Vertex clothing in quantum field theory
The problem of the vertex renormalization in quantum field theory is tackled via the implementation of the unitary clothing transformation method. In the model of charged spinless nucleon and scalar meson fields coupled by the Yukawa-type three-linear interaction the expression for the charge correc...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Cite this: | Vertex clothing in quantum field theory / V.Yu. Korda and I.V. Yeletskikh // Вопросы атомной науки и техники. — 2007. — № 3. — С. 42-46. — Бібліогр.: 7 назв. — рос. |
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| author | Korda, V.Yu. Yeletskikh, I.V. |
| author_facet | Korda, V.Yu. Yeletskikh, I.V. |
| citation_txt | Vertex clothing in quantum field theory / V.Yu. Korda and I.V. Yeletskikh // Вопросы атомной науки и техники. — 2007. — № 3. — С. 42-46. — Бібліогр.: 7 назв. — рос. |
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| description | The problem of the vertex renormalization in quantum field theory is tackled via the implementation of the unitary clothing transformation method. In the model of charged spinless nucleon and scalar meson fields coupled by the Yukawa-type three-linear interaction the expression for the charge correction in the first non-vanishing (third) order in the coupling constant is derived. Being the off-energy-shell quantity, the expression can be brought to the explicitly covariant form on the energy shell, providing the momentum independence of the charge renormalization.
Метод унітарних одягаючих перетворень застосовано в моделі квантової теорії поля, в якій нуклонне і нейтральне піонне поля взаємодіють через регуляризований зв’язок типу Юкави. В цьому підході масові контрчлени частково скорочуються з комутаторами генераторів одягаючих перетворень і операторів взаємодії, що формує піонні та нуклонні зсуви мас. Знайдені величини подаються тривимірними інтегралами від деяких коваріантних комбінацій відповідних імпульсів частинок, що забезпечує незалежність розрахованих поправок від імпульсів. Визначено умови, що висуваються до вершинних функцій, які здійснюють регуляризацію зв’язку полів.
Метод унитарных одевающих преобразований применен в модели квантовой теории поля, в которой нуклонное и нейтральное пионное поля взаимодействуют посредством регуляризованной псевдоскалярной связи типа Юкавы. В этом подходе массовые контрчлены частично сокращаются с коммутаторами генераторов одевающих преобразований и операторов взаимодействия, формируя пионные и нуклонные сдвиги массы. Найденные величины выражаются трехмерными интегралами от некоторых ковариантных комбинаций соответствующих импульсов частиц, что обеспечивает независимость рассчитанных поправок от импульсов. Определены условия, налагаемые на обрезающие вершинные функции, осуществляющие регуляризацию связи полей.
|
| first_indexed | 2025-12-07T18:50:31Z |
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| fulltext |
VERTEX CLOTHING IN QUANTUM FIELD THEORY
V.Yu. Korda and I.V. Yeletskikh
Institute of Electrophysics and Radiation Technologies NAS of Ukraine, Kharkov, Ukraine;
e-mail: kvyu@kipt.kharkov.ua
The problem of the vertex renormalization in quantum field theory is tackled via the implementation of the uni-
tary clothing transformation method. In the model of charged spinless nucleon and scalar meson fields coupled by
the Yukawa-type three-linear interaction the expression for the charge correction in the first non-vanishing (third)
order in the coupling constant is derived. Being the off-energy-shell quantity, the expression can be brought to the
explicitly covariant form on the energy shell, providing the momentum independence of the charge renormalization.
PACS: 21.45.+v; 21.40. Jv; 11.80.-m
1. INTRODUCTION
The unitary clothing transformation method pro-
posed by Greenberg and Schweber [1] relying upon the
penetrating analyses of the problems of quantum field
theory performed by Van Hove [2,3] allows to over-
come in a natural way some difficulties one faces in the
few-body physics (see, e.g., [4]). Namely, e.g., the pri-
mary interaction vertex usually include particles which
do not stay simultaneously on their mass shells, there-
fore the energies of intermediate states in some process
can take on arbitrary values. That is why the account for
the relativistic effects off the energy shell becomes of
high importance while interpreting experimental data
for the few-nucleon systems in wide range of energies,
including bound states (see, e.g., [5]).
