(k)t;+(-k) - b(-k)a(k)l)~0 >,

(56)

x @(x). fl(y) + fl(y). ~,(x)l,

17 November 1983

(58)

w h e r e 0 ( I x - y]) is the F o u r i e r transform of 0 ( k ) and the scalar p r o d u c t in the O ( N ) space is taken a m o n g fields and c o n j u g a t e m o m e n t a . T h e v a c u u m instability we have f o u n d should persist at higher o r d e r in the loop expansion since it is present also for A - 0. W e have not yet a t t e m p t e d to include interaction with the gauge fields. W e think that it would be very interesting to try to extend the formalism of the double L e g e n d r e transform to massless scalar electrodynamics and Y a n g - M i l l s theories. If the m e c h a n i s m works it could be possible to g e n e r a t e dynamically the masses of the vector b o s o n s without destroying gauge invariance and witmout giving any expectation value to the scalar fields. This should be possible by introducing a suitable generalized B o g o l i u b o v t r a n s f o r m a t i o n which mixes up scalar and vector doson degrees of freedom. T h e energetically f a v o u r e d v a c u u m would then contain (in the case of massless scalar electrodynamics) a neutral scalar and a massive vector boson. W h e n e v e r realized these phases would be very interesting and m o r e o v e r they should not suffer from the drawbacks related to the cosmological constant. W e acknowledge useful discussions with Professor A. Agodi, Professor M. Baldo, Professor N. Cabibbo, Professor F. Catara, Professor G. J o n a - L a s i n i o and Professor L. Maiani. References

zero m o m e n t u m collective states m a d e up using only & (or/~) o p e r a t o r s do not minimize the energy. T h e formalism can be e x t e n d e d to an O ( N ) invariant theory by defining 11)) = e x p ( - i P ) 1 0 ) ,

(57)

with ,3 In this context it is interesting the close relation with Dyson's argument [8] for the vacuum instability in QED in the case of a repulsive particle-antiparticle interaction.

[1] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888. [2] G. Jona-Lasinio, Nuovo Cimento 34 (1964) 1790; H.D. Damen and G. Jona-Lasinio, Nuovo Cimento 52A (1967) 807; K. Symanzik, Commun. Math. Phys. 16 (1970) 48. [3] J.M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428; see also: R. Jackiw, Quantum mechanical approximations in quantum field theory, Orbis Scientiae II (Coral Gables, Florida, 1975). [4] C. Itzykson and J.B. Zuber, Quantum field theory (McGraw-Hill, New York, 1980). 355

Volume 131B, number 4,5,6

PHYSICS LETTERS

[5] J.R. Finger and J.E. Mandula, Nucl. Phys. B199 (1982) 168. [6] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 348.

356

17 November 1983

[7] K. Huang, Statistical mechanics (Wiley, New York, 1963) p. 421. [8] F.J. Dyson, Phys. Rev. 85 (1952) 631.