Strong-coupling limits induced by weak-coupling expansions

V.I. Yukalov1,2 and E.P. Yukalova3

1Bogolubov Laboratory of Theoretical Physics,
Joint Institute for Nuclear Research, Dubna 141980, Russia

2Instituto de Fisica de São Carlos, Universidade de São Paulo,
CP 369, São Carlos 13560-970, São Paulo, Brazil

3Laboratory of Information Technologies,
Joint Institute for Nuclear Research, Dubna 141980, Russia

corresponding author<EMAIL_ADDRESS>

###### Abstract

A method is described for the extrapolation of perturbative expansions in
powers of asymptotically small coupling parameters or other variables onto the
region of finite variables and even to the variables tending to infinity. The
method involves the combination of ideas from renormalization group theory,
approximation theory, dynamical theory, and optimal control theory. The
extrapolation is realized by means of self-similar factor approximants, whose
control parameters can be uniquely defined. The method allows to find the
large-variable behavior of sought functions knowing only their small-variable
expansions. Convergence and accuracy of the method are illustrated by explicit
examples, including the so-called zero-dimensional field theory and anharmonic
oscillator. Strong-coupling behavior of Gell-Mann-Low functions in
multicomponent field theory, quantum electrodynamics, and quantum
chromodynamics is found, being based on their weak-coupling perturbative
expansions.

Keywords: Perturbative expansions, Divergent series, Extrapolation method,
Self-similar approximants, Strong-coupling limit

##  1 Introduction

The majority of interesting physical problems cannot be solved exactly, but
succumb only to perturbation theory resulting in asymptotic, usually
divergent, expansions. Such expansions have, as a rule, the form of truncated
series in powers of a parameter, which often is a coupling parameter, or some
other variable assumed to be asymptotically small. Usually such series are
divergent for any finite value of the considered variable. However, the
realistic values of these variables are usually finite, and in many cases the
most interesting region is that of asymptotically large variables. This is why
the extrapolation of divergent expansions from the region of asymptotically
small variables to that of finite variables has been the topic of intensive
research.

Several tools have been developed for such an extrapolation to the region of
finite variables, e.g. Padé approximants [1], Borel summation [2], change of
variables, such as conformal mapping [3], and series transformations, for
instance Schmidt-Shanks transformation [4]. These tools can allow for accurate
extrapolation to finite variables, but they cannot be used for the
extrapolation to asymptotically large variables.

Consider, for example, the Padé approximants constructed from a perturbative
expansion of order $k$ in powers of a variable $x$,

$P_{M/N}(x)=\frac{a_{0}+a_{1}x+\ldots+a_{M}x^{M}}{1+b_{1}x+\ldots+b_{N}x^{N}},$

where $M+N=k$. It can provide a reasonable approximation for finite $x$, but
at asymptotically large $x$, it behaves as

$P_{M/N}(x)\simeq B_{M/N}\;x^{\nu_{M/N}}\qquad(x\rightarrow\infty)\;,$

with the large-variable amplitude and exponent

$B_{M/N}=\frac{a_{M}}{b_{N}}\;,\qquad\nu_{M/N}=k-2N\qquad(0\leq N\leq k)\;.$

Since, for each order $k$, there exists the whole table of approximants for
different $M$ and $N$, the large-variable limit is not defined giving,
depending on $M$, $N$, and fixed $k$, the set of large-variable exponents

$\nu_{M/N}=-k,~{}-k+2,~{}-k+4,\ldots,~{}k-4,~{}k-2,~{}k\;,$

which varies with $k$.

Series transformations are usually composed of power-law expansions, thus
being reduced to the ratio of polynomials, similarly to Padé approximants.
Hence, in the same way as for the latter, the limit $x\rightarrow\infty$ is
not defined, although for finite $x$ a reasonable accuracy can be achieved.

When using some change of variables, one makes a mapping $x=x(z)$, like
conformal mapping, and re-expands the given truncated series in powers of the
new variable $z$. Then one again confronts the necessity of resorting to an
effective summation of the new series, with returning back to the old variable
$x$. Again, such a procedure can lead to a reasonable approximation for finite
$x$, but it is not uniquely defined for $x$ tending to infinity, being
dependent on the particular mapping and the summation method for the series in
terms of new variables.

Resorting to the Borel summation, one needs to find an effective sum for the
Borel transform

$B_{k}(x)=\sum_{n=0}^{k}\frac{a_{n}}{n!}\;x^{n}\;,$

say by using Padé approximations, $B_{M/N}(x)$, and to substitute it in the
inverse Borel transformation. For example, in the widely employed Padé-Borel
method, one has

$f_{PB}(x)=\int_{0}^{\infty}B_{M/N}(xt)\;e^{-t}\;dt\;.$

For large $x$, Padé approximants behave as

$B_{M/N}(x)\simeq C_{M/N}x^{M-N}\qquad(x\rightarrow\infty)\;,$

so that for the sought function, one has the large-variable form

$f_{PB}(x)\simeq C_{M/N}\;\Gamma(M-N+1)\;x^{M-N}\qquad(x\rightarrow\infty)\;,$

which is not uniquely defined, analogously to the case of Padé approximants.

In this way, it is clear that the existing methods of defining effective sums
of divergent perturbative expansions, although can yield good approximations
for finite values of variables, but are not able to correctly characterize the
limit of asymptotically large variables.

In the present paper, we describe a method allowing for the extrapolation of
perturbative expansions, derived for asymptotically small variables, to the
region of finite and even infinite variables. The sequence of steps is as
follows. First, we present the justification of the method. Then the approach
is applied to the problems with known solutions, so that the convergence and
accuracy of the approach could be explicitly illustrated. After that,
predictions are made for the problems whose large-variable behavior is not
known. This concerns the strong-coupling limit of the Gell-Mann-Low functions
in multicomponent $\varphi^{4}$ field theory, quantum electrodynamics and
quantum chromodynamics.

##  2 Controlled approximation cascade

Let us consider a complicated problem whose solution can be found only in the
region of an asymptotically small parameter or variable, where perturbation
theory yields the asymptotic truncated series

$f_{k}(x)=\sum_{n=0}^{k}a_{n}x^{n}\qquad(x\rightarrow 0)\;,$ (1)

where, without the loss of generality, we can set $a_{0}=1$. The more general
case, where

$f_{k}(x)=f_{0}(x)\sum_{n=0}^{k}a_{n}x^{n},$

with a known function $f_{0}(x)$, can be easily reduced to the above form by
considering $f_{k}(x)/f_{0}(x)$. For concreteness, we keep in mind a real
function of a real variable $x\in[0,\infty)$.

Suppose our problem of the most interest is to find the large-variable
behavior of a function knowing solely its small-variable expansion (1). Can we
find the behavior of a function $f(x)$ at $x\rightarrow\infty$ knowing only
its behavior at $x\rightarrow 0$?

The pivotal idea of the approach, we are presenting, is to reformulate the
sequence of perturbative truncated series into a dynamical system, with the
approximation order $k$ playing the role of discrete time [5, 6, 7, 8]. If we
are able to accomplish this, then the equation of motion for the dynamical
system can be analyzed for the purpose of finding a fixed point representing
an effective limit of the series. To our understanding, although expansion (1)
is obtained for $x\rightarrow 0$, but its coefficients $a_{n}$ contain
information on the whole function representing its effective limit, similarly
to how the coefficients of a Taylor expansion contain information of the whole
function which the expansion is obtained from. The goal is to extract the
information hidden in the coefficients $a_{n}$.

