





Summing Divergent Perturbative Series
in a Strong Coupling Limit.
The Gell-MannLow Function of the j4 Theory

I. M. Suslov

Kapitza Institute of Physical Problems, Russian Academy of Sciences, Moscow, 117334 Russia
e-mail: suslov@kapitza.ras.ru




Abstract--An algorithm is proposed for determining asymptotics of the sum of a perturbative series in the
strong coupling limit using given values of the expansion coefficients. Application of the algorithm is illus-
trated, methods for estimating errors are developed, and an optimization procedure is described. Applied to the
4 theory, the algorithm yields the Gell-MannLow function asymptotics of the type (g) 7.4g0.96 for large g.
The fact that the exponent is close to unity can be interpreted as a manifestation of the logarithmic branching
of the type (g) ~ g(lng) (with 0.14), which is confirmed by independent evidence. In any case, the 4
theory is self-consistent. The procedure of summing perturbative series with arbitrary values of the expansion
parameter is discussed.




1. INTRODUCTION approach allowed the critical indices of the phase tran-
sition theory to be determined to within the third deci-
This paper presents a systematic description of the mal position [68], thus rendering the intermediate
algorithm proposed previously in a brief communica- coupling region (g ~ 1) principally accessible. However,
tion [1]. Operation of the algorithm is illustrated by test this direction was not developed further because the prob-
examples, methods for estimating errors are developed, lem of renormalon contributions arose that cast doubt [9]
and an optimization procedure is described. Using this on the applicability of the Lipatov method. The interest in
algorithm, the Gell-MannLow function of the 4 the- this field had dropped sharply and no breakthrough into
ory--the main physical result of this study--can be the strong coupling region took place.
reconstructed with a tenfold greater precision.
The abstract formulation of the problem is as fol- Expanding the theory into the strong coupling
lows. Let some function W(g) be expanded into a series region is required in many fields of theoretical physics.
of the perturbation theory in powers of a coupling con- The most known cases, related to the dependence of the
stant g: effective coupling constant g on the spatial scale L, include
the problem of electrodynamics at very small distances
and the confinement problem. The dependence of g
W g
( ) = W 
( g)N. (1)
on L in renormalizable theories is determined by the
N
equation
N = 0

The first several expansion coefficients WN can be dg =
obtained by straightforward diagram calculations. The -------------- g
( ) = g2 g3
 + g4  ... . (3)
2 3 4
d L2
ln
high-order terms can be determined using the Lipatov
method [2], which is applicable to most of the impor- In the general case, this description requires informa-
tant problems and yields for WN an asymptotic behavior tion on the Gell-MannLow function (g) for arbitrary g.
of the type (see reviews [35]): The possible variants were classified by Bogolyubov
and Shirkov [10]. In the case of > 0, the situation
Was = caN(N + b) caN Nb  1N!. (2) 2
N reduces to the following. If the function (g) possesses
Matching asymptotics (2) to the first coefficients pro- a root at g
0, then g(L)
g0 as L 0. If (g) at large g
vides information about all terms of the series and behaves as g with 1, then g(L)
at small L;
allows the W(g) function to be approximately restored, should (g) grow as g with > 1, the theory is no
but this procedure requires using special methods for longer self-consistent and cannot describe the behavior
summing divergent series. Implementation of this of g(L) in the entire range of L.


0001





2 SUSLOV

The first attempt at restoring the function in the 4 Note that the interaction term in expression (4) corre-
theory with the Euclidean action sponds to the "natural" charge normalization, for which
the parameter a in asymptotics (2) is unity. It will be
demonstrated that the results obtained in [12, 13] are
4 1
S{} = d x ---
()2 162
-----------g4
+ (4)
not artifacts: they objectively reflect the behavior (g)
2 4!
in the interval 1 g 10. However, the true asymptot-
ics is manifested at still greater g and gives evidence of
was undertaken by Popov et al. [11]. The Shirkov group self-consistency of the 4 theory.
attempted to move into the strong coupling region [12]
and obtained for large g asymptotics of the type 0.9g2, 2. RELATIONSHIP BETWEEN W(g)
which differs only by a coefficient from a one-loop law ASYMPTOTICS AND EXPANSION
1.5g2 valid for g 0. A close asymptotic behavior COEFFICIENTS
(1.06g1.9) was obtained by Kubyshin [13], while the
more recently developed variational perturbation the- Let us formulate the problem of restoring the
ory of Sissakian et al. [14] yields 2.99g1.5. All these asymptotics
results give evidence that the 4 theory is not self-con-
W g
( ) = W , g , (7)
g
sistent.1 This is, however, rather strange from the stand-
point of condensed-matter applications, where a quite using the coefficients WN of the series (1). This coeffi-
reasonable disordered system model [16, 17] well cients, increasing at large N according to the factorial
defined in the continuum limit is mathematically law (2), are assumed to be set numerically. By analogy
strictly reduced to the 4 theory. Another argument fol- with the case of critical indices introduced in the phase
lows from the author's recent study [9] showing the 4 transition theory, the slow (logarithmic) corrections to (7)
theory to contain no renormalon singularities, which are considered as overstating the accuracy. For exponen-
can be considered as evidence of self-consistency. This tially growing W(g), which can be revealed by abnor-
situation makes revision of the above results an urgent mally large values of , the series (1) is considered
task. upon preliminarily taking the logarithm.

In this paper, an algorithm is proposed for restoring
asymptotics of the sum of a perturbative series in the 2.1. Standard (Conform-Borel) Summing Procedure
strong coupling limit using given values of the expan- Considering the sum of series (2) in the Borel
sion coefficients (Section 2). Application of the algo- sense [20], we use a modified definition of the Borel
rithm is illustrated by test examples with both known image B(g),
expansion coefficients (Section 4) and the coefficients
obtained by interpolation (Sections 5 and 6). Methods
b  1
for estimating errors and an optimization procedure are W g
( ) = dxex x 0 B g
( x),

developed (Sections 3 and 6). The problem of summing
the perturbative series with finite g is considered, and it 0 (8)

is demonstrated that knowledge of the W(g) asymptot-
ics significantly increases precision of the results (Sec- B g
( ) = B 
( g)N, B
W N
= -----------------------,
N N (N + b )
tion 7). The main physical result of this study consists 0
N = 0
in reconstructing the Gell-MannLow function of the
where b
4 theory (Section 8). The task is solved proceeding 0 is an arbitrary parameter (convenient for opti-
mization of the summation procedure [6]). It was sug-
from the same information as that used in [13], namely, gested by Le Guillou and Zinn-Justin [6] and recently
the first four coefficients of expansion of the (g) func- proved for the 4 by the author [9] that the Borel image
tion in the subtraction scheme [15, 18] is analytical in the complex plane g cut from 1/a to -
(Fig. 1a). The analytical continuation of B(g) from the
3 154.14
g
( ) = ---g2 17
------g3
 + ----------------g4 2338
------------g5
 + ..., (5) convergence circle |g| < 1/a to an arbitrary complex g
2 6 8 16 value is provided by a conformal mapping g = f(u) of
the plane with a cut into a unity circle |u| < 1 (Fig. 1b).
and their asymptotics according to Lipatov, taking into The re-expansion of B(g) into a series in u,
account the first-order correction [19]:

B g
( ) = B 
( g)N B u
( ) uN, (9)
= U

1.096 4.7 N g = f u
( ) N
= ------------- N7/2 N!1  ------- + ... . (6)
N 162 N
N = 0 N = 0
gives a series converging for any g. Indeed, all the pos-
1 sible singular points (P, Q, R, ...) of the B(g) function
It should be noted that Kazakov et al. [12] do not insist on this
conclusion, emphasizing the preliminary character of their results occur on the cut and their images (P, Q, Q', R, R', ...)
(see also [15]). fall on the boundary |u| = 1 of the circle. Therefore, the

2001


SUMMING DIVERGENT PERTURBATIVE SERIES 3

second series in (9) converges at any u < 1, but the inte- (a) g (b) u
rior of this circle is in a single-valued correspondence Q R
with the region of analyticity in the g plane.
1/a
The conformal mapping is defined by the formulas 0 g = 0 g =
P
R Q P 1 1
4 u (1 + ag)1/2  1
g = --- ------------------ or u = ----------------------------------, (10) Q'
a ( R'
1  u)2 (1 + ag)1/2 + 1

from which we readily find a relationship between UN
and BN: u u
(c) R (d)
N Q
4
K
U = B , U = B --- CN  K

0 0 N K a N + K  1 g =
(11)
K = 1 g = 0 g =
P g = 0 g = 0
(N 1). 0 1 0 1

In order to establish a relationship between asymptotics
(7) and the expansion coefficients, we will use the fact that
the behavior of UN at large N is determined by a sum of the Q' R'
contributions from singular points occurring on the
boundary |u| = 1. This can be readily checked by
expressing UN in terms of B(u), Fig. 1. (a) The Borel image is analytical in the complex
plane with (, 1/a) cut; (b) this analyticity region can be
du B u
( ) conformally mapped onto the unity circle; (c) restricting the
U = -------- -----------, (12) consideration to analytical continuation to the positive
N 2
 iuN+1 semiaxis, the conformal mapping is admitted onto any
C region in which the point u = 1 is the closest boundary point
to the origin; (d) in the extremal form (18) of this mapping,
and deforming the integration contour (enclosing the the analyticity region can be conformally mapped onto the
point u = 0) so as to make it passing around the cuts plane with (1, ) cut.
from all singular points to infinity. A singularity of the
type A(1  u/u0) at the point u0 = ei makes a contribu- and B
~ g
( ) corresponding to b
tion to U 0 and b1, we readily obtain
N of the type a recalculation formula
A eiN b  b  1
-------------- ------------. (13) 1 0
1 x g


( ) N1 + B
~ g
( ) = ----------------------- dx-------------------- B ------------ , (15)

b
(  b ) 1 + x
1 0 (1 + x)b1
0
Now we can readily find the contributions to UN from and a rule of singularity transformation at a finite (g
the singular points of the initial Borel image B(g). For 0)
or infinite points on the passage from b
power singularities at the points g = , g = 1/a, and 0 to b1:
g = g0 with g0 (, 1/a), the corresponding expres- g  g
0

sions are as follows: B g
( ) = A 
( ) --------------
B
~ g
( )
g
0

A 4
1  g +  b0
B g
( ) = Ag U = -------------- --- --------------, b1
= A(  b + b ) g0
-------------- , (16)
N 2
( )a 1 0
N1  2 g
0

