


Journal of Experimental and Theoretical Physics, Vol. 90, No. 4, 2000, pp. 571578.
Translated from Zhurnal ksperimental'nooe i Teoreticheskooe Fiziki, Vol. 117, No. 4, 2000, pp. 659667.
Original Russian Text Copyright  2000 by Suslov.

NUCLEI, PARTICLES,
AND THEIR INTERACTION


Structure of Higher Order Corrections
to the Lipatov Asymptotic Form
I. M. Suslov
Kapitza Institute of Physical Problems, Russian Academy of Sciences, Moscow, 117334 Russia
e-mail: suslov@kapitza.ras.ru
Received November 1, 1999


Abstract--High orders of perturbation theory can be calculated by the Lipatov method, whereby they are
determined by saddle-point configurations, or instantons, of the corresponding functional integrals. For most
field theories, the Lipatov asymptotic form has the functional form caN(N + b) (N is the order of perturbation
theory) and the relative corrections to it are series in powers of 1/N. It is shown that this series diverges facto-
rially and its high-order coefficients can be calculated using a procedure similar to the Lipatov one: the Kth
expansion coefficient has the form const[ln(S K
1/S0)] (K + (r1  r0)/2), where S0 and S1 are the values of the
action for the first and second instantons of this particular field theory, and r0 and r1 are the corresponding num-
ber of zeroth-order modes; the instantons satisfy the same equation as in the Lipatov method and are assumed
to be renumbered in order of their increasing action. This result is universal and is valid in any field theory for
which the Lipatov asymptotic form is as specified above.  2000 MAIK "Nauka/Interperiodica".


1. INTRODUCTION in which the saddle-point method can be used for large
Lipatov proposed a general method of calculating N. The functional form of the Lipatov asymptote is
high orders of perturbation theory whereby these are given by
determined by saddle-point configurations, or instan- IN = caN(N + b), N , (3)
tons, of the corresponding functional integrals [1]. On
its appearance, the Lipatov method stimulated major and the relative corrections to it have the form of a reg-
discussion (see the collection of articles [2]) but was ular expansion in terms of 1/N:
subsequently cast into doubt because of the possible
existence of additional renormalon contributions [3]. In IN = caN(N + b)
a recent work [4], the present author put forward a
detailed discussion of the existing argumentation in A A A (4)
1
+ ------ 2
+ ------ K
+ + ------- + .
support of renormalons and showed that this is untena-  1
... ...
N
N2 NK
ble in the broad philosophical sense and in the mathe-
arXiv: 8 Jun 2000 matical sense: this clears any obstacles from applying Calculation of the corrections to the asymptotic form
the Lipatov method to a wide range of problems in the- provides important information on the expansion coef-
oretical physics. ficients and is an alternative to the direct diagrammatic
The Lipatov method can be used to study any quan- calculations of the lower orders: for instance, instead of
tities [5] but the starting point is that it can be applied calculating the fourth or fifth orders [6, 7], it is more
to functional integrals having the form economical to calculate A1 or A2. So far, the first correc-
tions to the asymptotic form have only been calculated
I(g) = D(1)...(M)
4
(1) in theory [8] and in a few quantum-mechanical prob-
 lems [9, 10].
exp( S0{}  gSint{}), In the present paper, we study the behavior of the
where (1), ..., (M) is a certain sample coefficients A for large K. This topic has not been stud-
i , ..., K
1 iM
from the set of integration variables ied theoretically and the only available data has been
i contained within obtained by numerical methods: for a perturbation the-
the symbol D. The expansion coefficients IN of the ory series in the problem of an anharmonic oscillator,
integral (1) in terms of the coupling constant g are Bender and Wu [9] determined the first ten coeffi-
determined by the Cauchy integral cients AK:
dg ( ) A =  = 7.0142876,
I = -------- I g
----------- 1 1.3194444, A3
N , (2)
2
 igN+1 A =  = 40.118943,
C 2 1.9385609, A4