Keeping in mind that the sequence of expansions (1) is usually divergent, it
is clear that by itself it cannot form a stable dynamical system. In order to
stabilize a dynamical system, one needs to introduce control functions that
would transform the system so that to govern its stability and provide the
existence of a fixed point, similarly to how this is done in optimal control
theory [9, 10].

The transformation of expansions (1), incorporating control functions, can be
schematically represented as an operation

$F_{k}(x,u_{k})=\hat{T}[\;u_{k}\;]\;f_{k}(x)\;.$ (2)

Control functions $u_{k}=u_{k}(x)$ are to be defined so that to provide
convergence to the renormalized sequence of the approximants $F_{k}(x,u_{k})$,
which implies the validity of the Cauchy criterion, when for any
$\varepsilon>0$ there exists $k_{\varepsilon}$ such that

$|\;F_{k+p}(x,u_{k+p})-F_{k}(x,u_{k})\;|<\varepsilon$ (3)

for $k>k_{\varepsilon}$ and $p>0$.

The renormalized sequence $\\{F_{k}(x,u_{k})\\}$ does not compose yet a
dynamical system. For the correct mathematical definition of a dynamical
system, one has to construct a phase space, or state space, define an
endomorphism acting on that space, and formulate the evolution law [11]. For
this purpose, we impose a rheonomic constraint

$f=F_{0}(x,u_{k}(x))\;,\qquad x=x_{k}(f)$ (4)

and introduce the function

$y_{k}(f)=F_{k}(x_{k}(f),u_{k}(x_{k}(f)))\;.$ (5)

The sequences $\\{y_{k}(f)\\}$ and $\\{F_{k}(x,u_{k})\\}$ are bijective by
construction. The closed linear envelope

${\cal A}=\overline{\cal L}\\{y_{k}(f):~{}k=0,1,2,\ldots\\}$ (6)

constitutes the approximation space [12]. The function $y_{k}(f)$ is the
sought endomorphism acting on the approximation space. The dynamical system in
discrete time

$\\{y_{k}(f):~{}\mathbb{Z}_{+}\times{\cal A}~{}\longmapsto~{}{\cal A}\\}$ (7)

is called cascade. In the present case this is an approximation cascade, whose
trajectory

$y_{k}(f)~{}\longmapsto~{}y_{k+1}(f)~{}\longmapsto~{}\ldots~{}\longmapsto~{}y^{*}(f)$
(8)

is bijective to the sequence of the approximants $\\{F_{k}(x,u_{k})\\}$.

Finally, we have to formulate the evolution equation for the approximation
cascade. The Cauchy criterion now takes the form

$|\;y_{k+p}(f)-y_{k}(f)\;|<\varepsilon.$ (9)

By assumption, the control functions guarantee the existence of a limit

$y_{k+p}(f)\simeq y^{*}(f)\;,$ (10)

when $k$ increases. This limit corresponds to a fixed point that, by
definition, satisfies the equation

$y_{k}(y^{*}(f))=y^{*}(f))\;.$ (11)

From these equations, it follows that in the vicinity of a fixed point the
endomorphism obeys the evolution equation

$y_{k+p}(f)=y_{k}(y_{p}(f))\;$ (12)

that can be called self-similar relation [5, 6, 7, 8]. This is a group
property typical of the theory of dynamical systems as well as of
renormalization group theory [13, 14].

In this way, the approximation cascade (7) is defined, with the approximation
space (6), endomorphism (5), the self-similar evolution equation (12), and the
approximation order $k$ playing the role of discrete time. Since in practical
calculations, we cannot reach the limit of $k\rightarrow\infty$, but have to
deal with a finite approximation order $k$, the approximation for the sought
fixed point will depend on the considered order $k$, being denoted as
$y_{k}^{*}(f)$.

A fixed point of the cascade, $y^{*}_{k}(f)$, by construction, is bijective to
the effective limit of the perturbative sequence called the self-similar
approximation:

$f_{k}^{*}(x)=\hat{T}^{-1}[\;u_{k}\;]\;y_{k}^{*}(F_{0}(x,u_{k}(x)))\;.$ (13)

The details of techniques for determining fixed points of approximation
cascades can be found in Refs. [14, 15, 16].

##  3 Self-similar factor approximants

The practical realization of the above program can be accomplished as follows.
Control functions can be incorporated into the considered sequence in several
ways. Since the main idea is to extract information from the given sequence of
expansions in a self-similar manner, we use the fractal transform

$F_{k}(x,\\{n_{j}\\})=\prod_{j=1}^{k}x^{-n_{j}}\;f_{k}(x)\;,$ (14)

with the inverse transformation

$f_{k}(x)=\prod_{j=1}^{k}x^{n_{j}}\;F_{k}(x,\\{n_{j}\\})\;.$ (15)

Expansion (1) can be rewritten as

$f_{k}(x)=\prod_{j=1}^{k}(1+b_{j}x)\;,$ (16)

with $b_{j}$ expressed through $a_{n}$. Then the fractal transform (14) can be
represented in the form

$F_{k}(x,\\{n_{j}\\})=\prod_{j=1}^{k}x^{-n_{j}}\;(1+b_{j}x)\;.$ (17)

Following the scheme described in the previous section, we obtain [17, 18] the
fixed point of order $k$ giving the self-similar factor approximant

$f_{k}^{*}(x)=\prod_{j=1}^{N_{k}}(1+A_{j}x)^{n_{j}}\;,$ (18)

where, depending on whether the approximation order is even or odd,

$\displaystyle N_{k}=\left\\{\begin{array}[]{ll}k/2,{}&~{}k=2,4,\ldots\\\
(k+1)/2{}&~{}k=3,5,\ldots\end{array}\right.\;.$ (21)

In order to define explicit forms of the control quantities $A_{j}$ and
$n_{j}$, it is necessary to impose some optimization conditions. From the
Cauchy criterion of convergence, it is straightforward to derive the
optimization conditions in the form of minimal-difference and minimal-
derivative conditions [14, 15, 16] by comparing the approximants (18) of
different orders and by minimizing some cost functionals, as is customary in
determining control functions for controlled dynamical systems [19, 20].
Deriving such general forms of the control functions $A_{j}(x)$ and $n_{j}(x)$
in the present case is rather expensive requiring to deal with several
optimization conditions and making the overall use of the approach
complicated. Instead of trying to derive $A_{j}(x)$ and $n_{j}(x)$ for
arbitrary $x$, it is possible to resort to the learning techniques employed in
machine learning [21, 22]. Then one trains the considered system on the set of
known data by comparing the approximate expressions characterizing the system
with the given empirical data. In the present case, the known data are given
by the explicit expansion $f_{k}(x)$ at small $x$. Then we need to compare the
approximants (18) with $f_{k}(x)$ by considering the minimal-difference
condition

$f_{k}^{*}(x)-f_{k}(x)\simeq 0\qquad(x\rightarrow 0)\;.$ (22)

The advantage of this training procedure is that the control quantities
$A_{j}$ and $n_{j}$ become control parameters, but not functions. If the
asymptotic form of the approximant $f_{k}^{*}(x)$ is close to expansion (1),
then the training conditions are

$\lim_{x\rightarrow
0}\;\frac{1}{n!}\;\frac{d^{n}}{dx^{n}}\;f_{k}^{*}(x)=a_{n}\qquad(n=0,1,2,\ldots)\;.$
(23)

Explicitly, this leads to the equations

$\sum_{j=1}^{N_{k}}n_{j}A_{j}^{m}=\Lambda_{m}\qquad(m=1,2,\ldots,k)\;,$ (24)

where

$\Lambda_{m}=\frac{(-1)^{m-1}}{(m-1)!}\;\lim_{x\rightarrow
0}\;\frac{d^{m}}{dx^{m}}\;\ln\left(\sum_{n=0}^{m}a_{n}x^{n}\right)\;.$

Some general mathematical properties of the self-similar factor approximants
(18) are described in Appendix.

