Planar Diagrams E. Br6zin, C. Itzykson, G. Parisi*, and J. B. Zuber Service de Physique Th6orique, Centre d'I~tudes Nucl6aires de Saclay, F-91190 Gif-sur-Yvette, France

Commun. math. Phys. 59, 35--5I (1978)

Communications in

Mathematical Physics

© by Springer-Verlag 1978

Planar Diagrams

E. Br6zin, C. Itzykson, G. Parisi*, and J. B. Zuber

Service de Physique Th6orique, Centre d'I~tudes Nucl6aires de Saclay, F-91190 Gif-sur-Yvette, France

Abstract. We investigate the planar approximation to field theory through the limit of a large internal symmetry group. This yields an alternative and powerful method to count planar diagrams. Results are presented for cubic and quartic vertices, some of which appear to be new. Quantum mechanics treated in this approximation is shown to be equivalent to a free Fermi gas system.

1. Introduction

We present some investigations of the planar approximation to field theory calculated through a limit of a large internal symmetry. Part of the motivation for this work lies in the hope that it might ultimately provide a mean of performing reliable computations in the large coupling phase of non-abelian gauge fields in four dimensions. In addition there are some indications that such topological expansions are related to the dual string models [1]. To support these hopes we may quote the significant simplifications occuring in the large N-limit for the linear or non-linear a-models which indeed allow to discriminate the phases of broken and unbroken symmetry (even in two dimensions where the symmetry is never broken). On the other hand one has 't Hooft's solution to two-dimensional Q C D in this same limit [2]. These promising features suggest to pursue this line of reasoning and develop some new techniques.

A first part of this paper is devoted to preliminary combinatorial aspects [3].

Some of these have already been discussed by Koplik, Neveu and Nussinov [4].

The method that we have used for this "zero-dimensional" field theory, in which every propagator is set equal to unity, is not of combinatorial nature and hopefully allows for extension to genuine calculations of Green functions in a real field theory. This enabled us to solve a few counting problems the solution of which does not seem to be known.

In Section 5, we compute explicitly the contribution of all the planar Feynman diagrams to the ground state energy of a one dimensional gx4-anharmonic

* ENS, Paris. On leave of absence from INFN-Frascati

0010-3616/78/0059/0035/$03.40

oscillator. The solution may also be generalized to include the first non-planar corrections. Amazingly it is found that the problem can be restated as the one of finding the ground state energy of a one-dimensional uninteracting Fermi gas, which is of course trivial.

Let us note finally that in contrast with the true theory the planar sum is analytic near the origin in the complex coupling constant space, which reveals that the large field region of the Feynman path integral has been drastically mutilated.

2. Planar I~agrams and Large N Limit

It is known from the work of 't Hooft that the only diagrams which survive the large N limit of an SU(N) gauge field theory are planar. The planar topology maximizes the number of factors N associated to closed index loops for fixed number of vertices. This feature is not specific of Yang-Mills fields and similar ideas may be applied to study the planar approximation to a cp 3 or a cp 4 (or any interaction V(cp)) field theory. The method consists in introducing a field theory in which the field is an N x N matrix

M(x)

belonging to any of the following three sets characterized by an integer a taking the values 1, 2 or 4

(i) a = 1 real symmetric matrices, (ii) e - - 2 complex hermitian matrices, (iii) e = 4 complex matrices.

The (Euclidean) Lagrangian is chosen to be

= tr (0 uMt?u M ~) + tr

(MM t) + ~N

tr

(MM ~ MM*).

(1) The global invariance group is, respectively SO(N), SU(N), and SU(N) x SU(N).

