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\title{\bf Multiloop Calculations at BLTP:\\[0.4cm] 30 Years of Progress }

\author{D.\,I.\,Kazakov and A.\,A.\,Vladimirov}
\date{}
\maketitle
\vspace{-0.8cm}
\begin{center}
{\it Bogoliubov Laboratory of Theoretical Physics,
Joint Institute for Nuclear Research,
Dubna, Russia }
\end{center}

\begin{abstract}
We briefly review the history of multiloop calculations in modern quantum field
theory. They started, with our participation, in the Bogoliubov Laboratory of
Theoretical Physics, JINR, more than 30 years ago, produced a number of remarkable
results, and now developed into a worldwide computational industry.
\end{abstract}

A considerable progress achieved in multiloop calculations in 1970s-80s in the Laboratory
of theoretical physics at JINR has been stimulated by the renaissance of perturbative
quantum field theory (QFT) which followed the discovery of the asymptotic freedom in
non-Abelian gauge theories. After several years of despair caused by a seemingly
inevitable ghost-pole menace, the perturbation theory calculations were again given their
chance to contribute to our knowledge of the properties of elementary particles,
including strongly interacting ones -- hadrons.

The first period of our activity in this field (1973-78), looking now only preparatory,
was nevertheless very important. At that time, our main motivation was a problem of
ultraviolet fixed point for the ``running" coupling. In quantum field theory at high
energies, as a rule, we are interested in the dependence of observable quantities on
momenta $p$ of interacting particles. By means of the renormalization group, this
$p$-dependence can be effectively accumulated in one function $g(p)$ which is called
effective charge, or running coupling. The name is due to the fact that a charge
(coupling constant) $g$, being originally a true constant, now becomes a function
of momenta $p$, and has to vary, or ``run", when $p$ goes to infinity.

There are very few typical patterns of asymptotical behavior of $g(p)$. In the first
nontrivial order of perturbation theory (called ``one loop" order due to Feynman diagrams
with only one closed loop being involved) we have only two possibilities: $g(p)$
increases fast and reaches infinity at some finite value of $p$ (this corresponds to a
notorious ghost pole -- not a good thing at all from physical point of view), or $g(p)$
tends to zero, and this is called {\it asymptotic freedom}. Though the rate of decrease
of the running coupling in this case is rather slow, $g(p)\sim1/\ln p$, nevertheless
$g(p)$ asymptotically vanish, and the corresponding particles decouple at high momenta:
in effect, they become free. The phenomenon of asymptotic freedom discovered in 1973 in
gauge theories with non-commutative (non-Abelian) groups, such as SU(3), made it possible
to build a first reliable quantum field model of strong interactions -- Quantum
Chromodynamics (QCD). Logarithmically decreasing asymptotics of the QCD effective charge
appears to be in a good agreement with the present experimental data at high energies. On
the theory side, asymptotically free theories are favored because they develop (in a
strongly interacting model!) a new small parameter of the improved perturbation theory
expansion -- a running coupling $g(p)$.

However, taking into account the next (two-loop) order may drastically change the whole
picture: it can produce a third kind of high-energy asymptotics, a so-called
(ultraviolet) fixed point. This is the case when $g(p)$ tends to some finite value
$g^*\neq0$. In principle, this effect could occur in theories troubled by a ghost pole
(thus curing them) as well as in asymptotically free ones, keeping them manageable.
Needless to say that in the fixed-point situation our (dis)ability to compute, or at
least estimate the higher-loop corrections becomes of crucial importance.

In 1973, motivated and supervised by D.\,V.\,Shirkov, we started a long programme of such
calculations, firstly at the two- and three-loop orders, in a wide range of QFT models,
with and without asymptotic freedom. These were by no means the first multiloop
calculations known in the literature; rather, we tried to adapt the existing techniques
devised for evaluating radiative corrections to the case of high-energy asymptotics
treated by the renormalization group approach. Only a few aspects of our computational
practice were really new, such as essential use of dimensional regularization and easy
conversion from one renormalization scheme to another (Shirkov and Vladimirov, 1979).
Also, the great computational
facilities of the algebraic manipulation program SCHOONSCHIP were used throughout. Thus,
the problem of doing tedious algebra was more or less settled. Of course, individual
complicated integrals still required evaluation ``by hand".

