General properties of interaction currents of higher spin fermions and their consequences for πN–scattering
The model for the currents of interactions of higher spin fermions with the 0- and 1/2-spin particles is proposed. These currents obey to the theorem on currents and fields as well as the theorems on current asymptotics. The comparison of the proposed model with partial wave analysis of the πN–scatt...
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irk-123456789-1069752016-10-11T03:02:11Z General properties of interaction currents of higher spin fermions and their consequences for πN–scattering Kulish, Yu.V. Rybachuk, E.V. Section A. Quantum Field Theory The model for the currents of interactions of higher spin fermions with the 0- and 1/2-spin particles is proposed. These currents obey to the theorem on currents and fields as well as the theorems on current asymptotics. The comparison of the proposed model with partial wave analysis of the πN–scattering in Δ(1232)-region shows the validity of the theorem on currents and fields. It is shown that in consequence of the theorems on current asymptotics the contributions of higher spin nucleon resonances to πN–scattering amplitudes must decrease at high energy at least as s⁻⁶. Предложена модель для токов взаимодействий высокоспиновых фермионов с частицами, обладающими спином 0 и 1/2. Эти токи удовлетворяют теореме о токах и полях и теореме об асимптотике токов. Сравнение предложенной модели с парциально волновыми анализами πN–рассеяния в области Δ(1232) показывает справедливость теоремы о токах и полях. Показано, что вследствие теоремы об асимптотике токов вклады высокоспиновых нуклонных резонансов в амплитуды πN–рассеяния должны убывать при высоких энергиях по крайней мере как s⁻⁶. Запропоновано модель для струмів взаємодій високоспінових ферміонів з частинками, які мають спін 0 та 1/2. Ці струми задовольняють теоремі про струми та поля та теоремі про асимптотику струмів. Порівняння запропонованої моделі з парціально хвильовими аналізами πN–розсіювання в області Δ(1232) показує справедливість теореми про струми і поля. Показано, що внаслідок теореми про асимптотику струмів внески високоспінових нуклонних резонансів в амплітуди πN–розсіювання повинні спадати при високих енергіях по меншій мірі як s⁻⁶. 2012 Article General properties of interaction currents of higher spin fermions and their consequences for πN–scattering / Yu.V. Kulish, E.V. Rybachuk // Вопросы атомной науки и техники. — 2012. — № 1. — С. 27-31. — Бібліогр.: 17 назв. — англ. 1562-6016 PACS: 02.30 Jr, 03.65. Pm, 11.10. J, 11.10 L, 11.10 Q, 11.40 D, 11.80 E, 13.75 G, 14.20 G. http://dspace.nbuv.gov.ua/handle/123456789/106975 en Вопросы атомной науки и техники Ukrainian State Academy of Railway Transport |
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Section A. Quantum Field Theory Section A. Quantum Field Theory |
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Section A. Quantum Field Theory Section A. Quantum Field Theory Kulish, Yu.V. Rybachuk, E.V. General properties of interaction currents of higher spin fermions and their consequences for πN–scattering Вопросы атомной науки и техники |
| description |
The model for the currents of interactions of higher spin fermions with the 0- and 1/2-spin particles is proposed. These currents obey to the theorem on currents and fields as well as the theorems on current asymptotics. The comparison of the proposed model with partial wave analysis of the πN–scattering in Δ(1232)-region shows the validity of the theorem on currents and fields. It is shown that in consequence of the theorems on current asymptotics the contributions of higher spin nucleon resonances to πN–scattering amplitudes must decrease at high energy at least as s⁻⁶. |
| format |
Article |
| author |
Kulish, Yu.V. Rybachuk, E.V. |
| author_facet |
Kulish, Yu.V. Rybachuk, E.V. |
| author_sort |
Kulish, Yu.V. |
| title |
General properties of interaction currents of higher spin fermions and their consequences for πN–scattering |
| title_short |
General properties of interaction currents of higher spin fermions and their consequences for πN–scattering |
| title_full |
General properties of interaction currents of higher spin fermions and their consequences for πN–scattering |
| title_fullStr |
General properties of interaction currents of higher spin fermions and their consequences for πN–scattering |
| title_full_unstemmed |
General properties of interaction currents of higher spin fermions and their consequences for πN–scattering |
| title_sort |
