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N expansion of renormalizable and non-renormalizable scalar field theories
Nuclear Physics BI09 (1976) 297-321 © North-Holland Publishing Company
THE I/N EXPANSION OF RENORMALIZABLE AND NON-RENORMAL1ZABLE SCALAR FIELD THEORIES * Howard J. SCt|NITZER Department of Physics, Brandeis University, "Waltham, Massachusetts 02154 Received 26 I:ebruary 1976 The formalism of the 1/N expansion is extended to arbitrary O(N) symmetric scalar field theories with non-derivative couplings, to leading order N, and for any spact-time dimension. The resulting equations satisfied by the Green functions, which are the Hartree equations of the model, are renormalized and criteria formulated for the avoidan,:e of taehyons, with particular emphasis on two and three dimensions, extending earlier results in four dimensions. The generality of the approach permits examination of a nun bet of issues relevant to non-renormalizable theories.
1. Introduction The I/N expansion [1,2] is one of tile relatively few methods available in quantum field theory which avoids a perturbation expansion in the coupling constants. All attractive feature of the method is that it provides a systematic expansion whose first term is the Hartree approximation of tile theory. To date, progress in this subiect has been held up [2] by the incorrect idea that the expansion was not viable in ft;ur-dimensional theories due to the presence of tachyons. However, it has recently ~een shown [3,4] that the expansion is in fact a consistent approximation scheme "or ?,04 theory in four dimensions. A surprising, and dramatic, result of this new work is that the true ground state of the model does not go smoothly into the vacuum of lhe loop expansion in the weak-coupling (or/-z ~ 0) limit, due to an essential singularity in these parameters, reminiscent of superconductivity. Furthermore, there is no "signal" for this phase transition in any finite order of perturbation theory. Although the ~nodels studied to date are not of immediate physical interest to hadron physics, the methods and results might well provide insights into some of the pressing problems of tield theory, particularly in view of the richness of the results already exposed. It seems to us that further exploration of these ideas might be rewarding.
* Research supported in part by ERDA undcr contract no. E(I 1-1)3230. 297
298
H.J. Schnitzer / The I /N expansion
This paper is directed primarily to the further development of the formalism of the
l/N expansion, llowever the generality of our results allows us to explore some interesting issues of recent interest. In this paper we present an extension of the 1/N expansion to all 0(5/) symmetric scalar field theories without derivative couplings, i.e. to the class of theories described by the Lagrangian £(qb) : -~(Sud))2 -- NVo(ap2/N ) .
(1.1)
where VO(ap2/N) is a polynomial of either finite or infinite degree in the variable op2/N, wherc (I)a is an N component scalar field. We present the unrenormalized effective action, valid to leading order in N and arbitrary space-time dimension, for those theories described by (1.1). The resulting effective action generates the Green functions and gap equation of the model. The renormalization of the model, computation of Green functions, and formulation of criteria for the avoidance of tachyons is carried out in two and three space-time dimensions, supplementing available results [1,2] for four-dimensional renormalized theories. Due to the generality of our formalism, we are able to examine a class of non-renormalizable theories described by (1.1) in both three and four dimensions. (Extension to higher dimensions is straightforward, but not presented here.) It has frequently been hoped that a suitable resummation or expansion technique [5] for non-renormalizable theories will result in Green functions with "good" large p2 behavior. Although several attempts with this objective have been made in the past [5], none to date have been fully successful. Recently attention has been called to the possibility that the 1//V expansion might provide a systematic treatment [6] of a class of non-renormalizable theories which avoid the obvious difficulties of the usual coupling cofistant expansions. It is hoped that the I/N expansion will lead to Green functions with high-energy behavior no worse than that of renormalizable theories, and hence will avoid the usual proliferation of arbitrary parameters. If true this would probably mean that the Green functions of non-renormalizable theories were not i1~finitely differentiable in the coupling constant at g = 0 [5,6], which then disallows the usual coupling-constant expansions. We examine aspects of this issue in sects. 3 and 5, with some interesting results. We point out that terms which are non-leading in 1/N are likely to exhibit all the usual difficulties of non-renormalizable theories, due to induced effective couplings, which generate "bad" high-energy behavior, and an infinite number of arbitrary parameters. However, these problems may not be present in the Green functions computed in leading order in N, so that the non-renormalizable features of the theory may be suppressed by powers of I/N. We present a general formulation of the problem in sect. 2, and examine the renorrealization of these models in sect. 3. In sect. 4 we show that the spectrum found in this class of models in leading order is not sensitive to the detailed behavior of the interaction in eq. (1.1). We examine the large 02 dependence of the effective potential in sect. 5, and contrast the behavior of renormalizable and non-renormalizable theories.
H.J. Schnitzer / The I/N expansion
299
2. General f o r m u l a t i o n
2. 1. ttartree equations Consider the O(N) w m m e t r i c unrenormalized l,agrangian £(¢b) = ~ (au 4>) 2 - N V0(,.I,2./N ) ,
(2. l)
where 4~a(X) is an N component quantum field, and q)2 = xN=I q)a~a. The interaction term l/o(q)2/N) is a polynomial of either finite or infinite degree in the variable cb2/N. The re fore
NVo(
(2.2)
2 ~ )co) = .g,
(2.3)
with
being the conventional unrenormalized mass. Similarly the divergent constants lfl0k)(0) are the bare 2k point-function interaction parameters. We consider a ] the vgk;(0) to be of order one, which is the natural generalization of the requirel lent for a I/N expansion ore 4 or q~6 theory [I ,2,7]. The effective action for the complete theory can be expressed in terms of the two-particle irreducible (2PI) vacuum graphs [8,9] by the relation I'(4,, G) = I(~) + ~i/i Tr In G I + ~ ih T r ~ " 1 (¢)G + F2(¢, G) + constant. (2.4) where Tr denotes a trace over internal indices as well as a trace of the operatc,r, considered as an integral operator L:uclidean n-space. The classical action l(qS), obtained from the classical field Ca(X), is
y(,~)=fd"x~C(¢).
(2.5)
v.'here n is the dimension of space-time being considered. (Our derivation of the large N limit of the unrenormalized Green functions can be given for any n.) The free-particle propagator in the presence of the external field (~a(X) is obtained from [~-;,9]"
iCl)al (¢: x, ),) =
{
= -. [D] +
621(0)
6%(x)6¢,b(y) %(x)¢b'x)
2V'o(O2/N)]6ab + 4 . . . . TV . . . .
(2.6a)
Vt';(O2/N)
}
6"(x- )'),
(2.6b)
where
V~)(O2/N) -
a V0(¢2/N) Do2
(2.7a)
tLJ. Schnitzer I The 1IN expansion
300
V~(621N) -
a 2 Vo(62/N)
(2.7b)
0(62) 2
The two-particle irreducible vacuum functional [8,9] P2(6, G) is constructed from the interaction part of the Lagrangian £(6 + q~), but with propagators chosen as the exact propagator Gab(6; x, y), which is determined from 6|'(6, G) = O.
(2.8)
8Gab(X,Y)
The conventional effective action is obtained when the solution of (2.8) is substituted into eq. (2.4). The interaction part o f £ ( 6 + ~), denoted by £int(6; ~), is -N- l£int(6; qb) = VO((6 + (I))2/N) - VO(~2/N )
- fo&Vo((6+~)2/N)l.=oCl)a(X) I ~2 .... 2 [Ocba(x) ~cbb(x ) V0((6 + qb)21N1 a,=0 ¢Pa(X)*b(X) '
(2.9)
with repeated internal indices summed. The result is £int(6; q~) =- NVo((6 + q~)2/N) + NVo(621N) + 20a(X)aaa(x) V0(621 N)
To leading order in N, £int(@; ~) must be "spherical" [1o] in both classical fields and quantum fields, so that (2.10) can only depend on ~2(x) and dp2(x), but not Ea6a(X) ¢'a(x)in leading order. Thus, to leading order int(6; ~ ) ~---. N V0((6 2 +
2)lN) + N VO(¢21N) + -~dp2 [2 V~)(qa21N)]
1 tl ~ 2) \ k V~k)(62/N) = N . k~2 = ~..
(2.1 1a) (2.11b)
which gives the 6 2 dependence of the 4, 6 ..... 2k, ... point vertices appropriate to the large N limit. The corresponding 2PI graphs appropriate to leading order are shown in fig. 1. These results are a generalization of the I/N expansion of ¢ 6 theory studied by Townsend [7]. The Feynman rules to leading order in N are obtained from the basic 2k-point function
a % , ( x ) ... a%2k(X) *=0
X [~)ala= ... 8a2k... pa=k + all distinct permutations] .
(2.12)
H.J. Schnitzer / The I /N expansion
301
Fig. 1. Two-particle irreducible graphs which contribute to F2(~, G).
From these vertex functions, and the 2PI graphs shown in fig. 1, one can reuonstruct F2(¢, G) to leading order inN,
F2(cLG)=-N~ydnxlGaa(X,x)]k(~)kV(ok)(+21N)rr(k), oo
(2.13a)
k=2
= -N
dnxlGaafx, x)] k
)(462/N),
(2.13b)
where it(k)= (2k)!/2kk! represents the number of distinct permutations in (2.12), and where repeated internal indices are sumined. Since @~al (x, y) = No'/)- l(x, y) + O(1),
Goa(x, ),) = Ng(x, y) + 0(1 ) ,
(2.14)
we obtain ~' k F2(~,g ) = - N ~ ¢dnxFfig(x'x)l V(ok)(~b21N)+ O(1) s<:2 a
L
kr
J
- tig(x,x) IAO1)((~21N)J + O(1).
12.1 5a)
(2.15b)
Substituting (2.14) and (2.15) back into (2.4), one obtains to leading order F(4b, g) = 1(¢)+ ½Nih tr In g--I + ½NiX tr In e/)-lg
-- X f d n x [ V0 (¢2/N + #lg (x, x)) - V0 (462/N) -- h g (x, x) IA01)(4~2/N)I + con st, (2.16) where tr denotes a trace of the operator in Euclidean n-space. From (2.8)ant (2.16), one obtains g - l (,p; x, y ) -- ~ - i @; x, y ) + 2 i[ IAol)(¢2/N + ~g(x, x)) .- ~1 )(¢2/N)] 8n (x • y ) .
(2.17)
I1..I. Schnitzer / The l/N expansion
302 However, to leading order
q) -1(05;x, y) = i[UJ + 21"g 1~(052/N)]6''(x
y).
(2.18)
Thus thially g.1 (¢,';x, "~ 3') : i[IZl + 2V (01)(052/N +lTg(x, x ) ) l f n ( x • y ) ,
(2.19)
which is identical to the naive (unrenormalized) Hartree equations for the Green functions of the theory defined by the Lagrangian (2.1). One may multiply (2.1 7) by g(x, .v), take traces, and insert in (2.16), to obtain the effective action of the theory to leading order in N, 1"(05) = I(4,) + {Nih tr In g I
-. N fd"x[Vo(052/N + fig(x, x)) -
VO(052/N )
~g(x, x) IA01)(052/N + tTg(x, x))] + O ( I ) ,
(2.20)
where g - 1 (052, x, y) is given by' (2.1 9). All unrenormalized Green functions of the theory are obtained from (2.20) in conjunction with (2.19).
