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A chance for interaction in λø43+1, λ>0
Volume 197, number 3
PHYSICS LETTERS B
29 October 1987
A CHANCE FOR INTERACTION IN $q~+1,2> 0 Sbnia PABAN Departament d'Estructura i Constituents de la Matbria, University of Barcelona, E-08028 Barcelona, Spain
and Rolf T A R R A C H NORDITA, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark
Received 7 August 1987
By variational methods an approximate, nonperturbative, unique, renormalized, interacting effective action for 2q~4+ ~, 2 > 0, is obtained. It completes and corrects an earlier result for the effective potential. It opens a door towards interaction.
N o doubt the man in the street believes that 2 0 4 + 1 , 2 > 0, is a trivial theory. This idea goes back to Landau [ 1 ] and Wilson [ 2 ] (see footnote 8), and has received very strong, almost rigorous, support by Aizenman, Fr6hlich and collaborators and Sokal [ 3]. Numerical Monte Carlo work [4], and more recent Monte Carlo renormalization group methods [ 5 ], also all find that the interaction decreases on the way towards the continuum theory. Another general feeling is that if eventually the theory exists and interacts, it will have very little connection to renormalized perturbation theory [6]. For 2 < 0 the standard regularized definition of a q u a n t u m field theory does not allow to define a vacuum. Still, partially successful attempts have been made to define a sound theory anyhow. Stevenson, using an analytic variational approach, finds an effective potential which is only stable at the renormalized level, after removing the regulator [ 7 ]. There are indications, however, that the theory destabilizes when one goes beyond the gaussian approximation used in ref. [ 7] (see ref. [ 8 ]). In its euclidean version constructions have been performed by Greensite and Halpern [ 9], and recent rigorous ones by Gawedzki and Kupiainen [ 10]. None of them, however, has allowed continuation into Minkowski space. In spite of the attractive feature of perturbative asymptotic freedom [ 11 ], we consider "wrong-sign" 0 4 much more unlikely to lead to a bona fide q u a n t u m field theory in Minkowski space than the "right-sign" version. We will only consider 2 > 0. Recently, by using the variational definition of the effective potential, and within a gaussian ansatz, a bounded (from below), interacting, finite effective potential was obtained for 204+ ~ [ 12]. It required rescaling o f the classical field and a bare coupling which vanishes logarithmically when the UV cutoff, A, is removed, A ~ o o . This made this phase look asymptotically free [ 13 ], notwithstanding the lack of perturbative asymptotic freed o m o f a positive 2 theory. Its unrelatedness with perturbation theory, loop expansion and 1 / N expansion for the O (N) theory christened it the " a u t o n o m o u s " phase [ 13 ]. It might be unbroken, in which case the appearance of massless particles made it infrared singular. In its spontaneously broken phase particles were massive, and the variational approach seemed reliable. Finally, the first corrections to the gaussian approximation computed so far vanish [ 8 ], a further indication o f its reliability. Some features o f this phase were however poorly understood. Let us list them: Permanent address: Departament d'Estructura i Constituents de la Mat6ria, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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29 October 1987
(i) the connection of the classical field rescaling with field renormalization [ 14 ]; (ii) the nonperturbative asymptotic freedom [ 13 ]; (iii) the physical significance of the variational mass parameter 12, and a worrisome consequence for the two particle states of it being assumed to be the physical mass, a problem inherited from the negative 2 phase [ 7 ]; (iv) its unrelatedness to the large-N limit [I 3]. The variational calculation within the gaussian approximation of the effective action is performed here. It requires a thorough understanding of the non-perturbative field renormalization and it allows to compute directly the physical mass of the particle. This will completely clarify points ( i ) - ( i i i ) above. Point (iv) will have to wait for the analysis of the O ( N ) model. This is, to our knowledge, the first analytic, non-perturbative, continuum calculation of the effective action for 204+1. It leads to one unique interacting phase, which coincides basically with the autonomous one, but without sharing its worrisome features. Before starting, let us comment on asymptotic freedom and existence of continuum limits. It is generally believed that asymptotic freedom is a necessary condition for the existence of a quantum field theory. However, one should not forget that the strongest support for such a dogma comes from the almost proven triviality of ;tOn+ i. Studies of an interacting 2¢]+ l theory should thus not take into account the asymptotic freedom dogma; we would otherwise be assuming what should actually follow (or not) from the study. Variational methods are best formulated in the field representation of the Schrrdinger picture. Renorrnalization, however, is quite subtle there; it has only been worked out relatively recently by Symanzik [ 15 ]. Fortunately we only have to compute expectation values, i.e. we will integrate over field configurations, and this makes it basically unnecessary to keep track of the short distance singularities which appear when defining operators in the Schrrdinger representation. Standard renormalization theory is all what we will need. Jackiw and Kerman have given the master formula to compute the effective action variationally [ 16 ]. Although we will eventually not need it, it will prove pedagogically useful to apply it here to a free field theory. The effective action is given by
F[¢o]= f dt f ~O(x) ~*[Oo,t] (iot-f d3x{-½[~/6O(x)]2+½[VO(x)]2+½m202(x)})~[Oo,t],
(1)
-oo
where
q/[Oo,t]=Nexp(-fd3x[O(x)-Oo(X)](iO,+½x/cmS-V2) [O(x)-Oo(X)]),
(2)
and N is chosen such that
f~O(x)
(3)
I~'[0o, t ] 1 2 = l .
