Self-Consistent Renormalization as an Efficient Realization of Main Ideas of the Bogoliubov−Parasiuk R-operation
“...At the present time, the intimate connection between causality and the analytic continuation is revealed. So, it is not improbable to develop a subtraction procedure even in the most general case by the use of analytic continuation techniques.” (O.S. Parasiuk,, 1956 ). This possibility is reali...
Збережено в:
| Дата: | 2010 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Відділення фізики і астрономії НАН України
2010
|
| Назва видання: | Український фізичний журнал |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/56190 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Self-Consistent Renormalization as an Efficient Realization of Main Ideas of the Bogoliubov−Parasiuk R-operation / V.I. Kucheryavy // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 487-504. — Бібліогр.: 87 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-56190 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-561902014-02-14T03:10:30Z Self-Consistent Renormalization as an Efficient Realization of Main Ideas of the Bogoliubov−Parasiuk R-operation Kucheryavy, V.I. Поля та елементарні частинки “...At the present time, the intimate connection between causality and the analytic continuation is revealed. So, it is not improbable to develop a subtraction procedure even in the most general case by the use of analytic continuation techniques.” (O.S. Parasiuk,, 1956 ). This possibility is realized explicitly and efficiently in a body of our self-consistent renormalization (SCR). The self-consistency means that all formal relations between UV-divergent Feynman amplitudes are automatically retained as well as between their regular values obtained in the framework of the SCR. Self-consistent renormalization is efficiently applicable on equal grounds both to renormalizable and nonrenormalizable theories. The SCR furnishes new means for the constructive treatment of new subjects: i) UVdivergence problems associated with symmetries, Ward identities, and quantum anomalies; ii) new relations between finite bare and finite physical parameters of quantum field theories. The aim of this paper is to expose main ideas and properties of the SCR and to describe three mutually complementary algorithms of the SCR that are presented in the form maximally suited for practical applications. “...В зв’язку з тим, що в останнiй час знайдено тiсний зв’язок мiж причиновiстю та аналiтичнiстю, не виключена ймовiрнiсть побудови вiднiмальної операцiї навiть в самому загальному випадку методами аналiтичного продовження.” (О.С. Парасюк, 1956 ). Цю можливiсть реалiзовано явно та ефективно засобами нашої самоузгодженої ренормалiзацiї (СУР). Пiд самоузгодженiстю розумiють, що всi формальнi спiввiдношення мiж УФрозбiжними фейнмановими амплiтудами також автоматично зберiгаються мiж їхнiми регулярними значеннями, знайденими згiдно з процедурою СУР. Самоузгоджена ренормалiзацiя з однаковою ефективнiстю застосовна як до ренормовних, так i до неренормовних теорiй. СУР має ефективнi засоби для конструктивного розгляду нових задач: а) ренормалiзацiйних проблем, що пов’язанi з симетрiями, тотожностями Уорда та квантовими аномалiями; б) нових взаємозв’язкiв мiж скiнченними зародковими та скiнченними фiзичними параметрами квантово-польових теорiй. Наведено огляд головних iдей та властивостей СУР, а також чiтко описано три взаємодоповнювальнi алгоритми СУР, якi подано у виглядi, максимально пристосованому для практичних застосувань. 2010 Article Self-Consistent Renormalization as an Efficient Realization of Main Ideas of the Bogoliubov−Parasiuk R-operation / V.I. Kucheryavy // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 487-504. — Бібліогр.: 87 назв. — англ. 2071-0194 PACS 11.10.Gh http://dspace.nbuv.gov.ua/handle/123456789/56190 en Український фізичний журнал Відділення фізики і астрономії НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Поля та елементарні частинки Поля та елементарні частинки |
| spellingShingle |
Поля та елементарні частинки Поля та елементарні частинки Kucheryavy, V.I. Self-Consistent Renormalization as an Efficient Realization of Main Ideas of the Bogoliubov−Parasiuk R-operation Український фізичний журнал |
| description |
“...At the present time, the intimate connection between causality and the analytic continuation is revealed. So, it is not improbable to develop a subtraction procedure even in the most general case by the use of analytic continuation techniques.” (O.S. Parasiuk,, 1956 ).
This possibility is realized explicitly and efficiently in a body of our self-consistent renormalization (SCR). The self-consistency means that all formal relations between UV-divergent Feynman amplitudes are automatically retained as well as between their regular values obtained in the framework of the SCR. Self-consistent renormalization is efficiently applicable on equal grounds both to renormalizable and nonrenormalizable theories. The SCR furnishes new means for the constructive treatment of new subjects: i) UVdivergence problems associated with symmetries, Ward identities, and quantum anomalies; ii) new relations between finite bare and finite physical parameters of quantum field theories. The aim of this paper is to expose main ideas and properties of the SCR and to describe three mutually complementary algorithms of the SCR that are presented in the form maximally suited for practical applications. |
| format |
Article |
| author |
Kucheryavy, V.I. |
| author_facet |
Kucheryavy, V.I. |
| author_sort |
Kucheryavy, V.I. |
| title |
Self-Consistent Renormalization as an Efficient Realization of Main Ideas of the Bogoliubov−Parasiuk R-operation |
| title_short |
Self-Consistent Renormalization as an Efficient Realization of Main Ideas of the Bogoliubov−Parasiuk R-operation |
| title_full |
Self-Consistent Renormalization as an Efficient Realization of Main Ideas of the Bogoliubov−Parasiuk R-operation |
| title_fullStr |
Self-Consistent Renormalization as an Efficient Realization of Main Ideas of the Bogoliubov−Parasiuk R-operation |
| title_full_unstemmed |
Self-Consistent Renormalization as an Efficient Realization of Main Ideas of the Bogoliubov−Parasiuk R-operation |
| title_sort |
self-consistent renormalization as an efficient realization of main ideas of the bogoliubov−parasiuk r-operation |
| publisher |
Відділення фізики і астрономії НАН України |
| publishDate |
2010 |
| topic_facet |
Поля та елементарні частинки |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/56190 |
| citation_txt |
Self-Consistent Renormalization as an Efficient Realization of Main Ideas of the Bogoliubov−Parasiuk R-operation / V.I. Kucheryavy // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 487-504. — Бібліогр.: 87 назв. — англ. |
| series |
Український фізичний журнал |
| work_keys_str_mv |
AT kucheryavyvi selfconsistentrenormalizationasanefficientrealizationofmainideasofthebogoliubovparasiukroperation |
| first_indexed |
2025-07-05T07:25:21Z |
| last_indexed |
2025-07-05T07:25:21Z |
| _version_ |
1836790916930600960 |
| fulltext |
SELF-CONSISTENT RENORMALIZATION AS AN EFFICIENT REALIZATION OF MAIN IDEAS
SELF-CONSISTENT RENORMALIZATION
AS AN EFFICIENT REALIZATION OF MAIN IDEAS
OF THE BOGOLIUBOV–PARASIUK R-OPERATION
V.I. KUCHERYAVY
Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
(14b, Metrolohichna Str., Kyiv 03680, Ukraine; e-mail: vnkucher@ bitp. kiev. ua )
PACS 11.10.Gh
c©2010
Il libro della natura é scritto in lingua matematica.
Galileo Galilei, [ Il Saggiatore, 1623 ].
“...At the present time, the intimate connection
between causality and the analytic continuation
is revealed. So, it is not improbable to develop a
subtraction procedure even in the most general case
by the use of analytic continuation techniques.”
O.S. Parasiuk, [ [7], p.566, the last paragraph, 1956 ].
This possibility is realized explicitly and efficiently in a body of our
self-consistent renormalization (SCR). The self-consistency means
that all formal relations between UV-divergent Feynman ampli-
tudes are automatically retained as well as between their regular
values obtained in the framework of the SCR. Self-consistent renor-
malization is efficiently applicable on equal grounds both to renor-
malizable and nonrenormalizable theories. The SCR furnishes new
means for the constructive treatment of new subjects: i) UV-
divergence problems associated with symmetries, Ward identities,
and quantum anomalies; ii) new relations between finite bare and
finite physical parameters of quantum field theories. The aim of
this paper is to expose main ideas and properties of the SCR and
to describe three mutually complementary algorithms of the SCR
that are presented in the form maximally suited for practical ap-
plications.
1. Introduction
The keystone idea of a purely mathematical genesis of
the ultraviolet (UV) divergencies of Feynman amplitudes
(FAs) in quantum field theories is at the heart of the
Bogoliubov–Parasiuk R-operation [1–7]. Using this idea
along with related considerations of mathematicians of
the 19th and 20th centuries,1 the author has developed
1 It is appropriate to pointed out here that the first regularization
recipe to subtract infinities for turning a divergent integral into
a convergent one had been used in Cauchy’s “extraordinary inte-
gral” [9–11] and in d’Adhémar’s [12,13] and Hadamard’s [14–17]
“finite part of a divergent integral”. These recipes are similar
but not identical. But, in both cases, it was extended the va-
lidity of the usual rules of change of a variable, integration by
an universal, high-efficient, and self-consistent renormal-
ization (SCR) technique which is applicable for any di-
mension n = 2rn + δn, δn = 0, 1, rn ∈ {0 ∪ N+} of a
space-time that is endowed by a pseudo-Euclidean (p, q)
parts, and differentiation with respect to the upper limit of in-
tegration to these new objects. The Cauchy’s “extraordinary
integral” has been used for an efficient analytic continuation
of the Γ(z)-function to some noninteger real values Re z < 0
firstly by Cauchy himself [10] in 1827, and then in the strips
(−n − 1 < Re z < −n) by Saalschütz [18, 19] in 1887-1888.
The term “finite part of a divergent integral” was introduced
by d’Adhémar in his thesis presented at the Sorbonne Univer-
sity in December 1903 and defended in April 1904 (see [ [20],
p.477 ]). Referring to Hadamard’s article [14], d’Adhémar [ [13],
p.371 ] writes “...Independently of each other, we understood the
role of these finite parts...”. In d’Adhémar’s thesis and articles,
this notion was applied to the construction of solutions of the
equation for cylindrical waves [12, 13], whereas Hadamard used
finite parts for the solution of the Cauchy problem for second-
order equations with variable coefficients [14–16] and an arbi-
trary number of independent variables [17]. On the applications
of d’Adhemar’s and Hadamard’s “finite part of a divergent inte-
gral” in more details, see Hadamard’s book [21]. 40 years later
on, when analyzing the connections between the intuitive and
logical ways of mathematical inventions, Hadamard [22] wrote:
“...All mathematicians must consider themselves as logics. For
example, I have been asked by what kind of guessing I thought
of the device of the “finite part of a divergent integral”, which I
have used for the integration of partial differential equations.
Certainly, considering in itself, it looks typically like “think-
ing aside”. But, in fact, for a long while my mind refused to
conceive that idea until positively compelled to, I was led to
it step by step as the mathematical reader will easily verify
if he takes the trouble to consult my researches on the sub-
ject, especially my Recherches sur les solution fondamentales et
l’int’egration des ’equations lin’eaires aux d’eriv’ees partielles,
2nd Memoir, especially p.121 and so on (Annales Scientifiques
de l’Ecole Normale Superieure, Vol.XXII, 1905) [16]. I could
not avoid it any more than the prisoner in Poe’s tale The Pit and
Pendulum could avoid the hole at the center of his cell...”, see
[ [22], p.110, and p.104 or p.86 in two identical Russian trans-
lations from French edition of 1959 ]. About further develop-
ments see M. Riesz [23,24], F. Bureau [25], R. Courant [26], and
S.G. Samko, A.A. Kilbas, and O.I. Marichev [27].
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 487
V.I. KUCHERYAVY
metric gµν , where p+ q = n, and for an arbitrary topol-
ogy of Feynman graphs.
