On the strong coupling expansion of the Φd2k theory

Nuclear Physics B170 [FS1] (1980) 165-174 (~) North-Holland Publishing Company

ON THE STRONG COUPLING EXPANSION OF THE ~ k THEORY J.P. ADER, B. BONNIER and M. HONTEBEYRIE Laboratoire de Physique Th~orique*, Bordeaux, France

Received 4 January 1980

The strong coupling expansion of the fl-)2k field theory with either usual or gaussian propagators is investigated in a d-dimensional euclidean space. Some simple diagrams which occur at all orders of the vacuum energy density expansion are summed. This resummation is shown to give, in the zero lattice spacing limit, good results for all anharmonic oscillators, even for k = 0o, and in all dimensions in the case of a gaussian propagator.

1. Introduction In this w o r k we are c o n c e r n e d with a perturbative expansion of the G r e e n functions in negative powers of the coupling constant which is expected to provide useful results in the strong coupling regime and is thus usually called the " s t r o n g coupling e x p a n s i o n " (SCE). It results f r o m s o m e manipulations of the F e y n m a n path integral f o r m u l a t i o n of the generating functional which require a regularizing device, such as a lattice in configuration space. H o w e v e r , in contrast with o t h e r w e l l - k n o w n studies [1] d e v o t e d to discrete q u a n t u m field theory, the lattice remains in the present a p p r o a c h m e r e l y an intermediate tool [2]. I m p o r t a n t advances have b e e n recently accomplished in this f r a m e w o r k . First, using functional m e t h o d s , a simplified derivation [3, 4] of the expansion has b e e n carried out and its algebra r e d u c e d to a set of d i a g r a m m a t i c rules [4], avoiding in particular any reference to auxiliary fields [5]. Second, s o m e rules have b e e n p r o p o s e d [4, 6] to extrapolate the h i g h - t e m p e r a t u r e expansion to zero lattice spacing so as to obtain the true strong coupling expansion of the theory. T h e present w o r k is based on the m e t h o d given in refs. [3, 4], where the S C E technical details can be found. O u r main result concerns the e n e r g y density E of a (~2k euclidean t h e o r y with a usual or gaussian p r o p a g a t o r . T h e reason for considering E is that it provides a g o o d testing g r o u n d for any SCE, as most of its properties are rigorously k n o w n in the strong coupling regime, but we think that our analysis can be e x t e n d e d to other, m o r e general, G r e e n functions. W e p r o p o s e for E a * Equipe de Recherche Associ6e au CNRS. Postal address: Laboratoire de Physique Th6orique, Universit6 de Bordeaux I, Chemin du Solarium, 33170 Gradignan, France. 165

J.P. Ader et al. / Strong coupling expansion

166

perturbative representation which takes into account an infinity of simple diagrams appearing in the SCE and which allows a proper normalisation in the free case. Its strong coupling regime is then tested in one dimension, where the results for oscillators of x 2k anharmonicity are reproduced (including k = ~), and in any dimension when the propagator is gaussian. Again, known results are reproduced, but in addition new ones are predicted. Our plan is the following. Sect. 2 contains a functional derivation of the SCE with some comments. The expansion of the vacuum energy density is given in sect. 3, where the resummation formula is also given. Its zero lattice spacing limit is taken in sect. 4 for the above-mentioned examples, the results being briefly commented on in sect. 5.

2. Strong coupling expansion: a functional derivation We consider the field theory defined in a d-dimensional euclidean space by an interaction term A ~ 2k and a free action S(~), which we write as

S(cP) =½ 1 dx ~ ( x ) K ( x - y ) ~ ( y ) d y .

(2.1)

The generating functional W(J) of the connected, unamputated Green functions is then

The SCE originates [5, 7] in expanding the exponentiated free action of definition (2.2); that is to say

Z ( J ) = N e -s(8/SJ~ O(J) ,

(2.3)

where the functional O(J) containing the interaction is

[-I This formally defines O(J) as a non-gaussian functional integral from which some regularized form can be computed. A simple derivation is provided by introducing a d-dimensional hypercubical lattice of spacing a and index site i, with the following correspondence rules: A = a a,

f ~0(y) dy = lim A )~ ~0i, ./

A=O

8(y) -A-18io.

