A nonlocal extension of the Wentzel field model in the clothed-particle representation
The clothed particle approach is applied to express the total Hamiltonian of interacting fields in terms of clothed particles. In order to avoid ultraviolet divergences typical of many field theories we introduce some covariant cutoff functions in momentum space in the Wentzel field model. We will s...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2012 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2012
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| Цитувати: | A nonlocal extension of the Wentzel field model in the clothed-particle representation / A.V. Shebeko, P.A. Frolov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 64-69. — Бібліогр.: 5 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859659843870130176 |
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| author | Shebeko, A.V. Frolov, P.A. |
| author_facet | Shebeko, A.V. Frolov, P.A. |
| citation_txt | A nonlocal extension of the Wentzel field model in the clothed-particle representation / A.V. Shebeko, P.A. Frolov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 64-69. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The clothed particle approach is applied to express the total Hamiltonian of interacting fields in terms of clothed particles. In order to avoid ultraviolet divergences typical of many field theories we introduce some covariant cutoff functions in momentum space in the Wentzel field model. We will show how in the framework of the nonlocal meson-boson field model one can build interactions between the clothed mesons and bosons. Moreover, the mass renormalization terms, that are compulsory to ensure the relativistic invariance of the theory as a whole (in Dirac's sense), turn out to be expressed through certain covariant integrals. They are convergent in the field model with appropriate cutoff factors.
Для того чтобы избежать ультрафиолетовых расходимостей, типичных для многих полевых теорий, вводятся ковариантные обрезающие функции в импульсном пространстве в модели Вентцеля. Показано, каким образом в рамках нелокальной мезон-бозонной полевой модели можно построить взаимодействия между одетыми мезонами и бозонами. Кроме того, массовые перенормировочные члены, которые обязательны для обеспечения релятивистской инвариантности теории в целом (по Дираку), оказываются выраженными через определенные ковариантные интегралы. Эти интегралы сходятся в полевой модели с соответствующими обрезающими функциями.
Щоб уникнути ультрафіолетових розбіжностей, типових для багатьох польових теорій, вводяться коваріантні обрезаючі функції в імпульсному просторі в моделі Вентцеля. Показано, як в рамках нелокальної мезон-бозонної польової моделі можна побудувати взаємодії між одягненими мезонами і бозонами. Крім того, масові перенорміровочні члени, які обов'язкові для забезпечення релятивістської інваріантності теорії у цілому (за Діраком), виявляються вираженими через певні коваріантні інтеграли. Дані інтеграли збігаються в польовій моделі з відповідними обрезаючими функціями.
|
| first_indexed | 2025-11-30T09:40:46Z |
| format | Article |
| fulltext |
A NONLOCAL EXTENSION OF THE WENTZEL FIELD
MODEL IN THE CLOTHED-PARTICLE REPRESENTATION
A.V. Shebeko 1∗and P.A. Frolov 2
1National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
2Institute of Electrophysics & Radiation Technologies, NAS of Ukraine, 61002, Kharkov, Ukraine
(Received October 31, 2011)
The clothed particle approach is applied to express the total Hamiltonian of interacting fields in terms of clothed
particles. In order to avoid ultraviolet divergences typical of many field theories we introduce some covariant cutoff
functions in momentum space in the Wentzel field model. We will show how in the framework of the nonlocal
meson-boson field model one can build interactions between the clothed mesons and bosons. Moreover, the mass
renormalization terms, that are compulsory to ensure the relativistic invariance of the theory as a whole (in Dirac’s
sense), turn out to be expressed through certain covariant integrals. They are convergent in the field model with
appropriate cutoff factors.
PACS: 11.10.Ef, 11.10.Gh, 11.10.Lm, 11.30.Cp
1. INTRODUCTION
Following our recent work [1] we will show how an
algebraic approach proposed there for constructing
the generators of the Poincaré group can be realized
within a nonlocal extension of the so-called Wentzel
model. Our departure point is a nonlocal Hamil-
tonian for interacting fields, that can be built up
by introducing some “cutoff” function (shortly the
g-factor) in every vertex which is associated with par-
ticle creation and/or annihilation. As usually, such
g-factors are needed, first of all, to carry out finite in-
termediate calculations trying to remove ultraviolet
divergences inherent in local field models. However,
in the instant form of relativistic dynamics used here
it is very important to take into account certain con-
straints imposed upon such cutoffs to meet require-
ments of special relativity and other symmetries, e.g.,
with respect to charge conjugation, space inversion
and time reversal.
