LETTER]~ AL lqUOVO CIMENTO VOL. 35, N. 3 18 Set tembre 1982

The Nonlinear ~-Model; a Critical Analysis of Semi-Classical

A. PATRASCIOIU (*) and A. ROU~T (**)

The Inst i tute ]or Advanced Study - Princeton, N e w Jersey 08540, N . J .

(ricevuto il 24 Giugno 1982)

Results.

Summary . We calculate the contribution of all unit charge S 2 instantons to the expectation value of any Oa-invariant operator. I t can be represented as the contribu- tion of the 2-particle sector to the grand canonical part i t ion function of a gas, but the interaction is not of Coulomb type. The method used employs extensively the sym- metries presented in the problem.

The nonlinear 03 a-model in 2-dimensions, also known as C P 1, is described by the following action :

(1) S(n) ~- d 2 x ~ n . ~ n , n 2 = 1.

The classical action is manifestly scale invariant , thus the existence of a mass gap in the quantum theory cannot be taken for granted. Previous studies of the spectrum of the Hamil tonian conducted at the semi-classical level have produced the following apparently contradictory rcsults:

i) BERG and LT2SCHER and FA'r~,EV, FROLOV and SCHWARTZ (~), using the sphere S 2 as an infra-red cut-off, concluded tha t the contribution of all mult i- instanton solutions to the part i t ion function is identical to that of a 2-dimensional Coulomb gas. The lat ter model is known to have a mass gap.

it) :RICHARD and ROU~T (2) performed the same calculation using the torus T 2 as an infra-red cut-off. They obtained quite different results.

(*) A. P . S l o a n F o u n d a t i o n Fe l low. P e r m a n e n t a d d r e s s : U n i v e r s i t y of A r i z o n a , P h y s i c s D e p a r t - m e n t , T u c s o n , Ar iz . 85721. (**) A l b e r t E i n s t e i n P r o f e s s o r . P e r m a n e n t a d d r e s s : C . N . R . S . - L u m i n y - C a s c 907, C e n t r c de P h y s i q u e T h 6 o r i q u e F - 1 3 2 8 8 ; ~Iarsei l le C E D E X 9, F r a n c e . (1) B. BERG a n d M. L(~SCHER: C o m m u n . M a t h . P h y s . , 69, 57 (1979); V. A. FATEEV, I . V. FROLOV a n d A. S. SCHWARTZ: N u c l . Phys . B , 154, 1 (1979). (z) J . - R . RmHXRD a n d A. ROUET: The C P 1 model on the torus: contribution of instanlons, Marse i l l e p r e p r i n t C P T - 8 1 / P . 1 3 1 8 .

107

108 A. PATRiSCIOIU and A. ROU]~T

iii) I~EINAST and STACK (3) showed that , by a change of variables, one could rewrite the result in i) as the part i t ion function of a dilute gas of instantons of uni t topolo- gical charge.

The dichotomy in the above results is the following:

a) If the system has a mass gap, in the thermodynamic limit any infra-red cut-off should yield identical results.

b) The original motivat ion for taking into account all the mult i - instanton solu- tions was precisely the fact that the assumption that the ins tanton gas was dilute could be shown to be inconsistent (4).

In this letter we calculate the contribution to the part i t ion function of all classical solutions of uni t topological charge defined on a sphere S ~ of radius /{. We find that for all operators which are Oa-invariant this contribution can be represented as the contribution to the grand canonical ensemble of the 2-particle sector; their interaction though is not Coulomb, but rather the one shown in eq. (17), in contradiction with ref. (1). As we do not know yet whether this interpretat ion persists for arbi trary topological charge, we cannot say whether or not the nonlinear Oa ~-model has a mass gap.

We begin our analysis by recalling the main results derived in ref. (1). A convenient parameterization of the field n ( x ) is

2u 1 - I ~ l ~ (2) nl + ins - - na =

1 + [ul ~' 1 + [ul ~"

The uni t charge classical solutions can be parameterized as (z =-- x 1 -k ix~)

Z - - a (3) u(z ) = c - - a , b , c e C I

z--b'

In the semi-classical approximation their contribution to the expectation value of some operator 0 can be writ ten as (1)

(4)

where

(5)

(6)

(7)

1 fd d~c <o>1= ~ g2 2ad~b o + lr ~ O(a, b, c) exp [-- U(a, b, c)],

=- y e x p - 7 + In ~ R + 1 ,

U = log l a - - b12 + Y - - Y0,

2fd 1 loj jz- l + Iz--bl Y - - ](o = ~ ~ Z ( l + Izl2) ~l~ 1 + Iv?