The clothing procedure carried out via the unitary
transformation provides the transition from the repre-
sentation of the initial “bare” particles and interactions
towards the representation of the “clothed” particles
with observable properties and physical (observed) in-
teractions between them. As the byproducts of clothing,
the mass and vertex renormalization programs are per-
formed alongside the construction of the operators of
relativistic interactions being Hermitian, energy inde-
pendent and containing off-energy-shell structures in a
natural way.
2. UNITARY CLOTHING
TRANSFORMATION
The starting point of our consideration is the repre-
sentation of bare particles with physical masses [6]:
( ) ( ) ( )0 0F IH H Hα α= + 0α
)0ren
( ) ( ) ( ) (0 0 0F renH V M Vα α α α= + + + , (1) O W
where is the free part of Hamiltonian, V is the pri-
mary interaction operator, and V are the usual
mass and vertex renormalization counterterms. Symbol
denotes the set of creation/destruction operators of
bare particles with physical masses.
FH
renM ren
0α
By definition, the one-bare-particle states †
0α Ω
which are generated from the vacuum state Ω by bare
creation operators α are the eigenstates of the free part
of Hamiltonian: However, due to the presence of inter-
action, the same one-particle states are not the eigen-
states of the total Hamiltonian:
†
0
It is natural to question whether it is possible to find
a new set of creation/destruction operators α in terms
of which both free and total Hamiltonians would satisfy
the requirements:
c
( ) †
F c c cH Eα α αΩ = Ω† ; (2)
( ) †
c c c cH Eα α αΩ = Ω†
)α
)
)cα
. (3)
The set of operators α called clothed corresponds
to particles supposed to have observable properties.
Here we assume subscript “c” for the Hamiltonian in
terms of clothed particles to emphasize different de-
pendence of the same Hamiltonian on particle opera-
tors:
c
( ) (0 c cH Hα = . (4)
In order to keep observables unchanged (i.e., the S-
operator intact) Greenberg and Schweber assumed the
transformation which would carry out the transition to-
wards the representation of “clothed” particles to be one
of a unitary kind:
( ) (†
0 c c cW Wα α α α= , WW , † † 1W W= =
( ) ( )cR
cW e αα = , . (5) ( ) (†
cR Rα = −
The transition between bare and clothed particle rep-
resentations for an arbitrary operator O having polyno-
mial dependence on the creation/destruction operators is
fulfilled in the following manner:
( ) ( ) ( ) ( ) ( ) ( ) ( )†
0
c cR R
c c c cO W e O eα αα α α α α −= =
( ) ( ) ( )
1
1 ,
!
k
c c c
k
O R O
k
α α α
∞
=
= + ∑ , (6)
where we adopt the denotation for the multiple commu-
tator:
[ ], , ,... ,k
k
R O R R R O ≡
... . (7)
Applying the transition recipe (6) to the total Hamil-
tonian operator (1), we find:
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (1), p. 42-46. 42
( ) ( ) ( ) ( ) ( )c c F c c ren c ren cH H V M Vα α α α= + + + α
( ) ( ) ( )( )
1
1 ,
!
k
c F c c ren ren
k
R H V M V
k
α α α
∞
=
+ + + +∑
)
)
α
V
. (8)
If it is supposed that total Hamiltonian (8) satisfies
the requirements (2) and (3) the generator R has to be
chosen in such a way that the former does not contain
terms, called “bad”, which simultaneously do not con-
serve the number of particles (e.g., ) and prevent
the one-particle states to be the eigenstates of the total
Hamiltonian (e.g., V).
renM
Extracting, collecting and removing bad terms of the
increasing orders in g, we automatically derive the mass
and charge shifts and construct the operators of relativ-
istic interactions being Hermitian, energy independent
and containing off-energy-shell structures in a natural
way.
