Interactions of elementary particles similar to gravitational one
It is shown that in the non-relativistic limit the Newtonian gravity law can be derived at the equality of the coupling constants for the electron, proton, neutron interactions with the spin-2 massless particle (quasigraviton). The interactions of the photons and the gluons with quasigraviton in one...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Zitieren: | Interactions of elementary particles similar to gravitational one / Yu.V. Kulish // Вопросы атомной науки и техники. — 2001. — № 6. — С. 80-83. — Бібліогр.: 4 назв. — англ. |
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| citation_txt | Interactions of elementary particles similar to gravitational one / Yu.V. Kulish // Вопросы атомной науки и техники. — 2001. — № 6. — С. 80-83. — Бібліогр.: 4 назв. — англ. |
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| description | It is shown that in the non-relativistic limit the Newtonian gravity law can be derived at the equality of the coupling constants for the electron, proton, neutron interactions with the spin-2 massless particle (quasigraviton). The interactions of the photons and the gluons with quasigraviton in one-loop approximation are studied at the contributions of quarks, leptons, W, and the scalar fields. From the absence of the quadratic divergences it is derived that the sums of the coupling constants of the quasigravitational interactions of all the quarks and all the leptons vanish. Therefore the black holes do not appear.
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INTERACTIONS OF ELEMENTARY PARTICLES SIMILAR TO
GRAVITATIONAL ONE
Yu.V. Kulish
Kharkov State Academy of Railway Transport, Kharkov, Ukraine
e-mail: kulish@kart.kharkov.com
It is shown that in the non-relativistic limit the Newtonian gravity law can be derived at the equality of the
coupling constants for the electron, proton, neutron interactions with the spin-2 massless particle (quasigraviton).
The interactions of the photons and the gluons with quasigraviton in one-loop approximation are studied at the
contributions of quarks, leptons, W, and the scalar fields. From the absence of the quadratic divergences it is derived
that the sums of the coupling constants of the quasigravitational interactions of all the quarks and all the leptons
vanish. Therefore the black holes do not appear.
PACS: 04.25.Dm, 02.70.Bf, 04.04.70.Bw
INTRODUCTION
As is known the electric charges of the hadrons do
not renormalized by the strong interactions. In
particular, if the charges of the bare electron and proton
are equal then their charges induced by the
electromagnetic and the strong interactions are equal
also [1 – 4]. However, as the proton and the electron
have different masses their gravitational interactions
may be different. Therefore we can assume that the
gravitotional corrections to the electric charges of the
proton and the electron may be different (i. e. the values
of these charges may have a small difference. In
connection with this it is important to study the problem
of the anomaly similar to the axial Adler–Bell–Jackiw
anomaly in the electroweak theory. Well-known theory
of the gravitation – the general theory of relativity faces
with the problems for the elementary particles. For
example, how do the vacuum mean and the Dirac sea
affect on the gravitation? Progress in the problem of the
gravitational interactions of the elementary particles has
been achieved in the supergravity models, but this
problem is not solved yet. Therefore it takes interest in
the investigations in simple models too.
In this paper the interactions of the massive spin–0,
½, 1 particles induced by the spin–2 massless particle
exchange are considered in Minkowski space–time. We
study the quantities similar to the axial ABJ–anomaly in
the electroweak theory.
1. INTERACTIONS OF TENSOR FIELDS
The equation for the massless spin–2 field µ ν)(xG
we write as
ð µ νµ ν )()( xjxG = ,
0)(,0)()( ==∂=∂ µ µµ ννµ νµ xGxGxG , (1)
0)(,0)()( ==∂=∂ µ µµ ννµ νµ xjxjxj ,
where µ ν)(xj is the tensor of the interaction current.
This tensor can be derived from some current tensor
µ νη )(x as follows
µ ν)(xj = 1, 1)( νµµ νxΠ
11)( νµη x (2)
where 1, 1
)( νµνµxΠ is the projection operator. The
Fourier components of this operator are given by
( )
,,
3
1
2
1)(
2
,
2211
122121212211
q
qq
gddd
ddddq
νµ
νµνµνµνµ
νµνµννµµνµνµ
+−=
−+Π =
(3)
where q is the 4–momentum of the tensor particle.