We have managed to do it [1] by defining a covari-
ant generating function for the cutoffs in case of trilin-
ear Yukawa-type couplings. The function, being de-
pendent on some Lorentz scalars composed of the par-
ticle three-momenta, plays a central role when inte-
grating the Poincaré commutators to derive then the
clothed-particle representation (CPR) expressions for
the Hamiltonian, the boost operators, the mass renor-
malization terms and so on accordingly [2].
Moreover, it is expected that by choosing the g-
factors in a proper way (for instance, as square inte-
grable functions of particle momenta) one can get rid
of certain drawback of field models with local inter-
actions (see [1]).
2. METHOD OF UNITARY CLOTHING
TRANSFORMATIONS
As before (see, e.g., [3]), let us remind that the UCT
method exposed in [1–4] is aimed to express a given
field Hamiltonian
H ≡ H(α) = HF (α) + HI(α)
= W (αc)H(αc)W †(αc) ≡ K(αc), (1)
primarily dependent on the α set of “bare” parti-
cle creation and annihilation operators, through their
“clothed” counterparts αc via the unitary transfor-
mation W . The latter removes from the interaction
V (α) that enters HI(α) = V (α) + Vren(α) the so-
called “bad” terms. By definition, such terms prevent
the physical vacuum |Ω〉 (the H lowest eigenstate)
and the one-clothed-particle states |n〉c = a†
c(n)|Ω〉
to be the H eigenvectors for all n included. The bad
terms occur every time when any normally ordered
product
a†(1′)a†(2′)...a†(n′
C)a(nA)...a(2)a(1)
of the class [C.A] embodies, at least, one substructure
which belongs to one of the classes [k.0] (k = 1, 2, ...)
and [k.1] (k = 0, 1, ...). Our consideration is focused
upon various field models (local and nonlocal) in
which the interaction density HI(x) consists of scalar
Hsc(x) and nonscalar Hnsc(x) contributions:
HI(x) = Hsc(x) + Hnsc(x), (2)
where the property to be a scalar means
UF (Λ)Hsc(x)U−1
F = Hsc(Λx), ∀x = (t,x) (3)
for all Lorentz transformations Λ.
∗Corresponding author E-mail address: shebeko@kipt.kharkov.ua
64 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 64-69.
Therefore, we have
HI(α) =
∫
HI(x)dx = Hsc(α) + Hnsc(α), (4)
Hsc(nsc)(α) =
∫
Hsc(nsc)(x)dx,
Hsc(α) = Vbad(α) + Vgood(α)
to eliminate the bad part Vbad from the similarity
transformation
K(αc) = W (αc)[HF (αc) + HI(αc)]W †(αc)
= W (αc)[HF (αc) + Vbad(αc) (5)
+Vgood(αc) + Hnsc(αc)]W †(αc).
For the unitary clothing transformation (UCT) W =
expR with R = −R† it is implied that we will elimi-
nate the bad terms Vbad in the r.h.s. of
K(αc) = HF (αc) + Vbad(αc) + [R, HF ]
+[R, Vbad] +
1
2
[R, [R, HF ]] (6)
+
1
2
[R, [R, Vbad]] + ... + eRVgoode
−R + eRHnsce
−R
by requiring that
[HF , R] = Vbad (7)
for the operator R of interest.
One should note that unlike the original clothing
procedure we eliminate here the bad terms only from
Hsc interaction in spite of such terms can appear in
the nonscalar interaction as well (details in [5]).
Now, we get the division
H = K(αc) = KF + KI (8)
with a new free part KF = HF (αc) ∼ a†
cac and inter-
action
KI = Vgood(αc) + Hnsc(αc) + [R, Vgood]
+
1
2
[R, Vbad] + [R, Hnsc] +
1
3
[R, [R, Vbad]] + ..., (9)
where the r.h.s. involves along with good terms other
bad terms to be removed via subsequent UCTs.