Also the classical field ¢o(x) is assumed to vanish at the far past and the far future. It is given by
f ~o(x) ¢(x)
I~'[0o, t] 12 ~
0O(X)
(4)
•
The computation of (1) is now easy. The different terms lead to the following results
f dtf ~¢(x)~*[Oo,t]iOt~[Oo,t]=fd'x[OtOo(X)]2, -
fj
o o
d 4 x j f .~O(x) ~*[Oo, t] ½[8/8O(x)] 2 ~[¢o, t] = - ] [ d 4x [OtO0(x)] 2 +const.,
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29 October 1987
P
f d4x J ~0(X) ~*[0o,
t] {-- ½[V 0(X)] 2 - ½m202(x)} ~//[00, t]
= _ 12f d4x {[V0°(x)]2 + m202°(x)} +c°nst"
(5)
so that
if d4x{[O,Oo(x)]2- [V0o(x)] 2-m20~(x)},
F[0o]= ~
(6)
as expected. This exercise teaches us two things: first, it would have been easier to take a stationary 0o, Oo(x), solve the time-independent variational problem, and then obtain the action by Lorentz invariance, i.e. by the substitution [ V 0o ( x ) ] 2__+_ [ 0~,0o(x) ] [ 0~0o(x) ]; second, when the interaction is included the renormalization of [8/80(x)12 will have to be the same as the one of [V 0(x)l 2, again by Lorentz invariance. Our hamiltonian density is ~f(0) = - ½[8/80(x)12 + ½ [V 0(x)l 2 + ½m202(x) +204(x),
(7)
and the effective action (for stationary 00) is given by -F[0o] =min f
d4xf ~0(x) ~*Jf~',
(8)
with the constraints f ~O(x)1¢/[2=1,
f ~O(x) O(X)l~[2-----Oo(X).
(9)
We will work within the gaussian approximation, so that our trial wave functionals are ~/[ 00] = N e x p ( - I f d3x[ 0 ( x ) - 00(x)] x/[22(x)-V2 [ 0 ( x ) - 0o(X)] ).
(10)
N is such that (9) holds and f2(x) is the variational (mass) parameter, which will be given by minimization and which we choose positive. The gaussian effective action, a non-perturbative approximation to the true one, is then given by --FG[00] = m i n f
d4x f ~0(x)~[0o] Jr(0)~ff[0o].
(11)
The RHS of (10) does not make any sense, however. There are ultraviolet divergences which have to be renormalized. There is an ill-defined double functional derivative which requires point-splitting. There is the zero point energy, which requires subtraction. With Z lim 8 + ½Z[V 0(x)] 2 + ½m',Z02(x) +~.Z204(x), ~ z ( 0 ) = - -~ ,~x ____8 80(y) 80(x--~
(12)
the renormalized effective action is given by --FR[0°] =A~lim\(mann
Jf d4Xf ~0(x)~[Z-1/20°] ,)~z(0)~[Z-l/20o]-O),
where Z, m 2, 2 and D are now functions of the UV cutoffA such that the limit A ~ field is Zt/20 and its expectation value is 0o(X), as it should,
f~O(x) Zl/20(x ) ~//2[Z-1/20o ] =0o(X).