2. 2. Composite field Define the classical field function [71
f(x) = ~)2(x)/N + ~ g ( 0 2 ; x , x ) ,
(2.21)
st) that l'(qS) = 1(05) + ~Nitz tr In g-
- ~"x
I
[ Vo(f(x)) - VO(f~2/N)-- I f ( x ) - 052(x)/N] Vg I)(f(x)) } + O(1 ).
(2.22) It is tempting to think o f f ( x ) as a composition field with its own Feynman rules defined by tile Lagrangian of (2.22). But this is incorrect, because of the presence of non-quadratic./'3, f4 ..... terms means that the theory defined by the Lagrangian
£(05, x)=fdn)'{-~(a,~0) 2 N[V0(x)
(x
~)2/N)lAol)(x)}
(2.23)
is not equivalent to the original. It defines a different theory because of the presence of the X3, X4 .... terms. However, we generalize the argument of Townsend [71 to show that the theory defined by (2.23), which we call the comparison theory, is equivalent to the original one to leading order in N. The connected Green functions for the comparison theory is obtained from the
ILJ. Schnitzer/ The 1/N expansion
303
functional integral
expiW(f)/h
=fld9] [dx]exp{i/hfdny[£(G X) +fa(.V)ga(y)] ).
(2.24)
Shift the X functional integration by O2/N, so that X(Y) = X(Y) + 92(3')/N,
(2.25)
whereby
expiW( /)/~ =f [dg lexpg i/l~f d"y[~ (~ue~)2 --
NV
o
(92/N) +Jg]}
Xf[dx]exp{-iN/hfdnylVo(5( + 92/N)--VO(92/N)-
(2.26)
;~ V~)I)(~" + ~2/N)] },
which demonstrates that the comparison theory is distinct from that defined by (2. I ), since in general the 5( functional integration is not Gaussian. Comparing (2.25) with (2.21), we see that 5(0') in tile comparison theory plays a role similar to that of -Eg(x, x) in the original theory. Define L(~ 2, :~) = V0(5( + ~2/N) -
VO(92/N) -- X I"(01~(5(+ 92/:¥),
(2.27)
which has the property aL(92'3~ X)-]k=0 = 0 ,
(2.28)
so that the X functional integration in (2.26) does not generate tree diagrams;, but does generate a loop expansion with expansion parameter X/N, where the 5( loops have a 92 dependence, but are suppressed by at least one power of N. compared to the leading order in N obtained from (2.1). Therefore we have shown that fo- the comparison theory
expiW(j)/h
= Idg]exp
i/
dny {(3,~)2. NV0(~2/N ) +].~ +~(l×Ioop) + . . . .
(2.29) which is equivalent to the original theory to leading order in N. Thus, to leading order we may study the Green functions and effective potential defined by (2.33), which is much more convenient to work with than (2.20). Of course, if we desire I/N corrections we must go back to the original theory. Z3.
Effective action
The net result of this analysis is that the unrenormalized effective action, cgrrect to leading order in N, is
F(9, X)=fdn.Y{~(aug)2--NIJ/o(X)-. (X-N~----2)~O1)(X)I/
304
H.J. Schnitzer / The I /N expansion + ~Ni// tr In 112 + 2V~01)(X)] .
(2.30)
From this it follows directly that the unrenorma]ized effective potential, obtained
for constant fields 4~and X, is QY(c),X)=N[Vo(X).=(X.
~)vgl)(x)]+~Nh
f dnp ln[p2+2vgll(x)], ~ (2rr)" (2.31)
where the subscript denotes a divergent integration over EucLidean momenta. The gap equation of the model, obtained from aq~(¢, X) ax
= 0,
(2.32)
is IX IA2)(X)
4~2 N
~ drip 1 E (2/r)n p2 + 2]/(01)(X) = 0.
(2.33)
Since Ocl~(q~,X) _ ig01)(X) 0¢
(2.34)
spontaneous symmetry breakdown requires
!2.35)
-- O ,
where ×vac denotes the lowest energy solution to the gap equation (2.31). Note that (2.35) is essentially the classical condition for symmetry breaking, but with argument ×vat, which loosely speaking is Xvac ~ (~2/N). The unrenormalized Green functions to reading order are obtained from (2.30), with the two-point functions given by 62F(¢~, X) ~%(x)6~b(y )
62r(¢, x) _ 8%(x)~X(y)
(2.36)
[E~x + 2V(ol)(x)16ab6n(x • y)
2~Ao2)(x)%(v)6,,(x_.v) " '
62r(~, x) =NI~z)(X)+ (×-. (p2/N)Vg3)(X)]6"(x
(2.37) . 3,)
1 x a"(x. 34.
(2.38)
H.J. Sclmitzer / The I /N expansion
305
Higher order 1PI Green functions are obtained from successive functional derivatives. Note that 5 P(~b, X) _ NV(02)(X)[X fiX(y)
~2 N + i/~
1 .] • 127 + 2 v(oi )(x)
(2.39)
Therefore, on the "physical orbit" X(¢2), defined by (5F(qL X) _ 0 .
(2.40)
8x(.v)
one may simplify eq. (2.38) to read 82F(~b,X) ] 6X(x)fiX(.V)
=N[IA02)(X ) .. 2ihV~2)(X)_. - 1. ~)2)(X) x~¢,2) [] + 21AJ)(X)
X 6 n ( x • .v).
(2.41)
This has the same structure as q~4 theory [ 1,2,4] even though we have allow-~d arbitrary V0(X). The results of this section give a natural setting to the Hartree approximation of arbitrary O(N) symmetric scalar field theories without derivative couplings, in that the I / N expansion is a systematic approximation scheme in which the l-lartr.'e approximation is the leading term. Note that we have not specified the detailed nature of the interaction, no, the dimension of space-time. Examination of eqs. (2.34)(2.41) shows that this entire class of theories has a number of common featmes, iladependent of the details of the interaction.
3. Renormalization
Tile effective action and Green functions inwflve a number of divergent integrals which require renormalization. We shall show that eq. (2.30) is super-renorn alizable in two and three dimensions. In four dimensions if V0(X ) is (a) a quadratic polynomial in X the theory is renormalizable, as is already known; if (b) a polynomml in X of degree three or greater, the model is non-renormalizable even to leading o d e r in N, since a V0(X) of a finite degree in X induces logarithmic divergences in eff-~ctive vertices which are not present in the original Lagrangian. It is convenient to treat two, three, and four dimensions separately in what follows. 3.1. Two dimensh)ns
In two dimensions the theory described by eq. (2.1) is superrenormalizable, only requiring normal ordering, which is accomplished here by subtracting a logarithmically
306
H.J. Schnitzer / The I /N expansion
divergent constant from ×. We now fommlate our renormalization procedure and show that it leads to a finite theory. Consider oo
with the Vl0k)(0) independent infinite constants. Now define XR(Y) by ×(y) = XR(y ) + c ,
(3.2)
where c is a dimensionless infinite constant to be determined. Define VR(XR) = V0(X R + c)
=i~07(×R)SVIR)(0) "
(3.3)
As our renormalization condition we require that VIR/)(0) = finite for all J, which will make tile theory finite in leading order. That is =
= ~ k=0
1 ,.kv(]+k)(o-~
~ -0
"~"
for all/
•
= finite , (3.4) renders (2.30) finite in two (and three) dimensions. Let us begin with the effective potential. Our condition (3.4) trivially guarantees that (2.34) is finite. On the other hand, (2.33) becomes
3cl'°(4~'X)- NV(R2'(×R){×R- q$2/N+c--(h/4n)lnr 6X
A!
-l].
(3.5)
L2 V~)(XR)AJ
If we choose c - (h/4"n)In(A2/M 2) = CM = finite ,
(3.6)
then eq. (3.5) is also finite. Then carrying our the integrations in (2.31) one obtains, using (3.2) and (3.6) cl)(~, X) = N[ VR(XR) -- (XR + C M - oZ/N) V(1)(XR)] V(1)(XR) In
--1
+ constant,
(3.7)
H..I. Schnitzer ,/The I /N expansion
307
so that Q,~(0, X) = Q'~R(0, XR) + constant ,
(3.8)
where C~R(0, XR) is the finite, renormalized effective potential. Notice that CM and therefore XR depend additively on lnM 2, so that without loss of generality wc may choose Crf = 0, which we do in what follows. This renormalization scheme also makes the Green functions finite. First // In (A2/M 2 ) c=;~
fi
f
A2 d2p
d b
I
(2rr)2 p2 +M 2
,,)
....
.._,_ I-]v +/1,12 " _ '
which shows that choosing CM = 0 still leaves a dependence on M 2 in the theory. From (3.2), 1 X('Y)= I)'IXR - i/~ ~]'+~M "v~ .
(3.10)
We now rewrite (2.39), by using (3.3) and (3.10), us
_ ;5 ra(0, x~)
8xv,(),)
(3.11)
which is clearly finite in two- (and three-) dimensional space-time for any' XW y ) and 0 2 ( r ) . Similarly
81"(0, x)_
[K3~,+2IA~)(XR)I0~(y)
~o~,(),)
8 FR(0, XR) ~ --~0;(~'i -
(3.12)
which is also finite for all XR(Y) and 0 2 0 0 . Given the finiteness of (3.11) ant (3.12) ti)r any XR and 02, this establishes tile finiteness of tile renormalized effective action FR(0, XR). Since functional integration of (3.11) and (3.12) gives
|'R(qS, XR) = r(o, x) + constant,
(3.13)
H.J. Schnitzer / The I/N expansion
308
we have succeeded in showing the adequacy of our renormalization procedure. Of particular interest are the renormalized two-point functions, obtained from (2.36)-(2.41), and (3.11), (3.12). We find 82FR(0, XR) 8%(x)6¢bO, ) -
[E3x+ 2V(RI)(XR)]fab82(X - y),
(3.14)
621"R(qS, XR) 5¢)a(X) tSXR(.}')- _2V(R2)(XR)~a(X) 62(X . y),
(3.15)
62 I~R(qS,XR) 6XR(X) 6XR(Y) ' V(2)(XR)Uly+ 2V(RI)(XR) I =N ( V(2)(XR)II- 2i, I>, I [~>'+2V(R)(XR) ~2 /y [[]y +2V(1)(X 1 + V~3)(XR)i XR---~+il~ ) (3.16)
If we set (5 rR(~b , XR)
8XR(y)
(3.17)
--0,
and insert into (3.16), we arrive at the renormalized version of (2.41), namely 62rR(~b' XR) ]
=N(IAR2)(XR)_2ihI~R2)(XR)
I[]y+ 2 V(1)(XR)
I-qy+ 2 I/{1)(XR)
(3.18)
where the subscript indicates that (3.11) and (3.17) are satisfied. Note that (3.14), (3.15) and (3.18) have the same general structure as 04 theory, even though VR(×R) is arbitrary.