3. A NONLOCAL EXTENSION OF THE
WENTZEL FIELD MODEL
As an illustration, let us consider the field model of
“scalar nucleons” (more precisely, charged spinless
bosons) and neutral scalar bosons, in which
HI = Vnloc + Ms + Mb (10)
with the normally ordered interaction
Vnloc =
1
2[2(2π)3]1/2
∫
dp′
Ep′
∫
dp
Ep
∫
dk
ωk
×{δ(p′ − p − k)g11(p′, p, k)b†(p′)b(p)a(k)
+ δ(p′ + p − k)g12(p′, p, k)b†(p′)d†(p)a(k) (11)
+ δ(p′ + p + k)g21(p′, p, k)d(p′)b(p)a(k)
+ δ(p′ − p− k)g22(p′, p, k)d†(p′)d(p)a(k)} + H.c.
Adopting the convention
[b†(p′), d(p′)]
[
X11(p′, p) X12(p′, p)
X21(p′, p) X22(p′, p)
] [
b(p)
d†(p)
]
= F †
ε′(p′)Xε′ε(p′, p)Fε(p) ≡ F †
b (p′)X(p′, p)Fb(p)
(12)
we can write in more compact form
Vnloc = Vb + V †
b , Vb =
∫
dk
ωk
: F †
b G(k)Fb : a(k).
Matrix G(k) is composed of elements
Gε′ε(p′, p, k) =
1
2[2(2π)3]1/2
ḡε′ε(p′, p, k)
× δ(k− (−1)εp + (−1)ε′
p′), (ε′, ε = 1, 2), (13)
where ḡε′ε(p′, p, k) coincide with gε′ε(p′, p, k) except
ḡ22(p′, p, k) = g22(p, p′, k).
It is implied that operators a(a†), b(b†) and d(d†)
meet commutation relations
[a(k), a†(k′)] = k0δ(k − k′), (14)
[b(p), b†(p′)] = [d(p), d†(p′)] = p0δ(p − p′), (15)
with all the remaining ones being zero. Here k0 =
ωk =
√
k2 + μ2
s (p0 = Ep =
√
p2 + μ2
b) is the energy
of the neutral (charged) particle with mass μs(μb).
For our nonlocal model we will retain the property to
be Lorentz scalar assuming
UF (Λ)Vnloc(x)U−1
F (Λ) = Vnloc(Λx). (16)
It is readily seen that this relation holds if the coeffi-
cients gε′ε meet the condition
gε′ε(Λp′, Λp, Λk) = gε′ε(p′, p, k). (17)
On the mass shell with p′2 = p2 = μ2
b and k2 = μ2
s
the latter means that functions gε′ε(p′, p, k) can de-
pend only upon invariants p′p, p′k, pk.
These cutoffs are subject to other constraints im-
posed by different symmetries. For example, invari-
ance of the hermitian operator Vnloc with respect to:
i) space inversion; ii) time reversal and iii) charge
conjugation yields the relations:
gε′ε(p′, p, k) = gε′ε(p, p′, k), ε′ �= ε (18)
gε′ε(p′, p, k) = gε′ε(p′−, p−, k−), (19)
g11(p′, p, k) = g22(p′, p, k). (20)
“Mass renormalization” terms Ms and Mb can be rep-
resented in the form:
Ms =
∫
dk
ω2
k
{m1(k)a†(k)a(k)
+ m2(k)[a†(k)a†(k−) + a(k)a(k−)]} (21)
and
Mb =
∫
dp
E2
p
{m11(p)b†(p)b(p) + m12(p)b†(p)d†(p−)
+ m21(p)b(p)d(p−) + m22(p)d†(p)d(p)}, (22)
where the coefficients m1,2(k) and mε′ε(p′, p), being
for the time unknown, may be momentum dependent.
65
4. GENERATORS FOR CLOTHED
PARTICLES. ELIMINATION OF BAD
TERMS
At this point we will come back to our model with
Vbad = Vnloc, Vgood = 0 and R = Rnloc to calculate
the simplest commutator [Rnloc, Vnloc] in which the
clothing operator Rnloc is determined by
[HF , Rnloc] = Vnloc. (23)
From the equation it follows that its solution can be
given by
Rnloc =
∫
dk
ωk
: F †
b R(k)Fb : a(k) − H.c.
