Are even exact effective potentials triviality blind?

Volume 262, number 2,3

PHYSICS LETTERS B

20 June 1991

Are even exact effective potentials triviality blind? R. T a r r a c h Departament Estructura i Constituents de la MatOria, Facultat de Fisica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain

Received 9 April 1991

We give renormalizations for 204+ ~ which lead for any d to rigorous upper and lower bound effective potentials which are finite (apart from a common divergent constant) and interacting. This proves that the corresponding exact effective potential has the same features. As this is known to be wrong for d> 3 we conclude that even exact effective potentials can see interaction where there is none. They either have to be further qualified or should be traded for effective actions.

The question o f whether 20~+ l is a trivial theory is both a f u n d a m e n t a l one as well as relevant to the Higgs sector o f the standard model. Starting with the work o f A i z e n m a n [ 1 ] and Fr/Shlich [ 2 ] and finishing with the one of Liischer and Weisz [ 3 ], the evidence for triviality has been accumulating. On the other hand, analytical variational effective potential studies found, within the gaussian approximation and for 2 > 0 to which we will limit ourselves here, one renormalization flow which led to a r e n o r m a l i z e d interacting effective potential [4]. Although this result can and has been challenged from different points o f view [ 5,6], leading to the attitude that one should trade effective potential studies for effective action studies [ 6 ], the idea o f using the effective potential as a tool for studying the triviality issue survived by arguing that what is a good a p p r o x i m a t i o n for the effective potential might be a b a d one for the effective action. We will show here, to our own chagrin, that even exact effective potentials are too interaction prone to be useful for the study o f triviality. Very recently [ 7 ] an analytic lower b o u n d o f the effective potential for 20,~+1 was given. It corresponds to the interior envelop o f the effective potentials o f the h a r m o n i c oscillators tangent from below to the potential we are studying:

294

v(o)=½m202+aO

4

>~ -V(0) _ ~M202

( M z _ m 2) 2

162

'

(1)

where 2 > O, m 2 can have any sign and M 2 >~max ( m 2, 0). It is given by _Verr(~o) ----maxM _Verff0o, M )

(2)

with l IA2,/,2

V_eer(Oo' M ) = I I a ) ( M ) +

~1,, v o -

(M2-m2) 2 162

(3) being the effective potential o f _V(q~) and where da k

(2n)a.2~

I~a)(M) =

(ka+M2) n

(4)

The value o f M w h i c h maximizes _Veff(0o,M ) is given by the nonnegative root o f M2_m I~a)(M)+¢ 2

- 42

2

=0,

(5)

which does not exist when 2 2_ Oo <~Oc =

m2 42

i~a)(O)>~O,

(6)

and then M = 0. We called ( 2 ) the h a r m o n i c effective potential ( H E P ) . It allows, together with the wel' known up-

0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )

Volume 262, number 2,3

PHYSICS LETTERS B

per bound effective potentials given by variational methods, to constrain the exact effective potential Veff(Oo) from above and below. The most popular upper bound effective potential is the gaussian one (GEP) obtained by using gaussian wave functionals and which is given by lTefr(¢~o) = minn JTeff( q~O,~r2)

(7 )

with l?eff(~o, g2) =I~ a) (.Q) + ½(mZ-.Q2)I6 a) ((2) + ½mZ(b~

+204 +6202I~a)(g2)+ 32[I~d)(.Q) ]2 .

(8)

The value off2 which minimizes ITeff(~o,£2) is given by the nonnegative root of 12~

---0,

(9)

which does not exist when m 2

m2= th2'aA' ) ' = ' ~ ( ~ )

2

'

~°2=A7'~°'2

(13,

where rh 2 and/t > 0 are arbitrary scales and ~ is a finite number. Recalling

I~2)(M)= 1-~[A3+3M2A-M3+O(-~-)], I~2)(M)=I[A-M+O(-~)],

(14,

122

I6a)(O)>~O

ITCff(Cb)>t VCff(Oo)>/_Veff(0o) •

one obtains, for rh 2 +/z~Dr > 0,

M2= (~2+4,~o2 + --~-) M Au ' (lO)

and then ~ = 0 . One then has ( I 1)

+,(~4.

interacting, i.e., not just quadratic in the classical renormalized field (bo. Then, as the exact effective potential necessarily has to lie in the finite stripe between the GEP and the HEP, it cannot correspond to a free theory because it cannot be only quadratic in (it)o.