3.2. Coleman's theorem The renormalized effective potential in two dimensions is given by (3.8). From (3.8) or (3.12), ,9q~R(&, ×R) 0% - 2V(R1)(XR)~a ,
(3.19)
H.J. Schnitzer / The I /N expansion
309
where XR satisfies the gap equation (3.20) Spontaneous symmetry breaking requires =
o,
but (3.20)shows that this cannot be satisfied for any finite value of XR, independent of the choice of M 2. This is Coleman's theorem [11 ] as examplified by this model. Therefore the ground state occurs at ~ = 0. Since
(3.21)
m 2 = 2 IARI)(XR)IXR(0) > 0 is the elementary meson (mass) 2, x
(o) : -
In[ ''2
(3.22)
\M2]'
this exhibits the M 2 behavior of XR, since rn 2 is observable. If we combine (3.8) with (3.20), we obtian Cl~R(~2 ' XR (~2)) = N [ VR(XR) +/7/47r IARI)(XR) l ,
(3.23)
where of course XR satisfies (3.20). 3. 3. Three dimensions
To leading order in I]N, the model in three dimensions is also superrenorlralizable so that the results and discussion of subsect. 3.2 can be taken over directly to three dimensions. The only renormalization required to leading order in N is again r ormal ordering. In this section we only take note of the differences that distinguish lhe models in two and three dimensions. The renormalization procedure is that described by eqs. (3.1)-(3.4). Specializing (2.33) to three dimensions, we obtain
aq~(~, x) - -WV(R2)(XR){X ax
R - (a2/N
+c-/~/27r 2 [A - ½ 1 r ~ - ) ( X R ) ] } ,
(3.24)
which means that in three dimensions c = -h/2rr2A + finite constant
(3.25a) (3.25b)
310
t1..I. Schnitzer / The I /N expansion
which is the three-dimensional version of (3.9), (3.10), where M 2 is an arbitrary renormalization mass. The effective action is finite, as demonstrated by combining eqs. (3.11) (3.18) with (3.25b), since the resulting equations are finite in three dimensions. Thus, the model is superrenormalizable "in three dimensions in leading order in I / N , for any VR(XR). The renormalized effective potential is QPR(¢ 2, XIO = N {VR(XR) - (X n - 3t - O2/N)V(~)(XR) - t(127r[ 2V(1R)(XpO]312 }, (3.26)
so that 3 Q~R(O 2, XR) 3X R
-
_N(X R
31
02IN) V~2) (XR).--N/I/4n" Vg,2}(XR)[2 V~ )(XR)I 1/2 (3.27) (3.28)
~Oa The condition for spontaneous symmetry breakdown is
(3.29)
V~~)(×R) = 0 . which is permitted in three dimensions. The gap equation is XR(q~2) = 31 + ck2/N -
r,/47rv~IY~)(XR),
(3.30)
so that °PR(~2, XR(~b2)) = N { VR(XR) + ~'l/24rr[ 2 IAI)(XR) 13/23,
(3.31)
which should be compared with eqs. (3.20) and (3.23). 3.4. N o n - r e n o r m a l i z a b l e
theories
It has been suggested [6] that the 1/N expansion might be useful in making a systematic resummation of non-renormalizable theories. The idea is attractive, particularly if one believes that non-renormalizable theories are not flmdamentally flawed [5], but that the proliferation of arbitrary parameters is due to an invalid perturbation expansion in the coupling parameters of the model. Certainly the 1/N expansion offers a possible alternative. We found that to leading order in N, in three dimensions, the theory behaves like a superrenormalizable theory, for any VR(×R). This situation deteriorates in higlier orders of I / N in three dimensions, and no special advantage for non-renormalizable theories appears in four dimensions. However, in sect. 5, we shall show that the large 02 behavior of the effective potential qYR(02, XR(02)) is no worse than renormalizable theories in both three and four
ILL Schnitzer / The I/A" expansion
311
Fig. 2. A divergent contribution to an induced effective @10 coupling, appearing in or~ er I/N, in three-dimensional @8 theory.
g# Fig. 3. A divergent contribution to an induced effective 0 a coupling, appearing in lead ng order, in four-dimensional ¢~6 theory.
dimensions. Since there is generally a o n n e c t i o n between tire large <~2 beha ¢ior of the effective potential, and the large p2 behavior of the Green functions, our results might be construed as giving encouragement to this set of ideas. If
V(O/)((p2/N) = 0
{
for j ~> 4
ira three dimensions,
for/>~ 3
in four dimensions,
(3.32)
the O(N) model is in fact renormalizable to all orders in 1/N. (This depends :m naive power counting, and has been checked explicitly [12] to the first two order:; in I/N.) If (3.32) is not satisfied, but V0(q~2/,'V) is a polynomial ol limte degree, the next-toleading order m I/:V is non-renormalizable in both three and four dimension; due to induced couplings. As we have shown in subsect. 3.4, in three dimesnions, h. leading order the model is superrenormalizable. I lowever, as a specific example in fi~.:. 2, the next-to-leading order ira 1/N has a divergent effective 010 coupling induced, ,equiring counter terms which cannot be absorbed in tire original Lagrangian. If (3.32) is not satisfied in four dimensions, the model is non-renormaliza ~le even in leading order in 1/N. For example, in 06 theory in four dimensions, a divergent effective ~8 coupling is induced in leading order, as illustrated in fig. 3.
4. Particle spectrum The particle spectrum obtained from the Green functions computed to le~.ding order in N is the elementary meson, and whatever bound states or resonance,, appear in the X propagator. Tile renormalized Green functions presented in sect. 3 allow us to examine tile propagators for botmd state, resonance, or tachyon poles. It is convenient to analyze the two- and three-dimensional cases separately. An extendve analysis of the O(N) model in four dimensions is already available [4] and will not be repeated.
312
lt.J. Schnitzer
The 1/5I expansion
4.1. Two dimensions Tile renormalized q~ - X inverse matrix propagator, given as the momentum space version of (3.18) is k2+rn 2
0
D l ( - k 2 , ¢ , X R) = ( 0
- N { IflR2)(XR)+ 2~i [V(R2)(X R)]2B2(--k2, m2)} )
(4.1) for Euclidean momenta, as evaluated at the ground state. The quantity B 2 ( - k 2, m 2) is given by tile integral over Euclidean momenta, B2(-k2 , m 2) = f
d2p
l E (2rr) 2 [p2 + m 2] [(k +p)2
+m21
(4.2)
The elementary meson (mass) 2 is m 2 = 2 V(1)(XR) > 0 ,
(4.3)
which must be strictly positive in accord with Coleman's theorem. Let us define an effective four-poi:.: .'oupling parameter by X 3! - 2 V(R2)(XR).
(4.2)
Notice that two-point function and meson-meson scattering amplitudes only depend on two parameters, m 2 and X. The integration in (4.2) can be carried out, with the result B2( -k 2, m 2) = ~ ( k 2 + 4 m 2 ) V - - k 2
In
2m
for k 2 ~ 0 , (4.5)
which has thc properties for k 2 > 0, m 2 4: 0, B2(--k 2, rn 2) > 0 , B2(0 ' rn2) =
1 4rim2 '
B2(-k2, m 2) is monotonically decreasing for increasing k 2 > 0 .
(4.6)
Combining these results with (4.1), we have --12 _-1,.7~._~ l)xx (- k2, m2) = l + ~ X B 2 ( - k 2 ,
m2 ) .
(4.7)
H.J. Schnitzer / The I/N expansion
313
Since -12
Dx.×l(0 ' m 2 ) : I +
N~,
~
(4.8a)
24rrm 2'
- - 1 2 D - l ( - - k 2 m 2) /V'A --x'x" "" ' k2 ~
,1 ,
(4.8b)
. -1 (- k 2, m 2) is free of zeros for Euclidean k 2 > 0. tachyons are avoided in Dxx lfDxx This puts restrictions on the physical parameters of the theory, i.e. 1 + --
< 0
forbidden domain ,
(4.9a)
> 0
allowed domain .
(4.9b)
24rrrn 2 I + -
fix
24rrm 2 (In four dimensions there is an analogous forbidden domain [4].) If the gap equation is multivalued, one must choose real solutions satisfying (4.9b), which is similar to the situation in four dimensions [3,4]. Since 53r
8¢a(x)6~b(Y)SX(z)
-~, vac = -~.
6ab82(x -
Y)62(x - z ) ,
(4.10)
the singularities of Dxx are observable in the meson-meson scattering amplitt.,de
Tab, cd(S, t, u) = ~6 X2{Dxx(S, m2)Sab5cd + crossed terms} ,
(4.11)
where s, t and u are tile usual Mandelstam variables, and s = - k 2 is the Minkcwski continuation of k 2. Continuing (4.5) to Minkowski momenta 1 1 {2Vs-4m21nfX/s+x/s--4m21 2rrs_4m 2 s ~-
B2(s'm2)= --1T/'V
S -
/
2rrs_4m 2 -2
+rrl/4m~z-s I r s J
for s > 4m 2
for 0~
(4.12a) arctan (4.12b)
with limiting values
ReB2(s,m 2)
~ - ~ 1 [ I n s - in], £-÷ +oo "/IS
(4.13a)
tt.J. Schnitzer / The I / N expansion
3 14
B2(s,m 2)
(4.13b)
, +~, S +4m 2
( 4 . 1 3c)
1
4n-m 2
x-,4m2+
B2(0 , m 2) . . . . .
1
.
(4.1 3d)
47rm 2
We plot Re B2(s, m 2) versus s in fig. 4. From eqs. (4.12), (4.13), and fig. 4 we can deduce tim bound-state and resonance structure, which is restricted to the identity representation of the O(N) group. For the allowed domian of parameters (4.9b), we obtain 0 "< 3~-' ~< 47rm 2
no resonance or bound state ,
(4.14)
0 < 47rm 2 --~~
resonance ,
(4.1 5)
_ fix
0 > _--=->~ .-4rim 2 3~
bound state
(4.16)
l
Therefore, comparing (4.14)-(4.16) with (4.9a), we observe that the tachyon occurs for coupling parameters X which are too negative, generating a bound state whose binding energy e x c e e d s 2m. Tile masses of the bound state and resonance is trivially found from tile zero of Dxl(s, m 2) for s > 0; we do not elaborate. 4. 2. Three d i m e n s i o n s
Tile behavior of tile two-point function in three dimensions is very similar to that in two dimensions, but now symmetry breaking is possible if the parameters of the
Re{B,(]
_1/
7 4Tin e
4mz I
S ...--"
-I 4 ~r m-"i'-
Fig. 4. A graph of Re B21s, m 2 ) l,er~us s, where B2(s, m 2) is given by eqs. (4.1 2) and (4. I 3).
