Self-similar extrapolation in quantum field theory

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 e-mail<EMAIL_ADDRESS>

###### Abstract

Calculations in field theory are usually accomplished by employing some
variants of perturbation theory, for instance using loop expansions. These
calculations result in asymptotic series in powers of small coupling
parameters, which as a rule are divergent for finite values of the parameters.
In this paper, a method is described allowing for the extrapolation of such
asymptotic series to finite values of the coupling parameters, and even to
their infinite limits. The method is based on self-similar approximation
theory. This theory approximates well a large class of functions, rational,
irrational, and transcendental. A method is presented, resulting in self-
similar factor approximants allowing for the extrapolation of functions to
arbitrary values of coupling parameters from only the knowledge of expansions
in powers of small coupling parameters. The efficiency of the method is
illustrated by several problems of quantum field theory.

Keywords: Asymptotic series, self-similar approximation theory, extrapolation
problem, Gell-Mann-Low functions, large-variable limit

##  1 Introduction

The solution of almost all nontrivial problems resorts to the use of some kind
of perturbation theory yielding asymptotic series in powers of small
parameters. However the physical values of the parameters of interest are
usually not small and often are even quite large. Thus we come to the
necessity of being able to extrapolate the asymptotic series, that are usually
divergent, to the finite values of the parameters of interest. Moreover,
sometimes the main interest is in the behaviour of the studied characteristics
at asymptotically large parameters tending to infinity. Padé approximants can
sometimes extrapolate small-variable series to the finite-variable region.
However, as is well known, they cannot describe the large-variable behaviour
at the variable tending to infinity, if only a small-variable expansion is
available [1]. Let us emphasize that here we keep in mind the case where no
large-variable behaviour is known, because of which it is impossible to turn
to two-point Padé approximants requiring the knowledge of the large-variable
behaviour [1, 2]. Similarly, it is not possible to use other interpolation
methods needing the information on the large-variable asymptotic behaviour for
the quantity of interest. Our aim here is to consider not interpolation but
extrapolation, when only the small-variable expansions are available.

Padé approximants, as is known, provide the best approximation for rational
functions, but the reason why they cannot predict the large-variable behavior
for irrational functions is rather straightforward. Really a $P_{M/N}(x)$ Padé
approximant in the limit of $x\rightarrow\infty$ behaves as $x^{M-N}$, where
$M$ and $N$ are integers. Moreover, the difference $M-N$ depends on the used
Padé approximant, but not uniquely defines the limiting exponent.

The other method of extrapolation, Borel summation, requires the knowledge of
the large-order behaviour of series coefficients, so that the error of the
truncated series be bounded by $C^{n}n!|z|^{n}$, where $C$ is a constant [3,
4, 5]. However, this large-order behavior of the expansion coefficients not
always is known. There exist several variants of the approach involving Borel
summation, including the combination of the Borel transform, conformal
mapping, and Padé approximants [3, 4, 5, 6, 7, 8].

Among other methods allowing for the large-variable extrapolation, it is
possible to mention the approach based on the introduction of control
functions defined by fixed-point conditions optimizing the series convergence
[9, 10]. Several variants of this approach have been considered, e.g., [9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. The introduction of control
functions requires to rearrange the considered series by either a change of
the variable containing trial parameters [4] or by incorporating trial
parameters into an initial approximation [9, 10].

The series convergence depends on the choice of an initial approximation. In
many cases, one chooses as an initial approximation a Gaussian form
corresponding to free particles. More complicated forms for the initial
approximation can also be chosen. For example, one can start perturbation
theory with a non-Gaussian approximation [23, 24, 25, 26] or one can use for
the initial approximation nontrivial Hamiltonians, as in the method of
Hamiltonian envelopes [27].

The methods mentioned above are numerical, require rearrangements of
perturbation series and rather involved calculations. It would be good to have
a simple analytical method that could extrapolate the standard Taylor series
derived by the usual perturbation theory in powers of a parameter, say the
coupling parameter.

Here we describe such a simple general method allowing for the extrapolation
of asymptotic series in powers of a small variable to arbitrary values of this
variable, including infinity. As illustration, we accomplish the extrapolation
of the series for several functions met in quantum field theory, whose
behaviour at large coupling parameters is of interest by its own. The
advantages of the suggested approach, as compared to other methods, are as
follows.

(i) First of all, the suggested method is analytical allowing for the
derivation of explicit forms of the sought functions. This makes it
straightforward to analyse the results with respect to different parameters
entering the problem, which is not always easy in numerical methods.

(ii) Moreover, numerical methods in some cases are not applicable, while the
presented method of extrapolation of asymptotic series can always be applied,
provided at least several terms of perturbation theory are available.

(iii) Even if numerical simulations could be invoked, they usually require
powerful computational facilities and essential calculational time. On the
contrary, the suggested method is very simple and straightforward.

(iv) Numerical methods have their own problems and limitations. Therefore the
employment of simpler analytical methods can serve as a guide for numerical
calculations.

(v) Finally, the physics of the considered problem becomes much more
transparent when possessing an explicit, although approximate, formula
allowing for studying its behavior in limiting cases.

It is important to stress that the main aim of the article is to develop a
method of extrapolation that would be simple, analytical, and applicable even
for those cases where just a few terms of perturbative series are available.
This is why we concentrate our attention on these points.

When numerous terms of a series are available, there are several methods
allowing for accurate extrapolation. However this is not the point of our
interest. If we were interested in getting high accuracy of extrapolation for
series with numerous terms, we should resort to some modifications of our
method, using additional tricks, such as the introduction of control functions
into self-similar approximants [21], the combination of self-similar
approximations with Pade approximants or with Borel transforms, etc. Thus it
has been demonstrated [28, 29] that the combination of self-similar and Padé
approximants converges much faster and provides essentially higher accuracy
then the best Padé approximants of the same order. However all that is a quite
different problem requiring separate investigations and publications, some of
which we cite. We stress it again that the main aim of the present paper is to
suggest a simple analytical method providing reliable approximations when
other methods are not applicable.

##  2 Self-similar approximation theory

The suggested method is based on self-similar approximation theory advanced in
Refs. [30, 31, 32, 33, 34]. In Ref. [35], this theory is used for developing a
convenient approach to the problems of interpolation in high-energy physics,
when weak-coupling as well as strong-coupling expansions are known. Here we
extend the applicability of the approach for the essentially more complicated
problem of extrapolation in quantum field theory, when only the weak-coupling
asymptotic series are available, but the behavior at the strong-coupling limit
is not known, and even more, finding this behavior is the point of main
interest.

First we briefly recall the main ideas of self-similar approximation theory in
order that the reader could understand its justifications and would get the
feeling why it can successfully work. This approach is based on mathematical
techniques of renormalization group theory, dynamical theory, and optimal
control theory [36, 37, 38]. Note that these theories are closely interrelated
since the renormalization group theory, actually, is a particular case of
dynamical theory.

The pivotal idea is to reformulate perturbation theory to the language of
dynamical theory or renormalization-group theory. For this purpose, we treat
the approximation-order index as discrete time and the passage from one
approximation to another as the motion in the space of approximations.
Suppose, we can find the sought function only as a sequence of approximations
at a small variable, $f(x)\simeq f_{k}(x)$ for $x\rightarrow 0$, where
$k=0,1,2,\ldots$ is the approximation order. For concreteness, we consider
here real-valued functions of real variables. The extension to complex-valued
functions can be straightforwardly done by considering several functions
corresponding to real and imaginary parts of the sought function.

The sequence of the bare approximants $f_{k}(x)$ is usually divergent.
Therefore the first thing that is necessary to do is to reorganize this
sequence by introducing control functions $u_{k}=u_{k}(x)$ governing the
sequence convergence. Control functions can be incorporated in several ways,
through initial conditions, calculational algorithm, or by a sequence
transformation. Thus, instead of the bare approximants $f_{k}(x)$, we pass to
a transformed sequence of the approximants

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

For short, we write here one control function $u_{k}$, although there can be
several of them, so that $u_{k}$ can be understood as a set of the necessary
control functions. We assume that the used transformation is invertible, in
the sense that

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

The sequence $\\{F_{k}(x,u_{k})\\}$ is convergent if and only if it satisfies
the Cauchy criterion, when for each $\varepsilon>0$ there exists a number
$k_{c}$ such that

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

for all $k>k_{c}$ and $p>0$. In the language of optimal control theory, this
implies that control functions can be defined as the minimizers of the
convergence cost functional [38]

$C[u]=\sum_{k}|F_{k+1}(x,u_{k+1})-F_{k}(x,u_{k})|\;.$ (1)

In order to formulate the passage between different $F_{k}$ as the evolution
of a dynamical system, it is necessary to define an endomorphism in the space
of approximants

${\cal A}=\\{F_{k}(x):\;k=0,1,2,\ldots;\;x\in\mathbb{R}\\}\;.$

For this purpose, we introduce the expansion function $x=x_{k}(f)$ by the
reonomic constraint

$F_{0}(x,u_{k}(x))=f\;.$

The endomorphism in the approximation space is defined as

$y_{k}(f)\equiv F_{k}(x_{k}(f),u_{k}(x_{k}(f)))\;,$

with the inverse relation

$F_{k}(x,u_{k}(x))=y_{k}(F_{0}(x,u_{k}(x)))\;.$

By this construction, the approximation sequence $\\{F_{k}\\}$ is bijective
with the sequence of the endomorphisms $\\{y_{k}\\}$. Therefore, if the
sequence $\\{F_{k}\\}$ converges to a limit $F^{*}$, then the sequence of the
endomorphisms $\\{y_{k}\\}$ converges to a limit $y^{*}$. The limit $y^{*}$
plays the role of a fixed point for the endomorphism sequence $\\{y_{k}\\}$,
where $y_{k}(y^{*}(f))=y^{*}(f)$. In the vicinity of a fixed point, the
endomorphism enjoys the property of self-similarity

$y_{k+p}(f)=y_{k}(y_{p}(f))\;,$ (2)

with the initial condition $y_{0}(f)=f$. This is, actually, just the semi-
group property $y_{k+p}=y_{k}\cdot y_{p}$, with the unity element $y_{0}=1$.
The sequence of endomorphisms, with the above semi-group property, is called
cascade (or semi-cascade),

$\\{y_{k}(f):\;\mathbb{Z}_{+}\times\mathbb{R}~{}\rightarrow~{}\mathbb{R}\\}\;,$

where the role of time is played by the approximation order $k$.

A cascade , which is a dynamical system in discrete time, can be embedded [39]
into a flow that is a dynamical system in continuous time,

$\\{y_{k}(f):\;\mathbb{Z}_{+}\times\mathbb{R}\rightarrow\mathbb{R}\\}\subset\\{y(t,f):~{}\mathbb{R}_{+}\times\mathbb{R}~{}\rightarrow~{}\mathbb{R}\\}\,.$

The embedding implies that the flow enjoys the same group property

$y(t+t^{\prime},f)=y(t,y(t^{\prime},f))$

and the flow trajectory passes through all points of the cascade trajectory,

$y(t,f)=y_{k}(f)\qquad(t=k)\;,$

with the same initial condition $y(0,f)=f$.

The above group property can be rewritten as the Lie differential equation

$\frac{\partial}{\partial t}\;y(t,f)=v(y(t,f))\;,$ (3)

with the velocity

$v(y)\equiv\lim_{\tau\rightarrow
0}\;\frac{\partial}{\partial\tau}\;y(\tau,y)\;.$

Integrating the differential evolution equation (3) yields the evolution
integral

$\int_{y_{k}}^{y_{k}^{*}}\frac{dy}{v(y)}=t_{k}\;,$ (4)

in which the integration is from a point $y_{k}(f)$ to an approximate fixed
point $y_{k}^{*}(f)$, with $t_{k}$ being the effective time needed for
reaching the latter point. Here $y_{k}^{*}(f)$ is an approximate fixed point,
since in practice we always have to limit the consideration by a finite number
of steps. Taking in the evolution integral the cascade velocity represented in
the form of the Euler discretization

$v_{k}(f)=y_{k+1}(f)-y_{k}(f)\;,$

we come to the integral

$\int_{F_{k}}^{F_{k}^{*}}\frac{df}{v_{k}(f)}=t_{k}\;,$ (5)

in which

$F_{k}^{*}(x)=y_{k}^{*}(F_{0}(x,u_{k}(x))$

is the effective limit of the sequence $\\{F_{k}\\}$ corresponding to the
approximate fixed point $y^{*}_{k}$. Applying the inverse transformation, we
obtain the self-similar approximant

$f_{k}^{*}(x)=\hat{T}^{-1}[u]F_{k}^{*}(x)\;.$

These are the principal steps in deriving self-similar approximants. The
practical realization depends on the form of the bare approximants $f_{k}(x)$,
the concrete form of the transformation $\hat{T}$ introducing control
functions, and on the method of defining the latter. When the asymptotic
behaviour of the sought quantity is known for small as well as for large
coupling parameters, it is convenient to accomplish the interpolation with the
use of self-similar root approximants, as is demonstrated in Ref. [35]. But
for the problem of extrapolation we need to employ another type of
approximants.