As a result, we succeeded in finding several examples of ultraviolet fixed points in
different models (Belokurov, Kazakov, Slavnov, Shirkov and Vladimirov, 1973, 1974).
However, these were not the results of immediate physical importance. And not only
because most of the models involved were unphysical, toy models. There was a fundamental
defect in the whole project: if one does discover a fixed point in the $n$-loop
approximation, would it persist when $n+1$ loops are taken into account? As a rule, the
answer is: No! A general reason for this is the divergent nature of perturbative series
in quantum field theory. Paradoxically, the highest-precision physical predictions of
Quantum Electrodynamics (QED) are actually obtained by taking several first terms of a
divergent series! This seems to be a real mystery, even when an expansion parameter (a
charge) is small, as in QED or QCD. But in the situation with the fixed point, this
parameter (the asymptotical value $g^*$ of a running coupling) does not need to be small
at all. Therefore, we cannot be sure that the fixed points found perturbatively do
actually exist. So, we may say that our activity in 1973-78 produced a certain
methodical, rather than physical, progress. However, it provided a base for a near
breakthrough.

Our principal achievements in multiloop calculations, dated roughly to 1979-83, resulted
from a noticeable progress in the technique of analytic computation of higher-order
renormalization group parameters, and also in half-empirical methods of ``summation" of
divergent series. Let us retrace the key stages of this progress, paying main attention
to the results that have been obtained at JINR.

In the framework of the renormalization group method, effective charge $g(p)$ is
unambiguously determined by a certain function, $\beta(g)$, with $g$ being original (not
running) coupling constant of a quantum theory under consideration. This $\beta$-function
is actually an infinite power series in $g$. Of course, in reality we are able to find
only several first (lowest) terms of this series, by calculating the Feynman diagrams
with appropriate number of loops. Thus, the one-loop calculations produce the first term
of $\beta(g)$ (which already indicates the presence, or absence, of asymptotic freedom),
the two-loop diagrams account for the second term, and so on. The coefficients at the
specific powers of $g$ in the power series $\beta(g)$ are mere numbers: they cannot
depend on momenta or masses which enter into algebraic expressions for the Feynman
diagrams involved. This property has been used in the method elaborated by our group
(Vladimirov, 1978, 1980) and known now as the ``infrared rearrangement", which
provides highly efficient tools for computing coefficients of $\beta(g)$ by putting all,
except one, momenta and masses to zero, thus making fairly manageable analytic evaluation
of all the needed integrals. One cannot nullify all momenta and masses due to some subtle
effects of interplay between ultraviolet and infrared divergences. Nevertheless, this
method, supplemented by some very useful tricks (Gegenbauer polynomial expansion and
integration by parts in the momentum space), invented by the INR group in Moscow
(K.\,G.\,Chetyrkin, F.\,V.\,Tkachov, et al.) opened a possibility to compute analytically
the three- and four-loop contributions to $\beta(g)$ in any renormalizable model.

The other direction of activity that contributed much to the progress in multiloop
calculations proved to be the development of summation techniques for divergent power
series for $\beta(g)$ and the other renormalization-group functions. As mentioned above,
the problem is that in the QFT perturbation theory the corresponding power series in $g$
are generally divergent so that their radius of convergence is actually zero. More
precisely, these series are called ``asymptotic" in a sense of Poincar\'e, because their
numerical coefficients increase like $n!$, with $n$ being a number of loops. As a result,
the more terms are taken into account, the smaller are those values of the coupling
constant $g$ for which the expansion is still reasonable.

To overcome this difficulty and obtain a reliable information about a function from its
perturbative expansion, there exist several methods for ``summation" of such series.
These methods, as a rule, rely on some additional information about the relevant
function, like its analytical properties in the coupling constant plane, exact
asymptotics of the perturbation theory coefficients in higher orders, etc. We were not
inventors of these methods, but we modified them when needed and, which was the most
important, successfully applied them to extract physical predictions from the results of
multiloop calculations.