II.J. Schnitzer ,/]Tte I/V expansion model satsify eq. (3.29). The ~
315
X inverse propagator is
gk 2 + m 2
I ;~a x
\
~5:\'X[ 1 + ~hX133(-k 2. m2)]
O I( k 2,~,XR )=[
(4.17) -~0b ~
where we have made use of definitions (4.3) and (4.4), with B3( k 2, m 2) the threedimensional analogue of (4.2). One finds B3 ( k2 ' m2 ) _ 47r ~/k 1 2 sin 1 ]//4 n ~k2"+ k 2
1
for m 2 :¢: O,
for m 2 = 0 ,
8~/k2
(4.18a)
,'4.18b)
for Euclidean k 2 > O. On continuation to Minkowski momenta
1
r 2,,, +,/s q
B3(~,m2) GU,, l
4rr,,/s
L,/g,,i-
s
for 0 ~< s ~ 4m 2,
(lnr2m_+x/s']+,ilr} Lx/J-s -4-STriA ~
t
for s > 4m 2 ,
(4.1%)
(4.19b)
ifm 2 :¢: 0, while
B3(s, O) . . . . . . . 8x/
1
for s < 0 ,
(4.20a)
s
i
for s > 0 ,
---8,/2
(4.20b)
it" m 2 = 0. The following limits are useful: B3(0" m2 ) _
B 3 ( s ' m 2 ) -S
1
(4.21a)
g a"1;~1'
,4m
2 -
s-.4m:
1 inIAm2 .... .~] 'l--6zrt~ ~ 16m2 7 '
L \
(-l.21b)
16m 2 /
We plot ReB3(s, m 2) in fig. 5. The analysis of tile spectrum is separated into two cases: Case l: broken ,9'mmetrv. The requirement for spontaneously broken symmetry
316
H.J. Schnitzer / The I/N expansion
4 Im'
Fig. 5. A graph of Re B3(s, m 2) versuss, where
s
B3(s,rn2) is given by eqs. (4.19) and (4.21).
m2 = 0
(4.22)
as inferred from (3.29) and (4.3), which is Goldstone's theorem. In this case we may choose % = 0 for a = 2 .... , N and q~l :~ 0 for the vacuum expectation values of the scalar field, giving a residual O(N - I) symmetry. Since for m 2 = 0,
-12 NX
detD_l(_k2 ~1, ×R) ; k2 + - ~ X X / k 2 + ~X
'
(4.23)
i
the absence of tachyons requires det D -1 not to vanish for k 2 > 0, so that we have X> 0 ] m2 = 0 J
allowed d o m a i n ,
(4.24a)
forbidden d o m a i n .
(4.24b)
X< 0 / m 2 =0 J
In the Minkowski region, for s > 0,
7¢~2 det D-l(s, c~l, XR) = - s + ~ (C2~N) + ~tt~ ix/s ,
(4.25)
which develops a zero at 2 l ¢1 ..~ s o = -~X ~ . - - 0 ,
(4.26)
giving the (real part) of the (mass) 2 of both the o meson, and a resonance in the identity representation. Case 2: O(N) invariant vacuum. Here % = 0 at the ground state. Then D -1 (s, m 2)
ILJ. Schnitzer / The I /N expansion
317
is block diagonal and
D~(--k 2,m2)=-~NXII+~hXB3(
k 2,m2)1 ,
(4.27)
for k 2 Euclidean. From eqs. (4.18), (4.19) and (4.21), -12D_l(0, tzX N-X- xx" m2) = 1 + 4 - 8 ~ ,
(4.28)
and -12
NX D
( - k 2,m 2)
+1,
(4.29)
k 2 ~ +oo
we obtain 1+~ 1+~
fi?, hX
< 0
forbidden domain ,
(4.30a)
> 0
allowed domain .
(4.30b)
(Compare with eq. (4.11).) Further, from (4.27) continued to the Minkowsl.:i region, we have 0 K fix < oo
no resonance or bound state,
mix -87rm <---~-.~< 0
resonance and bound state.
(4.31 a) (4.31 b)
(Compare eqs. (4.14)--(4.16).) The masses of the bound state and resonance are obtained from the zeros o f D x l ( s , m2).
5. Large q~2 behavior We can obtain the large q52 behavior of the effective potential for this clas., of models, to leading order in N, from the gap equation. Although the limit (~2/5V >> I is outside the domain of tile validity of the 1/3,' expansion, a number of interesting results are obtained, particularly for non-renormalizable theories, which justiqes presentation here.
5.1. Two dimensions From (3.20) it is clear that XR(~2 ) -' @2 ,
* cp2/N + O(ln ~2/M2).
(5.1)
318
tt.J. Schnitzer / The 1/N exl~ansion
Combined with (3.23) this gives C)?R(O2. X R ( O 2 ) ) ~ ,¥ I/R(02/'V) 0 2 -. oo
+ ....
(5.2)
Thus lhe efl'ective potenlia[ increases with 4)2 like the classical interaclion. This is as expected ['or a superrenormalizable theory. 5.2. Three dimensions: rem~rnzalizal)le theories
The ,gap equation is given by (3.30). If we restrict ourselves to renormalizable lheories, then VR(XR)[V~)(XR) ] is a cubic (quadratic) polynomial in XR, so that ?
Vd b v
--R
'
t~'R "
3r/X 2
~
(5.3)
X.R-, oo
say, m which case
x|d~2)
, O2/;v 02
t~14~: x/~xR.
(5.4)
.
(5.5)
.~
Of
xR(~2)~=
,_ .....
. . ° 2. / N . . . .
1
+ r,/a,~v%
tlence from (3.31 ) -,.-I ,'~ t:n,.,,3R +/~/247r[3r/XR] 3/2} ÷
~ R ( ~ 2, xa(¢,2))-. ---~"
A"rt
[1 +///4rr(37r) 1/2 ]
(02/N) 3 + ....
...
(5.6a)
( 5.6b )
Notice that we may t~,ke that l i m i t / / ~ 0 without encountering a singularity, consistent with the superrenormalizability of the model in leading order. 5.3. Three dimensiotzs." mm-rettormalizable theories
If the theory is non-lenormalizable, then VR(XR) is quartic or higher in XR. We rewrite (3_]0) as (4n'/h)2 (XR
M
02/N) 2 = 2 q~I(XR) .
(5.7)
This equation can be solved for XR(¢2), but with the highest power of XR coming from IA,I)(XR)in this case. Thus
2 |AI~)(XR) ~ 2 7
(4rr/h)2(02'/'\;)2 "
{5.8)
I1..I. Schnitzer / The 1/N expansion
319
Since V~)(XR) is of degree cubic or higher, (5.8) implies XR(~02)/~52
0 2 _•
(5.9)
) 0.
Therefore,
VR(XR)
)"constant XR(.~2) V~)(XR) --* constant XR(O2) (02/N) 2 ,
(5.10)
Combining (5.8) -(5.1 0) with (3.31 ) we have tieR(02, XR(02).) -----+ N{const XR(02) (~2/N) 2 + fl/247rl(4Uh) 2 (~b2/N) 2 ] 3/2 +... }, -"N't - S tr3h2 r 2 . (~52/N)3 + . . .}.
(5. l 1a) /5.1 lb)
Therefore, the large ¢2 behavior of the effective potential also increases as (~b2/N)3, just as for a renormalizable thoery in three dimensions. Itowever there is a crJcial difference, since (5.1 l b) is independent of the parameters of the model, in con trast to (5.6b). Given a specific theo,y we can also give the behavior of the correction term to (5.1 lb). Suppose
xj VR(XR) : j~0 {2/~. (XR)/'
k ~> 4 .
(5.1 2)
Proceeding exactly as in (5.7)-(5.11), we find
O/YR(~52,XR(q52))~..
,~,[ 877
Ii l-~l(kl)~2] 2k/(k- I))
+} ,,.,3)
Note that one cannot take the limit Xk -,- 0, or ~ --+ 0 in (5.13), a feature wile i distinguishes this from renormalizable theories. Thus eq. (5.13)is dominated by the quantum corrections, in contrast to (5.6). 5. 4. I,'our dimensions." non-ren(~rmalizable theories
We obtain tile unrenormalized effective potential in four dinmnsions from 12.3 I), whereby
320
t1..I. Schnitzer / The I/N expansion +--[IA01)(×)12 32~v2
{ + 21n
(5.14)
with gap equation
xc~2) : ¢2/,v
~
ffo~)(x) } + 2
167r 2
L ~ dj
.
(5.15)
We now study the limit ~2/N ~ oo, with A 2 fixed. If the theory is non-renormalizable, the Vo(X) is of degree three or higher in X, so that the quantum corrections dominate the large 42 limit. Therefore,
167r2
V!,I)(X ) ~ - (*2/N) u - ~2..~ ~ + 21n[2V(ol)(X)/A2]
(5.16)
Since d°d (42, X(~2)) = IAOl)(X) d02
(5.17)
we obtain ----.-. dq~2
--16n2¢2/h/V 2
--8rr2 ¢2/FjV lnq~2
(5.18a) '
(5.18b)
This is identical to the result obtained for tile renormalizable O(N) model in four dimensions [4], where the large 42 behavior of the effective potential is also dominated by quantum corrections. The overall result of this section is that the effective potential always has the large 42 behavior of a renormalizable theory, at least to leading order in N, for all models considered, whether renormalizable or non-renormalizable. Since the renormalization group describes both the large ~b2 behavior of the effective potential, and tile large (non-exceptional) momentum behavior of the Green functions, our results might give some encouragement to the idea that the large p2 behavior of nonrenormalizable theories is better than naively expected.
6. Summary This paper was primarily devoted to an advance of the formalism of the 1/N expansion. Techniques were developed which allow tile derivation of the effective ac-
H.J. Schnitzer / The I /:V expansion
321
tion, to leading order in N, for all O(N) symmetric scalar field theories of the class defined by eq. (1.1), in any space-time dimension. The resulting equations for the Green functions are in fact the Hartree equations of the model. These equations were renormalized, and criteria were formulated for the avoidance of tachyons in twoand three-dimensional theories, extending results obtained earlier in four dimensions. The generality of the approach allows examination of the special class of non-renormalizable theories described by (1.1). Although the effective potential, to leading order in N, increases no faster than renormalizable theories at large ¢2, o'le suspects that the presence of induced divergent effective vertices in higher orde's in N leads to the same proliferation of arbitrary parameters that typically plague aon-renormalizable theories. We wish to thank Dr. J.S. Kang, Mr. L.F. Abbott and Mr. P.K. Townsend for conversations on this and related subjects.
References [I] K.G. Wilson, Phys. Rcv. D7 (1973) 2911; k. Dolan and R. Jackiw, Phys. Rev. D9 (1974) 3320; H.J. Schnitzer, Phys. Rev. D10 (1974) 1800, 2042. [2] S. Coleman, R. Jackiw and H.D. Politzer, Phys. Rev. DI0 (1974) 2491; D. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235; R.G. Root, Phys. Rev. DI0 (1974) 3322. [3] M. Kobayashi and T. Kugo, University of Kyoto preprint KUNS 316; R.W. Haymaker, Louisiana State University preprint. [41 L.F. Abbott, J.S. Kang and li.J. Schnitzer, Phys. Rev. I)13 (1976) 2212. [5] P.J. Redmond and J.L. Uretski, Phys. Rev. Letters I (1958) 145; Ann. of l'hys. 9 ~1960) 106; T.D.l.ee, Phys. Rev. 128 (1962) 899; G. Fcinberg and A. Pais, Phys. Rev. 132 (1963) 2724; J.R. Klauder and H. Narnhofer, Phys. Rev. DI3 (1976) 257. [61 K.G. Wilson, Phys. Rev. D4 ~1973) 2911; G. Parisi, Nucl. Phys. BI00 (1975) 368. [71 P.K. Townsend, Phys. Rev. D12 (1975) 2269. [81 T.D. Lee and C.N. Yang, Phys. Rev. 117 (1960) 22; J.M. Luttinger and J.C. Ward, Phys. Rev. 118 (1960) 1417; T.D. kce and M. Margulies, Phys. Rev. DI I (1975) 1591. [9] J.M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428. [10l ll.J. Schnitzer, Phys. Rev. DI0 (1974) 2042. [111 S. Coleman, Comm. Math. Phys. 31 (1973) 259. [121 R.G. Root, Phys. Rev. DI0 (1974) 3322; P.K. Townsend, Brandeis University preprint.