The limit of interest is to let N go to infinity with fixed g; this selects only planar diagrams. It may be useful to state the Feynman rules derived from the Lagrangian (1). The propagators for the M-fields may be represented by double lines each one corresponding to the separate propagation of its two indices. These lines carry two different colors (in order to distinguish the two SU(N) groups) and have the same orientation in the case c¢ = 4. For c¢--2 the lines must be oriented in opposite directions. No orientation is required for a = 1. The large N-limit may be described in terms of a simple @-theory with single lines in which all non planar diagrams are omitted. The remaining diagrams are all those which can be drawn on a plane from rigid vertices and fixed external lines. For completeness let us repeat here the original derivation o f ' t Hooft, establishing the connection between planarity and large N limit. A general diagram consists of P propagators, V vertices, I closed loops of internal index. If we take an arbitrary interaction

g3 tr M 3 + g4tr M 4 + ....

there will be V 3 three-point vertices, V~ four-point vertices etc..., and

v = v3 + + . . . .

If we consider for instance a connected vacuum diagram, a simple topological argument gives

2P=3V3+4V4+ ....

Each loop of internal index may be con- sidered as a face of a polyhedron, and the Euler relation gives

V - P + I = 2 - 2 H

Planar Diagrams 37

in which H is the number of holes of the surface on which the polyhedron is drawn (0 for a plane or a sphere, one for a torus, etc .... ). The contribution of the diagrams is proportional to

gV3wiV

N I = ( 6 j 3 N 1 / Z ) V 3 ( g a N ) V 4 N 2 - 2 I t

3 ~44 . . . . . . .

Thus provided one takes coupling constants

gp

proportional t o

N ~-p/2,

the vacuum energy divided by 2"] 2 has a finite limit for the diagrams which may be drawn on a planar ( H = 0 ) surface. Corrections of order

1IN 2

are given by diagrams which may be drawn on a torus. If E~a)(g) stands for the sum of the connected vacuum diagrams for any of the three theories (1) in d dimensions and if

E(a)(g)

is the same sum for the planar e4-theory then in any dimension

• 2 d d

h m ~-N~- E~ )(g) = E ()(g). (2)

The counting rules for the lowest orders are given in Table 1 and Equation (2) may be checked from the Lagrangian (1).

Table 1. C o u n t i n g rules for the v a c u u m amplitude E(°)(g) in the planar limit, up to order three

CC)

2g

£

64g 3

2g 2

C O C X D 128g 3

Cx:::>O

16g 2

o&

256 g3 3

8

32 g3 T

It is thus sufficient to study the simpler hermitian case a = 2, which is analyzed in the following. Note however that the corrections to the leading behaviour may be different in the various cases.

3. Combinatorics of Quartie Vertices

t) Vacuum Diagrams

Setting each diagram equal to unity, apart from the overall weight, is equivalent to treat a field theory in zero dimension, in which space-time is reduced to one or to a finite number of points• It means that

e x p -

NaE(°)(9)

= lim S dN2Mexp- ½tr M 2 + ~ tr M 4 . (3)

N--* oo

The integration measure on hermitian matrices is

dN2M = [-I dMu [I

d(Re Mu)d(Im

Mij )

(4)

i i < j

and it is convenient to express it in terms of the eigenvalues ,~ of M and of the unitary matrix U which diagonalizes the matrix M. This is a well-known problem [5] and the result is

dN~M = [-I d21 1-[ (Z~- 2j) 2dU"

(5)

i i < j

Since the integrand (3) depends only on the eigenvalues 2~, this allows us to integrate over U and, up of to a g-independent normalizing factor we obtain

I g r~'4]

exp -

N2E(°)(g)

= tim S 1] d2~ 1] (2i- 2.)2 e x p - ½ 2 22 + ~ ~ ~ ]- (6)

N ~ o o i i < j J

In the large N-limit the steepest descent method can be used to compute (6), noting that the factor I-I (2~-2~) 2 in the measure requires that the eigenvalues

i < j

repel each other and spread evenly around zero. To leading order we have

1 g 2 ~ ) __ ~,ln [21_ 2j[} (7)

(in which the primed sum runs over

i#j),

and the 2,~ are given by the stationary condition

½ 2 i + 2 9 2 ~ _ Z , 1

N - J 2 i _ 2 j. (8)

The eigenvalue Equation (8) may be solved in the large N limit by going to a continuous problem. Let us introduce a non decreasing function 2(x) such that

2, = ~/N2(i/U).