3. MASS AND VERTEX
RENORMALIZATION PROGRAM
To be more specific, we are going to consider the
bad terms elimination procedure in the few lowest or-
ders in g. To make the following derivations more
transparent, it is convenient to separate several types of
operators appearing in . We shall call “transi-
tion” the operators, denoted as and O of the
order, which consist of more than three crea-
tion/destruction operators of any kind. Subscripts “g”
and “b” mark “good” operators which refer to the physi-
cal processes and “bad” operators which prevent the
one-particle states to be the eigenstates of the total
Hamiltonian, respectively. The notations O and
will be used for the “mass-“ and “vertex-like” op-
erators of the order which replicate the structures of
the mass and vertex counterterms and V , re-
spectively. Assuming the latter being expanded in or-
ders of g: , , we
expect the mass and charge corrections to have the same
expansions.
(c cH α
( )2
1
k
renM M
( )
,
n
t gO
M
renV
( )
,
n
t b
(
r
n
M
re
(2k
renV +
ng
(
r
n
VO
)
n
1
)
ng
ren
ren
k
∞
=
= ∑
k
∞
=
= ∑
1
For example, in the model of interacting nucleons
and mesons, in which b ( ) and ( ) are the nu-
cleon and antinucleon creation (destruction) operators
while ( ) state for the mesonic creation (destruc-
tion) operators respectively, the term is the
bad transition operator, b d is of the good transi-
tion type, and are the mass- and vertex-like
operators respectively.
†
†a
b
† †
†d
b
d
†b
†a a
†b
† † †a a a
bd
b †d d
Taking explicitly few first terms from Hamiltonian
(8), we have:
( ) ( ) [ ] ( ),c c F c F cH H R H Vα α= + +
( ) [ ],ren cM Rα+ +
( ) [ ]21 , ,
2ren c renV R V R Mα + + + ...+
0=
. (9)
The Hamiltonian operator (9) is expected to contain
bad terms of all orders in g. Thus, the generator R of the
unitary clothing transformation, which is aimed at
eliminating them, is supposed to be expanded
in orders of g and to have the same struc-
tures as “bad” terms contained in Hamiltonian.
( )
1
k
k
R R
∞
=
= ∑
In the wide class of field-theoretical models the pri-
mary interaction operator V consists totally of the
order bad terms . Therefore, we are going to define
the generator in the following way:
1g
( )1
bH
( )1R
( ) ( )1 1 , FbH R H +
. (10)
Under this requirement, leaving terms up to the third
order in g and baring in mind the notation (7), we find:
( ) ( ) ( ) ( ) ( )2 11, ,
2c c F c F reH H R H R V Mα α = + + +
2
n
( ) ( ) ( ) ( ) ( )23 1 1 2 31, , ,
3F renR H R V R M V + + + + +
..ren
. . (11)
To proceed in defining R generator, it is necessary
now to collect all of good and bad terms of the or-
der. The order commutator contains the
transition good part the operators of which are respon-
sible for the physical (observable) interactions between
physical particles in the second order [4] and the mass-
like good part the operators of which replicate struc-
tures of the good part of . Besides, this commuta-
tor contains the transition bad part and the mass-like
bad part which replicates structures of the bad part of
.
