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 (
0" - f i e l d we r e p r e s e n t (1) as

and w e m a k e the Euclidean rotation

Here ~

is an incidental n o r m i n g f a c t o r ,

We denote

We r e m a r k that the a s s e r t i o n that the chiral field is the strong coupling limit f o r the )~~ ~ - i n t e r a c t i o n in the p s e u d o - E u c l i d e a n f o r m u l a t i o n follows d i r e c t l y f r o m (2). Indeed, neglecting the s u m m a n d the integration o v e r the

~

, after

~ - f i e l d we obtain ~ ( ~ - - ~ o ) . In the Euclidean formulation, however, this a s s e r t i o n

follows f r o m the d i r e c t Euclidean continuation of functional (1). F o r integral (3) to be w e l l - p o s e d we need to a s s u m e that the i n t e g r a t i o n o v e r the ff -field is t a k e n in the " i m a g i n a r y ~ direction, which c o r r e s p o n d s to the substitution with r e s p e c t t o

~4,0-.

Then, passing to the l i m i t

k--*-o~

0" , we obtain the ~ -function needed.

F o r the chiral field (henceforth we denote it respondenee:

d i r e c t l y in the integrand in (3) and, next, integrating

on

the one hand,

~z = _ ~

v e c t o r %(~) which can be z e r o .

is n o n z e r o , while on the other hand, as Iml----oo , a(~) tends to a

This l a c k of c o r r e s p o n d e n c e can be e l i m i n a t e d by o b s e r v i n g that the ~ -

function in generating functional (i) corresponds i n v a r i a n e e the r e l a t i o n n . ( c c ) ~ a o

~.(~) ) t h e r e is, at f i r s t glance, the following lack of c o t -

to the equality

:

c o r r e s p o n d s to the f a c t that < ~ > =

11,0

, while because of the t r a n s l a t i o n a l 9 Then the equality

<~>=(~>~

m a y not be fulfilled.

2.

Basic

Order

with

Respect

to

~/N

Stationarity

In the i s o s p a c e we take a coordinate s y s t e m such that the tor

and in (3) we i n t e g r a t e with r e s p e c t to the f i r s t

N- i

Equations ~t-th axis coincides with the direction of v e c -

components. The generating functional t a k e s the

form

(4)

1537

where

gT, where

7-c7,,...,7,-,). F r o m (4) we see that 4/N o c c u r s in the integrand in the s a m e way that the Planck constant ~ usually

o c c u r s [14]. T h e r e f o r e , the ;I/bI expansion is obtained analogously to the s t a n d a r d expansion with r e s p e c t to the n u m b e r of loops. We apply the Laplace method to integral (4). To do this we f i r s t find a point of stationariW of the action

$~(~N,(Y)[31]: (5)

A%r %c' %= O.

(6)

In integral (4) we can take into account the contributions only of the constant solutions. contains a divergence in case ~>~ 2

The s u m m a n d

which can be eliminated when ;b= 2,3, st by r e n 0 r m a l i z i n g

~0 and )~0 9 When ] ) = 2 , 3 the r e n o r m a l i z a t i o n s

~_~+

~-~

~ ?~'p c~3~-+m , D--%s,

a r e sufficient. A r e n o r m a l i z a t i o n of ])--~ as well is n e c e s s a r y when

%=~-

(7)

~0

p+-~-~m, ~'o ~'

:

'

Here M is the subtraction point. The s y s t e m of Eqs. (5) and (6) can have two constant solutions:

,

II. ~ r

0

, ~r

~Nc--- ~- ~ ~

L-~ - ~ - ~

(9)

is determined from the nonllrzear equation

c~)'>tp'+<~ p~+M~]-s-r (here we have c o r r e c t e d an e r r o r that crept in into Eq. (7) of [9], an e r r o r in the sign before

/~%-~=~

,

which led to c o r r e s p o n d i n g changes of sign in the succeeding f o r m u l a s ; see below). 3.