However, equations (24) uniquely define all control parameters $A_{j}$ and
$n_{j}$ only for even orders of $k$. This is because for even $k=2,4,\ldots$,
when $N_{k}=k/2$, there are $k/2$ parameters $A_{j}$ and $k/2$ parameters
$n_{j}$, hence $k$ unknowns, all of which are defined by $k$ equations (24).
Unfortunately, for odd orders $k=3,5,\ldots$, when $N_{k}=(k+1)/2$, there are
again $k$ equations (24), but $(k+1)/2$ parameters $A_{j}$ and $(k+1)/2$
parameters $n_{j}$, so that in total $k+1$ unknowns, which is larger by one
than the number of equations. To uniquely define all control parameters for
odd orders $k$, it is necessary to impose an additional constraint, for
example, by fixing one of the parameters $A_{j}$, which, actually, is a not
well justified ad hoc procedure [23]. Otherwise, one is forced to consider
only even approximants [24] neglecting at all odd orders.

In the present paper, we suggest a way of overcoming the problem of odd
approximation orders. To check the accuracy of the suggested method, we
consider the most difficult challenge of the extrapolation problem, when one
needs to evaluate the large-variable limit of the sought function. In the
following section, we formulate the method of dealing with odd orders.

##  4 Diff-log transformation

As is explained above, in odd orders the number of the sought parameters is
larger by one then the number of training equations, which requires to impose
an additional constraint for uniquely defining all control parameters. This
can be done as follows.

Let us define the diff-log transformation

$D[\;f(x)\;]\equiv\frac{d}{dx}\;\ln f(x)\;.$ (25)

For a function with a boundary condition $f(0)=1$, the inverse transformation
is

$f(x)=\exp\left\\{\int_{0}^{x}D[\;f(x)\;]\;dx\right\\}\;.$ (26)

If the large-variable limit of the function is

$f(x)\simeq Bx^{\nu}\qquad(x\rightarrow\infty)\;,$ (27)

then the large-variable exponent can be found from the limit

$\nu=\lim_{x\rightarrow\infty}x\;D[\;f(x)\;]\;.$ (28)

Accomplishing for the considered truncated series (1) the diff-log
transformation

$D[\;f_{k}(x)\;]=\frac{d}{dx}\;\ln f_{k}(x)\;,$ (29)

we expand the latter in powers of $x$ getting

$D_{k}(x)=\sum_{n=0}^{k}c_{n}x^{n}\qquad(x\rightarrow 0)\;,$ (30)

with $b_{0}=a_{1}$. For this expansion, a self-similar factor approximant of
odd order reads as

$D_{k}^{*}(x)=a_{1}\prod_{j=1}^{(k+1)/2}(1+M_{j}x)^{m_{j}}\;,$ (31)

with the control parameters $M_{j}$ and $m_{j}$ satisfying the system of
equations

$\sum_{j=1}^{(k+1)/2}m_{j}\;M_{j}^{p}=\frac{(-1)^{p-1}}{(p-1)!}\;\left[\;\frac{d^{p}}{dx^{p}}\;\ln\left(\sum_{n=0}^{p}c_{n}x^{n}\right)\;\right]_{x=0}\;,$
(32)

where $p=1,2,\ldots,k$. To uniquely define all control parameters for odd
orders $k$, we have to impose an additional constraint.

By condition (28), the self-similar approximation for the exponent is

$\nu_{k}=\lim_{x\rightarrow\infty}x\;D_{k}^{*}(x)\;,$ (33)

which requires that the approximant (31), in the large-variable limit, behaves
as

$D_{k}^{*}(x)\simeq D_{k}x^{-1}\qquad(x\rightarrow\infty)\;,$ (34)

with the additional constraint

$\sum_{j=1}^{(k+1)/2}m_{j}=-1\;.$ (35)

This allows to uniquely define the amplitude $D_{k}$, hence the large-variable
exponent

$\nu_{k}=D_{k}=a_{1}\prod_{j=1}^{(k+1)/2}M_{j}^{m_{j}}\;.$ (36)

The found exponent $\nu_{k}$ can be used as a fixed constraint for uniquely
defining the parameters $A_{j}$ and $n_{j}$ for odd orders of $k$ in equation
(24).

##  5 Self-similar Borel summation

When the coefficients of the given expansion (1) are growing fast, it is known
that Borel summation and its variants can help for defining an effective limit
of a series, provided that the Borel transform can be efficiently summed. As
is mentioned in the Introduction, the Padé-Borel summation is not appropriate
for finding the limiting behavior at $x\rightarrow\infty$. However, we can
employ the self-similar factor approximants for summing the Borel transform,
combining this with the Borel transformation. This way, for short, will be
called self-similar Borel summation.

For example, the given expansion (1) can be subject to the Borel-Leroy
transformation

$B_{k}(x)=\sum_{n=0}^{k}\frac{a_{n}x^{n}}{\Gamma(n+1+u)}\;,$ (37)

where $u$ is an additional control parameter. For $u=0$, we have the standard
Borel transform that we shall use. For the truncated series (37), we can
construct self-similar factor approximants

$B_{k}^{*}(x)=\frac{1}{\Gamma(1+u)}\prod_{j=1}^{N_{k}}(1+A_{j}x)^{n_{j}}\;,$
(38)

where the control parameters $A_{j}$ and $n_{j}$ are defined in the same way
as explained above, but for the expansion (37), so that $A_{j}$ and $n_{j}$
here are different from those in the approximant (18).

Applying the inverse Borel transformation gives the approximant

$f_{k}^{*}(x)=\int_{0}^{\infty}e^{-t}\;t^{u}\;B_{k}^{*}(xt)\;dt\;.$ (39)

With the large-variable limit

$B_{k}^{*}(x)\simeq C_{k}x^{\nu_{k}}\qquad(x\rightarrow\infty)$ (40)

for the self-similar approximant (38), where the amplitude is

$C_{k}=\frac{1}{\Gamma(1+u)}\prod_{j=1}^{N_{k}}A_{j}^{n_{j}}$ (41)

and the large-variable exponent is

$\nu_{k}=\sum_{j=1}^{N_{k}}n_{j}\;,$ (42)

we obtain the large-variable limit for the function of interest,

$f_{k}^{*}(x)\simeq B_{k}x^{\nu_{k}}\qquad(x\rightarrow\infty)\;,$ (43)

with the amplitude

$B_{k}=C_{k}\Gamma(1+u+\nu_{k})=\frac{\Gamma(1+u+\nu_{k})}{\Gamma(1+u)}\prod_{j=1}^{N_{k}}A_{j}^{n_{j}}\;.$
(44)

For even orders of $k$, all control parameters are uniquely defined. And for
odd orders of $k$, we follow the same trick as above, by accomplishing the
diff-log transformation

$D[\;B_{k}(x)\;]\equiv\frac{d}{dx}\;\ln B_{k}(x)$ (45)

for the Borel transform (37), constructing its factor approximant
$D_{k}^{*}(x)$, and finding the large-variable exponent $\nu_{k}$, which
serves as an additional constraint for uniquely determining all control
parameters $A_{j}$ and $n_{j}$.