A A
A (1)N B g
( ) = ----------------------g B
~ g
( ) = ----------------------g .
B g
( ) = A g
( + 1/a) U = ---------------------------------------------, ( + b ) ( + b )
N ( 0 1
4a) 
( 2) N1 + 2
(14) As is seen, an increase in b
0 weakens the singularities
B g
( ) = A g
(  g ) U at a finite point, while the character of singularity at
0 N
infinity remains unchanged. For sufficiently large b0,
2 A cos
(/2) cos(N  /2) the contributions from finite points to U
= N are sup-
-------------- -------------------------- ---------------------------------------,

( ) 3
+ pressed and the corresponding asymptotic behavior is
a sin (/2) N1 determined by the singularity of B(g) (and, hence, of
where = arccos(1 + 2/ag ) . W(g)) at g :
0
W
The singularities of B(g) change depending on the 4

U = -------------------------------------- --- N2  1, N . (17)
N
parameter b 2
( ) b
( + ) a
0 in formulas (8). For the Borel images B(g) 0


4 SUSLOV

This formula solves the problem: the coefficients UN W
are related by a linear transformation (11) to the initial U = ----------------------------------------. (21)
( +
coefficients W a ( ) b )
0
N (see Eq. (8)), while their asymptotic
behavior (17) determine the parameters W and of As a result, we arrive at a simple algorithm: calculate
asymptotics (7). coefficients BN by formula (8) using preset WN, recalcu-
Formulas (14) indicate that a contribution to U late B
N N to UN using relationship (19), and take the power
from the singular point g = is monotonic, while the limit (20) for large N to determine parameters W and
contributions from other points are oscillating. There- for asymptotics (7).
fore, increasing b0 leads to a change in the UN behavior
from oscillating to monotonic. This phenomenon was 2.3. Random Error Growth
observed in [6] and, albeit not given any satisfactory
explanation, regularly employed for improving the The above algorithms possess an implicit drawback
divergence of perturbative series. that significantly restricts the accuracy of description.
Let us introduce a reduced coefficient function:

2.2. Modified Conformal Mapping W W
F N
= --------- N
= ------------------------------
N as
A more effective algorithm is provided by using a W caN(N + b)
N
modified conformal mapping. (22)
A A
1 2
According to the Riemann theorem [21], the confor- = 1 + ------ + ------ + ... AK
+ ------- + ...,
N
mal mapping of a simply connected region into a unity N2 NK
circle is unique to within the so-called normalization, which varies within finite limits and admits a regular
which can be fixed by setting the images of two expansion in the powers of 1/N. The latter can be
(internal and boundary) points. Under the conven- checked by calculating sequential corrections to the
tion that the point g = 0 is imaged by u = 0 and g = by Lipatov asymptotics [19]. In practice, FN is set with a
u = 1, conformal mapping (10) is the only one that certain accuracy N (calculation or round-off error),
allows the Borel image to be analytically continued to which leads to a random error in U
arbitrary complex g values. However, this is not neces- N. The error disper-
sion for the algorithm considered in Section 2.2 is as
sary: to perform the integration in (8), the analytical follows:
continuation to positive semiaxis is sufficient. Then,
any conformal mapping into a region of the type N (K + b) 2
depicted in Fig. 1c is admissible, in which the point (U )2 = c -----------------------CK  1 . (23)

N K N  1
u = 1 is the boundary point closest to the origin. The (K + b )0
K = 1
second series in expansion (9) is convergent at u < 1
and, in particular, in the interval 0 < u < 1 imaging the For the round-off errors, the value of K = is indepen-
positive semiaxis. An advantage of this conformal map- dent of K. A sum calculated by the steepest descent
ping is that the contributions from singular points P, Q, method for large N,
Q', R, R' ... to UN are exponentially suppressed and the U 2N, (24)
U N
N asymptotics for all b0 is determined by a contribu-
tion of the singular point at u = 1 related to the singu- demonstrates a catastrophic growth of the error. Calcu-
larity of W(g) at g . lation with a double computer accuracy yields ~ 1014,
Let us use an extremal form of such mapping, imag- so that UN is on the order of unity for N 45.2 This
ing the plane with cut (, 1/a) into the plane with cut restricts the accuracy of determining the parameters of
(1, ) (Fig. 1d). This mapping is given by the formula asymptotics (7) to approximately 1%. According to
expression (23), an increase in b
u 0 decreases the error so
g = ------------------, (18) that the permissible N level grows. However, large b
a 1
(  u) 0
values delay the process of attaining the asymptote (20),
which leads to the following relationship between U so that no advantages are eventually gained.
N
and B For the algorithm considered in Section 2.1, the
N:
error grows at a still higher rate,
U = B ,
0 0 U ( 2 + 1)2N 5.8N, (25)
N N
B (19)
U K
= ------(1)KCK  1
(N 1).
N N  1 and the requirement of using sufficiently large b0 signif-
aK
k = 1 icantly restricts the possibility of optimization (see Sec-
tion 3). Nevertheless, this algorithm may still be useful
The asymptotic behavior of UN for large N is
2 This error growth is observed in fact in the form of rapidly
U = U  1 , N , (20)
N N increasing irregular oscillations.





SUMMING DIVERGENT PERTURBATIVE SERIES 5

UN 2
(a) (b)


~n

n
U N  1

N N opt
min Nmin

Fig. 2. The UN treatment according to the power law: (a) a typical situation whereby large N correspond to a large statistical error
and small N, to a large systematic error; (b) the plot of 2 versus Nmin at a constant number of points n.


to increase the accuracy of calculations in the region of suited to obtaining a reliable zero-order approxima-
small g (Section 7). Below we dwell on the algorithm tion.4
of Section 2.2 based on a modified conformal mapping, Treating U
which offers indisputable advantages in the region of N by the power law can involve a standard
strong coupling. procedure of minimization of 2 [22]:
The above considerations indicate that the computer 2 y  y(x )
i i
2
= ---------------------

round-off errors restrict the accuracy of the algorithm to , (27)

i
~1% even for test examples where the W i
N values are
precisely known. In real cases, the accuracy of WN cal- where yi are the values set at the points xi with a statis-
culations is much worse and the situation might appear tical error i and fitted to the theoretical function y(x).
as hopeless. However, this is not so in fact because we In this process, it is important to select properly the
mostly deal with interpolation errors, the influence of interval Nmin N Nmax for the UN treatment. Indeed,
which has a quite different character. The linear rela- large N values lead to large statistical errors determined
tionship (19) known in mathematics as the Hausdorff by formula (23), while small N values increase the role
transformation [20] possesses a remarkable property of a systematic error related to the fact that UN still did
N not attain asymptotics (20) (Fig. 2a). The
K upper limit N
Km 
( 1) CK  1 = = , , ,  (26)
0, m 0 1 ... N 2, max can be chosen using the condition
N  1 UN ~ UN, since the points with greater N provide no
K = 1 additional information; this choice is not very critical
that makes smooth errors (well approximated by poly- since the procedure of 2 minimization automatically
nomials) insignificant even despite their large magni- discriminates the points with large statistical errors,
tude.3 Of course, limitations related to the computer 2
which are used in averaging with a weight of 1/ . The
i
round-off error are still valid, but a 1% accuracy is quite lower limit N
sufficient for real problems and this level can hardly be min has to be selected taking into account
improved for the level of information accessible at the 2 value, which reaches an extremely high level for
present. small Nmin but attains a "normal" level of n  const n
Strictly speaking, the problem of round-off errors is (n is the number of points) with increasing Nmin (Fig. 2b).
purely technical and can be solved by means of special The optimum value of Nmin corresponds to the left end
precise arithmetic programs which allow the calcula- of the "plateau," where a systematic error becomes
tions to be performed with arbitrary number of signifi- smaller than the statistical error and the available infor-
cant digits [22], however, the accuracy of and W res- mation is most completely employed.
toration logarithmically depends on the computation In fact, the conditions for a strict statistical treat-
accuracy. Algorithms that are more perfect in this ment of 2 were not fulfilled because the errors UN for
respect do exist, but their consideration falls outside the various N were not independent (see Eq. (23)). This was
scope of this paper; such methods, albeit providing for
a high accuracy in the test examples, are insufficiently 4 This situation is well known in computational mathematics [22].
robust and work unsatisfactorily under conditions of All algorithms can be roughly divided into two groups: those in
restricted information. The algorithm under consider- the first group possess moderate accuracy and convergence rate
but are highly reliable (an example is offered by seeking for a
ation is quite stable and, in the author's opinion, ideally root of equation through segment halving); algorithms of the sec-
ond group show high accuracy and ensure rapid convergence but
3 This implies that, in the case when many WN values are known pose stringent requirements with respect to the function smooth-
with low precision, the data should be used upon approximation ness (e.g., in seeking a root with the forecast for several deriva-
by a smooth function rather than directly. tives).


6 SUSLOV

Accordingly, the asymptotic behavior of UN written by
2 (a)
analogy with (20) and (21) is described as

W
U = ---------------------------------------- N  1
N
a
( ) b
( + )
0 (29)
W'
+ '  1 + ...