Choice

of Phase

Different boundary conditions in the original integral (1) c o r r e s p o n d to the solutions I and E: to s o l u tion I c o r r e s p o n d s a nondecreasing boundary condition; to solution II, a d e c r e a s i n g one. F o r p s e u d o - E u c l i d e a n ftmztionals a different choice of boundary conditions c o r r e s p o n d s to different vacua (i.e., to different phases) [16]. T h e r e f o r e , in o r d e r to choose one of the solutions it is n e c e s s a r y to determine which one of the vacua is

1538

r e a l i z e d f o r the p r e s c r i b e d p a r a m e t e r values. F o r this it is n e c e s s a r y , as was already noted in Sec. 1, to ffl]d the vacuum's e n e r g y density in the s y s t e m where all the G r e e n ' s functions are specified with n o n d e c r e a s iag boundary conditions and to c o m p a r e it with the vacuum's energy density in a s y s t e m with d e c r e a s i n g boundary conditions. The solutien with s m a l l e r e n e r g y density will be r e a l i z e d . In the Appendix it is shown that the e n e r g y density V(~) is a s y s t e m where the G r e e n ' s functions are p r e s c r i b e d by a generating functional with boundary condition ~(~c)----~F , where F=(0,, ,~) is given by a Legendre t r a n s f o r m of the generating functional of the G r e e n ' s functions. In the basic o r d e r with r e s p e c t to 4/N we compute the e x p r e s s i o n

%7J.

where the Lagrangian

"

has, in the Euclidean formulation, the form:

In the basic o r d e r with r e s p e c t to

4/N t h e r e

holds the equality

(11) f r o m which it is convenient to find V(:~) as follows (compare with the second r e f e r e n c e in [11]). F r o m (ii) we derive a differential equation for ~-C~) and, then, we solve it. F r o m (11) it follows that ~ =

i ' but on

the other hand, ~ is d e t e r m i n e d by the motion equation

i--

>,

Consequently,

(12) In order to integrate

(12) we find the connection

and, in the basic o r d e r with r e s p e c t to

between

mz

and ~ 9 It follows f r o m the condition

~/N ' is given by the nonlinear equation

(13) Comparing (12) and (13) we see that

4.

Computation

o f V(~) W h e n D = 2 , 3

It is of i n t e r e s t to compute ~-(~) at the strong coupling limit since it is p r e c i s e l y this limit that describes the chiral field when

]~ < ~ .

1539

4.1. D=~

9 In a t w o - d i m e n s i o n a l s p a c e - t i m e Eq. (13), with due r e g a r d to r e n o r m a l i z a t i o n (7), takes

the f o r m

~

~

~

~.

Hence at the s t r o n g coupling l i m i t ~--~oo , we have

Using the r e l a t i o n obtained, it is e a s y to integrate (14). To within an additive constant

~((~)

is:

M~

this e x p r e s s i o n is m i n i m a l when

~=0

. F o r finite

k it is not difficult to be eo,~vinced that

V(~)

as wetl

takes a m i n i m a l value when ~----0 , while the m a s s is d e t e r m i n e d f r o m the n o n l i n e a r equation

4.2.

] ) = S . In this case u

has been computed in [21]. We p r e s e n t the final answer:

,

It has meaning f o r all ~ when ~- ; ~ , w h e r e ~c = -~~'~ , and for ~ ; ~ -

when ~ U

' %f(~) is minimal for ~ = ~ -

~

, while when ~ > ~

~

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~ <_~_~ -~-~z s~z Fig. 5

~]'I,,~=~.~

Consequently, corresponds 6.

~rr~#, and ~ >rr~.

for all admissible values of ~ , i.e., when

to decreasing boundary

Spectrum

~-~

of Particles

f~/- ~-T M~

, we must choose solution II, which

conditions. in the

Basic

Order

with

Respect

to

~/N

From the results of Secs. 4 and 5 it follows that the boundary conditions in (1) must be chosen in the following manner: ])~ 2~ , T ])=5 , ~ > ~ ]~

, 7"

is arbitrary

/ decreasing boundary ] conditions for ~

nondecreasing boundary conditions for G',

are any admissible ones

] ) = 5,

~4 ~r

nondecreasing boundary conditions for ~u

decreasing boundary conditions for ff

In the basic order the generating functional for the case of boundary conditions decreasing in r and nondecreasing in ff has the form

Hence it follows that ff~_~m~ in the basic order with respect to 4/N is the physical mass of the ~ -field.

1542

7.