We have found renormalizations for any d which just do this. For d-- 1 it is fine to prove that the theory is finite and interacting. Also for d = 2. But for d = 3 it is a surprise, as it goes against the almost proven triviality of 2¢~4+~ [1-3]. And what about d = 4? There triviality is rigorously proven [ 2 ]. What is wrong? Let us first see how it works.

~2+

2 (16)

Equally, from (9) 3 .Q2 = (rh 2 + 1 2 , ~ 2 + --~--~ ~/t

(17)

for dt2+3pJ~/n>0 and from (7) and (8) the GEP comes out to be

(12)

such that apart form the same UV-divergent constant, the GEP and the HEP turn out to be finite and

(15)

and from (2) and (3) the HEP _V(~)(0o)=~+~+~+

Of course, both the GEP and the HEP contain, for d> 0, ultraviolet (UV) divergences because the integrals (4) are UV divergent for n>_-½( 1 - d ) . Now, imagine that we find a renormalization, i.e., a functional dependence o f m 2, 2 and Z in the UV cutoffA, Z being the field renormalization constant 02=Zq~2

We will be somewhat cavalier and dispose of the details of the search of the renormalization flows which satisfy the above conditions, but instead give them offhand. We will also skip the case d = 1. Consider the renormalizations

~Q2_m2

I~d)(Q)+02

¢~°2< 0-2 ---

20 June 1991

JT(~f) ( 0 0 ) =

A3 flrh 2 3/t2,( ( 3 / t ~ (bo2 ~ "]- ~ "]- ~ -]- Fh2-~2

+~¢,~.

(18)

It now follows from ( 11 ) A3 V~(~)(0o) = ~ + V~,2,)(~o)

(19)

and, for large (bo, V~2) ((bo) ~ ) ~ 4 ,

(20)

so that the theory is interacting, for d = 2. The case rh 2+ 3 / ~ / n ~<0 offers no difficulty either, although it presents situations for which (6) and/or (10) apply. One might be surprised to see that we have needed coupling constant and field renormalization for d = 2, 295

Volume 262, number 2,3

PHYSICS LETTERSB

but one can convince oneself that our conditions, i.e., that the HEP and the GEP are finite, except for a common UV-divergent constant, and interacting, require it. And there is nothing wrong with it, nothing forbids these renormalizations in d= 2. Let us now turn to d= 3. Consider the renormalizations m2=(~-~--~) z, 2 = , ~ ( ~ ) 4,

0 o 2 = ( A ) 2 + o2.

(21,

Recalling

I}3)(M)=I~__~[A4+AEM2+I.ar4 11/4

21

M6

'~3)(M): -~2 [ A2 + ½M2-Mz ln 2AM M4 one obtains, for rh 2+//2 (/2~22> 0, M2= (rh2+4,(+o2 +/~2(] (].L~ 2

2nzJ \A} '

(23)

and from (2) and (3) the HEP A4 ]/2ff/2 ]24A¢ ( ~//2A¢~ + 02 -v~??(+°1 = F6~ + l-i-~ + 6 - ~ 4+ ~=+ T~V--T+2"+~.

(24)

Equally, from (9) 3/22~ (g'~ 2 0 2 = ( rh2+ 12~'+°2 + 2 n 2 / \ A J

(25)

for rh 2+ 3/)2~'/2n 2> 0 and from (7) and (8) the GEP comes out to be

].~2/~/2

h4 17-(~')(0o) = ~

+ ~

+ (thE+ -~T~2 3U2-2~ }~

3~4X + 64~r-----~

+;(+4.