(9)

Then the large N-limit may be explicitly performed and the Equations (7) and (8) are replaced by

1 1 1

E(°)(g) = I dx

[½22(x)+ g24(x)] - ~ ~

dxdyln

[2(x)- 2(y)] (10)

0 0 0

(up to a constant g-independent term) and

dy

(11)

½2(x) + 2g23(x) = io 2(x) - 2(y)

in which ~ stands for the principal part of the integral.

The condition (11) on 2(x) suggests to introduce the density of eigenvalues u(2) defined as

dx

~-~ = u(2). (12)

The function u(2) should be positive, even, and normalized to

+ 2 a

S d,~u(~)= 1. (13)

- - 2 a

The condition (11) becomes an equation for u(2)

+2° u(/~)

121 __<2a (14)

½,t+2g,~3= ~ du~_ ,

- 2 a

Planar Diagrams 39 and u(#) should vanish outside some support ( - 2a, 2a), otherwise the equation is inconsistent for large 2. The solution is easily obtained by introducing the analytic function

+2~ u(~)

F(2)= 5 d / ~ - - (15)

- 2 a 2 - - ] /

defined for complex 2 outside the real interval ( - 2a, 2a). Clearly F(2) enjoys the following properties :

(i) it is analytic in the complex 2 plane cut along the interval ( - 2 a , 2a), (ii) it behaves as 1/2 when 121 goes to infinity, as a consequence of (13), (iii) it is real for 2 real outside ( - 2 a , 2a),

(iv) when 2 approaches the interval ( - 2 a , 2a),

1¢(2 + ie) = 32 + 2923 ~- ircu(2). (16)

There is a unique function which satisfies these requirements which is

F(2) = 32 + 2023 - (½ + 40a 2 + 2922) ~ (17a)

with

120a 4 + a 2 - - 1 = 0. (17b)

The square root is defined in the cut 2-plane and is chosen to be positive for 2 real larger than 2a. The odd function F(2) has an even discontinuity

u(.;t) = 1 (3 + 49 a2 + 2022) ~ 121 _-< 2a (18)

;g

with a given by (17b).

In order to obtain E(°)(O), we first transform (10) into

+ 2a 2a

Em}(g) = ~ d2u(2)[322 + 924] - ~ d2d~u(2)u(#)tn 12- #1. (19)

- 2 a - 2 a

Then, integrating (14) with respect to 2 we replace this expression by

2a

E(°)(g)- E(°)(0) = ~ d2u(2)(½22 + 9 2 4 - 2 In 2 ) - (g = 0)

0

= ~ ( a z - 1)(9 - a 2 ) - 31o8 a z . (20)

This is supplemented by Equation (17b) giving a2 = 1

~99 [(1 + 489) 1/2 -- 1] = 1 -- 120 + 2(120) 2 -- 5(120) 3 + .... (21) Correspondingly the first perturbative terms of Equation (20) give

E(°)(g)- E(°)(0) = 2 0 - 1802 + 28803 - 604894 + 0(g 5) in agreement with Table 1.

The formulae (20) and (21) count the connected planar vacuum diagrams of the q)4-theory. It is interesting to note that it yields an expression for E(°)(g) which is

analytic in the neighborhood of g = 0 . Its nearest singularity occurs for real negative g at

gc = - 1/48 (22)

which is the singularity of a2(g); the branch point of the logarithm,

a2=0,

corresponds to g = oo. It is easy to derive from (20) the large order behaviour of E(°)(g)

c o

E(°)(g) = ~ A~(-- g)k-1

o (23)

A~ "~ 1 _ (48)kk- 7/2.

*~co 2]//rc

Finally let us note that the solution (18) for u(2) generalizes Wigner's semi- circle law for the spacing of eigenvalues of hermitian r a n d o m matrices with gaussian distributions, which corresponds to the simple case g = 0 , and hence a 2 = 1, u(2) = _1 I f ] - 22/4 [5-]. F o r g not equal to zero u(2) may be interpreted as the distribution of eigenvalues of a non-gaussian r a n d o m set of hermitian matrices.