The main ingredient of our summation procedure (Kazakov, Shirkov and Tarasov, 1979;
Kazakov and Shirkov, 1980)  was a due account of analytical properties of the desired
function through the use of the Laplace, or Borel, transformation. All this is usually
called the Borel summation technique. After the transformation, a Borel-transformed
power series contains extra $n!$ in the denominator and thus becomes convergent inside
some circle. It follows from the known behaviour of the perturbation theory coefficients
in higher orders which was estimated earlier for several QFT models. In particular, as
was first calculated by L.\,N.\,Lipatov, in the $g\varphi^4$ theory, the coefficients of
the $\beta$-function  $\beta(g)=\sum \beta_n g^n$ behave like $\beta_n \sim (-a)^n
n!n^{7/2}$, where $a=1/16\pi^2$. This implies that the corresponding Borel-transformed
series is convergent inside the circle of radius $1/a$. The next important step is the
analytical continuation from this circle to the entire integration domain which is done
with the help of a special conformal map. The resulting series in a new conformal
variable are to be cut at the highest known perturbation-theory coefficient.

Miraculously, these almost empirical recipes did really work. We calculated  the
four-loop contributions (the first four terms) for $\beta(g)$ and some related functions
of the scalar $g\varphi^4$ model in the framework of so-called $\varepsilon$-expansion
formalism, carried out the summation procedure described above, and then applied all this
to analysis of phase transitions of the second kind (Kazakov, Tarasov and Vladimirov, 1979).
This is reasonable because certain characteristics of condensed matter near critical
temperature, such as critical exponents, etc., are known to allow description in terms of
appropriate QFT models. The results were impressive: an agreement with experiment and
with other theoretical approaches was very good. In fact, this was the most precise
calculation of the critical exponents reached so far, which was cited in the Nobel
lecture by Kenneth Wilson.

This activity found its continuation after several years when a group of people from BLTP
and INR (Moscow) (K.\,G.\,Chetyrkin, S.\,G.\,Gorishny, S.\,A.\,Latin and F.\,V.\,Tkachov)
developed computer codes for multiloop calculations and, to check their efficiency,
performed five-loop calculations in the $g\phi^4$ theory. They also suggested a new
method for evaluating complicated integrals in addition to the infrared rearrangement
technique developed earlier in our laboratory. This is the so-called $R^*$-operation
which combines the conventional $R$-operation developed by Bogoliubov in 1950s for
eliminating ultraviolet divergences, with a similar infrared operation. The
$R^*$-operation proved to be an extremely powerful computational tool because it allows
to simplify the integrals by sending all external momenta to zero. The above-mentioned
subtleties are circumvented here by preventive subtraction of the spurious infrared
divergences. And though rigorous proof and full development of the $R^*$-operation method
took several years (K.\,G.\,Chetyrkin and V.\,A.\,Smirnov; F.\,V.\,Tkachov) , its first
applications were very exciting. The 5-loop calculation in $g\phi^4$ model was the top of
the art at that time (and still is), though some integrals were evaluated in the form of
infinite convergent series. It was natural then to look for a closed analytical answer in
the form of Euler gamma functions and further Riemann zeta functions.

It so happened that this task was partially fulfilled with the help of a new technique
called the ``method of uniqueness", first suggested by A.\,N.\,Vasiliev with
collaborators in St.\,Petersburg and then elaborated at BLTP (Kazakov, 1983, 1984). This
method uses conformal properties of massless theories together with the so-called
star-triangle relation (also called the uniqueness relation) for evaluating Feynman
diagrams by either transforming them to exactly solvable ones or expanding them into
power series in $\varepsilon$ with known coefficients. (Recently an elegant interpretation
of the uniqueness  relation in terms of noncommutative algebra was proposed by 
A.\,P.\,Isaev.) With the help of this method it became possible for the first time 
to complete the 5-loop calculation (Kazakov, 1985) of $\beta$-function 
in the $g\phi^4$ theory in explicit analytical form. Later on, this
result was used for the calculation of critical exponents by the method discussed above.
These achievements stimulated further activity in BLTP focused on analytical evaluation
of multiloop Feynman diagrams (Gorishny and Isaev, 1985\,; Kotikov, 1991)\,.