(13) exists. The renormalized (14)
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The computation of - F ~ [ 0 o , Q]----J d4x J ~O(x) vtZ-l/20o] )ffz(0) ~[Z-'/20o]
(15)
is not difficult. It gives - F ~ [00, t2] = J d4x[ ½ (V 00) 2 + (Z/96rt 2) (V 0) 2 + ZII (t2) + Z ½( m 2 - 0 2) Io(t2) 2i_1
~m 2 20o +2004 + 62Z020Io(t2) + 32Z2120(£2)],
(16)
where /~(Q)__ f
d3k (21t)32COk (O0~)n,
(17)
with tok = ~ and a symmetric UV cutoffA understood in the k-integral./1 is quartically, Io quadratically and I_ 1 logarithmically divergent. The following formulae are relevant to the following calculations: Io(I2) =Io(0) - ½£2211(Q) -t22/16n 2, I_~ (12) --I1 (#) - (l/Sn 2) In (122//z2),
(18)
where terms vanishing when A~oo have been omitted. The £2(x) which minimizes --FG[0o, £2], O(x), is given by OI_j(O) [½(0 2 -m2)-6202-62ZIo(~2)] = (1/48n 2) V2~.
(19)
Using (18) and with rh2-m2+ 122Zio(0), (19) can be written as O 2 = rh 2 + 122020- 62Zs~2I_ ~(~) - ( 3/4zt 2) 2ZO 2 + (1/24~t 2) V 2ff2/ff2I_i (12).
(20)
On the other hand, from (16) and with the help of (20), Qt3
8Fc[0o, ~] -
80o(X)
~ dt[ - V 200 + 0 0 ( 0 2 - 8 2 0 2 ) ] ,
(21)
--oo
where a term which vanishes for A~oo has been neglected. Recall that the RHS of (21) has to be UV finite. The study of how 2, rh 2 and Z have to depend on A so that the RHS of (21 ) is finite and not linear (interacting) can be performed following the steps of ref. [ 12 ], with suitable modifications. After some work one can find the unique UV renormalization flows of 2, rh 2 and Z. They are
2=~I_l(~)+(l/96rtZ)lnI_l(g),
Z=I-2(lt),
r h 2 = 3 m 2,
(22)
where # and mo are two finite arbitrary scales. The t92 equation (20) can now be solved by iteration. It gives ~2 = 2 I_, (/z) 02 + (1/12n 2) 020 In 1_1 (#) + m 2 + (02/36rc 2) [1n(202/3# 2) - 1 ],
(23)
and is thus UV logarithmically divergent. Plugging back into (21) one finds
8Fc[0o, t~] -lim
0o(X)
d t ( _ V 20o +Oo{mE + (o20/36rcE)[ln(202/31t2)- l ]} ),
-
(24)
--co
which upon functional integration and Lorentz covariantization leads to our final result F c [ 0 o ] = J d4x{ ½(0~0o) (0~0o) - ½m2020 - (0g/144n2) [ln(2020/a/fl) - 3 ]}.
(25 )
The potential part of (25) reproduces the result of ref. [ 12]. But there the analogy ends. Let us shortly comment on the differences. First, ~ diverges. Thus, it is certainly not a physical mass. We can, however, obtain immediately the physical 386
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mass from the inverse propagator, which comes out of the effective action at once. It reads M E = mE + (02m/12n 2) ln(2O2m/3~2) - 0 2 / 3 6 n 2 ,
(26)
where ~m is the (constant) vacuum expectation value of the quantum field. In the case of spontaneous symmetry breaking, which occurs when m E < (x//e/487~E)fl 2, 02 > 0 is given by m E + (~2m/36n2) [ln (2~2m/3~2) -- 1 ] --0.
(27)
In the symmetric phase, 0m=0 and M : = mE. The fact that ~ is not the physical mass solves an old problem (see point (iii) above). Traditionally [17] one used creation operators of mass ~ to construct two-particle states, and then work out the corresponding scattering amplitude. Strange things happened [ 7 ]. Either one ran into UV problems or otherwise there was no scattering, in spite of the effective potential not being quadratic. Although this problem was encountered in the 2 < 0 theory, it persisted in the 2 > 0 theory. It is clear now that the paradox is no paradox. ~ is not physical, scattering amplitudes should be obtained from Green functions, and these ones from the effective action, and not with the help of creation operators which do not even create the correct one-particle states. Unfortunately the kinetic energy obtained in the gaussian approximation is just the classical one. Thus no m o m e n t u m dependence can be studied for the four-point proper vertices. This would require going somehow beyond the approximation. It should be mentioned here that in the unbroken phase the four-, six-, etc. point