F o r any g real, and greater than the critical gc of Equation (22), u(2) has a square root behaviour near the end points of the interval ( - 2 a , 2a). When g reaches the value go, a has increased from 1 to V~, and u(2) now vanishes as (2T-2 ]/~2)3/2 (Fig. 1).

/ \

/ \

/ \

/ \~

_ , . . . o . . .

Fig. 1. The level spacing u(2) as a function of 2/2a; solid line, g = 0, the semi-circle law nu(2)= [/~ Z)~g;

23,~/ ~)3/2

dotted line, the critical curve go= -1/48, nu(2)=~-- ~1- , normalized to the same area 2) Green Functions

The same method may be applied to derive the :planar limit of the (zero dimensional) Green functions. They are given by the moments of the distribution u(2) since

2 2

G2p(g ) = ( t r M 2p) = f d•u(,,],)• 2p . (24)

- 2 a

Consequently the generating function

co

~b(j) = ZjZPG2p (25)

0

Planar Diagrams 41

may be expressed in terms of the function F(2) of Equation (17) by noting that u(2) is the discontinuity of F(2).

The result is

¢(j) = 1/jF( I/j)

1 2g

¹

[1 2 2 )

= 2 j ~ + ~ - y l g + 4 g a + ~ .

⁽²⁶⁾

It then follows that

(2p)! a2p[2p+ 2 - p a 2 ] . (27)

Gzp(g) - p !(p + 2)!

Clearly the singularity of Gzp in the complex g plane is again given by the analytic structure of

a(g),

i.e. a branch point at g =go = - 1/48. Note also that

Gzp

does not involve any logarithm and is purely algebraic in g.

An explicit check of the formula (27) for p = 2 and 4 can be made at the first few orders in g with the help of the Table 2.

Table 2. Counting factors for G 2 and G 4 to the first few orders

0 o

0

0 o o

1 89 1692 6492 6492

A _o 8 0 o o o t - .

v

2 7 93 2893 2~09 a 2993 2893

e g

2993 27g 3 2793 29g s

X 0 > o <

2 49 329 3292 t28cj 2

3) Connected Green Functions

The usual exponential relation between the generating functionals of connected and disconnected diagrams is invalid in the planar theory. We have rather to use a Lagrange relation expressed as follows. Let ~p(]) generate the connected Green functions

oo

q~{j) = 1 + t~

G2{~)j2P.

(28)

The Green functions may be obtained in terms of the connected ones if the source

j

is replaced in tp(/) by the solution of the implicit equation

z(j) =jq~(z(])). (29)

Consequently, if we solve for z(j), then

~(/) = ~(z(/)), (30)

This defines ~v through a purely algebraic procedure and before giving the solution, it may be useful to note that (29) and (30) summarize the following relations between the connected and disconnected Green functions:

~G ~ )r~

(2p)! (G~) r~ (G]) r~ . . . ' 2q ... (31)

G 2 p ' = r ~ T r21 r~/!

Zzq~:_2;(2p+l--q~rq)I r l . .

or explicitly G z = G~

G 4 = GC4-I- 2(Ge2) 2

_ _ C C C C 3

G 6 - G 6 q- 6G4G 2 + 5(G z)

G 8 = G~ + 8G6G 2 c c + 4(G4 )c z + 28G4(G2)C c z + 14(G~)4 etc . . .

The expression (31) is the solution to the following combinatorial problem:

label 2p points on the boundary of a circle and join them in non overlapping clusters of r I pairs, r 2 quadruplets,..., rq2q-plets . . . . in all possible ways. This gives the coefficients of Equation (31). The algebraic relations (29) and (30) are of course much more tractable. The solution z(j) is very simple, since z(j)=j~b(j), and thus from (26)

1 2g

^{1 / 1 --}

2 20)

zU)= +7-7i +4ga + 7 l/fZaa2j 2. (32)

We then solve for j in terms of z and from (29) we obtain

tp(z)=z/j(z) (33)

Substituting (33) into (32) and using the (g, a) relation (17b) we end up with the cubic equation for ~p(z)