THE I/N EXPANSION OF RENORMALIZABLE AND NON-RENORMAL1ZABLE SCALAR FIELD THEORIES * Howard J. SCt|NITZER Department of Physics, Brandeis University, "Waltham, Massachusetts 02154 Received 26 I:ebruary 1976 The formalism of the 1/N expansion is extended to arbitrary O(N) symmetric scalar field theories with non-derivative couplings, to leading order N, and for any spact-time dimension. The resulting equations satisfied by the Green functions, which are the Hartree equations of the model, are renormalized and criteria formulated for the avoidan,:e of taehyons, with particular emphasis on two and three dimensions, extending earlier results in four dimensions. The generality of the approach permits examination of a nun bet of issues relevant to non-renormalizable theories.
1. Introduction The I/N expansion [1,2] is one of tile relatively few methods available in quantum field theory which avoids a perturbation expansion in the coupling constants. All attractive feature of the method is that it provides a systematic expansion whose first term is the Hartree approximation of tile theory. To date, progress in this subiect has been held up [2] by the incorrect idea that the expansion was not viable in ft;ur-dimensional theories due to the presence of tachyons. However, it has recently ~een shown [3,4] that the expansion is in fact a consistent approximation scheme "or ?,04 theory in four dimensions. A surprising, and dramatic, result of this new work is that the true ground state of the model does not go smoothly into the vacuum of lhe loop expansion in the weak-coupling (or/-z ~ 0) limit, due to an essential singularity in these parameters, reminiscent of superconductivity. Furthermore, there is no "signal" for this phase transition in any finite order of perturbation theory. Although the ~nodels studied to date are not of immediate physical interest to hadron physics, the methods and results might well provide insights into some of the pressing problems of tield theory, particularly in view of the richness of the results already exposed. It seems to us that further exploration of these ideas might be rewarding.
* Research supported in part by ERDA undcr contract no. E(I 1-1)3230. 297
298
H.J. Schnitzer / The I /N expansion
This paper is directed primarily to the further development of the formalism of the
l/N expansion, llowever the generality of our results allows us to explore some interesting issues of recent interest. In this paper we present an extension of the 1/N expansion to all 0(5/) symmetric scalar field theories without derivative couplings, i.e. to the class of theories described by the Lagrangian £(qb) : -~(Sud))2 -- NVo(ap2/N ) .
(1.1)
where VO(ap2/N) is a polynomial of either finite or infinite degree in the variable op2/N, wherc (I)a is an N component scalar field. We present the unrenormalized effective action, valid to leading order in N and arbitrary space-time dimension, for those theories described by (1.1). The resulting effective action generates the Green functions and gap equation of the model. The renormalization of the model, computation of Green functions, and formulation of criteria for the avoidance of tachyons is carried out in two and three space-time dimensions, supplementing available results [1,2] for four-dimensional renormalized theories. Due to the generality of our formalism, we are able to examine a class of non-renormalizable theories described by (1.1) in both three and four dimensions. (Extension to higher dimensions is straightforward, but not presented here.) It has frequently been hoped that a suitable resummation or expansion technique [5] for non-renormalizable theories will result in Green functions with "good" large p2 behavior. Although several attempts with this objective have been made in the past [5], none to date have been fully successful. Recently attention has been called to the possibility that the 1//V expansion might provide a systematic treatment [6] of a class of non-renormalizable theories which avoid the obvious difficulties of the usual coupling cofistant expansions. It is hoped that the I/N expansion will lead to Green functions with high-energy behavior no worse than that of renormalizable theories, and hence will avoid the usual proliferation of arbitrary parameters. If true this would probably mean that the Green functions of non-renormalizable theories were not i1~finitely differentiable in the coupling constant at g = 0 [5,6], which then disallows the usual coupling-constant expansions. We examine aspects of this issue in sects. 3 and 5, with some interesting results. We point out that terms which are non-leading in 1/N are likely to exhibit all the usual difficulties of non-renormalizable theories, due to induced effective couplings, which generate "bad" high-energy behavior, and an infinite number of arbitrary parameters. However, these problems may not be present in the Green functions computed in leading order in N, so that the non-renormalizable features of the theory may be suppressed by powers of I/N. We present a general formulation of the problem in sect. 2, and examine the renorrealization of these models in sect. 3. In sect. 4 we show that the spectrum found in this class of models in leading order is not sensitive to the detailed behavior of the interaction in eq. (1.1). We examine the large 02 dependence of the effective potential in sect. 5, and contrast the behavior of renormalizable and non-renormalizable theories.
H.J. Schnitzer / The I/N expansion
299
2. General f o r m u l a t i o n
2. 1. ttartree equations Consider the O(N) w m m e t r i c unrenormalized l,agrangian £(¢b) = ~ (au 4>) 2 - N V0(,.I,2./N ) ,
(2. l)
where 4~a(X) is an N component quantum field, and q)2 = xN=I q)a~a. The interaction term l/o(q)2/N) is a polynomial of either finite or infinite degree in the variable cb2/N. The re fore
NVo(
(2.2)
2 ~ )co) = .g,
(2.3)
with
being the conventional unrenormalized mass. Similarly the divergent constants lfl0k)(0) are the bare 2k point-function interaction parameters. We consider a ] the vgk;(0) to be of order one, which is the natural generalization of the requirel lent for a I/N expansion ore 4 or q~6 theory [I ,2,7]. The effective action for the complete theory can be expressed in terms of the two-particle irreducible (2PI) vacuum graphs [8,9] by the relation I'(4,, G) = I(~) + ~i/i Tr In G I + ~ ih T r ~ " 1 (¢)G + F2(¢, G) + constant. (2.4) where Tr denotes a trace over internal indices as well as a trace of the operatc,r, considered as an integral operator L:uclidean n-space. The classical action l(qS), obtained from the classical field Ca(X), is
y(,~)=fd"x~C(¢).
(2.5)
v.'here n is the dimension of space-time being considered. (Our derivation of the large N limit of the unrenormalized Green functions can be given for any n.) The free-particle propagator in the presence of the external field (~a(X) is obtained from [~-;,9]"
iCl)al (¢: x, ),) =
{
= -. [D] +
621(0)
6%(x)6¢,b(y) %(x)¢b'x)
2V'o(O2/N)]6ab + 4 . . . . TV . . . .
(2.6a)
Vt';(O2/N)
}
6"(x- )'),
(2.6b)
where
V~)(O2/N) -
a V0(¢2/N) Do2
(2.7a)
tLJ. Schnitzer I The 1IN expansion
300
V~(621N) -
a 2 Vo(62/N)
(2.7b)
0(62) 2
The two-particle irreducible vacuum functional [8,9] P2(6, G) is constructed from the interaction part of the Lagrangian £(6 + q~), but with propagators chosen as the exact propagator Gab(6; x, y), which is determined from 6|'(6, G) = O.
(2.8)
8Gab(X,Y)
The conventional effective action is obtained when the solution of (2.8) is substituted into eq. (2.4). The interaction part o f £ ( 6 + ~), denoted by £int(6; ~), is -N- l£int(6; qb) = VO((6 + (I))2/N) - VO(~2/N )
- fo&Vo((6+~)2/N)l.=oCl)a(X) I ~2 .... 2 [Ocba(x) ~cbb(x ) V0((6 + qb)21N1 a,=0 ¢Pa(X)*b(X) '
(2.9)
with repeated internal indices summed. The result is £int(6; q~) =- NVo((6 + q~)2/N) + NVo(621N) + 20a(X)aaa(x) V0(621 N)
To leading order in N, £int(@; ~) must be "spherical" [1o] in both classical fields and quantum fields, so that (2.10) can only depend on ~2(x) and dp2(x), but not Ea6a(X) ¢'a(x)in leading order. Thus, to leading order int(6; ~ ) ~---. N V0((6 2 +
2)lN) + N VO(¢21N) + -~dp2 [2 V~)(qa21N)]
1 tl ~ 2) \ k V~k)(62/N) = N . k~2 = ~..
(2.1 1a) (2.11b)
which gives the 6 2 dependence of the 4, 6 ..... 2k, ... point vertices appropriate to the large N limit. The corresponding 2PI graphs appropriate to leading order are shown in fig. 1. These results are a generalization of the I/N expansion of ¢ 6 theory studied by Townsend [7]. The Feynman rules to leading order in N are obtained from the basic 2k-point function
a % , ( x ) ... a%2k(X) *=0
X [~)ala= ... 8a2k... pa=k + all distinct permutations] .
(2.12)
H.J. Schnitzer / The I /N expansion
301
Fig. 1. Two-particle irreducible graphs which contribute to F2(~, G).
From these vertex functions, and the 2PI graphs shown in fig. 1, one can reuonstruct F2(¢, G) to leading order inN,
F2(cLG)=-N~ydnxlGaa(X,x)]k(~)kV(ok)(+21N)rr(k), oo
(2.13a)
k=2
= -N
dnxlGaafx, x)] k
)(462/N),
(2.13b)
where it(k)= (2k)!/2kk! represents the number of distinct permutations in (2.12), and where repeated internal indices are sumined. Since @~al (x, y) = No'/)- l(x, y) + O(1),
Goa(x, ),) = Ng(x, y) + 0(1 ) ,
(2.14)
we obtain ~' k F2(~,g ) = - N ~ ¢dnxFfig(x'x)l V(ok)(~b21N)+ O(1) s<:2 a
L
kr
J
- tig(x,x) IAO1)((~21N)J + O(1).
12.1 5a)
(2.15b)
Substituting (2.14) and (2.15) back into (2.4), one obtains to leading order F(4b, g) = 1(¢)+ ½Nih tr In g--I + ½NiX tr In e/)-lg
-- X f d n x [ V0 (¢2/N + #lg (x, x)) - V0 (462/N) -- h g (x, x) IA01)(4~2/N)I + con st, (2.16) where tr denotes a trace of the operator in Euclidean n-space. From (2.8)ant (2.16), one obtains g - l (,p; x, y ) -- ~ - i @; x, y ) + 2 i[ IAol)(¢2/N + ~g(x, x)) .- ~1 )(¢2/N)] 8n (x • y ) .
(2.17)
I1..I. Schnitzer / The l/N expansion
302 However, to leading order
q) -1(05;x, y) = i[UJ + 21"g 1~(052/N)]6''(x
y).
(2.18)
Thus thially g.1 (¢,';x, "~ 3') : i[IZl + 2V (01)(052/N +lTg(x, x ) ) l f n ( x • y ) ,
(2.19)
which is identical to the naive (unrenormalized) Hartree equations for the Green functions of the theory defined by the Lagrangian (2.1). One may multiply (2.1 7) by g(x, .v), take traces, and insert in (2.16), to obtain the effective action of the theory to leading order in N, 1"(05) = I(4,) + {Nih tr In g I
-. N fd"x[Vo(052/N + fig(x, x)) -
VO(052/N )
~g(x, x) IA01)(052/N + tTg(x, x))] + O ( I ) ,
(2.20)
where g - 1 (052, x, y) is given by' (2.1 9). All unrenormalized Green functions of the theory are obtained from (2.20) in conjunction with (2.19).