Multiloop results described above were obtained in scalar models, like $g\phi^4$.
Proceeding now to our calculations in gauge theories we firstly have to mention, as our
major achievement of that time, an evaluation  of the three-loop $\beta$-function in QCD
(Tarasov, Vladimirov and Zharkov, 1980). For technical reasons, this task is even more
difficult than to compute four loops in $g\varphi^4$. The final formula (with $n_f$ being
the number of flavors, i.e., sorts of quarks),
$$
\beta_{QCD}(g^2) = \frac{g^4}{(4\pi)^2}(-11 + \frac{2}{3} n_f)
 + \frac{g^6}{(4\pi)^4} (-102 + \frac{38}{3} n_f)
 + \frac{g^8}{(4\pi)^6} (-\frac{2857}{2} + \frac{5033}{18} n_f
  - \frac{325}{54} n_f^2)\,,
$$
obtained in early 1980, is actively exploited up to now in high-energy calculations based
on Quantum Chromodynamics. It appeared to be one of the most frequently cited results
produced in the Laboratory of theoretical physics. The next, fourth, term of
$\beta_{QCD}$ has been computed only in 1997 by S\,.A.\,Larin and J.\,Vermaseren.

The other example of application of the developed technique to QCD was the calculation of
the structure functions of the deep inelastic scattering process in three loops. It
required modification of the method of uniqueness as applied to integrals with arbitrary
powers of momenta (Kazakov and Kotikov, 1987). As a result the complete quartic
($\alpha^2_s$)  correction to the deep inelastic longitudinal structure function $F_L$ in
QCD earlier considered by other groups was calculated (Kazakov and Kotikov, 1990;
Kazakov, Kotikov, Parente, Sampayo and Sanchez-Guillen, 1990), which was the highest
precision at that time.

An additional problem, intimately related to any multiloop calculations, but fortunately
not so serious as the calculations themselves, is a possible renormalization-scheme
dependence of their results. To cope with ultraviolet divergences inevitably present in
the Feynman diagrams of quantum field theory, one makes use of (a certain variant, or
scheme, of) the renormalization procedure. Do final results of the corresponding
computations depend on the scheme we choose? Generally they don't, provided that all
orders of the perturbation expansion are taken into account. However, in practical
situations it seems to be unmanageable. So, the results of calculations performed at any
finite order of the perturbation theory do depend on the renormalization scheme. In the
above case of $\beta_{QCD}$, such dependence starts from the three-loop order.
Specifically, a numerical coefficient at $g^8$ in this formula has been found in the MS
(minimal subtraction) renormalization scheme. But another calculations, using some other
scheme, could produce different three- and higher-loop results. The situation looks quite
unsatisfactory, but actually it is not that bad. In several papers (our paper (Vladimirov,
1979) being one of the first of them) it has been shown that the whole problem of
scheme dependence can be substantially reduced by a clever choice of an expansion
variable in the final expressions for appropriate physical quantities, where the
$\beta$-function is only one of the ingredients. Despite the principal problem being thus
fixed, a certain activity in which we also took part (Kazakov and Shirkov, 1985;
Vladimirov, 1989) around the scheme dependence was noticeable during 1980s. It produced a
couple of new, mostly empirical, recipes which proved to be quite useful in treating
multiloop perturbative results.

Almost as a by-product of our three-loop QCD calculations, we were lucky to obtain
another result of great importance (maybe, rather theoretical than practical). It was
known for several years that two first terms in the $\beta$-function of a special
supersymmetric model, so-called $N=4$ SUSY Yang-Mills theory, surprisingly vanish.
Without very great effort, by adding a set of diagrams required by supersymmetry, we
succeeded in three-loop calculation (Avdeev, Tarasov and Vladimirov, 1980) of $\beta_{N=4}$
with the result being exactly zero! Of course, this still was not a proof that
$\beta$-function in this model vanishes to all orders, but it was a true incentive to
think so. And indeed, our result, obtained in its final form on 21 April 1980, has
triggered much activity in this field. The result itself (three-loop zero) has been
immediately verified by another computation. Moreover, a few years later, the all-order
proof of vanishing of $\beta_{N=4}$ was presented by several groups. Thus, our 1980 work
gave birth to studying the so-called finite quantum field models.