3(1 - a Z ) l p 2 ( t p - 1) + 9a4zZtp - aZz z [9aZz z + (2 + a2) 23 = 0 . (34) This equation can be further simplified if we define a new variable

yZ = ~-z u t,u l _ 2 ~ z , ~ Z _ l) , (35)

then it follows from (34) that (a 2 - 1)~p(z) can be written uniquely as

(a a - 1)~p(z) = aZ)~(y) + #(y) (36)

with 2 and # independent of a z. Indeed inserting (35) and (36) into (34) leads to four equations for the two unknown functions ).(y) and #(y), which reduce to the

Planar Diagrams 43

two equations

23 -- ~2 _~_ y2 = 0 (37a)

9y 2 - 22 2(0) = -,u(0) = 1 .

# = ~ (37b)

Solving these equations we obtain

G~p = - ~i-(a2-1)P- lAp [3p(a2- a2p

2 ) - 2 ( a 2 - t)]

A 1 = 1

( - 1)P2-p

(p+q) V

Ap+~=(p+l)!(3p+l) p/2~q<=p

~ (--4) q

(2q-p)!(p-q)!

• .

It is gratifying to observe that due to the factor

(a z

- 1) p- 1 the lowest order term of

G~2p

is indeed in 8P- 1 Once again we verify that G~p is a simple polynomial in a 2.

4) One Particle Irreducible Functions

Finally we may define one particle irreducible vertex-functions. F o r convenience we set

F(x)~.r2 X2 4- ~ F'2pX 2p ,

(38)

p = 2

_ _ c c - 4

- F 4 - G 4 [ G 2 ]

- F 6 = G ; [ G ~ ] - 6 _ 3 [ G ~ , ] 2 [ G ~ ] - 7 ( 3 9 )

- F 8 - G

_ c

8 [ G ~ ]

- _8G6G4[G2]

s ~ ~ ~ - 9 + 1 2 [ G ~ ] 3 [ G ~ ] - l o

Of course the unusual weights appearing in (39) are consequences of the planar topology. The Legendre transformation defined on a generating function ~p which would include the division by the cyclic symmetry factor 2p is equivalent to the following relations

~(])

= 1 + r ( x )

1 (40)

x = : [~p(j)- 1] . J

This is easily seen to be a summary of the previous Equation (39). Thus

F(x)

is obtained by substituting in (34) 1 + F for 74? and F for z. The result is the cubic

x equation

3x2(1-a2)(1+ F)2 + 9a4F(l + F - ~ ) -aZF(2 +a2)2=O

(41) which gives

F(x)

by an algebraic formula.

4. Combinatorics for a Cubic Interaction

For completeness we shall briefly present some results for the case of a cubic interaction which makes sense for real symmetric (~= 1) or hermitian matrices (~ = 2). Calculations will be presented in the latter case. The case of a combined cubic and quartic interaction is then a simple extension.

The integral

exp -X2e~°~(g)=

j'nd,~i

i<j[-I (21- 2i)2exp - +

1/~

^{, ] (42)}

is only meaningful for complex-g due to the instability of the cubic interaction.

However the power series in g is well defined and is the quantity of interest. We repeat the steps of the previous section in the limit of N large using the steepest descent method

dx v(2(x))= 2 d2(x ~ 2 + 392 z = ~ v(#)d#

2 - #

with the normalization condition

(43)

d2v(2) = 2.

The symmetry property 2 - - , - 2 is lost and v has a non-vanishing support in an interval 2a _< 2 < 2b. The solution is defined in terms of the analytic function

F(X)=

~, 2 - # = 2 + 3g22 - [392 + 1 + 3g(a + b)] [(2 - 2a)(2 - 2b)] 1/z (44) which has been determined by the fact that its real part on the interval (2a, 2b) should be given by (43), and by demanding that the coefficients of 22 and 2 vanish at infinity. The function v(2) is related to the imaginary part of F(2) since

F(2 + is) = 2 + 3922 N ircv(2) 2a < 2 < 2b

v(2) = - [t + 1 3g(a + b) + 392] 1 / ( 2 - 2a)(2b- 2).