general properties of interaction currents of higher spin fermions and their consequences for πn–scattering |
| publisher |
Ukrainian State Academy of Railway Transport |
| publishDate |
2012 |
| topic_facet |
Section A. Quantum Field Theory |
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http://dspace.nbuv.gov.ua/handle/123456789/106975 |
| citation_txt |
General properties of interaction currents of higher spin fermions and their consequences for πN–scattering / Yu.V. Kulish, E.V. Rybachuk // Вопросы атомной науки и техники. — 2012. — № 1. — С. 27-31. — Бібліогр.: 17 назв. — англ. |
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Вопросы атомной науки и техники |
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2025-07-07T19:16:32Z |
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| fulltext |
GENERAL PROPERTIES OF INTERACTION CURRENTS OF
HIGHER SPIN FERMIONS AND THEIR CONSEQUENCES
FOR πN-SCATTERING
Yu.V. Kulish and E.V. Rybachuk ∗
Ukrainian State academy of Railway Transport, 61050, Kharkov, Ukraine
(Received October 26, 2011)
The model for the currents of interactions of higher spin fermions with the 0- and 1/2-spin particles is proposed. These
currents obey to the theorem on currents and fields as well as the theorems on current asymptotics. The comparison
of the proposed model with partial wave analysis of the πN-scattering in Δ(1232)-region shows the validity of
the theorem on currents and fields. It is shown that in consequence of the theorems on current asymptotics the
contributions of higher spin nucleon resonances to πN-scattering amplitudes must decrease at high energy at least
as s−6.
PACS: 02.30 Jr, 03.65. Pm, 11.10. J, 11.10 L, 11.10 Q, 11.40 D, 11.80 E, 13.75 G, 14.20 G.
1. INTRODUCTION
At present a lot of higher spin particles (the spin
J ≥ 1) is known. It is known that the higher spin par-
ticles as well as the nucleons, the pions and nuclei are
not the elementary particles. But the approximation
of elementary particles gives rather good description
of reactions at low and intermediate energies. There-
fore we can assume that the higher spin particles may
be considered approximately as elementary particles
also, similarly to the nuclei and the pions. The non-
elementarity of the particles can be taken into ac-
count by means of the form factors in the interaction
currents. As a rule the Rarita-Schwinger formalism
[1, 2] for the higher spin particles is used in the cal-
culations of the reaction amplitudes. We can write
for the amplitude of any interaction of higher spin
fermion (HSF)
V = U(p)μ1...μrη(p)μ1...μl
, (1)
where η(p)μ1...μl
is the HSF interaction current. The
U(p)μ1...μl
= U(p)l
μ is the spin-tensor of HSF with
the spin J = l + 1/2 and the momentum p. This
spin-tensor is symmetric and traceless and its convo-
lutions with momenta p and γ-matrices vanish. As
usual we assume that the HSF interactions are de-
scribed by the system of the non-homogeneous Dirac
equations. The field spin-tensors U(p)l
μ and U(x)l
μ
have got 2J + 1 = 2l + 2 independent components.
As a rule for the current spin-tensors η(p)μ1...μl
and
η(x)μ1...μl
it is assumed that they are the symmet-
ric only. Therefore they have got Nl = 4 · 4 · 5 ·
... · (l + 3)!/l! = 2(l + 1)(l + 2)(l + 3)/3 indepen-
dent components. We name the approaches with such
currents as the common (or conventional) ones. Un-
fortunately the common approaches have got some
shortcomings [3]: 1) the inconsistences of equation
systems (as Nl > 2J + 1); 2) power divergences due
to the higher spin particle propagators and the in-
teraction currents; 3) the ambiguities of the vertex
functions; 4) contradictions to the experimental data
in wide energy regions. Therefore we conclude that
the common approaches must be modified. As the
shortcomings of common approaches exist for differ-
ent higher spin particles we may propose that the in-
teraction currents for higher spin particles must obey
some general properties in addition to the symmetry
property. In Refs. [3-6] it is shown that the interac-
tion currents for higher spin particles must obey the
theorem on currents and fields as well as the theorem
on current asymptotics. In Refs. [7, 8] the model for
the interaction of higher spin boson with two spin-
less particles is proposed in the agreement with these
theorems. The calculation of the virtual higher spin
boson contributions to the self-energy operator of the
spinless particle shows that these contributions are fi-
nite in the one-loop approximation [3]. These finite
values must be compared with the logarithmic diver-
gences for two spinless particle contribution to the
self-energy operator.