Usually, the asymptotic behaviour at small coupling parameters $x\rightarrow
0$, is of the form

$f_{k}(x)=f_{0}(x)\left(1+\sum_{n=1}^{k}a_{n}x^{n}\right)\;,$ (6)

where $f_{0}(x)$ is a given function. The above sum is usually divergent for
finite values of $x$, hence makes no sense for finite $x$. Moreover, often it
is necessary to find the behaviour of the sought function $f(x)$ at
asymptotically large $x\rightarrow\infty$.

By the fundamental theorem of algebra [40], a polynomial of any degree of one
real variable over the field of real numbers can be split in a unique way into
a product of irreducible first-degree polynomials over the field of complex
numbers. This implies that the finite series (6) can be represented as the
product

$f_{k}(x)=f_{0}(x)\prod_{j}\left(1+b_{j}x\right)\;,$ (7)

with $b_{j}$ expressed through $a_{n}$.

Control functions can be explicitly incorporated by employing fractal
transforms [38, 41], which can be written in the form

$\hat{T}[s,u]f_{k}(x)=x^{s_{k}}f_{k}(x)+u_{k}\;.$ (8)

Then, following the scheme described above, we obtain the self-similar factor
approximants

$f_{k}^{*}(x)=f_{0}(x)\prod_{j=1}^{N_{k}}\left(1+A_{j}x\right)^{n_{j}}\;,$ (9)

with $A_{j}$ and $n_{j}$ playing the role of control parameters [42, 43, 44].

The number of factors $N_{k}$ equals $k/2$ for even $k$ and $(k+1)/2$ for odd
$k$. A factor approximant (9) represents the sought function, therefore their
asymptotic expansions should coincide. Then the parameters $A_{j}$ and $n_{j}$
are to be chosen so that the asymptotic expansion of approximant (9) of order
$k$ be equal to the asymptotic form (6), that is, $f_{k}^{*}(x)\simeq
f_{k}(x)$ for $x\rightarrow 0$. This condition yields the equations

$\sum_{j=1}^{N_{k}}n_{j}A_{j}^{n}=D_{n}\qquad(n=1,2,\ldots,k)\;,$ (10)

where

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

When $k$ is even, hence $N_{k}=k/2$, we have $k$ equations for $k$ unknown
parameters $A_{j}$ and $n_{j}$, uniquely defining these parameters [44].
However if $k$ is odd, and $N_{k}=(k+1)/2$, we have $k$ equations for $k+1$
parameters. Then, to make the system of equations complete, it is necessary to
add one more condition. For instance, resorting to scaling arguments [44], it
is possible to set one of $A_{j}$ to one, say fixing $A_{1}=1$. This method
gives for odd approximants the results close to the nearest even-order
approximants. However below we prefer to deal with uniquely defined even
orders. Sometimes it may happen that the solutions for the parameters $A_{j}$
and $n_{j}$ are complex-valued. But this does not lead to any problem, since
such complex solutions for the parameters appear in complex conjugate pairs,
so that the whole expression remains real valued.

When we are interested in predicting the large-variable behaviour of a
function $f(x)$, we study the self-similar factor approximant (9) at
$x\rightarrow\infty$. If the function $f_{0}$ behaves as $f_{0}(x)\simeq
Ax^{\alpha}$ for $x\rightarrow\infty$, then the self-similar factor
approximant (9) for large $x$ is

$f_{x}^{*}(x)\simeq B_{k}x^{\gamma_{k}}\qquad(x\rightarrow\infty)\;,$ (11)

with the amplitude and the exponent

$B_{k}=A\prod_{j=1}^{N_{k}}A_{j}^{n_{j}}\;,\qquad\gamma_{k}=\alpha+\sum_{j=1}^{N_{k}}n_{j}\;.$
(12)

In those cases, where the large-variable asymptotic behaviour of the sought
function is known, say being $f(x)\simeq Bx^{\gamma}$ for
$x\rightarrow\infty$, it is straightforward to determine the accuracy of the
prediction by calculating the percentage errors
$\varepsilon(B_{k})\equiv((B_{k}-B)/B)\times 100\%$ and
$\varepsilon(\gamma_{k})\equiv((\gamma_{k}-\gamma)/\gamma)\times 100\%$. When
the exact large-variable asymptotic behaviour is not available, one usually
presents the difference between the subsequent approximations for the quantity
of interest. This difference characterizes the variation bar or dispersion of
the obtained results, which is related to the stability of the calculational
procedure [45]. If the subsequent results strongly differ from each other,
this induces suspicion of the procedure stability.

Often, the most important hard case is the prediction of the exponent in the
strong-coupling limit, since the value of the exponent essentially defines the
physics of the problem. This hard case is the main study in the present
article.

It is important to stress that the derivation of self-similar factor
approximants is based on the Cauchy criterion of convergence, so that these
approximants are expected to converge by construction. Moreover the numerical
convergence of self-similar factor approximants has been confirmed by a number
of problems enjoying many terms in their asymptotic expansions, when a long
sequence of the factor approximants could be considered [38, 42, 43, 44, 46].
It has been shown that for finite values of the considered variable the
accuracy of self-similar approximants is comparable with that of heavy
numerical calculations. In the present paper we concentrate on the most
difficult and interesting challenge of finding strong-coupling limits of
functions, especially their exponents, from the knowledge of only a few terms
in their asymptotic weak-coupling expansions. This type of problems is
difficult even for numerical methods. The approach is illustrated by several
problems of quantum field theory.

##  3 Convergence of self-similar factor approximants

When a number of terms in a weak-coupling expansion are available and the
asymptotic behavior in the strong-coupling limit is known, it is possible to
study numerical convergence of the approximants. Below we illustrate this by
several examples.

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

Let us start with the simple example of the so-called zero-dimensional
$\varphi^{4}$ theory characterized by the generating functional (partition
function)

$Z(g)=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-\varphi^{2}-g\varphi^{4}}\;d\varphi\;,$
(13)

with the coupling parameter $g\geq 0$. The weak-coupling asymptotic expansion
reads as

$Z_{k}(g)=\sum_{n=0}^{k}a_{n}g^{n}\qquad(g\rightarrow 0)\;,$ (14)

where the coefficients are

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

Using only this weak-coupling expansion, we construct the self-similar factor
approximants and study their strong-coupling limit, which gives

$Z_{k}^{*}(g)\simeq B_{k}g^{\gamma_{k}}\qquad(g\rightarrow\infty)\;.$ (15)

The accuracy of the obtained strong-coupling exponents can be found by
comparing the above limit with the known strong-coupling behavior

$Z(g)\simeq 1.022765\;g^{-1/4}\qquad(g\rightarrow\infty)\;.$ (16)

The results, shown in Table 1, demonstrate monotonic convergence to the exact
limiting value $-0.25$. In the $16$-th order, we have

$Z_{16}^{*}(g)\simeq 0.828\;g^{-0.187}\qquad(g\rightarrow\infty)\;.$ (17)

Table 1: Strong-coupling exponents and their percentage errors for the generating functional of zero-dimensional $\varphi^{4}$ theory. $k$ | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16
---|---|---|---|---|---|---|---|---
$\gamma_{k}$ | $-0.09$ | $-0.13$ | $-0.15$ | $-0.16$ | $-0.17$ | $-0.18$ | $-0.18$ | $-0.19$
$\varepsilon(\gamma_{k})\%$ |  $-$63 |  $-$48 |  $-$41 |  $-$36 |  $-$32 |  $-$29 |  $-$27 |  $-$25

###  3.2 One-dimensional anharmonic oscillator

The one-dimensional anharmonic oscillator with the Hamiltonian

$H=-\;\frac{1}{2}\;\frac{d^{2}}{dx^{2}}+\frac{1}{2}\;x^{2}+gx^{4}\;,$ (18)

where $g\geq 0$ and $-\infty<x<\infty$, imitates the one-dimensional
$\varphi^{4}$ theory. The weak-coupling expansion of the ground-state energy
is

$E_{k}(g)\simeq\frac{1}{2}+\sum_{n=0}^{k}a_{n}g^{n}\qquad(g\rightarrow 0)\;,$
(19)

with the coefficients that can be found in Refs. [47, 48].

Constructing the factor approximants and looking for their strong-coupling
limit

$E_{k}^{*}(g)\simeq B_{k}g^{\gamma_{k}}\qquad(g\rightarrow\infty)$ (20)

yields the results for the exponents shown in Table 2. The accuracy is found
by the comparison with the known strong-coupling asymptotic behavior

$E(g)\simeq 0.667986\;g^{1/3}\qquad(g\rightarrow\infty)\;.$ (21)

In the $16$-th order, we get

$E_{16}^{*}(g)\simeq 0.736\;g^{0.298}\qquad(g\rightarrow\infty)\;.$ (22)

Table 2 demonstrates monotonic numerical convergence.

Table 2: Strong-coupling exponents and their percentage errors for the ground-state energy of one-dimensional ahharmonic oscillator. $k$ | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16
---|---|---|---|---|---|---|---|---
$\gamma_{k}$ | 0.18 | 0.23 | 0.26 | 0.27 | 0.28 | 0.29 | 0.29 | 0.30
$\varepsilon(\gamma_{k})\%$ |  $-$47 |  $-$31 |  $-$23 |  $-$18 |  $-$16 |  $-$13 |  $-$12 |  $-$11

###  3.3 Massive Schwinger model in lattice theory

One of the simplest nontrivial gauge-theory models is the Schwinger model
[49]. This is a lattice model of quantum electrodynamics in $1+1$ space-time
dimensions. The model exhibits several phenomena typical of quantum
chromodynamics, such as confinement, chiral symmetry breaking with an axial
anomaly, and a topological vacuum [50, 51, 52, 53, 54]. The spectrum of
excited states for a finite lattice, calculated by means of perturbation
theory, is expressed through the series

$f_{k}(z)\simeq 1+\sum_{k=1}^{k}a_{n}z^{n}\qquad(z\rightarrow 0)$ (23)

in powers of the variable $z\equiv 1/(ga)^{4}$, where $g$ is the coupling
parameter and $a$ is the lattice spacing [55]. The coefficients for the vector
boson are

$a_{1}=2\;,\qquad a_{2}=-10\;,\qquad a_{3}=78.66667\;,\qquad
a_{4}=-7.362222\times 10^{2}\;,$ $a_{5}=7.572929\times 10^{3}\;,\qquad
a_{6}=-8.273669\times 10^{4}\;,\qquad a_{7}=9.428034\times 10^{5}\;,$
$a_{8}=-1.108358\times 10^{7}\;,\qquad a_{9}=1.334636\times 10^{8}\;,\qquad
a_{10}=-1.637996\times 10^{9}\;.$

Constructing factor approximants and considering their large-$z$ limit, we
have

$f_{k}^{*}(z)\simeq B_{k}z^{\gamma_{k}}\qquad(z\rightarrow\infty)\;,$ (24)

with the results for the large-$z$ exponent listed in Table 3. This is to be
compared with the known limiting behavior

$f(z)\simeq 1.1284z^{1/4}\qquad(z\rightarrow\infty)\;.$ (25)

For example, in the $10$-th order

$f_{10}^{*}(z)\simeq 1.519\;z^{0.2}\qquad(z\rightarrow\infty)\;.$ (26)

Table 3: Large-$z$ exponents and their percentage errors for the function $f(z)$ of the finite-lattice Schwinger model. $k$ | 2 | 4 | 6 | 8 | 10
---|---|---|---|---|---
$\gamma_{k}$ | 0.167 | 0.185 | 0.193 | 0.198 | 0.200
$\varepsilon(\gamma_{k})\%$ |  $-$33 |  $-$26 |  $-$23 |  $-$21 |  $-$20

###  3.4 Ground-state energy of Schwinger model

The ground-state energy of the Schwinger model with a vector boson, in the
continuum limit, can be found [51, 52, 53, 54] as an expansion in powers of
the dimensionless variable $x=m/g$, where $m$ is the electron mass and $g$ is
the coupling parameter,

$\frac{E(x)}{g}\simeq 0.5642-0.219x+0.1907x^{2}\qquad(x\rightarrow 0)\;.$ (27)

This short series allows us to construct only the second-order factor
approximant

$\frac{E_{2}^{*}(x)}{g}=\frac{0.5642}{(1+1.35339x)^{0.286805}}\;.$ (28)

In the large-$x$ limit this gives

$\frac{E_{2}^{*}(x)}{g}\simeq 0.5173x^{-0.287}\qquad(x\rightarrow\infty)\;.$
(29)

Comparing the large-$x$ exponent, with the known asymptotic behavior [54, 55,
56, 57]

$\frac{E(x)}{g}\simeq 0.6417x^{-1/3}\qquad(x\rightarrow\infty)\;,$ (30)

we find that the percentage error of the predicted exponent is
$\varepsilon(\gamma_{2})=-14\%$.