Comparison

the

Basic

of the

Order

Mass

of the

Spectra

Resulting

in the

Constructive

Order

and

in

4/N E x p a n s i o n

Let us c o m p a r e the s p e c t r u m of p a r t i c l e s obtained in the basic o r d e r with r e s p e c t to

~/t~ with the s p e c -

t r u m of p a r t i c l e s in the f r a m e w o r k of the constructive approach to the 0(N) q~- model [23]. In the c o n s t r u c tive approach, f o r the Lagrangian - z

:

(20)

(: : denotes a n o r m a l ordering relative to the m a s s ra~>O ) it is usual to examine two modes: 1. Weak coupling,

/~<}$0(m~o), ~ 4~0(~%),

2. Strong coupling, ~

is finite, while ~

7

is sufficiently small;

is sufficiently large. By comparing (1) and (20) we see that

our notation and that adopted in [23] are r e l a t e d as follows: the interaction constants coincide to within N , i e.,

=

, w h i t e the b a r e m a s s i s c o n n e c t e d w i t h

t

by the r e l a t i o n :

notation the weak coupling mode c o r r e s p o n d s to the case: ~'< 0 ~ c o r r e s p o n d s to a firdte

--- #.

C o n s e q u e n t l y , in our

%-[~-~_~=-]~<< ~ . The strong coupling mode

k/t ~ (here k can tend to infinity simultaneously with N ), while

sufficiently small. Hence we see that as ~ - - - ' 0

~ is positive and

we have a w e a k coupling, while as ~----+0 , a strong coupling.

7.1. We c o n s i d e r the two-dimensional case. It is well known [23] that in the weak coupling mode, for any ~ , only a s y m m e t r i c phase is r e a l i z e d and t h e r e are with respect to symmetric

4/N equals

m~-)~ 9 In the strong coupling mode

phase exists when

N -~ Z,5 . Regarding the spectrum

that to the extent of the approximation and that the gap is absent when Thus, as ~-0 mass

to the critical point ~cr

there are

~ particles whose

of particles when the mass

N ~- ~ , while one 0(N)-

N >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<~ we ex-

)~0 9 Obviously, it did

not depend on the finite r e n o r m a l i z a t i o n and, in this s e n s e , was singled out. However, in the case of ]3-= ~ we were compelled to make an infinite r e n o r m a l i z a t i o n of the charge, and the strong coupling limit could be cons i d e r e d only with r e s p e c t to the r e n o r m a l i z e d constant

t-----k(M) 9 As follows f r o m (16'), the answers depend 5Z~ z only on a quantity invariant to renormalization, to be precise, on ~ ~ M ~ ~ ~ [7],ami the limit ~(~)--~-~

is not signled out. As a matter of fact, the quantity X can be m a d e an arbitrary finite n u m b e r without changing ~ at the expense of changing the subtraction point. Therefore, the quantum chirality condition, valid for ~@

, is not fulfilled in the strong coupling limit when I)<@

, and all the properties typical of

preserved. In particular, a phase transition is absent in the model and only an

0(~) ~

are

0(~) -symmetric v a c u um is

realized. 1545

ii.

Applicability We

remark

Goldstone

of the

4IN

that the

symmetry

~/N Expansion

theory of perturbations

breakin~ mechanism

does not correspond

generally accepted for small

the existence of a certain critical Nor, below which the

to the qualitative picture of the

N 9 Apparently this is connected with

~/N expansion is not applicable.

And, conversely, the

usual Goldstone picture cannot be valid when N > ~ cr 9 This phenomenon can have important consequences for realistic models of the Weinberg type, in which the Higgs mechanism, based on the Goldstone picture, is used. In particular, the existence of such an

Ncr would enable us to estimate the number of scalar Higgs bosons.