##  6 Zero-dimensional $\varphi^{4}$ theory

In this and the following section, we apply the developed method to the test
problems, whose large-variable behavior is known. This will explicitly
illustrate the efficiency of the method and its convergence properties. Such
an illustration is necessary before applying the method to the problems whose
large-variable behavior is not available.

Let us start with the consideration of the generating functional

$Z(g)=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}\exp(-\varphi^{2}-g\varphi^{4})\;d\varphi$
(46)

of the so-called zero-dimensional $\varphi^{4}$ theory, which serves as a
touch-stone for checking almost any novel approach. Expanding the integrand in
powers of the coupling $g\rightarrow 0$ yields the expansion

$Z_{k}(g)=\sum_{n=0}^{k}a_{n}g^{n}\qquad(g\rightarrow 0)\;,$ (47)

with the coefficients

$a_{n}=\frac{(-1)^{n}}{\sqrt{\pi}\;n!}\;\Gamma\left(2n+\frac{1}{2}\right)\;.$
(48)

This expansion diverges for any finite value of $g$.

Our aim is to predict, by employing the developed approach, the large-variable
limiting behavior

$Z_{k}^{*}(g)\simeq B_{k}g^{\nu_{k}}\qquad(g\rightarrow\infty)$ (49)

and to compare our predictions with the known exact numerical limit

$Z(g)\simeq 1.022765\;g^{-1/4}\qquad(g\rightarrow\infty)\;.$ (50)

As is described in the previous sections, we employ two variants of the
method, first, by directly constructing self-similar factor approximants and,
second, by resorting to self-similar Borel summation. The results are shown in
Table 1 and Table 2, where the large-variable amplitudes $B_{k}$ and exponents
$\nu_{k}$ are given, with the corresponding percentage errors. As is seen,
knowing only the small-variable expansion for $g\rightarrow 0$, we are able to
find the large-variable behavior for $g\rightarrow\infty$. The use of the
self-similar Borel summation improves the accuracy of the predicted results.
The accuracy is to be accepted as rather good, if we remember that the most
difficult case is considered, when the strong-coupling behavior for
$g\rightarrow\infty$ is found being based solely on the knowledge of several
terms of weak-coupling expansion.

$k$ | $B_{k}$ | $\varepsilon(B_{k})\%$ | $\nu_{k}$ | $\varepsilon(\nu_{k})\%$
---|---|---|---|---
2 | 0.823 | -19.5 | -0.094 | -62.5
3 | 0.805 | -21.3 | -0.137 | -45.3
4 | 0.806 | -21.2 | -0.129 | -48.4
5 | 0.808 | -21.0 | -0.159 | -36.6
6 | 0.806 | -21.2 | -0.148 | -40.6
7 | 0.815 | -20.3 | -0.172 | -31.2
8 | 0.810 | -20.8 | -0.161 | -35.6
9 | 0.823 | -19.6 | -0.181 | -27.5
10 | 0.814 | -20.4 | -0.170 | -32.0
11 | 0.830 | -18.9 | -0.188 | -24.7
12 | 0.819 | -19.9 | -0.177 | -29.3
13 | 0.836 | -18.3 | -0.193 | -22.6
14 | 0.824 | -19.4 | -0.182 | -27.1
Table 1: Zero-dimensional $\varphi^{4}$ theory. Strong-coupling amplitudes and exponents, with the related errors in percents, predicted by self-similar factor approximants. $k$ | $B_{k}$ | $\varepsilon(B_{k})\%$ | $\nu_{k}$ | $\varepsilon(\nu_{k})\%$
---|---|---|---|---
2 | 0.898 | -12.2 | -0.207 | -17.2
3 | 0.936 | -8.53 | -0.231 | -7.47
4 | 0.924 | -9.62 | -0.226 | -9.73
5 | 0.955 | -6.63 | -0.238 | -4.86
6 | 0.941 | -7.96 | -0.233 | -6.80
7 | 0.967 | -5.43 | -0.241 | -3.56
8 | 0.953 | -6.83 | -0.237 | -5.23
10 | 0.961 | -6.00 | -0.239 | -4.26
11 | 0.978 | -4.39 | -0.243 | -2.62
12 | 0.968 | -5.38 | -0.241 | -3.59
13 | 0.983 | -3.87 | -0.246 | -2.19
14 | 0.973 | -4.88 | -0.242 | -3.10
Table 2: Zero-dimensional $\varphi^{4}$ theory. Strong-coupling amplitudes and
exponents, with the related errors in percents, predicted by self-similar
Borel summation.

##  7 Anharmonic oscillator

The other touch-stone example is the anharmonic oscillator with the
Hamiltonian

$H=-\;\frac{1}{2}\;\frac{d^{2}}{dx^{2}}+\frac{1}{2}\;x^{2}+gx^{4}\;,$ (51)

in which $x\in(-\infty,\infty)$ and $g\geq 0$. The ground-state energy of the
oscillator can be expressed [25, 26] in the form of the weak-coupling
expansion

$E_{k}(g)=\sum_{n=0}^{k}a_{n}g^{n}\qquad(g\rightarrow 0)\;,$ (52)

with the coefficients shown in Table 3. This series is also divergent for any
finite $g$.

$n$ | $a_{n}$
---|---
0 | 1/2=0.5
1 | 3/4=0.75
2 | -0.2625$\times 10$
3 | 0.208125$\times 10^{2}$
4 | -0.2412890625$\times 10^{3}$
5 | 0.358098046875$\times 10^{4}$
6 | -0.639828134766$\times 10^{5}$
7 | 0.132973372705$\times 10^{7}$
8 | -0.314482146928$\times 10^{8}$
9 | 0.833541603263$\times 10^{9}$
10 | -0.244789407028$\times 10^{11}$
11 | 0.789333316003$\times 10^{12}$
12 | -0.277387769635$\times 10^{14}$
13 | 0.105564665831$\times 10^{16}$
14 | -0.432681068354$\times 10^{17}$
Table 3: Anharmonic oscillator. Coefficients of weak-coupling expansion for
the ground-state energy.

Our aim is to predict, being based solely on this expansion, the strong-
coupling behavior

$E_{k}^{*}(g)\simeq B_{k}g^{\nu_{k}}\qquad(g\rightarrow\infty)$ (53)

and to compare it with the known exact limit

$E_{k}(g)\simeq 0.667986\;g^{1/3}\qquad(g\rightarrow\infty)\;.$ (54)

The results are given in Table 4 for the direct self-similar factor
approximants and in Table 5 for the self-similar Borel summation. As we see,
the Borel summation improves the accuracy by an order. Again we have to accept
that even directly applying self-similar factor approximants, we get not bad
accuracy, if to keep in mind that solely several terms of the weak-coupling
expansion are used.