These models have a twofold interest. First of all, finite field theories, or theories
without ultraviolet divergences (the usual illness of any quantum field theory) are very
attractive by themselves. In a sense, they realize a dream of a self-consistent quantum
field theory where all quantities and parameters are directly meaningful, where we
do not encounter infinite self energies, infinite polarizations, etc. Remarkable
successes of the renormalization program in QFT helped us to get used to infinities
implicitly absorbed into redefinitions of fields, masses, coupling constants and
other quantities. But this was always considered as a weak point of a theory, if we
think of it as a prescription to get meaningful results, without any divergences.
Not criticizing this point of view, we talk now about a very restricted set
of supersymmetric field theories which, happily, are free from these problems.
These models give us an example of field theories with special properties, perhaps
selected by Nature. Mathematically, these models might allow exact solutions.
Thus, the already mentioned N=4 supersymmetric Yang-Mills theory is believed to be
an integrable model, the property found so far exclusively in two-dimensional world.

After those thrilling years of a quest for N=4 finiteness it was realized that N=2
supersymmetry was enough to get a finite model. Unfortunately these N=2 models are not
explicitly applicable to high energy physics since they contain the so-called mirror
particles creating severe problems for phenomenology. We were the first to propose how
one can construct models with the ordinary (or N=1) supersymmetry which are also finite
(Ermushev, Kazakov and Tarasov, 1987). Using the method developed earlier we performed
the three-loop calculations in N=1 supersymmetric gauge theories and demonstrated that
the realization of our algorithms really yields finite models. Later on we generalized
the method to the theories with softly broken supersymmetry (Avdeev, Kazakov and
Kondrashuk, 1998; Kazakov, 1998). These are not exactly the models that generalize the
Standard Model of fundamental interactions, but rather supersymmetric Grand Unified
theories with very special properties. Whether these models are chosen by Nature is the
matter of future experiments.

The second attractive aspect of finite theories is related to string theories. Here the
multiloop calculations also played their role in determining the consistent theory.
Formulation of the string theory on the world sheet leads to a two dimensional sigma
model with the world-sheet supersymmetry. To be reparametrization invariant, this theory
should have conformal invariance at the quantum level. This invariance is usually
violated by conformal anomalies which are proportional to the renormalization group
$\beta$-function. Hence, in a conformally invariant quantum theory the $\beta$-function
should vanish. As mentioned above, it is ordinarily evaluated within the perturbation
theory. The leading (one-loop) calculation yields the $\beta$-function proportional to
the Ricci tensor of the corresponding target space manifold. Thus, the requirement of
conformal symmetry, and hence of vanishing $\beta$-function, distinguishes the Ricci-flat
manifolds (those with zero Ricci tensor), usually called Calabi-Yau manifolds, as
possible compact components of multidimensional space-time. Of course,  this leading
order result is to be confirmed in higher orders. The first nontrivial contribution
appears only at the four-loop level and produces a certain correction to the
$\beta$-function (Grisaru, Milewski and Zanon). In this puzzling situation we
participated in the 5-loop calculation which used all the skills and techniques invented
earlier (Grisaru, Kazakov and Zanon, 1987). This was also a top of the art calculation
that demonstrated that higher orders do not break the general statement concerning
Ricci-flat manifolds.

When dealing with perturbative calculations in supersymmetric theories, we encounter a
bothersome problem of invariant regularization. In quantum field theory, some kind of
regularization procedure is required to give a sense to (generally divergent) Feynman
integrals. Actually, the regularization modifies original theory to a certain extent. In
particular, it can destroy, or rather deform, its symmetry properties. But very often it
is these symmetry features that made the theory to be worth dealing with! Of course, all
the miraculous cancellations of divergences in supersymmetric models are due to this very
supersymmetry. And we would be greatly disappointed if this symmetry do not persist at
the regularized (and renormalized) level. Unfortunately, it appears to be uneasy task to
adapt dimensional regularization (which is absolutely indispensable tool for multiloop
calculations) to supersymmetry. Moreover, there was a wide spread opinion that it is
quite impossible, and the most popular approach in this direction, so-called
``regularization by dimensional reduction", is contradictory. In 1981-82 we elaborated a
method (Avdeev, Chochia and Vladimirov, 1981; Avdeev and Vladimirov, 1983)
which enabled one to perform the
calculations in a manifestly supersymmetric (and consistent) manner through a certain
number of loops. However, at some higher order of perturbation theory (fourth or fifth,
as a rule) the supersymmetry of the results could be lost. We managed to roughly estimate
these potentially dangerous orders from below for the models of interest, and found, for
example, that our three-loop zero result was not under suspicion (Avdeev and Vladimirov,
1983). Furthermore, the above-mentioned general arguments in favor of all-order
finiteness of extended supersymmetric Yang-Mills theories seem not to be liable to this
trouble at all.