However, in a relativistic model with due regard to the long-range Yang-Mills field it can happen that such an

Ncr does not exist, and spontaneous symmetry breaking obtains. Indeed, as the example of a twvo-

dimensional model shows, the qualitative picture can change when the Tang-Mills field is taken into account [25]. To be precise, in the two-dimensional model of the chiral field there is no phase transition within the framework of the 4/N theory of perturbations; however, a phase transition is possible in the framework of the 4/N expansion for the inclusion of the Yang-Mills field. An

0(N) -asymmetric phase is realized for small

coupling constants; however, in accordance with Coleman's theorem, Goldstone bosons are absent, and the chiral field acquires mass. It is interesting to compare the picture of the spectrum obtained in the 4/N expansion for the 0(N) ~

-

theory when D-~2,3 with the exact results of the constructive approach still in the following aspect. In the framework of the constructive approach to the 0(N) (~~-interaction it is well known that a phase transition exists when

N-~ ~

and the mass ~

is fixed: only a symmetric (relative to ~--~-~

small ~ , but two asymmetric phases are realized as k increases [26, 27]. For N)~

) phase exists for because of the

presence of continuous symmetry, a phase transition is impossible and only a symmetric phase must be realized [28]. A comparison of these results alzo points to the fact that at least for ])~-~ a critical value of N does exist and equals i. The existence of instanton solutions when ~)=Z

and

~-3

[30] also indicates that apparently an ])=2

equal to 3 exists for Ncr APPENDIX Let us consider the m i n i m u m energy problem with a fixed mean value of the ~ -th component of field ~ , equal to ~ (in other words, let us find the energy of the vacuum in a system where the Green's functions are specified by a g~~177

functional with an arbitrary boundary condition ~0a[~i_, ~ ). W e are interested in the

m i n i m u m of the functional (~[H[~/) under the conditions <x~ [q~ IV >----~, II~[l-~ ~" W e denote this m i n i m u m

by ~r(~) . The formula subsequently

(A.6) for ~(~)

derived in the literature.

We introduce a Lagrange

1546

obtained below has been so well known

[29, 32] that, apparently, it was never

Therefore, we present its proof.

multiplier and we consider an auxiliary Hamiltonian

be the normalized eigenvector of Hamiltonian ~

L~t

with eigenvalue

Kti ) 9 Applying the variation prin-

cipie

we obtain ])-I

and, consequently,

(An) Thus, the determination of ~F(~) is reduced to the computation of the expression

])-t where ~,j)

is the minimal

eigenvalue of the auxiliary Hamiltonian

~

and i is connected with ~ by r e l a -

tion (9). Usually the E(i) are computed by a f o r m u l a of the G e t l - M a n n - L o w type

where E~ is the d e s i r e d minimum energy and tions give

~v(0) is determined as a path integral over the set of func-

on

2:

We prove formula (A.2), assuming that the eigenvector _0_ corresponding to the minimum eigenvalue of the complete Hamiltonian H9 is not orthogonal to the vacuum ~0

of the unperturbed Hamiltonian [33],

(2, %)~ 0, We use the following representation for :~r(0)

~(0)= (%, (rH~%) and then we prove that T._,,+_~ *

(A,3)

This formula follows from the bound (A.4) where ~-~ ~r~I (~o,~)i

~>-c<~ . The right-hand inequality in (A.4) is obvious since ~.~ =6m~(i~,H~t~), while the

left-hand one follows f r o m the inequality

~ltfl,%)i ~ (%,.d~Q%),

(A.5)

1547

under the a s s u m p t i o n that (O, ~)0)~ 0

. Inequality (A.5) s t e m s f r o m the following chain

<

j@%t

=

C%,

Applying the f o r m u l a p r o v e d to our c a s e , we obtain

and, consequently,

or

F o r the case when ~ is independent of ~

we have (A.6)

Thus, we have obtained the v e r y well-known f o r m u l a f o r the effective potential V(~) in t e r m s of the Legendre t r a n s f o r m of the g e n e r a t i n g functional of the G r e e n ' s functions. LITERATURE 1. 2.

3.

4. 5.

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* T r a n s l a t o r ' s note. In keeping with the usual c u s t o m in the p h y s i c s l i t e r a t u r e , the p r e s e n t author does not give titles of the p a p e r s cited. However, t h e r e w e r e s e v e r a l e r r o r s in the citations the author does give, and, so, since I did have to t r a c e all the citations, I have, f o r the r e a d e r s ' convenience, given the titles (both h e r e and in the other a r t i c l e s in this issue, w h e r e n e c e s s a r y ) . ~ T r a n s l a t o r ' s note. This citation is c o m p l e t e l y e r r o n e o u s . P o h l m e y e r does not have an a r t i c l e in any issue of this journal in 1973. 1548

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21. 22. 23.

24. 25. 26. 27. 28. 29. 30.

31.

32.

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