$k$ | $B_{k}$ | $\varepsilon(B_{k})\%$ | $\nu_{k}$ | $\varepsilon(\nu_{k})\%$
---|---|---|---|---
2 | 0.729 | 9.20 | 0.176 | -47.1
3 | 0.757 | 13.4 | 0.241 | -27.8
4 | 0.755 | 13.1 | 0.231 | -30.6
5 | 0.754 | 12.9 | 0.267 | -19.8
6 | 0.756 | 13.2 | 0.257 | -22.9
7 | 0.748 | 12.0 | 0.282 | -15.5
8 | 0.752 | 12.6 | 0.272 | -18.4
9 | 0.742 | 11.1 | 0.291 | -12.7
10 | 0.748 | 11.9 | 0.282 | -15.5
11 | 0.737 | 10.3 | 0.297 | -10.9
12 | 0.743 | 11.3 | 0.289 | -13.4
13 | 0.732 | 9.61 | 0.302 | -9.51
14 | 0.739 | 10.7 | 0.294 | -11.9
Table 4: Anharmonic oscillator. Strong-coupling amplitudes and exponents, with the related percentage errors, predicted by self-similar factor approximants. $k$ | $B_{k}$ | $\varepsilon(B_{k})\%$ | $\nu_{k}$ | $\varepsilon(\nu_{k})\%$
---|---|---|---|---
2 | 0.727 | 8.87 | 0.300 | -10.0
3 | 0.727 | 8.89 | 0.289 | -13.2
4 | 0.727 | 8.90 | 0.289 | -13.3
5 | 0.713 | 6.78 | 0.310 | -6.88
6 | 0.712 | 6.57 | 0.312 | -6.44
7 | 0.702 | 5.04 | 0.319 | -4.32
10 | 0.698 | 4.53 | 0.322 | -3.42
11 | 0.695 | 4.08 | 0.324 | -2.92
13 | 0.690 | 3.23 | 0.326 | -2.06
14 | 0.688 | 3.00 | 0.327 | -1.85
Table 5: Anharmonic oscillator. Strong-coupling amplitudes and exponents, with
the related percentage errors, predicted by self-similar Borel summation.

##  8 $O(N)$ symmetric $\varphi^{4}$ field theory

Now let us consider the problems, where the answer for the strong-coupling
limit is not known, so that the found results are to be considered as
predictions. We start with weak-coupling expansions whose coefficients are
given with the accuracy not more than six digits. Therefore rounding up the
numbers, we can set zero the quantities of order or less than $10^{-6}$. We
shall employ both ways of defining the strong-coupling amplitudes and
exponents, by directly constructing self-similar factor approximants and by
using the self-similar Borel summation.

An interesting problem in field theory is the behavior of Gell-Mann-Low
functions. Let us start with the $O(N)$ symmetric $\varphi^{4}$ field theory,
whose Gell-Mann-Low function $\beta(g)$ is a function of the effective
coupling parameter $g=\lambda/(4\pi)^{2}$ entering the equation

$\mu\;\frac{\partial g}{\partial\mu}=\beta(g)\;,$ (55)

where $\mu$ is the renormalization scale. This function can be calculated,
within the minimal subtraction scheme $\overline{\rm MS}$, by means of
perturbation theory yielding

$\beta_{k}(g)=g^{2}\sum_{n=0}^{k}b_{n}g^{n}\qquad(g\rightarrow 0)\;.$ (56)

The coefficients $b_{n}$ are known in six-loop approximation [27] for
arbitrary $N$ and for $N=1$, in the seven-loop approximation [28]. In Table 6,
we give the explicit values of the coefficients for $N=0,1,2,3,4$.

$N$ | $0$ | $1$ | $2$ | $3$ | $4$
---|---|---|---|---|---
$b_{0}$ | 2.66667 | 3 | 3.33333 | 3.66667 | 4.0
$b_{1}$ | -4.66667 | -5.66667 | -6.66667 | -7.66667 | -8.66667
$b_{2}$ | 25.4571 | 32.5497 | 39.9478 | 47.6514 | 55.6606
$b_{3}$ | -200.926 | -271.606 | -350.515 | -437.646 | -532.991
$b_{4}$ | 2003.98 | 2848.57 | 3844.51 | 4998.62 | 6317.66
$b_{5}$ | -23314.7 | -34776.1 | -48999.1 | -66242.7 | -86768.4
$b_{6}$ |  | 474651 |  |  |
Table 6: Coefficients of weak-coupling expansion for the Gell-Mann-Low
function of the $N$-component $\varphi^{4}$ field theory.

Employing our method, we find the large-variable amplitudes and exponents for
the limit

$\beta_{k}^{*}(g)\simeq B_{k}g^{\nu_{k}}\qquad(g\rightarrow\infty)\;,$ (57)

for different $N$ from $0$ to $4$, by the direct self-similar approximation
and using the self-similar Borel summation. The results are presented in Table
7 ($N=0$), Table 8 ($N=1$), Table 9 ($N=2$), Table 10 ($N=3$), and Table 11
($N=4$).

$k$ | $B_{k}$ | $\nu_{k}$ | $B_{k}(Borel)$ | $\nu_{k}(Borel)$
---|---|---|---|---
2 | 1.747 | 1.809 | 2.42 | 1.53
3 | 1.696 | 1.764 | 1.57 | 1.77
4 | 1.699 | 1.769 | 1.42 | 1.83
5 | 1.698 | 1.764 |  |
Table 7: Strong-coupling amplitudes and exponents for the Gell-Mann-Low function of the $N$-component $\varphi^{4}$ field theory, predicted by self-similar factor approximants and self-similar Borel summation for $N=0$. $k$ | $B_{k}$ | $\nu_{k}$ | $B_{k}(Borel)$ | $\nu_{k}(Borel)$
---|---|---|---|---
2 | 1.922 | 1.803 | 2.69 | 1.51
3 | 1.856 | 1.750 | 1.84 | 1.72
4 | 1.859 | 1.757 | 1.69 | 1.77
5 | 1.857 | 1.749 |  |
6 | 1.857 | 1.750 | 1.70 | 1.77
Table 8: Strong-coupling amplitudes and exponents for the Gell-Mann-Low function of the $N$-component $\varphi^{4}$ field theory, predicted by self-similar factor approximants and self-similar Borel summation for $N=1$. $k$ | $B_{k}$ | $\nu_{k}$ | $B_{k}(Borel)$ | $\nu_{k}(Borel)$
---|---|---|---|---
2 | 2.102 | 1.800 | 2.96 | 1.49
3 | 2.015 | 1.735 | 2.16 | 1.67
4 | 2.020 | 1.744 | 2.03 | 1.71
5 | 2.017 | 1.735 | 2.18 | 1.67
Table 9: Strong-coupling amplitudes and exponents for the Gell-Mann-Low function of the $N$-component $\varphi^{4}$ field theory, predicted by self-similar factor approximants and self-similar Borel summation for $N=2$. $k$ | $B_{k}$ | $\nu_{k}$ | $B_{k}(Borel)$ | $\nu_{k}(Borel)$
---|---|---|---|---
2 | 2.286 | 1.798 | 3.21 | 1.49
3 | 2.175 | 1.718 | 2.49 | 1.63
4 | 2.182 | 1.732 | 2.93 | 1.66
5 | 2.178 | 1.719 | 2.50 | 1.63
Table 10: Strong-coupling amplitudes and exponents for the Gell-Mann-Low function of the $N$-component $\varphi^{4}$ field theory, predicted by self-similar factor approximants and self-similar Borel summation for $N=3$. $k$ | $B_{k}$ | $\nu_{k}$ | $B_{k}(Borel)$ | $\nu_{k}(Borel)$
---|---|---|---|---
2 | 2.474 | 1.797 | 3.45 | 1.49
3 | 2.336 | 1.701 | 2.82 | 1.60
4 | 2.345 | 1.721 | 2.74 | 1.62
5 | 2.340 | 1.702 | 2.81 | 1.61
Table 11: Strong-coupling amplitudes and exponents for the Gell-Mann-Low
function of the $N$-component $\varphi^{4}$ field theory, predicted by self-
similar factor approximants and self-similar Borel summation for $N=4$.