1063-7761/00/9004-0571$20.00  2000 MAIK "Nauka/Interperiodica"




572 SUSLOV

t t strated by calculating the corrections to the Stirling for-
mula:
(a) (b)
2i 2i C'
C C (N + 1) 2NeNNN
=
0 0 C
2i ''
2i A A A (9)
 1 1
+ ------ 2
+ ------ +
... K
+ ------- + ... .
N
N2 NK
Fig. 1. (a) Saddle points and integration contour in integral
(11). (b) In calculations of the asymptotic form of A The result is well known for the logarithmic form of
K the
contour must be deformed since the point t = 0 corresponds expression (9): in this case it is possible to find a gen-
to a singularity not a saddle point. eral term of the series known as the Stirling series [11].
By calculating the exponential function of the Stirling
series using factorial series algebra [5], we can easily
A find the asymptotic form of A
5 = 305.5223, A8 = 3.65  105, (5) K:

A 2(K)
6 = 2808.09, A9 = 4.4  106, A =  ----------------------
K Re , K . (10)
(2i)K + 1
A7 = 2.995 104
 , A10 = 1  108. We shall subsequently show how this result is obtained.
The rapid growth of these coefficients indicates that the Using the definition of a gamma function and mak-
series in (4) diverges. ing the substitutions x Nx and t = lnx, we have
Another example which can easily be studied is the
zero-dimensional limit of 4 theory. In this case, the
functional integral in fact reduces to a single one (N + 1) dxxNex
NeN NN
= =

0

I(g) = dM exp(2 g4
 ), (6)

0  dxexp{N[x  1  ln x]} (11)

and its expansion coefficients can be calculated in the 0
explicit form:

M + 3
M + 1 = Ne N NN dtet exp{N[et  1  t]}.
N + -------------- N + --------------
2M/2 4 4
I = ------------------------------------------------------------------------------------ 
N (4)N. (7)
2 2 (N + 1) For large N, the saddle-point condition has the form
By isolating the asymptotic form for N and et  1 = 0 so that there is a set of saddle points ts = 2is,
expressing the result in the form (4), we obtain the fol- s = 0, 1, 2, ..., lying on an imaginary axis (Fig. 1a).
lowing expression for the coefficients A The integration contour in (11) passes through the sad-
K for K
(see Appendix) dle point t = 0 and satisfies all the conditions for valid-
ity of the saddle-point method [12]. Thus, its deforma-
2(1 eiM
+ ) tion is not required and the other saddle points can be
A = ---------------------------
K Re (K), (8) neglected. Calculating the integral in the saddle-point
(2i)K + 1 approximation yields the Stirling's formula.
whose functional form is similar to the Lipatov asymp- Formally isolating the asymptotic form, we identi-
totic form (3) but with the complex parameters a and c. cally set
In the present study, we shall show that factorial 
divergence of the series in (4) also occurs in the general (N + 1) = 2Ne NNNF(1/N) (12)
case and a universal result [see formula (47)] valid for
any field theory using the Lipatov asymptotic form (3) and making the substitution
can be obtained for the asymptotic form AK. = 1/N, (13)

we have for the function F( ) introduced by us
2. SIMPLE EXAMPLE
AND QUALITATIVE PICTURE   t
The qualitative aspects involved in the calculation of F( ) 1
= ------------- dtet et 1
exp -------------------- . (14)