------------------------------------------- N .
 ' a '
b
( ') b
( + ')
0
0
eff (b) First, let us neglect the correction terms indicated by
dots in expansion (29). A formal treatment of this
expression according to the power law (20) yields quite
satisfactory results because the truncated function (29)
' in the double logarithmic scale varies smoothly and is
well approximated by a straight line. However, this
approximation only leads to certain effective values of
 ' b0
U and U
.
(c)
Uexact
Note that, because of the poles of the gamma function,
the first and second terms in (29) become zero for b0 = 
and b0 = ', respectively. These b0 values correspond
 ' b0 to the purely power laws, UN N'  1 and UN N  1,
which results in increasing quality of the approxima-
tion and a sharply decreasing 2 value. Within a fixed
Fig. 3. Theoretical plots of (a) 2, (b) eff, and (c) U ver- working interval Nmin N Nmax, the pattern is as follows
sus b0 constructed with neglect of the correction terms indi- (Fig. 3): the 2 versus b0 curve exhibits sharp minima at
cated by dots in expression (29). b0 = ' and b0 = ; the effective index eff drops down
to ' in the vicinity of b0 =  and is close to outside
manifested by the fact that 2 values decreased below this region (being exactly equal to at b0 = '); the
the "normal" level (dashed curve in Fig. 2b), while the effective parameter U corresponds to exact W at
statistical uncertainty of and W became very small b0 = ' and crosses the zero level in the vicinity of the
and did not reflect real errors even in the order of mag- point b0 = -. The slope of a linear portion of the curve
nitude. For this reason, we considered the choice of near this root is
Nmin as satisfactory when the 2 values were on the cor-
rect order of magnitude (~n); small changes in N W
min did U -----------------(b + ), (30)

not significantly influence the results. 0
a
( )

which provides for an W estimate not too sensitive
3. DEPENDENCE ON THE PARAMETER b with respect to errors. The rejected terms in (29) may
0
AND ACCURACY ESTIMATION only slightly perturb this pattern.
The pattern outlined above was actually observed,
Direct application of the algorithm described in Sec- but the behavior of
tion 2.2 is insufficiently effective since the results eff and U in the vicinity of b0 =  is
usually discontinuous (as indicated by dashed branches
depend on the arbitrary parameter b0, which implies in the curves of Fig. 3). However, this circumstance is
that an additional investigation is necessary to select the not physically significant and only reflects features of
optimum value. the mathematical procedure involving taking logarithm
It is naturally expected that corrections to asymp- of the UN modulus,
tote (7) have the form of a regular expansion with ln U = ln U + (  1) ln N, (31)
respect to 1/g. However, even the simplest examples N
show that, in the general case, this assumption is not followed by using a linear fitting algorithm [22]. The
valid: in the zero-dimensional case, the corrections fol- sign of U is determined by calculating 2 for U = |U|
low the powers of g1/2; for an anharmonic oscillator, and |U| and selecting a variant with the minimum
the corrections follow the powers of g2/3 (see Section 4). value. This procedure leads to rather senseless results in
For this reason, we admit the power corrections in the the case of U
general form: N changing sign, but this is only possible
in a small vicinity of the point b0 = , while the sign
of UN outside this narrow interval is determined by the
W g
( ) = W + W ' + ... . (28)
g 'g sign of the first term in the right-hand part of Eq. (29).


SUMMING DIVERGENT PERTURBATIVE SERIES 7

Smoothness of the U(b0) function is restored when which can be considered as a zero-dimensional limit of
the treatment according to power law (20) is performed the functional integral in the n-component 4 theory.
by varying only U at a fixed (approximate) value. Here, it is easy to calculate the expansion coefficients
Small variations of virtually do not affect the position
of the root of U n + 2
n

(b0), while significantly influencing the N + ------------
N + ---

W 4 4
value determined from the slope of the linear rela- W = caN -------------------------------------------------------- (33)
tionship (30). The above considerations suggest four N (N + 1)
different methods for estimating the index, based on (i) and their behavior for large N:
the eff value at the first minimum of 2 (counting from
large b0), (ii) the position of the second 2 minimum, (iii) A
1
the change in the sign of U upon the logarithmic treat- W = caN(N + b)1 + ------ + ... , (34)
N N
ment, and (iv) the change in the sign of U
upon treatment
at a fixed value (taken equal to a preliminary estimate). where
The first two estimates ensure, in the general case, a
higher precision, since their uncertainty is determined n  1
a = 4, b = -----------,
by the ratio of rejected terms in the right-hand part of 2
expansion (29) to the characteristic value of the first (35)
term outside the narrow vicinity of b 2n/2 (n  2) 4
(  n)
0 . The accu- c = ----------, A = --------------------------------.
1
racy of the last two estimates is determined by the ratio 4 16
of the second term to the first term. When the rejected
terms in (29) are comparable with the second term (this Asymptotic behavior of the integral at g is
condition can be monitored by reproducibility of the ' described by the following relationships:
value), all four methods are on the same footing. In
W g
( ) = W , = n/4,
practice, it is always important to monitor the change in g (36)
the sign of U because this point reliably indicates the W = n
( /4)/4,
minimum in 2 corresponding to b0 =  (the number- with the corrections having the form of a series in pow-
ing of minima may change because of their disappear-
ance, appearance of spurious minima, etc. (see below). ers of g1/2. In the test, the required number of coeffi-
cients W
There are three possible estimates of W N was set with a double computer accuracy
, which use ( ~ 1014), after which the and W
either (i) the U values were
value at the first minimum of 2 or restored assuming their Lipatov asymptotics to be
(ii, iii) the slope of a linear portion of the U(b0) curve known.
in the vicinity of the root for the treatment at a fixed
(variation of the latter parameter within the interval of (i) n = 1. Figure 4 shows the U
~ N against N curves
uncertainty obtained by the four methods indicated above calculated for various values of the parameter b0
provides the upper and lower estimates for W, respec- (points) and the results of treatment according to the
tively). power law (solid curves). For better illustration, the
As can be readily shown, a difference between vari- data are presented in the form of coefficients,
ous estimates of and W is on the same order of mag- U
~ =
N U b
( + N ), (37)
nitude as the deviation of each estimate from the exact N 0 0
value. This correlation can be used for estimating normalized so as to tend to a finite limit for b0 ;
errors. The availability of several estimates is of great N
significance: while any two estimated values can acci- 0 is the lower limit of summation in relationship (19),
which can differ from unity when several first terms of
dentally be close to each other (leading to understated the series (1) are zero. As is seen, all curves in fact
value of the predicted error), the accidental proximity exhibit a power asymptotic behavior for large N.
of three or four estimates is hardly probable. Attaining the asymptote is delayed for b0 1 and
b0 N0, because of the existence of the correspond-
4. TEST EXAMPLES ing large parameters in relationship (19). In contrast,
The operation of the proposed algorithm can be the power law holds even for small N for b0 = 0.82 cor-
illustrated by application to several test systems. responding to the first minimum of 2.

Figure 5 shows the plots of 2, U
~
4.1. Zero-Dimensional Case eff, and =
U(b0 + N0) versus b0 calculated in the interval
The first example is offered by the integral 24 N 50. For the first minimum of 2 corresponding
to b0 = 0.82, estimates obtained according to Section 3
are as follows:
W g
( ) = dn  1 exp( 2
 g4
 ), (32)
= 0.247, W = 0.892, ' = 0.82. (38)
0


8 SUSLOV

The second minimum of 2 taking place at b0 = 0.26
 ~
UN yields
b0 = 5 = 0.26, ' = 0.67. (39)
4

3 The U
0.02 value changes sign at b0 = 0.210 and 0.215 for
the treatment with taking a logarithm and at a fixed
2 index, which yields the estimates = 0.210 and
1.5 -0.215, respectively. The slope of a linear portion in the
0.01 U(b0) curve in the vicinity of the root (dashed line in
0.5 Fig. 5 constructed upon treatment at a fixed index)
0.82 yields the W values depending on the preselected
0.25 value: for = (0.210.26), the estimates range within
0 W = 0.8830.933. Summarizing all these estimates,
0.1 we obtain the set of estimates
0.2 =  0.235  0.025, W = 0.908  0.025, (40)
0.01 0.6 ' = 0.75  0.08,
which are consistent with the exact values
= 0.25, W = 0.9064, ' = 0.75. (41)

0.02
Since the ' values in (38) and (39) agree satisfactorily,
we may conclude that the rejected terms in expansion
10 20 30 40 (29) are small as compared to the second term. There-
N fore, the best estimates for are provided (see Section 3)
by relationships (38) and (39). Restricting to these esti-
Fig. 4. The plots of U
~ = U
N
N (b0 + 1) versus N mates, we obtain
at fixed b 0 (points and dashed curves) for integral =  0.253  0.007, W = 0.887  0.005 (42)
(32) with n=1. Solid curves show the results of treatment
according to the power law. instead of set (40). Here, the accuracy of determining
really increased, but the error of W is somewhat under-
estimated.
log2 The shape of the 2 curves is highly sensitive to
8 selection of the lower boundary of the working interval
Nmin N Nmax. As the Nmin value decreases, the 2
6 minima tend to smear, while an increase in Nmin leads to
4 flattening of the curves and the appearance of small-
scale fluctuations hindering identification of the min-
2 ima. In attempts at obtaining the clearest minima corre-
sponding to 2 values of the correct order in magnitude,
0 0.5 1.0 1.5 b0 the choice was usually made between twothree Nmin
 ~
U eff values.5 A change in the working interval most signifi-
0.25 cantly affects the estimates (39), with the and ' vari-
0.2 eff ~
U ations approximately corresponding to a difference
0 between (38) and (39).
0.1
(ii) n = 2. The 2 plots in Fig. 6 exhibit sharp minima
0
~ 0.5 1.0 1.5 at b
b 0 = 1.26 and 0.50. The first 2 minimum yields
0
U
0.1
0.50
= 0.4996, W = 0.442, ' = 1.26, (43)

eff
0.2 ' while the other three methods give = 0.5000 accu-
rate to within the last digit. An estimate for ' obtained
using the second 2 minimum amounts to about 20,
Fig. 5. The plots of 2, eff, and U~ = U
(b0 + 1) versus
5
b0 for integral (32) with n = 1 in the averaging interval of It should be noted that, in displaying the results of calculations
24 N 50. Dashed line shows a portion of the U(b0) with fixed decimal point, the 2 minima are well distinguished by
curve in the vicinity of the root, obtained by the treatment at a the configuration of digits even in the course of a rapid on-screen
constant index = 0.25. computer survey.