(2.5)

i

This regularization of 8(y) induces a cut-off in Fourier space

I d q f ( q ) = lim f dqf(q), x=oo J

Iq, l<-m~X,

l~l<-d;

x=

1

am

,

(2.6)

J.P. Ader et al. / Strong coupling expansion

167

and leads to the form

Q(J)=Q(O) 12=m ° {l~i Et(tzJi)}~Q(O)exp[-~ I lnFk(la,J(x))dx ] ,

(2.7)

where

Fk(U) =--I-

t* = a ( a a ) -1/2k,

exp [-- yZk + uy] dy

oo

exp [ - y2k] d y . oo

(2.8) If the numbers {B2m} are now defined by B2m

In ek(u) = ,,,~,X ~

u

2m

,

(Fk(U) is even and Ft(O) = 1),

(2.9)

then the representation

Z(J) = N O ( 0 ) exp [-S(6/6J)] II O,n(J) ,

(2.10)

m>>-I

with Az"-IB2"

f

Ore(J) = exp [(2m)!(AA).,/k j J2"(x)

dx},

generates a series in powers of (A?t)-1/k whose terms can be interpreted as diagrams exactly as is the case for the Feynman perturbative series. In fact, for the free case (k = 1) where F l ( U 2) = e u2/4,

B2ra =

1 ~61,,,

(2.11)

all these diagrams can be summed up since from (2.10) 6Z aJ(x)

~(Q(0) exp [ - S(6/6J)] ~

Ol(J)

(2.12)

= J(x)Z(J),

(2.13)

which ensures that Z(J) fulfills I dy (K(x - y) + 2A6(x - y))

6Z

where we have used the functional identity 6 " exp [ - S(6/6J)]J(x) exp [S(6/6J)1 = J(x)- I d r K(x - y) 6J(y)

(2.14)

As the solution of eq. (2.13) is

I d(x - y)[K(z - y) + 2A6(z - y)] dy = 8(x - z),

(2.15)

168

J.P. Ader et al. / Strong coupling expansion

this shows that the terms of the expansion (2.10) sum up to the free functional with the correct free propagator. This is also true for the normalisation Z(0) as will be shown in sect. 4. Of course, in non-trivial cases the resummation of terms is no longer obvious. All the Qz~ contribute, the series conserves a A dependence, and its singular nature is manifest in view of the following facts. (i) Only m-plets of coinciding points in configuration space appear in Qm(J). Thus the operator exp [ - S(~/6J)] generates powers of K(0), i.e., powers of 8(0) - A -1. (if) The definition (2.9) simplifies the combinatorial rules given elsewhere [3] to compute the numbers {B2m}, but it also shows their large m divergent behaviour,

n2m - ( - 1)"+1F(2m + 1)IzkJ"/m,

(2.16)

where Z~ 1 is the smallest zero of Fk(U). (Although these functions have an infinity of zeros, as entire functions of non-integer order p = 2 k / ( 2 k - 1), the approximation (2.16) is good as soon as m/>2.) In spite of these drawbacks, it has been shown possible [3, 4] to extract some useful information about the strong coupling regime (h -->0o) of the series (2.10). For that purpose one has to perform the A = 0 (or X = co) limit, which is usually taken by requiring some dimensional properties of a given Green function to be held fixed [6]. We consider this program in the next sections, where for the reasons previously explained we focus on the vacuum energy density.

3. Strong coupling expansion ot the vacuum energy density The dimensionless vacuum energy density E is E = - W ( J = O ) / m d V d,

Vd = (2~')d6(0) = (2zr) dA-1 ,

(3.1)

and from the representation (2.10) it can be split into a perturbative (Ep) and a normalisation (EN) contribution, according to E = Ep- EN,

(3.2)

° ,

m a V a E N = l n { W O ( O ) } = l n I @~ exp [ - A I ~2k d x ] - l n I ~qO exp [-S(qO)] .

(3.4) We first evaluate EN, which from (3.4) is obviously infinite. It must be computed with the lattice rules (2.5), where in addition the functional measure must reduce at d = 1 to the quantum mechanical one. This gives mdEN = A -1 1.~ - [ A(k-1)/2kF(l/2k ~ 1~S~- )]~ +½ Tr In {K}

(3.5)

J.P. Ader et al. / Strong coupling expansion

169

Thus, introducing the Fourier transform/~(q2) of g ( x ) , K ( x ) = f (2d~q)d eiq~ I((q2) ,

(3.6)

and a dimensionless coupling constant g through A = gm 2k-a(k-1) ,

(3.7)

we are left with E N = 2 - - ~ f ( - [~d) qd In ~ F2(1/2k)I((q2)Xd(1-k)/k~~

J'

(3.8)

where the infinite nature of EN is explicit due to the presence of X as defined in relation (2.6) and implicit through the d-dimensional q integration where the rule [Iqtl <~ mTrX, 1 ~< l ~< d} is understood. Our next task is to compute the perturbative series Ep from its definition (3.3). Up to the fifth order, to which we restrict ourselves, only two kinds of integrals involving the inverse propagator appear, namely (3.9)

Irn--

m 2n+(n-1)d(27"/')d I

i=ll~ { ~ / ~ ( q 2 ) }

~(~/ql)

.