###  3.5 Summary for considered examples

The above examples show that the knowledge of only a small-variable asymptotic
expansion makes it possible to extrapolate the small-variable expansion to
finite-values of the variable and even to predict the behavior of the
corresponding function at asymptotically large values of the variable. When a
number of terms in the small-variable series are known, the sequence of the
related self-similar approximants is shown to converge. The self-similar
extrapolation allows for sufficiently accurate evaluation of the large-
variable exponent even when just a few terms of the small-variable expansion
are available.

The reason why a small-variable expansion can be extrapolated to the finite
and even infinite values of the variable lays in the following. The
coefficients of the expansion contain hidden information on the whole function
which they are derived from. Separate coefficients do not allow for noticing
this hidden information. However this information can be extracted by
analyzing the relations between the coefficients. Self-similar approximation
theory provides an instrument revealing the relations between the expansion
coefficients and thus allowing for the reconstruction of the whole sought
function.

##  4 Exact reconstruction of Gell-Mann-Low functions

Gell-Mann-Low functions in quantum field theory are usually calculated by
means of loop expansions yielding series in powers of asymptotically small
coupling parameters. However, the behaviour of these functions at strong
coupling is of special interest. Below we consider the extrapolation of these
functions to the arbitrary values of coupling parameters, including the limit
to $\infty$, by employing self-similar factor approximants. A special
attention will be paid to the study of the strong-coupling limit. In the
present section, we demonstrate that in some cases, having just a few
perturbative terms, self-similar approximants can reconstruct the sought Gell-
Mann-Low function exactly.

For this purpose, let us turn to the $N=1$ supersymmetric pure Yang-Mills
theory whose exact beta function is known [58, 59, 60, 61, 62, 63]:

$\beta(g)=-\;\frac{3g^{3}N_{c}}{16\pi^{2}(1-g^{2}N_{c}/8\pi^{2})}\;.$ (31)

If one resorts to perturbation theory with respect to the coupling $g$, one
gets

$\beta_{k}(g)=-\;\frac{3g^{3}N_{c}}{16\pi^{2}}\sum_{n=0}^{k}b_{n}g^{2n}\;,$
(32)

with the coefficients

$b_{n}=\left(\frac{N_{c}}{8\pi^{2}}\right)^{n}\;.$

For the second-order factor approximant, we have

$\beta_{2}^{*}(g)=-\;\frac{3g^{3}N_{c}}{16\pi^{2}}\;\left(1+A_{1}g^{2}\right)^{n_{1}}\;.$
(33)

Expanding Eq. (33) and comparing the expansion with series (32) yields

$n_{1}=-1\;,\qquad A_{1}=-\;\frac{N_{c}}{8\pi^{2}}\;.$

Thus the second-order factor approximant results in the exact expression (31).
It is easy to check that all approximants of orders $k\geq 2$ give the exact
beta function (31).

##  5 Gell-Mann-Low function in $\varphi^{4}$ field theory

Let us consider the $O(N)$ symmetric $\varphi^{4}$ field theory. The Gell-
Mann-Low function is defined as

$\beta(g)=\mu\;\frac{\partial g}{\partial\mu}\;,$ (34)

where $g=\lambda/(4\pi)^{2}$ is the coupling parameter and $\mu$ is
renormalization scale. This function has been found [64], within minimal
subtraction scheme, in the six-loop approximation:

$\beta(g)\simeq g^{2}\sum_{n=0}^{k}b_{n}g^{n}\qquad(g\rightarrow 0)\;,$ (35)

with the coefficients

$b_{0}=\frac{N+8}{3}\;,\qquad b_{1}=-\;\frac{3N+14}{3}\;,$
$b_{2}=\frac{1}{216}\left[96(5N+22)\zeta(3)+33N^{2}+922N+2960\right]\;,$
$b_{3}=-\;\frac{1}{3888}\left[1920\left(2N^{2}+55N+186\right)\zeta(5)-288(N+8)(5N+22)\zeta(4)+\right.$
$+\left.96\left(63N^{2}+764N+2332\right)\zeta(3)-\left(5N^{3}-6320N^{2}-80456N-196648\right)\right]\;,$
$b_{4}=\frac{1}{62208}\left[112896\left(14N^{2}+189N+526\right)\zeta(7)-768\left(6N^{3}+59N^{2}-446N-3264\right)\zeta^{2}(3)-\right.$
$-9600(N+8)\left(2N^{2}+55N+186\right)\zeta(6)+256\left(305N^{3}+7466N^{2}+66986N+165084\right)\zeta(5)-$
$-288\left(63N^{3}+1388N^{2}+9532N+21120\right)\zeta(4)-$
$-16\left(9N^{4}-1248N^{3}-67640N^{2}-552280N-1314336\right)\zeta(3)+$
$+\left.13N^{4}+12578N^{3}+808496N^{2}+6646336N+13177344\right]\;,$
$b_{5}=-\;\frac{1}{41990400}\left[204800\left(1819N^{3}+97823N^{2}+901051N+2150774\right)\zeta(9)+\right.$
$+14745600\left(N^{3}+65N^{2}+619N+1502\right)\zeta^{3}(3)+$
$+995328\left(42N^{3}+2623N^{2}+25074N+59984\right)\zeta(3,5)-$
$-20736\left(28882N^{3}+820483N^{2}+6403754N+14174864\right)\zeta(8)-$
$-5529600\left(8N^{3}-635N^{2}-9150N-25944\right)\zeta(3)\zeta(5)+$
$+11520\left(440N^{4}+126695N^{3}+2181660N^{2}+14313152N+29762136\right)\zeta(7)+$
$+207360(N+8)\left(6N^{3}+59N^{2}-446N-3264\right)\zeta(3)\zeta(4)-$
$-23040\left(188N^{4}+132N^{3}-93363N^{2}-862604N-2207484\right)\zeta^{2}(3)-$
$-28800\left(595N^{4}+20286N^{3}+277914N^{2}+1580792N+2998152\right)\zeta(6)+$
$+5760\left(4698N^{4}+131827N^{3}+2250906N^{2}+14657556N+29409080\right)\zeta(5)+$
$+2160\left(9N^{5}-1176N^{4}-88964N^{3}-1283840N^{2}-6794096N-12473568\right)\zeta(4)-$
$-720\left(33N^{5}+2970N^{4}-477740N^{3}-10084168N^{2}-61017200N-117867424\right)\zeta(3)-$
$-\left.45\left(29N^{5}+22644N^{4}-3225892N^{3}-88418816N^{2}-536820560N-897712992\right)\right]\;.$

Here $\zeta(3,5)$ denotes the double zeta function

$\zeta(3,5)=\sum_{1\leq n<m}\frac{1}{n^{3}m^{5}}=0.037707673\;.$

The numerical values of the coefficients for the number of components from
$N=0$ to $N=4$ are given in Table 4.

Table 4: Coefficients of weak-coupling expansion for the Gell-Man-Low function of the $N$-component $\varphi^{4}$ field theory, in the six-loop approximation. $N$ | 0 | 1 | 2 | 3 | 4
---|---|---|---|---|---
$b_{0}$ | 2.66667 | 3.0 | 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

In the case of $N=1$, the Gell-Mann-Low function is known in the seven-loop
approximation [65] having the coefficients

$b_{0}=3\;,\qquad b_{1}=-5.66667\;,\qquad b_{2}=32.5497\;,\qquad
b_{3}=-271.606\;,$ $b_{4}=2848.57\;,\qquad b_{5}=-34776.1\;,\qquad
b_{6}=474651\;.$

We construct self-similar factor approximants for different $N$. Thus for
$N=0$, we have

$\beta_{2}^{*}(g)=\frac{2.66667g^{2}}{(1+9.16021g)^{0.191044}}\qquad(N=0)\;,$
$\beta_{4}^{*}(g)=\frac{2.66667g^{2}}{(1+6.06377g)^{0.194059}(1+15.5161g)^{0.0369468}}\;,$
(36)

which yields the strong-coupling limit

$\beta_{2}^{*}(g)\simeq 1.747g^{1.809}\;,\qquad\beta_{4}^{*}(g)\simeq
1.699g^{1.769}\qquad(N=0,~{}g\rightarrow\infty)\;.$ (37)

For $N=2$, we find

$\beta_{2}^{*}(g)=\frac{3.33333g^{2}}{(1+9.98433g)^{0.200314}}\qquad(N=2)\;,$
$\beta_{4}^{*}(g)=\frac{3.33333g^{2}}{(1+5.79973g)^{0.206354}(1+16.2195g)^{0.0495207}}\;,$
(38)

with the strong-coupling limit

$\beta_{2}^{*}(g)\simeq 2.102g^{1.7997}\;,\qquad\beta_{4}^{*}(g)\simeq
2.020g^{1.7441}\qquad(N=2,~{}g\rightarrow\infty)\;.$ (39)

For $N=3$, we obtain

$\beta_{2}^{*}(g)=\frac{3.66667g^{2}}{(1+10.3399g)^{0.202218}}\qquad(N=3)\;,$
$\beta_{4}^{*}(g)=\frac{3.66667g^{2}}{(1+5.53502g)^{0.212617}(1+16.526g)^{0.055311}}\;,$
(40)

giving the limit

$\beta_{2}^{*}(g)\simeq 2.286g^{1.798}\;,\qquad\beta_{4}^{*}(g)\simeq
2.182g^{1.732}\qquad(N=3,~{}g\rightarrow\infty)\;.$ (41)

And for the case of $N=4$, we get

$\beta_{2}^{*}(g)=\frac{4g^{2}}{(1+10.6781g)^{0.202908}}\qquad(N=4)\;,$
$\beta_{4}^{*}(g)=\frac{4g^{2}}{(1+5.2767g)^{0.219241}(1+16.8661g)^{0.0598713}}\;,$
(42)

with the strong-coupling limit

$\beta_{2}^{*}(g)\simeq 2.474g^{1.797}\;,\qquad\beta_{4}^{*}(g)\simeq
2.345g^{1.721}\qquad(N=4,~{}g\rightarrow\infty)\;.$ (43)

For $N=1$, we use the seven-loop expansion obtaining the functions

$\beta_{2}^{*}(g)=\frac{3g^{2}}{(1+9.59923g)^{0.196775}}\qquad(N=1)\;,$
$\beta_{4}^{*}(g)=\frac{3g^{2}}{(1+6.01378g)^{0.200415}(1+15.9204g)^{0.0429409}}\;,$
$\beta_{6}^{*}(g)=\frac{3g^{2}}{(1+5.35692g)^{0.186798}(1+13.7203g)^{0.0596501}(1+22.0958g)^{0.0031595}}\;,$
(44)

whose strong-coupling behavior is

$\beta_{2}^{*}(g)\simeq 1.922g^{1.803}\;,\qquad\beta_{4}^{*}(g)\simeq
1.859g^{1.757}\;,$ $\beta_{6}^{*}(g)\simeq
1.857g^{1.750}\qquad(N=1,~{}g\rightarrow\infty)\;.$ (45)

Summarizing, we present in Table 5 the averaged results

$B=\frac{1}{k}\sum_{n=1}^{k}B_{2n}\;,\qquad\gamma=\frac{1}{k}\sum_{n=1}^{k}\gamma_{2n}$
(46)

for the amplitudes and exponents characterizing the strong-coupling limit

$\beta(\gamma)\simeq Bg^{\gamma}\qquad(g\rightarrow\infty)\;,$ (47)

together with the dispersion between the subsequent approximants.

The overall behaviour of the Gell-Mann-Low function of $\varphi^{4}$ field
theory for $N=1$ is shown in Fig. 1, where the convergence of the approximants
is evident. The behaviour of the Gell-Mann-Low functions for other $N$ is
similar, only slightly differing from that for $N=1$.

In literature, it is possible to find the estimates for the strong-coupling
exponent $\gamma$ in the case of $N=1$. Thus Borel summation with conformal
mapping gives $\gamma=2$ [66] or $\gamma=1.9$ [67]. A variational estimate
[68] yields $\gamma=1.5$. Our result of $\gamma=1.77$ for $N=1$ is between
those given by the Borel summation and variational calculations.