SUMMING DIVERGENT PERTURBATIVE SERIES 9

which is inconsistent with (43). Therefore, the rejected log2
terms in (29) are comparable with the second, so that all
four possible estimates are on the same footing. Treat- 6
ment of a linear portion of the U(b0) curve near the 4
root yields W = 0.460. As a result, we obtain
2
=  0.5000  0.0004, W = 0.451  0.009 , (44)


in good agreement with the exact values 0 0.5 1.0 1.5 b0
~
U eff
= 0.50, W = 0.4431. (45)
0.45
0
(iii) n = 3. Here, the 2(b 0.5 1.0 1.5 0.55
0) plots exhibit minima at ~
b 0.2 eff U
0 = 1.07 and 0.77, which yield 0.65
= 0.704, W = 0.192, ' = 1.07 (46)

and Fig. 6. The plots of 2, eff, and U~ = U
(b0 + 1) versus
b
0 for integral (32) calculated with n = 2 in the averaging
= 0.77, ' = 1.42, (47) interval of 20 N 50. Dashed line shows a portion of the
U(b
respectively. Estimates obtained using U 0) curve in the vicinity of the root, obtained by the
changing treatment at a constant index = 0.5. The eff for b0 = 0.5
sign are = 0.86 for the treatment with taking a loga- falls far outside the diagram boundaries.
rithm and = 0.84 for the treatment at a fixed index.
Determining W from the slope of a linear portion in
the U(b function exhibits a pole at the exact value of the index
0) curve in the vicinity of the root yields 0.311,
0.420, and 0.751 for = 0.704, 0.77, and 0.86, = 1 (see Eq. (29)), so that the next term of the expan-
respectively. Since the two values of ' reasonably sion becomes significant with the parameters
agree with each other, the estimates (46) and (47) for
must be more precise. Taking only these estimates into ' = 1.50, W =  /8 = 0.2216. (52)

'
account, we obtain
= 0.737  0.033, W Therefore, the proposed algorithm is incapable of restor-
= 0.306  0.114, (48) ing correct asymptotics described by Eq. (7) in the case of
' =  1.25  0.18, nonpositive integer values. In order to avoid these prob-
lems, the algorithm has to be supplemented by the follow-
in good agreement with the exact values ing rule: if the treatment yields a negative value, the
= 0.75, W = 0.3063, ' = 1.25. (49)
result must be checked by taking a negative or frac-
tional power of series (1) and summing the reexpanded
An allowance for all four estimates of yields series.
= 0.78  0.08, W = 0.47  0.28 (50)


with markedly greater errors. 4.2. Anharmonic Oscillator
In this case, we may also point out difficulties aris-
ing due to an additional "spurious" minimum appearing The second example is offered by the problem of
at b determining the ground sate E
0 = 1.90. However, this minimum can be excluded 0(g) of an anharmonic
from consideration upon identifying the minimum at oscillator described by the Schrdinger equation
b0 = 0.77 as corresponding to b0 =  (by U changing
sign) and the minimum at b0 = 1.07 as corresponding to d2 x2 gx4
b -------- + ----- + -------- (x) = E(x). (53)
0 = ' (by the consistent ' values). In the general 4 4
case, the process of identifying useful minima resem- dx2
bles the situation in spectroscopy under high noise con-
ditions: selecting informative signals requires certain This problem can be reduced to a one-dimensional 4
skill. theory. Consider E0(g) as the W(g) function with the
(iv) n = 4. In this case, application of the algorithm initial terms of the perturbative series having the fol-
encounters the "hidden rock" of this method. Based on lowing form:
the usual estimates, we obtain a quite precise result:
1 3 21 333
= 1.500  0.004, W = 0.222  0.005. (51)
W g
( ) = --- + ---g ------g2
 + ---------g3 30885
---------------g4
 + ... . (54)
2 4 8 16 128
However, these values do not agree with (36). The dis-
crepancy is caused by the fact that the main contribu- Bender and Wu [23] calculated the first 75 coefficients
tion to the UN asymptotics vanish because the gamma WN up to the 12th decimal digit and obtained an expres-


10 SUSLOV

For g , the last term in braces is insignificant and
 ~
UN 0 tends to a constant value of 0.6679863 that can be
0.2 determined by the variational method [24]. Thus, the
W(g) asymptotics is described by power series (7) with
b0 = 10 the parameters

0.1 5 = 1/3, W = (57)
0.668,
1 2 and the corrections having the form of a series in pow-
ers of g2/3.
0
0 0.6 Figure 7 presents the plots of U
~ N against N and the
0.8 results of their treatment according to the power law.

0.9 Figure 8 shows the plots of 2, eff, and U
~ versus b0.
0.1 As is seen, 2 exhibits minima b0 = 1.30 and 0.34 corre-
sponding to

= 0.349, W = =  (58)
0.602, ' 1.80
0 10 20 30 N

and

Fig. 7. The plots of U
~ = U
N
N (b0 + 1) versus N for an = 0.34, ' 20, (59)
anharmonic oscillator. The notations are the same as in Fig. 4.
respectively. Estimates obtained using U
changing sign are = 0.285 for the treatment
log2 with taking logarithm and = 0.337 for the treatment
at a fixed index. Determining W from the slope of a
3 linear portion in the U(b0) curve in the vicinity of the
2 root yields values in the interval from 0.616 to 0.883.
The two values of ' having nothing in common indi-
1 cates that all estimates are on the same footing. As a
1 0 1 2 b0 result, we obtain
 ~ eff
U
0.2 ~ = 0.317  0.032, W = 0.74  0.14, (60)

U 0.4
0.1 eff in good agreement with the exact values (57).
0 The above examples show that the accuracy of
0 1 2 restoring the W(g) asymptotics, while depending sig-
0.1 0.3
nificantly on the particular problem, is generally corre-
lated with the character of corrections to the UN asymp-
totics described by relationship (20). An average accuracy
Fig. 8. The plots of 2, eff, and U~ = U
(b0 + 1) versus on the order of 102 is attained in the zero-dimensional
b0 for an anharmonic oscillator in the averaging interval of case with odd n, where the corrections to (20) have the
24 N 45. Dashed line shows the result of treatment at a
constant index = 0.34. form of power series in N1/2. For even n, every other cor-
rection vanishes due to the poles of the gamma function
to leave a regular expansion in 1/N, which markedly
sion describing behavior of the expansion coefficients increases the resulting accuracy. A relatively low accu-
with large N: racy in the case of an anharmonic oscillator is related to
the fact that corrections have the form of series in pow-
6 1
95/72
W = --------3N N + --- 1  -------------- + ... . (55) ers of N1/3.6 It is important to note, however, that the
N 3/2 2 N
algorithm automatically yields an estimate of the error.
The estimate is rather reliable when all four possible
The asymptotics of E methods for evaluating are employed.
0(g) for g is revealed by
substituting E0(g) = 0g1/3and x xg1/6, after which
6
Eq. (53) transforms into The first term in (28) gives, in addition to the main contribution

to U  1  2
N proportional to N , the regular corrections N ,
 3 '  1 '  2
d2 x2 x2 N , ...; the second term contributes by N , N , ..., etc.
-------- + ----- + ----------- (x) = (x). (56)
0 As a result, the expansion in g2/3 converts into the expansion in
d x2 4 4g2/3 N1/3.


SUMMING DIVERGENT PERTURBATIVE SERIES 11

5. ALGORITHM OPERATING Table 1. Comparison of UN values calculated for b0 = 1
WITH INTERPOLATED COEFFICIENT using exact and interpolated coefficients WN
FUNCTION
U
The importance of interpolation was strongly under- N
estimated, although this method can obviously provide N Interpolation Interpolation
for an increase in the accuracy of calculations. In most Exact WN values with L0 = 1, L = 5 with L0 = 1, L = 1
investigations in the field under consideration, the algo-
rithms were formulated so as to avoid mentioning the 30 2.911  103 2.911  103 2.868  103
coefficients WN at intermediate N values. This approach 35 2.408  103 2.409  103 2.369  103
is conceptually incorrect since, using a finite number of
the initial coefficients and their asymptotics, it is possi- 40 2.038  103 2.041  103 2.004  103
ble to construct a function with preset behavior in infin-
ity.7 A reasonable problem formulation corresponds to
approximately setting all WN, after which W(g) can be Table 2. The parameters of asymptotics for integral (32)
reconstructed with certain precision. with n = 1 calculated using exact and interpolated coeffi-
cients W
Thus, a necessary stage in solving the problem con- N
sists in interpolating the coefficient function, which Interpolation Interpolation
naturally implies that this function is analytical (see Estimates Exact WN with L0 = 1, with L0 = 1,
Section 8.2). The interpolation stage allows the param- based on values L = 5 L = 1
eter c in the Lipatov asymptotics (essentially not used
in the standard conform-Borel procedure [6]) to be First 2 mini- = 0.246 = 0.245 = 0.269
effectively employed. In addition, it is possible to take mum ' = 0.827 ' = 0.830 ' = 0.761
into account smoothness of the reduced coefficient
function, its regularity with respect to 1/N, and (eventu- W = 0.893 W = 0.892 W = 0.912
ally) the information concerning asymptotics of the AK Second 2 = 0.249 = 0.245 = 0.271
coefficients in expansion (22) [25]. minimum ' = 0.792 ' = 0.849 ' = 0.747
In Section 2.3, some qualitative considerations were
presented suggesting that the influence of the interpola- U changing = 0.210 = 0.210 = 0.218
tion errors is not as significant as that of the round-off sign
errors. Unfortunately, no particular estimates illustrat- U(b0) linear = 0.215 = 0.215 = 0.225
ing this were obtained. Validity of this statement will be W
experimentally demonstrated for the zero-dimensional = 0.889 W = 0.887 W = 0.885
test example with n = 1.
With a view to modeling a situation for the 4 the- tions indicate that no catastrophic consequences take
ory, let us assume that several coefficients in the expan- place up to N = 40, when the influence of the round-off
sion of series (1) are known, errors becomes significant. This can be seen in Table 1
presenting the values of some coefficients UN calcu-
W , W , ..., W , (61)
L L + 1 L lated for b
0 0 0 = 1 using the exact and interpolated coeffi-
cients of WN. An increase in the b0 value improves the
together with the Lipatov asymptotics (2) and the cor- accuracy; when b0 decreases, the accuracy drops some-
responding first correction in 1/N. The interpolation is what, although the resulting deviations would be indis-
conveniently performed for the reduced coefficient tinguishable on the scale of Fig. 4.
function, retaining a finite number of terms in expan-
sion (22) and selecting coefficients AK by correspon- The curve of 2(b0) is analogous to (albeit not fully
dence to set (61). coinciding with) that depicted in Fig. 5. Estimates of
Let us consider in detail two examples of the inter- the asymptotic parameters are listed in Table 2; for bet-
polation procedure, which correspond to (i) L0 = 1, L = 5 ter illustration, all values refer to the same working
and (ii) L interval of 23 N 45 and the value = 0.25 used for
0 = 1, L = 1. Owing to a slow character of vari-
ation of the coefficient function, the accuracy of inter- the treatment of a linear portion of the U(b0) curve. As
polation in both cases is very high: ~109 and ~104, is seen from these data, changes in and W caused by
respectively. A random error of such amplitude should the interpolation fall within the scatter of various esti-
lead to large fluctuations in UN for N 30 in the mates and virtually do not influence the accuracy of res-
former case and N 13 in the latter case. Real calcula- toration of asymptotics (7). Therefore, interpolation
using a single expansion coefficient W1 allowed the
7 A function of the factorial series possesses the same asymptotics W(g) asymptotics to be restored with an accuracy not
of coefficients (2) but with a different parameter c [17]; the last
statement in the text can be readily proved by taking an appropri- worse than that achieved with the exact coefficients WN.
ate linear combination of several functions. Of course, this is by no means a typical situation.