(3.10)

Then Ep can be written as

Ep = -½ ~ (-Y)" E,, n~l

y = xd[(1-k)/k)]g-1/kB2,

(3.11)

/'1

where ..t- -11~ t /-2 ~---d

E1 =11,

E2 = I 2 - 2x~,4-llX 3

t

Ea=Ia+~BjII2

X

,

--d t _ l _ l : ~ t l 3 v ~ - 2 d

- 8~6"1-~

,

Eg = i4 + B,a ( 2il i3 + i ~ ) x - d ± o ,6~trZ1Jtr 2.,'X v - Ed _~ o , 24 zkitr21Jr2 r ± Xv - 2d ± & o ' 8Jrr411~, v-3d ~ !2LJ ~ !2Jt~ ~ !r, 3~t 4 l~x ~ 48z~

,

Es=Is+5B~(I114+I213)X-a+ zB4 5 t2 (1112 + I113 + 71213 ) X - 2 d 2

2

2

5 t 2 t t 1 3 + ~B6 (Ili3 + IlI~ ) X -2a + 5B4B6 (igllI2 + ~1I i 1 4t 1 lt~t lr5w---4d x 3 ~ L ~ 10~t 1.,-x ,

+ 51t~tT3 T v~-3dj_

~Jt.i 8Jr 1 1 2 ~

t

)x-3d

(3.12)

with !

m

B2,~ = BEm/B2 •

(3.13)

J.P. Ader et al. / Strong coupling expansion

170

A m o n g the set of contributions appearing in the sequence (3.12), one can single out the following series:

Eo=---1 ,~1 ~ (--y)nI n =2--~m l dd I (2d~q)d l n ( l + y I ~ ( q 2 ) / m 2 ) , n

(3.14)

which from relation (2.11) is the only one surviving in the free case and whose structure is reminiscent of the one found for EN in expression (3.8). In fact, the whole series Ep can only be in an analogous form since one can recursively determine the coefficients of a polynomial p(y) such that the representation 1 7 1 (2@)~ In (1 + Tp(2/)ff~(q2)/m 2) Ep = 2m----

(3.15)

and the perturbative series (3.11) match at all orders in y. One finds, for example, at the first orders n

1Dtlr

p(y)= 1+ Z P,Y,

,it,- -- d

Pa=-~,-,4-11~

,

n~>l

P2 =

l l.~tT ,~--d

- - 4 - u 4Jt2-,'x

(3.16)

1 D I T 2 ,ld---2d - - 2 4 L ~ 6 ~ t 1,,'x •

Then, noticing from the rule (2.6) and dimensional analysis that

I , , X -a ~ I72"(q2M),

I ' X (1-')a ~ I~"(q2 ),

q u ~ mX,

X-->

oo ~

(3.17) one can rescale the coefficients (3.16) according to

p, = P,I~"(q 2 ) / m 2" ,

(3.18)

where Pn is now a dimensionless constant (not dependent on X or g). Thus one can write

e(z) =p(y),

Z = yI~(q2)/m z ,

(3.19)

and subtracting EN from El,, one ends up with

_ 1_1.__f E-2m a j

dq

~Cpizx+CkI((qzM)l k , , ~ - ~(q2)

(27r)dIn [

j,

(3.20)

Ck = 2¢rk2B2/r2(1/2k) = 2 ~ ' k 2 F ( 3 / 2 k ) / F 3 ( 1 / 2 k ) . The representation (3.20) is thus a closed form for the SCE of the vacuum energy density in the free case where P ( Z ) --- 1, and it will be checked later that it trivially gives the true strong coupling regime of E in the infinite limit of the cut-off. In the case of some interaction, P ( Z ) is known up to a finite order, nevertheless all the "free graphs" (3.14) are summed up. Thus the relation (3.20) represents a perturbation around the free energy density and we hope to obtain reasonable results from it, like

J.P. Ader et al. / Strong coupling expansion

171

a rapid decrease of P , with n and an accelerated convergence of the zero lattice spacing limits (for each n). We investigate these points in sect. 4.