= Rnloc −R†
nloc. (24)
The matrix R(k) is composed of the elements:
Rε′ε(p′, p, k) = − ḡε′ε(p′, p, k)
ωk + (−1)ε′Ep′ − (−1)εEp
× δ(k + (−1)ε′
p′ − (−1)εp) (ε′, ε = 1, 2). (25)
Such a solution is valid if μs < 2μb. In other words,
under such an inequality the operator Rnloc has the
same structure as Vnloc itself. After the normal or-
dering of meson and boson operators in commutator
[Rnloc, Vnloc] one can obtain the 2 → 2 interactions
of the type b†a†ba, d†a†da, b†d†aa, a†a†bd and b†b†bb,
b†d†bd, d†d†dd.
For example, the boson-boson interaction opera-
tor can be represented as
1
2
[Rnloc, Vnloc](bb → bb)
= −1
4
∫
dp′
2
Ep′
2
∫
dp2
Ep2
∫
dp′
1
Ep′
1
∫
dp1
Ep1
× δ(p′
1 + p′
2 − p1 − p2)
× g11(p′1, p1, k)g11(p′2, p2, k)
×
{
1
(p1 − p′1)2 − μ2
s
+
1
(p2 − p′2)2 − μ2
s
}
× b†c(p
′
2)b
†
c(p
′
1)bc(p2)bc(p1) (26)
with k = p′
1 − p1. In these equations we meet a
covariant (Feynman-like) “propagator”
1
2
{
1
(p1 − p′1)2 − μ2
s
+
1
(p2 − p′2)2 − μ2
s
}
, (27)
which on the energy shell
Ep1 + Ep1 = Ep′
1
+ Ep′
2
(28)
is converted into the genuine Feynman propagator for
the corresponding S matrix.
5. MASS RENORMALIZATION AND
RELATIVISTIC INVARIANCE
We have seen how in the framework of the nonlo-
cal meson-boson model one can build the 2 → 2 in-
teractions between the clothed mesons and bosons.
They appear in a natural way from the commuta-
tor 1
2 [Rnloc, Vnloc] as the operators b†a†ba, d†a†da,
b†b†bb, b†d†bd, d†d†dd, b†d†aa, a†a†bd of the class
[2.2]. Moreover, this commutator is a spring of the
good operators a†a, b†b and d†d of the class [1.1] to-
gether with the bad operators aa and bd of the class
[0.2] and their hermitian conjugates a†a† and b†d† of
the class [2.0]. These operators may be cancelled by
the respective counterterms from
Hnsc(α) = Ms(α) + Mb(α). (29)
Let us show that such a cancellation gives rise to cer-
tain definitions of the mass coefficients. Indeed, one
can show that
1
2
[Rnloc, Vnloc](a†a)
= −1
2
∫
dk
ω2
k
∫
dp
EpEp−k
[
g2
21(p, q−, k−)
Ep + Ep−k + ωk
+
g2
12(p, q−, k)
Ep + Ep−k − ωk
]a†(k)a(k), (30)
where q = (Ep−k,p−k). In the same way we obtain
1
2
[Rnloc, Vnloc](aa)
=
∫
dk
ω2
k
∫
dp
Ep
g12(p, q−, k)g21(p, q−, k−)
×[
1
μ2
s + 2p−k
+
1
μ2
s − 2pk
]a(k)a(k−). (31)
Furthermore, assuming that
M (2)
s (α) +
1
2
[Rnloc, Vnloc]2mes = 0 (32)
with
[Rnloc, Vnloc]2mes = [Rnloc, Vnloc](a†a)
+[Rnloc, Vnloc](aa) + [Rnloc, Vnloc](a†a†),
we find
m
(2)
1 (k) =
1
2
∫
dp
EpEp−k
[
g2
21(p, q−, k−)