the asymptotic form of A 2
K can be conveniently demon- 


JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 4 2000


STRUCTURE OF HIGHER ORDER CORRECTIONS TO THE LIPATOV ASYMPTOTIC FORM 573

Expanding (14) as a series in terms of gives the mining the parameters a and b, it is possible to have
required coefficients AK which are calculated by anal- simple structural calculations which reduce to a formal
ogy with (2): expansion near the saddle point and isolate the depen-
dence on N. We shall demonstrate these calculations for
d 1 the case of 4 theory; however, we do not need the
A = ---------------------
K  1 2
 i explicit form of the action and we shall only use its
2
C characteristic properties of homogeneity
(15)
S0() = 2S0(), Sint() = 4Sint(). (18)
   t
dtet et 1
exp --------------------  K ln
.
Similar properties of homogeneity are encountered in
 other field theories and, with slight modifications, the
This is an exact expression which for large K can be scheme put forward subsequently also holds for the
calculated using the saddle-point approximation. The general case.
saddle-point conditions yield the set of solutions According to (1) and (2), the expansion coefficients
t are given by
t s
= = ----
s 2is, s , s = integer, (16)
K dg ( ) ( )
so that in the complex plane t the saddle points are for- I = --------
N  1 D 1 ... M
2
 i
mally the same as those in the calculations of the lead- (19)
C
ing asymptotic term. However, for the integrand at the  exp( S { }  gS { }  N lng).
sth saddle point we can easily obtain the estimate 0 int
We introduce the new variable
~ exp{K  K ln K
s} ts K!, (17)
= g (20)
from which it is clear that the solution with s = 0 does
not in fact correspond to a saddle point but to a singu- and set
larity.1 Hence, the integration contour over t cannot S{} = S0{} + Sint{}. (21)
pass through the point t = 0 but must be deformed and
pass through one of the neighboring saddle points, 2i In terms of the new variable , the saddle-point condi-
or 2i (Fig. 1b) which, because of (17), gives the tions have the form
required asymptotic form of AK ~ (2)KK! [see (10)].2 S{c}
A similar situation is encountered in the general S '{ = = --------------
c} 0, gc , (22)
N
case. When the coefficients AK are calculated in the sad-
dle-point approximation, the instanton equation is the and expanding the expression in the exponential func-
same as that used to calculate the Lipatov asymptotic tion (19) as far as quadratic terms in =  c and
form. However, using the same solution as in the last g = g  gc gives
case yields a singularity rather than a saddle point
(because of the additional integration over ). There is N (, S''{ }) N
N  N lng  ---- c
--------------------------------------  --------
c (g)2. (23)
thus a need to consider other solutions of the instanton 2 S{ 2
c} 2g
equation which can be numbered in order of increasing c
action corresponding to them. If the Lipatov asymp- We use a symbolic notation, denoting the first and sec-
totic form is determined by the first instanton, having ond functional derivatives by single and double primes
the smallest action, the principal contribution to the and taking these to be a vector and a linear operator,
asymptotic form of A respectively; the variables of integration
K is made by the second instanton. i contained
within D are taken to be components of the vector .
Bearing in mind that because of (20)
3. GENERAL CASE
Calculations of the Lipatov asymptotic form (3) are = g + --------
c g c , =  c, (24)
fairly cumbersome if the aim is to find all its parameters 2g
c
a, b, and c. However, if the analysis is confined to deter- and shifting the origin , we have
1 The solutions (16) are written assuming 0 which does not hold dt
for s = 0. A similar observation must be made with reference to I = eNg N + 1  M/2 ------ D(1)...(M)
formula (38) below. N  1 c 2 c c

2 The integration contour over in (15) is conveniently drawn  (25)
slightly to the right of the imaginary axis, enveloping the left half- N
plane over an infinitely distant contour; in this case for Im < 0  1

exp  ---(, S''{ ----
c} ) t2
+ ,
the integration contour over t is shifted upward and passes 2 2
through the point 2i whereas for Im > 0 it is shifted downward
and passes through the point 2i. where we have set g = igct.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 4 2000