)z

(45)

It remains now to express that F(2) behaves as 2/2 for 121 going to infinity. This determines the interval (2a, 2b) in terms of g by the conditions

3g(b - a) 2 + 2(a + b)[1 + 3g(a + b)] = 0

( b - a) 2 [ 1 + 6g(a + b)] = 4. (46)

Planar Diagrams 45 Thus for g small

b = 1 - 3g + 18g 2 - 162g 3 + . . .

(47) a-- - 1 - 3 g - 18g 2 - 162g 3 + ....

It is convenient to introduce the single parameter

a = 3g(a + b)

(48)

which, as a consequence of Equation (46) is the solution of

18g 2 + a(t + a)(1 + 2a) = 0 . (49)

The expansion of a as a power series in g 2 reads

1 m~ (7292)k

F(~(3k--

1))

(50)

The vacuum diagrams are then readily obtained and their generating function

E(°)(9 )

is given as

2 b

E(O)(g) = 1 ~ d).v(2)[½22 + 9).3] _ 15 f d2d#v(2)v(,u)ln t2-,u1.

2a

With the explicit expression (45) we obtain

1 a( 3a2 + 6 a + 2 ) +½1n (1 + 2 a ) , (51)

Et°)(g)

= - 5 (1 + a)(1 + 2a) 2 in which a is given by (49) and (50).

After tedious calculations, one obtains from (49)-(51)

(72fy r(3k/2)

E(°)(g)=

(k+2)!

F(k/2+

1)

k = l

which gives explicitly the number of connected vacuum diagrams with 2k vertices.

The function E(°)(g) is thus analytic near g = 0 since o- is itself analytic, up to the value of

g2

for which Equation (49) acquires a double root. This gives the closest singularity, of

E(°)(g)

for a value

2 1/(108 V~ ) (52)

gc

which gives the radius of convergence of the planar perturbation series. Again for this value go,

v(2),

instead of vanishing as a square root at both ends of its support, vanishes as a power 3/2 at one of these limits.

By the same techniques we can obtain the generating function for the planar Green functions

2 b

S (53)

2 a oo

4)(/)=

y f f @ = 1 F(1]

(54)

o 2j \ j ]

which from (44) yields

This gives the following expressions for the Gp

[(p+1)121 1 l + a k p[

+ 1 - ~ a - ( a / S y ( k - 1)!(k + 1)!(p- 2k + 1)!"

The connected Green functions are generated by

¢x)

tp(]) = 1 + Z j P G ~ ) (57)

1

given by the quadratic equation + a)(2 + 3a)

3gtP2 + ( J - 3g)~v-- ~J (1 1+2o- j s = 0 " (58) The one-particle irreducible function are more delicate to derive due to the presence of tadpole graphs. To cope with this difficulty we generalize the initial model by including an extra term Q ~ 2 i in Equation (42). By adjusting Q it enables

i

one to cancel all tadpole insertions. Consequently the expectation value of 2 vanishes and in this new theory the bare propagator remains equal to unity. This has the effect of modifying the Green functions. The new connected ones are generated by lp' satisfying

3g~v '2 + ( j - 3g)~v'-j(1 - 0j +j2) = 0. (59)

The demand that G1 vanishes leads after some algebra to the parametric relation between ~ and g

396 = - ~(1 - 3~)

9g 2 = z(1 - 2z) 2 . (60)

The expansion of z is

® ( 3 n - 2)!

z = Z 2" - ~ (n)!(2n- 1)! (9g27" (61)

1

One then deduces the one-particle irreducible functions Fv(g )

c~

r(x)= y~rp(a)x p (62)

2

Planar Diagrams 47 from the same equations as in Section 3. This gives

F 2 ( x ) - F ( x ) [ o x + x z + 39x 3] - 39x 3 = 0. (63)

Note that the coefficients of the expansion of Fp(g) in powers of 9 give correctly the number of irreducible diagrams with p external lines and a given number of vertices. For instance

(1 -- 2"C) 2 F2 = 1-- 3z

= 1 -- 9g 2 -- 3(9y2) 2 - 17(9gz) 3 ...