(26)

Again it follows from ( 11 ) A4 V~3)(0o) = ~ + V~3.)(q~o)

296

(27)

20 June 1991

and, for large +o,

v~)(+o) ~£+4

(28)

and the theory is interacting, for d= 3! This is certainly a surprising result, as it is generally believed that ;tO]+ 1 is trivial. But, even worse, it must be clear by now that one can go on to d> 3 by a straightforward generalization of our previous renormalization flows and prove interaction for any ~ However, for d> 3, 20~+ ~ is rigorously known to be trivial. The HEP and the GEP are, for a given (and the same! ) renormalization flow, rigorous lower and upper bounds of the exact effective potential corresponding to this flow. There is nothing to object here. The renormalization flows chosen are certainly far from the ones corresponding to perturbation theory, or even far from the ones found usually in analytic nonperturbative studies. The flows relevant here all lead to UV-vanishing (evanescent) O's and M's, which were explicitly discarded in refs. [4,6], because they were not considered faithful. But they do not contradict any basic principles. Thus our conclusion seems correct: the true effective potential corresponding to the given renormalization flow is finite (apart from an UV-divergent constant) and interacting, as it grows faster than quadratically for large values of the renormalized classical field. There are two possible escape routes. The first is that the exact effective potential is quadratic for small +o and at some critical value becomes interacting for larger values of +o. Our results allow this, and we do not know whether this scenario is forbidden for all d by some theorem along the lines of the Lee-Yang theorem [ 8 ]. This would mean that the theory is free, in the absence of external sources, but becomes interacting in a strong enough classical background. If this were so, the knowledge of the large +o behaviour of the effective potential would be irrelevant and the use of an approximate effective potential for studying the presence, or absence, of interaction very limited, short of having the exact result. The effective potential would be triviality short-sighted. The second escape route is to scrutinize the effective potential more closely and to require something else beyond finiteness and interaction: notice that both the HEP and the GEP given here are somewhat sick as they lead to nontrivial four-point functions but not to nontrivial six-point functions. This is related

Volume 262, number 2,3

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to the evanescent character o f M a n d f2 but on the other h a n d the sickness o f the b o u n d s does not necessarily affect the exact result. It might, however, but we will not know, as the exact result is not known in detail. Thus it seems difficult to impose further qualifications on the effective potential. Then the question is: if the exact effective potential is interacting, a n d the theory is not, what is the effective potential good for? Well, as given, we do not know. In fact, it is known [ 6 ] that having an approximate r e n o r m a l i z e d effective potential is a far shot from having an a p p r o x i m a t e r e n o r m a l i z e d effective action, but up to now to have an exact r e n o r m a l i z e d interacting effective potential was considered a first step towards an interacting theory; now, our result shows that it is a somewhat useless first step, because we find an exact interacting potential where there is no interaction. The unqualified effective potential is triviality blind. This is actually not so surprising, as an effective potential is a generating functional for z e r o - m o m e n t u m G r e e n functions, and finiteness a n d interaction at one, usually unphysical, p o i n t is o f no significance unless the same finiteness a n d interaction holds for all the physical values o f the m o m e n t a .

20 June 1991

We conclude that either one adds further requirements ( a p a r t from finiteness a n d interaction) to the effective potential studies or one trades t h e m for effective action studies. One might be seeing mirages otherwise. I thank R a m o n Mufioz-T/tpia, Jos6 Ignacio Latorre a n d especially Joan Soto for interesting discussions. Financial support by C I C Y T u n d e r contract no. AEN90-0033 is acknowledged.

References [ 1] M. Aizenman, Phys. Rev. Lett. 47 ( 1981 ) 1. [2] J. FriShlich,Nucl. Phys. B 200 (1982) 281. [3] M. Liischer and P. Weisz, Nucl. Phys. B 240 [FS 20] (1987) 25. [4] P.M. Stevenson and R. Tarrach, Phys. Lett. B 176 (1986) 436. [5] J. Wudka, Phys. Rev. D 37 (1988) 1465; U. Ritschel, Z. Phys. C 47 (1990) 457. [6] J. Soto, Nucl. Phys. B 316 (1989) 141; B. Rosenstein and A. Kovner, Phys. Rev. D 40 (1989) 504; S. Paban and R. Tarrach, Phys. Lett. B 213 (1988) 48. [7] R. Mufioz-T/lpia and R. Tarrach, Phys. Lett. B 256 (1991) 50. [ 8 ] T.D. Lee and C.N.Yang, Phys. Rev. 87 ( 1952) 410.

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