In writing the result (eq. (4)) we have scaled a and b by the radius of tile sphere R,

(3) R . NEINAST a n d J . STACK: Nucl . Phy s . B , 185 , 101 (1981). (4) F o r a d i s cus s ion of t h e f a i lu re of t h e d i l u t e g a s a p p r o x i m a t i o n , see sec t . I of t h e p a p e r b y BERG a n d LOSOHER, re f . Q).

THE NONLINEAR (;-MODEL ETC. 109

which f rom now on appears only in the runn ing coupl ing g. I n ref. (1) i t is a rgued tha t , as R--~ c~, Y - - Y 0 - ~ 0, and thus eq. (4) is a p p r o x i m a t e d as

1 ~ f d% 1 (4') <0>1 = Z g d 2 a d 2 b (1 b, c) - - . + ]cl~)~ ~ l~--b[~

This is t he resul t analysed by NEINAST and STACK and shown to lead to t he d i lu te ins tan ton gas.

I n fac t , the resul t in eq. (4') is incorrect . Th is can be seen by di rec t e x a m i n a t i o n of eq. (7) g iv ing Y - - I / 0 ; R does no t en ter th is equa t ion , hence the re canno t be any s impli f icat ion as R--> c~. Moreover , t he resul t in eq. (4') v io la tes one of t he basic invar iances present in t h e problem, which we shall discuss next .

a) I n v a r i a n c e u n d e r i s o s p i n v o t a t i o n s . Rota t ions of t h e O a axis can be used to induce the fol lowing t rans format ions of the in s t an ton pa ramete r s , which should leave the pa r t i t i on func t ion unchanged :

r o t a t i on b y ~ about the 3-axis (Ta):

(8)

Ta :

b - - + b ,

c --~ c exp [i~] ;

ro t a t ion by 0 about t he 2-axis (Tz):

(9)

ac - - ),b T.~ : a ---). a~ - -

).ac -]- b tg b --~ b.~ = - - i : - - - - ,

i c + l ' 2

c - - t C " - ~ C 2 - -

i t + l "

The following three independen t func t ions of a, b and c are i n v a r i a n t unde r T 8 and T~:

(10)

A - - alcl2 + b

1 + l~l 2 '

1 -{-" Iol ~ '

B = [ a - - b l 2 (1 + [c[~) ~"

A n y o ther func t ion of a, b and c i n v a r i a n t under T 8 and T~ can be expressed in t e rms of A, A and B.

II0 A. PATRASCIOIU and A. ROUET

b) I n v a r i a n c e u n d e r ro ta t ions o] the p a r a m e t e r space S ~. The co-ordinates (xl, x2) are the sterographie co-ordinates of points on a sphere of radius 1. Reparameterizations of the sphere must leave the part i t ion function invariant. I t is convenient to consider the following transformations:

Rotat ion about an axis perpendicular to the projection plane Ra:

R~ : a --~ a exp [ iv] ,

b --~ b exp [i~p],

e - - ~ c ;

infinitesimal rotations about an axis parallel to the projection plane R2:

b - - ~ b ~ e ( l ~ b e ),

c - ~ e ~- e c ( b - - a ) ;

inversion of the projection point I :

] / . : a - - ) - - - - -

a

1 b ----)- - - -

b '

a

e - - o - e - .

b

I t can be verified tha t the only combination of the isospin rotation invariants A, .4 and B defined in eq. (10), invariant under the parameter space t r ans fo rmat ions /~ , R 2 and I , is

B lel2l~-bl2 (11) ~ ~ (A.A + B + 1) 3 = (]aHc[ 2 + ]5 3 + Iv[ 2 + 1) 3.