In present paper we propose the model for the
vertex of the HSF interaction with 0- and 1/2-spin
particles (e.g., πN ↔ N∗ ), which obeys the theorem
on currents and fields as well as the theorem on cur-
rent asymptotics. We study the application of this
model to the elastic πN -scattering.
2. CONSEQUENCES OF THEOREM ON
CURRENTS AND FIELDS
In accordance with the theorem on currents and
fields [6] the system of the algebraic equations for
∗Corresponding author E-mail address: rybachuk l@mail.ru
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 27-31.
27
the Fourier components is consistent only in the case
when the current spin-tensors have got the same
properties as the field spin-tensors. We name the cur-
rent which obey the theorem on currents and fields
as the physical ones and denote j(p)l
μ = j(p)μ1...μl
.
Thus for physical currents we have
j(p)μ1 ...μl
pμi = 0, ∂μij(x)μ1 ...μl
= 0, (2)
j(p)μ1 ...μl
gμiμk
= 0, j(x)μ1 ...μl
gμiμk
= 0, (3)
j(p)μ1 ...μl
γμi = 0, γμij(x)μ1 ...μl
= 0, (4)
momentum coordinate
representation representation
where i, k = 1, 2, ..., l. The physical currents have got
2J + 1 independent components and can be derived
using the projection operator Π(p)μ1...μl,ν1,...νl
=
Π(p)l
μ,ν [9], modified in Refs. [6, 10]:
j(p)l
μ = (p2)lΠ(p)l
μ,νη(p)l
μ. (5)
As example, for J = 3/2 we derive
P (p)μν =
p̂ + M
p2 −M2
[
dμν − 1
3
γ̃μγ̃ν
]
=
=
[
dμν − 1
3
γ̃μγ̃ν
]
p̂ + M
p2 −M2
,
γ̃μ = γ5(γμ − pμp̂/p2), dμν = −gμν + pμpν/M2. (6)
This propagator differs from usual propagator for
J = 3/2 [2, 11,12].
We can see several distinctions of the HSF propa-
gators in our and common approaches: 1) In our ap-
proach the convolutions of the HSF propagator with
the p momentum, the γ-matrices, and the metric ten-
sors vanish at any p and J , but in common ap-
proaches they vanish only at p̂ = M (i. e. on the
mass shell); 2) In our approach the operators p̂ + M
and Π(p)l
μν commute; 3) As a consequence of the
current conservation (2) and the condition (4) the
power divergences due to the HSF propagator dis-
appear in our approach; 4) The scale dimension of
our HSF propagators equals −1 for any J , whereas
in common approach it equals 2J − 2. This allows
to eliminate the one source of the power divergences
existing in common approaches.
3. CONSISTENT MODEL FOR
INTERACTION CURRENTS OF HIGHER
SPIN FERMIONS
We consider the simplest HSF interactions deter-
mined by one amplitude. Using the definition(5) we
may write for the J(p)→ O(q2) +1/2(p2) transition:
j(p, q)μ1...μl
= glFl(p, q)(p2)lϕ+(q2)u(p2)·
·
{
1
iγ5
}
Π(p)μ1...μl,ν1...νl
q
′
ν1
...q
′
νl
, (7)
where q
′
= q2 − p2, gl is the coupling constant;
Fl(p, q
′
) is the form factor providing the necessary
asymptotic decrease in agreement with the theorem
on current asymtotics. In particular for J = 3/2 we
have
j(p, q)
′
μ = g1F1(p, q)ϕ+(q2)u(p2) ·[
−Q
′
μ +
1
3
γ5γ̃μQ̂
′
]{
1
iγ5
}
, (8)
where Q
′
μ = −dμνp2q
′
ν = p2q
′
μ − pμ(pq′), (pQ′) = 0.
The 1 and iγ5 matrices in the currents (7), (8) corre-
spond to different sets of the particle parities. Note
that different common currents lead to the physical
currents (5) with the same momentum dependencies,
as the convolutions of the projection operators with
the p momentum and the γ-matrixes vanish in our ap-
proach. Therefore the physical currents correspond-
ing to different common currents in Eq. (5) differ by
the coupling constant only. Thus the ambiguities of
vertex functions do not appear in our approach.