Ep + Ep−k + ωk
+
g2
12(p, q−, k)
Ep + Ep−k − ωk
], (33)
m
(2)
2 (k) = −
∫
dp
Ep
g12(p, q−, k)g21(p, q−, k−)
×[
1
μ2
s + 2p−k
+
1
μ2
s − 2pk
]. (34)
The operators that conserve the boson (antiboson)
number can be written as:
1
2
[Rnloc, Vnloc](b†b)
=
∫
dk
ωk
∫
dp
E2
pEp−k
[
g2
11(p, q, k)
Ep − Ep−k − ωk
− g2
21(p, q−, k−)
Ep + Ep−k + ωk
]b†(p)b(p), (35)
66
1
2
[Rnloc, Vnloc](d†d)
=
∫
dk
ωk
∫
dp
E2
pEp−k
[
g2
22(p, q, k)
Ep − Ep−k − ωk
− g2
21(p, q−, k−)
Ep + Ep−k + ωk
]d†(p)d(p). (36)
One can show that from the condition
M
(2)
b (α) +
1
2
[Rnloc, Vnloc]2bos = 0, (37)
where
[Rnloc, Vnloc]2bos
= [Rnloc, Vnloc](b†b) + [Rnloc, Vnloc](b†d†)
+[Rnloc, Vnloc](db) + [Rnloc, Vnloc](d†d),
it follows
m
(2)
11 (p) = −
∫
dk
ωkEp−k
[
g2
11(p, q, k)
Ep − Ep−k − ωk
− g2
21(p, q−, k−)
Ep + Ep−k + ωk
], (38)
m
(2)
22 (p) = −
∫
dk
ωkEp−k
[
g2
11(p, q, k)
Ep − Ep−k − ωk
− g2
21(p, q−, k−)
Ep + Ep−k + ωk
]. (39)
Similarly one can obtain the non-diagonal coefficients
m
(2)
12 (p) = m
(2)
21 (p)
= −
∫
dk
ωkEp−k
g11(p, q, k)g21(p, q−, k−)
×[
1
Ep − Ep−k − ωk
− 1
Ep + Ep−k + ωk
] (40)
or
m
(2)
12 (p) = m
(2)
21 (p)
= −
∫
dk
ωk
g11(p, q, k)g21(p, q−, k−)
×[
1
μ2
s − 2pk
+
1
μ2
s + 2p−k
]
−
∫
dq
Eq
g11(p, q, u)g21(p, q−, u−)
×(
1
2[μ2
b − pq] − μ2
s
+
1
2[μ2
b + pq−] − μ2
s
), (41)
where u = (Ep−q,p− q).
Thus the clothing procedure has allowed us to get
analytical expressions for the interaction operators
between the clothed particles. Moreover, we have
obtained some prescriptions when finding the coef-
ficients in the “mass renormalization” operators.
At last, one should emphasize that if one starts
from expansion
Hnsc(x) =
∞∑
p=2
H(p)
nsc(x) (42)
with the second-order contribution H
(2)
nsc = M
(2)
s +
M
(2)
s = 0, then the RI would be violated at the be-
ginning because of the obvious discrepancy between
[HF ,D(2)] = [NF , H(2)
nsc] + [NB, Hsc], (43)
and
[Pk, D
(p)
j ] = iδkjH
(p)
nsc, (p = 2, 3, ...). (44)
By using previous equations, we obtain
−
∫
x[HF , Hsc(x)]dx
= [HF ,NI ] + [HI ,NI ] + [Hnsc,NF ]. (45)
Evidently, this equation is fulfilled if we put
NI = NB ≡ −
∫
xHsc(x)dx, (46)
[Hsc,NI ] = −
∫
xdx
∫
dx′[Hsc(x′), Hsc(x)]
= [NF + NI , Hnsc]. (47)
In a model with Hnsc = 0 the latter reduces to∫
e−iPXIeiPXdX = 0, (48)
where
I =
1
2
∫
rdr[Hsc(
1
2
r), Hsc(−1
2
r)]. (49)
One should note that we have arrived to previous
equation being inside the Poincaré algebra itself with-
out addressing the Noether integrals.
