N expansion for scalar fields

1/N

EXPANSION I.

Ya.

FOR

SCALAR

FIELDS UDC 530.145;539.12

Aref'eva

Models of s c a l a r field t h e o r i e s with a l a r g e n u m b e r N of isotopic d e g r e e s of f r e e d o m a r e c o n s i d e r e d . A t h e o r y of p e r t u r b a t i o n s with r e s p e c t to a s m a l l p a r a m e t e r is developed in the f o r m a l i s m of path i n t e g r a t i o n f o r a s p a c e - t i m e dimension ~D = 2, 3, 4. The p a r t i c l e s p e c t r u m obtained in basic o r d e r with r e s p e c t to N -1 is c o m p a r e d with the s p e c t r u m in the path a p p r o a c h . It is shown that when ~) = 4 the chiral field model is turned, as a r e s u l t of r e n o r m a l i z a t i o n , into a model with four i n t e r a c t i o n s . The limitations of the applicability of the 1 / N expansion* a r e d i s c u s s e d .

In r e c e n t t i m e s the nontrivial models in field t h e o r y w e r e e x a m i n e d by m e a n s of the s t a n d a r d theory of p e r t u r b a t i o n s with r e s p e c t to the coupling constant. Nowadays it is p o s s i b l e to go outside the f r a m e w o r k of this p e r t u r b a t i o n t h e o r y . On the one hand, this b e c a m e p o s s i b l e thanks to the a p p e a r a n c e of a c o m p a r a t i v e l a r g e n u m b e r of exactly solvable models both c l a s s i c a l [1, 2] as well as quantum [3]. On the other hand, the n e c e s s i t y of doing away with the expansion with r e s p e c t to the c h a r g e f o r c e r t a i n i n t e r e s t i n g physical models [4] is connected with the f a c t t h a t they a r e n o n r e n o r m a l i z a b l e under the c l a s s i c a l a p p r o a c h , and this, possibly, is connected with the inapplicability of the t h e o r y of p e r t u r b a t i o n s with r e s p e c t to the i n t e r a c t i o n constant [5]. We can s c a r c e l y expect that s o m e of the r e a l i s t i c models in f o u r - d i m e n s i o n a l s p a c e - t i m e will t u r n out to be e x a c t l y solvable in the quantum c a s e . T h e r e f o r e , it is d e s i r a b l e to c o n s t r u c t a p e r t u r b a t i o n theory s c h e m e whose p a r a m e t e r would be other than the c h a r g e . to p o w e r s of

4/~ , w h e r e

One such s c h e m e is the t h e o r y of p e r t u r b a t i o n s with r e s p e c t

N is the n u m b e r of components of the field.

The 4/t~ t h e o r y of p e r t u r b a t i o n s was f i r s t e x a m i n e d in s t a t i s t i c a l p h y s i c s [6]. In quantum field theory h

the b a s i c o r d e r with r e s p e c t to 1/~ f o r s c a l a r t h e o r i e s with ~N) - i n v a r i a n t interaction was e x a m i n e d in a n u m b e r of p a p e r s [7]. A f t e r this the p r o b l e m was p o s e d of accounting f o r the following o r d e r s with r e s p e c t to

4/~ . The o r d e r

t u r b a t i o n s f o r the

J/Ha was c o m p u t e d in [8], while a r e g u l a r method f o r constructing the ~

-interaction (I)

4/~ theory of p e r -

is the s p a c e - t i m e dimension) was p r o p o s e d in [9]. A p e r t u r b a t i o n

theory was c o n s t r u c t e d in [10] f o r models with f e r m i o n s .

The basic o r d e r with r e s p e c t to

field was e x a m i n e d in [11], while a D4~ t h e o r y of p e r t u r b a t i o n s , r e n o r m a l i z a b l e wherL

4/N f o r a chiral

~/N

, was c o n s t r u c t e d

in [9, 12, 13]. In [9, 12] the c h i r a l field with D ~ 2 , 5 was t r e a t e d as the l i m i t of a s t r o n g coupling with r e s p e c t to a nonr e n o r m a l i z e d i n t e r a c t i o n constant f o r the 0(N) ~

model.