4. TEST OF THEOREM ON CURRENTS
AND FIELDS IN Δ(1232)-REGION
HSF in the 0 + 1/2 →← J(p) -transitions can be
N∗(J) resonances in the s-channel of the elastic
πN -scattering (πN → N∗(J) → πN). In elas-
tic πN -scattering the S31(W )−, P31(W )−, P33(W )-
amplitudes correspond to isospin I = 3/2 and the to-
tal angular momentum JπN = 1/2, 1/2, 3/2 , respec-
tively (where W is the total energy in the c. m. s.).
The Δ(1232) on the mass shell (at W = MΔ, where
MΔ is the Δ(1232) mass) contributes to the ampli-
tude P33 only. But at W �= MΔ the Δ(1232) con-
tribute to other amplitudes of the πN -scattering too.
These contribution are different in our and common
approaches.
In common approach Δ(1232) contribute to the
amplitudes S31, P31, P33, and D33 at W �= MΔ. We
denote the possible contributions of Δ(1232) to these
amplitudes as SΔ
31, PΔ
31, PΔ
33, and DΔ
33, respectively.
To derive these contributions in common approach
we use common propagator of the 3/2-spin particle
[2] and the Breit-Wigner formula [13]. The calcula-
tions of the SΔ
31−, PΔ
31−, DΔ
33-amplitudes show that
among these amplitudes SΔ
31 achieves largest values
and DΔ
33 have smallest values for MΔ − Γ/2 ≤ W ≤
MΔ + Γ/2 (where Γ = 112 MeV is the total with of
Δ(1232) [13]). In common approach energy depen-
dences are fairly sharp for S31(W ).
In our approach SΔ
31 = PΔ
31 = 0 , as consequence
of (2), (4). From comparison with the partial wave
analysis [13, 14] we may conclude that SΔ
31 = 0 in
agreement with the consequence of the theorem on
current and fields.
5. CONSEQUENCES OF THEOREMS ON
CURRENT ASYMPTOTICS
We consider HSF which moves along the z-axis (i.e.,
p = (p0, 0, 0, p3) ). Then the physical currents in mo-
mentum representation j(p)l
μ depend on p0 and p3,
whereas j(x)l
μ depend on x0 and x3. As the compo-
nents of j(p)l
μ are the Fourier components of j(x)l
μ
28
these currents in coordinate representation are the
improper integrals depending on the parameters x0
and x3. In Ref [6] it is shown, that the physical
currents j (x)l
μ and some their derivatives must be
continuous functions. We use the Weierstrass test to
study the continuity of the currents. Therefore, we
consider the integrals
+∞∫
−∞
dp0
+∞∫
−∞
dp3
∣∣∣j (p)l
μ
∣∣∣ |p0|m 0 |p3|m 3 , (9)
where m0, m3, m(j) are integer non-negative num-
bers (m0 + m3 = 0, 1, 2, ..., m(j)). The theorem on
current asymptotics may be formulated as:
If the currents J(x)l
μ and their partial deriva-
tives of upper degree m(j) are continuous functions
then their Fourier components j(p)l
μ must decrease at
|pν | → ∞ to provide the convergence of all integrals
(9).
Note that the integrals (9) must be convergent in
all kinematic regions. The powers of the decrease for
j(p)l
μ are determined by the number m(j) . In Ref.
[6] it is derived that m(j) = 2 , as consequence of the
condition that 2J + 1 equations must be for U(x)l
μ.
Now in addition we demand that the double Fourier
transformations for the function of two variables con-
verges to the value of this function in any space-time
point. But it is possible if this function, first deriva-
tive and its mixed derivative of the second degree are
continuous [15]. It allows to derive m(j) = 4.