2. 2. Composite field Define the classical field function [71
f(x) = ~)2(x)/N + ~ g ( 0 2 ; x , x ) ,
(2.21)
st) that l'(qS) = 1(05) + ~Nitz tr In g-
- ~"x
I
[ Vo(f(x)) - VO(f~2/N)-- I f ( x ) - 052(x)/N] Vg I)(f(x)) } + O(1 ).
(2.22) It is tempting to think o f f ( x ) as a composition field with its own Feynman rules defined by tile Lagrangian of (2.22). But this is incorrect, because of the presence of non-quadratic./'3, f4 ..... terms means that the theory defined by the Lagrangian
£(05, x)=fdn)'{-~(a,~0) 2 N[V0(x)
(x
~)2/N)lAol)(x)}
(2.23)
is not equivalent to the original. It defines a different theory because of the presence of the X3, X4 .... terms. However, we generalize the argument of Townsend [71 to show that the theory defined by (2.23), which we call the comparison theory, is equivalent to the original one to leading order in N. The connected Green functions for the comparison theory is obtained from the
ILJ. Schnitzer/ The 1/N expansion
303
functional integral
expiW(f)/h
=fld9] [dx]exp{i/hfdny[£(G X) +fa(.V)ga(y)] ).
(2.24)
Shift the X functional integration by O2/N, so that X(Y) = X(Y) + 92(3')/N,
(2.25)
whereby
expiW( /)/~ =f [dg lexpg i/l~f d"y[~ (~ue~)2 --
NV
o
(92/N) +Jg]}
Xf[dx]exp{-iN/hfdnylVo(5( + 92/N)--VO(92/N)-
(2.26)
;~ V~)I)(~" + ~2/N)] },
which demonstrates that the comparison theory is distinct from that defined by (2. I ), since in general the 5( functional integration is not Gaussian. Comparing (2.25) with (2.21), we see that 5(0') in tile comparison theory plays a role similar to that of -Eg(x, x) in the original theory. Define L(~ 2, :~) = V0(5( + ~2/N) -
VO(92/N) -- X I"(01~(5(+ 92/:¥),
(2.27)
which has the property aL(92'3~ X)-]k=0 = 0 ,
(2.28)
so that the X functional integration in (2.26) does not generate tree diagrams;, but does generate a loop expansion with expansion parameter X/N, where the 5( loops have a 92 dependence, but are suppressed by at least one power of N. compared to the leading order in N obtained from (2.1). Therefore we have shown that fo- the comparison theory
expiW(j)/h
= Idg]exp
i/
dny {(3,~)2. NV0(~2/N ) +].~ +~(l×Ioop) + . . . .
(2.29) which is equivalent to the original theory to leading order in N. Thus, to leading order we may study the Green functions and effective potential defined by (2.33), which is much more convenient to work with than (2.20). Of course, if we desire I/N corrections we must go back to the original theory. Z3.
Effective action
The net result of this analysis is that the unrenormalized effective action, cgrrect to leading order in N, is
F(9, X)=fdn.Y{~(aug)2--NIJ/o(X)-. (X-N~----2)~O1)(X)I/
304
H.J. Schnitzer / The I /N expansion + ~Ni// tr In 112 + 2V~01)(X)] .
(2.30)
From this it follows directly that the unrenorma]ized effective potential, obtained
for constant fields 4~and X, is QY(c),X)=N[Vo(X).=(X.
~)vgl)(x)]+~Nh
f dnp ln[p2+2vgll(x)], ~ (2rr)" (2.31)
where the subscript denotes a divergent integration over EucLidean momenta. The gap equation of the model, obtained from aq~(¢, X) ax
= 0,
(2.32)
is IX IA2)(X)
4~2 N
~ drip 1 E (2/r)n p2 + 2]/(01)(X) = 0.
(2.33)
Since Ocl~(q~,X) _ ig01)(X) 0¢
(2.34)
spontaneous symmetry breakdown requires
!2.35)
-- O ,
where ×vac denotes the lowest energy solution to the gap equation (2.31). Note that (2.35) is essentially the classical condition for symmetry breaking, but with argument ×vat, which loosely speaking is Xvac ~ (~2/N). The unrenormalized Green functions to reading order are obtained from (2.30), with the two-point functions given by 62F(¢~, X) ~%(x)6~b(y )
62r(¢, x) _ 8%(x)~X(y)
(2.36)
[E~x + 2V(ol)(x)16ab6n(x • y)
2~Ao2)(x)%(v)6,,(x_.v) " '
62r(~, x) =NI~z)(X)+ (×-. (p2/N)Vg3)(X)]6"(x
(2.37) . 3,)
1 x a"(x. 34.
(2.38)
H.J. Sclmitzer / The I /N expansion
305
Higher order 1PI Green functions are obtained from successive functional derivatives. Note that 5 P(~b, X) _ NV(02)(X)[X fiX(y)
~2 N + i/~
1 .] • 127 + 2 v(oi )(x)
(2.39)
Therefore, on the "physical orbit" X(¢2), defined by (5F(qL X) _ 0 .
(2.40)
8x(.v)
one may simplify eq. (2.38) to read 82F(~b,X) ] 6X(x)fiX(.V)
=N[IA02)(X ) .. 2ihV~2)(X)_. - 1. ~)2)(X) x~¢,2) [] + 21AJ)(X)
X 6 n ( x • .v).
(2.41)
This has the same structure as q~4 theory [ 1,2,4] even though we have allow-~d arbitrary V0(X). The results of this section give a natural setting to the Hartree approximation of arbitrary O(N) symmetric scalar field theories without derivative couplings, in that the I / N expansion is a systematic approximation scheme in which the l-lartr.'e approximation is the leading term. Note that we have not specified the detailed nature of the interaction, no, the dimension of space-time. Examination of eqs. (2.34)(2.41) shows that this entire class of theories has a number of common featmes, iladependent of the details of the interaction.
3. Renormalization
Tile effective action and Green functions inwflve a number of divergent integrals which require renormalization. We shall show that eq. (2.30) is super-renorn alizable in two and three dimensions. In four dimensions if V0(X ) is (a) a quadratic polynomial in X the theory is renormalizable, as is already known; if (b) a polynomml in X of degree three or greater, the model is non-renormalizable even to leading o d e r in N, since a V0(X) of a finite degree in X induces logarithmic divergences in eff-~ctive vertices which are not present in the original Lagrangian. It is convenient to treat two, three, and four dimensions separately in what follows. 3.1. Two dimensh)ns
In two dimensions the theory described by eq. (2.1) is superrenormalizable, only requiring normal ordering, which is accomplished here by subtracting a logarithmically
306
H.J. Schnitzer / The I /N expansion
divergent constant from ×. We now fommlate our renormalization procedure and show that it leads to a finite theory. Consider oo
with the Vl0k)(0) independent infinite constants. Now define XR(Y) by ×(y) = XR(y ) + c ,
(3.2)
where c is a dimensionless infinite constant to be determined. Define VR(XR) = V0(X R + c)
=i~07(×R)SVIR)(0) "
(3.3)
As our renormalization condition we require that VIR/)(0) = finite for all J, which will make tile theory finite in leading order. That is =
= ~ k=0
1 ,.kv(]+k)(o-~
~ -0
"~"
for all/
•
= finite , (3.4) renders (2.30) finite in two (and three) dimensions. Let us begin with the effective potential. Our condition (3.4) trivially guarantees that (2.34) is finite. On the other hand, (2.33) becomes
3cl'°(4~'X)- NV(R2'(×R){×R- q$2/N+c--(h/4n)lnr 6X
A!
-l].
(3.5)
L2 V~)(XR)AJ
If we choose c - (h/4"n)In(A2/M 2) = CM = finite ,
(3.6)
then eq. (3.5) is also finite. Then carrying our the integrations in (2.31) one obtains, using (3.2) and (3.6) cl)(~, X) = N[ VR(XR) -- (XR + C M - oZ/N) V(1)(XR)] V(1)(XR) In
--1
+ constant,
(3.7)
H..I. Schnitzer ,/The I /N expansion
307
so that Q,~(0, X) = Q'~R(0, XR) + constant ,
(3.8)
where C~R(0, XR) is the finite, renormalized effective potential. Notice that CM and therefore XR depend additively on lnM 2, so that without loss of generality wc may choose Crf = 0, which we do in what follows. This renormalization scheme also makes the Green functions finite. First // In (A2/M 2 ) c=;~
fi
f
A2 d2p
d b
I
(2rr)2 p2 +M 2
,,)
....
.._,_ I-]v +/1,12 " _ '
which shows that choosing CM = 0 still leaves a dependence on M 2 in the theory. From (3.2), 1 X('Y)= I)'IXR - i/~ ~]'+~M "v~ .
(3.10)
We now rewrite (2.39), by using (3.3) and (3.10), us
_ ;5 ra(0, x~)
8xv,(),)
(3.11)
which is clearly finite in two- (and three-) dimensional space-time for any' XW y ) and 0 2 ( r ) . Similarly
81"(0, x)_
[K3~,+2IA~)(XR)I0~(y)
~o~,(),)
8 FR(0, XR) ~ --~0;(~'i -
(3.12)
which is also finite for all XR(Y) and 0 2 0 0 . Given the finiteness of (3.11) ant (3.12) ti)r any XR and 02, this establishes tile finiteness of tile renormalized effective action FR(0, XR). Since functional integration of (3.11) and (3.12) gives
|'R(qS, XR) = r(o, x) + constant,
(3.13)
H.J. Schnitzer / The I/N expansion
308
we have succeeded in showing the adequacy of our renormalization procedure. Of particular interest are the renormalized two-point functions, obtained from (2.36)-(2.41), and (3.11), (3.12). We find 82FR(0, XR) 8%(x)6¢bO, ) -
[E3x+ 2V(RI)(XR)]fab82(X - y),
(3.14)
621"R(qS, XR) 5¢)a(X) tSXR(.}')- _2V(R2)(XR)~a(X) 62(X . y),
(3.15)
62 I~R(qS,XR) 6XR(X) 6XR(Y) ' V(2)(XR)Uly+ 2V(RI)(XR) I =N ( V(2)(XR)II- 2i, I>, I [~>'+2V(R)(XR) ~2 /y [[]y +2V(1)(X 1 + V~3)(XR)i XR---~+il~ ) (3.16)
If we set (5 rR(~b , XR)
8XR(y)
(3.17)
--0,
and insert into (3.16), we arrive at the renormalized version of (2.41), namely 62rR(~b' XR) ]
=N(IAR2)(XR)_2ihI~R2)(XR)
I[]y+ 2 V(1)(XR)
I-qy+ 2 I/{1)(XR)
(3.18)
where the subscript indicates that (3.11) and (3.17) are satisfied. Note that (3.14), (3.15) and (3.18) have the same general structure as 04 theory, even though VR(×R) is arbitrary.