2g
2g ( )1 ,R V
( )2
renM
( )2
renM
Collecting good mass-like operators, we assume the
following equation from which the mass corrections
(contained in the ) can be obtained:
2g
( )2
,ren gM
( ) ( )2 1
,
,
1 ,
2 r
ren g
M g
M R V +
0= . (12)
At the same time, applying the result for the mass
shifts to the bad mass-like operators (in order), we
find, that in general it appears:
2g
( ) ( )2 1
, ,, ,
1 , 0
2 r
ren b restren b M b
M R V M + ≡
≠ , (13)
see Ref. [6,7] Using the outcome of the first step of
mass renormalization (12) and (13), we can rewrite
Hamiltonian (11) in the form containing only those bad
operators of the order which are intended to be
eliminated via the second clothing:
2g
( ) ( ) ( )1
,
1 ,
2c c F c
t g
H H R Vα α = +
43
( ) ( ) ( )22 1
, ,,
1, ,
2F ren b restt b
R H R V M + + +
( ) ( ) ( ) ( ) ( )23 1 1 21, , , ...
3F renR H R V R M V + + + +
3
ren +
0=
. (14)
The generator can be defined now in the simi-
lar way as :
( )2R
R( )1
( ) ( )2 2 , FbH R H +
, (15) i
where ( ) ( ) ( )2 1
, ,,
1 ,
2b ren b restt b
H R V M = +
2 . Thus, after
the second clothing the total Hamiltonian reaches the
form which contains only good transition operators in
the order: 2g
( ) ( ) ( ) ( ) ( ) 21 3 1
,
1 1, ,
2 3c c F c F
t g
H H R V R H R Vα α = + + +
( ) ( ) ( ) ( ) ( ) ( )1 2 3 4 4 4, , ...ren ren F renR M V R H M T + + + + + +
,
, (16)
where operators of the fourth order in g are extracted:
( ) ( ) ( ) ( )34 1 2 1
,
1 1, , ,
8 2 t g
T R V R R V
= +
( ) ( ) ( ) ( ) ( ) ( )222 1 2 11 1, ,
2 2 ren renbR H R M R V + + +
3,
,
.(17)
To define the generator we have to collect op-
erators of the third order in g. The commutator
can be expanded as:
while is assumed
to have only the vertex-like part. The charge shift in the
order can be obtained via collecting vertex-like op-
erators:
( )3R
( ) 21 ,R V
( )R V
3g
( ) 21 ,R V
( ) ( )1 , renR M
( ) 21
,
,
t g
R V =
2
( )2 21 1
,
,
rt b V
R V+ +
( ) ( ) ( ) ( ) ( )2 31 1 2 3
,
1 , ,
3 r r
ren ren ren rest
V V
R V R M V V + + ≡
0≠ .(18)
After renormalizing the charge, we are allowed to
extract all of the bad terms in the order 3g
( ) ( ) ( )23 31
,
,
1 ,
3 ren restb t b
H R V V = +
, and define the : ( )3R
( ) ( )3 3 , FbH R H +
0=
kk
. (19)
Contrary to the Dyson-Feynman approach, the illus-
trated clothing procedure has a recursive character. It
means that the structure of Hamiltonian in some n-th
order in g can not be specified until all the corrections
of physical constants of n lower orders are fixed and all
the bad operators of all n lower orders are removed.
Thus, depending on how we determine the operators to
remove (those are “bad” after Ref. [1] in our case) and
choose the primary interaction, the operators corre-
sponding to physical (observed) processes can acquire
quite different forms.
4. FIELD THEORETICAL MODEL
Let us implement the developed technique in the
simple model of quantum field theory including scalar
mesons and spinless charged nucleons. The interaction
operator is chosen in the form of the Yukawa-type
three-linear interaction. In this model the explicit de-
pendencies of the operators entering the total Hamilto-
nian on the creation/destruction operators are as fol-
lows:
††
1,1
i i
FH d E F F d a aω
=−
= +∑∫ ∫q q q k kq ; (20)
( ) ( )
( ) † †
3/ 2 1/ 2
,2 8
. .
i j
i j
g d d dV F
E E
H c
δ
π ω
= −
+
∑∫ p q k
p q k
p q k p-q+k F a
,
;(21)
,ren ren mes ren nuclM M M= + ; (22)
( )
2
† † †
, . .