The 4/t~ expansion f o r the 0(N) ~

model is of in-

t e r e s t p r e c i s e l y in connection with the f a c t that in each o r d e r with r e s p e c t to 4/~ we can p a s s , as was shown * T r a n s l a t o r ' s note. In the English l i t e r a t u r e it is m o r e c u s t o m a r y to use the lower c a s e , i.e., " l / n expansion." T r a n s l a t e d f r o m Zapiski Nauchnyl~h S e m i n a r o v Leningradskogo Otdeleniya M a t e m a t i c h e s k o g o Instituta im. V. A. StePdova AN SSSR, Vol. 77, pp. 3-23, 1978.

0090-4104/83/2205-1535507'50

1983 Plenum Publishing C o r p o r a t i o n

1535

in [9], to the l i m i t of the strong coupling. H e r e , f o r D = ~ , 3

the p e r t u r b a t i o n t h e o r y is r e n o r m a l i z a b l e with both

a finite as well as an infinite coupling constant. The situation when D-- ~ is v e r y i n t e r e s t i n g . We dwell on it in the p r e s e n t p a p e r .

It t u r n s out that if

we s t a r t with a chiral L a g r a n g i a n with a fixed u l t r a v i o l e t r e g u l a r i z a t i o n , then at the expense of a n e c e s s a r y finite n u m b e r of r e n o r m a l i z a t i e n s we obtain the ~

-theory.

The p a p e r is planned as follows: in Sec. 1 we introduce the notation and d e s c r i b e the g e n e r a t i n g functional of the G r e e n ' s functions; in Sec. 2 we compute the basic o r d e r with r e s p e c t to 4/~ by m e a n s of the Laplace method; in S e c s . 3, 4, and 5 we analyze the question of s e l e c t i n g t h e boundary conditions in a path integral; in Secs. 6 and 7 we d i s c u s s the p a r t i c l e s p e c t r u m obtained in the basic o r d e r with r e s p e c t to

4/N and we c o m -

p a r e it with the s p e c t r u m in the c o n s t r u c t i v e approach; in Sec. 8 we p r e s e n t a s c h e m e f o r constructing a t h e o r y of p e r t u r b a t i o n s f o r the

0(N) - s y m m e t r i c phase; in Sec. 9 we p r o v e that when

])~

~/N

tachyons a r e ab-

sent in the strong coupling limit under a p r o p e r choice of the p h a s e in the basic o r d e r with r e s p e c t to ~/N ; in Sec. 10 we explain why, when ])-~4

a ~

model is obtained as a r e s u l t of r e n o r m a l i z a t i o n s c a r r i e d out f o r

the c h i r a l field; and, finally, in Sec. 11 we d i s c u s s the applicability of the

4/~ expansion.

In this p a p e r we shall make wide use of the path integration method [14-19]. This method was s u c c e s s fully applied to p r o b l e m s w h e r e it m a k e s s e n s e to u s e the s t a n d a r d p e r t u r b a t i o n t h e o r y f o r obtaining f i b e r r e sults. We hope that the s c h e m e of s u c c e s s i v e i n t e g r a t i o n s with r e s p e c t to f a s t and slow v a r i a b l e s , developed in [18], can p r o v e useful a l s o in connection with the

~

expansion.