574 SUSLOV

We make the linear substitution S^ with  1

exp ---[(, S''{
c} ) Nt2
 ] ,
detS^ = 1, which diagonalizes the matrix of the operator 2
S''{c} and we set where f '{c} is an operator whose matrix consists of the
r columns f 'i {
D = D' d~ c} and [...]P is its projection onto the sub-
i, (26)
space of the zero-order modes. The integral over D'
i = 1 and dt is determined by the determinant of the quadratic
where we have isolated r variables of integration form in brackets in the exponential function (31) given
(denoted by the tilde) which correspond to zero eigen- by (N)det[S''{c}]P', although caution must be exer-
values of the operator S''{ cised when reducing this to a sum of squares [13]; the
c} and do not in fact appear
in the exponential function (25). In order to ensure cor- subscript P' indicates a projection on a subspace com-
rect integration over zero-order modes, the following plementary to the subspace of the zero-order modes.
partition of unit is introduced below in the integrand (25) Performing elementary transformations in (31), we
obtain a result having the form (3) where
r
1 = di(i  f i{}), (27)
1 M + r
a = --------------, b = -------------,
i = 1 S{c} 2
where i are collective variables. An example of such a
variable is the instanton center x (   2)/2
0 defined as (2) r det[ f '{ }]
c = (S{ ( + )/2 c P
----------------------------------------------------------------
c} ) M r
ddx (x) 4(x  x det[S''{c}]P'
0) = (28)
0, (32)
r
for which integration of the type (27) has the form  d (1) (M)
ic (i)...c (i),

i = 1
ddx (x) 4x
= 
1 ddx0 x0 ------------------------------- . (29)
and is the number of variables of integration con-
ddx (x) 4
tained within D [this disappears from the answer on
going over to a ratio of integrals of the type (1)].
By introducing collective variables (which can also be
the instanton "orientation," its radius, and so on [5]), Similar structural calculations can be made for the
we can confine ourselves to homogeneous functions asymptotic form of the coefficients AK. Making the sub-
fi{} [compare with (29)] where the degree of homoge- stitution in (19)
neity can be considered to be zero without limiting the
generality: if f S{ }
i {} = pfi{}, the substitution i g c
--------------g (33)
pi eliminates the factor p from (27). We linearize the N
arguments of the -functions in (27) near the saddle-
point configuration and isolating the dependence on corresponding to
the asymptotic form (3), we have
r
1 = d (  ( '
i i  f i{c } f i{c}, ))
I  + 1
N  1 = (S{c}) N exp( N + N ln N)
i = 1 (30) (34)
r  1
N(M + r  3)/2F---- ,
= d (  ( ' N
i i  f i{c } gc f i{c}, )),

i = 1 where
and select the instanton such that i  fi{c} = 0 (in (28)
this corresponds to a choice of solution symmetric rel-
ative to the point x = 0); then 1 dg
c becomes a function of F---- = N(M + r  1)/2 -------- D(1)...(M)
N 2
 i
i. Substituting (26) and (30) into (25) and eliminating
the -functions by integrating over ~ C (35)
i , we have
 S
{}
exp  N------------------ + N  N lng .
I   N + 1  (M + r)/2 S{ }g
N  1 = e N gc det[ f '{c}]P c = S{c}g/N

r Setting = 1/N and expanding F( ) as a series
 dt
------ D' d (1) (M)
ic
( i)...c
( i) (31)
2 2 K
i
 = 1
F( ) = A~0 + A~1 + A~2 + ... + A~K + ..., (36)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 4 2000


STRUCTURE OF HIGHER ORDER CORRECTIONS TO THE LIPATOV ASYMPTOTIC FORM 575

we have by analogy with (2) If c(x) is a localized solution of the equation S'{} = 0,
we know that there also exists a solution c(x) corre-
d sponding to two infinitely distant instantons
A~ ( +  )/2 dg ( ) ( ) c(x) for
K  1 = -------- M r 1 -------- D 1 ... M
2
 i 2
 i which S{c} = 2S{c}; in general a solution c can
C C (37) exist such that S{c} < S{c} < 2S{c} which yields
(43). Since the contribution (43) is larger than (41), in
 1