4ren mes
dM a a a aδµ
ω
= +∫ k k k -k
k
k H c+ ; (23)
2
†
,
, 1,1
.
8
i j
ren nucl
i j
m dM F
E
δ
=−
= +∑∫ q q
q
q . ; (24) F H c
( )
( )
( )3/ 2 1/ 22 8
ren
gV d d d
E E
δδ
π ω
= − ∫
p q k
p - q + k
p q k
† †
, 1,1
.i j
i j
F F a H c
=−
× ∑ p q k . , (25) +
2
2
where δµ states for the mesonic mass shift
with as the physical (observable) mass and as
the bare (trial) one; δ = is the nucleonic
mass shift, and m are the physical (observable)
and bare (trial) nucleonic masses, respectively;
is the charge shift depending on the
physical charge and the trial one .
2 2
0µ µ= −
µ
m
0 g−
g
0µ
g
2 2
0m m m−
0
g gδ =
0
2 2E m= +p
p
p is the energy of a nucleon with the
momentum , 2 2ω µ= +kk is the energy of a
meson with the momentum k .
In Eqs. (20)–(25) we adopt the denotations:
†
† 1,
1,
i b i
F
d i−
==
= −
q
q
q
(26) †
1,
1,
i
b i
F
d i−
==
= −
q
q
q
where and are the creation (destruc-
tion) operators of nucleon and antinucleon with the
momentum . Operators and satisfy the fol-
lowing commutation relations:
(†b bq q
q
) )
)
)
)
(†d dq q
iFp
† iFq
(†,i j
ijF F iδ δ = p q p - q , i, j = 1, –1, (27)
which follow from the usual commutation relations for
the creation/destruction operators of bosons:
and .
is the creation (destruction) operator of a meson with
the momentum :
(†,b b δ = p q p - q
k
( )†,d d δ = p q p - q ( )†a akk
(†,a a δ′
′= k k k - k . (28)
44
5. CHARGE CORRECTION
With help of Eq. (10) the generator acquires
the following form replicating the structure of :
( )1R
( )1
bH
( )
( ) ( )
( )1
3/ 2 1/ 22 8
g d d dR
E E
δ
π ω
= ∫
p q k
p q k p - q + k
† †
, 1, 1
1 . .i j
i j
F F a H c
iE jE ω= −
× −
− +∑ p q k
p q k
(29)
Calculating the commutators and
in our model and separating their vertex-
like parts which enter the Eq. (18), we can derive the
expression for the charge shift in the order:
( ) 21 ,R V
3g
( ) ( )1 2, renR M
( )
( )
3
3
9 / 28 2
g dg
E E
δ
ωπ ′ ′
′
= ∫
p-k q-k k
k
{ 1, 1 1, 1 1, 1, ,D D D− − −
′ ′ ′ ′× ∆ −q-k ,q,k p-k ,q-k ,k p-k ,p
′
)1, 1 1, 1 1, 1− − − − −
)
)
( ) ( , ,D D D′ ′ ′ ′ ′ ′ ′ ′+ ∆ − −,k q-k ,q,k p-k ,p,k p-k ,q-k ,k
( 1, 1 1, 1 1,1, ,D D D− − − −
′ ′ ′ ′ ′ ′− p-k ,p,k q-k ,q,k p-k ,q-k ,k
( )1,1 1,1 1,1
p-k ,p,k, ,D D D′ ′ ′ ′ ′ ′+∆ − + ∆q-k ,q,k p-k ,q-k ,k
( ) ( }1,1 1,1 1, 1 1,1 1,1 1,1, , , ,D D D D D D− − − −
′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′+∆ − − + ∆ −p-k ,p,k q-k ,q,k p-k ,q-k ,k q-k ,q,k p-k ,p,k p-k ,q-k ,k , (30)
where we adopt the denotations: ,
, ,
1i jD
iE jE ω
=
+ +p q k
p q k
and ( ) 1 1 1 2, ,
3
a b c
ab bc ac
∆ = + −
.