Looking back, we see the 30-years-long history of multiloop calculations, which included
the development of cunning computational techniques, the record calculations performed in
various models, as well as many successful applications to the quantum field theory,
particle physics and condensed matter phenomena. In 1970s-80s we were at the center of
this activity. It should be mentioned that the later development added a set of dedicated
computer packages which are able not only to draw all the relevant Feynman diagrams, but
also to perform automatically all the necessary Lorentz algebra, and even to evaluate
multiloop integrals. In fact, we were among the first who started this activity which now
has developed into a worldwide computational industry, extensively exploited in the high
energy (especially QCD) calculations. The other major contribution of a more recent time
period is related to the problem of evaluating Feynman diagrams with masses, where some
new methods appeared. The breakthrough was achieved with the help of a mass expansion,
which combines the power of asymptotic expansions with the $R^*$-operation. The very last
achievement here was the analytical evaluation of a new class of integrals, the so-called
boxes, which appear in many QCD problems and also in some SUSY models.

Today there exists a very active and ambitious computational community around the globe
(but mostly of Russian origin) which, as in the early days of QED, can perform unique
calculations (and really does), thus pushing the theoretical predictions to the limit.

\newpage

%\begin{thebibliography}{99}
\section*{References}
\begin{itemize}
\item[-] L.\,V.\,Avdeev, O.\,V.\,Tarasov and A.\,A.\,Vladimirov, \,
Vanishing of the three loop charge renormalization function in a
supersymmetric gauge theory, \,Phys.\,Lett. {\bf B96} (1980) 94.
\item[-] L.\,V.\,Avdeev, G.\,A.\,Chochia and A.\,A.\,Vladimirov, \,
On the scope of supersymmetric dimensional regularization, \,
Phys.\,Lett. {\bf B105} (1981) 272;
\item[-] L.\,V.\,Avdeev and A.\,A.\,Vladimirov, \,
Dimensional regularization and supersymmetry, \, Nucl.\,Phys. {\bf B219} (1983) 262.
\item[-] L.\,V.\,Avdeev, D.\,I.\,Kazakov and I.\,N.\,Kondrashuk, \,
Renormalizations in Softly Broken SUSY Gauge Theories, \,
Nucl.\,Phys. {\bf B510} (1998) 289.
\item[-] V.\,V.\,Belokurov, D.\,I.\,Kazakov, D.\,V.\,Shirkov, A.\,A.\,Slavnov
and A.\,A.\,Vladimirov, \,Ultraviolet behaviour of spontaneously broken gauge theories,
\,Phys.\,Lett.\,{\bf B47} (1973) 359.
\item[-] V.\,V.\,Belokurov, D.\,I.\,Kazakov, D.\,V.\,Shirkov, A.\,A.\,Slavnov
and A.\,A.\,Vladimirov, \,Ultraviolet asymptotics in the presence of nonabelian
gauge fields, \,Theor.\,Math.\,Phys. {\bf 19} (1974) 149.
\item[-] A.\,V.\,Ermushev, D.\,I.\,Kazakov and O.\,V.\,Tarasov, \,Finite $N\!=\!1$
supersymmetric grand unified theories, \,Nucl.\,Phys. {\bf B281} (1987) 72.
\item[-] S.\,G.\,Gorishny and A.\,P.\,Isaev, 
\,On an approach to the calculation of multiloop massless Feynman integrals, 