exp  --- S{}
------------------  1 + lng  K ln
.
S{ any real field theory the second of these mechanisms is
c}g = g S{c} the principal one; the first mechanism is only important
in various nondegenerate cases such as zero-dimen-
The coefficients A~K are simply related to the unknown sional theory [see (8)] when the solution of the instan-
coefficients AK but differ from them (see below). ton equation is unique.
For large K in (37) we can use the saddle-point Expanding the expression in the exponential func-
method, for which the conditions have the form tion (37) near the second instanton as far as quadratic
terms in , g, and and making the substitutions
S{} lng
S'{} = 0, g = -------------- c
= ---------- g = ig
c , c , (38) ct, = i c, we have
S{c} K
A~ = g  K + (r + 1)/2(S{ })M/2eK
and the function in the integrand for the saddle-point K  1 c c c
configuration is determined by the factor
(1) (M)
exp(K  K ln   dt
------ d
------ Dc c (44)

s ) ( ln gc) K K!. (39) 2 2
 
Taking into account the substitution (33), the first two
equations (38) coincide with (22) but using the solution 1
exp ---
(, S''{ }) t2
 ---- K2
 .
=  c
c, gc = 1 (40) 2 c

leads to a singularity not to a saddle point because of The number of zero-order modes r' for the second
(39). Thus, other solutions of the system of the first two instanton generally differs from r; these are isolated as
equations (38) must be sought for which two possibili- before by introducing a partition of unit of the type
ties exist. (27), giving a dependence on K having the form
1. Using other branches of the logarithm. In
accordance with (38), c is determined by the logarithm r'  r
A~ 
K ( ln gc) K K + ----------- . (45)
of g
c and thus the substitution gc with integer gc 2
gcexp(2is) is not an identity transformation: in this
case we have lngc lngc + 2is. For = c, gc = In order to find the relationship between A~K and AK, we
exp(2is) we have c = 2is/K and the contribution to substitute (36) with = 1/N into (34), make the substi-
the asymptotic form of AK is determined by the saddle tution N N + 1, and use the series reexpansion rule
points with s = 1: given in the Appendix. As a result, we obtain

A  A~
K ( 2 ) K K! . (41) A K
= -------------------
K , (46)
This is exactly the same mechanism as that used to cal- 2cgc
culate the corrections to the Stirling formula: the g where c is a coefficient appearing in the Lipatov asymp-
dependence of the function in the integrand of (19) is totic form (3) and determined by formula (32). Taking
similar to the x dependence in (11). into account (46) and performing trivial transforma-
2. Using other instantons. Let us assume that c is tions in (44), we obtain
a solution of the equation S'{} = 0 which differs from
S{ }  r'  r
c; then on account of (38) and (39) we have the contri- A c
K
K = c1 ln ---------------- K + ----------- , (47)
bution to the asymptotic form S{
c} 2

S{ } K where
A c
K K! ln ---------------- , (42)
S{c} S{ } (  )/2
c = ( { })(M + r')/2 c
ln---------------- r r'
which is larger the smaller S{ 1 S c
c}. The principal contri- S{c}
bution comes from the second instanton (see end of
Section 2) and has the lower estimate (2)(  r'  4)/2det[ f '{ }]
 c P
------------------------------------------------------------------ (48)
A  c det[S''{ }]
K ( ln 2 ) K K!. (43) c P'


JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 4 2000


576 SUSLOV

r' series (51) is much larger than the imaginary one and
 d (1) (M) should originate from contributions which are higher in
ic (i)...c (i).
the hierarchy than the last term in (50); only
i = 1 N can ful-
fill this role. Combining the second and third terms in
In these structural calculations, we used the form of brackets (50), we obtain3
the functional integral (1) and the homogeneity rela-
tions (18) typical of 4 theory; thus, the parameter M
appearing in the result (32) for b was determined by the IN = Re caN(N + b) 1 + const
(K + b~  b)