12 SUSLOV

asymptotic parameter c [17]. The new series is summed
Q (a) 1 2 upon selecting the value so as to provide for the best
convergence of the second series in expansion (9). The
3
Qexact optimization procedure is employed, bringing both
advantages and troubles, in most investigations in the
3 field under consideration. On the one hand, the princi-
2 1 pal possibility of improving the convergence is defi-
nitely valuable. On the other hand, the results become
opt
dependent of an arbitrary parameter and it is difficult
Q (b) to get rid of the feeling that any result can be obtained.
Theoretically, the use of series (62) is fully equiva-
lent to the study of initial series (1) and the value of any
Q quantity Q obtained upon summation must be indepen-
exact
dent of the parameter . However, under the conditions
of restricted information concerning coefficients WN,
the Q value begins to depend on the choice of , this
dependence weakening as the amount of information
increases. In the general case, no uniform convergence
opt
with respect to takes place and an approximate Q
value is close to the exact one only within a certain
Fig. 9. Schematic diagrams illustrating the optimization
procedure: (a) theoretically, any quantily Q obtained upon "plateau" region (Fig. 9a), the deviations rapidly grow-
summation of the series must be independent of the optimi- ing outside this region. As the amount of necessary
zation parameter ; however, such dependence arises under information increases, the plateau expands and flattens
the conditions of restricted information and weakens (on the (see, e.g., [26]). Apparently, the best convergence takes
passage from curve 1 to 2, 3, etc.) as the amount of informa- place at the center of the plateau. However, this point is not
tion increases (the optimum value = opt occurs at the cen- always unambiguously selected, since the plateau may be
ter of the plateau); (b) the choice of affects both the
approximate Q value (thick solid curve) and the error of asymmetric or poorly pronounced, the center may shift in
determination (cross-hatched area), so that a correct estima- the course of convergence, etc. Therefore, selecting the
tion of this error must provide for the exact value Qexact best approximation for Q and estimating the approxima-
being compatible with all data. In the "ideal" situation tion uncertainty are rather subjective procedures.
depicted, optimization with respect to consists in select-
ing the result characterized by a minimum error. In the author's opinion, the optimization problem can
nevertheless be solved objectively. Indeed, since the
choice of affects both the approximate Q value and the
6. OPTIMIZATION error of determination, a correct estimation of this error
OF THE INTERPOLATION PROCEDURE must provide for the exact value Qexact being compatible
Considering an example in the preceding section, with the approximate values obtained for any (Fig. 9b).
we were lucky to see that the most natural method of This criterion eliminates the problem of an apparent
interpolation may give good results. In the general case, dependence of Q on . Once such an "ideal" situation
the interpolation procedure requires optimization that is attained, optimization of the procedure with respect
will be demonstrated in the case of an anharmonic to reduces to selecting the result characterized by a
oscillator. Let us first discuss the general strategy of minimum error.
optimization, which has been significantly modified in The optimization procedure is expediently per-
comparison to that used in the previous works. formed in the interpolation stage, since all the final
errors arise essentially from the uncertainties in WN.
Rewriting expansion (22) in the equivalent form
6.1. General Strategy of Optimization

On an abstract level, the optimization consists in W = caN Nb~( N + b b~
 )
N
introducing a certain variation of the summation proce-
dure characterized by a parameter , the latter value (63)

 A
~ A
~
1 1
+ -------------- 2
+ --------------------- + ... A
~ K
+ ---------------------- + ...
being eventually selected in a "optimum manner." For
N N
~
 (N N~
 )2 (N N~
 )K
example, the initial series (1) can be raised to the
power and reexpanded to yield and using the interpolation by truncating the series and

W g
( ) = W
~ selecting coefficients A
~ K , we obtain a manifold of real-
0  W
~ 1g + W~2g2  ...
(62) izations of the interpolation procedure characterized by
+ c~aN(N + b)(g)N + .... two parameters, b~ and N
~ . An analysis of the test exam-
The properties of this series are analogous to those of ples shows this parametrization to be sufficiently effec-
the initial one, except for a change in the Lipatov tive: the accuracy of interpolation achieved for the opti-


SUMMING DIVERGENT PERTURBATIVE SERIES 13


mum b~ and N
~ values can be higher by several orders where A1 is the value of A1 for b~ = b  1/2. In all known
of magnitude as compared to that for a random choice cases, A1 < 0 (see [19, 23, 27, 28]) and a minimum cor-
of these parameters. Below, the optimization with rection corresponds to parametrization (65), which
respect to b~ is based on theoretical consideration, while favors a good matching between the high-order asymp-

the optimum N
~ value is selected based on the results of totics and the low-order behavior. Note that the asymp-
tote according to the Lipatov method [2] is
numerical calculations.8
2c(a/e)N Nb  1/2NN.

6.2. Optimization with Respect to b~ The above parametrization (65) corresponds to approx-
imation
Optimization with respect to b~ 
is related to the problem of selecting parametrization for 2e N NN (N + 1/2)
the Lipatov asymptotics which can be written in various whose accuracy is 4% even for N = 1; so, this paramet-
forms: caN(N + b), caNNb  1N!, etc. This problem was risation is close to "natural" one. For an anharmonic os-
actively discussed (see, e.g., [11, 12]), but no satisfac- cillator, the optimum parametrization coincides with (55),
tory solutions were proposed. while in the zero-dimensional case with n = 1 it is close
to (34) and (35).
Note that the values b~ = b and b~ = b  1 lead to
identical results:
6.3. Optimization with Respect to N
~

Nb~(N + b b~
 ) The case of an anharmonic oscillator was studied in

(64)
detail using the interpolation with L
Nb(N), b~ = b 0 = 1, L = 9 (i.e.,
= using the first nine WN coefficients), which corresponded
Nb  1(N + 1) = Nb(N), b~ = b  1. to an accuracy of ~103. The interpolation based on
expression (22) was unsatisfactory: the 2 values obtained
Therefore, the approximate values of any quantity Q by treatment according to the power law (20) were abnor-
obtained upon summation of the series will coincide for mally large even for reasonable averaging intervals and
b~ = b and b  1. As the amount of information concern- gave no clear pattern with minima. The reason for this
behavior is revealed by comparison of the UN coeffi-
ing the coefficients WN increases, the Q(b~ ) function cients (obtained by interpolation) to the exact values.
varies more and more slowly. When the characteristic As is seen from Fig. 10a, the difference is very large,
scale L of this variation increases, the kth derivative of making treatment by the power law practically impos-
the function drops as 1/Lk. As a result, an extremum at sible. Deviations increase by approximately the same
the point b~ = b  1/2 appears in the general situation, law as those for the random errors, but the variation is
with a plateau between the Q values corresponding to rather smooth and is analogous for different b0 values.
It appears that these deviations can be compensated in
b~ = b and b  1and the point b~ = b  1/2 being the nat-
ural center of this plateau. The error of restoring Q, like a broad range of b0 by optimization with respect to N~ .
any other value, exhibits an extremum (which is naturally This is really so and the region of optimum N
~ val-
expected to be minimum) at b~ = b  1/2 (see Section 8). ues can be determined without knowledge of the exact
Thus, the optimum choice is b~ = b  1/2; this corre- result. Figure 11 shows the behavior of 2 in the interval
sponds to the following parametrization of the Lipatov of 20 N 40 depending on N
~ for integer b0 values.
asymptotics: As is seen, small 2 values are immediately obtained for
b N
~
Was = caN Nb  1/2(N + 1/2). (65) 0 = 0, 1, 2, 3 in the interval of = (5.05.5). This is
N evidence that the error of UN can be compensated for all
The first correction A b
1/N to this asymptotics (see expan- 0 0, since greater b0 correspond to still smaller errors
(see Section 2.3). As is seen from Fig. 10b, deviations
sion (22)) depends on b~ as
of the resulting UN for N~ = 5.4 from exact values for
A = A  ( )2 b
1 b  1/2 b~
 /2, (66)
1 0 0 are in fact virtually indistinguishable.
The possibility of more refined optimization is based
8 Further increase in the number of optimization parameters seems on the fact that the interpolation errors in formula (29)
to be inexpedient: this way may lead to absurd results. In particu- play the same role as do the high-order scaling correc-
lar, a large number of parameters allows imitation of a rapid con-
vergence of the algorithm to an erroneous result. In the framework tions indicated by dots. As N
~ is changed, the interpola-
of the proposed approach, it is possible to ensure coincidence of
four estimates of and to obtain a zero value for estimated error. tion errors smoothly vary and (for a certain N
~ value)