In Tables $7$ to $12$, we give the predicted values for the strong-coupling
amplitudes and exponents of the Gell-Mann-Low functions. Since the exact
behavior of the Gell-Mann-Low functions in the strong-coupling limit is not
known, the actual errors of the predicted values cannot be defined. What one
could do is to compare our results with calculations accomplished by other
methods.

Estimates for the strong-coupling exponent $\nu$ of the Gell-Mann-Low function
of the $N$-component $\varphi^{4}$ field theory have been done by other
methods only for $N=1$. Thus, Borel-type summation with conformal mapping
gives [29, 30] the estimates around $\nu=2$, although this value essentially
depends on the type of the Borel transform and on the used conformal mapping.
Nevertheless, it is in agreement with our prediction.

The large-variable exponent can also be estimated by resorting to optimized
perturbation theory based on the introduction of control functions or
parameters in perturbative series and defining these parameters by some
optimization conditions, like minimal difference or minimal derivative
conditions. This approach, advanced in [31, 32], has been used for numerous
problems (see reviews [15, 16]), including quantum field theory (e.g. [33, 34,
35, 36, 37, 38, 39]). The strong-coupling behavior of the Gell-Mann-Low
function for $\varphi^{4}$ theory with $N=1$ is estimated in [40], were it is
found that for $g\rightarrow\infty$, in the second order of perturbation
theory, one gets $\nu\approx 1.5$, which agrees with the second order
approximation in Table 8.

##  9 Quantum electrodynamics

The Gell-Mann-Low function $\beta(\alpha)$ in quantum electrodynamics is a
function of the coupling parameter $\alpha$ in the $\overline{\rm MS}$ scheme,
satisfying the equation

$\mu^{2}\;\frac{\partial}{\partial\mu^{2}}\;\left(\frac{\alpha}{\pi}\right)=\beta(\alpha)\;.$
(58)

The weak-coupling expansion, in five-loop approximation, taking into account
the electron, but neglecting the contributions of leptons with higher masses,
that is, muons and tau leptons, reads as

$\beta_{k}(\alpha)=\left(\frac{\alpha}{\pi}\right)^{2}\sum_{n=0}^{k}b_{n}\left(\frac{\alpha}{\pi}\right)^{n}\qquad(\alpha\rightarrow
0)\;,$ (59)

with the coefficients [41]

$b_{0}=\frac{1}{3}\;,\qquad b_{1}=\frac{1}{4}\;,\qquad b_{2}=-0.107639\;,$
$b_{3}=-0.523614\;,\qquad b_{4}=1.47072\;.$

For the large-coupling behavior

$\beta_{k}^{*}(\alpha)\simeq
B_{k}\left(\frac{\alpha}{\pi}\right)^{\nu_{k}}\qquad(\alpha\rightarrow\infty)\;,$
(60)

we find the amplitude and exponent shown in Table 12. Notice that the large-
variable exponents diminish with increasing order.

$k$ | $B_{k}$ | $\nu_{k}$ | $B_{k}(Borel)$ | $\nu_{k}(Borel)$
---|---|---|---|---
2 | 0.416 | 2.466 | 0.333 | 2.64
3 | 0.469 | 2.167 | 0.587 | 2.12
4 | 0.476 | 2.096 | 1.560 | 1.64
Table 12: Strong-coupling amplitudes and exponents for the Gell-Mann-Low
function of quantum electrodynamics, predicted by self-similar factor
approximants and self-similar Borel summation.

It is useful to note that the Borel summation is not necessarily preferable
than the direct self-similar approximation. The Borel summation is supposed to
work well, when the coefficients of the asymptotic series factorially grow,
while the Borel transform compensates this growth. However, when the
coefficients $b_{n}$ of the perturbative series do not exhibit persistent
growth close to factorial, but vary rather chaotically, Borel summation can
result in sharp variations of the calculated quantity. This happens for the
series representing the Gel-Mann-Low function of quantum electrodynamics. In
the case of factorial growth, the ratio $|b_{n+1}/(n+1)b_{n}|$ would be
approximately constant, but here we have rather different abruptly varying
ratios $|b_{2}/2b_{1}|=0.215$, but $|b_{3}/3b_{2}|=1.622$, and
$|b_{4}/4b_{3}|=0.702$. Because of this, the Borel summation in different
orders essentially varies, as is seen from Table 12. On the contrary, the
direct self-similar approximation is not so sensible to the coefficient
variation, which results in more smooth changes of the quantities in different
orders.

Self-similar factor approximants do not strongly depend on the used scheme.
Thus for the coefficients $b_{n}$, calculated [41] in the on-shell scheme,
employing the self-similar factor approximants, we get $B=0.43$ and
$\nu=2.19$, while in the momentum subtraction scheme, $B=0.49$ and $\nu=2.31$.

Since the analytic expression for the whole $\beta$ function can be explicitly
represented by self-similar factor approximants (18), it is possible to solve
the equation for the running coupling (58), taking for the boundary condition
the value $\alpha(m_{Z})=0.007815$ at the Z-boson mass $m_{Z}=91.1876$ GeV.
The resulting running coupling grows from zero to divergence at
$\mu_{0}\approx 10^{261}$ GeV by the law

$\alpha\simeq\frac{2.74}{[\;\ln(\mu_{0}/\mu)\;]^{0.68}}\qquad(\mu\rightarrow\mu_{0}-0)\;.$
(61)

This value of $\mu_{0}$ is incomparably larger than the Landau pole that is of
the order of $10^{30}-10^{40}$ GeV [42, 43]. Because of the so large value of
the divergence point $\mu_{0}\approx 10^{261}$ GeV, it is hardly achievable.
Also notice that, with the increasing order of the approximants, the pole
shifts closer to infinity.

The self-similar factor approximants, as has been checked by numerous
examples, do possess the power of well extrapolating perturbative series up to
large-variable limit, including the variable tending to infinity, provided the
considered perturbative expansion is correctly derived taking into account all
physics of the problem, even if the influence of some effects seems to be
negligible for asymptotically small variables. Therefore the occurrence of the
divergence of the running coupling at a finite value of the scale $\mu_{0}$ is
not caused by the extrapolation as such. To our mind, this is the result of
incomplete account of physical effects at small couplings, which leads to the
loss of information and to not completely correct coefficients of the
perturbative expansions. Self-similar approximation theory, by its
construction, has the ability of extracting the information contained in the
perturbative coefficients, even when the influence of some effects is small,
and allows for the self-similar extrapolation of this information to large
variables, such as coupling. However if some physical effects have been
neglected in the derivation of a perturbative expansion, then the expansion
coefficients contain no information on them, hence there is nothing to be
extrapolated. For instance, in the derivation of the weak-coupling expansion
(59), the contributions of leptons with higher masses, that is, muons and tau
leptons, have been neglected, hence the coefficients $b_{n}$ contain no
information on the effects that can become important at large coupling. Not
correctly defined coefficients can lead to the appearance of unphysical
divergences.