Green functions are divergent (at zero momenta), while the mass, M 2 = m E is positive. This might indicate the existence of a zero-mass bound state. Second, 2 increases with increasing A. Thus the theory is not asymptotically free, exactly as happens in its perturbative version. The r-function is positive, but here the analogy with perturbation theory ends. Our result is certainly nonperturbative and it is unrelated to the Coleman-Weinberg result [ 18 ]. We do not know of any other analytic, nonperturbative, continuum result for the effective action (nor effective potential) to compare with. We find an interacting theory within a variational approximation which at least satisfies the most immediate criterion of acceptance: the variational parameter is not pushed to the border points of its range of variation, for finite A it stays finite and nonvanishing. Furthermore, and assuming that the understanding of the field renormalization gained here does not alter Yotsuyanagi's result, going beyond the gaussian approximation with a BCS ansatz does not lead to any corrections (for 2 > 0) [ 8 ]. Of course, corrections will be found eventually: in the symmetry broken phase Symanzik-convexity is not satisfied [ 19 ]. The question is" will these corrections lead back to triviality? In the meanwhile our result differs from constructive (or rather destructive) theory and numerical computations, which both strongly hint on triviality. Not having the expertise to critically compare our result to theirs, let us here only offer some vague thoughts on how our result might slip through the current triviality evidence. Rigorous proofs of triviality have been given only for more than four dimensions [ 3 ]. Then variational methods also lead to triviality [20]. Furthermore, we understand that those proofs have only been given in the symmetric phase. Our symmetric phase is somewhat peculiar: it seems to have a zero-mass bound state; a conventional theory is only obtained in the symmetry-broken phase. We do not know whether then triviality proofs go through. All numerical Monte Carlo results hint on triviality [ 4,5 ]. To mention only the more powerful Monte Carlo renormalization group studies [ 5 ], we remind that our result requires a very specific renormalization flow, given in (22), and that any other leads at best to a trivial theory. This indicates that the search o f a non-gaussian fixed point might be an extremely difficult undertaking, and random starts with three block spin transformations might just not be enough. We hope that (22) might give some indication of where to look for a nongaussian fixed point. If2~34+~ is trivial, variational methods should confirm triviality. So far, they leave one single door open to interaction.
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O n e o f us ( R . T . ) wishes to t h a n k m a n y colleagues for c o n s t r u c t i v e a n d d e s t r u c t i v e r e m a r k s c o n c e r n i n g ~4. A m o n g t h e m C. Bender, M. Braun, C. C a r a c c i o l o , J . M . C e r v e r 6 , E. d ' E m i l i o , A. G o n z ~ l e z - A r r o y o , D.I. K a z a kov, M. M o s h e , H. N e u b e r g e r , E. de R a f a e l a n d P. S t e v e n s o n , J. A m b j ~ r n a n d J. G r e e n s i t e h a v e b e e n so k i n d to r e a d the m a n u s c r i p t . O n e o f us ( R . T . ) wishes to t h a n k P. D i Vecchia a n d N O R D I T A for h o s p i t a l i t y a n d partial f i n a n c i a l s u p p o r t , as well as C I R I T ( G o v e r n m e n t o f C a t a l o n i a ) for c o m p l e m e n t a r y f i n a n c i a l support. S.P. enjoys a g r a n t f r o m t h e M E C ( S p a n i s h G o v e r n m e n t ) . O u r research is s u p p o r t e d by C A I C Y T c o n t r a c t no. A E 8 6 - 0016. T h e h o r n y a t m o s p h e r e at N O R D I T A a n d N i e l s B o h r I n s t i t u t e t h a v e c o n t r i b u t e d to the p l e a s u r e o f f i n i s h i n g this work.