At this point, we put NI = NB + D,
[HF ,D] = [NB + D, Hsc] + [NF + NB + D, Hnsc],
(50)
that replaces commutator [H,N] = iP and deter-
mines displacement D. Assuming that scalar density
Hsc(x) is of the first order in coupling constants in-
volved and putting
Hnsc(x) =
∞∑
p=2
H(p)
nsc(x), (51)
we will search operator D in the form:
D =
∞∑
p=2
D(p), (52)
i.e., as a perturbation expansion in powers of the in-
teraction Hsc. Here label (p) denotes the p-th order
in these constants. One should keep in mind that
higher (p ≥ 2) terms are usually associated with per-
turbation series for mass and vertex counterterms.
By substituting Hnsc and D we get the chain of
relations:
[HF ,D(2)] = [NB, Hsc] + [NF , H(2)
nsc], (53)
[HF ,D(3)] = [D(2), Hsc] + [NF , H(3)
nsc] + [NB, H(2)
nsc],
(54)
67
[P k, D(p)j ] = 0, (p = 2, 3, . . .) (55)
. . . . . . . . . . . .
Further, after such substitutions into the commuta-
tors
[Pk, Nj ] = iδkjH, [Jk, Nj ] = iεkjlNl,
[Nk, Nj ] = −iεkjlJl
we deduce, respectively, the following relations:
[Pk, D
(p)
j ] = iδkjH
(p)
nsc, (p = 2, 3, ...) (56)
[Jk, D
(p)
j ] = iεkjlD
(p)
l , (57)
[NFk, NBj ] + [NBk, NFj] = 0, (58)
[NFk, D
(2)
j ] + [D(2)
k , NFj] + [NBk, NBj ] = 0, (59)
[NFk, D
(3)
j ] + [D(3)
k , NFj ]
+ [NBk, D
(2)
j ] + [D(2)
k , NBj ] = 0,
(60)
(p = 2, 3, . . .).
6. DISCUSSION. TOWARDS WORKING
FORMULAE
We see that our algebraic approach in combination
with the UCT method makes our consideration more
and more appropriate for practical applications (in
particular, as one has to work with the vertex cut-
offs). The formulae for the 2 → 2 interactions become
more tractable if we assume that
gε′ε(p′, p, k)
= vε′ε([k+(−1)ε′
p′− (−1)εp][k− (−1)ε′
p′+(−1)εp]).
(61)
One can verify the nonlocal model with such cutoffs
possesses necessary properties. In terms of the vε′ε
functions we get
m
(2)
1 (k) =
1
2
∫
dp
EpEp−k
[
v2
21(ω
2
k − (Ep + Ep−k)2)
Ep + Ep−k + ωk
+
v2
12(ω
2
k − (Ep + Ep−k)2)
Ep + Ep−k − ωk
], (62)
m
(2)
2 (k) = −
∫
dp
Ep
v21(ω2
k − (Ep + Ep−k)2)
× v12(ω2
k − (Ep + Ep−k)2)
× [
1
μ2
s + 2p−k
+
1
μ2
s − 2pk
]. (63)
Now, by handling the charge-independent cutoffs,
v12(x) = v21(x) = f(x), (64)
we obtain
m
(2)
1 (k) = m
(2)
2 (k)
= −
∫
dp
Ep
f2(ω2
k − (Ep + Ep+k)2)
μ2
s + 2pk
−
∫
dp
Ep
f2(ω2
k − (Ep + Ep−k)2)
μ2
s − 2pk
. (65)
In other words, the option (64) yields the momentum-
independent coefficients m
(2)
1 (k) = m
(2)
2 (k) ≡ m
(2)
s .