14 SUSLOV

become approximately compensated by the scaling cor-
 ~
UN ~ rections. This point can be detected by the maximum
N = 0
= 10 proximity of various estimates obtained for the
b and
0
0.1 W
5 values.
2 A systematic treatment with determination of the
1
0 and W values was carried out for N~ in the interval
0 from 5.0 to 5.6 at a step of 0.1. A "correct" pattern of
0.6 10 20 30 N 2 minima was observed for N~ = 5.5, while for N~ = -5.6
the first minimum disappeared and for N
~ -5.4 it was
0.8 split in two. The reason for this splitting is qualitatively
0.1 (a)
0.9 evident: Figs. 10 and 11 show that, at a fixed N
~ , there
is a certain b0 value for which the effect of the interpo-
 ~
UN lation error upon UN is virtually compensated. This
b0 = 10 ~
0.1 N = 5.4 very b
5 0 corresponds to an "extra" minimum of 2 in
1 2 comparison with the pattern of Fig. 8. Since it is diffi-
cult to decide a priori which of the two minima is true,
0 the estimates were obtained for both (and proved to be
0 very close to each other).
0.6 10 30 N
The results of these numerical calculations are sum-
marized in Table 3 and depicted in Fig. 12. The scatter
0.8 of and W values allows the error to be evaluated by
0.1 (b) the order of magnitude. In order to obtain an "ideal"
0.9 pattern according to Fig. 9b, the error interval should be
expanded by a factor of 1.3 and 1.1 for and W,
respectively (dotted curves in Fig. 12). Then the values
Fig. 10. Optimization of the interpolation procedure for an of = 0.38 and W = 0.52 (dashed curves in Fig. 12)
anharmonic oscillator: (a) a comparison of the UN values
are compatible with the results for all N
~ . Selecting the
obtained by interpolation for N
~ = 0 using the first nine WN
coefficients (solid curves) to exact values (dashed curves); N
~ values in each particular case so as to minimize the
vertical bars indicate the N values above which behavior of one-side error (as indicated by arrows in Fig. 12), we
the exact UN values is visually indistinguishable from that obtain the following estimates:
according to the power law; (b) an analogous pattern after the

optimization with respect to N
~ (for N~ = 5.4). = 0.38  0.05, W = 0.52  0.12. (67)


A comparison to the set (57) shows that the error is esti-
mated adequately, while the average values are some-
2 what displaced; the shift in W
log is induced by the shift
in .
12

7. SUMMING PERTURBATIVE SERIES
FOR AN ARBITRARY g
10 b0 = 5
When the amount of information concerning the W
4 N
coefficients suffices for restoring the W(g) asymptotics
8 3 as g , summing series (1) for an arbitrary g
2 encounters no problems: the coefficients UN for N 40 are
calculated by formula (19) and the subsequent terms can
6 1 be obtained according to the UN  1 asymptotics, so that
0 all coefficients of the converging series (9) are known.
The summation error is determined by the accuracy of
4 restoring the asymptotics,

0 2 4 6 8 10
U U
N
= ---------- = ---------- + ln N, (68)
as
 ~
N UN N 1 U



Fig. 11. The plots of 2 versus N
~ for an anharmonic oscil- which varies logarithmically with N and can be consid-
lator in the interval of 20 N 40 at various fixed b values. ered as constant with a restricted interval. Introducing a
0


SUMMING DIVERGENT PERTURBATIVE SERIES 15

Table 3. Asymptotic parameters for an anharmonic oscillator obtained by the interpolation with L0 = 1, L = 9 (the values in
parentheses for N
~ = 5.6 were estimated at the point b0 = 2.20 where the first 2 minimum disappears)

for N~
Estimates based on
5.0 5.1 5.2 5.3 5.4 5.5 5.6

First 2 minimum 0.398 0.396 0.393 0.390 0.385 0.378 (0.373)
0.476 0.452 0.422 0.399 0.384

Second 2 minimum 0.50 0.47 0.42 0.37 0.33 0.29 0.34
U changing sign 0.585 0.535 0.485 0.445 0.405 0.365 0.335
U(b0) linearization 0.495 0.445 0.40 0.36 0.32 0.29 0.26

W for N~

5.0 5.1 5.2 5.3 5.4 5.5 5.6

First 2 minimum 0.490 0.495 0.500 0.505 0.513 0.529 (0.540)
0.356 0.390 0.440 0.487 0.517

U(b0) slope 0.226 0.290 0.373 0.463 0.572 0.675 0.712
0.502 0.538 0.568 0.698 0.885 1.09 0.953




characteristic scale Nc on which the relative error is Substituting these expressions into (8) and using the
comparable with as and using the approximation steepest descent method for ag N , we obtain
c


U 0, N < N
N c , ag N
---------- = (69) W g
( ) as c
U --------------- (71)
N , N N ,
as c W g
( ) exp{ 2(N /ag)1/2
 }, ag N
as c c

we obtain for ag 1 (where some preexponential factors are omitted for
clarity). For negative , the results for ag Nc are
N

B g
( ) = U c
exp ------
somewhat different. In particular, for 1 < < 0 we
as N ag obtain W(g) =
N = N as(W(g)  W(gc)), where agc ~ Nc.
c (70) A natural scale for Nc is provided by the middle of the
B g
( ), ag N
as c working interval (Nmin, Nmax), that is, Nc 30; however,
= U agexp(N /ag), ag
N . deviations from this value may be quite large because the
as N c c
c corresponding equality holds in fact on the logarithmic


W
0.6 (a) (b)
3 1.0 3
2
0.5 1 0.8 2
1 4 1
0.4 0.6 1
1 2 3
0.3 4 1
0.4
= 0.38  0.05 1 2 W = 0.52  0.12
0.2 0.2
5.0 5.2 5.4 5.6 5.0 5.2 5.4 5.6
 ~
N  ~
N

Fig. 12. The plots of and W values estimated for an anharmonic oscillator by various methods (see Section 3): (a) estimates

based on the (1) first 2 minimum, (2) second 2 minimum, (3) U
changing sign, and (4) U(b0) linearization; (b) W estimates based on
the (1) first 2 minimum and (2, 3) U(b0) slope (upper and lower bounds, respectively). Small-dash lines indicate the error interval
expanded by a factor of 1.3 and 1.1 for and W values, respectively.




16 SUSLOV

Table 4. Comparative data for the exact integral (32) with n = 1 and the results obtained by summing the perturbative series

W(g)  10
g Summing with exact Summing upon interpo- Summing upon interpo-
Exact value WN lation with L0 = 1, L = 5 lation with L0 = 1, L = 1

1 6.842134 6.842135 6.842134 6.8436

2 6.183453 6.183454 6.183452 6.1867

4 5.497111 5.497110 5.497105 5.5034

8 4.820615 4.820608 4.820594 4.832

16 4.181699 4.181669 4.181637 4.200

32 3.597297 3.59720 3.59714 3.624

64 3.075230 3.07500 3.07490 3.113

128 2.616802 2.61633 2.61617 2.668

256 2.219222 2.2184 2.2182 2.285

512 1.877472 1.8761 1.8758 1.959

1024 1.585578 1.5835 1.5831 1.68

g 9.064g0.25 8.95g0.247 8.95g0.247 9.12g0.269



scale (lnNc ln30). In practice, approximation (69) with Table 4 presents the results of calculations for the
a constant N zero-dimensional case. Here, the first column gives the
c is expedient only for large g. In the gen-
eral case, estimate (71) is valid with an effective N exact values of integral (32) with n = 1, while the col-
c
value, which is determined by the number N of the max- umns from second to fourth present the results of sum-
imum term U mation obtained using exact WN coefficients and inter-
NuN in the series for B(u) (for small g, this
value is close to L + 1, e.g., to the number of the first polated values (with L0 = 1, L = 5 or L0 = 1, L = 1),
unknown coefficient W respectively. In each case, the calculations were per-
N). formed for b0 corresponding to the first 2 minimum.
A comparison to (71) indicates that Nc ~ 200 for the
Table 5. Comparative data for the exact ground state energy second and third columns and Nc ~ 10 for the fourth
E0(g) of an anharmonic oscillator and the results obtained by column.
summing the perturbative series (the 2E0(g) and 2g values are
given in order to provide for the correspondence with the data Table 5 presents the analogous data for an anhar-
reported in most other papers using a different normaliza- monic oscillator. Here, the first column gives the exact
tion) E0(g) values taken from [24], while the second and third
columns present the results of summation obtained
2E0(g) using exact WN coefficients and interpolated values
Summing upon (with L0 = 1, L = 9), respectively. In this case, the esti-
interpolation
2g Summing with mates give Nc ~ 200 for the second column and about
Exact value exact W with L0 = 1, L = 9
N 50 for the third column.
(b0 = 1.30) ( N
~ = 5.3, Information concerning the W(g) asymptotics can
b0 = 3.55) also be taken into account within the framework of the
standard conform-Borel procedure (Section 2.1) by inter-
0.5 1.241854 1.241854 1.241857
polating the UN coefficients (with the known asymptotics
1 1.392352 1.392352 1.392396 (17)) calculated using formula (11). For approximation
2 1.607541 1.607545 1.60790 (69), we obtain by analogy with (71)
3 1.769589 1.769605 1.7706
4 1.903137 1.903178 1.9051 2
W g
( ) , ag N
as c
--------------- (72)
5 2.018341 2.018418 2.0214 W g
( ) exp 3(N2/ag)1/3
{ }, ag N2.
as c c
10 2.449174 2.44961 2.4599
This procedure is preferred in the case of sufficiently
20 3.009945 3.0117 3.040
small g values (when Nc is close to L + 1), leading to
50 4.003993 4.0115 4.096 smaller errors as compared to those obtained for (71).
100 4.999418 5.018 5.19 For greater g, the attaining of Nc values indicated above
g 2  0.668g1/3 2  0.602g0.349 2  0.511g0.387 seems to be impossible.


SUMMING DIVERGENT PERTURBATIVE SERIES 17

According to the standard procedure of calculating log2
the critical indices [6], the second series (9) is truncated
on the Lth term that corresponds to the error given by b0 = 5
15
(72) with Nc = L + 1 and as ~ 1. In the three-dimen- 3
sional case, a large number of expansion coefficients 2
are known (for L = 6). These values are well matched
with (2), which gives hope for restoring the asymptotics 1
10
of scaling functions with an accuracy of as ~ 102 and 0
for increasing Nc at the expense of interpolation. Thus, 1
it is apparently possible to increase the accuracy of cal-
culation of the critical indices by twothree orders of 5
magnitude even for the currently available information.
Using the modified conformal mapping may lead to a
further increase in the accuracy, provided that the scale 2 0 2 4 6 8
of Nc 20 would be accessible in the corresponding  ~
N
region of ag ~ 0.2.