2"c)z(1 -- 4~) = 3g[1 + 992 + 6(992) 2 + . , ,].

F3 = 39(1 -(1 - 3z) 3

(64)

As noticed by the authors of [4] it is also simple to count one-particle irreducible diagrams without self-energy insertions. This can be achieved using a similar modification of the theory. Not only do we add the 02 term but we also change the

22 ~2

quadratic part from ~- to (1 + m ) ~ - . Again 0 and m are chosen as functions of 9 2

to give G 1 = 0 and the complete propagator G 2 -= 1. The resulting equations for the connected functions are

39~ "z + V/'[ - 39 + (1 + rn)] - j ( 1 + m) - 39j 2 - - 9 = 0 with

m = e(1 - 2c~) 902 =~(1-003 .

This generates modified irreducible functions d e n o t e d Fp(9) with

F(x) = ~ ?.(g)x"

2

F2(x) + F ( x ) [ 3 9 x - x2(1 + m ) - 39x 3] -- 39x 3 = O.

Of course by construction /~2 m 1, while for instance

m /~3 = 39"

Using the parametric relations (66) we obtain

co

3gF3 = 2 2(992) ~ ( 4 k - 3)!

(3k- 1)~k~

1

Analogous formulae hold for higher functions. They all have convergence equal to

3 g,2_ 256

(65)

(66)

(67)

(68)

(69) radius of

(70)

This is of course larger than the value given by Equation (52). Similar techniques can also be applied to the computation of the vacuum energy without tadpole and self energy insertions.

5. One Dimensional Planar Theory

We must now go beyond the mere counting problem and try to sum the planar theory corresponding to interacting systems. The simplest problem, the only one which will be solved in this article, corresponds to one dimension, where we put on the lines a propagator

1/(p 2 +

t) and integrate the momenta from - ~ to + ~ . This corresponds to coupled anharmonic oscillators, and as before we introduce hermitian N × N matrices (~ =2). The extension to real symmetric or complex matrices is straightforward. We thus consider the problem of determining in the large N limit the vacuum diagrams, i.e., the ground state energy of the Hamiltonian

H = - ½ A + V

~2 ~2 C~2

A =Z~M~-'2-'i ~ u +½,~jctRe. • M~ + glmM~

(71) and for a ~b 4 interaction

V=½trM 2+

9 t r M 4 .

(72)

We set in this limit

H,p = NZE(1)(g)~ (73)

and look for a ground state wave function symmetric under the U(N) group. Thus is a symmetric function of the eigenvalues 2 i of M, if we note that the Laplaeian is invariant under the group. The energy E (I) is then the minimum

E(1)(g) = lim I Min

i

dN2M(½(&p) 2 +

V,p 2 )

Iv-- oo 1"4 v ~ dN2Mtp2

(74)

over functions invariant under the transformation

,p(M)-, tp(UM U - i),

We rewrite (74) by eliminating the "angular" variables U as

E(1)(~) = lira ~ Min ~ I-[ d2, [-[ ( 2 , - 2j) 2 [2 ~ t~-2~] + V(2,)~ z

i i<~ (75a)

~ d~i H (]~i-

"~j)21j )2

i i<j

where we have noted that the gradient term reduces to . k ~ j and the potential l/also appears as a symmetric function of the eigenvatues 2 v This relation suggests to introduce the antisymmetric function

~()l, ...,).N)= {i~<I (t~i-- )~j)} ~])(fi~l, ...,}.N)

(75b)

Planar Diagrams 49

as if we were dealing with a fermionic problem with N degrees of freedom. The corresponding Schr/Sdinger equation

2 2

could of course also be derived directly from (71) and (72) by going over to "polar"

coordinates. Note that the substitution (75) has not generated any new terms in the potential, and as a result we have a non interacting Fermi gas where each

-~ g 24

"particle" is only submitted to the central potential + ~ .