References [ 1] L.D. Landau, Fundamental problems, in: Collected Papers of L.D. Landau, ed. D. Ter Haar (Pergamon, Oxford, 1965). [2] K.G. Wilson, Phys. Rev. D 6 (1972) 419. [3] M. Aizenman, Phys. Rev. Len. 47 (1981) 1; Commun. Math. Phys. 86 (1982) 1; J. Fr6hlich, Nucl. Phys. B 200 (1982) 281; C. Arag~o de Carvalho, C.S. Caracciolo and J. Fr6hlich, Nucl. Phys. B 215 (1983) 209; A. Sokal, Ann. Inst. Henri Poincar6 37 (1982) 317. [4] B. Freedman, P. Smolensky and D. Weingarten, Phys. Len. B 113 (1982) 481. [5] D.J.E. Callaway and R. Petronzio, Nucl. Phys. B 240 (1984) 577; I.A. Fox and I.G. Halliday, Phys. Lett. B 159 (1985) 148; C.B. Lang, Phys. Lett. B 155 (1985) 399; Nucl. Phys. B 265 (1986) 630. [6] G. Gallavoni and V. Rivasseau, Ann. Inst. Henri Poincar6 40 (1984) 185. [7] P.M. Stevenson, Z. Phys. C 24 (1984) 87; Phys. Rev. D 32 (1985) 1389. [ 8 ] I. Yotsuyanagi, The ~ ÷j theory with infinitesimal bare coupling constant, Z. Phys. C, to be published. [9] J. Greensite and M.B. Halpern, Nucl. Phys. B 242 (1984) 167. [ 10] K. Gawedzki and A. Kupiainen, Nucl. Phys. B 257 (1985) 474. [ 11 ] K. Symanzik, Lett. Nuovo Cimento 6 (1973) 77. [ 12] P.M. Stevenson and R. Tarrach, Phys. Len. B 176 (1986) 436. [ 13] P.M. Stevenson, B. Allbs and R. Tarrach, Phys. Rev. D 35 (1987) 2407. [ 14] J. Soto, Phys. Lett. B 188 (1987) 340. [ 15 ] K. Symanzik, Nucl. Phys. B 190 (1981 ) 1. [ 16 ] R. Jackiw and A. Kerman, Phys. Lett. A 71 (1979) 158. [ 17] L.I. Schiff, Phys. Rev. 130 (1963) 458; T. Barnes and G.I. Ghandour, Phys. Rev. D 22 (1980) 924. [ 18] S. Coleman and E. Weinberg, Phys. Rev. D 7 (1973) 1888. [19] K. Symanzik, Commun. Math. Phys. 16 (1970) 48; 18 (1970) 227. [20] P.M. Stevenson, Z. Phys. C, to be published; S. Paban, J. Taron and R. Tarrach, Z. Phys. C 34 (1987) 85.
388
PHYSICS LETTERS B
29 October 1987
A CHANCE FOR INTERACTION IN $q~+1,2> 0 Sbnia PABAN Departament d'Estructura i Constituents de la Matbria, University of Barcelona, E-08028 Barcelona, Spain
and Rolf T A R R A C H NORDITA, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark
Received 7 August 1987
By variational methods an approximate, nonperturbative, unique, renormalized, interacting effective action for 2q~4+ ~, 2 > 0, is obtained. It completes and corrects an earlier result for the effective potential. It opens a door towards interaction.
N o doubt the man in the street believes that 2 0 4 + 1 , 2 > 0, is a trivial theory. This idea goes back to Landau [ 1 ] and Wilson [ 2 ] (see footnote 8), and has received very strong, almost rigorous, support by Aizenman, Fr6hlich and collaborators and Sokal [ 3]. Numerical Monte Carlo work [4], and more recent Monte Carlo renormalization group methods [ 5 ], also all find that the interaction decreases on the way towards the continuum theory. Another general feeling is that if eventually the theory exists and interacts, it will have very little connection to renormalized perturbation theory [6]. For 2 < 0 the standard regularized definition of a q u a n t u m field theory does not allow to define a vacuum. Still, partially successful attempts have been made to define a sound theory anyhow. Stevenson, using an analytic variational approach, finds an effective potential which is only stable at the renormalized level, after removing the regulator [ 7 ]. There are indications, however, that the theory destabilizes when one goes beyond the gaussian approximation used in ref. [ 7] (see ref. [ 8 ]). In its euclidean version constructions have been performed by Greensite and Halpern [ 9], and recent rigorous ones by Gawedzki and Kupiainen [ 10]. None of them, however, has allowed continuation into Minkowski space. In spite of the attractive feature of perturbative asymptotic freedom [ 11 ], we consider "wrong-sign" 0 4 much more unlikely to lead to a bona fide q u a n t u m field theory in Minkowski space than the "right-sign" version. We will only consider 2 > 0. Recently, by using the variational definition of the effective potential, and within a gaussian ansatz, a bounded (from below), interacting, finite effective potential was obtained for 204+ ~ [ 12]. It required rescaling o f the classical field and a bare coupling which vanishes logarithmically when the UV cutoff, A, is removed, A ~ o o . This made this phase look asymptotically free [ 13 ], notwithstanding the lack of perturbative asymptotic freed o m o f a positive 2 theory. Its unrelatedness with perturbation theory, loop expansion and 1 / N expansion for the O (N) theory christened it the " a u t o n o m o u s " phase [ 13 ]. It might be unbroken, in which case the appearance of massless particles made it infrared singular. In its spontaneously broken phase particles were massive, and the variational