Algorithmically, the SCR is an efficient realization of
the Bogoliubov–Parasiuk R-operation as some special
analytical extension of the UV-divergent FAs in two pa-
rameters ωG and νG by means of recurrence, compati-
bility, and differential relations fixing a renormalization
arbitrariness of the R-operation in some universal way
based on the mathematical properties of FAs only. The
parameters ωG and νG depend on a space-time dimen-
sion n, a graph-topological invariant |C| determining a
number of independent circuits of a graph G, and two
FAs characteristics λL and dG. The numbers λL and dG
determine the maximal degree of polynomials of the de-
nominator, dden = 2λL, and the numerator, dnum = dG,
respectively in the integrand. As a result, the SCR is
efficiently applicable on equal grounds both to renor-
malizable and nonrenormalizable theories, which is very
important for quantum gravity.
The self-consistency means that all formal relations
between UV-divergent FAs are automatically retained,
as well as between their regular values obtained in the
framework of the SCR. The SCR furnishes new means
for the constructive treatment of new subjects: i) UV-
divergence problems associated with symmetries, Ward
identities, reduction identities, and quantum anomalies;
ii) new relations between finite bare and finite physical
parameters of quantum field theories.
The aim of this article is to expose main ideas and
properties of the SCR (see Sections 2 and 3) and to de-
scribe three mutually complementary algorithms of the
SCR (see Sections 3–5) which are presented in the form
maximally suited for practical applications.
2. The Bases and Possibilities of the SCR
2.1. The SCR is an efficient realization of the
Bogoliubov-Parasiuk R-operation [1–8] which is supple-
mented with recurrence, compatibility, and differential
relations fixing a renormalization arbitrariness of the R-
operation in some universal way based on mathemati-
cal properties of Feynman amplitudes (FAs) only. In its
turn, the Bogoliubov-Parasiuk approach is rested on an
idea that the nature of UV-divergences is purely math-
ematical and, per se, the R-operation is a constructive
form of the Hahn–Banach theorem on extensions of lin-
ear functionals (see, for e.g., [28–30]).
2.2. Elaborating this idea, the author [31–45] has
obtained the high-efficiency realization of this renormal-
ization scheme (renormscheme). In this realization:
• Properties of special functions of the hypergeometric
type are essentially used.2
• Combinatorics is simplified considerably. Our in-
vestigations confirm the very important assertion by
D.A. Slavnov [52] that the combinatorics of the R-
operation is overcomplicated considerably and can be
simplified essentially.
• Renormalization arbitrariness of the R-operation is
fixed in such a way that the basic functions (Rν0F)sj ≡
(Rν0F)sj(ω;Mε, A) of renormalized FAs obey the same
recurrence relations as the basic functions Fsj ≡
Fsj(ω;Mε, A) of convergent or dimensionally regularized
FAs:
Mε Fs−2,j−1 −AFs,j−1 + (ω + j)Fsj = 0,
Mε (Rν0F)s−2,j−1 −A (Rν0F)s,j−1+
+ (ω + j) (Rν0F)sj = 0. (2.1)
The explicit form of Fsj and (Rν0F)sj are given below
by Eqs. (3.30)–(3.31). On the self-consistent version
of the Clifford aspect of the dimensional regularization
which efficiently overcomes the known difficulties con-
nected with n-dimensional generalization of the Dirac
γ5 matrix, see [39, 53, 54].
• Compatibility relations of the first kind:
(Rν0F)sj = Fsj , if νsj := [(ν − s)/2] + j ≤ −1,
(Rν+1
0 F)s+1,j = (Rν0F)sj , (2.2)
and the compatibility relations of the second kind:
Fs−2,j−1(ω;Mε, A) = Fs,j−1(ω;Mε, A) =
= Fsj(ω − 1;Mε, A),
2 The connection of particular FAs with the hypergeometric func-
tions are well known. See, for example, the investigations of
analytic properties of convergent scalar FAs by using of alge-
braic topology methods [46–48], or calculations of some classes
of FAs for needs of phenomenological physics, by using the dif-
ferential equation method [49–51]. But, in our case, this connec-
tion is established for general divergent FAs in any space-time
dimension n and the (p, q) pseudo-Euclidean metric, p + q = n.
Apart from, this connection suggests some simple method of fix-
ing a renormalization arbitrariness of the Bogoliubov–Parasiuk
R-operation in some universal way based on the mathematical
properties of FAs only. As a result, we obtain the self-consistent
renormalization with new valuable properties and possibilities.
488 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
SELF-CONSISTENT RENORMALIZATION AS AN EFFICIENT REALIZATION OF MAIN IDEAS
(Rν0F)s−2,j−1(ω;Mε, A) = (Rν0F)sj(ω − 1;Mε, A),
(Rν0F)s,j−1(ω;Mε, A) = (Rν−2
0 F)sj(ω − 1;Mε, A), (2.3)
are satisfied automatically. From the first of Eqs. (2.2),
it follows that the formulae for regular values obtained
in the framework of the SCR describe uniformly both di-
vergent and convergent FAs.
• Differential relations for Fsj and (Rν0F)sj with respect
to mass-damping variables µl := (m2
l − iεl), l ∈ L,
∂m
∂µl1 · · · ∂µlm
[
Fsj(ω)
(Rν0F)sj(ω)
]
=
= (−1)mαl1 · · ·αlm
[
Fsj(ω −m)
(Rν0F)sj(ω −m)
]
(2.4)
are the same, and the differential relations for ones with
respect to external momenta ke, e ∈ E ,
∂ σ1
e1 · · · ∂
σm
em
[
Fsj(ω)
(Rν0F)sj(ω)
]
= 2m
[m/2]∑
κ=0
Aσ1···σm
e1 ··· em(κ)×
×
[
Fsj(ω −m+ κ)
(Rν−2m+2κ
0 F)sj(ω −m+ κ)
]
(2.5)
are almost the same. Here, ∂ σiei ≡ ∂/∂(kei)σi , and
Aσ1···σm
e1 ··· em(κ) ≡ Aσ1···σm
e1 ··· em(κ|α, k) are special homo-
geneous polynomials of degree m − 2κ in Aσiei ≡
Aσiei(α, k) :=
∑
e∈E Aeie(α)kσie and of degree κ in
( σiσjei ej ) := Aeiej (α)gσiσj , where Aee′(α) are matrix ele-
ments of the quadratic Kirchhoff form in external mo-
menta ke, e ∈ E . The polynomials Aσ1···σm
e1 ··· em(κ|α, k)
have an algebraic structure of quantities generated by
the Wick formula, which represents a T -product ofm bo-
son fields in terms of some set of N -products of m− 2κ
boson fields with κ primitive contractions. Here, the
quantities Aσiei and ( σiσjei ej ) play the role of boson fields
and their contractions, respectively.
• It is essential that Fsj and (Rν0F)sj as functions of
two variables Mε and A are the homogeneous functions
of the same degree ω+ j. From this, it follows that they
are solutions to the same partial differential equations,
namely to the Euler equation for the homogeneous func-
tions[
Mε∂Mε +A∂A − (ω + j)
] [ Fsj(ω)
(Rν0F)sj(ω)
]
= 0, (2.6)
and to some family of second-order equations emerging
from Eq.(2.6), for example,[
Mε∂
2
MεMε
± (Mε ±A)∂2
MεA ±A∂
2
AA−
−(ω + j − 1)(∂Mε
± ∂A)
] [ Fsj(ω)
(Rν0F)sj(ω)
]
= 0, (2.7)
that can be again represented as the Euler equation[
Mε∂Mε
+A∂A − (ω + j − 1)
]
×
×
[
(∂Mε
± ∂A)Fsj(ω)
(∂Mε ± ∂A) (Rν0F)sj(ω)
]
= 0. (2.8)
So, an important role of the quantities (∂Mε±∂A)Fsj(ω)
and (∂Mε ± ∂A)(Rν0F)sj(ω) is revealed in our problem.
After repeating this procedure N + 1 times, one obtains[
Mε∂Mε
+A∂A − (ω + j −N − 1)
]
×
×
[
(∂Mε ± ∂A)FN±sj (ω −N)
(∂Mε
± ∂A) (Rν0F)N±sj (ω −N)
]
= 0, (2.9)
where we define FN±sj (ω−N) := (∂Mε±∂A)NFsj(ω) and
(Rν0F)N±sj (ω−N) := (∂Mε
± ∂A)N (Rν0F)sj(ω). If N such
that (ω −N + j) ≤ − 1 then both (∂Mε
+ ∂A)FN±sj (ω −
N) = 0 and (∂Mε
+ ∂A)(Rν0F)N±sj (ω − N) = 0. As a
result, Eq. (2.9) with the plus sign is degenerated into
the identical zero, and the equation with the minus sign
is reduced to the Euler–Poisson–Darboux equation[
∂2
∂Mε∂A
+
(ω + j −N − 1)/2
Mε −A
( ∂
∂Mε
− ∂
∂A
)]
×
×
[
FN±sj (ω −N)
(Rν0F)N±sj (ω −N)
]
= 0. (2.10)
The consistency of solutions to Eqs. (2.9)–(2.10) for dif-
ferent preassigned asymptotics of (Rν0F)sj at the vicinity
of A = 0 leads to the relations
∂MεFsj(ω) = −Fsj(ω − 1),
∂AFsj(ω) = Fsj(ω − 1),
∂Mε
(Rν0F)sj(ω) = −(Rν0F)sj(ω − 1),
∂A(Rν0F)sj(ω) = (Rν−2
0 F)sj(ω − 1) (2.11)
which are also followed from the explicit form of the
basic functions Fsj and (Rν0F)sj , see Eqs. (3.30)–(3.31)
below.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 489
V.I. KUCHERYAVY
2.3. Relations (2.1)–(2.11) manifest the mutual con-
sistency of asymptotic properties of different terms of
FAs with respect to external momenta and masses. It is
precisely these recurrence, compatibility, and differential
relations that are of great importance for investigating
the problems of symmetries and anomalies and for turn-
ing the developed renormscheme into a self-consistent
one.
In addition, there exist some obvious identities of the
generic nature which are called as the reduction iden-
tities (RIs) [40, 41], which leads in another way to the
recurrence relations (2.1). The simple idea of cancelling
the equal factors in factorized polynomials in a numera-
tor and a denominator of integrands is used in RIs. The
RIs also are of great importance for applications as an
origin of new nontrivial identities. Some of them have
been used essentially in our investigations [39–41, 43–
45, 55–58].
2.4. Equations (2.1)–(2.11) and the explicit form of
the basic functions Fsj , (Rν0F)sj (see Eqs. (3.30)–(3.33)
and (3.36)–(3.40)) imply the following important prop-
erties of the SCR:
Algorithmic universality. The SCR is a special an-
alytic continuation of any FA firstly given by an UV-
divergent integral. In so doing, the divergence indices ν
of FAs may be as large as one needs. Hereafter, this con-
tinuation will be named as the regular (i.e., finite) value
of a FA. As a result, the regular values of FAs respect
certain recurrence, compatibility, and differential prop-
erties of an universal character and have already been re-
alized efficiently as convergent integrals. Therefore, the
calculations of FAs corresponding to renormalizable and
nonrenormalizable theories do not differ from each other
in the framework of this renormscheme. Actually, the
problem is reduced to calculations of the characteristic
numbers, ω, νsj , and λsj determining the basic functions
(Rν0F)sj .
Separation of problems. The SCR clearly and effi-
ciently separate the problem of evaluating the regular
values of UV-divergent quantities of quantum field theo-
ries from that of the relations between bare and physical
parameters of these theories, i.e., the SCR realizes, in
practice, this very important potential possibility of the
Bogoliubov–Parasiuk R-operation.
Conservation of relations. Any formal relation be-
tween UV-divergent quantities will be retained also be-
tween regular values of those if the regular values of all
quantities involved in this relation are calculated by the
same renormalization index ν (the maximum one since,
otherwise, we cannot guarantee the finiteness of all terms
in the relation). So, the SCR is automatically consistent
with the correspondence principle. As a result, the reg-
ular values obtained in the framework of the SCR do
satisfy the vector and axial-vector canonical Ward iden-
tities (CWIs) simultaneously.