##  10 Quantum chromodynamics

The running quark-gluon coupling $\alpha_{s}$, as a function of the
renormalization scale $\mu$, satisfies the equation

$\mu^{2}\;\frac{\partial\alpha_{s}}{\partial\mu^{2}}=\beta(\alpha_{s})\;.$
(62)

Keeping in mind the realistic case of three colors ($N_{c}=3$), the beta
function can be written [44] as the weak-coupling expansion

$\beta_{k}(\alpha_{s})=-\;\frac{\alpha_{s}^{2}}{4\pi}\sum_{n=0}^{k}b_{n}\left(\frac{\alpha_{s}}{4\pi}\right)^{n}\qquad(\alpha_{s}\rightarrow
0)\;.$ (63)

In the five-loop approximation, with the $\overline{\rm MS}$ renormalization
scheme, the coefficients are [45, 46, 47]

$b_{0}=11-\;\frac{2}{3}\;n_{f}\;,\qquad
b_{1}=102-\;\frac{38}{3}\;n_{f}\;,\qquad
b_{2}=\frac{2857}{2}\;-\;\frac{5033}{18}\;n_{f}\;+\frac{325}{54}\;n_{f}^{2}\;,$
$b_{3}=\frac{149753}{6}+3564\zeta_{3}-\left(\frac{1078361}{162}+\frac{6508}{27}\zeta_{3}\right)\;n_{f}\;+\;\left(\frac{50065}{162}+\frac{6472}{81}\zeta_{3}\right)\;n_{f}^{2}\;+\;\frac{1093}{729}\;n_{f}^{3}\;,$
$b_{4}=\frac{8157455}{16}+\frac{621885}{2}\;\zeta_{3}-\;\frac{88209}{2}\;\zeta_{4}-288090\;\zeta_{5}\;+$
$+\;\left(-\;\frac{336460813}{1944}-\;\frac{4811164}{81}\zeta_{3}+\frac{33935}{6}\;\zeta_{4}+\frac{1358995}{27}\;\zeta_{5}\right)\;n_{f}\;+$
$+\;\left(\frac{25960913}{1944}+\frac{698531}{81}\zeta_{3}-\;\frac{10526}{9}\;\zeta_{4}-\;\frac{381760}{81}\;\zeta_{5}\right)\;n_{f}^{2}\;+$
$+\;\left(-\;\frac{630559}{5832}-\;\frac{48722}{243}\zeta_{3}+\frac{1618}{27}\;\zeta_{4}+\frac{460}{9}\;\zeta_{5}\right)\;n_{f}^{3}\;+\;\left(\frac{1205}{2916}-\;\frac{152}{81}\;\zeta_{3}\right)n_{f}^{4}\;,$
(64)

where $\zeta_{n}$ is the Riemann zeta function. For the physically realistic
case of six flavors ($n_{f}=6$), we have

$b_{0}=7\;,\qquad b_{1}=26\;,\qquad b_{2}=-32.5\;,\qquad b_{3}=2472.28\qquad
b_{4}=-11783.7\;.$

Using the self-similar factor approximants gives in second order

$\beta_{2}^{*}(\alpha_{s})=-\;\frac{7}{4\pi}\;\alpha_{s}^{2}(1+A_{1}\alpha_{s})^{n_{1}}\;,$
(65)

where the control parameters $A_{1}$ and $n_{1}$ are found from the training
conditions (24), yielding

$A_{1}=0.494517\;,\qquad n_{1}=0.597701\;.$

In third order, the beta function reads as

$\beta_{3}^{*}(\alpha_{s})=-\;\frac{7}{4\pi}\;\alpha_{s}^{2}(1+A_{1}\alpha_{s})^{n_{1}}(1+A_{2}\alpha_{s})^{n_{2}}\;.$
(66)

The control parameters $A_{1}$, $n_{1}$, and $n_{2}$ are prescribed by the
training conditions (22), as is described in Sec. 3, and $A_{2}$ is treated as
an additional control parameter defined by the minimal-derivative optimization
condition $|\partial\nu_{3}/\partial A_{2}|=\delta$, where $\delta=10^{-6}$,
as is explained in Appendix. Thus we obtain

$A_{1}=0.490298\;,\qquad n_{1}=0.602838\;,\qquad A_{2}=412.291\;,\qquad
n_{2}=0.734531\times 10^{-8}\;.$

This leads to the strong-coupling limit

$\beta_{2}^{*}(\alpha_{s})\simeq-0.366\;\alpha_{s}^{2.598}\;,\qquad\beta_{3}^{*}(\alpha_{s})\simeq-0.362\;\alpha_{s}^{2.603}\qquad(\alpha_{s}\rightarrow\infty)\;.$
(67)

Similarly, employing the self-similar Borel summation of Sec. 5, we get the
strong-coupling behavior

$\beta_{2}^{*}(\alpha_{s})\simeq-0.255\;\alpha_{s}^{2.748}\;,\qquad\beta_{3}^{*}(\alpha_{s})\simeq-0.254\;\alpha_{s}^{2.751}\qquad(\alpha_{s}\rightarrow\infty)\;.$
(68)

As is mentioned in Appendix, self-similar approximants are uniquely defined
for the functions in the complex range. However, when we are interested only
in real-valued functions, we have to discard occasionally appearing complex-
valued approximants. In the case of the Gell-Mann-Low function of quantum
chromodynamics, the fourth order approximant is complex and is discarded. So,
we deal with the second and third orders that are close to each other.

Having in hands the beta function, it is straightforward to solve equation
(62) for the running coupling. As a boundary condition, we can take the value
$\alpha(m_{Z})=0.1181$ at the Z-boson mass $m_{Z}=91.1876$ GeV. The running
coupling, with decreasing $\mu$, grows from zero to the logarithmic divergence

$\alpha_{s}\simeq\frac{0.91}{[\;\ln(\mu/\mu_{c})\;]^{0.63}}\qquad(\mu\rightarrow\mu_{c}+0)$
(69)

at the point $\mu_{c}=0.1$ GeV, which is an order lower than the Landau pole
being about $1$ GeV [44].

Similarly to quantum electrodynamics, the appearance of the divergence of the
running coupling is not caused by the extrapolation as such, but it can be
induced by the absence of information on physical effects that could become
important at large couplings, while they where completely ignored in the
derivation of the expansion (63). For example, the possibility of arising
bound states is ignored in the derivation of expansion (63), which makes the
coefficients $b_{n}$ incompletely defined, thus making impossible the precise
extrapolation to large couplings. Self-similar approximation theory is able to
extrapolate functions to arbitrary values of their variables, provided the
perturbative expansion contains information for being extrapolated. When there
is no information, there is nothing to be extrapolated, and unphysical
divergences can happen. It is not the method of extrapolation that is guilty
but incorrect expansion coefficients.

##  11 Conclusion

We have addressed an important but rather complicated problem of the
possibility to reconstruct the whole functions being based only on their
asymptotic expansions over coupling parameters or other variables. Of special
interest is the problem of finding the strong-coupling or large-variable
behavior of physical quantities from the knowledge of only several terms of
their weak-coupling or small-variable expansions. The method is presented
allowing for resolving this problem. The approach is based on the ideas of
renormalization group theory, approximation theory, dynamical theory, and
optimal control theory.

The method allows for the extrapolation of small-variable expansions because
the coefficients of such expansions contain information on the whole functions
to be restored. This information can be extracted by observing self-similarity
between the subsequent approximations and considering the passage from one
approximation to another as the motion in the approximation space, with the
order of approximation playing the role of discrete time. The sequence of
approximations can be transformed to a dynamical system called approximation
cascade. The fixed point of the cascade represents the effective limit of the
approximation sequence. The extrapolation is efficient even for the most
difficult case of finding large-variable or strong-coupling limits.