The author is s i n c e r e l y g r a t e f u l to L. D. Faddeev, V. N~ Popov, A. D. Linde, and P. P. Kulish f o r useful dis c u s s i o n s .

i. G e n e r a t i n g

Functional

for

the

O(N)~

Model

The g e n e r a t i n g functional of the G r e e n ' s functions f o r the 0(N) ~

model with spontaneous s y m m e t r y

b r e a k i n g has the f o r m :

(1) where

_

_~

~

~-$

^

In o r d e r to d e t e r m i n e (1) it is n e c e s s a r y to indicate the v a c u u m ~ with r e s p e c t to which the a v e r a g i n g is c a r r i e d out, or, what is the s a m e , to m a k e c o n c r e t e the boundary conditions in the path integral in the right hand side of f o r m u l a (1). In what follows, by examining a v e r a g i n g with r e s p e c t to v a r i o u s vacua, we shall s p e a k of the different p h a s e s in which the s y s t e m can be found. Such a concept of the phase c o r r e s p o n d s to the one accepted in s t a t i s t i c a l m e c h a n i c s . We can c o n s i d e r only the p h a s e s which c o r r e s p o n d to the i n t e g r a tion v a r i a b l e ~(~) going onto a constant (possibly, zero) v e c t o r

1536

~ as t - - ~ z ~

. As the c h a r a c t e r i s t i c of a

phase we shall use the m e a n (l-l, ~(~)/l~. In the c a s e of i n v a r i a n c e r e l a t i v e to t i m e shifts the m e a n (ft, ~(oc)~l) is the s a m e as the a s y m p t o t i c b e h a v i o r of the c l a s s i c a l field r

as t

*-+ ~

, in whose neighborhood the

integration in (1) is c a r r i e d out, i.e., (~, q ( ~

~

when ~'< ~

Hence we see that

, for ~ = 0 . Thus, a phase transition

takes place in the model (see [21] for details). For finite X a phase transition also takes place when ~c =~c[k). 5.

Computation

o f V(~)

When

~)~t"

It is i n t e r e s t i n g to c o n s i d e r the s t r o n g coupling l i m i t a l s o f o r D= ~ . H e r e we shall have in mind the strong coupling l i m i t with r e s p e c t to a r e n o r m a l i z e d constant >~ . A f t e r r e n o r m a l i z a t i o n (8), in the l i m i t ~-4~

Eq. (13) takes the f o r m

~

~

l • _ [

~

~

~)~ p~+,~ p~+C~+-~] § ~- ~-~0.

Hence we obtain the d e s i r e d r e l a t i o n between ~ ~

and ~ at the s t r o n g coupling limit: ~ ~~~

~

m~ ~e-~M

(16)

The equation

coinciding with the " m a s s equation" f o r the d e t e r m i n a t i o n of %== ~(~)

f r o m [7], holds f o r finite X . r~

rrL~,

Relation (16) i m p o s e s c o n s t r a i n t s on the a d m i s s i b l e values of ~ .* Indeed, the function -~k-~ ~ - ~ z with m~>0 (Fig. 1) t a k e s the m a x i m u m value -~-+ ~ at the point m z = B ~ , and, consequently, the c o n s t r a i n t

* T h e r e is a point of view that we m u s t e x a m i n e V(~) f o r a r b i t r a r y 4 . H e r e V(~) b e c o m e s c o m p l e x - v a l u e d [7], which can be i n t e r p r e t e d as the p o s s i b i l i t y of the g e n e r a t i o n of p a r t i c l e s [22].

1540

V(m,]

If z

• m~

Fig. I

Fig. 2

N~4 holds.

+~

M~ (17)

F r o m (17) follows a c o n s t r a i n t on the a d m i s s i b l e values of

+ , viz.,

On the o t h e r hand, since ~ > / 0 , f r o m r e l a t i o n (16) a l s o follows a c o n s t r a i n t on the a d m i s s i b l e values of mz :

, w h e r e mzo is defined as follows:

Differentiating (16) with r e s p e c t to ~ , we have

whence with due r e g a r d to (14) we obtain

9

and, consequently, V, ra~, _

N

~~

raz

(19)