Extraction of anomalies (quantum corrections).
In the SCR, owing to the analytic continuation tech-
nique, quantum anomalies (i.e., quantum corrections
(QCs) more exactly) are automatically accounted for in
quantities satisfying the CWIs. More specifically, the
quantum anomalies (i.e., QCs) reveal themselves either
as the oversubtraction effect for a non-chiral case and for
the chiral limit case (in these cases, the Schwinger terms
contributions (STCs) of current commutators are zero)
or as the nonzero STCs for the chiral case. If necessary,
the explicit form of quantum anomalies (i.e., QCs) can
be easily extracted as a difference between two regular
values of the same UV-divergent quantity calculated for
proper and improper divergence indices.
2.5. Algorithmically, the SCR is a union of three effi-
cient algorithms of finding:
i) the convergent α-parametric integral representations
of renormalized FAs with a compact domain of integra-
tion of the simplex type and with the self-consistent basic
functions (Rν0F)sj , s = 0, . . . , dG, j = 0, . . . , [s/2];
ii) the homogeneous k-polynomials P G
sj (m,α, k), j =
0, 1, . . . , [s/2], of degree (s − 2j) in external momenta
ke, e ∈ E , being as α-parametric images of homogeneous
p-polynomials P G
s (m, p), s = 0, . . . , dG, of degree s in
internal momenta pl, l ∈ L;
iii) the α-parametric functions Δ(α), A(α, k), Yl(α, k),
Xll′(α), l, l′ ∈ L.
3. Parametric Integral Representations and
Basic Functions of FAs in the SCR
3.1. From the mathematical point of view, any Feynman
amplitude associated with an oriented graph G,
G := 〈V,L ∪ E | eil = 0, ±1, vi ∈ V, l ∈ L ∪ E〉,
in which V is a set of vertices; L is a set of internal lines;
E is a set of external lines; and eil is an incidence matrix
(i.e., a vertex-line incidence matrix) such that eil = 0 if
the line l ∈ L ∪ E is nonincident to the vertex vi ∈ V;
eil = 1 if the line l ∈ L ∪ E is outgoing from the vertex
vi ∈ V; eil = −1 if the line l ∈ L ∪ E is incoming to the
vertex vi ∈ V, can be always represented by the integral
IG(m, k)ε := cG
∞∫
−∞
(dnp)LδG(p, k)
PG(m, p)
QG(m, p)ε
,
490 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
SELF-CONSISTENT RENORMALIZATION AS AN EFFICIENT REALIZATION OF MAIN IDEAS
(dnp)L := dnp1 · · · dnp|L|, dnpl :=
∏n
σ=1dp
σ
l ,
l ∈ L, m := (m1, . . . ,m|L|), (3.1)
p := (p1, . . . , p|L|), k := (k1, . . . , k|E|).
Here, PG(m, p) and QG(m, p) are polynomials in the
numerator and the denominator,
PG(m, p) :=
∏
vi∈V Pi(m, p)
∏
l∈L Pl(m, p) =
=
∑dG
s=0 P
G
s (m, p), (3.2)
QG(m, p)ε :=
∏
l∈L (µlε − p2
l )
λl ,
µlε := m2
l − iεl, ml ≥ 0, εl > 0, λl ∈ N+, ∀l ∈ L,
δG(p, k) is a product of vertex δ-functions
δG(p, k) :=
∏
vi∈V
δi(p, k),
δi(p, k) := δ
(∑
l∈L
eilpl +
∑
e∈E
eieke
)
; (3.3)
|A| is a number of elements of some finite set A; N+ is
the set of positive integers; PGs (m, p), s = 0, . . . , dG, are
s-degree homogeneous polynomials in internal momenta
pl, l ∈ L; Pi(m, p), and Pl(m, p) are multiplicative gen-
erating polynomials of the numerator PG(m, p) that cor-
respond to the vertex vi-contribution Vi(m, p, k) and to
the internal line l-contribution Δl(m, p)ε, respectively:
Vi(m, p, k) := Pi(m, p)δi(p, k),
degpPi(m, p) =:di ≥ 0, ∀vi ∈ V,
Δl(m, p)ε :=
Pl(m, p)
(µlε − p2
l )λl
, (3.4)
degpPl(m, p) =:dl ≥ 0, ∀l ∈ L.
The non-degenerate metric form
diag gµν := ( 1, . . . , 1︸ ︷︷ ︸
p
,−1, . . . ,−1︸ ︷︷ ︸
q
), (3.5)
p+ q = n = 2rn + δn, δn = 0, 1, rn ∈ {0 ∪ N+},
is used for each n-dimensional pl-integration in Eq. (3.1).
3.2. Two characteristics
νG := 2ωG + dG, ωG := (n/2)|C| − λL,
|C| = |L| − |V|+ 1, λL :=
∑
l∈L λl,
dG := dV + dL =
∑
vi∈V di +
∑
l∈L dl, (3.6)
of integral (3.1) are especially important. Here, |C| is the
number of independent circuits of the graph G. There
exist analogous characteristics for all one-particle irre-
ducible (1PI) subgraphs G ⊂ G. If νG ≥ 0 or νG ≥ 0
for some 1PI G ⊂ G, the integral is UV-divergent and a
renormalization is needed [28, 30].
While Eqs. (3.1)–(3.3) are identical to the well-known
representation in terms of vertex-line contributions,
δG(p, k)
PG(m, p)
QG(m, p)ε
=
∏
vi∈V
Vi(m, p, k)
∏
l∈L
Δl(m, p)ε,
they are more suited for practical calculations. The uni-
versal decomposition of PG(m, p) in terms of s-degree
homogeneous p-polynomials PGs (m, p) is very useful.
3.3. We use of the Fock–Schwinger exponential α-
representation (see, for example, [59–61]) along with the
Hepp regularization [30] that introduces parameters rl >
0 in the vicinity of αl = 0, ∀l ∈ L,
1
(µlε − p2
l )λl
= lim
rl→0
∞∫
rl
dαl α
λl−1
l iλl
Γ(λl)
e−iαl(µlε−p
2
l ),
pτl = (−i∂/∂qlτ )ei(pl·ql)|ql=0, (3.7)
(pl · ql) := plτqlσg
τσ, 0 < rl ≤ αl ≤ ∞, ∀l ∈ L.
Then the ratio of polynomials PG(m, p)/QG(m, p)ε in
Eqs. (3.1)–(3.2) can be represented in the form
PG(m, p)
QG(m, p)ε
= lim
rl→0
∀l∈L
{ ∫
R
|L|
+ (r )
dvG(α) iλL×
×
dG∑
s=0
PGs (m,−i∂/∂qL) e−iMε + iW
qL
pL
}∣∣∣∣ql=0
∀l∈L
,
W qL
pL := (pTL · αLLpL) + (pTL · qL) = (3.8)
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 491
V.I. KUCHERYAVY
=
∑
l∈L
αlp
2
l +
∑
l∈L
(pl · ql), [αLL]ll′ := αl δll′ .
In Eq. (3.8), pL and qL are (|L|×n)-dimensional actual
and auxiliary internal momenta column-vectors associ-
ated with the set of internal lines, L, of a graph G; T is
the transpose sign, so pTL is the row-vector; αLL is the
|L|-dimensional diagonal matrix of α-parameters; and
λL is defined in Eq.(3.6). Here, the integration mea-
sure dvG(α), the integration region R
|L|
+ (r ), and the α-
parametric function Mε ≡ M(m,α)ε which is the linear
form in the square of internal masses with iε-damping
are defined as
dvG(α) :=
∏
l∈L
(
dαl α
λl−1
l
Γ(λl)
)
,
R
|L|
+ (r ) := {αl| 0 < rl ≤ αl ≤ ∞, ∀l ∈ L, },
Mε :=
∑
l∈L
αlµlε, µlε := (m2
l − iεl). (3.9)
Now, substituting Eq. (3.8) in Eq. (3.1) and interchang-
ing the order of integration in pl and αl, ∀l ∈ L, we ob-
tain the very useful representation of the regularized-by-
Hepp integral IG(m, k)rε . Its integrand is the (|L| × n)-
dimensional pseudo-Euclidean Gaussian-like expression
but in the mutually dependent variables pl,∀l ∈ L,
IG(m, k)rε := cG
∫
R
|L|
+ (r )
dvG(α)×
×
dG∑
s=0
PGs (m,−i∂/∂qL)
∞∫
−∞
(dnp)LδG(pL, kE) iλL×
×e−iMε + iW
qL
pL
∣∣∣∣ql=0
∀l∈L
. (3.10)
The set of internal lines, L, can be always decom-
posed (as a rule, in more than one way) into two mu-
tually disjoint subsets, L = M ∪ N and M ∩ N = ∅
which determine some skeleton tree, i.e., 1-tree subgraph
G(V,M∪E), with |M| = |V|−1, and the corresponding
co-tree subgraph G(V,N ∪E), with |N | = |L|−|V|+1 =
|C| of the graph G. Supports of all δi(pL, kE)-functions,
∀vi ∈ V, (see Eq. (3.3)) are defined by Eqs. (3.11)
and are equivalent to the matrix relations given in Eqs.
(3.12) and Sec. 5,∑
l∈L eilpl +
∑
e∈E eieke = 0, ∀vi ∈ V, (3.11)
e{V/j}MpM + e{V/j}N pN + e{V/j}EkE = 0{V/j},
ejMpM + ejN pN + ejEkE = 0, vj – the basis vertex,
pL = eLN pN + eLE(j)kE ,
pM = eMN pN + eME(j)kE . (3.12)
Thus, the (|M| × n)-dimensional integration by means
of δi(pL, kE)-functions, ∀vi ∈ V/j (this is equivalent to
make use of Eqs. (3.12)), gives rise to the intermediate
α-parametric representation
IG(m, k)rε := δG(kE)cG
∫
R
|L|
+ (r )
dvG(α)×
×
dG∑
s=0
PGs (m,−i∂/∂qL)
∞∫
−∞
(dnp)N iλL×
×e−iMε + iW
qL
N ,E
∣∣∣∣ql=0
∀l∈L
, (3.13)
W qL
N ,E := (pTN · CNN (α)pN ) + 2(fTN · pN )+
+(kTE · EEE(j|α)kE) + (qTL · eLE(j)kE),
fN := ΠT
EN (j|α)kE + 1
2e
T
LN qL,
δG(kE) := δ
(∑
e∈E
e(v∗)eke
)
.
The explicit forms and some properties of the matri-
ces eLN , eLE(j), CNN (α), EEE(j|α), and ΠEN (j|α) are
given in Eqs. (5.1)-(5.4).
The change of the variables pN by means of a nonde-
generate linear transformation such that
pN = BNN (α)p̃N −BNN (α)B T
NN (α)fN ,
492 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
SELF-CONSISTENT RENORMALIZATION AS AN EFFICIENT REALIZATION OF MAIN IDEAS
B T
NN (α)CNN (α)BNN (α) = 1NN ,
BNN (α)B T
NN (α) = C−1
NN (α),
detBNN (α) = [detCNN (α)]−1/2 =:Δ(α)−1/2,
(dnp )N = (dnp̃ )N |detBNN (α)|n |det g| |N |
= (dnp̃ )N /Δ(α)n/2, det g = (−1)q, (3.14)
reduces Eqs. (3.13)–(3.14) to the form
IG(m, k)rε := δG(kE)cG
∫
R
|L|
+ (r )
dvG(α)
Δn/2
×
×
dG∑
s=0
PGs (m,−i∂/∂qL)
∞∫
−∞
(dnp̃ )N ei(p̃
T
N ·p̃N ) iλL×
×e−iMε + iW̃
qL
E
∣∣∣∣ql=0
∀l∈L
, (3.15)
W̃ qL
E := −(fTN · C−1
NN (α)fN )+
+(kTE · EEE(j|α)kE) + (qTL · eLE(j)kE) =
= (kTE ·AEE(j|α)kE)+
+(qTL · YLE(j|α)kE)− 1
4 (qTL ·XLL(α)qL).
