LETTERE AL NUOVO CIMENTO VOL. 10, ~. 13 27 Lugl io 1974

I d e n t i t i e s in L a g r a n g i a n F i e l d T h e o r y .

J . DES CLOIZEAUX

Service de Physique Thdorique Centred'Etudes Nucldaires de Saclay - Gi]-sur-Yvette (Saclay)

( r i cevu to il 24 Apr i l e 1974)

L e t us cons ider t he L a g r a n g i a n (~)

(1) [: :] if(x) = 2 J~l ~ [(Vq~J)~ + m~(q~)~] + go8_ ~=1" ~ ~2

in a space of d i m e n s i o n d = 4 = e. F o r ~ ~ 0, a f t e r a t r i v i a l mass s u b t r a c t i o n , we m a y define (( u n r e n o r m a l i z e d ~) Green ' s

f u n c t i o n s a n d for s >~ 0, a f t e r a mass s u b t r a c t i o n a n d r e n o r m a l i z a t i o n s of t h e wave func- t i o n of t h e 4 -po in t v e r t e x f u n c t i o n , we m a y also define r eno rma l i zed Green ' s func t ions .

2 2 2 -~ I n t h e c r i t i ca l r eg ion ( g o ( t o o - moo ) >> 1), t h e r e n o r m a l i z e d t w o - b o d y Green ' s func- t i o n G(k, m) (m = r e n o r m a l i z e d mass) can be w r i t t e n

(2) G(k, m) = m-~](k/m).

The n o r m a l i z a t i o n cond i t i on is def ined b y t h e f irst two t e r m s of t h e e x p a n s i o n of [](x)] -1 w i t h r e spec t to x

(3) [/(x)] -1 = 1 + x 2 + . . . .

W e w a n t to p r o v e t h a t for e > 0

co

G, - ~ d x x ~ - l [ x / ' ( x ) - ~ (7/v)/(z)] = O. &

( 4 )

0

W e sha l l ca lcu la te , in two d i f fe ren t ways , t h e de r i va t i ve s G~)(mo, k) ~ ~SGe(mo, k)/(~m~) s of t h e u n r e n o r m a l i z e d Green ' s f u n c t i o n G~(m o, k) and show t h a t G 1 m u s t v a n i s h for reasons of cons i s tency .

W e se t

(5) go = asl-e , ml = tt .

(1) K . G . WILSo~r an d 5I. E. FISHER: Phys. Rev. Left., 28, 240 (1972); K. G. WILSON: Phys. Rev. Left., 28, 548 (1972); E. BRI~ZIN, J . C. LE GUILLOU a nd J . ZINN-JUSTIN: Phys. Rev. D, 8, 434 (1973).

552

I D E N T I T I E S IN LAGRANGIAN FIELD THEORY 5 5 3

The renormal iza t ion cons tan ts of the fields ~ and ~ m a y be defined b y

{ ~(r) = z l (~ )~ ( r ) ,

(6) [mo~ _ m~,] -x = z~(~) m-2 .

We m a y set

{ zl(/~ ) = bl(/~)/~a,~-a,,,

(7) z2(l~) = b2(l~) #d. , -d .~ ,

where bl(/~ ) and b2(#) have finite l imits bl and b 2 when #--> 0. By definition, the anormalous and canonica l dimensions of the fields are

2 d a l = d - - ? / v , d a ~ = d - - 1 / v ,

(8) 2dcl = de2 = d - - 2 .

Thus

(9) G~(m o, k) ~_ 12(bl)21x-rl~/(kl//~)

and for /~-+ 0

Keeping the most s ingular terms when ~ - + 0, we find

(11) f ddk G~)(ra, k) "" --GflZ-a-~S(bl)2(b2/v)s# ~-r ,

where G s is a cons tan t (see eq. (4)) provided tha t the in tegra l converges, i.e. if

(12) S > v d - - ? ~ _ 1 - - 3 e / ( 4 ~ - 8) . . . .

On the other hand a direct ca lcula t ion shows tha t

(13) fd~k a~'(~0, k) = 2-,fd% ... d%<~(0)r ~(~.)> o

The correlat ion l eng th is m 1. Therefore, we can evaluate the order of magn i tude of eq. (13):

(14) f d'~b G~)(mo, k) oc 12-2S(ml)(5+l)a,, -s~ = l~-~s #a-(s+l)/v $

Since da2--2dal= (? - -1 ) / v> 0, this expression is less s ingular t h a n can be expected from eq. (11). In order to avoid cont radic t ion , we mus t admi t t ha t G s = 0 for S ~ 1 and tha t the behaviour predic ted b y eq. (14) is no t related to the scaling law of eq. (2), bu t comes from correct ions to this law.

However, an object ion can be immedia te ly raised. For S = 1 and e = 0, we see, by power count ing , t h a t the in tegra l ffd% 8G~(mo, k)/Sm~ diverges and i t does even in the zeroth-order approx imat ion (see Fig. I). How is i t compat ible wi th the s t a t emen t t ha t G ~ = 0 for S = 1 in the l imi t e -+0?

5~/~ J . DES CLOIZEAUX

The answer is s imple: the in tegra l defining G~ does no t converge un i fo rmly wi th respect to ~ and the anomaly is t he resul t of an invers ion of l imits .

O Fig. 1. - Diagram contributing to ffd4k OGy(mo, k)/Om~ in the zeroth-order approximation.

Indeed, for x >> 1 we know t h a t (~)

(15) [](x)] -~ _~ xr/~[A + Bx-~/~ + Ox-(d-*/:)]

and for 0 ~ e < < l

A _ ~ 1 , B _ ~ 6 / ( 4 - - n ) , C _ - - ( n + 2 ) / ( 4 - - n ) ,

y _~ 1 , v-~ _~ 2 - - e (n + 2 ) / (n + 8 ) .

Thus, for x >> 1, e <~ 1,

(16) fd: x<,-,r:/'<=) + 2 (4- +

I f we pu t e = 0 in this expression, the in tegra l diverges when x-> ~ , bu t for ~ ) 0 the in t eg ra l converges to a va lue which remains finite when e ~ 0 . This amusing fact solves the paradox.

Since, usual ly, t he anomalous d imension of a p roduc t of fields is not equal to the sum of the anomalous dimensions of the cons t i tu t ing fields, we see tha t the phenomenon descr ibed here is qui te general. Thus, s imilar ident i t ies can be der ived for o ther Green's func t ions and o ther types of Lagrangians .

Similar considera t ions have been appl ied (a) to expla in in teres t ing features of the shape of po lymers (chain wi th exc luded volume) and results obta ined by mach ine calculat ions p rov ide an ~ exper imenta l ~} verif icat ion of eq. (3). The na ture of the re- s is t ive anomalies occurr ing in magne t i c mater ia ls (4) at T~ depends also impl ic i t ly on the va l id i ty of th is equat ion .

The au thor wishes to t hank Dr. E. BR~ZIN, whose adv ice and explana t ions have been v e r y helpful , and J . ZINN-JusTIN for discussions.

(i) M. E. FISheR and ~. A~ARONr: Van der Waals Centennial Co~]erence on Statistical Mechanics (Amsterdam, 1973}; E. BRI~ZIN, D. J. AMIT and J. ZINN-JUSTXN: Saclay larelorint (1973). (a) J. DES CLOIZEAUX: S~clay prePrint. ~') M. E. FISHER and J. E. LANOER: Phys. Rev. Lett., 20, 665 (1968).