We propose that the form factors Fl(p, q) in the
currents (7) have a form:
F̃l(p, q) = (pq)2n1
[
(p2 −M2)2n2 + a4n2
]−1
[
(pq)2n3 + b4n3
]−1
, (10)
where n1, n2, n3 are integer non-negative numbers,
a and b are positive constants. Using the method of
Refs. [7, 8] we derive at n1 = 1: n2 ≥ m1(η)/4 +
1, n3 ≥ m1(η)/2 + 2, where even number m1(η) =
m(j) + 2l for even m(j) and m1(η) = m(j) + 2l + 1
for odd m(j). For m(j) = 4 we have the restrictions
n2 ≥ l/2 + 2, n3 ≥ l + 4
Now we consider the convergence of the integrals
(9) for the contributions of N∗(J) to s-channel am-
plitudes of the πN -scattering. In c. m. s. we have
p = (W, 0, 0, 0) , p2 = W 2 = s, p · q′
= m2
π −m2
N
q
′
= (q0 − E, 2�q2) , where q0, E, and �q2 are the pion
energy, the nucleon energy, and the 3-momentum of
the final pion, respectively. The form factor (10) may
be written as F (p, q′) = A/
(
W 4n2 + a4n2
)
, where A
is the constant. The asymptotic behavior of the cur-
rent (7) is given by j(p)l
μ ∼ CjW
3lFl(p, q′), where
Cj is some constant.
The p3-dependence of the current (7) in c. m. s.
is determinated by the factor δ(p3). Then the inte-
grals (9) converge in the case of the convergence of
the integral
∞∫
0
W 3l+4dW
W 4n2 + a4n2
. (11)
This integral converges at n2 ≥ 3/4 · l+3/2. Thus we
derive two restrictions. For better convergence of the
integrals we must choose larger n2. Both restrictions
give the same values of the number n2 : n2 ≥ 3, 3, 4
for J = 3/2, 5/2, 7/2, respectively. At l ≥ 4 sec-
ond restriction gives larger integer number n2. It is
of interest to study the asymptotic behaviour of the
N∗(J) contributions to the πN -scattering amplitudes
including the physical currents (7) with the form fac-
tors (10)
T(πN → N∗(J)→ πN) =
= Al
ū2(p̂±M)u1
W 2 −M2 + iMΓ
[
W 3l
W 4n2 + a4n2
]2
, (12)
where Al are the constants. In general we derive
that the asymptotic contributions of N∗(J) to in-
variant amplitudes decrease at least as s−6. But we
can derive stronger decrease at integer n2. Indeed,
for J = 3/2, 5/2, 7/2 we have n2 = 3, 3, 4 and
T (πN → N∗(J)→ πN)/Ãl ≤ s−9, s−6, s−7 respec-
tively.
6. EQUATIONS FOR GENERATIONS OF
HIGHER SPIN FERMIONS
In Ref.[16] it is shown that the integrals correspond-
ing to the Green functions of the Klein-Gordon and
Dirac equations diverge. To derive the convergent
integrals for the Green functions we may study the
partial differential equations of higher degrees. We
consider some sets (kinds, families, dynasties) of par-
ticles, which have different masses but the same val-
ues of the electric charge, the spin, and the pari-
ties. The members of such kinds belong to gener-
ations. We can consider the electron kind (e1 = e,
e2 = μ, e3 = τ, . . .), the neutrino kind (ν1 = νe,
ν2 = νμ, ν3 = ντ , . . .), three colored up-quark kinds
(u1 = u, u2 = c, u3 = t, . . .) and three colored down-
quark kinds (d1 = d, d2 = s, d3 = b, . . .). We propose
the equations for the 1/2-spin particles as the gener-
alization of Dirac equation:(
M1 − i∂̂
) (
M2 − i∂̂
)
...
(
MN − i∂̂
)
= χ (x) , (13)
where M1, M2, ..., MN are the particle masses
(M1 < M2 < M3 < ... < MN ). The number N is the
degree of the differential equation, which is equal to
the number of the generations in the kind. It fol-
lows from the convergence of the integrals for the
Green function of Eq. (13) that N ≥ 5. We de-
note the minimal N for the elementary fermions
(the leptons and the quarks) as Nf . The num-
bers of the generations for composite particles are
larger then ones for elementary particles. For ex-
ample, the minimal numbers for the proton and
the neutron kinds are equal to 75, as we may de-
rive N(proton kind)min = N(neutron kind)min =
N2
f (Nf + 1)/2 [16]. The proton kind includes
p,
∑
+(1189), Λ+
c (2285) [17]. The neutron kind in-
cludes n, Λ0(1115),
∑
0(1193), Ξ0(1315).