With regard for the formula
∞∫
−∞
dte±it
2
= π1/2e±iπ/4,
which is followed from [62], Ch. 1.5., Eqs. (31) and (32),
we find
∞∫
−∞
dnp̃le
ip̃2l = πn/2ei(p−q)π/4 = πn/2i(p−n/2),
∞∫
−∞
(dnp̃ )N ei(p̃
T
N ·p̃N )iλL = (πn/2ip)|N |i−ω. (3.16)
So, all the integrations over n-dimensional pseudo-
Euclidean momenta in the (p, q)-metric are performed.
Thus, any FA (3.1) leads to the α-parametric represen-
tation in the fully exponential form,
IG(m, k)rε := (2π)nδG(kE)aG
∫
R
|L|
+ (r )
dvG(α)
Δn/2
×
×
dG∑
s=0
PGs (m,−i∂/∂qL) e−iMε + iW̃
qL
E
∣∣∣∣ql=0
∀l∈L
, (3.17)
W̃ qL
E := (kTE ·AEE(j|α)kE)+
+(qTL · YLE(j|α)kE)− 1
4 (qTL ·XLL(α)qL),
aG := cG(πn/2ip)|N |(2π)−ni−ω, |N | = |C|,
where the n-dimensional auxiliary momenta, ql, l ∈ L,
are still used. The explicit form and important prop-
erties of matrices AEE(j|α), YLE(j|α), and XLL(α) are
given in Eqs. (5.5)–(5.13). Some properties of them are
new.
Next, the following two operations must be car-
ried out: i) to differentiate the exponential func-
tion exp{i(qTL · YLE(j|α)kE) − i/4 (qTL · XLL(α)qL)} by
means of the s-homogeneous differential polynomials
PGs (m,−i∂/∂qL) in −i∂/∂qlσ, l ∈ L, σ ∈ {1, . . . , n},
1 ≤ s ≤ dG; ii) to put qlσ = 0,∀l ∈ L, ∀σ ∈ {1, . . . , n},
and ∀s ∈ {0, 1, . . . , dG}. Finally, we obtain the im-
portant α-parametric representation for the general FA
(3.1),
IG(m, k)rε := (2π)nδG(kE)bG
∫
R
|L|
+ (r )
dvG(α)×
× 1
Δn/2
dG∑
s=0
[s/2]∑
j=0
PGsj(m,α, k)i−ω−je−iMε+iA, (3.18)
A ≡ A(α, k) := (kTE ·AEE(j|α)kE),
bG := cG(πn/2ip)|C|(2π)−n, |C| = |N |,
PGsj(m, ρα, τk) = ρ−jτs−2j PGsj(m,α, k).
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 493
V.I. KUCHERYAVY
Here, [s/2] means the largest integer ≤ s/2, i.e., the in-
teger part of s/2; the quadratic Kirchhoff form A(α, k)
in external momenta ke, e ∈ E , and the Kirchhoff deter-
minant Δ(α) := detCNN (α) are defined by the topolog-
ical structure of a graph G and are homogeneous func-
tions of the first and |C|th degrees in α, respectively,
see Sec. 5 for more details. The quantities PGsj(m,α, k)
are homogeneous k-polynomials in external momenta ke,
e ∈ E , of the degree s − 2j, j = 0, 1, . . . , [s/2]. They
are α-parametric images of homogeneous polynomials
PGs (m, p). Each monomial of PGsj(m,α, k) is a product of
s − 2j linear Kirchhoff forms Yl(α, k) :=
∑
e∈E Yle(α)ke
and j line-correlator functions Xll′(α), l, l′ ∈ L, of a
graph G. The parametric functions Yl(α, k) and Xll′(α)
are homogeneous functions of the 0th and (-1)st degree
in α, respectively (see Sections 4 and 5 for more details).
By introducing the new variables,
αl = ρα′l, ∀l ∈ L/j, αj = ρ
(
1−
∑
l∈L/jα
′
l
)
,
∑
l∈L
α′l = 1,
∏
l∈L
dαl = ρ|L|−1dρ
∏
l∈L/j
dα′l,
R
|L|
+ (r )→ R1
+(r)× Σ|L|−1, r > 0, (3.19)
and assuming that rl = r > 0, ∀l ∈ L, we can perform
the integration over the variable ρ, 0 < r ≤ ρ ≤ ∞ by
using [63], see Ch. 6.3., eq.(3) or Ch. 6.5., Eq. (29),
IG(m, k)rε := (2π)nδG(kE)bG
∫
Σ|L|−1
dµG(α)
Δn/2
×
×
dG∑
s=0
[s/2]∑
j=0
PGsj(m,α, k)F r
sj(ω;Vε),
F rsj(ω;Vε) := i−ω−j
∞∫
r
dρ ρ−ω−j−1e−iρ Vε =
= V ω+j
ε Γ(−ω − j; irVε), Vε := Mε −A, (3.20)
ω := (n/2)|C| − λL, bG := cG(πn/2ip)|C|(2π)−n.
Here, p involved in bG is the number of positive squares
in the space-time metric gµν . The integration measure
dµG(α) and the integration domain Σ|L|−1 of the simplex
type are defined as
dµG(α) := δ
(
1−
∑
l∈L
αl
)∏
l∈L
(
dαl α
λl−1
l
Γ(λl)
)
,
Σ|L|−1 := {αl|αl ≥ 0, ∀l ∈ L,
∑
l∈L
αl = 1}. (3.21)
In Eq. (3.20), Γ(α;x) is one of the two incom-
plete gamma-functions appearing in the decomposition,
Γ(α) = Γ(α;x)+ γ(α;x), see [64], Ch. 9.1., Eqs. (1)–(2),
such that, at Reα > 0, Γ(α; 0) = Γ(α), γ(α; 0) = 0,
where Γ(α) is an ordinary gamma-function.
It is worth noting that we actually have the regular-
ization which combines three ones: i) the Hepp regular-
ization [30], (due to a change in the region of integration
over the auxiliary variable ρ); ii) the analytic regulariza-
tion [80], (due to the complexification of the parameter
λL, the half-degree of the denominator polynomial); iii)
the dimensional regularization [39,53,54] and other suit-
able references in them, (due to the complexification of
the parameter n, the space-time dimension). Recall that
λL and n are constituents of ω.
For convergent FAs, the quantities ω + j < 0, ∀j ∈
{0, 1, . . . , [dG/2}, and there exists the limit r → 0. After
passing to the limit r → 0 in Eq. (3.20), we obtain
IG(m, k)ε := (2π)nδG(kE)bG
∫
Σ|L|−1
dµG(α)
Δn/2
×
×
dG∑
s=0
[s/2]∑
j=0
PGsj(m,α, k)Fsj(ω;Mε, A),
Fsj(ω;Mε, A) := i−ω−j
∞∫
0
dρ ρ−ω−j−1e−iρ (Mε−A) =
= M ω+j
ε (1− Zε)ω+jΓ(−ω − j) (3.22)
= M ω+j
ε
∞∑
k=0
Γ(−ω − j + k)
Zkε
k!
, Zε := A/Mε.
It is easy to verify that the basic functions Fsj(ω;Mε, A)
satisfy Eqs. (2.1), (2.3), (2.6)–(2.8), and (2.11).
In the case of divergent FAs, for which ω + j ≥ 0 at
least for one j ∈ {0, 1, . . . , [dG/2}, the limit r → 0 does
494 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
SELF-CONSISTENT RENORMALIZATION AS AN EFFICIENT REALIZATION OF MAIN IDEAS
not exist. In this case, expressions (3.18)–(3.21) strictly
defined in the region R|L|+ (r) := R1
+(r)×Σ|L|−1 must be
made meaningful in a wider region R|L|+ := R1
+×Σ|L|−1,
where R1
+ := R1
+(r)|r=0.
The Bogoliubov–Parasiuk subtraction procedure used
for this purpose replaces IG(m, k)rε by
(Rν0 I)
G(m, k)ε = (2π)nδG(kE)bG×
×
∫
R
|L|
+
dvG(α)(Rν0I)G(m,α, k)ε, (3.23)
(Rν0I)G(m,α, k)ε :=
:= IG(m,α, k)ε −
ν∑
β=0
1
β !
∂ β
∂τβ
IG(m,α, τk)ε
τ=0
=
=
1
ν !
1∫
0
dτ(1− τ)ν ∂
ν+1
∂τν+1
IG(m,α, τk)ε, (3.24)
where the subtraction operations under the integral
sign are performed by using the Schlömilch integro-
differential formula (see the 2nd line of Eq. (3.24)) for
the remainder term of Maclaurin’s series. First, this
formula was applied explicitly to FAs in the Parasiuk
paper [8]. Although this expression guarantees a com-
pact representation of the subtraction procedure, it is,
nevertheless, inconvenient for computational purposes,
because it involves the additional integration and differ-
entiations in the integrand. The expression in the 1st
line of Eq. (3.24) is all the more inconvenient for these
purposes, since every term on the right-hand side of it
may be associated with a divergent integral.
At the same time, the algorithm proposed and applied
in [31–39] is based on the observation (see [64], Ch. 9.2.,
Eqs. (16, 17, 18)) that,
ex −
νsj∑
k=0
xk
k !
= exγ̃(1 + νsj ;x), γ̃(α;x) :=
γ(α;x)
Γ(α)
,
νsj∑
k=0
xk
k !
= exΓ̃(1 + νsj ;x), Γ̃(α;x) :=
Γ(α;x)
Γ(α)
,
γ̃(α;x) + Γ̃(α;x) = 1. (3.25)
Now, if we use: the explicit form of the integrand in
Eq. (3.18); the homogeneous properties for parametric
functions in ke, e ∈ E , see Eqs. (5.8); the 1st line of Eq.
(3.24); the 1st line of Eq. (3.25); and the relation
ν∑
β=0
1
β !
∂ β
∂τβ
{
τs−2jeiτ
2A
}
τ=0
=
νsj∑
k=0
(iA)k
k !
,
νsj := [(ν − s)/2] + j, (3.26)
we arrive at the multiplicative realization of the sub-
traction procedure in the integrand of Eq. (3.23) for the
regular value of the general FA (3.1),
(Rν0I)G(m,α, k)ε =
1
Δn/2
×
×
dG∑
s=0
[s/2]∑
j=0
PGsj(m,α, k)i−ω−je−iVε γ̃(1 + νsj ; iA). (3.27)
The integral in Eq. (3.23) with integrand (3.27) at
ν ≥ νG is now well-defined in the domain R
|L|
+ . The
substitution of (3.27) into integral (3.23) and the change
of integration variables according to Eq. (3.19) give rise
to the expression
(Rν0I)
G(m, k)ε = (2π)nδG(k) bG
∫
Σ|L|−1
dµG(α)
Δn/2
×
×
dG∑
s=0
[s/2]∑
j=0
PGsj(m,α, k)(Rν0F)sj(ω;Mε, A),
(Rν0F)sj(ω;Mε, A) :=
:= i−ω−j
∞∫
0
dρ ρ−ω−j−1e−iρ(Mε−A)γ̃(1 + νsj ; iρA) =
= M ω+j
ε
Γ(λsj)
Γ(2 + νsj)
Z 1+νsj
ε 2F1(1, λsj ; 2 + νsj ; Zε),
νsj := [(ν − s)/2] + j, λsj := −ω − j + 1 + νsj . (3.28)
The integration over ρ in (3.28) is performed with the
use of the formula (see [65], Ch. 17.3., Eq. (15))
∞∫
0
dxxµ−1e−vxγ̃(ν; a x) =
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 495
V.I. KUCHERYAVY
=
aνΓ(µ+ ν)
(a+ v)µ+νΓ(1 + ν) 2F1
(
1, µ+ ν; 1 + ν;
a
a+ v
)
,
Re (a+ v) > 0, Re v > 0, Re (µ+ ν) > 0.