Indeed, along with the Lorentz invariant denomina-
tors the integrand in the r.h.s. of (65) contains func-
tion f(I) whose argument
I(p,k) ≡ ω2
k − (Ep + Ep−k)2
= μ2
s − 2μ2
b − 2EpEp−k − 2p(p − k)
does not change under the simultaneous transforma-
tion p ⇒ p′ = Λp and p−k ⇒ Λ(p− k) on the mass
shells p2 = μ2
b and k2 = μ2
s. Now, we can reduce the
triple integral to the simple one:
m(2)
s = 8π
∫ ∞
0
t2dt√
t2 + μ2
b
f2(μ2
s − 4t2 − 4μ2
b)
4t2 + 4μ2
b − μ2
s
. (66)
Furthermore, it has turned out:
m
(2)
11 (p) = m
(2)
22 (p)
= −
∫
dk
ωkEp−k
[
v2
11(ω
2
k − (Ep − Ep−k)2)
Ep − Ep−k − ωk
−v2
21(ω
2
k − (Ep + Ep−k)2)
Ep + Ep−k + ωk
], (67)
m
(2)
12 (p) = m
(2)
21 (p)
= −
∫
dk
ωkEp−k
v11(ω2
k − (Ep − Ep−k)2)
×v21(ω2
k − (Ep + Ep−k)2)
×[
1
Ep − Ep−k − ωk
− 1
Ep + Ep−k + ωk
]. (68)
Evaluation of these coefficients is simplified once we
put
v11(ω2
k − (Ep − Ep−k)2) = v21(ω2
k − (Ep + Ep−k)2)
= f(ω2
k − (Ep + Ep−k)2), (69)
m
(2)
b (p) ≡ m
(2)
11 (p) = m
(2)
21 (p)
= 2
∫
dk
ωk
f2(ω2
k − (Ep + Ep−k)2)
E2
p−k − (Ep − ωk)2
+2
∫
dk
Ep−k
f2(ω2
k − (Ep + Ep−k)2)
ω2
k − (Ep + Ep−k)2
(70)
or
m
(2)
b (p) = C1(p) + C2(p),
C1(p) = 2
∫
dk
ωk
f2(ω2
k − (Ep + Ep−k)2)
2pk − μ2
s
,
C2(p) = 2
∫
dq
Eq
f2(μ2
s − 2μ2
b − 2pq)
μ2
s − 2μ2
b − 2pq
.
Evidently, the second integral does not depend upon
p so
C2(p) = C2(0) = 2
∫
dq
Eq
f2(μ2
s − 2μ2
b − 2μbEq)
μ2
s − 2μ2
b − 2μbEq
68
= 8π
∫ ∞
0
q2dq
Eq
f2(μ2
s − 2μ2
b − 2μbEq)
μ2
s − 2μ2
b − 2μbEq
. (71)
It is not the case for integral C1(p). Thus the boson
“mass renormalization” coefficients may be momen-
tum dependent.
7. CONCLUSIONS
In order to avoid ultraviolet divergences typical of
many field theories we have introduced some co-
variant cutoff functions in momentum space in the
Wentzel field model, that makes our model nonlocal.
For this model we retain the property of the interac-
tion density to be Lorentz-scalar.
We have shown how in the framework of the non-
local meson-boson field model one can build interac-
tions between the clothed mesons and bosons. More-
over, the mass renormalization terms, that are com-
pulsory to ensure the relativistic invariance of the the-
ory as a whole (in Dirac’s sense), turn out to be ex-
pressed through certain covariant integrals. They are
convergent in the field model with appropriate cutoff
factors.
References
1. A.V. Shebeko and P.A. Frolov. A possible way
for constructing generators of the Poincaré group
in quantum field theory // Few-Body Syst. 2011,
DOI: 10.1007/s00601-011-0262-5.
2. A.V. Shebeko and M.I. Shirokov. Unitary Trans-
formations in Quantum Field Theory and Bound
States // Phys. Part. Nuclei. 2001, v. 32, p. 31-
95.
3. A.V. Shebeko. The method of unitary clothing
transformations in quantum field theory: the
bound-state problem and the S-matrix // Prob-
lems of Atomic Science and Technology. 2007,
N 3, p. 61-65.
4. V.Yu. Korda, L. Canton, and A.V. Shebeko. Rel-
ativistic interactions for the meson-two-nucleon
system in the clothed-particle unitary represen-
tation // Ann. Phys. 2007, v. 322, p. 736-768.
5. E.A. Dubovyk and A.V. Shebeko. The method
of unitary clothing transformations in the theory
of nucleon-nucleon scattering // Few-Body Syst.
2010, v. 48, p. 109-142.