4. Strong coupling regime of the vacuum energy density: examples We now turn to specific examples to test the relevance of the expression (3.20). As X and g go to infinity, all divergences must be displayed, especially the cut-off dependence of the q integration. Thus

E =½ dXe lol Oa-l ln { CkP(Z)+ =2x/-~[~dF~2 ~j

Cki((q2 = a2m2X2) I 21~(q2 = og2m20222) J dO,

(4.1)

•

The first example we consider is d = 1,/~(q2) = q2. Then the infinite limit of E must reduce to the ground-state energy for the anharmonic oscillator of hamiltonian Hk: 1

d2 +

Hk----~x2

gm

k+l 2k

x

,

(4.2)

g+oo.

(4.3)

which in the strong coupling regime behaves [8] as

E~gl/(l+k)Ek,

After integration and elimination of X in favour of Z, which from definition (3.19) is 2 = "lr2B2x(k+l)/kg -1/k ,

(4.4)

the representation (4.1) gives, in accordance with (4.3),

E = gl/(k+l)Ek(Z), Ek(Z) =12

Z

k/(k+l'{ln[CkP(Z)+Ck/Z]+ ~ _2_ _ _ _ a r c t a n ~

I"

(4.5)

One must then compute Ek from the infinite Z limit of Ek (Z). This is trivially done in the free case, where Ck = P(Z)= 2B2 = 1. The result is, as it should be for the hamiltonian (4.2),

EI(Z

= oo) = x/~ = E1 •

(4.6)

In the interaction case P(2) is perturbatively computed as shown in sect. 3. Its coefficients are found conveniently decreasing, since typically, k = 2:

P(Z)

k = oo: P(Z)

= 1 +0.0676 Z - 0.0233 2 2 + 0 . 0 0 6 0

2 3 - 0.0033 Z 4 + 0 . 0 0 4 2

Z s,

-- 1 + 0 . 1 0 0 Z - 0.0282 2 2 + 0 . 0 0 0 4

z a - 0.0033 Z4 + 0.0090

Z s .(4"7)

The limit Ek(Z = oo) is then extracted by "brute force", as one requires that P(2) have an ad hoc Z = oo behaviour in addition to the perturbative information (4.7).

172

J.P. Ader et al. / Strong coupling expansion

This is the normal procedure in the field and various methods [4, 6], which we have found roughly equivalent, have been proposed to achieve this program. A simple and efficient one is the following. (i) The Z plane cut for Z <~- 1 / Z o < 0 is m a p p e d onto the unit disc of the t plane through

t=(41+ZoZ-1)/(41+ZoZ+l),

Z=4t/Zo(1-t)2=Z(t);

(4.8)

then the points ( Z = 0, t = 0) and ( Z = ~ , t = 1) are in correspondance with

{Z -k/(k+l), Z

= oo}--{(1

-

t) 2k/(k+l), t = 1}.

(4.9)

(ii) A polynomial S(t) is built by identification in powers of t of the expansions around t - 0 of ( 1 - t)2k/¢k+l)s(t) and of P(Z(t)), where 1 2 arctan x/ZP(Z). _P(Z) =--P(Z)----~kk + x/ZP(Z)

(4.10)

This determination of S(t) uses the perturbative expansion (4.7). (iii) The previous identification (1 - t)2k/~k+l)S(t)=-- P(Z(t)) is assumed to hold up to t = 1. Then P(Z) at Z = ~ is well-behaved and Ek(Z=oO) is found to be proportional to S(t = 1). The mapping p a r a m e t e r Zo is fixed so as to stabilize the result. (One finds Zo ~ --Ck, which maps the singularity of arctan x/Z-ff(oo) at t = - 1 . ) We have p e r f o r m e d such a computation for k = 2, 3, 4 and k = oo, which is possible here since E is known in absolute normalisation (for k = oo, E no longer depends upon the coupling constant) and since all quantities converge as k ~ ~ . Notice for example that

Foo(u) = sinh (u)/u,

B2m = (-1)m÷122"~-lB'~m/m,

(4.11)

where B~,~ are the Bernoulli numbers. In such a limit the anharmonic potential is an infinitely deep square well and Eoo = ~zr2 - 1.2337. (4.12) This value, together with the ones associated with other values of k, is well reproduced in spite of the small perturbative input, as can be seen from table 1. Our second example is an attempt to go beyond quantum mechanics. In order to deal with a well-defined field theory in all dimensions we consider the class of gaussian propagators /~(q2) = m E exp

[(qE/m2)V],

"y ~> 1.