3.4. So, using the properties of special functions sub-
stantially, the author has obtained [31, 32, 34, 36, 37, 39–
41, 44, 45] high-efficiency formulas which realize an an-
alytical continuation (in the variables ωG and νG) of
the FAs which are represented first in Eqs. (3.1)–(3.3)
by UV-divergent integrals and are given finally in Eqs.
(3.28)–(3.30) as convergent ones. As a result, we have
the following α-parametric integral representation:[
IG(m, k)ε
(Rν0I)
G(m, k)ε
]
= (2π)nδG(k) bG
∫
Σ|L|−1
dµG(α)
Δn/2
×
×
dG∑
s=0
[s/2]∑
j=0
PGsj(m,α, k)
[
Fsj(ω;Mε, A)
(Rν0F)sj(ω;Mε, A)
]
, (3.29)
for the convergent or dimensionally regularized value
IG(m, k)ε and for the regular value (Rν0I)
G(m, k)ε of in-
tegral (3.1). The subscripts 0 and superscript ν on R
indicate that (Rν0I)
G(m, k)ε is the regular function in a
vicinity of zero values of the external momenta ke, e ∈ E ,
and is evaluated for an renormalization index ν = νG.
The explicit forms of the basic functions Fsj(ω;Mε, A)
and (Rν0F)sj(ω;Mε, A) are as follows:
Fsj(ω;Mε, A) := M ω+j
ε (1− Zε)ω+j Γ(−ω − j) =
= M ω+j
ε
∞∑
k=0
Γ(−ω − j + k)
Zkε
k!
, Zε := A/Mε,
(Rν0F)sj(ω;Mε, A) := M ω+j
ε Γ(λsj)/Γ(2 + νsj)×
×Z 1+νsj
ε 2F1(1, λsj ; 2 + νsj ; Zε) =
= M ω+j
ε
∞∑
k=1+νsj
Γ(−ω − j + k)
Zkε
k!
, (3.30)
νsj := [(ν − s)/2] + j = [ω] + j + σs,
λsj := −ω − j + 1 + νsj = 1− δnδ|C|/2 + σs,
[ω] := rn|C|+ δnr|C| − λL, ω = [ω] + δnδ|C|/2,
σs := [(δnδ|C| + d− s)/2],
|C| = 2r|C| + δ|C|,
ν = νG, ω = ωG, d = dG.
The quantities [(ν − s)/2], [(ν + 1 − s)/2], and [ω] in
Eqs. (3.28)–(3.31) are the integer parts of (ν − s)/2,
(ν + 1 − s)/2, and ω, respectively. The subscripts
(s, j) on Fsj and (Rν0F)sj just mean that these func-
tions are attached to the homogeneous k-polynomials
PGsj(α,m, k) of the degree s− 2j, j = 0, . . . , [s/2], in the
external momenta ke, e ∈ E . The latter are α-images
of the homogeneous p-polynomials PGs (m, p) of the de-
gree s appearing in PG(m, p), see Eqs. (3.2). The k-
polynomials PGsj(α,m, k) are constructed by means of
the α-parametric functions Yl(α, k) and Xll′(α), l, l′ ∈
L. The efficient and universal algorithm of building
PGsj(α,m, k) is presented in Sec. 4. The α-parametric
functions Mε ≡M(m,α)ε and A ≡ A(α, k), incoming in
Eqs. (3.30) are defined in Eqs. (3.9) and (3.18), respec-
tively. The quantity M(m,α)ε is the linear form in the
square of internal masses with iε-damping. The func-
tions A(α, k) and Yl(α, k) are known as the quadratic
and linear Kirchhoff forms in the external momenta, ke,
e ∈ E . The function Δ ≡ Δ(α) is the Kirchhoff determi-
nant, and the Xll′(α) are the line-correlator functions.
The high-efficiency and universal algorithm of finding
the α-parametric functions A(α, k), Yl(α, k), Xll′(α),
and Δ(α) is given in Sec. 5.
3.5. The investigation of a complicated tangle of prob-
lems associated, on the one hand, with renormalization
methods and, on the other hand, with conserved and
broken symmetries, the Ward identities behavior, the
Schwinger terms contributions, and quantum anomalies
requires finding the renormalized FAs for different di-
vergence indices. For example, the amplitudes involved
in the Ward identities have divergence indices νG and
νG + 1.
The regular values (Rν+1
0 I)G(m, k)ε calculated for the
renormalization index νG + 1 once again have form of
Eq. (3.29), but with another basic functions (Rν+1
0 F)sj :
(Rν+1
0 F)sj := M ω+j
ε Γ(λ1
sj)/Γ(2 + ν1
sj)×
×Z 1+ν1
sj
ε 2F1(1, λ1
sj ; 2 + ν1
sj ; Zε), (3.31)
496 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
SELF-CONSISTENT RENORMALIZATION AS AN EFFICIENT REALIZATION OF MAIN IDEAS
ν1
sj := [(ν + 1− s)/2] + j = [ω] + σ1
s + j,
λ1
sj := −ω − j + 1 + ν1
sj = 1 + σ1
s − δnδ|C|/2,
σ1
s := [(δnδ|C| + d+ 1− s)/2].
In general, (Rν+1
0 F)sj 6= (Rν0F)sj , as far as ν1
sj 6= νsj .
The difference between them is the important quantity
(Δν+1,ν)
0 F)sj := (Rν+1
0 F)sj − (Rν0F)sj =
= −Θ(ν+1,ν)
sj
Γ(λsj)
Γ(2 + νsj)
M ω+j
ε Z 1+νsj
ε , (3.32)
Θ(ν+1,ν)
sj := H+(ν1
sj)θ
(ν+1,ν)
s ,
θ(ν+1,ν)
s := ν1
sj − νsj = σ1
s − σs =| δν − δs |,
ν = 2rν + δν , s = 2rs + δs, ν, s ∈ {0 ∪ N+},
where H+(x) is the Heaviside step function such that
H+(x) = 0, x < 0, H+(x) = 1, x ≥ 0, and δν , δs :=
ν(mod 2), s(mod 2) = 0, 1. It is this quantity that allows
one to obtain some efficient formulas for calculating the
quantum corrections (QCs) (i.e., quantum anomalies) to
the canonical Ward identities (CWIs) of the most gen-
eral kind, for example, to those involving canonically
non-conserved vector and (or) axial-vector currents for
nondegenerate fermion systems (i.e., for systems with
different fermion masses). Another very useful quantity
that is produced by differences
(Δ(ν+2,ν)
0 F)sj := (Rν+2
0 F)sj − (Rν0F)sj =
=(Rν0F)s−2,j − (Rν−2
0 F)s−2,j=(Rν0F)s−2,j − (Rν0F)sj=
= −H+(1 + νsj)
Γ(λsj)
Γ(2 + νsj)
M ω+j
ε Z 1+νsj
ε (3.33)
is closely related to (Δ(ν+1,ν)
0 F)sj .
3.6. The expressions given by Eqs. (3.28)–(3.30) have
two very important properties.
First, they describe both divergent and convergent
FAs in the unified manner. Really, due to the prop-
erties [62] Ch. 2.8, eqs.(4, 19), i.e., 2F1(α, β;α; z) = (1−
z)−β and
lim
c→ 2−l, l=1,2,...
2F1(a, b; c; z)/Γ(c) =
=
(a)l−1(b)l−1
(l − 1)!
zl−1
2F1(a+ l − 1, b+ l − 1; l; z), (3.34)
in the case a = 1, b = λsj = −ω− j + 1− l, c = 2− l, it
follows from Eqs. (3.30) and (3.34) that
(Rν0F)sj = M ω+j
ε Γ(−ω − j)2F1(l,−ω − j; l; Zε) =
= Fsj , if νsj = − l, l ∈ N+ , (3.35)
i.e., the first relation in Eqs. (2.2).
Second, the basic functions (Rν0F)sj ≡
(Rν0F)sj(ω;Mε, A) of the self-consistently renor-
malized FAs obey the same recurrence relations as the
basic functions Fsj ≡ Fsj(ω;Mε, A) of convergent or
dimensionally regularized FAs. Really, let us multiply
the recurrence relation (see [62] Ch. 2.8, Eq. (42))
(c− b− 1) 2F1(a, b; c; z) + b 2F1(a, b+ 1; c; z)−
−(c− 1) 2F1(a, b; c− 1; z) = 0, (3.36)
between the contiguous Gauss hypergeometric functions
2F1 in the case a = 1, b = λsj , c = 2 + νsj , by the
quantity M ω+j
ε Z
1+νsj
ε Γ(λsj)/Γ(2 + νsj). By using the
relations νs−2,j−1 = νsj , λs−2,j−1(ω) = λsj(ω) + 1, and
νs,j−1 = νsj − 1, λs,j−1(ω) = λsj(ω), we obtain the
recurrence relations
Mε (Rν0F)s−2,j−1 −A (Rν0F)s,j−1+
+(ω + j) (Rν0F)sj = 0, (3.37)
between the basic functions (Rν0F)sj ≡
(Rν0F)sj(ω;Mε, A), i.e., the second relation in Eqs.
(2.1).
3.7. Transformation formulae (see [62] Ch. 2.1.4, Eqs.
(22) and (23)) of 2F1 give rise to the representations
(Rν0F)sj =
(−1)Γ(λsj)Aνsj
Γ(2 + νsj)M
λsj−1
ε
(
Zε
Zε − 1
)
×
×2F1
(
1, ω + j + 1; 2 + νsj ;
Zε
Zε − 1
)
, (3.38)
(Rν0F)sj = (Mε −A)ω+j Γ(λsj)
Γ(2 + νsj)
Z 1+νsj
ε ×
×2F1(1 + νsj , ω + j + 1; 2 + νsj ; Zε). (3.39)
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 497
V.I. KUCHERYAVY
Equation (3.38) and the behavior of 2F1(a, b; c; z) in a
vicinity z → 1− determine completely the asymptotics
of the basic functions (Rν0F)sj for A < 0 in a vicinity
Mε → 0, i.e., the chiral limit
(Rν0F)sj
Mε→0' (−1)Γ(λsj − 1)Aνsj
Γ(1 + νsj)M
λsj−1
ε
,
if νsj ≥ 0 and λsj − 1 > 0;
(Rν0F)sj
Mε→0' (−1)Aνsj
Γ(1 + νsj)
ln (1−A/Mε) , (3.40)
if νsj ≥ 0 and λsj − 1 = 0;
(Rν0F)sj
Mε→0' Γ(−ω − j)(−A)ω+j ,
if νsj ≥ 0 and λsj − 1 < 0 or νsj ≤ −1,
which is equivalent also to the asymptotic behavior of the
basic functions in the case A→ −∞, Mε 6= 0. Equations
(3.30) yield four different series of values for λsj − 1:
λsj − 1 = −δnδ|C|/2+
+(rd − rs) + [(δnδ|C| + δd − δs)/2], (3.41)
λsj − 1 = (rd − rs)− 1/2, δnδ|C| = 1 & δs ≥ δd;
= (rd − rs) + 1/2, δnδ|C| = 1 & δd > δs;
= (rd − rs), δnδ|C| = 0 & δd ≥ δs;
= (rd − rs)− 1, δnδ|C| = 0 & δs > δd; (3.42)
d = 2rd + δd, s = 2rs + δs, δn, δ|C|, δd, δs = 0, 1.
It is evident that Eq. (3.39) presents a multiplicative
realization of the subtraction procedure explicitly,
(Rν0F)sj := Fsj − (Sν0F)sj =
= Fsj(ω;Mε, A) (Πν
0F)sj(ω;Zε),
(Πν
0F)sj(ω;Zε) :=
(−ω − j)1+νsj
Γ(2 + νsj)
Z 1+νsj
ε ×
×2F1(1 + νsj , ω + j + 1; 2 + νsj ;Zε),
(Sν0F)sj := M ω+j
ε
νsj∑
k=0
Γ(−ω − j + k)
Zkε
k!