[in (19) an unessential constant has been omitted]. The g r a p h of function V = Y ( r a ~) is shown in Fig. 2. The function V(m z) is m a x i m a l when the d o m a i n ' s boundary, i.e., when

~=~0

0*~=-kr , v a n i s h e s when m ~ 0 and ~z = M~f~ 9 When ~ > / - ~

thepoint

and is m i n i m a l at

~t~ lies to the right of the point

Mz'~-,

~ ~r~e-~ raz has a positive value f o r the ~- being examined. Since runesince when ~----M~/e- the function ~ _ 46~ tion ra=m.(~)

is t w o - v a l u e d (Fig. 1), the function ~/=V(~)

a m i n i m u m value when ~ - 0

, and, consequently, when

The g r a p h s of functions ~ ' = V (ra z)

too is two-valued (Fig. 3).

e6 > ~Mz 3~z

Function

V(~) takes

a s y m m e t r i c p h a s e is r e a l i z e d .

, ~z= ~(mz) , and Y=V(~)

f o r values of

l d /. such that ~M~ .~ -~-~

~_z --

3%~z a r e shown in F i g s . 4 and 5. Hence we see that a s y m m e t r i c phase is r e a l i z e d also when - ~ - ~ 4 ~ ~ $2~

The c u r v e b in Fig. 5

This follows f r o m the f a c t that the d e r i v a t i v e of function V(~) has no z e r o s on the s e g -

lies below c u r v e and

1541

M :&

89

?>0

t

rI

I

I

I

=_

1/\I

/

Fig, 3

f~M ~

Fig. 4

~2

rna~

~ ~4y~ O ) it is usual to examine two modes: 1. Weak coupling,

/~2

it has been conjectured

is reduced proportionally to (~-~c~)

mass

roughly equals

-~

, while as

~--~-+0 there is no

gap.

mined from Eq. (15). At first we consider the case solute value and it is mainly compensated

pensated by the second since absolute value.

Thus,

m~--~0

~-! > 0

for by the second.

. Consequently,

which corresponds

7.2. About the three-dimensional coupling case there are case the

0(N) -symmetry

obtained in the framework Higher

Orders

~{ particles

h particles whose mass

Here the third summand Consequently,

we have m~ ~ -~,

~ + 0 , the third summand

it must be compensated

which is con-

in (15) cannot be com-

by a negative first term large in

precisely to the conjecture on the absence of the mass

(N={,2,5) whose

is broken, and there are

is deter-

in (15) is large in ab-

case in the constructive approach we know the following: mass

approximately

equals

gap.

in the weak

m0~-)~ , in the strong coupling

N-~ Goldstone bosons [24]. This coincides with the picture

of the 4/N expansion. with

Respect

t o 4/N

Let us consider the phase corresponding ing to an asymmetric

~ there are

~-0

sistent with the result of the constructive approach. As ~

~ 0

two phases exist when

~ >/)~cr.

As has been shown in Secs. 4 and 6, for all admissible

8.

N p a r t i c l e s whose physical m a s s in the basic o r d e r

boundary

to symmetric

boundary conditions in (I).

The case correspond-

condition will be considered in detail in a separate article. In this case,

in general (i). Integrating with respect to the variable

(~N (~) in (4), we obtain

1543

In order to obtain the ~/

e~-ipansionwe e.~oand the exponent $C~) of the i~egrand exponential function in (2D

into a s e r i e s in a neighborhood of the s t a t i o n a r i t y point ( ~

ca~ which is d e t e r m i n e d f r o m (10) [the question

of the e x i s t e n c e and uniqueness of the solution of Eq. (10) is d i s c u s s e d below]. We have (22) where .