Each of the items in Eq. (30) corresponds to one of the six mechanisms responsible for the charge renormaliza-
tion in the third order in g (see Fig.).
(a) (d)
(b) (e)
(c) (f)
Six mechanisms responsible for the charge renormalization in the third order in g
Namely, the first item refers to the diagram a, the
second item refers to the diagram b, etc. The directions
of arrows on these graphs differ particles from antipar-
ticles.
Each of the non-covariant propagators cor-
responds to the vertex on the respective diagram where
the energy conservation is not assumed. Thus, the
,
, ,
i jDp q k
45
charge shift, which is determined via the cancellation of
the vertex-like operators being the off-energy-shell
quantities, appears off the energy shell too.
Therefore, it is important to note that the expression
for the charge shift can be presented as the following
decomposition:
( ) ( ) ( )3 33
Feynman like off energy shellg g gδ δ δ− −= + − , (31)
where the “off-energy-shell” part goes to zero on the
energy shell.
The “Feynman-like” part can be brought to the ex-
plicitly covariant form on the energy shell, providing
the momentum independence of the charge shift derived
and giving another representation for that shift obtained
within the Dyson-Feynman approach:
( )
( ) ( )
3
3
3 2 2 2
1 1
2 (2 ) 2 2 2Feynman like
g dg
E p k m p p
δ
π µ µ−
′
′
= ′ ′− − −
∫
p
p
( ) ( ) ( ) ( )
3 3
2 2 2 2 2
1 1
2 2 2 2 2
d d
Ek p k q p k m p qω µ µ µ µ′ ′
′ ′ − +
′ ′ ′ ′+ + + − +
∫ ∫
k p
k p
, (32)
where , , , ,
. The momentum conservation for that vertex
has the form .
( ),qq E= q
( ),k′ ′k
q p
( ),pp E= p
= + k
( ),pp E ′′ ′= p ( ),kk ω= k
k ω′=
6. CONCLUSION
The charge shift in the third order in the coupling con-
stant g is obtained as the byproduct of the clothing proce-
dure by means of collecting operators off the energy shell.
Six mechanisms of the charge renormalization in the third
order are generated by the products of non-covariant
propagators, typical of the old-fashioned perturbation the-
ory, forming the expression for the charge correction.
Each of these propagators marks the vertex on the respec-
tive diagram in which the energy conservation is not as-
sumed.
Being an object off the energy shell, the expression
for the charge shift acquires the explicitly covariant
form on the energy shell, giving another representation
to the respective Dyson-Feynman result and providing
the momentum independence of the charge shift.
Having a recursive feature, the clothing procedure
gives an expectation that the account for operators off
the energy shell in the Hamiltonian could lead to new
physical results in higher orders in the coupling con-
stant, just to mention the problem of calculating the
πNN form-factors in nuclear physics.
REFERENCES
1. S. Greenberg, O. Schweber. Clothed particle operators
in simple models of quantum field theory //Nuovo
Cim. 1958, v. 8, p. 378-406.
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ОДЕВАНИЕ ВЕРШИНЫ В КВАНТОВОЙ ТЕОРИИ ПОЛЯ
В.Ю. Корда, И.В. Елецких
С помощью метода унитарного одевающего преобразования изучена проблема перенормировки верши-
ны в квантовой теории поля. В модели, описывающей заряженное бесспиновое нуклонное и скалярное ме-
зонное поля, взаимодействующие посредством трилинейной связи типа Юкавы, получено выражение для
сдвига заряда в третьем порядке по константе связи. Будучи величиной вне энергетической оболочки, най-
денное выражение может быть представлено в явно ковариантной форме на энергетической оболочке, обес-
печивая независимость перенормировки заряда от импульсов частиц.