. (3.43)
4. Homogeneous k-Polynomials PG
sj(m,α, k) of
α-Parametric Representation of FAs
4.1. It is evident from Eq. (3.29) that the basic
functions (Rν0F)sj and the homogeneous k-polynomials
PGsj(m,α, k) in external momenta ke, e ∈ E , of degree
s − 2j, j = 0, 1, . . . , [s/2], are two closely coupled im-
portant universal ingredients of the SCR representation
of FAs. The latter are α-images of the homogeneous p-
polynomials PGs (m, p) in the internal momenta pl, l ∈ L,
of degree s, s = 0, 1, . . . , dG, appearing in the numerator
polynomial PG(m, p) (see Eqs. (3.1)–(3.2)).
Each monomial of PGsj(m,α, k) is a product of s − 2j
linear Kirchhoff forms Yl(α, k) :=
∑
e∈E Yle(α)ke and j
line-correlator functions Xll′(α), l, l′ ∈ L, of a graph G.
The efficient algorithm of finding these expressions from
the initial homogeneous p-polynomials PGs (m, p) in the
internal momenta pl, l ∈ L, of degree s = 0, 1, . . . , dG,
has been elaborated in [31–34]. It resembles Wick rela-
tions between time-ordered and normal products of bo-
son fields in quantum field theory. The main steps of
this algorithm are as follows.
• The polynomials PGs0(m,α, k) are determined as
PGs0(m,α, k) := PGs (m, p)|pl=Yl(α,k),
j = 0, s = 0, 1, . . . , dG, (4.1)
i.e., by the straightforward substitution pl → Yl(α, k),
∀l ∈ L, in the polynomials PGs (m, p).
• The polynomials PGsj(m,α, k), j = 1, . . . , [s/2], have
the algebraic structure of quantities generated by the
Wick formula which represents a T -product of s boson
fields in terms of some set of N -products of s − 2j bo-
son fields with j primitive contractions. In this case,
the linear Kirchhoff forms Y σl (α, k) and their primitive
correlators
Y σ1
l1
· · · Y σ2
l2︸ ︷︷ ︸ := (−1/2)Xl1l2(α)gσ1σ2 ≡
≡ (−1/2)( σ1σ2
l1 l2
). (4.2)
play a role of boson fields and contractions, respectively.
4.2. As a result, we come to the following general
formulae. As far as the homogeneous p-polynomials
PGs (m, p) can be always represented as
PGs (m, p) =
∑
(i)∈G
a(i)
s (m) pσ
(i)
1
l
(i)
1
p
σ
(i)
2
l
(i)
2
· · · pσ
(i)
s
l
(i)
s
,
498 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
SELF-CONSISTENT RENORMALIZATION AS AN EFFICIENT REALIZATION OF MAIN IDEAS
l(i)a ∈ L, a = 1, . . . , s, (4.3)
where the coefficients a(i)
s (m) are functions of the masses
ml, l ∈ L, it is sufficient to find the image of some general
monomial entering into the sum over (i) ∈ G in Eq.
(4.3). The calculation according to the above-mentioned
Wick-type rule yields
p
σ
(i)
1
l
(i)
1
p
σ
(i)
2
l
(i)
2
· · · pσ
(i)
s
l
(i)
s
→
[s/2]∑
j=0
Pσ
(i)
1 ··· σ
(i)
s
(l
(i)
1 ··· l
(i)
s );j
(α, k),
Pσ
(i)
1 ··· σ
(i)
s
(l
(i)
1 ··· l
(i)
s );j
(α, k) =
= (−2)−j
∑
d∈(1s−2j2j)
P
σ
(i)
d(1)··· σ
(i)
d(s)
(l
(i)
d(1)··· l
(i)
d(s));j
(α, k),
P
σ
(i)
d(1)··· σ
(i)
d(s)
(l
(i)
d(1)··· l
(i)
d(s));j
(α, k) :=
s−2j∏
Y
σ
(i)
d(a)
l
(i)
d(a)
(α, k)×
×
j∏(
X
l
(i)
d(b)l
(i)
d(c)
(α)gσ
(i)
d(b)σ
(i)
d(c)
)
, (4.4)
where the summation in the second equation in (4.4)
is extended over all partitions d of (l(i)1 , l
(i)
2 , . . . , l
(i)
s ) ac-
cording to the Young scheme (1s−2j2j). Then the image
of homogeneous p-polynomials PGs (m, p) given by Eq.
(4.3) is
PGs (m, p)→
[s/2]∑
j=0
PGsj(m,α, k),
PGsj(m,α, k) =
∑
(i)∈G
a(i)
s (m)Pσ
(i)
1 ··· σ
(i)
s
(l
(i)
1 ··· l
(i)
s );j
(α, k). (4.5)
In so doing, we arrive at special j-degree homogeneous
polynomials in the variables ( σ1σ2
l1 l2
) involved in primitive
correlators (see Eq. (4.2)). Polynomials of this type was
introduced and named as hafnians by Caianiello [66, 67]
in the course of his QED investigations. Hafnians are
the counterparts of phaffians and closely connected with
permanents. The simplest nontrivial hafnian ( σ1σ2σ3σ4
l1 l2 l3 l4
)
of degree 2 is given below in two last lines of Eq. (4.6).
4.3 In view of the very important applied signifi-
cance of the algorithm of constructing a family of ho-
mogeneous k-polynomials PGsj(m,α, k) from the initial
p-polynomials PGs (m, p), we give some examples:
1→ j = 0 : 1;
pσl → j = 0 : Y σl =:[ σl ];
pσ1
l1
pσ2
l2
→ j = 0 : Y σ1
l1
Y σ2
l2
=: [ σ1σ2
l1 l2
],
j = 1 : (− 1
2 ){Xl1l2g
σ1σ2 =: ( σ1σ2
l1 l2
)};
pσ1
l1
pσ2
l2
pσ3
l3
→ j = 0 : Y σ1
l1
Y σ2
l2
Y σ3
l3
=: [σ1σ2σ3
l1 l2 l3
],
j = 1 : (− 1
2 )
{
(σ1σ2
l1 l2
)[σ3
l3
]+( σ1σ3
l1 l3
)[ σ2
l2
] + ( σ2σ3
l2 l3
)[ σ1
l1
]
}
;
pσ1
l1
pσ2
l2
pσ3
l3
pσ4
l4
→ j = 0 : Y σ1
l1
Y σ2
l2
Y σ3
l3
Y σ4
l4
=: [σ1σ2σ3σ4
l1 l2 l3 l4
],
j = 1 : (− 1
2 )
{
( σ1σ2
l1 l2
)[ σ3σ4
l3 l4
]+( σ1σ3
l1 l3
)[ σ2σ4
l2 l4
]+( σ1σ4
l1 l4
)[ σ2σ3
l2 l3
]+
+( σ2σ3
l2 l3
)[σ1σ4
l1 l4
]+(σ2σ4
l2 l4
)[σ1σ3
l1 l3
]+(σ3σ4
l3 l4
)[σ1σ2
l1 l2
]
}
,
j = 2 : (− 1
2 )2
{
( σ1σ2
l1 l2
)( σ3σ4
l3 l4
) + ( σ1σ3
l1 l3
)( σ2σ4
l2 l4
)+
+( σ1σ4
l1 l4
)( σ2σ3
l2 l3
)
}
=:(− 1
2 )2(σ1σ2σ3σ4
l1 l2 l3 l4
). (4.6)
5. Parametric Functions of FAs
5.1. We now formulate an algorithm of finding the para-
metric functions
Δ(α), A(α, k), Yl(α, k), Xll′(α), l, l′ ∈ L,
of Feynman amplitudes. Of course, it is to be men-
tioned that we can use, in principle, any one of
the available approaches. Contributions to this sub-
ject have been made by many authors. We give a
very incomplete list of quotes here, namely, the pa-
pers by Chisholm [68], Nambu [69], Symanzik [70],
Nakanishi [71], Schimamoto [72], Bjorken and Wu [73],
Peres [74], Lam and Lebrun [75], Stepanov [76], Liu
and Chow [77], Cvitanovic and Kinoshita [78], and
the books by Todorov [79], Speer [80], Nakanishi [81],
Zav’yalov [82], Smirnov [83], in which many other refer-
ences can be found. Nevertheless, our algorithm seems
to be very simple, but universal enough. It is named by
the author [32, 84, 85] as circuit-path algorithm.
5.2. Suppose we have a connected graph G(V,L ∪ E)
with sets of vertices, V, internal lines, L, and external
lines, E , and with a certain relation of incidence be-
tween V and Λ ≡ L ∪ E described by an oriented in-
cidence matrix eil ≡ [eVΛ]il = 0, ±1, vi ∈ V, l ∈ Λ.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 499
V.I. KUCHERYAVY
In particular, eil = 0, if line l is nonincident to the
vertex vi; eil = 1, if line l is outgoing from the ver-
tex vi; and eil = −1, if line l is incoming to the ver-
tex vi. The fact that the set of all lines Λ is separated
from the very beginning into two mutually disjoint sub-
sets L and E (their incident properties are different) is
very important both from the algorithmic point of view
and from potential possibilities. In so doing, we need
not to replace here the set of external lines (incident
to some vertex) by some effective line or to assign the
same orientation to all external lines, as is usually done.
Therefore, we can pose the task of constructing the para-
metric functions of the whole graph via the parametric
functions of its subgraphs. As a result, the circuit-path
approach is naturally arose, and the recursive struc-
ture of the parametric functions of FAs has been ob-
tained [85, 86].
5.3. The set of external lines, E , induces the single-
valued decomposition of the set of all vertices, V, into
the subset of external vertices, Vext, and the subset of
internal vertices, Vint. The set of internal lines, L, can
be always decompose (as a rule, in more than one way)
into two mutually disjoint subsets, M and N , which
determine some skeleton tree and the corresponding co-
tree subgraphs of the graph G. So, we have the following
decomposition of the set Λ = E∪N ∪M of all lines of the
graph G into mutually disjoint subsets, E , N , and M.
Then the circuit-path algorithm requires the following
steps:
• Let us choose a subsetN ⊂ L such that the subgraph
G(V,M∪ E), where M := L/N , is a skeleton-tree-type
graph and the subgraph G(V,N ∪ E) is a co-tree-type
graph. It is clear that this choice is ambiguous. It is
shown in [84], however, that the parametric functions
are independent of any choice of N .
• Let us choose a vertex vj ∈ V which will be referred
as a basis vertex, (or reference vertex, or zero point). It is
clear that this choice is also ambiguous. But it is shown
in [84], that the parametric functions are again indepen-
dent of any given choice of vj . From the viewpoint of
practical calculations, it seems reasonable to choose the
basis vertex vj as such a vertex, to which the largest
number of external lines of the graph are incident.
• The choice ofN ⊂ L and the basis vertex vj uniquely
defines the notions of basis circuits C(n), n ∈ N and
basis paths P (j|e), e ∈ E .
The basis circuit C(n) generated by the line n ∈ N
is a union of the line n with the subset M(n) ⊂ M
which forms a chain in M between vertices incident to
the line n, i.e. C(n) := {n} ∪M(n). The orientation in
the circuit C(n) is defined by the orientation of the line
n ∈ N .
The basis path P (j|e) generated by the line e ∈ E and
the basis vertex vj is a union of the line e with the subset
M(j|e) ⊂M which forms a chain inM between a vertex
incident to the line e ∈ E and the basis vertex vj , i.e.
P (j|e) := {e} ∪ M(j|e). The orientation in the path
P (j|e) is defined by the orientation of the line e ∈ E .