~

4

9

,I

d,Do;,

,

~,

~rx

,

_

+

_~

~:~m

~'

F r o m such a r e p r e s e n t a t i o n follows the d i a g r a m m a t i c technique d e s c r i b e d in [9]. Its e l e m e n t s a r e the p r o p a g a t o r (23) of ff -lines and the g e n e r a l i z e d v e r t i c e s c o r r e s p o n d i n g to the second and t h i r d s u m m a n d s in (22). D i a g r a m s with

(t/N)~, Z ~ - l o o p s contribute to the

ff - t a i l of o r d e r ~ - ~ + ~ . However, it is convenient

to i n t e r p r e t these d i a g r a m s with the aid of d i a g r a m s having ~ - and ~" -lines; m o r e o v e r , to the

So - l i n e s c o r -

r e s p o n d the p r o p a g a t o r

])~'(P')-~ ~ , where

r~~ is d e t e r m i n e d f r o m (10), and to the

(24)

~ - l i n e s c o r r e s p o n d p r o p a g a t o r (23). The d i a g r a m s contain

the v e r t i c e s - ~ : ~z; 6- (a n o r m a l o r d e r i n g r e l a t i v e to m a s s m ~ ) and the v e r t i c e s that it is u n n e c e s s a r y to take into account the i n s e r t i o n s into the loop d i a g r a m

~-(~) (Fig. 7).

9.

of S o l u t i o n

Choice

of Eq.

(10).

Absence

(~,n)

(Fig. 6). We r e m a r k

ff - l i n e s c o r r e s p o n d i n g to the s i m p l e s t o n e -

of a Tachyon

We t u r n to the question on the choice of the solution of Eq. (10). Its solutions c o r r e s p o n d p r e c i s e l y to the z e r o s of the function ~ - - - ~ ( m ~) . F o r

])----2,5 i n t h e l i m i t as

~o

the function ~(m~) h a s only one z e r o .

In the f o u r - d i m e n s i o n a l c a s e , since the function ~ ( m z) has two z e r o s , two solutions of Eq. (10) a r e p o s sible. H o w e v e r , as we saw, the value of the effective potential when the second z e r o is chosen is less than when the f i r s t is chosen. Consequently, we m u s t choose the l a r g e r z e r o (compare with the l a s t two r e f e r e n c e s in [7]). Let us show that with such a choice of m z the d e n o m i n a t o r of the G r e e n ' s functions of the not vanish. 1544

f f - f i e l d does

Fig. 6

Fig. 7

Indeed,

where

]~(R~, ~ )

is a growing function. T h e r e f o r e , if m ~ > M ~ , then the denominator does not vanish. This condition

is always fulfilled for the l a r g e r z e r o of the function ~Z=~z(m~). Consequently, with the p r o p e r choice of the m a s s of the T -field a t a c h y o n d o e s not appear in the strong coupling limit in the Euclidean domain. 10.

Chiral

Interaction

w h e n D=4

In this section we wish to dwell specially on an a s s e r t i o n that is a simple c o r o l l a r y of our work in [9], and at the s a m e t i m e , as it s e e m s to us, one that should be highlighted s e p a r a t e l y . To be p r e c i s e , the chiral interaction in a f o u r - d i m e n s i o n a l s p a c e - t i m e is r e n o r m a l i z a b l e and after r e n o r m a l i z a t i o n turns out to be equivalent to i n t e r a c t i o n q~

. Indeed, we begin with the generating functional f o r the chiral field.

and we reckon that the ultraviolet cutoff has been made. We make the a b o v e - d e s c r i b e d

4/N expansion. The

basic o r d e r r e q u i r e s a counter t e r m , viz., - h ~ %~ [cf. (8)]. Standard arguments on the index of the diagrams d e s c r i b e d above lead to the counter t e r m s _

N~

Thus we obtain a r e n o r m a l i z e d generating functional f o r the model

~

%o44 [of. (3)]. Hence follows the a s s e r t i o n

stated above. This s a m e result is obtained by treating the chiral field as a strong coupling limit. amined the s t r o n g coupling limit with r e s p e c t to a n o n r e n o r m a l i z e d interaction constant

For

D