ОДЯГАННЯ ВЕРШИНИ В КВАНТОВІЙ ТЕОРІЇ ПОЛЯ
В.Ю. Корда, І.В. Єлецьких
За допомогою методу унітарного одягаючого перетворення досліджено проблему перенормування вер-
шини в квантовій теорії поля. У моделі, яка описує заряджене безспінове нуклонне і скалярне мезонне поля,
що взаємодіють через трилінійний зв’язок типу Юкави, знайдено вираз для зсуву заряду в третьому порядку
за константою зв’язку. Розрахований вираз визначено поза енергетичною оболонкою, проте його можна по-
дати в явно коваріантній формі на енергетичній оболонці, що забезпечує незалежність перенормування за-
ряду від імпульсів частинок.
46
|
| id | nasplib_isofts_kiev_ua-123456789-110914 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:50:31Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Korda, V.Yu. Yeletskikh, I.V. 2017-01-06T19:48:25Z 2017-01-06T19:48:25Z 2007 Vertex clothing in quantum field theory / V.Yu. Korda and I.V. Yeletskikh // Вопросы атомной науки и техники. — 2007. — № 3. — С. 42-46. — Бібліогр.: 7 назв. — рос. 1562-6016 PACS: 21.45.+v; 21.40. Jv; 11.80.-m https://nasplib.isofts.kiev.ua/handle/123456789/110914 The problem of the vertex renormalization in quantum field theory is tackled via the implementation of the unitary clothing transformation method. In the model of charged spinless nucleon and scalar meson fields coupled by the Yukawa-type three-linear interaction the expression for the charge correction in the first non-vanishing (third) order in the coupling constant is derived. Being the off-energy-shell quantity, the expression can be brought to the explicitly covariant form on the energy shell, providing the momentum independence of the charge renormalization. Метод унітарних одягаючих перетворень застосовано в моделі квантової теорії поля, в якій нуклонне і нейтральне піонне поля взаємодіють через регуляризований зв’язок типу Юкави. В цьому підході масові контрчлени частково скорочуються з комутаторами генераторів одягаючих перетворень і операторів взаємодії, що формує піонні та нуклонні зсуви мас. Знайдені величини подаються тривимірними інтегралами від деяких коваріантних комбінацій відповідних імпульсів частинок, що забезпечує незалежність розрахованих поправок від імпульсів. Визначено умови, що висуваються до вершинних функцій, які здійснюють регуляризацію зв’язку полів. Метод унитарных одевающих преобразований применен в модели квантовой теории поля, в которой нуклонное и нейтральное пионное поля взаимодействуют посредством регуляризованной псевдоскалярной связи типа Юкавы. В этом подходе массовые контрчлены частично сокращаются с коммутаторами генераторов одевающих преобразований и операторов взаимодействия, формируя пионные и нуклонные сдвиги массы. Найденные величины выражаются трехмерными интегралами от некоторых ковариантных комбинаций соответствующих импульсов частиц, что обеспечивает независимость рассчитанных поправок от импульсов. Определены условия, налагаемые на обрезающие вершинные функции, осуществляющие регуляризацию связи полей. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Quantum field theory Vertex clothing in quantum field theory Зображення одягнених частинок в квантовій теорії поля: перенормування маси Представление одетых частиц в квантовой теории поля: перенормировка массы Article published earlier |
| spellingShingle | Vertex clothing in quantum field theory Korda, V.Yu. Yeletskikh, I.V. Quantum field theory |
| title | Vertex clothing in quantum field theory |
| title_alt | Зображення одягнених частинок в квантовій теорії поля: перенормування маси Представление одетых частиц в квантовой теории поля: перенормировка массы |
| title_full | Vertex clothing in quantum field theory |
| title_fullStr | Vertex clothing in quantum field theory |
| title_full_unstemmed | Vertex clothing in quantum field theory |
| title_short | Vertex clothing in quantum field theory |
| title_sort | vertex clothing in quantum field theory |
| topic | Quantum field theory |
| topic_facet | Quantum field theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110914 |
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