Table 5: Strong-coupling amplitudes and exponents for the $N$-component $\varphi^{4}$ field theory, predicted by self-similar factor approximants. $N$ | B | $\gamma$
---|---|---
0 | $1.723\pm 0.02$ | $1.789\pm 0.02$
1 | $1.879\pm 0.03$ | $1.770\pm 0.03$
2 | $2.061\pm 0.04$ | $1.772\pm 0.03$
3 | $2.235\pm 0.05$ | $1.765\pm 0.03$
4 | $2.410\pm 0.06$ | $1.759\pm 0.04$

##  6 Gell-Mann-Low function in quantum electrodynamics

In quantum electrodynamics, the Gell-Mann-Low function in the renormalized
minimal subtraction scheme $(\overline{\rm MS})$ reads as

$\beta(\alpha)=\mu^{2}\;\frac{\partial}{\partial\mu^{2}}\left(\frac{\alpha}{\pi}\right)\;,$
(48)

where $\alpha$ is the renormalized $(\overline{\rm MS})$ scheme coupling
parameter and $\mu$ is the $(\overline{\rm MS})$ scale parameter. 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, has the form [69]

$\beta(\alpha)\simeq\left(\frac{\alpha}{\pi}\right)^{2}\sum_{n=0}^{k}b_{n}\left(\frac{\alpha}{\pi}\right)^{n}\;,$
(49)

with the coefficients

$b_{0}=\frac{1}{3}=0.333333\;,\qquad b_{1}=\frac{1}{4}\;,\qquad
b_{2}=-\;\frac{31}{288}=-0.107639\;,$
$b_{3}=-\;\frac{2785}{31104}\;-\;\frac{13}{36}\;\zeta(3)=-0.523614\;,$
$b_{4}=-\;\frac{195067}{497664}\;-\;\frac{25}{96}\;\zeta(3)\;-\;\frac{13}{96}\;\zeta(4)\;+\;\frac{215}{96}\;\zeta(5)=1.47072\;.$

From here, we find the factor approximants

$\beta_{2}^{*}(\alpha)=\frac{1}{3}\left(\frac{\alpha}{\pi}\right)^{2}\left(1+1.61111\;\frac{\alpha}{\pi}\right)^{0.465517}\;,$
$\beta_{4}^{*}(\alpha)=\frac{1}{3}\left(\frac{\alpha}{\pi}\right)^{2}\left(1+A_{1}\;\frac{\alpha}{\pi}\right)^{n_{1}}\left(1+A_{2}\;\frac{\alpha}{\pi}\right)^{n_{2}}\;$
(50)

where

$A_{1}=1.394295+2.70199797\;i=A_{2}^{*}\;,\qquad
n_{1}=0.047762-0.11413974\;i=n_{2}^{*}\;.$

Therefore in the strong-coupling limit, we have

$\beta_{2}^{*}(\alpha)\simeq
0.4162\left(\frac{\alpha}{\pi}\right)^{2.4655}\;,$
$\beta_{4}^{*}(\alpha)\simeq
0.4759\left(\frac{\alpha}{\pi}\right)^{2.0955}\qquad\left(\frac{\alpha}{\pi}\rightarrow\infty\right)\;.$
(51)

Defining average quantities, we see that the strong-coupling behaviour of the
Gell-Mann-Low function

$\beta(\alpha)\simeq
B\left(\frac{\alpha}{\pi}\right)^{\gamma}\qquad\left(\frac{\alpha}{\pi}\rightarrow\infty\right)$
(52)

can be characterized by the amplitude and exponent

$B=0.446\pm 0.03\;,\qquad\gamma=2.281\pm 0.19\;.$ (53)

The dependence of the Gell-Mann-Low function for quantum electrodynamics is
presented in Fig. 2. Its overall behaviour only slightly depends on the chosen
scheme. Thus, accepting the coefficients of the weak-coupling expansion found
in the on-shell scheme or in the momentum subtraction scheme [69] results in
$B=0.433,\;\gamma=2.189$ in the on-shell scheme and $B=0.493,\;\gamma=2.309$
in the momentum subtraction scheme.

The running coupling defined by equation (48), with the beta function
represented by a factor approximant, increases from zero to infinity. As the
boundary condition, we can take the value $\alpha(m_{Z})=0.007815$ at the
$Z$-boson mass $m_{Z}=91.1876$ GeV. Then the logarithmic divergence occurs at
$\mu_{0}=8.584\times 10^{260}$ GeV, where

$\alpha\simeq\frac{2.743}{(\ln(\mu_{0}/\mu))^{0.682}}\qquad(\mu\rightarrow\mu_{0}-0)\;.$

The value of $\mu_{0}$ is much larger than the point of the simple Landau pole
that is of the order of $10^{30}-10^{40}$ GeV [70]. The value of $\mu_{0}$ is
so large that practically can be considered as infinity.

##  7 Gell-Mann-Low function in quantum chromodynamics

The Gell-Mann-Low function in quantum chromodynamics is defined by the
equation

$\beta(\alpha_{s})=\mu^{2}\;\frac{\partial
a_{s}}{\partial\mu^{2}}\qquad\left(a_{s}\equiv\frac{\alpha_{s}}{\pi}\right)\;,$
(54)

where $\alpha_{s}$ is the quark-gluon coupling and $\mu$ is the normalization
scale. We keep in mind the realistic case of three colours $(N_{c}=3)$. In the
five-loop approximation, the weak-coupling expansion is

$\beta(\alpha_{s})\simeq-
a_{s}^{2}\sum_{n=0}^{k}b_{n}a_{s}^{n}\qquad(a_{s}\rightarrow 0)\;.$ (55)

Within the minimal subtraction scheme $(\overline{\rm MS})$, the coefficients
are [71, 72, 73]

$b_{0}=2.75-0.166667\;n_{f}\;,\qquad b_{1}=6.375-0.791667\;n_{f}\;,$
$b_{2}=22.3203-4.36892\;n_{f}+0.0940394\;n_{f}^{2}\;,$
$b_{3}=114.23-27.1339\;n_{f}+1.58238\;n_{f}^{2}+0.0058567\;n_{f}^{3}\;,$
$b_{4}=524.56-181.8\;n_{f}+17.16\;n_{f}^{2}-0.22586\;n_{f}^{3}-0.0017993\;n_{f}^{4}\;,$

with $n_{f}$ being the number of quark flavors.

It turns out that factor approximants as real functions exist not for all
$n_{f}$. But they do exist for the physically realistic number of flavors
$n_{f}=6$. For this case, the available factor approximants are

$\beta_{2}^{*}(\alpha_{s})=-1.75a_{s}^{2}(1+1.5536a_{s})^{0.597706}\;,$
$\beta_{3}^{*}(\alpha_{s})=-1.75a_{s}^{2}(1+a_{s})^{0.91227}(1+32.5316a_{s})^{0.0005011}\;.$
(56)

This gives the strong-coupling limit

$\beta_{2}^{*}(\alpha_{s})\simeq-2.2772a_{s}^{2.5977}\;,\qquad\beta_{3}^{*}(\alpha_{s})\simeq-1.7531a_{s}^{2.9128}\qquad(a_{s}\rightarrow\infty)\;.$
(57)

In that way, the strong-coupling limit of the Gell-Mann-Low function

$\beta(\alpha_{s})\simeq-Ba_{s}^{\gamma}\qquad(a_{s}\rightarrow\infty)$ (58)

is characterized by the amplitude and exponent

$B=-2.015\pm 0.26\;,\qquad\gamma=2.755\pm 0.16\;.$ (59)

We are not aware of other reliable estimates of these characteristics that
could be compared with our result for the physically interesting case of the
flavour number $n_{f}=6$. The behaviour of the Gell-Mann-Low function of
quantum chromodynamics for this flavour number, as a function of the coupling
parameter, is shown in Fig. 3. For varying $n_{f}$, the Banks-Zaks [74] fixed
point exists in the region $8.05\leq n_{f}\leq 16.5$.

The running coupling, defined by Eq. (54), with the boundary condition
$\alpha_{s}(m_{Z})=0.1181$ logarithmically grows when $\mu$ tends to
$\mu_{c}=0.1$ GeV from above,

$\alpha_{s}\simeq\frac{0.907}{(\ln(\mu/\mu_{c}))^{0.626}}\qquad(\mu\rightarrow\mu_{c}+0)\;.$

The appearance of a pole at $\mu_{c}$ characterizes the scale at which
perturbative QCD breaks down, so that the series (55) as such, and hence their
extrapolation, become invalid. The value of $\mu_{c}$ can be associated with
the confinement scale, or equivalently the hadronic mass scale and
nonperturbative effects, such as the arising bound states [70]. The smaller
the value of $\mu_{c}$, i.e., the smaller the momentum scale at which the
divergence occurs, the slower the increase of $\alpha_{s}(\mu)$ as $\mu$
decreases. This would imply the effective extrapolation of perturbative
expressions to smaller momentum scales [70]. The value of $\mu_{c}$ is really
much smaller than the point of the Landau pole which in the $(\overline{\rm
MS})$ scheme for $n_{f}=6$ happens at the point $0.9$ GeV [75].

##  8 Discussion

In the above examples, we have considered the cases where the large-variable
behavior is of power-law. As is demonstrated, for these cases, self-similar
factor approximants can provide good extrapolation of small-variable Taylor-
type asymptotic expansions to the range of finite variables and even for the
large-variable limit. Moreover, in some cases these approximants, using only a
small-variable expansion, are able to reconstruct the sought function exactly,
as in the case of the Gell-Mann-Low function for the supersymmetric pure Yang-
Mills theory and in some other cases to be considered below.

Two related natural questions arise: How well the self-similar factor
approximants could extrapolate the functions with the large-variable behavior
different from the power-law, such as exponential and logarithmic behavior?
And the other question is: What would be other examples of the exact function
reconstruction by means of self-similar factor approximants?

###  8.1 Class of exactly reproducible functions

First of all, let us notice that there exists a class of real-valued functions
exactly reproducible by factor approximants. These are the functions having
the form

$R_{k_{M}}(x)=\prod_{i=1}^{M}\;P_{m_{i}}^{\alpha_{i}}(x)$ (60)

of the product of polynomials

$P_{m_{i}}(x)=c_{i0}+c_{i1}x+c_{i2}x^{2}+\cdots+c_{im_{i}}x^{m_{i}}\;,$

where $m_{i}$ are integers; the powers $\alpha_{i}$ and coefficients $c_{ij}$
can be complex-valued numbers entering $R_{k_{M}}(x)$ in complex conjugate
pairs so that $R_{k_{M}}(x)$ be real, and

$k_{M}=\sum_{i=1}^{M}m_{i}+M.$

This follows from the fact that a polynomial $P_{m_{i}}(x)$ can be represented
as

$P_{m_{i}}(x)=c_{i0}\prod_{j=1}^{m_{i}}\;(1+b_{ij}x)\;,$

with $b_{ij}$ being expressed through $c_{ij}$. Then the function
$R_{k_{M}}(x)$ can be reduced to the form

$R_{k_{M}}(x)=\prod_{i=1}^{M}\;c_{i0}\;\prod_{j=1}^{m_{i}}\;(1+b_{ij}x)^{\alpha_{i}}\;,$
(61)

which is nothing but a particular case of a factor approximant possessing the
same asymptotic expansion as the given function (60).

If all powers $\alpha_{i}$ were $\pm 1$, then the function $R_{k}(x)$ would
reduce to a Padé approximant that is a rational function. However the powers
$\alpha_{i}$ are not necessarily integers. Hence factor approximants also
include irrational functions that can be reproduced exactly.

###  8.2 Exact reconstruction of exponential functions

Moreover, factor approximants can exactly reproduce transcendental functions,
such as the exponential function $\exp(x)$, where $x$ can take any complex
value. Let us consider the standard $k$-order expansion of the exponential
function

$e_{k}(x)=\sum_{n=0}^{k}\frac{x^{n}}{n!}\;.$ (62)

The second-order factor approximant is

$e_{2}^{*}(x)=(1+Ax)^{n}\;.$

Expanding this in powers of $x$ and comparing with $e_{k}(x)$ yields the
equations

$An=1\;,\qquad A^{2}n(n-1)=1\;.$

The sole solution to these equations is $A=1/n$ with $n\rightarrow\infty$.
Therefore already the second-order factor approximant gives exactly the
exponential function

$e_{2}^{*}(x)=\lim_{n\rightarrow\infty}\;\left(1+\frac{1}{n}\;x\right)^{n}=e^{x}\;.$
(63)

It is easy to check that all factor approximants of the order $k\geq 2$
reconstruct the exponential function exactly.

###  8.3 Exact solution of nonlinear equations

Some nonlinear differential equations can be solved exactly by looking for
solutions in the form of asymptotic series and then constructing factor
approximants. For instance, let us consider the nonlinear singular problem

$(\varepsilon y+t)\;\frac{dy}{dt}+y-1=0$ (64)

with the initial condition $y(0)=2$. This kind of equation is met in different
applications [76, 77]. It is called singular since it does not allow for the
use of perturbation theory in powers of the parameter $\varepsilon$.

The parameter $\varepsilon$ can be hidden in the renotation

$t=\varepsilon x\;,\qquad y=z-x$

resulting in the equation

$z\;\frac{dz}{dt}-x-1=0\;,$

with the initial condition $z(0)=2$. Looking for the solution at
asymptotically small $x$ implies the consideration of the series

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

whose coefficients can be found by substituting these series into the
equation. Thus $a_{0}=2$, $a_{1}=1/2$, $a_{3}=3/16$, and so on. Constructing
the fourth-order factor approximant gives

$z_{4}^{*}(x)=2(1+A_{1}x)^{n_{1}}(1+A_{2}x)^{n_{2}}\;,$

with the parameters

$A_{1}=\frac{1}{4}\;(1-i\sqrt{3})=A_{2}^{*}\;,\qquad
n_{1}=n_{2}=\frac{1}{2}\;.$

Returning to the initial variables yields the function

$y_{4}^{*}(t)=\sqrt{4+\frac{2t}{\varepsilon}+\frac{t^{2}}{\varepsilon^{2}}}\;-\;\frac{t}{\varepsilon}$
(65)

that is the exact solution of the given equation. The same exact solution
results for any factor approximant of $k\geq 4$.