We denote e~ <e2__<e 3 ... the individual energies corresponding to the one 1 t~2 22 g 2o 4 and define ee to be the Fermi level particle Hamiltonian h = - ~ O ~ y 2 + ~- + ~-

Then

NzE(1)= ~ ekO(eF-- ek)

(77)

N = ~ O(e F - ek).

k

This involves no approximation whatsoever. In the limit of large N we may use the semi classical approximation which reads

p2 22

f d£dp 0 e v

NaE°) = N e v - ~ 2re 2 2 N ,.d2dp,,( p2 22 g~4)

= j - ~ - v

e v 2 2 - - - "

Integration over p yields

-~)~eF 2 2 N )

(78)

"d2 [2e 2 24]3/2 [ ~2 /~4\

N 2 E ( 1 ) : N e r - - J ~ 7 [

F - 2 2 - g~-j 012eF--'~--2g-~)

td2[,, ~ 2 ~

2411/20(2eF_22__2g

(79)

rescaling 2 as I/N2 and e~ as Ne, we obtain a pair of equations giving

E(1)(g)

in the planar limit as

(~ d2 _ 22

E )(g) = e - ~ ~-n [2~ - 2g 24]

3/zO(2e - 22 -

2g 24) (80a)

1 = ~ ~ [2e - 22 - 2924] 1/20(2e -/~2 - - 2g) 4). _(80b) It is to some extent surprising to find in such an explicit manner the ground state energy for this approximation. We observe that equation (80b) may be given the following obvious interpretation from the statistical point of view. If the equation (80a) is considered as defining E °) as a function ofe and g then the Fermi level e is

obtained from the requirement of stationarity 0E ~1)

~3e (e' g) = 0.

The expression (80) defines an analytic function of g in the neighborhood of g = 0.

The nearest singularity occurs for negative g when the Fermi level just reaches the degenerate maximum of the potential

1 (go < 0) 16go

1 = ~d~2(- 8g--~l _)c2_2gj.4) 1/2

0( -18gc )~2- 2gc24) (81)

o r

g o - 3~r "

The approximate formulae (80) can be compared with the results of accurate numerical computations on the true anharmonic oscillator [6].

Table 3

g Epja..~ E

0.01 0.505 0.507

0.1 0.542 0.559

0.5 0.651 0.696

1.0 0.740 0.804

50 2.217 2.500

1000 5.915 6.694

For very large value of g the agreement gets worse. Asymptotically one finds from (80)

E(1) ~,,,~ 3,¢/313_3_3_3_~F2(3/4)]4/3=O.5899391/3

⁽⁸²⁾

planart~/] ~ 7 V L2 VTc J

whereas the exact result is E (t)(g) ~ 0.66799 g 1 / 3.

The planar approximation is therefore at most 12% wrong.

References

t. 't Hooft,G. : Nucl. Phys. B72, 461---473 (I974) 2. 'tHooft,G.: NucL Phys. B75, 461--470 (1974) 3. Tutte,W.T.: Can. J. Math. 14, 21--38 (1962)

4. Koplik, J., Neveu, A., Nussinov,S. : NucL Phys. B123, 109--131 (1977)

Planar Diagrams 51

5. Mehta, M.L. : Random matrices. New-York and London: Academic Press 1967 6. Hioe, F.T., Montroll, E.W. : J. Math. Phys. 16, 1945--1955 (1975)

7. 't Hooft, G.: Private communication

Communicated by R. Stora

Received December 8, 1977

Note Added in Proof

We collect, some explicit formulas completing the text.

Quartic Vertices. Vacuum diagrams, Equation (20) ( 2 k - 1)!

E~o~(o)_ E~o~(o)= E ( - 12g) ~

Green functions, Equation (27)

2p! ~ . . . . k ( 2 k + p - 1)!

Connected Green functions, following Equation (37) ( 3 p - l ) !31-p ~ ( - - 1 2 9 ) k ( 2 k + p - - 1 ) ! G~p(g) = (p _ t)!(2p- 1)! p ~ (k - p + 1)!(k + 2p)! "

Cubic Vertices. Generating functions of connected Green functions, Equations (57) and (58)

~ p = l + • jE ~ (_3g)V (2/~ 2)[

(E = number of external lines; V = number of vertices).

This formula was known ~o G. 't Hooft [7].