(4.13)

We apply the recipes which have been found successful in sect. 3 to give the strong coupling regime of E : the leading order in the coupling is provided by the (g, X) infinite limit with Zfixed. Afterwards one must take the Z = oo limit. Thus, X ~ - ~ 1 l n ( 1 / E v ) ( Z g l / k ) + O(ln (In g)), O/71"

(4.14)

J.P. Ader et al. / Strong coupling expansion

173

TABLE 1 Numerical results found for Ek, as defined in relation (4.3), at different perturbative orders using the m e t h o d of the sect. 4 k

2

3

4

oo

2nd order

0.925

1.000

1.053

0.725

3rd order

0.740

0.708

0.748

0.918

4th order

0.685

0.679

0.711

1.033

5th order

0.669

0.666

0.702

1.107

Exact value

0.668

0.680

0.745

1")T2 --

1.234

This quantity is linked to the strong coupling regime of the ground-state energy level for the oscillators with anharmonicity X 2k. The underlined figures are closest to the exact ones, taken from ref. [8], to emphasize the asymptoticlike character of the convergence.

R(q~) /~(q2~--exp E~Ek

(In

[(1 - 0 2v) In zgl/k],

g) l+(d/2~,),

g.->O0 ,

(4.15) (4.16)

with 1 E k = 2d

3a/2i,(ld)kl+a/2w

0d-1(1 -- 0 2v) dO.

(4.17)

The relation (4.16) is indeed exact, as has been shown [9] at least for k = 2. 5. Comments One must admit that the results given in table 1 are not fully satisfactory, since although they remain close to the true answer, their convergence looks typically like an asymptotic one. This tendency has been previously observed and remedies proposed [6], which we expect to work here, but we don't use them since we do not have sufficient orders at hand. Nevertheless they show that the representation (4.1) exact in the free case, remains a good approximation even with a strong (k--> oo) interaction. In the second example, the expression (4.1) has no "numerical" role since P ( Z ) appears only in non-leading orders, but the role of the logarithm in it is crucial for getting the correct answer (4.12), since the dimensional factor X d gives only (In g)d/2-r. These facts are thus in favour of this representation, from which some "analytic" predictions can be made, like those of equations (4.16), (4.17). One can thus hope

174

J.P. Ader et al. / Strong coupling expansion

that analogous resummation formulae can be constructed for general Green functions so as to cure some of the diseases of the original SCE concerning especially their behaviour in the external variables, as this has already shown to happen for the 2-point function [3, 4]. We want also to emphasize that a more complete study of ~4 with a gaussian propagator would be very useful for a deeper understanding of the strong coupling expansion, since the Feynman series of this regularized field theory is well-known [ 10] and can probably be summed up even in the strong coupling regime. In addition, usual ~4 theory can be reached in some limit, and this can help the investigations linked to the ultraviolet renormalisation, that we have avoided here. We thank Dr. J.T. Donohue for his careful reading of the manuscript.

References [1] [2] [3] [4] [5] [6] [7] [8]

J.M. Droutte and C. Itzykson, Phys. Reports 38 (1978) 133. R. Benzi, G. Martinelli and G. Parisi, Nucl. Phys. B135 (1978) 429. P. Castoldi and C. Schomblond, Phys. Lett. 70B (1977) 209; Nucl. Phys. B139 (1978) 269. C.M. Bender, F. Cooper, G.S. Guralnik and D.H. Sharp, Phys. Rev. D19 (1979) 1865. B.F.L. Ward, Nuovo Cim. 45A (1978) 1. C.M. Bender, F. Cooper, G.S. Guralnik, R. Roskies and D.H. Sharp, Phys. Rev. Lett. 43 (1979) 537. E.R. Caianiello, M. Marinaro and G. Scarpetta, Nuovo Cim. 44B (1978) 299. F.T. Hioe and E.W. Montroll, J. Math. Phys. 16 (1975) 1945. F.T. Hioe, D. MacMillen and E.W. Montroll, J. Math. Phys. 17 (1976) 1320. [9] G.V. Efimov, Comm. Math. Phys. 65 (1979) 65. [10] C. Bervfllier, J.M. Droufte, C. Godr~che and J. Zinn-Justin, Phys. Rev. D17 (1978) 2144.