Exact solutions of several other nonlnear differential equations can also be
found by employing the summation of asymptotic series by means of the self-
similar factor approximants [78].

###  8.4 Exponential behavior: Bose-Einstein distribution

As is shown above, the purely exponential behavior is reproduced by factor
approximants exactly. Then we should expect that a behavior close to the
purely exponential could be well approximated by these approximants. Let us
consider the well known Bose-Einstein distribution

$f_{B}(x)=\frac{1}{e^{x}-1}\;.$ (66)

Suppose, only the expansion at small $x$,

$f_{B}(x)\simeq\frac{1}{x}\;-\;\frac{1}{2}\;+\;\frac{x}{12}\;-\;\frac{x^{3}}{720}\;+\;\frac{x^{5}}{30240}\;-\;\frac{x^{7}}{1209600}\;+\;\ldots\qquad(x\rightarrow
0)\;,$ (67)

is available for us, and we do not know what function it represents.

In the standard way, we construct factor approximants $f_{k}^{*}(x)$ that
extrapolate expansion (67) from asymptotically small $x$ to finite values of
the latter. Our major interest is in the approximants providing the
extrapolation to the large variable behavior with respect to
$x\rightarrow\infty$. By their structure, the factor approximants give the
power-law behavior at infinity, for instance

$f_{4}^{*}(x)\simeq 2.94\cdot\frac{10^{-3}}{x^{3}}\;,\qquad f_{8}^{*}(x)\simeq
1.1\cdot\frac{10^{-7}}{x^{5}}\;,\qquad f_{12}^{*}(x)\simeq
2.97\cdot\frac{10^{-14}}{x^{7}}\qquad(x\rightarrow\infty)\;.$

As we see, these values quickly diminish, telling us that the real behavior at
large $x$ is faster then of power law, probably, of exponential type. If we
expect that the large-variable behavior is exponential, we can employ another
variant of self-similar approximants, i.e. the self-similar exponential
approximants [79]. However, since, as is proved above, the factor approximants
well approximate the exponential behavior, they should provide rather good
accuracy for the extrapolation of the distribution (45), which is demonstrated
in Fig. 4. The exact Bose-Einstein distribution (66) and its factor
approximants practically coincide in a large range of $x$, because of which in
Fig. 4 we show their difference.

###  8.5 Exponential behavior: Fermi-Dirac distribution

Similarly, we can consider the Fermi-Dirac distribution

$f_{F}(x)=\frac{1}{e^{x}+1}\;.$ (68)

Again assume that we possess only the small-variable expansion

$f_{F}(x)\simeq\frac{1}{2}\;-\;\frac{x}{4}\;+\;\frac{x^{3}}{48}\;-\;\frac{x^{5}}{480}\;+\;\frac{17x^{7}}{80640}\;-\;\frac{31x^{9}}{1451520}\;+\;\ldots\qquad(x\rightarrow
0)$ (69)

and are not aware of the function it corresponds to. Factor approximants
$f_{k}^{*}(x)$ that extrapolate the series (69) to the large-variable range
again quickly diminish, for instance

$f_{4}^{*}(x)\simeq 6.7\cdot\frac{10^{-2}}{x^{2}}\;,\qquad f_{8}^{*}(x)\simeq
3.4\cdot\frac{10^{-5}}{x^{4}}\;\qquad f_{12}^{*}(x)\simeq
1.05\cdot\frac{10^{-10}}{x^{6}}\qquad(x\rightarrow\infty)\;.$

This hints that the large-variable behavior should be faster than of power
law. Nevertheless, since factor approximants well approximate the purely
exponential behavior, they should provide quite accurate approximation for the
considered distribution. Again, the exact distribution (68) and its factor
approximants practically coincide in a large region of $x$, because of which
in Fig. 5 the related differences are shown.

###  8.6 Logarithmic behavior: no additional information

Now let us turn to functions with logarithmic asymptotic behavior at large
values of a variable. Let us take the function

$f(x)=1+\ln\;\left(\frac{1+\sqrt{1+x}}{2}\right)\;,$ (70)

with the logarithmic asymptotic behavior at large $x$,

$f(x)\simeq\frac{1}{2}\ln\;x\qquad(x\rightarrow\infty)\;.$ (71)

Let us pretend that we know neither the function itself nor its behavior at
large $x$, but what available is only the asymptotic expansion at small $x$,

$f(x)\simeq
1\;+\;\frac{1}{4}\;x\;-\;\frac{3}{32}\;x^{2}\;+\;\frac{5}{96}\;x^{3}\;-\;\frac{35}{1024}\;x^{4}\;+\;\frac{63}{2560}\;x^{5}\;-\;\frac{77}{4096}\;x^{6}\;+\ldots\qquad(x\rightarrow
0)\;.$ (72)

Constructing factor approximants from these series, we find that their large-
variable behavior demonstrates not so fast variation of the amplitude and
power:

$f_{2}^{*}(x)\simeq x^{0.25}\;,\qquad f_{4}^{*}(x)\simeq
1.195\;x^{0.182}\;,\qquad f_{6}^{*}(x)\simeq 1.137\;x^{0.197}\;,$
$f_{8}^{*}(x)\simeq 1.237\;x^{0.1747}\;,\qquad f_{10}^{*}(x)\simeq
1.235\;x^{0.1749}\;,\qquad f_{12}^{*}(x)\simeq
1.310\;x^{0.162}\qquad(x\rightarrow\infty)\;.$

Although, as is typical of the factor approximants, the large-variable
dependence is of power law, the factor approximants provide reasonable
accuracy in a wide region of the variable $x$, as is seen in Fig. 6(a).

###  8.7 Logarithmic behavior: known character of large-variable limit

A different situation develops when, although the function itself is not
known, but there is information that the large-variable behavior is expected
to be logarithmic. Then it is reasonable to deal not with series (72), but
with the exponential of it. This implies that a finite series $f_{k}(x)$,
corresponding to the truncated series (72), is exponentiated considering

$g_{k}(x)=\exp\\{f_{k}(x)\\}$ (73)

and then expanding the latter up to the $k$-th order. In the case of series
(72), we have

$g(x)\simeq
e\left(1+\frac{1}{4}\;x\;-\;\frac{1}{16}\;x^{2}\;+\;\frac{1}{32}\;x^{3}\;-\;\frac{5}{256}\;x^{4}\;+\;\frac{7}{512}\;x^{5}\;-\;\frac{21}{2048}\;x^{6}\;+\;\ldots\;\right)\quad(x\rightarrow
0)\;.$ (74)

The latter series is used for constructing the factor approximants
$g_{k}^{*}(x)$, after which, inverting transformation (73), we return to the
expressions

$F_{k}^{*}(x)=\ln\;g_{k}^{*}(x)$ (75)

approximating the sought function. Here we denote the final approximants as
$F_{k}^{*}(x)$ to distinguish them from the approximants $f_{k}^{*}(x)$
obtained directly from series (72).

The large-variable behavior of the approximants $F_{k}^{*}(x)$ is of correct
logarithmic form, for example

$F_{2}^{*}(x)\simeq 0.333\ln x\;,\qquad F_{4}^{*}(x)\simeq 0.4\ln x\;,\qquad
F_{6}^{*}(x)\simeq 0.429\ln x\;,$ $F_{8}^{*}(x)\simeq 0.444\ln x\;,\qquad
F_{10}^{*}(x)\simeq 0.456\ln x\;,\qquad F_{12}^{*}(x)\simeq 0.462\ln
x\qquad(x\rightarrow\infty)\;.$

The coefficient at the logarithm converges to the exact value $0.5$, in
agreement with limit (71). The overall behavior of the approximants
$F_{k}^{*}(x)$ is shown in Fig. 6(b). As it should be expected, the additional
information on the behavior of the function essentially improves the accuracy
of the approximation.

##  9 Conclusion

We have suggested a method, based on self-similar approximation theory,
allowing for the extrapolation of expressions from weak-coupling asymptotic
expansions to the region of arbitrary values of coupling parameters, including
their asymptotically large values. The region of large coupling parameters is
of special interest because of its physical importance and because
mathematically this region is the most difficult for the extrapolation that
uses only the coefficients of weak-coupling expansions.

In those cases, where a number of terms in the small-variable expansion are
available, the method is shown to posses numerical convergence. Good accuracy
can be obtained even for the expansions with a few terms. The extrapolation of
perturbative series for the Gell-Mann-Low functions of the $O(N)$ symmetric
$\varphi^{4}$ field theory, quantum electrodynamics, and quantum
chromodynamics is demonstrated.

In some cases, the method can transform perturbative series to the exact
expression valid for arbitrary values of the variable. Such an exact
reconstruction is illustrated for the Gell-Mann-Low function of a
supersymmetric pure Yang-Mills theory and for several other examples.

By their construction, self-similar factor approximants, for the variable
tending to infinity, give a power-law behavior. The possibility is discussed
of extrapolating functions with different types of the large-variable limits,
including not only power-law behavior, but also logarithmic and exponential
behavior. The purely exponential function is shown to be reconstructed exactly
by the factor approximants of any order starting from second. More complicated
functions with exponential or logarithmic large-variable behavior can be well
approximated in a wide range of the variable. If the additional information on
the character of the large-variable limit is available, the accuracy of the
approximants can be essentially improved.