number of cofactors in the preexponential function (1). K
In other field theories, several fields of various types (52)
generally occur and the homogeneity relations differ [ ln( ) + i0]K
from (18); nevertheless, for a wide range of problems  a/a~
------------------------------------- + ... .
the result for b has the previous form (32) but the NK
parameter M has a different meaning. However, the
parameter M does not appear in the asymptotic formula The singularity on the left-hand side of (51) is associated
with high-order terms of the series [see the discussion of
(47), indicating that its validity is not confined to 4 formula (4.10) in [5]] and the form of the general term
theory: this is confirmed by the reasoning put forward given in (52) is in fact only valid for large K. Bearing in
in the following section. mind that the parameters a and b of the instanton contri-
bution have the form (32) for a wide range of field the-
4. HEURISTIC DERIVATION ories, we return to the result (47) where, however, the
OF FORMULA (47) coefficient c1 no longer has the specific form (48).
In general, factorial series have an asymptotic form We shall explain the meaning of these manipula-
with complex parameters [see (8) and (10)] and the tions. As we know, the expansion of the function f ( ) as
expansion coefficients I a power series in has a radius of convergence equal to
N are determined by the real
part of some complex expression. We shall specify this the distance between the origin and the nearest singular
for large N, point f( ) on the complex plane. For a factorial series
the radius of convergence is zero and a singularity
I should be found for = 0. Characteristic singularities
N = Re{caN ( N + b )(1 + N ) (49) generating factorial series have the form of branch cuts
+ c~a~N(N b~
+ ) + ... }, at which the discontinuity decays exponentially for
| | 0 [14] [see (51)]. It is deduced from the qualita-
taking into account the Lipatov asymptotic form tive pattern established above that (a)
caN(N + b), the unknown power corrections to it N has the form
of a factorial series in 1/N; (b) the divergence of this
denoted by N, and the contribution of the next instan- series is determined by the second instanton; (c) the
ton c~a~N (N + b~ ); the corrections to the latter and the contribution of the latter in (50) contains a characteris-
contribution of higher order instantons are shown by tic singularity generating these series. From this it is
the ellipses. Removing the Lipatov asymptotic form logical to conclude that the second and third terms in
from the brackets, we have brackets (50) form a single entity, being related to the
real and imaginary parts of the same analytic function.
This reasoning is merely based on the fact that the
IN = Re caN
(N + b)--- instanton contribution to the asymptotic form has the
functional form (3). Thus, the result (47) is universal: it
(50) is not related to the specific field theory nor to the form
of the quantity being studied (for example, single-par-
 c~
1 + + -- a~
--- N 
N Nb~ b + ... .
ca ticle or two-particle Green's function).

Assuming = 1/N, we can see that the last term has an 5. QUANTITATIVE RESULTS
intrinsic singularity for = 0 which may be attributed
to the imaginary part of some factorial series [5] We shall apply these results to the problem of an
anharmonic oscillator [9]. This can be reduced to one-
 dimensional 4 theory [15] in which the instantons can

e/ = --------Im (K + )--- + i0 K, (51)
easily be investigated (in particular, by using a mechan-
K ical analogy [16]); the localized solution of the instan-
whose substitution into (50) leads to an expansion in ton equation is unique and all other solutions are
reciprocal powers of N. It is natural to assume that the exhausted by multi-instanton configurations containing
expression in brackets (50) is an analytic function 3 The term
whose imaginary and real parts can only appear in a N also contains similar contributions from higher
instantons which are small compared with those contained in
strictly determined combination. The real part of the (52).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 4 2000


STRUCTURE OF HIGHER ORDER CORRECTIONS TO THE LIPATOV ASYMPTOTIC FORM 577

several infinitely distant instantons. Thus, as c in (47) logA
K
we need to take the two-instanton solution for which
S{c} = 2S{c}, r' = r + 1 (an extraneous zero mode 8
appears corresponding to the motion of two instantons
relative to each other). Consequently, for an anhar-
monic oscillator we have 6

1 1
A
= -------- K 4
K c1 K + --- . (53)
ln2 2

The dependence (53) can be compared with the results 2
of Bender and Wu (5) using c1 as the fitting parameter;
results are plotted in Fig. 2 for c1 = 1.4.
Higher order instantons have been little studied in 0 2 4 6 8 10
multidimensional 4 theory. An exception is the four- K
dimensional case for which an infinite series of instan- Fig. 2. Comparison between the asymptotic formula (53) for
ton solutions was obtained analytically by Ushveridze c1 = 1.4 and the coefficients AK determined numerically in
[17]. The second instanton in this series, following the [9]. The value of |A10| is given in [9] with a single significant
Lipatov one (S{c} = 162/3) has the action S{c} = digit (1  108) and the error corresponding to the range
92 which gives the result for the asymptotic form of AK (0.5-1.5)  108 is indicated in the figure.