• By analogy with the incidence matrix eVΛ which can
be referred, more precisely, as the vertex-line incidence
matrix, one introduces topologically the line-circuit
eΛN [77, 78, 81, 84, 85], and the line-path eΛE(j) [84, 85]
incidence matrices, namely:
[eΛN ]ln :=
{
0, l /∈ C(n),
±1, l ∈ C(n);
[eΛE(j)]le :=
{
0, l /∈ P (j|e),
±1, l ∈ P (j|e).
(5.1)
Here, the plus or minus sign depends on whether the ori-
entation of the line l ∈ Λ coincides or not with the orien-
tation of the circuit C(n) for eΛN or the path P (j|e) for
eΛE(j). As a result, the column-vector pΛ of all momenta
pl, l ∈ Λ and submatrices of eΛN and eΛE(j), whose rows
are associated with the partition Λ = E ∪ N ∪M, can
be represented as follows [84, 85]:
pΛ = pext
Λ + pint
Λ , pext
Λ = eΛE(j)kE , pint
Λ = eΛN pN ;
eEE(j) = 1EE , eNE(j|N ) = 0NE ,
eME(j|N ) = −e−1
{V/j}Me{V/j}E ;
eEN = 0EN , eNN = 1NN ,
eMN = −e−1
{V/j}Me{V/j}N . (5.2)
From now on, kE and pN are the column-vectors of the
external momenta ke, e ∈ E , and the independent in-
tegration momenta pn, n ∈ N , respectively; 0AB is the
|A|×|B|-rectangular matrix of zeros, and 1AA is the |A|-
dimensional unit matrix. The matrices e{V/j}E , e{V/j}N ,
and e{V/j}M are submatrices of eVΛ. Their rows are de-
fined by the set (V/vj) ⊂ V, and their columns are de-
fined by the subsets E ,N ,M, respectively. The (|V|−1)-
dimensional square matrix e{V/j}M is nonsingular, and
det[e{V/j}M] = ±1. In submatrices of the second and
500 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
SELF-CONSISTENT RENORMALIZATION AS AN EFFICIENT REALIZATION OF MAIN IDEAS
third lines of Eqs. (5.2), the subset N is pointed out ex-
plicitly, because of eN ′E(j|N) 6= 0N ′E , and eM′E(j|N) 6=
eME(j|N) if N ′ 6= N , L = N ∪ M = N ′ ∪ M′, but
eEE(j|N ) = eEE(j|N ′) = 1EE .
• There exist the following very important “orthogo-
nality” relations [84, 85, 87]:
eVΛeΛN = eVLeLN = 0VN ,
e{V/j}ΛeΛN = e{V/j}LeLN = 0{V/j}N ,
[eVΛeΛE(j)]ie = δij [e(V∗)E ]e,
e{V/j}ΛeΛE(j) = 0{V/j}E , (5.3)
where e(V∗)E is the vertex-line incidence matrix of the
“star”-type graph G∗ :=< V∗, E > with the one vertex
V∗ and the set of external lines E of the graph G. The
graph G∗ :=< V∗, E > is a result of the shrinking of all
vertices vi ∈ V, and all internal lines l ∈ L, of the graph
G to the single “black-hole” vertex V∗.
• By assigning the parameter αl ≥ 0 to every internal
line l ∈ L, we define the circuit CNN (α), path EEE(j|α),
and path-circuit ΠEN (j|α) matrices [84, 85], according
to:
[CNN (α)]nn′ := [eTLNαLLeLN ]nn′ = ±
∑
l∈C(n)∩C(n′)
αl,
[EEE(j|α)]ee′ := [eTLE(j)αLLeLE(j)]ee′ = ±
∑
l∈P (j|e)∩P (j|e′)
αl,
[ΠEN (j|α)]en := [eTLE(j)αLLeLN ]en = ±
∑
l∈P (j|e)∩C(n)
αl.
(5.4)
Here, the plus or minus sign depends on the mutual ori-
entations of the sets, over which the summation is per-
formed, on their intersection. The plus sign corresponds
to the case of coinciding orientations. It is clear that
the explicit form of these matrices in any given case
can be easily obtained by inspecting the graph. From
now on, αLL is the diagonal |L|-dimensional matrix, i.e.,
[αLL]ll′ = αl δll′ .
• The parametric functions are derived by means of
the use of the following matrices [84, 86]:
AEE(j|α) := EEE(j|α)−ΠEN (j|α)C−1
NN (α)ΠT
EN (j|α),
YLE(j|α) := eLE(j)− eLNC−1
NN (α)ΠT
EN (j|α),
XLL(α) := eLNC
−1
NN (α)eTLN ,
Δ(α) := detCNN (α). (5.5)
So, the quadratic A(α, k) and linear Yl(α, k), l ∈ L,
Kirchhoff forms in the external momenta ke, e ∈ E , and
the line-correlator functions Xll′(α), l, l′ ∈ L, are defined
as [84, 86]
A(α, k) := (kTE ·AEE(j|α)kE) =
=
∑
e,e′∈E [AEE(j|α)]ee′(ke · ke′),
Yl(α, k) := YlE(j|α)kE =
∑
e∈E [YlE(j|α)]eke,
YlE(j|α) = elE(j)− elNC−1
NN (α)ΠT
EN (j|α),
Xll′(α) = elNC
−1
NN (α)eTl′N . (5.6)
Here, elN and elE(j) are the row-vectors of matrices
(5.1)–(5.2) for the line l ∈ L.
5.4. It should be mentioned that the functions Δ(α)
and A(α, k) do not depend on the orientation of inter-
nal lines. However, when the orientation of line l is
changed, the parametric functions Yl(α, k) and Xll′(α)
reverse their signs.
It is also useful to represent the quantities A(α, k) and
YL(α, k) in a form exhibiting a special role of the matri-
ces XLL(α) and XNN (α) [78, 84]:
A(α, k) =
(
pext
L (k)T · [αLL − αLLXLL(α)αLL ] pext
L (k)
)
,
pext
L (k) = eLE(j)kE , pext
E (k) = kE , pext
N (k) = 0N ,
YL(α, k) = [ 1LL −XLL(α)αLL ] pext
L (k) =
= pext
L (k)− Y int
L (α, k),
Y int
L (α, k) := XLL(α)αLL pext
L (k),
XLL(α) := eLNXNN (α)eTLN ,
XNN (α) := C−1
NN (α), (5.7)
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 501
V.I. KUCHERYAVY
where kE is the column-vector of the external momenta
ke, e ∈ E . The following homogeneous properties hold:
Δ(ρα) = ρ |C|Δ(α), Xll′(ρα) = ρ−1Xll′(α),
A(ρα, τk) = ρτ2A(α, k), Yl(ρα, τk) = τYl(α, k),
PGsj(m, ρα, τk) = ρ−jτs−2j PGsj(m,α, k). (5.8)
5.5. Now we exhibit some important properties of
α-parametric functions [45]. Let us introduce the quan-
tities
Kr
LL := XLLαLL, Kl
LL := αLLXLL,
LiLL := 1LL −Ki
LL, i = r, l. (5.9)
Using Eqs. (5.3)–(5.5), we find that the matrices
Ki
LL(α) and LiLL(α) are projectors with the properties
Ki
LLK
i
LL = Ki
LL, LiLLL
i
LL = LiLL, i = r, l,
Ki
LLL
i
LL = 0LL, Kl
LLαLLL
r
LL = 0LL, (5.10)
From Eqs. (5.10), we get some relations between prod-
ucts of XLL, αLL, and YLE :(
XLLαLL
)m
XLL = XLLαLLXLL = XLL,
(
LrLL
)m
XLL = 0LL,
(
XLLαLL
)m
YLE = XLLαLLYLE = 0LE , (5.11)
(
LrLL
)m
YLE = YLE ,
Tr
[(
Ki
LL
)m] = Tr
[
Ki
LL
]
=
∑
l∈L αlXll(α) = |N |,
Tr
[(
LiLL
)m] = Tr
[
LiLL
]
= |M|, i = r, l,
and the relations between the quadratic A(α, k) and lin-
ear Yl(α, k), l ∈ L, Kirchhoff forms:
A(α, k) = (Y T
L · αLLpext
L ) = (pextT
L · αLLYL) ≡
≡
∑
l∈L
αl(pext
l (k) · Yl(α, k)) =
= (Y TL · αLLYL) ≡
∑
l∈L
αlY
2
l (α, k). (5.12)
The following relations are also satisfied:
eVEkE + eVLYL(α, k) = 0V , eTLNαLLYL(α, k) = 0N ,
eVLXLL(α) = 0VL,
Kr
LLeLE(j) = −eLNYNE(j|α) = eLNK
r
NLeLE(j),
Kr
LLeLN = eLN ,(
Y intT
L · αLLYL
)
=
=
(
Y intT
L · αLL pext
L
)
−
(
Y intT
L · αLLY int
L
)
= 0. (5.13)
In our case of α-parametric functions, two relations in
the first line of Eqs. (5.13) are analogs of the first and
second Kirchhoff laws in electric networks. Similarly, in
the third line of Eqs. (5.12), we find an analog of the
well-known expression for a power dissipated in electric
networks.
The present author wishes to express his sincere
thanks to the anonymous referee for the constructive re-
marks and the valuable suggestions. The paper is based
on the report presented at the Bogolyubov Kyiv Confer-
ence “Modern Problems of Theoretical and Mathemati-
cal Physics”, September 15-18, 2009, Kyiv, Ukraine.
1. N.N. Bogoliubov and O.S. Parasiuk, Dokl. AN SSSR
100, 25 (1955).
2. N.N. Bogoliubov and O.S. Parasiuk, Dokl. AN SSSR
100, 429 (1955).
3. O.S. Parasiuk, Dokl. AN SSSR 100, 643 (1955).
4. N.N. Bogoliubov and O.S. Parasiuk, Izv. AN SSSR, Ser.
Mat. 20, 585 (1956).
5. O.S. Parasiuk, Izv.AN SSSR, Ser. Mat. 20, 843 (1956).
6. N.N. Bogoliubov und O.S. Parasiuk, Acta Math. 97, 227
(1957).
7. O.S. Parasiuk, in Proc. of The Third All-Union Math-
ematical Congress, Moscow, June 25–July 4, 1956,
Vol.III, Plenary Reports, (Izd. AN SSSR, Moscow, 1958),
p. 558.
8. O.S. Parasiuk, Ukr.Mat. Zh. 12, 287 (1960).
9. A.L. Cauchy, Exercices de Mathématiques (Impr. Royale,
Paris, 1826). See also, Oeuvres Complètes (Gauthier–
Villars, Paris, 1887), 6, 78. In Cauchy’s 1844 memoir [11],
he wrote that the paper on the intégrale extraordinaire
had been presented in its initial form to the Acad’emie
des Sciences on January 2, 1815.
502 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
SELF-CONSISTENT RENORMALIZATION AS AN EFFICIENT REALIZATION OF MAIN IDEAS
10. A.L. Cauchy, Exercices de mathématiques (Impr. Royale,
Paris, 1827). See also, Oeuvres Complètes (Gauthier–
Villars, Paris, 1887), 7, 121.
11. A.L. Cauchy, J. École Polytechnique 18, 147 (1844).
12. R. d’Adhemar, J.Math. Pures et Appl., Sér. 5 10, 131
(1904).
13. R. d’Adhemar, J. Math. Pures et Appl., Sér. 6 2, 357
(1906).
14. J. Hadamard, C.R. Acad. Sci., Paris 137, 1028 (1903).
See also Oeuvres de Jacques Hadamard (Centre National
de la Recherche Sci., Paris, 1968), 3, p. 1111.
15. J. Hadamard, Ann. Sci. École Norm. Sup., Sér. 3 21, 535
(1904). See also Oeuvres de Jacques Hadamard (Centre
National de la Recherche Sci., Paris, 1968), 3, P. 1173.
16. J. Hadamard, Ann. Sci. École Norm. Sup., Sér. 3 22, 101
(1905). See also Oeuvres de Jacques Hadamard (Centre
National de la Recherche Sci., Paris, 1968), 3, p. 1195.