Convergence and accuracy of the method are illustrated by explicit examples,
including the so-called zero-dimensional field theory and anharmonic
oscillator, which serve as typical touchstones for testing summation methods
of divergent series. Strong-coupling behavior of Gell-Mann-Low functions in
multicomponent field theory, quantum electrodynamics, and quantum
chromodynamics is found, being based on their weak-coupling perturbative
expansions.

##  Appendix A Appendix. Properties of self-similar factor approximants

Self-similar factor approximants

$f_{k}^{*}(x)=\prod_{j=1}^{N_{k}}(1+A_{j}x)^{n_{j}}\;,$ (A.1)

generally, can be considered on the complex plane [48] of the variable
$x\in\mathbb{C}$, although in applications one is more often interested in
real variables $x\in\mathbb{R}_{+}=[0,\infty)$. The form (A.1) is real-valued
provided that $x\in\mathbb{R}_{+}$ and either $A_{j}x>-1$, with $A_{j}$ and
$n_{j}$ being real, or in the product (A.1) there occur complex conjugate
pairs of $A_{j}$ and $n_{j}$, so that the product of two factors is real,
where

$\left|\;(1+A_{j}x)^{n_{j}}\;\right|^{2}=|\;1+A_{j}x\;|^{2{\rm
Re}\;n_{j}}\exp\\{-2{\rm Im}(n_{j}){\rm arg}(1+A_{j}x)\\}\;.$

The control parameters $A_{j}$ and $n_{j}$ are defined by the training
equations (23). When we are looking for a real-valued function, occasional
complex-valued solutions are discarded.

Considering the sequence of the terms $\\{f_{k}^{*}(x)\\}$, since
$N_{k}\rightarrow\infty$ as $k\rightarrow\infty$, one comes to the limiting
form

$f^{*}(x)=\prod_{j=1}^{\infty}(1+A_{j}x)^{n_{j}}\;.$ (A.2)

Definition 1. The sequence $\\{f_{k}^{*}(x)\\}$ converges to $f^{*}(x)$ at a
point $x$ if there are not more than a finite number of zero factors
$(1+A_{j}x)$ and

$0\leq|\;f^{*}(x)\;|<\infty\;.$ (A.3)

If the sequence $\\{f_{k}^{*}(x)\\}$ converges at all points $x$ of a given
domain $\mathbb{D}\subset\mathbb{R}_{+}$, then one says that it uniformly
converges on the given domain.

Cauchy criterion. The sequence $\\{f_{k}^{*}(x)\\}$ converges if and only if
for any $\varepsilon>0$ there is $j_{\varepsilon}>0$ such that

$\left|\;\prod_{j=j_{\varepsilon}}^{m}(1+A_{j}x)^{n_{j}}-1\;\right|<\varepsilon$
(A.4)

for all $m>j_{\varepsilon}$.

Theorem 1. The sequence $\\{f_{k}^{*}(x)\\}$ converges if and only if the
sequence of the sums

$S_{k}(x)=\sum_{j=1}^{N_{k}}n_{j}\ln(1+A_{j}x)$ (A.5)

converges.

Proof. The proof is clear from the identity

$f_{k}^{*}(x)=\exp\\{\ln f_{k}^{*}(x)\\}=e^{S_{k}(x)}\;.$

Corollary 1. From the Cauchy criterion and Theorem 1 it follows that

$\lim_{j\rightarrow\infty}(1+A_{j}x)^{n_{j}}=1$ (A.6)

and

$\lim_{j\rightarrow\infty}n_{j}\ln(1+A_{j}x)=0\;.$ (A.7)

This implies that either $A_{j}\rightarrow 0$ or $n_{j}\rightarrow 0$. In any
case

$\lim_{j\rightarrow\infty}n_{j}A_{j}=0\;.$ (A.8)

Theorem 2. If the sequence of the terms

$f_{k}(x)=\sum_{n=0}^{k}a_{n}x^{n}$ (A.9)

converges on a domain $\mathbb{D}\subset\mathbb{R}_{+}$ to a function

$f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}\;,$ (A.10)

then the sequence $\\{f_{k}^{*}(x)\\}$ converges on the same domain to
$f^{*}(x)$ coinciding with $f(x)$,

$f^{*}(x)=f(x)\;.$ (A.11)

Proof. The proof is given in Ref. [49].

Theorem 3. If $A_{j}x\geq-1$ and the sequence of the sums

$s_{k}=\sum_{j=1}^{N_{k}}|\;n_{j}A_{j}\;|$ (A.12)

converges, then the sequence of the approximants $\\{f_{k}^{*}(x)\\}$
converges.

Proof. For the approximants $f_{k}^{*}(x)$, we have

$|\;f_{k}^{*}(x)\;|\leq\exp\\{|\;S_{k}(x)\;|\\}\;,$

where

$|\;S_{k}(x)\;|\leq\sum_{j=1}^{N_{k}}|\;n_{j}\ln(1+A_{j}x)\;|\;.$

Since

$\ln(1+z)\leq z\qquad(z\geq-1)\;,$

then

$|\;S_{k}(x)\;|\leq\sum_{j=1}^{N_{k}}|\;n_{j}A_{j}\;|x=s_{k}x\;.$

Therefore

$|\;f_{k}^{*}(x)\;|\leq e^{s_{k}x}\;.$

The sequence $\\{s_{k}\\}$, by assumption converges, hence
$\\{\exp(s_{k}x)\\}$ converges, because of which $f_{k}^{*}(x)$ converges.

Corollary 2. If the sequence $\\{s_{k}\\}$ converges, then the scaling

$\overline{A}_{j}=\lambda_{j}A_{j}\;,\qquad\overline{n}_{j}=\frac{n_{j}}{\lambda_{j}}$
(A.13)

does not change the convergence of $s_{k}$, as far as
$n_{j}A_{j}=\bar{n}_{j}\bar{A}_{j}$.

Thus, when one of the parameters $A_{j}$ is not defined, as it happens for odd
approximants, it is possible to fix one of $A_{j}$ by treating it as a control
parameter prescribed by one of known optimization conditions [15, 16]. For
instance, being the most interested in the exponent of the large-variable
behavior

$f_{k}^{*}(x)\simeq B_{k}x^{\nu_{k}}\qquad(x\rightarrow\infty)\;,$ (A.14)

in which

$B_{k}=\prod_{j=1}^{N_{k}}A_{j}^{n_{j}}\;,\qquad\nu_{k}=\sum_{j=1}^{N_{k}}n_{j}\;,$
(A.15)

it is possible to resort to the minimal-derivative condition with respect to
$\partial\nu_{k}/\partial A_{j}$, with a fixed $j$. Very often, the solution
to the equation $\partial\nu_{k}/\partial A_{j}=0$ does not exist, then it is
sufficient to employ the optimization condition

$\min\left|\;\frac{\partial\nu_{k}}{\partial A_{j}}\;\right|=\delta\;,$ (A.16)

where $\delta$ is the required accuracy, say $\delta=10^{-6}$. However,
condition (A.16) makes calculations cumbersome, this is why it can be used
only in low orders and when the method of determining the exponent $\nu_{k}$
by means of the diff-log transformation of Sec. 4 does not have real-valued
solutions.

For realistic complicated problems, no general form of the coefficients in the
weak-coupling expansions are usually known, but merely a few terms of the
expansion are available. Therefore the general convergence properties cannot
be checked, but only numerical convergence can be observed, when the
subsequent approximations are close to each other.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Author contributions

V.I. Yukalov: Conceptualization, Methodology, Writing - Original Draft,
Writing - Review $\&$ Editing E.P. Yukalova: Software, Formal analysis,
Writing - Review $\&$ Editing

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