\,Theor.\,Math.\,Phys. {\bf 62} (1985) 232.
\item[-] M.\,T.\,Grisaru, D.\,I.\,Kazakov and D.\,Zanon,\, Five-loop divergences for
the $N\!=\!2$ supersymmetric non-linear sigma-model, \,Nucl.\,Phys. {\bf B287} (1987) 189.
\item[-] D.\,I.\,Kazakov, D.\,V.\,Shirkov and O.\,V.\,Tarasov, \,Analytical continuation
of perturbative results of the $g\phi^4$ model into the region $g\geq 1$, \,
Theor.\,Math.\,Phys. {\bf 38} (1979) 15.
\item[-] D.\,I.\,Kazakov, O.\,V.\,Tarasov and A.\,A.\,Vladimirov, \,On the calculation
of critical exponents by the methods of quantum field theory, \,JETP {\bf 77} (1979) 1035.
\item[-] D.\,I.\,Kazakov and D.\,V.\,Shirkov, \,Asymptotic series in quantum field theory
and their summation, \,Fortsch. der Phys. {\bf 28} (1980) 465.
\item[-] D.\,I.\,Kazakov, \,Calculation of Feynman diagrams by the method of uniqueness,
\, Theor.\,Math.\,Phys. {\bf 58} (1984) 357.
\item[-] D.\,I.\,Kazakov, \,The method of uniqueness, a new powerful technique for
multiloop calculations, \,Phys.\,Lett. {\bf B138} (1983) 406.
\item[-] D.\,I.\,Kazakov, \,Multiloop calculations: the  method of uniqueness and
functional equations, \,Theor.\,Math.\,Phys. {\bf 62} (1985) 127.
\item[-] D.\,I.\,Kazakov and D.\,V.\,Shirkov, \,The account of quark masses in
scheme-invariant perturbation theory in QCD, \,Sov.\,J.\,Nucl.\,Phys. {\bf 42} (1985) 768.
\item[-] D.\,I.\,Kazakov, \,Finite SUSY field theories and dimensional regularization, \,
Phys.\,Lett. {\bf B179} (1986) 352.
\item[-] D.\,I.\,Kazakov, \,Finite $N=1$ SUSY gauge field theories,
Mod.\,Phys.\,Lett. {\bf A2} (1987) 663.
\item[-] D.\,I.\,Kazakov and A.\,V.\,Kotikov, \,The method of uniqueness: multiloop
calculations in QCD, \,Theor.\,Math.\,Phys. {\bf 73} (1987) 348.
\item[-] D.\,I.\,Kazakov and A.\,V.\,Kotikov, \,Total $\alpha_s$-correction to the
deep-inelastic cross-section ratio $R=\sigma_L/\sigma_T$ in QCD, \,Nucl.\,Phys.
{\bf B307} (1988) 721, {\bf B345} (1990) 299 (E).
\item[-] D.\,I.\,Kazakov, A.\,V.\,Kotikov, G.\,Parente, O.\,A.\,Sampayo and
J.\,Sanchez Guillen, \,
Complete quartic ($\alpha^2_s$) correction to the deep inelastic longitudinal structure
function $F_L$ in QCD, \,Phys.\,Rev.\,Lett. {\bf 65} (1990) 1535, {\bf 65} (1990) 2921 (E).
\item[-] D.\,I.\,Kazakov, \,Finiteness of Soft Terms in Finite $N\!=1\!$ SUSY Gauge
Theories, \,Phys.\,Lett. {\bf B421} (1998) 211.
\item[-] A.\,V.\,Kotikov, \,
Differential equations method: new technique for massive Feynman diagrams calculation, \,
Phys.\,Lett. {\bf B254} (1991) 158.
\item[-] D.\,V.\,Shirkov and A.\,A.\,Vladimirov,
The renormalization group and ultraviolet asymptotics, \,
Sov.\,Phys.\,Usp. {\bf 22} (1979) 860.
\item[-] O.\,V.\,Tarasov, A.\,A.\,Vladimirov and A.\,Yu.\,Zharkov, \,
The Gell-Mann-Low function of QCD in the three loop approximation, \,
Phys.\,Lett. {\bf B93} (1980) 429.
\item[-] A.\,A.\,Vladimirov, \, Methods of multiloop calculations and the
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