## References

  * [1] G.A. Baker and P. Graves-Moris, Padé Approximants (Cambridge University Press, Cambridge, 1996).
  * [2] G.A. Baker and J.L. Gammel, The Padé approximant, J. Math. Anal. Appl. 2, 21–30 (1961).
  * [3] G.H. Hardy, Divergent Series (Chelsea, Rhode Island, 1992).
  * [4] H. Kleinert, Path Integrals (World Scientific, Singapore, 2003).
  * [5] S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, 2005).
  * [6] O. Costin and G.V. Dunne, Resurgent extrapolation: Rebuilding a function from asymptotic data, Painlevé I, J. Phys. A 52, 445205 (2019).
  * [7] O. Costin and G.V. Dunne, Physical resurgent extrapolation, Phys. Lett. B 808, 135627 (2020).
  * [8] O. Costin and G.V. Dunne, Uniformization and constructive analytic continuation of Taylor series, arXiv:2009.01962 (2020).
  * [9] V.I. Yukalov, Theory of perturbations with a strong interaction, Moscow Univ. Phys. Bull. 31, 10–15 (1976).
  * [10] V.I. Yukalov, Model of a hybrid crystal, Theor. Math. Phys. 28, 652–660 (1976).
  * [11] V.I. Yukalov, Quantum crystal with jumps of particles, Physica A 89, 363–372 (1977).
  * [12] V.I. Yukalov, Quantum theory of localized crystal, Ann. Phys. (Berlin) 491, 31–39 (1979).
  * [13] W.E. Caswell, Accurate energy levels for the anharmonic oscillator and a summable series for the double well potential in perturbation theory, Ann. Phys. (N.Y.) 123, 153–184 (1979).
  * [14] I.G. Halliday and P. Suranyi, Anharmonic oscillator: A new approach, Phys. Rev. D 21, 1529–1537 (1980).
  * [15] J. Killingbeck, Renormalized perturbation series, J. Phys. A 14, 1005–1008 (1981).
  * [16] P.M. Stevenson, Optimized perturbation theory, Phys. Rev. D 23, 2916–2944 (1981).
  * [17] I.D. Feranchuk and L.I.Komarov, The operator method of the approximate solution of the Schrödinger equation, Phys. Lett. A 88, 211–214 (1982).
  * [18] V.I. Yukalov and V.I. Zubov, Localized-particles approach for classical and quantum crystals, Fortschr. Phys. 31, 627–672 (1983).
  * [19] M. Dineykhan, G.V. Efimov, G. Gandbold, and S.N. Nedelko, Oscillator Representation in Quantum Physics (Springer, Berlin, 1995).
  * [20] A.N. Sissakian and I.L. Solovtsov, Variational expansions in quantum chromodynamics, Phys. Part. Nucl. 30, 1057–1119 (1999).
  * [21] H. Kleinert and V.I. Yukalov, Self-similar variational perturbation theory for critical exponents, Phys. Rev. E 71, 026131 (2005).
  * [22] I. Feranchuk, A. Ivanov, V.H. Le, and A. Ulyanenkov, Nonperturbative Description of Quantum Systems (Springer, Cham, 2015).
  * [23] B.S. Shaverdyan and A.G. Ushveridze, Convergent perturbation theory for the scalar $\varphi^{2p}$ field theories: The Gell-Mann-Low function, Phys. Lett. B 123, 316–318 (1983).
  * [24] A.G. Ushveridze, Superconvergent perturbation theory for Eucledean scalar field theories, Phys. Lett. B 142, 403–406 (1984).
  * [25] A.V. Turbiner and A.G. Ushveridze, Anharmonic oscillator: Constructing the strong coupling expansions, J. Math. Phys. 29, 2053–2063 (1988).
  * [26] V. Sazonov, Convergent series for polynomial lattice models with complex actions, Mod. Phys. Lett. A 34, 1950243 (2019).
  * [27] V.I. Yukalov and E.P. Yukalova, Degenerate trajectories and Hamiltonian envelopes in the method of self-similar approximations, Can. J. Phys. 71, 537–546 (1993).
  * [28] S. Gluzman and V. I. Yukalov, Self-similarly corrected Padé approximants for the indeterminate problem, Eur. Phys. J. Plus. 131, 340 (2016).
  * [29] S. Gluzman and V. I. Yukalov, Self-similarly corrected Padé approximants for nonlinear equations, Int. J. Mod. Phys. B 33, 1950353 (2019).
  * [30] V.I. Yukalov, Statistical mechanics of strongly nonideal systems, Phys. Rev. A 42, 3324–3334 (1990).
  * [31] V.I. Yukalov, Self-similar approximations for strongly interacting systems, Physica A 167, 833–860 (1990).
  * [32] V.I. Yukalov, Method of self-similar approximations, J. Math. Phys. 32, 1235–1239 (1991).
  * [33] V.I. Yukalov, Stability conditions for method of self-similar approximations, J. Math. Phys. 33, 3994–4001 (1992).
  * [34] V.I. Yukalov and E.P. Yukalova, Self-similar perturbation theory, Ann. Phys. (N.Y.) 277, 219–254 (1999).
  * [35] V.I. Yukalov and S. Gluzman, Self-similar interpolation in high-energy physics, Phys. Rev. D 91, 125023 (2015).
  * [36] L.R. Foulds, Optimization Techniques (Springer, New York, 1981).
  * [37] L.M. Hocking, Optimal Control (Clarendon, Oxford, 1991).
  * [38] V.I. Yukalov and E.P. Yukalova, Self-similar structures and fractal transforms in approximation theory, Chaos Solit. Fract. 14, 839–861 (2002).
  * [39] A.H. Cliffordand G.B. Preston, The Algebraic Theory of Semigroups (American Mathematical Society, Providence, 1967).
  * [40] S. Lang, Algebra (Addison-Wesley, Reading, 1984).
  * [41] M.F. Barnsley, Fractal Transform (AK Peters Ltd., Natick, 1994).
  * [42] V.I. Yukalov, S. Gluzman, and D. Sornette, Summation of power series by self-similar factor approximants, Physica A 328, 409–438 (2003).
  * [43] S. Gluzman, V.I. Yukalov, and D. Sornette, Self-similar factor approximants, Phys. Rev. E 67, 026109 (2003).
  * [44] V.I. Yukalov and E.P. Yukalova, Calculation of critical exponents by self-similar factor approximants, Eur. Phys. J. B 55, 93–99 (2007).
  * [45] N.J. Higham, Accuracy and Stability of Numerical Algorithms (SIAM, Philadelphia, 2002).
  * [46] V.I. Yukalov and E.P. Yukalova, Describing phase transitions in field theory by self-similar approximants, Eur. Phys. J. Web Conf. 204, 02003 (2019).
  * [47] C.M. Bender and T.T. Wu, Anharmonic oscillator, Phys. Rev. 184, 1231–1260 (1969).
  * [48] F.T. Hioe, D. MacMillen, and E.W. Montroll, Quantum theory of anharmonic oscillators: Energy levels of a single and a pair of coupled oscillators with quartic coupling, Phys. Rep. 43, 305–335 (1978).
  * [49] J. Schwinger, Gauge invariance and mass, Phys. Rev. 128, 2425–2428 (1962).
  * [50] T. Banks, L. Susskind, and J. Kogut, Strong-coupling calculations of lattice gauge theories: $(1+1)$-dimensional exercises, Phys. Rev. D 13, 1043–1053 (1976).
  * [51] A. Carrol, J. Kogut, D.K. Sinclair, and L. Susskind, Lattice gauge theory calculations in $1+1$ dimensions and the approach to the continuum limit, Phys. Rev. D 13, 2270–2277 (1976).
  * [52] J.P. Vary, T.J. Fields, and H.J. Pirner, Chiral perturbation theory in the Schwinger model, Phys. Rev. D 53, 7231–7238 (1996).
  * [53] C. Adam, The Schwinger mass in the massive Schwinger model, Phys. Lett. B 382, 383–388 (1996).
  * [54] P. Striganesh, C.J. Hamer, and R.J. Bursill, A new finite-lattice study of the massive Schwinger model, Phys. Rev. D 62, 034508 (2000).
  * [55] C.J. Hamer, Z. Weihong, and J. Oitmaa, Series expansions for the massive Schwinger model in Hamiltonian lattice theory, Phys. Rev. D 56 , 55–67 (1997).
  * [56] S. Coleman, More about the massive Schwinger model, Ann. Phys. (N.Y.) 101, 239–267 (1976).
  * [57] C.J. Hamer, Lattice model calculations for SU(2) Yang-Mills theory in $(1+1)$-dimensions, Nucl. Phys. B 121, 159–175 (1977).
  * [58] V.A. Novikov, M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov, Exact Gell-Mann-Low function of supersymmetric Yang-Mills theories from instanton calculus, Nucl. Phys. B 229, 381–393 (1983).
  * [59] V.A. Novikov, M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov, The beta function in supersymmetric gauge theories: Instantons versus traditional approach, Phys. Lett. B 166, 329–333 (1986).
  * [60] M.A. Shifman and A.I. Vainshtein, Solution of the anomaly puzzle in SUSY gauge theories and the Wilson operator expansion, Nucl. Phys. B 277, 456–486 (1986).
  * [61] N. Arkani-Hamed and H. Murayama, Renormalization group invariance of exact results in supersymmetric gauge theories, Phys. Rev. D 57, 6638–6648 (1998).
  * [62] N. Arkani-Hamed and H. Murayama, Holomorphy, rescaling anomalies and exact $\beta$ functions in supersymmetric gauge theories, J. High Energy Phys. 2000, 030 (2000).
  * [63] I.O. Goriachuk and A.L. Kataev, Exact $\beta$-function in Abelian and non-Abelian $N=1$ supersymmetric gauge models and its analogy with the QCD $\beta$-function in the C-scheme, JETP Lett. 111, 663–667 (2020).
  * [64] M.V. Kompaniets and E. Panzer, Minimally subtracted six-loop renormalization of $O(n)$-symmetric $\varphi^{4}$ theory and critical exponents, Phys. Rev. D 96, 036016 (2017).
  * [65] O. Schnetz, Numbers and functions in quantum field theory, Phys. Rev. D 97, 085018 (2018).
  * [66] D.I. Kazakov, O.V. Tarasov, and D.V. Shirkov, Analytic continuation of the results of perturbation theory for the model $g\varphi^{4}$ to the region $g\geq 1$, Theor. Math. Phys. 38, 9–16 (1979).
  * [67] K.G. Chetyrkin, S.G. Gorishny, S.A. Larin, and F.V. Tkachev, Five-loop renormalization group calculations in the $g\varphi^{4}$ theory, Phys. Lett. B 132, 351–354 (1983).
  * [68] A.N. Sissakian, I.L. Solovtsov, and O.P. Solovtsova, $\beta$-function for the $\varphi^{4}$-model in variational perturbation theory, Phys. Lett. B 321, 381–384 (1994).
  * [69] A.L. Kataev and S.A. Larin, Analytical five-loop expressions for the renormalization group QED $\beta$-function in different renormalization schemes, JETP Lett. 96, 61–65 (2012).
  * [70] A. Deur, S.J. Brodsky, and G.F. de Teramond, The QCD running coupling, Prog. Part. Nucl. Phys. 90, 1–74 (2016).
  * [71] T. Luthe, A. Maier, P. Marquard, and Y. Schröder, Towards the five-loop beta function for a general gauge group, J. High Energy Phys. 2016, 127 (2016).
  * [72] P.A. Baikov, K.G. Chetyrkin, and J.H. Kühn, Five-loop running of the QCD coupling constant, Phys. Rev. Lett. 118, 082002 (2017).
  * [73] F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren, and A. Vogt, The five-loop beta function of Yang-Mills theory with fermions, J. High Energy Phys. 2017, 090 (2017).
  * [74] T. Banks and A. Zaks, On the phase structure of vector-like gauge theories with massless fermions, Nucl. Phys. B 196, 189–204 (1982).
  * [75] M. Tanabashi et al., Review of particle physics (Particle Data Group), Phys. Rev. D 98, 030001 (2018).
  * [76] A.H. Nayfeh, Perturbation Methods (Wiley, New York, 1973).
  * [77] E.J. Hinch, Perturbation Methods (Cambridge University, Cambridge, 1991).
  * [78] E.P. Yukalova, V.I. Yukalov, and S. Gluzman, Self-similar factor approximants for evolution equations and boundary-value problems, Ann. Phys. (N.Y.) 323, 3074–3090 (2008).
  * [79] V.I. Yukalov ans S. Gluzman, Self-similar exponential approximants, Phys. Rev. E 58, 1359–1382 (1998).

Figure Captions

Figure 1. Gell-Mann-Low function of $\varphi^{4}$ $O(1)$ symmetric field
theory as a function of the coupling parameter $g$. The convergence of the
approximants of second, fourth, and sixth order is evident.

Figure 2. Gell-Mann-Low function of quantum electrodynamics as a function of
the coupling parameter $\alpha$.

Figure 3. Gell-Mann-Low function of quantum chromodynamics for the flavour
number $n_{f}=6$ as a function of the coupling parameter $\alpha_{s}$.

Figure 4. Difference between the Bose-Einstein distribution (66) and its self-
similar factor approximants: (a) $f_{B}(x)-f_{4}^{*}(x)$; (b)
$f_{B}(x)-f_{8}^{*}(x)$; (c) $f_{B}(x)-f_{12}^{*}(x)$.

Figure 5. Difference between the Fermi-Dirac distribution (68) and its self-
similar factor approximants: (a) $f_{F}(x)-f_{4}^{*}(x)$; (b)
$f_{F}(x)-f_{8}^{*}(x)$; (c) $f_{F}(x)-f_{12}^{*}(x)$.

Figure 6. (a) Function (70) (solid line) and its self-similar factor
approximants obtained from the direct series (72): $f_{4}^{*}(x)$ (dotted
line), $f_{8}^{*}(x)$ (dash-dotted line), and $f_{12}^{*}(x)$ (dashed line).
(b) Function (70) (solid line) and its self-similar factor approximants
obtained from expansion (74): $F_{4}^{*}(x)$ (dotted line), $F_{8}^{*}(x)$
(dash-dotted line), and $F_{12}^{*}(x)$ (dashed line).

Figure 1: Gell-Mann-Low function of $\varphi^{4}$ $O(1)$ symmetric field
theory as a function of the coupling parameter $g$. The convergence of the
approximants of second, fourth, and sixth order is evident.

Figure 2: Gell-Mann-Low function of quantum electrodynamics as a function of
the coupling parameter $\alpha$.

Figure 3: Gell-Mann-Low function of quantum chromodynamics for the flavour
number $n_{f}=6$ as a function of the coupling parameter $\alpha_{s}$.

Figure 4: Difference between the Bose-Einstein distribution (66) and its self-
similar factor approximants: (a) $f_{B}(x)-f_{4}^{*}(x)$; (b)
$f_{B}(x)-f_{8}^{*}(x)$; (c) $f_{B}(x)-f_{12}^{*}(x)$.

Figure 5: Difference between the Fermi-Dirac distribution (68) and its self-
similar factor approximants: (a) $f_{F}(x)-f_{4}^{*}(x)$; (b)
$f_{F}(x)-f_{8}^{*}(x)$; (c) $f_{F}(x)-f_{12}^{*}(x)$.

Figure 6: (a) Function (70) (solid line) and its self-similar factor
approximants obtained from the direct series (72): $f_{4}^{*}(x)$ (dotted
line), $f_{8}^{*}(x)$ (dash-dotted line), and $f_{12}^{*}(x)$ (dashed line).
(b) Function (70) (solid line) and its self-similar factor approximants
obtained from expansion (74): $F_{4}^{*}(x)$ (dotted line), $F_{8}^{*}(x)$
(dash-dotted line), and $F_{12}^{*}(x)$ (dashed line).