27  3
A K
K = c1 ln ------ K + --- (54)
16 2 lations of the asymptotic form can be used to determine
three parameters a1, b1, and c1 characterizing the coef-
(we assumed that r' = r + 3 because in view of the ficient function which is equivalent in efficiency to cal-
absence of spherical symmetry for the second instan- culating the next three orders of perturbation theory.
ton, three zero modes are added corresponding to its The calculations of a1 and b1 do not require functional
rotations in four-dimensional space). Unfortunately, integrals and can be reduced to solving nonlinear dif-
there is no evidence that the Ushveridze series exhausts ferential equations: the calculations of c
all the solutions; thus, the result (54) should be under- 1 are of approx-
imately the same complexity as the calculations of the
stood as a preliminary or lower estimate. leading Lipatov asymptotic form. This is incomparably
A method of determining the complete series of easier than calculating the successive terms of a pertur-
higher order instantons numerically was proposed in bation theory series where progression to the next order
[18]. It would be desirable to use this method to check takes, on average, ten years
the result (54) and to find the second instantons in all
existing field theories. The author thanks L.N. Lipatov for discussions of
preliminary results of this work and participants at sem-
inars at the Institute of Physical Problems and the Phys-
6. CONCLUSIONS ics Institute of the Russian Academy of Sciences for
their interest in this work and useful discussions.
Expression (4) can be used to interpolate the coeffi-
cient function, by truncating the series at a finite num-
ber of terms and selecting the parameters AK to ensure ACKNOWLEDGMENTS
agreement with the lowest orders of perturbation theory
known from direct diagrammatic calculations. This This work was supported by INTAS (grant no. 96-
procedure is highly accurate and can reliably estimate 0580) and the Russian Foundation for Basic Research
the error but nevertheless is unsatisfactory in many (project no. 00-02-17129).
respects. This is because when diverging series are
summed, the analytic properties of the coefficient func-
tion [19] are significant and these are reproduced quite APPENDIX
incorrectly in this procedure: the coefficient function is Derivation of Formula (8)
assigned a multiple pole at N = 0 but the intrinsic sin-
gularity is lost at N = because of the factorial diver- Let us assume that two expansions exist:
gence of the series in (4) [see formula (51)]. Qualitative
allowance for the functional form of the asymptotic A A1 A2 A
K F K
N = 1 + ------ + ------ + ... + ------- + ..., (A.1)
N
in the form c K N2 NK
1 a1 (K + b1) enables us to select basis
functions exhibiting correct behavior for N B B B
which should give a positive effect even when the num- F 1 2 K
N = 1 + -------------- + --------------------- + ... + --------------------- + ... .
ber of fitting parameters is constant. Quantitative calcu- N + p (N + p)2 (N + p)K (A.2)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 90 No. 4 2000


578 SUSLOV

If the second series is factorial, 5. I. M. Suslov, Usp. Fiz. Nauk 168, 503 (1998) [Phys. Usp.
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A 
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(1995).
  11. H. B. Dwight, Tables of Integrals and Other Mathemat-
(N + ) N 1 2Ne NNN
= ical Data (Macmillan, London, 1961, 4th ed.; Nauka,
 (A.5) Moscow, 1977).

 ( ) e 2i
1 +
... 2 K
 ----------------Re---------------------- + ... 12. Yu. V. Sidorov, M. V. Fedoryuk, and M. I. Shabunin, Lec-
NK (2i)K + 1 tures in the Theory of Complex-Variable Functions
(Nauka, Moscow, 1976).
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