Fig. 13. The plots of 2 versus N
~ for the 4 theory in the
8. THE 4 THEORY interval of 20 N 40 at various fixed b0 values.

8.1. Restoration of the Gell-MannLow Function
~
Now let us turn to a real physical problem of restor- UN
ing the Gell-MannLow function in the 4 theory, con- b b0 = 10
0 =
sidering (g) as W(g) and proceeding from the informa-
tion contained in relationships (5) and (6). 5
The interpolation was based on formula (63) with an 20
optimum value of b~ = 4. Figure 13 presents the plots of 2
2(
N
~ ) versus
N
~ calculated in the interval 20 N 40 1
for several fixed b0 values. As is seen, promising results can 10 0.5
be expected for N
~ values close to zero, where the curves 0
obtained at b 0.5
0 = 1, 0, 1 and 2 exhibit sharp minima. The

interval 0.5 N
~ 0.5 was studied in more detail. 0.9
0 1.1
Figure 14 shows the behavior of the coefficients U
~ N = 1.3
U
N (b0 + 2) in the case of a nearly optimum interpola- 1.5
tion with N
~ = 0. If the curves for b 1.7
0 1 and b0 2 10
(attaining the asymptote with delay) are rejected, the
data for large N asymptotically tend to a constant level,
which correspond to a critical index close to unity. 1.9
This conclusion is consistent with the position of the
second 2 minimum and with the change of sign in U 20
0 10 20 30 N
(Fig. 15). A clear pattern with 2minima was observed
for
N
~ 0.2; when the N~ value increased, the first 2 Fig. 14. The plots of U
~ = U
N
N (b0 + 2) versus N for vari-
minimum approached to and eventually merged with ous b0 (points and dashed curves) and the results of treat-
the second minimum. For this reason, no estimates ment according to the power law (solid curves) for the 4
using the first minimum could be obtained for N
~ 0.3. theory. The calculations were performed using a nearly
optimum interpolation with b~ = 4, N
~ = 0.
The results of determining the and W values are
presented in Table 6 and Fig 16. The ideal pattern for ,
corresponding to Fig. 9b, is obtained upon expanding related to a weak dependence on the averaging interval.
the error interval by a factor of two (dashed lines in Fig. With an allowance for the double error, we finally obtain
16a), after which the value of = 0.96 is compatible
with the results for all N
~ . In the fixed interval of 20 = 0.96  0.01. (73)

N < 40, all four estimates of coincide for N
~ = 0.12 For W (Fig. 16b), the ideal pattern is obtained imme-
on an accuracy level of 103; the main uncertainty is diately and the corresponding value of W = 7.4 is com-


18 SUSLOV

patible with all data. Here, the one-side error is mini-
log2
mum at N
~ = 0.08, which yields
10
W =  . (74)
7.4 0.4
8 Correctness of the optimization with respect to b~
(which was carried out in Section 6.2 in somewhat heuristic
6 manner) can be demonstrated now. For an optimum value of
~ N
~ = 0.12 and the b~ value varied in an interval from 0
N = 0 to 6, a clear pattern of
4 2 minima was obtained in the
middle of the interval. On approaching the boundaries,
the first
(a) 2 minimum approached to and merged with
2 the second minimum exactly as it was observed on

increasing N
~ . These corresponding results for and
1 0 1 b0
~ W are presented in Fig. 17; expanding the error inter-
U eff val by a factor of 2 and 1.1 for and W, respectively,
~
5 U makes the values (73) and (74) compatible with almost all

data (except for a narrow interval at b~ = 5.5, where the
proximity of all estimates is obviously accidental. As is
0 1
~ seen, the minimum errors also agree with (73) and (74).
U Summation of the perturbative series for the Gell-
MannLow function at finite g values was performed
eff
5 using a procedure analogous to that described in Sec-
tion 7. The accuracy was evaluated by variation with
(b) respect to b0 and N~ . The variation with respect to b0
10 0 gave a markedly greater Nc values and allowed the W(g)
asymptotics to be modified without significantly affect-
Fig. 15. (a) The pattern of minima in 2 for the 4 theory in ing the results for g ~ 1. On varying the N
~ value, with
the averaging interval of 20 N 40. (b) The plots eff and b0 adjusted so as to maintain a constant value of =
U
~ versus b
0 for N
~ = 0. The dashed curve shows the 0.96, the most probable value of W = 7.4 is obtained
U(b0) curve for fixed = 1.
for N
~ = 0.067; the uncertainty range indicated in (74)
corresponds to the interval 0.09 N
~ 0.05. Table 7

3 lists the data for N
~ = 0.067, with the error estimated
(a)
1.3 4 by comparison to the results for N
~ = 0.05 and 0.09.
2
4 Note that asymptote (7) is attained rather slowly, the
0.9 1
1 deviation amounting to about 15% even for g = 100.
3 = 0.96  0.01 Figure 18 presents a comparison of the results
0.5 2 obtained for g 20 to the data reported by other
researchers.
W
17 (b) 1 2
2 8.2. The Possibility of Logarithmic Branching
13 Since the value of differs only slightly from unity,
a question arises as to whether the accuracy is sufficient
9 to consider this deviation significant. Formally speak-
ing, this is really so because the error was estimated objec-
3
1 tively and there is no ground to expect it to be significantly
5 W = 7.4  0.4
3 understated. Nevertheless, the possibility that the equality
0.4 0.2 0 0.2 0.4 = 1 is strict is not excluded, since asymptotics (7) may
~
N contain logarithmic corrections of the type


Fig. 16. The plots of various (a) and (b) W estimates ver- W g
( ) = W (ln g) , g . (75)
g
sus b0 for the 4 theory. The notations are the same as in
Fig. 12. Small-dash lines indicate the error interval for For > 0, these corrections may inspire a small
expanded by a factor of two. decrease in . In this case, formula (20) contains an


SUMMING DIVERGENT PERTURBATIVE SERIES 19


Table 6. Asymptotic parameters for the 4 theory obtained for b~0 = 4 and various N
~ values by the interpolation with L0 = 2,
L = 5

for N~
Estimates based on
0.5 0.3 0.2 0.12 0.1 0.0 0.1 0.2 0.3 0.5

First 2 0.863 0.920 0.945 0.962 0.964 0.975 0.974 0.931  
minimum 0.005
Second 2 0.54 0.78 0.90 0.960 0.970 1.00 1.01 1.01 0.97 1.16
minimum

U changing 0.795 0.865 0.915 0.960 0.973 1.035 1.105 1.175 1.255 1.415
sign 0.961

U(b0) 0.907 0.90 0.929 0.961 0.971 1.022 1.082 1.147 1.218 1.371
linearization 0.001

W for N~

0.5 0.3 0.2 0.12 0.1 0.0 0.1 0.2 0.3 0.5

First 2 4.67 5.22 5.75 6.36 6.63 8.26 11.82 30.9  
minimum 0.16
U(b0) 3.02 5.58 6.55 7.35 7.34 7.18 6.78 6.45 5.91 5.05
slope 15.9 10.0 7.85 7.55 7.61 9.07 11.3 16.5 17.3 12.3



additional factor (lnN) with unchanged W, so that the Note that the first term 0 is absent in the expansion
results for U of the function (5) simply by its definition, while van-
N can be treated according to Eq. (75) with
the parameters ishing of the next coefficient 1 is accidental. Indeed, in
the (4  )-dimensional 4 theory, the latter term is non-
= 1, 0.14, W (76)
7.7 zero and has a magnitude on the order of ; accordingly,

without any increase in 2. Actually, the possibility of
such a logarithmic branching seems to be quite proba-
ble for the following reasons. 1.7 (a) 3
1. It can be ascertained that the logarithmic branch- 4
ing in the case of strict equality = 1 is unavoidable. 3
1.3
Indeed, let us write series (1) in the form of the Som- 4 2
merfeldWatson integral [2, 13]: 0.9 2 1 1

z
( )
W g
( ) = W 
( g)N = ----- dz-------------gz, (77)
1 W
N
2i sinz 3 1 (b)
N = N 20
0 C 3
where (z) is the analytical continuation of WN onto
the complex plane ( (N) = W 15
N) and C is the contour
containing the points N0, N0 + 1, N0 + 2, ... (Fig. 19). If
z = is the extreme right-hand singularity of 10
(z)/sinz, we can modify the contour into the posi- 2
tion C' and show that this singularity determines the
behavior of W(g) as g . The purely power law (7) 1
5
corresponds to the presence of a simple pole at z = , 2
while the law described by Eq. (75) corresponds to a ~
0 1 2 3 4 5 6 b
singularity of the (z  )  1 type.9
Fig. 17. The plots of various (a) and (b) W estimates ver-
9 It is clear from the above considerations that the assumption of
analyticity of the coefficient function on the real axis for N N sus b~ for the 4 theory. The notations are the same as in
0,
which is necessary for interpolation, is confirmed in all cases by Fig. 12. Small-dash lines indicate the error interval expanded
the results obtained. by a factor of 2 and 1.1 for and W values, respectively.