# Crossover from dynamical percolation class to directed percolation class on
a two dimensional lattice

M. Ali Saif
Department of Physics
University of Amran, Amran,Yemen.
<EMAIL_ADDRESS>
###### Abstract

We study the crossover phenomena from the dynamical percolation class (DyP) to
the directed percolation class (DP) in the model of diseases spreading,
Susceptible-Infected-Refractory-Susceptible (SIRS) on a two-dimensional
lattice. In this model, agents of three species S, I, and R on a lattice react
as follows: $S+I\rightarrow I+I$ with probability $\lambda$, $I\rightarrow R$
after infection time $\tau_{I}$ and $R\rightarrow I$ after recovery time
$\tau_{R}$. Depending on the value of the parameter $\tau_{R}$, the SIRS model
can be reduced to the following two well-known special cases. On the one hand,
when $\tau_{R}\rightarrow 0$, the SIRS model reduces to the SIS model. On the
other hand, when $\tau_{R}\rightarrow\infty$ the model reduces to SIR model.
It is known that, whereas the SIS model belongs to the DP universality class,
the SIR model belongs to the DyP universality class. We can deduce from the
model dynamics that, SIRS will behave as an SIS model for any finite values of
$\tau_{R}$. SIRS will behave as SIR only when $\tau_{R}=\infty$. Using Monte
Carlo simulations we show that as far as the $\tau_{R}$ is finite the SIRS
belongs to the DP university class. We also study the phase diagram and
analyze the scaling behavior of this model along the critical line. By
numerical simulation and analytical argument, we find that the crossover from
DyP to DP is described by the crossover exponent $1/\phi=0.67(2)$.

##  1 Introduction

Crossover phenomena are well-known for the equilibrium phase transitions [1,
2]. From the point of view of the renormalization group theory, crossover from
one class to another class occurs if more than one fixed point is embedded in
the critical surface in which each fixed point corresponds to a different
class. This crossover is generally interpreted by the competition of those
fixed points. Through that, if there is a system characterized by irrelevant
scaling fields and one relevant scaling field, then any change in the value of
the relevant scaling field will cause the movement of the system from one
fixed point to another fixed point. Crossover phenomena are useful for
understanding the critical phenomena in phase transitions [1, 2, 3, 4]. In
nonequilibrium phase transitions with absorbing states the crossover phenomena
between the universality classes of those kinds of transitions, has been
observed in many models. For example the crossover from directed percolation
universality class (DP) to compact directed percolation universality class
(CDP) [5, 6], from parity conserving class (PC) (directed Ising (DI) class) to
DP class [7], pair contact process with diffusion (PCPD) to PC [8], DP to DI
[8], PCPD to DP [8, 9], mean field PCPD to the PCPD [10], DI to DP [11],
isotropic percolation (IP) to DP [12], DP to mean field [13], Mean field to DP
[14], Manna universality class to mean field [15], non-DI critical behavior of
the branching annihilating attracting walk (BAAW) with infinite range of
attraction to DI [16] and DyP to DP [17].

In this work, we are going to study the crossover phenomena from dynamical
percolation class (DyP) to directed percolation class (DP) in the model of
diseases spreading, Susceptible-Infected-Refractory-Susceptible (SIRS) on a
two-dimensional lattice. In this direction, DyP universality class (general
epidemic process) has been found in some models of epidemics spreading with
immunization (no reinfection) such as the SIR model [18, 19] or forest fire
model with immune trees [20] or even in prey-predator model [21, 22]. DyP
class considers a generalization of DP class including the effect of
immunization. Geometrical interpretation of the DyP class as an ordinary
percolation system (isotropic percolation) has been introduced by Stauffer and
Aharony [23]. In the other direction, the DP universality class is the most
famous in the field of nonequilibrium phase transition from the active phase
to the absorbing phase [3, 24, 25, 26, 27]. This class is robust concerning
microscopic changes and many models have been found to be within this class.
According to Janssen and Grassberger [28, 29], the models which display a
continuous phase transition into a unique absorbing state with a positive
single component order parameter and short-range interactions without quenched
disorder or additional symmetries should belong to the DP class. By any way
this class has been found to be more general, systems with more than one
absorbing state or long-range interaction have been found to be within the DP
class (see Refs. [3, 24, 25, 26] and references therein).

##  2 Model and methods

The model of epidemics spreading SIRS on the networks can be described as
follows [30, 31]: consider a population of $N$ agents residing on the sites of
a two-dimensional square lattice (of size $L\times L$), in which each agent is
connected to $k$ of its first neighbors. The agents can exist in one state of
the following three states, susceptible $(S)$, infected $(I)$, and refractory
$(R)$. The interaction between the agents on the lattice occurs as follows:
the agents in state $I$ on the network can infect any one of their neighbors
which are in state $S$ with probability $\lambda$ at each time step. The
agents in state $I$ pass to the state $R$ after an infection time $\tau_{I}$.
The agents in state $R$ cross to the state $S$ after a recovery time
$\tau_{R}$. During the $R$ phase, the agents are immune and do not infect. The
dynamics of the model are summarized as

$\displaystyle
S+I\stackrel{{\scriptstyle\lambda}}{{\rightarrow}}I+I,I\stackrel{{\scriptstyle\tau_{I}}}{{\rightarrow}}R,R\stackrel{{\scriptstyle\tau_{R}}}{{\rightarrow}}I$
(1)

The SIRS model is a general model of disease spreading which we can map into
the following two extremist special cases. The first case occurs when
$\tau_{R}\rightarrow 0$, in this case, the SIRS model reduces to the SIS
model. On the other hand second case occurs when $\tau_{R}\rightarrow\infty$,
in this case SIRS model reduces to the SIR model. The SIS model is the model
of disease spreading without immunization, however SIR model is the model of
disease spreading with a perfect immunization (vanishing reinfection
probability). It is well-known that, whereas the SIS model belongs to the DP
universality class [32], the SIR model belongs to the DyP universality class
[18, 23]. In this work, we are interested in studying the crossover behavior
from the DyP universality class to the DP universality class of this model on
a two-dimensional lattice, when the value of $\tau_{R}$ is changed from $0$ to
$\infty$. Thus the crossover control parameters here are $\lambda$ and
$\tau_{R}$. However, for technical reasons we modify the control parameter
$\tau_{R}$ to be $\tau=\frac{\tau_{I}}{\tau_{R}}$ and for simplicity we fix
the value of $\tau_{I}$ to be $\tau_{I}=1$.

From the model dynamics we can deduce that, initially and during the time
which is $t\leq\tau_{R}$, the SIRS model will behave as if it is an SIR model.
During this time the agents pass only from the state $S$ to $I$ followed by
moving from the state $I$ to $R$, no one of the agents on the lattice can move
from the state $R$ to $S$. However as the time reaches the value of
$t>\tau_{R}$, there will be agents in the state $R$ that are ready to move
from the state $R$ to $S$. This moving for the agents from the state $R$ to
$S$ will alter the behavior of the model from SIR-like to SIS-like. So we can
say that SIRS will behave temporarily (for a time equal to $\tau_{R}$) as the
SIR model at the beginning of the time evolution of this model. Higher values
of $\tau_{R}$ will put the model for a longer time in the SIR-like regime, but
eventually, the SIRS will change its behavior to become similar to SIS
behavior. Because of that, we can conclude that, for any finite values of
$\tau_{R}$, soon or later SIRS will change its behavior from SIR-like to SIS-
like. This model will behave exactly as SIR only when $\tau_{R}=\infty$
($\tau=0$), in this case, the critical value of $\lambda$ is exactly given by
$\lambda_{c}=0.5$ [23].

To determine the critical points of our model as well as the critical
exponents we use the dynamical Monte Carlo method [25, 33, 34]. In the seed
simulations of this method, the time evolution of quantities such as the
average number of active sites $N(T)$ and survival probability $P(t)$ exhibit
a power-law behavior at criticality. Mathematically we can write this as
follows:

$\displaystyle N(t)\sim t^{\theta}$ (2)

and

$\displaystyle P(t)\sim t^{-\delta}$ (3)

where $\theta$ and $\delta$ are the critical exponents. Those critical
exponents in $d=2$ are $\delta=0.4505$, $\theta=0.2295$ for DP class [25, 35]
and $\delta=0.092$, $\theta=0.586$ for DyP class [35]. Using Eqs. 2 and 3 we
can determine the critical points and the critical exponents, however for a
more accurate determination we adopt here the local slope analysis with the
effective exponent which is defined as follows [25]

$\displaystyle\theta(t)=\frac{\ln[N(t)]-\ln[N(t/m)]}{\ln[m]}$ (4)

where $m$ is a constant larger than one. Same definition we can find for the
effective exponent of $\delta(t)$.

We simulate our model on a two-dimensional lattice with periodic boundary
conditions. We fix the value of infection time to be $\tau_{I}=1$. Each agent
on the lattice is connected to its first four nearest neighbors, and the
system updates synchronously. In Fig. 1 we show two examples of the results
which we find from the seed simulations. In our Monte Carlo simulations, we
use a lattice of size $4000\times 4000$ and we take the average over $2000$
configurations. Fig. 1 shows the value of the effective exponent Eq. 4 as a
function of inverse time $1/t$ for the value of $\tau=0.004$ (left) and
$\tau=0.5$ (right). The corresponding critical points are
$\lambda_{c}(0.004)=0.4755(3)$ and $\lambda_{c}(0.5)=0.4009(3)$. As it is
clear from the figure in both cases the critical exponent $\theta$ converges
to its value in the DP universality class $\theta=0.229$. We assert that, our
results using the critical exponents $\theta$ or $\delta$ confirm the phase
transition to be of DP universality class along the phase transition line
$\lambda_{c}(\tau)$ except when $\tau=0$ (see Fig. 2). We have checked the
validity of this conclusion up to small values of $\tau$, i. e. $\tau=0.001$
which corresponds to $\lambda_{c}(0.001)=0.4986(4)$. Fig. 2 shows the value of
$\theta$ as a function of $1/t$ for the case when $\tau=0$ in which the
critical point is exactly $\lambda_{c}(0)=0.5$. As it is clear from the figure
the critical exponent $\theta$ approaches its value in DyP class
$\theta=0.586$.

Figure 1: Time-dependent behavior of the effective exponent for the average
number of active sites exponent $\theta$ as a function of $1/t$ for the values
of $\tau=0.004$ (left) and $\tau=0.5$ (right), for a lattice of size
$4000\times 4000$. In both figures $\theta$ converges to its value in the DP
class. Inset shows the number of infected agents $N(t)$ as a function of time.
The data are averaged over $2000$ realization. Figure 2: Time-dependent
behavior of the effective exponent for the average number of active sites
exponent $\theta$ as a function of $1/t$ when $\tau=0$. $\theta$ converges to
its value in the DyP class. Inset shows the number of infected agents $N(t)$
as a function of time.

Fig. 3 shows the phase diagram of this model, in which we plot the values of
the critical points $\lambda_{c}(\tau)$ as a function of $\tau$. The phase
transition line separates the active phase (above) from the absorbing phase
(below). Each point in Fig. 3 is due to the Monte Carlo simulations of lattice
of size $4000\times 4000$ and, has been determined using the effective
exponent Eq. 4 as we have described previously. Figure shows clearly that, the
critical line approaches the DyP class at $\tau=0$ with infinite slope (see
inset of Fig. 3). The critical points we have found are summarized in Table 1.

Figure 3: Phase diagram of the model in the $\tau$-$\lambda$ plane. The active
(absorbing) phase is above (below) the curve. The error in the critical points
is smaller than the symbol size. Inset: shows the critical line beside
$\tau=0$.

Table 1: Critical points of the SIRS for various values of $\tau_{R}$.
---
$\tau_{R}$ | $\lambda_{c}$
2 | 0.4015(2)
10 | 0.4627(3)
50 | 0.4865(3)
100 | 0.4916(4)
200 | 0.4947(4)
350 | 0.4964(3)
500 | 0.4972(2)
1000 | 0.4983(4)
$\infty$ | 0.5

##  3 Crossover behavior

We are going here to study the crossover behavior of this model from the DyP
class to the DP class. The control parameters of this model are $\lambda$ and
$\tau$. Let us consider the case when $\tau=0$, in this case, the SIRS model
reduces to the SIR model which belongs to DyP class [18]. The critical point
for the system in this case is $\lambda_{c}=0.5$ [23]. Now from
renormalization methods, we know that the phase transitions into absorbing
states are characterized usually by simple scaling laws. For example, the
scaling behavior for the average number of active sites $N(t)$ near the
critical point is described by the function [36, 37]

$\displaystyle
N(\Delta_{\lambda};t)=t^{\theta}\psi(\Delta_{\lambda}t^{1/\nu_{\|}})$ (5)

where $\psi(\Delta_{\lambda}t^{1/\nu_{\|}})$ is a scaling function,
$\Delta_{\lambda}=\lambda_{c}-\lambda$ denotes the distance from criticality
and $\nu_{\|}$ is critical exponent corresponding to temporal correlation
length which can be given by the relation

$\displaystyle\xi_{\|}\approx\left|\Delta_{\lambda}\right|^{-\nu_{\|}}$ (6)

Now when $\tau>0$, we can modify the function Eq. 5 to include the additional
control parameter $\tau$ to be

$\displaystyle
N(\Delta_{\lambda},\tau;t)=t^{\theta}\Psi(\Delta_{\lambda}t^{1/\nu_{\|}},\tau
t^{1/\mu_{\|}})$ (7)

where $\mu_{\|}$ is the critical exponent of temporal correlation length
corresponding to the parameter $\tau$. Like $\nu_{\|}$ this exponent can be
given by the relation

$\displaystyle\xi_{\|}\approx\tau^{-\mu_{\|}}$ (8)

Comparing Eq. 6 with Eq. 8 we can define the crossover exponent by the shape
of phase boundary $\lambda_{c}(\tau)$ approaches the critical point
$\lambda_{c}(0)$ as the relevant scaling field $\tau\rightarrow 0$ as follows

$\displaystyle\Delta_{\lambda}(\tau)\approx\tau^{1/\phi}$ (9)

where $\Delta_{\lambda}(\tau)=\lambda_{c}(\tau)-\lambda_{c}(0)$ and the
crossover exponent will be

$\displaystyle 1/\phi=\frac{\mu_{\|}}{\nu_{\|}}$ (10)

In Fig. 4 we show a double-logarithmic plot of Eq. 9 near the DyP critical
point $\lambda_{c}(0)=0.5$. The points given in Fig. 4 are the same as those
given in Fig. 3 when the values of $\tau<<1$. For the best estimating, the
value of the crossover exponent is $1/\phi=0.67(2)$.