17. J. Hadamard, Acta Math. 31, 333 (1908). See also
Oeuvres de Jacques Hadamard (Centre National de la
Recherche Sci., Paris, 1968), 3, p. 1249.
18. L. Saalschütz, Zeitschr. Math. und Phys. 32, 246 (1887).
19. L. Saalschütz, Zeitschr. Math. und Phys. 33, 362 (1888).
20. V. Maz’ya and T. Shaposhnikova, Jacques Hadamard,
a Universal Mathematician (Amer. Math. Soc., Provi-
dence, RI, 1998).
21. J. Hadamard, Le Problème de Cauchy et les Équations
aux Dérivées Partielles Linéaires Hyperboliques (Her-
mann, Paris, 1932).
22. J. Hadamard, The Psychology of Invention in the Math-
ematical Field, (Princeton University Press, Princeton,
1945).
23. M. Riesz, Acta Math. 81, 1 (1949).
24. M. Riesz, Canad. J.Math. 13, 37 (1961).
25. F.J. Bureau, Comm. Pure and Apll.Math. 8, 143 (1955).
26. R. Courant, Partial Differential Equations (Interscience,
New York, 1962).
27. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional
Integrals and Derivatives. Theory and Applications (Gor-
don and Breach, New York, 1993).
28. B.M. Stepanov, Abstract theory of the R-operation,
Izv.AN SSSR, Ser. Mat. 27, 819 (1963).
29. B. M. Stepanov, Izv.AN SSSR, Ser.Mat. 29, 1037
(1965).
30. K. Hepp, Comm. Math. Phys. 2, 301 (1966).
31. V.I. Kucheryavy, Theor. Math. Phys. 20, 29 (1974);
Preprint ITP-73-46E, Kiev, 1973, 45p.
32. V.I. Kucheryavy, Nucl. Phys. B 127, 66 (1977); Preprint
ITP-76-132R, Kiev, 1976.
33. V.I. Kucheryavy, Preprint ITP-79-107R, Kiev, 1979;
Preprint ITP-79-112R, Kiev, 1979.
34. V.I. Kucheryavy, Theor.Math. Phys. 51, 355 (1982).
35. V.I. Kucheryavy, Preprint ITP-82-91R, Kiev, 1982.
36. V.I. Kucheryavy, Dopov. AN Ukr. SSR, Ser. A, No. 5, 62
(1983).
37. V.I. Kucheryavy, Dopov. AN Ukr. SSR, Ser. A, No. 7, 59
(1983).
38. V.I. Kucheryavy, Preprint ITP-87-130R, Kiev, 1987.
39. V.I. Kucheryavy, J.Nucl. Phys. 53, 1150 (1991).
40. V.I. Kucheryavy, Ukr. Fiz. Zh. 36, 1769 (1991).
41. V.I. Kucheryavy, Ukr.Mat. Zh. 43, 1445 (1991).
42. V.I. Kucheryavy, in Non-Euclidean Geometry in Modern
Physics, edited by L. Jenkovszky (Bogolyubov Institute
for Theoretical Physics of NAS of Ukraine, Kyiv, 1997),
p. 224.
43. V.I. Kucheryavy, in The Centenary of Electron (EL-100),
edited by A. Zavilopulo, Yu. Azhniuk, and L. Bandurina
(Karpaty Publ., Uzhgorod, 1997), p. 252.
44. V.I. Kucheryavy, in The Ukrainian Mathematical
Congress – 2001, Mathematical Physics, Section 5,
Proceedings, edited by A.G. Nikitin and D.Ya. Petrina
(Institute of Mathematics of the NAS of Ukraine, Kyiv,
2002), p. 54.
45. V.I. Kucheryavy, in Proceedings of the IVth Interna-
tional Hutsulian Workshop on Mathematical Theories
and Their Applications in Physics and Technology, edited
by S. S. Moskaliuk (TIMPANI, Kyiv, 2004), p. 97.
46. Tullio Regge, in Battelle rencontres. 1967 Lectures in
mathematics and physics, edited by C.M. DeWitt and
J.A. Wheeler, (Benjamin, New York, 1968), p. 433. On
page 442, we find the following text: “...Very helpful in
this direction is the remark that property III is equiv-
alent to a set of differential equations which reduces to
the ordinary differential equation for the hypergeometric
functions case. This equivalence has been pointed out to
us by Parasiuk in Kiev and we think that it was known
as far back as Schläfli...”
47. V.Z. Enolsky and V.A. Golubeva, Mat. Zamet. 23, 113
(1978); Preprint ITP-75-15E, Kiev, 1975.
48. V.A. Golubeva, Uspekhi Math.Nauk 31, 135 (1976).
49. A.V. Kotikov, Phys. Lett. B 254, 158 (1991); Preprint
ITP-90-45E, Kiev, 1990.
50. A.V. Kotikov, arXiv:hep-ph/0102178v1.
51. M. Argeri and P. Mastrolia, arXiv:hep-ph/0707.4037v1.
52. D.A. Slavnov, Theor. Math. Phys. 17, 342 (1973).
53. V.I. Kucheryavy, Preprint ITP-92-66E, Kiev, 1993.
54. V.I. Kucheryavy, Ukr. Fiz. Zh. 38, 1862 (1993).
55. V.I. Kucheryavy, in Proceedings of Institute of Mathemat-
ics of NAS of Ukraine, 30, Part 2, edited by A.G. Nikitin,
and V.M. Boyko, (Institute of Mathematics of the NAS
of Ukraine, Kyiv, 2000), p. 493.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 503
V.I. KUCHERYAVY
56. V.I. Kucheryavy, Nucl. Phys. B (Proc. Suppl.) 102–103,
377 (2001).
57. V.I. Kucheryavy, in Proceedings of Institute of Math-
ematics of NAS of Ukraine”, 43, Part 2, edited by
A.G. Nikitin and V.M. Boyko, (Institute of Mathemat-
ics of the NAS of Ukraine, Kyiv, 2002), p. 629.
58. V.I. Kucheryavy, in Proceedings of Institute of Math-
ematics of NAS of Ukraine”, 50, Part 2, edited
by A.G. Nikitin, V.M. Boyko, R.O. Popovych and
I.A. Yehorchenko, (Institute of Mathematics of the NAS
of Ukraine, Kyiv, 2004), p. 862.
59. V.A. Fock, Phys. Zs. Sowjetunion 12, 404 (1937).
60. J. Schwinger, Phys. Rev. 82, 664 (1951).
61. N.N. Bogoliubov and D.V. Shirkov, Introduction to the
Theory of Quantized Fields (Wiley, New York, 1980).
62. H. Bateman and A. Erdelyi, Higher Transcendental
Functions. Vol. I. Hypergeometric Function. Legendre
Functions (Nauka, Moscow, 1965) (in Russian).
63. H. Bateman and A. Erdelyi, Tables of Integral Trans-
forms. Vol. I. Fourier, Laplace, and Mellin Transforma-
tions (Nauka, Moscow, 1969) (in Russian).
64. H. Bateman and A. Erdelyi, Higher Transcendental
Functions. Vol. II. Bessel functions. Parabolical cylin-
der functions. Orthogonal polynomials (Nauka, Moscow,
1974) (in Russian).
65. H. Bateman and A. Erdelyi, Tables of Integral Trans-
forms. Vol. II. Bessel Transformations. Integrals of Spe-
cial Functions (Nauka, Moscow, 1970) (in Russian).
66. E.R. Caianiello, Nuovo Cim. 10, 1634 (1953).
67. E.R. Caianiello, Combinatorics and Renormalization in
Quantum Field Theory (Benjamin, London, 1973).
68. J.S.R. Chisholm, Proc. Cambr. Phil. Soc. 48, Part 2, 300
(1952); Part 3, 518, corrigendum.
69. Y. Nambu, Nuovo Cim. 6, 1064 (1957).
70. K. Symanzik, Progr. Theor. Phys. 20, 690 (1958).
71. N. Nakanishi, Progr. Theor. Phys. Suppl. 18, 1 (1961).
72. Y. Schimamoto, Nuovo Cim. 25, 1292 (1962).
73. J.D. Bjorken and T.T. Wu, Phys.Rev. 130, 2566 (1963).
74. A. Peres, Nuovo Cim. 38, 270 (1965).
75. C.S. Lam and J.P. Lebrun, Nuovo Cim. A 59, 397 (1969).
76. B.M. Stepanov, Theor.Math. Phys. 5, 356 (1970).
77. C.J. Liu and Y. Chow, J. Math. Phys. 11, 2789 (1970).
78. P. Cvitanovic and T. Kinoshita, Phys. Rev. D10, 3978
(1974).
79. I. T. Todorov, Analytical Properties of Feynman Dia-
grams in Quantum Field Theory (Bulgarian Acad. Sci.,
Sofia, 1966).
80. E.R. Speer, Generalized Feynman Amplitudes (Princeton
Univ. Press, Princeton, 1969).
81. N. Nakanishi, Graph Theory and Feynman Integrals
(Gordon and Breach, New York, 1971).
82. O.I. Zav’yalov, Renormalized Feynman Diagrams
(Nauka, Moscow, 1979) (in Russian).
83. V.A. Smirnov, Renormalization and Asymptotic Expan-
sions of Feynman Amplitudes (Moscow Univ. Press,
Moscow, 1990) (in Russian).
84. V.I. Kucheryavy, Dopov. AN Ukr. SSR, Ser. A, No. 10,
884 (1972); Preprint ITP-71-46R, Kiev, 1971.
85. V.I. Kucheryavy, Theor.Math. Phys. 10, 182 (1972);
Preprint ITP-71-39R, Kiev, 1971.
86. V.I. Kucheryavy, Dopov. AN Ukr. SSR, Ser. A, No. 9, 785
(1972); Preprint ITP-71-55E, Kiev, 1971.
87. P.R. Bryant, in Graph Theory and Theoretical Physics,
edited by F. Harary, (Academic Press, New York, 1967),
p. 111.
Received 30.10.09
САМОУЗГОДЖЕНА РЕНОРМАЛIЗАЦIЯ
ЯК ЕФЕКТИВНА РЕАЛIЗАЦIЯ ГОЛОВНИХ
IДЕЙ R-ОПЕРАЦIЇ БОГОЛЮБОВА–ПАРАСЮКА
В.I. Кучерявий
Р е з ю м е
Книга природи написана мовою математики.
Галiлео Галiлей, [ Il Saggiatore, 1623 ].
“... В зв’язку з тим, що в останнiй час знайдено тi-
сний зв’язок мiж причиновiстю та аналiтичнiстю,
не виключена ймовiрнiсть побудови вiднiмальної
операцiї навiть в самому загальному випадку
методами аналiтичного продовження.”
О.С. Парасюк, [ [7], с.566, останнiй абзац, 1956 ].
Цю можливiсть реалiзовано явно та ефективно засобами на-
шої самоузгодженої ренормалiзацiї (СУР). Пiд самоузгодже-
нiстю розумiють, що всi формальнi спiввiдношення мiж УФ-
розбiжними фейнмановими амплiтудами також автоматично
зберiгаються мiж їхнiми регулярними значеннями, знайдени-
ми згiдно з процедурою СУР. Самоузгоджена ренормалiза-
цiя з однаковою ефективнiстю застосовна як до ренормовних,
так i до неренормовних теорiй. СУР має ефективнi засоби
для конструктивного розгляду нових задач: а) ренормалiзацiй-
них проблем, що пов’язанi з симетрiями, тотожностями Уорда
та квантовими аномалiями; б) нових взаємозв’язкiв мiж скiн-
ченними зародковими та скiнченними фiзичними параметра-
ми квантово-польових теорiй. Наведено огляд головних iдей
та властивостей СУР, а також чiтко описано три взаємодопов-
нювальнi алгоритми СУР, якi подано у виглядi, максимально
пристосованому для практичних застосувань.
504 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
|