approach seemed reliable. Finally, the first corrections to the gaussian approximation computed so far vanish [ 8 ], a further indication o f its reliability. Some features o f this phase were however poorly understood. Let us list them: Permanent address: Departament d'Estructura i Constituents de la Mat6ria, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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(i) the connection of the classical field rescaling with field renormalization [ 14 ]; (ii) the nonperturbative asymptotic freedom [ 13 ]; (iii) the physical significance of the variational mass parameter 12, and a worrisome consequence for the two particle states of it being assumed to be the physical mass, a problem inherited from the negative 2 phase [ 7 ]; (iv) its unrelatedness to the large-N limit [I 3]. The variational calculation within the gaussian approximation of the effective action is performed here. It requires a thorough understanding of the non-perturbative field renormalization and it allows to compute directly the physical mass of the particle. This will completely clarify points ( i ) - ( i i i ) above. Point (iv) will have to wait for the analysis of the O ( N ) model. This is, to our knowledge, the first analytic, non-perturbative, continuum calculation of the effective action for 204+1. It leads to one unique interacting phase, which coincides basically with the autonomous one, but without sharing its worrisome features. Before starting, let us comment on asymptotic freedom and existence of continuum limits. It is generally believed that asymptotic freedom is a necessary condition for the existence of a quantum field theory. However, one should not forget that the strongest support for such a dogma comes from the almost proven triviality of ;tOn+ i. Studies of an interacting 2¢]+ l theory should thus not take into account the asymptotic freedom dogma; we would otherwise be assuming what should actually follow (or not) from the study. Variational methods are best formulated in the field representation of the Schrrdinger picture. Renorrnalization, however, is quite subtle there; it has only been worked out relatively recently by Symanzik [ 15 ]. Fortunately we only have to compute expectation values, i.e. we will integrate over field configurations, and this makes it basically unnecessary to keep track of the short distance singularities which appear when defining operators in the Schrrdinger representation. Standard renormalization theory is all what we will need. Jackiw and Kerman have given the master formula to compute the effective action variationally [ 16 ]. Although we will eventually not need it, it will prove pedagogically useful to apply it here to a free field theory. The effective action is given by
F[¢o]= f dt f ~O(x) ~*[Oo,t] (iot-f d3x{-½[~/6O(x)]2+½[VO(x)]2+½m202(x)})~[Oo,t],
(1)
-oo
where
q/[Oo,t]=Nexp(-fd3x[O(x)-Oo(X)](iO,+½x/cmS-V2) [O(x)-Oo(X)]),
(2)
and N is chosen such that
f~O(x)
(3)
I~'[0o, t ] 1 2 = l .
Also the classical field ¢o(x) is assumed to vanish at the far past and the far future. It is given by
f ~o(x) ¢(x)
I~'[0o, t] 12 ~
0O(X)
(4)
•
The computation of (1) is now easy. The different terms lead to the following results
f dtf ~¢(x)~*[Oo,t]iOt~[Oo,t]=fd'x[OtOo(X)]2, -
fj
o o
d 4 x j f .~O(x) ~*[Oo, t] ½[8/8O(x)] 2 ~[¢o, t] = - ] [ d 4x [OtO0(x)] 2 +const.,
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f d4x J ~0(X) ~*[0o,
t] {-- ½[V 0(X)] 2 - ½m202(x)} ~//[00, t]
= _ 12f d4x {[V0°(x)]2 + m202°(x)} +c°nst"
(5)
so that
if d4x{[O,Oo(x)]2- [V0o(x)] 2-m20~(x)},
F[0o]= ~
(6)
as expected. This exercise teaches us two things: first, it would have been easier to take a stationary 0o, Oo(x), solve the time-independent variational problem, and then obtain the action by Lorentz invariance, i.e. by the substitution [ V 0o ( x ) ] 2__+_ [ 0~,0o(x) ] [ 0~0o(x) ]; second, when the interaction is included the renormalization of [8/80(x)12 will have to be the same as the one of [V 0(x)l 2, again by Lorentz invariance. Our hamiltonian density is ~f(0) = - ½[8/80(x)12 + ½ [V 0(x)l 2 + ½m202(x) +204(x),
(7)
and the effective action (for stationary 00) is given by -F[0o] =min f
d4xf ~0(x) ~*Jf~',
(8)
with the constraints f ~O(x)1¢/[2=1,
f ~O(x) O(X)l~[2-----Oo(X).
(9)
We will work within the gaussian approximation, so that our trial wave functionals are ~/[ 00] = N e x p ( - I f d3x[ 0 ( x ) - 00(x)] x/[22(x)-V2 [ 0 ( x ) - 0o(X)] ).
(10)
N is such that (9) holds and f2(x) is the variational (mass) parameter, which will be given by minimization and which we choose positive. The gaussian effective action, a non-perturbative approximation to the true one, is then given by --FG[00] = m i n f
d4x f ~0(x)~[0o] Jr(0)~ff[0o].