Figure 4: Double-logarithmic plot of the phase transition line. The dashed
line is power-law fitting on the numerical data points using Eq. 9 with the
crossover exponent $1/\phi=0.67(2)$.

At this point, we are going to suggest the following method to estimate the
value of crossover exponent $1/\phi$ for this model. As we have mentioned
previously, initially and during the time which is proportional to
$\tau_{R}=\tau^{-1}$, the SIRS model will behave as the SIR model. During this
time the agents on the lattice pass from the state $S$ to $I$ and from the
state $I$ to $R$, no one of the agents can pass from the state $R$ to $S$.
After this time the first infected agents on the lattice will start departing
from the state $R$ to $S$ consequently, the behavior of the system changes to
be similar to the behavior of the SIS model. In Fig. 5 we manifest this
behavior of this system where we plot the average value of active sites $N(t)$
(left) and the average value of survival probability $P(t)$ (right) as a
function of time for values of $\tau=0,0.004,0.04$ and $0.5$ at the
corresponding critical points $\lambda_{c}(\tau)$. In both figures, the upper
line represents the DyP behavior ($\tau=0$), whereas the lower line
corresponds to the DP class ($\tau=0.5$). In between those two lines, the
figures show how the system starts as DyP class for some time after that
reaches DP class.

Figure 5: Number of active sites $N(t)$ (left) and survival probability $P(t)$
(right) as a function of time for values of $\tau=0,0.004,0.04$ and $0.5$ at
the corresponding values of critical points $\lambda_{c}(\tau)$.

Therefore we can conclude that, initially the process behaves as a critical
DyP until the first refractory agents on the lattice move from the state $R$
to $S$ at a typical time $\xi_{\|}$. Here we assume that $\xi_{\|}$ scales in
the same way as the typical time at which the first refractory agents on the
lattice start moving from the state $R$ to $S$ [17]. Whereas the time at which
the first refractory agents on the lattice start moving from the state $R$ to
$S$ is proportional to $\tau^{-1}$, so we can conjecture also that $\xi_{\|}$
is proportional to $\tau^{-1}$

$\displaystyle\xi_{\|}\propto\tau^{-1}$ (11)

comparing this relation with the relation in Eq. 8 we can deduce the value of
$\mu_{\|}$ to be $\mu_{\|}=1.0$.

To check the validity of this value of $\mu_{\|}$, we use here the scaling
function of the system besides its criticality. For that, we set $\lambda=0.5$
in the scaling function Eq. 7 to get

$\displaystyle N(0,\tau;t)t^{-\theta}=\Psi(0,\tau t^{1/\mu_{\|}})$ (12)

Fig. 6 shows the collapse of data into a single curve using the exponent
$\mu_{\|}=1.00(4)$ as the Eq. 12 suggests. Inset of Fig. 6 shows the average
number of active sites $N(t)$ as a function of time when $\lambda=0.5$, for
the values of $\tau=0.001,0.002$ and $0.004$. This result confirms what we
found previously for this value of $\mu_{\|}$.

Now, using this value of the exponent $\mu_{\|}=1.0$ and the value of the
critical exponent $\nu_{\|}=1.506$ for DyP class in $d=2$ [35], and the Eq.
10, we can estimate the value of $1/\phi$ to be equal to

$\displaystyle 1/\phi=\frac{\mu_{\|}}{\nu_{\|}}=\frac{1}{1.506}=0.664$ (13)

this value of $1/\phi$ coincides very well with its the value that we have
estimated using the Monte Carlo simulations of this model.

Figure 6: Semilogarithmic plot of $Nt^{-\theta}$ as a function of
$t\tau^{\mu_{\|}}$ with $\mu_{\|}=1.0$ for $\tau=0.001,0.002$ and $0.004$.
Curves collapse well into a single curve.

Finally, we mention that crossover from DyP class to DP class has been
observed in the model of epidemic spreading with immunization and mutations in
two dimensions [17]. However, the value of crossover exponent $1/\phi$ has
been found in Ref. [17] is different from the value we find here. For
comparison Table 2 gives the values of $1/\phi$ and $\mu_{\|}$ for both
models.

Table 2: Comparison of values $1/\phi$ and $\mu_{\|}$.
---
| $1/\phi$ | $\mu_{\|}$
This work | 0.67(2) | 1.00(4)
Ref. [17] | 0.41(3) | 0.63(4)

Such this difference in the value of the crossover exponent $1/\phi$ has been
observed in the case of the crossover from DI class to DP in one dimension
[11] and the crossover between directed percolation models [38]. According to
S.- C. Park and H. Park [11, 38] the difference in the value of the crossover
exponent reflects the difference in the route that leads to the crossover from
one class to another class. Using that, the crossover behavior from one fixed
point to another is related to the crossover operator, which forces the system
to cross from one to another. In our model, the route to crossover implies
only one kind of pathogen and the reinfection occurs due to the temporary
immunization111We can consider the time $\tau_{R}$ at which the agent is in
state $R$ as the agent memory (finite memory) of agents. In case the agents
are perfectly immune (infinite memory) after the first infection, the
reinfection will not occur, in which the system will behave as SIR. This leads
to the conclusion that memory, is crucial in establishing the crossover in our
model. In the model of Ref. [17] the agents become perfectly immune after each
infection however, reinfection occurs due to the pathogen mutation. Hence in
this model, the pathogen mutation establishes the crossover.

##  4 Conclusion

Depending on the value of infection time $\tau_{R}$ the model of epidemics
spreading SIRS can show the following two extremist special cases. In the case
of $\tau_{R}=0$ SIRS model reduces to the SIS model which belongs to the DP
universality class. On the other hand when $\tau_{R}=\infty$ SIRS reduces to
the SIR model which belongs to the DyP universality class. In this work, we
have studied the crossover phenomena for this model from the DyP universality
class to the DP universality class. we have found that the SIRS with finite
$\tau_{R}$ does belong to the DP universality class. Only the case when
$\tau_{R}=\infty$ the model is in the DyP universality class. Here we mention
that similar behavior to what is found here has been observed in the BAAW
model in one dimension when the range of attraction $R$ is varied [16]. In the
BAAW model only the case when $R=\infty$ is different from the DI class.
Anyway, Monte Carlo simulations of our model suggest the crossover exponent
from the DyP class to the DP class to be $1/\phi=0.67(2)$. We have confirmed
this result with an analytical argument using the fact that SIRS will behave
initially as critical SIR for time which is proportional to
$\tau_{R}=\tau^{-1}$. This suggests the temporal correlation length $\xi_{\|}$
also is proportional with $\tau^{-1}$ which leads us to estimate the value of
the temporal correlation exponent $\mu_{\|}$ to be $\mu_{\|}=1.0$. With this
value of $\mu_{\|}=1.0$ we have determined the value of crossover exponent
using the relation $1/\phi=\frac{\mu_{\|}}{\nu_{\|}}=0.664$ which agrees very
well with the numerical estimation.

Furthermore, we found that the crossover from DyP class to DP class has been
observed in the model of epidemics spreading with immunization and mutations
in a two dimension [17]. However, the crossover exponent we found in this work
is different from its value that has been calculated in Ref. [17]. The
difference in the value of the crossover exponent for both models backs to the
difference in route (operator) that leads to crossover from DyP class to DP
class [11]. By inspecting the dynamics of both models we can understand the
difference in the route which leads to crossover. In our model, there is only
one kind of pathogen and the agents become immune for a time $\tau_{R}$ before
they pass again to $S$ state. Hence the memory induces the crossover in our
model from DyP with perfect memory to DP class with temporary memory. In the
model of Ref. [17], the agents become perfectly immune after each infection
however the pathogen mutation leads to reinfection. Thus the pathogen mutation
is the route to crossover from the DyP class without mutation to the DP class
with mutation.

## References

  * [1] P. Pfeuty and G. Toulouse, Introduction to the renormalization group and critical phenomena John Wiley $\&$Sons, Chichester (1994).
  * [2] A. Aharony, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green Academic Press, London, Vol. 6 (1976).
  * [3] M. Henkel, H. Hinrichsen and S. L$\ddot{u}$beck, Non-equilibrium phase transitions, Volume1: Absorbing phase transitions, Canopus Academic Publishing Limited (2008).
  * [4] M. E. Fisher, Phys. Rev. Lett. 57, 1911 (1986).
  * [5] J. F. F. Mendes, R. Dickman, and H. Herrmann, Phys. Rev. E 54, R3071 (1996).
  * [6] S. L$\ddot{u}$beck, J. Stat. Mech. P09009, (2006).
  * [7] G. $\acute{O}$dor and N. Menyh$\acute{a}$rd, Phys. Rev. E 78, 041112 (2008).
  * [8] S. -C. Park and H. Park, Phys. Rev. E 79, 051130 (2009).
  * [9] B. Daga, and P. Ray, Phys. Rev. E 99, 032104 (2019).
  * [10] S. -C. Park, Phys. Rev. E 90, 052115 (2014).
  * [11] S. -C. Park and H. Park, Phys. Rev. E 78, 041128 (2008).
  * [12] Z. Zhou, J. Yang, R. M. Ziff2 and Y. Deng, Phys. Rev. E 86, 021102 (2012).
  * [13] A. Messer and H. Hinrichsen, J. Stat. Mech. P04024, (2008).
  * [14] T. B. dos Santos, C. I. N. S. Filho, N. A. M. Araujo, C. L. N. Oliveira, and A. A. Moreira, Physica A 512, 352 (2018).
  * [15] S. L$\ddot{u}$beck, Phys. Rev. Lett. 90, 210601 (2003).
  * [16] S. -C. Park, Phys. Rev. E 101, 052103 (2020).
  * [17] S. M. Dammer, and H. Hinrichsen, Phys. Rev. E 68, 016114 (2003).
  * [18] P. Grassberger, Mathematical Biosciences 63, 157 (1983).
  * [19] J. L. Cardy and P. Grassberger, J. Phys. A: Math. Gen. 18, L267 (1985).
  * [20] E. V. Albano, J. Phys. A: Math. Gen. 27, L881 (1994).
  * [21] T. Antal, M. Droz, A. Lipowski and G. $\acute{o}$dor, Phys. Rev. E 64, 036118 (2001).
  * [22] E. Arashiro and T. Tom$\acute{e}$, J. Phys. A: Math. Theor. 40, 887 (2007).
  * [23] D. Stauffer and A. Aharony, Introduction to Percolation Theory, Taylor $\&$ Francis, London, 1992.
  * [24] G. $\acute{O}$dor, Rev. Mod. Phys. 76, 663 (2004).
  * [25] H. Hinrichsen, Adv. Phys. 49, 815 (2000).
  * [26] S. -C. Park and H. Park, Eur. Phys. J. B 64, 415 (2008).
  * [27] M. A. Saif and P. M. Gade, J. Stat. Mech. P03016, (2010).
  * [28] H. K. Janssen, Z. Phys. B 42, 151 (1981).
  * [29] P. Grassberger, Z. Phys. B 47, 365 (1982).
  * [30] M. Kuperman, and G. Abramson, Phys. Rev. Lett. 86, 2909 (2001).
  * [31] M. A. Saif, Physica A 535, 122251 (2019).
  * [32] P. Grassberger and Y.-C. Zhang, Physica A 224, 169 (1996).
  * [33] P. Grassberger and A. de la Torre, Ann. Phys. (N.Y.) 122, 373 (1979).
  * [34] I. Jensen, Phys. Rev. A 43, 3187 (1991).
  * [35] M. A. Mu$\tilde{n}$oz, R. Dickman, A. Vespignani, and S. Zapperi, Phys. Rev. E 59, 6175 (1999).
  * [36] I. D. Lawrie and S. Sarbach, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz, Academic Press, London, Vol. 9 (1984).
  * [37] N. Menyh$\acute{a}$d and G. $\acute{O}$dor, J. Phs. A: Math. Gen. 31 6771 (1998).
  * [38] S.-C. Park and H. Park, Phys. Rev. E 76, 051123 (2007).