29
We propose that the generalization of the equa-
tion system for the HSF generations may be written
as
(−�)lΠ(x)μ1...μl,ν1...νl
(−i∂̂ + M1)(−i∂̂ + M2) · · ·
· · · (−i∂̂ + MN )U(x)ν1...νl
= j̃(x)μ1...μl
, (14)
where the physical currents j(x)μ1...μl
must obey the
conditions (2)-(4). We may rewrite Eq. (14) in the
form :
(−�)lΠ(x)μ1...μl,ν1...νl
(� + M2
1 )(� + M2
2 ) · · ·
· · · (� + M2
N)U(x)ν1...νl
=
= (−i∂̂ + M1)(−i∂̂ + M2) · · ·
· · · (−i∂̂ + MN )j̃(x)μ1...μl
. (15)
We see that the physical currents j̃ (x)μ1...μl
must
have the continuous derivatives of the degree N + 3,
(i. e. , m(j̃) = N + 3). Then for the number n2 in
the form factor (10) we derive n2 ≥ (2l + N + 7)/4.
For these n2 we have
T (πN → N∗(J)→ πN) ≤ Ãl/sN+7−l, (16)
where Ãl is constant. We may find Nmin
for Δ-isobar kinds. We assume that the
Δ++- kind includes Δ++(1232) and different
three-quark systems with Jp = 3/2+ consist
of u−, c−, t-quarks . The Δ−-kind includes
Δ−(1232),
∑−
δ (1385), Ξ−
δ (1531), Ω−(1672) and dif-
ferent three-quark systems with Jp = 3/2+
consist of d−, s−, b-quarks. We may derive
that N(Δ++ − kind)min = N(Δ− − kind)min =
Nf (Nf + 1)(Nf + 2)/6 ≥ 35. We may expect
that N(Δ+ − kind)min = N(Δ0 − kind)min =
N(proton kind)min = N(neutron kind)min ≥ 75.
From Eq. (16) we can derive T(πp→ Δ++ → π+p) ≤
Ãl/s41.
7. CONCLUSIONS
We proposed the model for the currents of HSF in-
teractions with the 0- and 1/2-spin particles. These
physical currents obey the general properties formu-
lated in Ref. [6] for the currents of the HSF interac-
tions. All physical currents must obey the theorem on
currents and fields as well as the theorem on current
asymptotics. We consider the nucleon resonances
N∗(J) as example of HSF. In consequence of the theo-
rem on currents and fields the virtual HSF can change
the parity but they do not contribute to the ampli-
tudes corresponding to the values of the angular mo-
mentum less than J , whereas in common approaches
such contributions exist. We have tested the predic-
tions of our and common approaches for the virtual
Δ(1232) in the elastic πN -scattering. The calcula-
tions performed in common isobar model show sharp
energy dependence of the Δ(1232)-contributions to
the S31- and P31−amplitudes at W ≈ MΔ. It
turned out that the S31−amplitude is the most sen-
sitive. According to the partial wave analyses the
energy dependences of the amplitudes are approxi-
mately linear in the Δ(1232) region, i. e., they dif-
fer from the predictions of common isobar model. It
means that the predictions of our approach are valid.
Thus we have examined the validity of the condi-
tions (2), (4).To examine the validity of the condi-
tions (2)-(4) we must consider HSF with J > 3/2.
For example, it is of interest to study the contri-
butions of F15(1680) [17] to the S11−, P11−, P13−,
D13−, F15− amplitudes; F35(1905) to the S31−,
P31−,P33−,D33−,F35-amplitudes; F37(1950) to the
S31−,P31−, P33−, D35−, F35-amplitudes.
We propose the form-factor (10), which allows to
obey the theorem on current asymptotics. The re-
strictions for the integer number n2 lead to the high-
energy decrease of the HSF contributions to the πN -
scattering amplitudes. This decrease explains the ab-
sence of the N∗(J) contributions at high energies.
In the cases for the interactions of several higher
spin particles (e. g., the ρΔN∗(J)-interaction) the
theorems on currents and fields as well as the the-
orem on current asymptotics must be valid for the
interaction currents of each higher spin particle. The
theorem on current and fields can be satisfied using
the projection operators. The theorem on current
asymptotics can be satisfied by consideration of the
product of the form factors (such as (10) or from
Refs. [7,8]) for each higher spin particle. Therefore
we may expect that the contributions of the vertex
functions for several higher spin particle interaction
to the amplitudes ought to decrease at high energies,
in comparison with similar vertex functions for the 0-
and 1/2-spin particle interactions.
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