(11)
The RHS of (10) does not make any sense, however. There are ultraviolet divergences which have to be renormalized. There is an ill-defined double functional derivative which requires point-splitting. There is the zero point energy, which requires subtraction. With Z lim 8 + ½Z[V 0(x)] 2 + ½m',Z02(x) +~.Z204(x), ~ z ( 0 ) = - -~ ,~x ____8 80(y) 80(x--~
(12)
the renormalized effective action is given by --FR[0°] =A~lim\(mann
Jf d4Xf ~0(x)~[Z-1/20°] ,)~z(0)~[Z-l/20o]-O),
where Z, m 2, 2 and D are now functions of the UV cutoffA such that the limit A ~ field is Zt/20 and its expectation value is 0o(X), as it should,
f~O(x) Zl/20(x ) ~//2[Z-1/20o ] =0o(X).
(13) exists. The renormalized (14)
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The computation of - F ~ [ 0 o , Q]----J d4x J ~O(x) vtZ-l/20o] )ffz(0) ~[Z-'/20o]
(15)
is not difficult. It gives - F ~ [00, t2] = J d4x[ ½ (V 00) 2 + (Z/96rt 2) (V 0) 2 + ZII (t2) + Z ½( m 2 - 0 2) Io(t2) 2i_1
~m 2 20o +2004 + 62Z020Io(t2) + 32Z2120(£2)],
(16)
where /~(Q)__ f
d3k (21t)32COk (O0~)n,
(17)
with tok = ~ and a symmetric UV cutoffA understood in the k-integral./1 is quartically, Io quadratically and I_ 1 logarithmically divergent. The following formulae are relevant to the following calculations: Io(I2) =Io(0) - ½£2211(Q) -t22/16n 2, I_~ (12) --I1 (#) - (l/Sn 2) In (122//z2),
(18)
where terms vanishing when A~oo have been omitted. The £2(x) which minimizes --FG[0o, £2], O(x), is given by OI_j(O) [½(0 2 -m2)-6202-62ZIo(~2)] = (1/48n 2) V2~.
(19)
Using (18) and with rh2-m2+ 122Zio(0), (19) can be written as O 2 = rh 2 + 122020- 62Zs~2I_ ~(~) - ( 3/4zt 2) 2ZO 2 + (1/24~t 2) V 2ff2/ff2I_i (12).
(20)
On the other hand, from (16) and with the help of (20), Qt3
8Fc[0o, ~] -
80o(X)
~ dt[ - V 200 + 0 0 ( 0 2 - 8 2 0 2 ) ] ,
(21)
--oo
where a term which vanishes for A~oo has been neglected. Recall that the RHS of (21) has to be UV finite. The study of how 2, rh 2 and Z have to depend on A so that the RHS of (21 ) is finite and not linear (interacting) can be performed following the steps of ref. [ 12 ], with suitable modifications. After some work one can find the unique UV renormalization flows of 2, rh 2 and Z. They are
2=~I_l(~)+(l/96rtZ)lnI_l(g),
Z=I-2(lt),
r h 2 = 3 m 2,
(22)
where # and mo are two finite arbitrary scales. The t92 equation (20) can now be solved by iteration. It gives ~2 = 2 I_, (/z) 02 + (1/12n 2) 020 In 1_1 (#) + m 2 + (02/36rc 2) [1n(202/3# 2) - 1 ],
(23)
and is thus UV logarithmically divergent. Plugging back into (21) one finds
8Fc[0o, t~] -lim
0o(X)
d t ( _ V 20o +Oo{mE + (o20/36rcE)[ln(202/31t2)- l ]} ),
-
(24)
--co
which upon functional integration and Lorentz covariantization leads to our final result F c [ 0 o ] = J d4x{ ½(0~0o) (0~0o) - ½m2020 - (0g/144n2) [ln(2020/a/fl) - 3 ]}.
(25 )
The potential part of (25) reproduces the result of ref. [ 12]. But there the analogy ends. Let us shortly comment on the differences. First, ~ diverges. Thus, it is certainly not a physical mass. We can, however, obtain immediately the physical 386
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mass from the inverse propagator, which comes out of the effective action at once. It reads M E = mE + (02m/12n 2) ln(2O2m/3~2) - 0 2 / 3 6 n 2 ,
(26)
where ~m is the (constant) vacuum expectation value of the quantum field. In the case of spontaneous symmetry breaking, which occurs when m E < (x//e/487~E)fl 2, 02 > 0 is given by m E + (~2m/36n2) [ln (2~2m/3~2) -- 1 ] --0.