Moreover, one can check that , whereas the exact answer eq. (4) is invariant under all the transformations exhibited above, the approximate answer given in eq. (4 ~) is not (see, for instance, transformation I) . Thus not only is the approximation of eq. (4) unjustified, but i t also violates some symmetries manifestly present in the problem.

In the remainder of this paper we will employ these symmetries and rewrite the answer in eq. (4) in such a way as to exhibit the analogy with a gas on S 2 and identify the potential. The main idea is that typically one is interested only in Oa-invariant oper- ators. Thus in eq. (4) we will consider tha t

O(a, b, c) = O ( A , A , B ) .

Since the measure mult ipl ied by the integrand is 03-invariant, the volume of the group can easily be factored by employing the Faddeev-Popov technique. We define

THE NONLINEAR g-MODEL ETC. III

AF(a, b, c) by

(12) 1 ) = l ,

where /2(2) is the invar iant measure

I t follows that

(13)

with

2 ~(~) 1 § ~t 2

AF(a, b, c) = ~V/D 2 § ( E - - ~ ) 2

D = c(1 + ab) § ~(1 + ~b),

E = I § lb[ 2,

.F = Ic]2(1 § la[2).

The integral in eq. (4) has the following general form:

(14) fd ad2b]ol dIc I dq 2*(a, b, big.

We use the &function to integrate over ]c], then carry out the t r ivial ~ integrat ion and obtain

(1 + I@)e ~, b, ~ ] .

Following this procedure one can rewrite eq. (4) as

(16) 1 f d~a d2b ( b , l+lb l2~ <0>1 = ~ 4~g 2 (1 + la19 ~ (1 + Ib19 ~ exp [-- V(o)]O a, ~ 1

where V(q) can be shown to be equal to

1 1 40 (17) V(0) = 2 log 0 - - ~ log (1 - - 40) log

~ / v = ~ (l + ~ / v = ~ ) ~ '

1 la--bl 2 (18) 0 = ~ (1 + [al~)(1 + Ib]9"

This answer, eq. (16), is val id for any R; in fact, the radius of the sphere enters only into the running coupling g (eq. (5)), consistent wi th the s t ra ightforward dimensional analysis.

112 A. PATRASCIOIU and A. R0V~T

W e wou ld l ike to po in t ou t t ha t our gauge condi t ion eq. (12) is u n i q u e l y chosen by the r equ i r emen t s t h a t i t be i nva r i an t unde r Rs, R~ and I and under the in te rchange of a and b. T h e last c r i te r ion is necessary if one is to in te rp re t eq. (16) as t he cont r ibut ion to t he g rand canonica l ensemble of two (Bose) par t ic les m o v i n g on the surface of a sphere of rad ius R and in t e r ac t ing wi th a po ten t i a l V(~). T h e po ten t i a l we find (eq. (17)) is no t t he Coulomb po ten t i a l

(19) Vcoul ~ log 0.

I n pr inciple , one could ca r ry ou t t he same analysis for t he h igher topologie charge sectors of t he field t h e o r y and der ive a gas analogy. There is no reason to expec t the po ten t i a l ident i f ied in th is m a n n e r to cor respond to two-body forces; indeed the results ob ta ined on t h e sphere S 2 (eq. (16)) and on the torus T 2 (eq. ( IV. l ) ref. (2)) exhibi t d i f ferent s ingular i t ies . (The s ingula r i ty at c ~ 0 found on T 2 can be re in te rpre ted th rough a Ta t r ans fo rma t ion as an ex t r a s ingula r i ty a t coinciding points.) Th is makes i t un l ike ly t h a t in th is a p p r o x i m a t i o n the Oa non l inear a -model develops a mass gap.

W e wou ld l ike to t h a n k the I n s t i t u t e for A d v a n c e d S tudy and our colleagues here for the i r hospi ta l i ty . One of us (AR) acknowledges t he suppor t of a g ran t f rom the F e de ra l Repub l i c of Germany .

by Societa Italiana di Fis ica

Proprieta letteraria riservata

Direttore responsabile: RENATO ANGELO RICGI

Stampato in Bologna daUa Tipografla Compositori ooi tipi della Tipografla Monograf Questo fascicolo b stato licenziato dai torchi il 14-IX-1982