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| id | nasplib_isofts_kiev_ua-123456789-106984 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-30T09:40:46Z |
| publishDate | 2012 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Shebeko, A.V. Frolov, P.A. 2016-10-10T15:52:56Z 2016-10-10T15:52:56Z 2012 A nonlocal extension of the Wentzel field model in the clothed-particle representation / A.V. Shebeko, P.A. Frolov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 64-69. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 11.10.Ef, 11.10.Gh, 11.10.Lm, 11.30.Cp https://nasplib.isofts.kiev.ua/handle/123456789/106984 The clothed particle approach is applied to express the total Hamiltonian of interacting fields in terms of clothed particles. In order to avoid ultraviolet divergences typical of many field theories we introduce some covariant cutoff functions in momentum space in the Wentzel field model. We will show how in the framework of the nonlocal meson-boson field model one can build interactions between the clothed mesons and bosons. Moreover, the mass renormalization terms, that are compulsory to ensure the relativistic invariance of the theory as a whole (in Dirac's sense), turn out to be expressed through certain covariant integrals. They are convergent in the field model with appropriate cutoff factors. Для того чтобы избежать ультрафиолетовых расходимостей, типичных для многих полевых теорий, вводятся ковариантные обрезающие функции в импульсном пространстве в модели Вентцеля. Показано, каким образом в рамках нелокальной мезон-бозонной полевой модели можно построить взаимодействия между одетыми мезонами и бозонами. Кроме того, массовые перенормировочные члены, которые обязательны для обеспечения релятивистской инвариантности теории в целом (по Дираку), оказываются выраженными через определенные ковариантные интегралы. Эти интегралы сходятся в полевой модели с соответствующими обрезающими функциями. Щоб уникнути ультрафіолетових розбіжностей, типових для багатьох польових теорій, вводяться коваріантні обрезаючі функції в імпульсному просторі в моделі Вентцеля. Показано, як в рамках нелокальної мезон-бозонної польової моделі можна побудувати взаємодії між одягненими мезонами і бозонами. Крім того, масові перенорміровочні члени, які обов'язкові для забезпечення релятивістської інваріантності теорії у цілому (за Діраком), виявляються вираженими через певні коваріантні інтеграли. Дані інтеграли збігаються в польовій моделі з відповідними обрезаючими функціями. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Section A. Quantum Field Theory A nonlocal extension of the Wentzel field model in the clothed-particle representation Нелокальное расширение модели Вентцеля в представлении одетых частиц Нелокальне розширення моделі Вентцеля у зображенні одягнених частинок Article published earlier |
| spellingShingle | A nonlocal extension of the Wentzel field model in the clothed-particle representation Shebeko, A.V. Frolov, P.A. Section A. Quantum Field Theory |
| title | A nonlocal extension of the Wentzel field model in the clothed-particle representation |
| title_alt | Нелокальное расширение модели Вентцеля в представлении одетых частиц Нелокальне розширення моделі Вентцеля у зображенні одягнених частинок |
| title_full | A nonlocal extension of the Wentzel field model in the clothed-particle representation |
| title_fullStr | A nonlocal extension of the Wentzel field model in the clothed-particle representation |
| title_full_unstemmed | A nonlocal extension of the Wentzel field model in the clothed-particle representation |
| title_short | A nonlocal extension of the Wentzel field model in the clothed-particle representation |
| title_sort | nonlocal extension of the wentzel field model in the clothed-particle representation |
| topic | Section A. Quantum Field Theory |
| topic_facet | Section A. Quantum Field Theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106984 |
| work_keys_str_mv | AT shebekoav anonlocalextensionofthewentzelfieldmodelintheclothedparticlerepresentation AT frolovpa anonlocalextensionofthewentzelfieldmodelintheclothedparticlerepresentation AT shebekoav nelokalʹnoerasšireniemodeliventcelâvpredstavleniiodetyhčastic AT frolovpa nelokalʹnoerasšireniemodeliventcelâvpredstavleniiodetyhčastic AT shebekoav nelokalʹnerozširennâmodelíventcelâuzobraženníodâgnenihčastinok AT frolovpa nelokalʹnerozširennâmodelíventcelâuzobraženníodâgnenihčastinok AT shebekoav nonlocalextensionofthewentzelfieldmodelintheclothedparticlerepresentation AT frolovpa nonlocalextensionofthewentzelfieldmodelintheclothedparticlerepresentation |