(27)
In the symmetric phase, 0m=0 and M : = mE. The fact that ~ is not the physical mass solves an old problem (see point (iii) above). Traditionally [17] one used creation operators of mass ~ to construct two-particle states, and then work out the corresponding scattering amplitude. Strange things happened [ 7 ]. Either one ran into UV problems or otherwise there was no scattering, in spite of the effective potential not being quadratic. Although this problem was encountered in the 2 < 0 theory, it persisted in the 2 > 0 theory. It is clear now that the paradox is no paradox. ~ is not physical, scattering amplitudes should be obtained from Green functions, and these ones from the effective action, and not with the help of creation operators which do not even create the correct one-particle states. Unfortunately the kinetic energy obtained in the gaussian approximation is just the classical one. Thus no m o m e n t u m dependence can be studied for the four-point proper vertices. This would require going somehow beyond the approximation. It should be mentioned here that in the unbroken phase the four-, six-, etc. point Green functions are divergent (at zero momenta), while the mass, M 2 = m E is positive. This might indicate the existence of a zero-mass bound state. Second, 2 increases with increasing A. Thus the theory is not asymptotically free, exactly as happens in its perturbative version. The r-function is positive, but here the analogy with perturbation theory ends. Our result is certainly nonperturbative and it is unrelated to the Coleman-Weinberg result [ 18 ]. We do not know of any other analytic, nonperturbative, continuum result for the effective action (nor effective potential) to compare with. We find an interacting theory within a variational approximation which at least satisfies the most immediate criterion of acceptance: the variational parameter is not pushed to the border points of its range of variation, for finite A it stays finite and nonvanishing. Furthermore, and assuming that the understanding of the field renormalization gained here does not alter Yotsuyanagi's result, going beyond the gaussian approximation with a BCS ansatz does not lead to any corrections (for 2 > 0) [ 8 ]. Of course, corrections will be found eventually: in the symmetry broken phase Symanzik-convexity is not satisfied [ 19 ]. The question is" will these corrections lead back to triviality? In the meanwhile our result differs from constructive (or rather destructive) theory and numerical computations, which both strongly hint on triviality. Not having the expertise to critically compare our result to theirs, let us here only offer some vague thoughts on how our result might slip through the current triviality evidence. Rigorous proofs of triviality have been given only for more than four dimensions [ 3 ]. Then variational methods also lead to triviality [20]. Furthermore, we understand that those proofs have only been given in the symmetric phase. Our symmetric phase is somewhat peculiar: it seems to have a zero-mass bound state; a conventional theory is only obtained in the symmetry-broken phase. We do not know whether then triviality proofs go through. All numerical Monte Carlo results hint on triviality [ 4,5 ]. To mention only the more powerful Monte Carlo renormalization group studies [ 5 ], we remind that our result requires a very specific renormalization flow, given in (22), and that any other leads at best to a trivial theory. This indicates that the search o f a non-gaussian fixed point might be an extremely difficult undertaking, and random starts with three block spin transformations might just not be enough. We hope that (22) might give some indication of where to look for a nongaussian fixed point. If2~34+~ is trivial, variational methods should confirm triviality. So far, they leave one single door open to interaction.
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O n e o f us ( R . T . ) wishes to t h a n k m a n y colleagues for c o n s t r u c t i v e a n d d e s t r u c t i v e r e m a r k s c o n c e r n i n g ~4. A m o n g t h e m C. Bender, M. Braun, C. C a r a c c i o l o , J . M . C e r v e r 6 , E. d ' E m i l i o , A. G o n z ~ l e z - A r r o y o , D.I. K a z a kov, M. M o s h e , H. N e u b e r g e r , E. de R a f a e l a n d P. S t e v e n s o n , J. A m b j ~ r n a n d J. G r e e n s i t e h a v e b e e n so k i n d to r e a d the m a n u s c r i p t . O n e o f us ( R . T . ) wishes to t h a n k P. D i Vecchia a n d N O R D I T A for h o s p i t a l i t y a n d partial f i n a n c i a l s u p p o r t , as well as C I R I T ( G o v e r n m e n t o f C a t a l o n i a ) for c o m p l e m e n t a r y f i n a n c i a l support. S.P. enjoys a g r a n t f r o m t h e M E C ( S p a n i s h G o v e r n m e n t ) . O u r research is s u p p o r t e d by C A I C Y T c o n t r a c t no. A E 8 6 - 0016. T h e h o r n y a t m o s p h e r e at N O R D I T A a n d N i e l s B o h r I n s t i t u t e t h a v e c o n t r i b u t e d to the p l e a s u r e o f f i n i s h i n g this work.
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