20 SUSLOV

Table 7. The Gell-MannLow function for the 4 theory 2. Lipatov [29] considered the class of field theories
(values in parentheses indicate the error estimated in units of (generalizing the four-dimensional 4 theory) with a
the last decimal digit) nonlinearity of the n type and a space dimension of
d = 2n/(n  2), for which a logarithmic situation takes
g (g) g (g)
place. For all such theories, 1 = 0; however, this coef-
0.2 0.04993(2) 30 138.7(50) ficient differs from zero when d decreases. Therefore,
0.4 0.18518(26) 40 193.2(75) (1) = 0 by analogy with the cases considered above.
0.6 0.3939(10) 50 248.3(100) In the limit n , the Gell-MannLow function is
exactly calculated [29] and the extreme right-hand sin-
0.8 0.6667(27) 60 303.9(127) gularity of (z) has the form of (z  1)3/2, which leads
1 0.9952(51) 70 359.7(155) to asymptotics of the type (g) g(lng)3/2. From the
2 3.272(33) 80 415.6(182) continuity considerations, we may expect for large but
3 6.278(85) 90 471.7(212) finite n values that a nonanalytical zero of the type (z  1)
4 9.758(157) 100 527.7(240) is retained and the singularity at z = 1 is still the extreme
right-hand one. Therefore, asymptotics (78) is natural
5 13.57(25) 150 808.1(389) for such field theories and it is not surprising that it may
6 17.64(36) 200 1087(54) be retained up to n = 4. Note that W is negative when
7 21.90(47) 250 1366(70) n , so that the Gell-MannLow function pos-
8 26.32(60) 300 1644(86) sesses a zero; a direct extrapolation of the results to n = 4
9 30.87(75) 350 1920(101) leads to an analogous conclusion for the 4 theory [29].
In fact, with this extrapolation we must take into
10 35.53(90) 400 2196(127) account that the index changes from 3/2 to small values
15 59.95(175) 450 2471(133) such as in (76); then the change in sign of the asymptot-
20 85.59(275) 500 2745(149) ics naturally takes place according to (78) at = 1. The
25 111.9(38) g 7.41g0.96 positiveness of 0 follows from the matching of
(2) ~ 0 and the positiveness of 2 [29].
Anyhow, we have to select between two possibili-
(1) ~ . The limiting transition 0 shows that, in ties: (i) a purely power law (7) with a critical index
the four-dimensional case, (1) = 0 and a simple pole slightly below unity and (ii) an asymptotics of the type
cannot take place at = 1. If the function tends to (78) with > 0. In both cases, the 4 theory turns out to
zero as z 1 by the law (z) = 0(z  1), then be self-consistent.


g
( ) 0
= (ln ), g (78)
( 8.3. On the Results Obtained in [12, 13]
1  )
--------------------g g

The curves in Fig. 14 display for N < 10 a linear por-
and the positive definiteness of has a quite clear origin. tion where U
~ N 1.1(N  1), which is stable with
respect to changes both in b0 and in the extrapolation
procedure. This region might be considered as a true
FN 0.9g2 asymptotics for U
~ N (assuming the results for N > 10 to
1.0 be the interpolation artifacts), corresponding to the
dependence (g) 1.1g2, which is close to the result
200 0.5 obtained in [12, 13].
1.96g1.9
In fact, stability of the above region has a different ori-
gin. This behavior is related to a characteristic "trough" in
0 50 1002.99g1.5
N
100

7.4g0.96 z
1.5g2 C'

0 5 10 15 g C

Fig. 18. A comparison of the Gell-MannLow function for N0
the 4 theory calculated in this work (solid curve) to the
results reported by other researchers (dashed curves top to
bottom corresponding to [12, 13, 14], respectively). The
inset shows a reduced coefficient function (in this scale, dif-
ferences between the data obtained using various interpola-
tion methods are insignificant). Fig. 19. Integration contour for Eq. (77).


SUMMING DIVERGENT PERTURBATIVE SERIES 21

the reduced coefficient function FN at N 10 (see the physical charge at large distances, the 4 theory is
inset in Fig. 18). Modeling this trough by assuming inconsistent: the effective charge g(L) turns into infinity
F3 = F4 = ... = F10 = 0 and taking into account Eqs. (19) at a certain Lc (Landau pole), while for L < Lc the g(L)
and (22), we obtain is undetermined. Considering the field theory as a lim-
iting case of the lattice theories, the 4 theory is "triv-
U
~ = ( )
N c b + 2
0 ial": the physical charge tends to zero for any value of
N (79) the bare charge.
 ( + )
F (1)K K b
-----------------------CK  1.
K In recent years, the problems related to the concept
(K + b ) N  1
0
K = 1 of triviality were actively discussed by several
From this we obtain for N 10 and all b researchers (see [30, 31] and references therein). On the
0 the result one hand, the existing indications of triviality of the 4
U
~ N = 1.5(N  1), which is determined by the first non- theory were emphasized; on the other hand, the 4 the-
vanishing coefficient F2 (see the curve for b0 = in ory was declared verified (with a positive result) by
Fig. 14) and is close to the real situation. For the numerical modeling on a lattice. Let us briefly discuss
function, this result implies that a single-loop law 1.5g2 this problem as well.
is valid up to g ~ 10. The 4 theory is strictly proven to be trivial in a
Upon modeling the trough in FN more precisely by space with the dimensionality d > 4 and nontrivial for
assuming F3 = F4 = ... = F10 = and using (26) we d < 4 [32, 33]. In the case of d = 4, the obtained inequal-
obtain, in the case when the ratio of gamma functions ities were only slightly insufficient for the statement of
in (79) reduces to a polynomial for b0 = b  p with an triviality. This fact was considered by
integer p and N in the interval p + 2 N 10, mathematicians as annoying unpleasanty
and triviality of the 4 theory was stated
1 + b
U
~ = ( ) 0 as "virtually proved." From the standpoint
N W 1  ----- N  1 + ----- -------------- . (80)
2 F F 1 + b
2 2
of physics, this optimism is by no means justified: on
the modern level, the aforementioned results for d 4 are
This result indicates that the linear slope varies but rather primitive, being simple consequences of renor-
remains independent of b0. More complicated calcula- malizability and a one-loop renormalization group.
tions show that Eq. (80) is valid for arbitrary b0 to On the contrary, the situation with d = 4 is physically
within corrections on the order of /(N + b highly complicated and no analytical approaches to
0)b + 1; for = 0.2
solving this problem have been developed so far.
(see Fig. 18), we obtain U
~ N = 1.1(N  1) + const, where
the last constant depends on b In the author's opinion, the results of numerical
0 but does not exceed a few
tenths in the interval 0 < b experiments on the lattice revealed nothing unexpected.
0 < 10. Thus, a notion of the qua- In view of the absence of zeros of the function, the
dratic law with modified coefficient (g) = 1.5(1  /F2)g2 effective charge g(L) always decreases with the dis-
is really meaningful in the interval 1 g 10 but is a tance. However, the numerical methods cannot answer
consequence of the trough in FN.10 The limited width of the question as to whether the "charge zero" does exist,
the trough indicates that this law is not related to a real which is explained by limited lattice dimensions. There
asymptotics (whatever it is). are many cases of misunderstanding related to the
The above considerations clearly indicate that the charge normalization: even in the "natural" normaliza-
result obtained in [12, 13] is by no means a computa- tion used in this work, the quadratic law is extended to
tional error and objectively reflects the behavior of the g ~ 10 (see Section 8.3); traditional normalizations extend
function for g 10. This result is unavoidably this interval even greater, for example, up to g ~ 600 when
obtained upon summing a series with a small number of the interaction term is written in the form of g4/8. There-
expansion coefficients, since no other portion obeying the fore, behavior of any quantities is indistinguishable
power law can be found in Fig. 14 for N < 7 (the points on from trivial in a broad range of parameters.
the curves for b0 < 0 are omitted for clarity, because their Among old publications, only the paper of Freed-
sharp oscillations would overload the pattern). man et al. [34] is worth of mentioning where it was
stated that g(L) uniformly decreases in g0, which is
8.4. The Question of "Triviality" actually indicative of the "charge zero." However, judg-
of the 4 Theory ing by the results, the charge normalization employed
in [34] differed by a factor of about 100 (an expression
The situation when the function possesses asymp- for the action obviously contains a misprint) from that
totics of the ga type with > 1 can be given a two-fold used in this work and all results for finite g
interpretation. From the standpoint of finiteness of a 0 fell within
a region where the quadratic law is operative. Nontriv-
10 ial results were only obtained for g
This law is more clearly pronounced for the Borel image and is 0 = by reduction to
somewhat distorted for the function as a result of integration in the Ising model. Although this reduction is apparently
Eq. (8); however, (g) remains downward-convex up to g ~ 100. possible, there is no method (except for extrapolation)


22 SUSLOV

to establish a correspondence between normalization of test examples showed that the algorithm is stable under
the field variable in the Ising model and that in the ini- conditions of strongly restricted information and con-
tial 4 theory. This leads to uncertainty in the charge firmed reliability of the error estimation. The main
normalization, an allowance for which makes unjusti- physical result of this study consists in restoring the
fied any conclusions concerning uniform convergence. Gell-MannLow function of the 4 theory and demon-
Now let us turn to the original results of [30, 31]. strating its self-consistency. The latter conclusion
The main idea was illustrated by the example of a non- agrees with the absence of renormalon singularities
ideal Bose gas possessing a well-known spectrum of established previously [9].
the Bogolyubov type: (k) ~ k for small k and (k) ~ k2 The proposed algorithm can be applied to solving
for k . Let us pass to the "continuum limit" by many other problems as well, in particular, to restoring
allowing two characteristic scales of the problem (scat- the Gell-MannLow functions in quantum electrody-
tering length and interparticle distance) to tend to zero. namics and quantum chromodynamics. At present,
If the first value tends to zero rather rapidly, a "quite solving this task is complicated by the absence of cal-
trivial theory" appears and a quadratic spectrum of the culations of the full-scale Lipatov asymptotics in these
ideal gas is restored. If the limiting transition is per- theories, although the basis for such calculations is
formed so as to maintain a certain relationship (ensur- fully prepared [27, 3539]. Application of the proposed
ing constant sound velocity) between the two scales, a algorithm to the theory of phase transitions may
"trivial theory with nontrivial vacuum" appears and the increase the accuracy of calculation of the critical indi-
spectrum becomes strictly linear (i.e., strongly different ces by at least twothree orders of magnitude.
from that of the ideal gas), although no interaction of
quasiparticles (phonons) takes place. The latter sce- ACKNOWLEDGMENTS
nario was suggested for the continuum limit of the 4
theory, stating that it is logically self-consistent. This study was supported by the INTAS foundation
Even if the last statement is accepted, a question still (grant no. 99-1070) and by the Russian Foundation for
remains unanswered as to why this limiting transition Basic Research (project no. 00-02-17129).
does physically take place. For a Bose gas of neutral
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