3.2. Coleman's theorem The renormalized effective potential in two dimensions is given by (3.8). From (3.8) or (3.12), ,9q~R(&, ×R) 0% - 2V(R1)(XR)~a ,
(3.19)
H.J. Schnitzer / The I /N expansion
309
where XR satisfies the gap equation (3.20) Spontaneous symmetry breaking requires =
o,
but (3.20)shows that this cannot be satisfied for any finite value of XR, independent of the choice of M 2. This is Coleman's theorem [11 ] as examplified by this model. Therefore the ground state occurs at ~ = 0. Since
(3.21)
m 2 = 2 IARI)(XR)IXR(0) > 0 is the elementary meson (mass) 2, x
(o) : -
In[ ''2
(3.22)
\M2]'
this exhibits the M 2 behavior of XR, since rn 2 is observable. If we combine (3.8) with (3.20), we obtian Cl~R(~2 ' XR (~2)) = N [ VR(XR) +/7/47r IARI)(XR) l ,
(3.23)
where of course XR satisfies (3.20). 3. 3. Three dimensions
To leading order in I]N, the model in three dimensions is also superrenorlralizable so that the results and discussion of subsect. 3.2 can be taken over directly to three dimensions. The only renormalization required to leading order in N is again r ormal ordering. In this section we only take note of the differences that distinguish lhe models in two and three dimensions. The renormalization procedure is that described by eqs. (3.1)-(3.4). Specializing (2.33) to three dimensions, we obtain
aq~(~, x) - -WV(R2)(XR){X ax
R - (a2/N
+c-/~/27r 2 [A - ½ 1 r ~ - ) ( X R ) ] } ,
(3.24)
which means that in three dimensions c = -h/2rr2A + finite constant
(3.25a) (3.25b)
310
t1..I. Schnitzer / The I /N expansion
which is the three-dimensional version of (3.9), (3.10), where M 2 is an arbitrary renormalization mass. The effective action is finite, as demonstrated by combining eqs. (3.11) (3.18) with (3.25b), since the resulting equations are finite in three dimensions. Thus, the model is superrenormalizable "in three dimensions in leading order in I / N , for any VR(XR). The renormalized effective potential is QPR(¢ 2, XIO = N {VR(XR) - (X n - 3t - O2/N)V(~)(XR) - t(127r[ 2V(1R)(XpO]312 }, (3.26)
so that 3 Q~R(O 2, XR) 3X R
-
_N(X R
31
02IN) V~2) (XR).--N/I/4n" Vg,2}(XR)[2 V~ )(XR)I 1/2 (3.27) (3.28)
~Oa The condition for spontaneous symmetry breakdown is
(3.29)
V~~)(×R) = 0 . which is permitted in three dimensions. The gap equation is XR(q~2) = 31 + ck2/N -
r,/47rv~IY~)(XR),
(3.30)
so that °PR(~2, XR(~b2)) = N { VR(XR) + ~'l/24rr[ 2 IAI)(XR) 13/23,
(3.31)
which should be compared with eqs. (3.20) and (3.23). 3.4. N o n - r e n o r m a l i z a b l e
theories
It has been suggested [6] that the 1/N expansion might be useful in making a systematic resummation of non-renormalizable theories. The idea is attractive, particularly if one believes that non-renormalizable theories are not flmdamentally flawed [5], but that the proliferation of arbitrary parameters is due to an invalid perturbation expansion in the coupling parameters of the model. Certainly the 1/N expansion offers a possible alternative. We found that to leading order in N, in three dimensions, the theory behaves like a superrenormalizable theory, for any VR(×R). This situation deteriorates in higlier orders of I / N in three dimensions, and no special advantage for non-renormalizable theories appears in four dimensions. However, in sect. 5, we shall show that the large 02 behavior of the effective potential qYR(02, XR(02)) is no worse than renormalizable theories in both three and four
ILL Schnitzer / The I/A" expansion
311
Fig. 2. A divergent contribution to an induced effective @10 coupling, appearing in or~ er I/N, in three-dimensional @8 theory.
g# Fig. 3. A divergent contribution to an induced effective 0 a coupling, appearing in lead ng order, in four-dimensional ¢~6 theory.
dimensions. Since there is generally a o n n e c t i o n between tire large <~2 beha ¢ior of the effective potential, and the large p2 behavior of the Green functions, our results might be construed as giving encouragement to this set of ideas. If
V(O/)((p2/N) = 0
{
for j ~> 4
ira three dimensions,
for/>~ 3
in four dimensions,
(3.32)
the O(N) model is in fact renormalizable to all orders in 1/N. (This depends :m naive power counting, and has been checked explicitly [12] to the first two order:; in I/N.) If (3.32) is not satisfied, but V0(q~2/,'V) is a polynomial ol limte degree, the next-toleading order m I/:V is non-renormalizable in both three and four dimension; due to induced couplings. As we have shown in subsect. 3.4, in three dimesnions, h. leading order the model is superrenormalizable. I lowever, as a specific example in fi~.:. 2, the next-to-leading order ira 1/N has a divergent effective 010 coupling induced, ,equiring counter terms which cannot be absorbed in tire original Lagrangian. If (3.32) is not satisfied in four dimensions, the model is non-renormaliza ~le even in leading order in 1/N. For example, in 06 theory in four dimensions, a divergent effective ~8 coupling is induced in leading order, as illustrated in fig. 3.
4. Particle spectrum The particle spectrum obtained from the Green functions computed to le~.ding order in N is the elementary meson, and whatever bound states or resonance,, appear in the X propagator. Tile renormalized Green functions presented in sect. 3 allow us to examine tile propagators for botmd state, resonance, or tachyon poles. It is convenient to analyze the two- and three-dimensional cases separately. An extendve analysis of the O(N) model in four dimensions is already available [4] and will not be repeated.
312
lt.J. Schnitzer
The 1/5I expansion
4.1. Two dimensions Tile renormalized q~ - X inverse matrix propagator, given as the momentum space version of (3.18) is k2+rn 2
0
D l ( - k 2 , ¢ , X R) = ( 0
- N { IflR2)(XR)+ 2~i [V(R2)(X R)]2B2(--k2, m2)} )
(4.1) for Euclidean momenta, as evaluated at the ground state. The quantity B 2 ( - k 2, m 2) is given by tile integral over Euclidean momenta, B2(-k2 , m 2) = f
d2p
l E (2rr) 2 [p2 + m 2] [(k +p)2
+m21
(4.2)
The elementary meson (mass) 2 is m 2 = 2 V(1)(XR) > 0 ,
(4.3)
which must be strictly positive in accord with Coleman's theorem. Let us define an effective four-poi:.: .'oupling parameter by X 3! - 2 V(R2)(XR).
(4.2)
Notice that two-point function and meson-meson scattering amplitudes only depend on two parameters, m 2 and X. The integration in (4.2) can be carried out, with the result B2( -k 2, m 2) = ~ ( k 2 + 4 m 2 ) V - - k 2
In
2m
for k 2 ~ 0 , (4.5)
which has thc properties for k 2 > 0, m 2 4: 0, B2(--k 2, rn 2) > 0 , B2(0 ' rn2) =
1 4rim2 '
B2(-k2, m 2) is monotonically decreasing for increasing k 2 > 0 .
(4.6)
Combining these results with (4.1), we have --12 _-1,.7~._~ l)xx (- k2, m2) = l + ~ X B 2 ( - k 2 ,
m2 ) .
(4.7)
H.J. Schnitzer / The I/N expansion
313
Since -12
Dx.×l(0 ' m 2 ) : I +
N~,
~
(4.8a)
24rrm 2'
- - 1 2 D - l ( - - k 2 m 2) /V'A --x'x" "" ' k2 ~
,1 ,
(4.8b)
. -1 (- k 2, m 2) is free of zeros for Euclidean k 2 > 0. tachyons are avoided in Dxx lfDxx This puts restrictions on the physical parameters of the theory, i.e. 1 + --
< 0
forbidden domain ,
(4.9a)
> 0
allowed domain .
(4.9b)
24rrrn 2 I + -
fix
24rrm 2 (In four dimensions there is an analogous forbidden domain [4].) If the gap equation is multivalued, one must choose real solutions satisfying (4.9b), which is similar to the situation in four dimensions [3,4]. Since 53r
8¢a(x)6~b(Y)SX(z)
-~, vac = -~.
6ab82(x -
Y)62(x - z ) ,
(4.10)
the singularities of Dxx are observable in the meson-meson scattering amplitt.,de
Tab, cd(S, t, u) = ~6 X2{Dxx(S, m2)Sab5cd + crossed terms} ,
(4.11)
where s, t and u are tile usual Mandelstam variables, and s = - k 2 is the Minkcwski continuation of k 2. Continuing (4.5) to Minkowski momenta 1 1 {2Vs-4m21nfX/s+x/s--4m21 2rrs_4m 2 s ~-
B2(s'm2)= --1T/'V
S -
/
2rrs_4m 2 -2
+rrl/4m~z-s I r s J
for s > 4m 2
for 0~
(4.12a) arctan (4.12b)
with limiting values
ReB2(s,m 2)
~ - ~ 1 [ I n s - in], £-÷ +oo "/IS
(4.13a)
tt.J. Schnitzer / The I / N expansion
3 14
B2(s,m 2)
(4.13b)
, +~, S +4m 2
( 4 . 1 3c)
1
4n-m 2
x-,4m2+
B2(0 , m 2) . . . . .
1
.
(4.1 3d)
47rm 2
We plot Re B2(s, m 2) versus s in fig. 4. From eqs. (4.12), (4.13), and fig. 4 we can deduce tim bound-state and resonance structure, which is restricted to the identity representation of the O(N) group. For the allowed domian of parameters (4.9b), we obtain 0 "< 3~-' ~< 47rm 2
no resonance or bound state ,
(4.14)
0 < 47rm 2 --~~
resonance ,
(4.1 5)
_ fix
0 > _--=->~ .-4rim 2 3~
bound state
(4.16)
l
Therefore, comparing (4.14)-(4.16) with (4.9a), we observe that the tachyon occurs for coupling parameters X which are too negative, generating a bound state whose binding energy e x c e e d s 2m. Tile masses of the bound state and resonance is trivially found from tile zero of Dxl(s, m 2) for s > 0; we do not elaborate. 4. 2. Three d i m e n s i o n s
Tile behavior of tile two-point function in three dimensions is very similar to that in two dimensions, but now symmetry breaking is possible if the parameters of the
Re{B,(]
_1/
7 4Tin e
4mz I
S ...--"
-I 4 ~r m-"i'-
Fig. 4. A graph of Re B21s, m 2 ) l,er~us s, where B2(s, m 2) is given by eqs. (4.1 2) and (4. I 3).
II.J. Schnitzer ,/]Tte I/V expansion model satsify eq. (3.29). The ~
315
X inverse propagator is
gk 2 + m 2
I ;~a x
\
~5:\'X[ 1 + ~hX133(-k 2. m2)]
O I( k 2,~,XR )=[
(4.17) -~0b ~
where we have made use of definitions (4.3) and (4.4), with B3( k 2, m 2) the threedimensional analogue of (4.2). One finds B3 ( k2 ' m2 ) _ 47r ~/k 1 2 sin 1 ]//4 n ~k2"+ k 2
1
for m 2 :¢: O,
for m 2 = 0 ,
8~/k2
(4.18a)
,'4.18b)
for Euclidean k 2 > O. On continuation to Minkowski momenta
1
r 2,,, +,/s q
B3(~,m2) GU,, l
4rr,,/s
L,/g,,i-
s
for 0 ~< s ~ 4m 2,
(lnr2m_+x/s']+,ilr} Lx/J-s -4-STriA ~
t
for s > 4m 2 ,
(4.1%)
(4.19b)
ifm 2 :¢: 0, while
B3(s, O) . . . . . . . 8x/
1
for s < 0 ,
(4.20a)
s
i
for s > 0 ,
---8,/2
(4.20b)
it" m 2 = 0. The following limits are useful: B3(0" m2 ) _
B 3 ( s ' m 2 ) -S
1
(4.21a)
g a"1;~1'
,4m
2 -
s-.4m:
1 inIAm2 .... .~] 'l--6zrt~ ~ 16m2 7 '
L \
(-l.21b)
16m 2 /
We plot ReB3(s, m 2) in fig. 5. The analysis of tile spectrum is separated into two cases: Case l: broken ,9'mmetrv. The requirement for spontaneously broken symmetry
316
H.J. Schnitzer / The I/N expansion
4 Im'
Fig. 5. A graph of Re B3(s, m 2) versuss, where
s
B3(s,rn2) is given by eqs. (4.19) and (4.21).
m2 = 0
(4.22)
as inferred from (3.29) and (4.3), which is Goldstone's theorem. In this case we may choose % = 0 for a = 2 .... , N and q~l :~ 0 for the vacuum expectation values of the scalar field, giving a residual O(N - I) symmetry. Since for m 2 = 0,
-12 NX
detD_l(_k2 ~1, ×R) ; k2 + - ~ X X / k 2 + ~X
'
(4.23)
i
the absence of tachyons requires det D -1 not to vanish for k 2 > 0, so that we have X> 0 ] m2 = 0 J
allowed d o m a i n ,
(4.24a)
forbidden d o m a i n .
(4.24b)
X< 0 / m 2 =0 J
In the Minkowski region, for s > 0,
7¢~2 det D-l(s, c~l, XR) = - s + ~ (C2~N) + ~tt~ ix/s ,
(4.25)
which develops a zero at 2 l ¢1 ..~ s o = -~X ~ . - - 0 ,
(4.26)
giving the (real part) of the (mass) 2 of both the o meson, and a resonance in the identity representation. Case 2: O(N) invariant vacuum. Here % = 0 at the ground state. Then D -1 (s, m 2)
ILJ. Schnitzer / The I /N expansion
317
is block diagonal and
D~(--k 2,m2)=-~NXII+~hXB3(
k 2,m2)1 ,
(4.27)
for k 2 Euclidean. From eqs. (4.18), (4.19) and (4.21), -12D_l(0, tzX N-X- xx" m2) = 1 + 4 - 8 ~ ,
(4.28)
and -12
NX D
( - k 2,m 2)
+1,
(4.29)
k 2 ~ +oo
we obtain 1+~ 1+~
fi?, hX
< 0
forbidden domain ,
(4.30a)
> 0
allowed domain .
(4.30b)
(Compare with eq. (4.11).) Further, from (4.27) continued to the Minkowsl.:i region, we have 0 K fix < oo
no resonance or bound state,
mix -87rm <---~-.~< 0
resonance and bound state.
(4.31 a) (4.31 b)
(Compare eqs. (4.14)--(4.16).) The masses of the bound state and resonance are obtained from the zeros o f D x l ( s , m2).
5. Large q~2 behavior We can obtain the large q52 behavior of the effective potential for this clas., of models, to leading order in N, from the gap equation. Although the limit (~2/5V >> I is outside the domain of tile validity of the 1/3,' expansion, a number of interesting results are obtained, particularly for non-renormalizable theories, which justiqes presentation here.
5.1. Two dimensions From (3.20) it is clear that XR(~2 ) -' @2 ,
* cp2/N + O(ln ~2/M2).
(5.1)
318
tt.J. Schnitzer / The 1/N exl~ansion
Combined with (3.23) this gives C)?R(O2. X R ( O 2 ) ) ~ ,¥ I/R(02/'V) 0 2 -. oo
+ ....
(5.2)
Thus lhe efl'ective potenlia[ increases with 4)2 like the classical interaclion. This is as expected ['or a superrenormalizable theory. 5.2. Three dimensions: rem~rnzalizal)le theories
The ,gap equation is given by (3.30). If we restrict ourselves to renormalizable lheories, then VR(XR)[V~)(XR) ] is a cubic (quadratic) polynomial in XR, so that ?
Vd b v
--R
'
t~'R "
3r/X 2
~
(5.3)
X.R-, oo
say, m which case
x|d~2)
, O2/;v 02
t~14~: x/~xR.
(5.4)
.
(5.5)
.~
Of
xR(~2)~=
,_ .....
. . ° 2. / N . . . .
1
+ r,/a,~v%
tlence from (3.31 ) -,.-I ,'~ t:n,.,,3R +/~/247r[3r/XR] 3/2} ÷
~ R ( ~ 2, xa(¢,2))-. ---~"
A"rt
[1 +///4rr(37r) 1/2 ]
(02/N) 3 + ....
...
(5.6a)
( 5.6b )
Notice that we may t~,ke that l i m i t / / ~ 0 without encountering a singularity, consistent with the superrenormalizability of the model in leading order. 5.3. Three dimensiotzs." mm-rettormalizable theories
If the theory is non-lenormalizable, then VR(XR) is quartic or higher in XR. We rewrite (3_]0) as (4n'/h)2 (XR
M
02/N) 2 = 2 q~I(XR) .
(5.7)
This equation can be solved for XR(¢2), but with the highest power of XR coming from IA,I)(XR)in this case. Thus
2 |AI~)(XR) ~ 2 7
(4rr/h)2(02'/'\;)2 "
{5.8)
I1..I. Schnitzer / The 1/N expansion
319
Since V~)(XR) is of degree cubic or higher, (5.8) implies XR(~02)/~52
0 2 _•
(5.9)
) 0.
Therefore,
VR(XR)
)"constant XR(.~2) V~)(XR) --* constant XR(O2) (02/N) 2 ,
(5.10)
Combining (5.8) -(5.1 0) with (3.31 ) we have tieR(02, XR(02).) -----+ N{const XR(02) (~2/N) 2 + fl/247rl(4Uh) 2 (~b2/N) 2 ] 3/2 +... }, -"N't - S tr3h2 r 2 . (~52/N)3 + . . .}.
(5. l 1a) /5.1 lb)
Therefore, the large ¢2 behavior of the effective potential also increases as (~b2/N)3, just as for a renormalizable thoery in three dimensions. Itowever there is a crJcial difference, since (5.1 l b) is independent of the parameters of the model, in con trast to (5.6b). Given a specific theo,y we can also give the behavior of the correction term to (5.1 lb). Suppose
xj VR(XR) : j~0 {2/~. (XR)/'
k ~> 4 .
(5.1 2)
Proceeding exactly as in (5.7)-(5.11), we find
O/YR(~52,XR(q52))~..
,~,[ 877
Ii l-~l(kl)~2] 2k/(k- I))
+} ,,.,3)
Note that one cannot take the limit Xk -,- 0, or ~ --+ 0 in (5.13), a feature wile i distinguishes this from renormalizable theories. Thus eq. (5.13)is dominated by the quantum corrections, in contrast to (5.6). 5. 4. I,'our dimensions." non-ren(~rmalizable theories
We obtain tile unrenormalized effective potential in four dinmnsions from 12.3 I), whereby
320
t1..I. Schnitzer / The I/N expansion +--[IA01)(×)12 32~v2
{ + 21n
(5.14)
with gap equation
xc~2) : ¢2/,v
~
ffo~)(x) } + 2
167r 2
L ~ dj
.
(5.15)
We now study the limit ~2/N ~ oo, with A 2 fixed. If the theory is non-renormalizable, the Vo(X) is of degree three or higher in X, so that the quantum corrections dominate the large 42 limit. Therefore,
167r2
V!,I)(X ) ~ - (*2/N) u - ~2..~ ~ + 21n[2V(ol)(X)/A2]
(5.16)
Since d°d (42, X(~2)) = IAOl)(X) d02
(5.17)
we obtain ----.-. dq~2
--16n2¢2/h/V 2
--8rr2 ¢2/FjV lnq~2
(5.18a) '
(5.18b)
This is identical to the result obtained for tile renormalizable O(N) model in four dimensions [4], where the large 42 behavior of the effective potential is also dominated by quantum corrections. The overall result of this section is that the effective potential always has the large 42 behavior of a renormalizable theory, at least to leading order in N, for all models considered, whether renormalizable or non-renormalizable. Since the renormalization group describes both the large ~b2 behavior of the effective potential, and tile large (non-exceptional) momentum behavior of the Green functions, our results might give some encouragement to the idea that the large p2 behavior of nonrenormalizable theories is better than naively expected.
6. Summary This paper was primarily devoted to an advance of the formalism of the 1/N expansion. Techniques were developed which allow tile derivation of the effective ac-
H.J. Schnitzer / The I /:V expansion
321
tion, to leading order in N, for all O(N) symmetric scalar field theories of the class defined by eq. (1.1), in any space-time dimension. The resulting equations for the Green functions are in fact the Hartree equations of the model. These equations were renormalized, and criteria were formulated for the avoidance of tachyons in twoand three-dimensional theories, extending results obtained earlier in four dimensions. The generality of the approach allows examination of the special class of non-renormalizable theories described by (1.1). Although the effective potential, to leading order in N, increases no faster than renormalizable theories at large ¢2, o'le suspects that the presence of induced divergent effective vertices in higher orde's in N leads to the same proliferation of arbitrary parameters that typically plague aon-renormalizable theories. We wish to thank Dr. J.S. Kang, Mr. L.F. Abbott and Mr. P.K. Townsend for conversations on this and related subjects.
References [I] K.G. Wilson, Phys. Rcv. D7 (1973) 2911; k. Dolan and R. Jackiw, Phys. Rev. D9 (1974) 3320; H.J. Schnitzer, Phys. Rev. D10 (1974) 1800, 2042. [2] S. Coleman, R. Jackiw and H.D. Politzer, Phys. Rev. DI0 (1974) 2491; D. Gross and A. Neveu, Phys. Rev. D10 (1974) 3235; R.G. Root, Phys. Rev. DI0 (1974) 3322. [3] M. Kobayashi and T. Kugo, University of Kyoto preprint KUNS 316; R.W. Haymaker, Louisiana State University preprint. [41 L.F. Abbott, J.S. Kang and li.J. Schnitzer, Phys. Rev. I)13 (1976) 2212. [5] P.J. Redmond and J.L. Uretski, Phys. Rev. Letters I (1958) 145; Ann. of l'hys. 9 ~1960) 106; T.D.l.ee, Phys. Rev. 128 (1962) 899; G. Fcinberg and A. Pais, Phys. Rev. 132 (1963) 2724; J.R. Klauder and H. Narnhofer, Phys. Rev. DI3 (1976) 257. [61 K.G. Wilson, Phys. Rev. D4 ~1973) 2911; G. Parisi, Nucl. Phys. BI00 (1975) 368. [71 P.K. Townsend, Phys. Rev. D12 (1975) 2269. [81 T.D. Lee and C.N. Yang, Phys. Rev. 117 (1960) 22; J.M. Luttinger and J.C. Ward, Phys. Rev. 118 (1960) 1417; T.D. kce and M. Margulies, Phys. Rev. DI I (1975) 1591. [9] J.M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428. [10l ll.J. Schnitzer, Phys. Rev. DI0 (1974) 2042. [111 S. Coleman, Comm. Math. Phys. 31 (1973) 259. [121 R.G. Root, Phys. Rev. DI0 (1974) 3322; P.K. Townsend, Brandeis University preprint.