The interaction Lagrangian may be written in several
general forms
L νµνµνµνµ η )()()()(int xxGxjxG == . (4)
In particular, we choose the next interactions
Lagrangians:
a) for the scalar particles
L ϕϕ νµνµ ∂∂= +Ggss , (5)
b) for the spin –
2
1
particles
L )( ψγψψγψ µννµνµ ∂−∂= GigFF , (6)
c) for the vector particles
L ( )++ ∂−∂∂−∂−= νλλνµλλµνµν AAAAGgv )( , (7)
d) for the VS ↔ – transitions
L ( )[ +∂∂−∂∂= +ϕνλµµλνµλ AAGgvsvs )(
]ϕνλµµλ ∂∂−∂+ ++ )( AA . (8)
As the Lagrangian (5)–(8) must be hermitian the
constants vsvFs gggg ,,, must be real. The
interactions (5)–(8) are C–even.
80 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 80-83.
2. NON–RELATIVISTIC LIMIT
Consider the elastic scattering of the particles in the
one spin–2 massless particle exchange approximation
(Fig. 1).
In the cms the spin–2 particle momentum is
2
sin2,0,0,0(12
θpppq =−= ), where p is the value of
the 3–momentum of the first particle, θ is the scattering
angle. At low energies we derive from (3) that the
leading components of the currents are next only
3
2)(2)(2)( 221100 === qjqjqj 00)(qη (9)
For the spin– 2
1 - particles
1210200 22)()(
2
)( χχγη ∗≈⋅= F
F mgEpupu
E
gq , (10)
where 1p and 2p are the 4–momenta of the particles in
the initial and the final states, respectively; E is their
energy ( 2
1p = 2
2p = 2m ). The matrix element
corresponding to the Fig. 1 is given by
)1()( µ νqjT fi = )2()( µ νqj
εiq +2
1 =–
3
2
⊗)1(
00)(qη
),()()()(1)( )1()2(
00
)2()1(
002
)2(
00 qVqqVq
iq
q ηη
ε
η ==
−
⊗
1
*
2
)1()1( )()( χχqUqV = , (11)
where )()1( qV is the Fourier component of the gravity
potential induced by the first particle. In the coordinate
space we derive the gravity potential and the interaction
force of two particles
,r
3
2
)(2F,
3
1)(
3
212
)1(
2
1)1(
r
mmg
xgradUgm
r
mgxU
F
FF
−=
=−=
π
−=
(12)
where r
is the distance vector between two particles.
This consideration is similar to the derivation of the
Coulomb law in the electrodynamics [3]. But in contrast
with the electrodynamics the one spin–2 massless
particle exchange leads to the attractions.
Fig. 1
Now we take into account that the Earth, Moon, Sun
and the planets consist of the spin–
2
1
particles: protons,
neutrons, electrons. If we propose that the couplings
constants are equal
1Felectronneutronproton ggggg
neutrinol
====
− (13)
and take into account the linearity of the eq.(1), then we
may obtain Newton gravity law. For this we must put
2/32
1 NF Gg π= , (14)
where 3910707,6 −⋅=NG ħc(GeV/c2)–2 is the
gravitational constant. If for the coupling constants sg
and vg in (5), (7) we put
protonFvs gggg 44 1 === (15)
then as consequence of the relations (13)–(15) the
attractive force will be valid for the interactions of the
scalar and vector particles between them and with the
protons, neutrons, electrons.
As the one spin–2 massless particle exchange leads
to the attractive long–rang force we will call these
interactions as quasigravitational and the spin–2
massless particle as the quasigraviton.
The quasigravitational interactions have the next
interesting properties. (a) The sources of these
interactions are just the elementary particles, but not the
masses or the energies of the particles. (b) The intensity
of these interactions is determined by the particle
coupling constant, which may, be positive, negative and
be equal to zero. (c) These interactions are covariant and
the consideration is carried out in the flat Minkowski
space–time. (d) We may regard that the inertial and
gravitational masses are equal. Indeed, for the
quasigravitational interactions the origins of the masses
in the interaction forces (12) (these masses are just
gravitational) are the particles energies, which are
related to the masses in the Lagrangian (i.e. inertial
masses)
In consequence of the properties (a) –(c) the
quasigravitational interactions stay similar to the
electromagnetic interactions.
Note that the long–range attractive property of force
induced by the spin–2 massless particle exchange was
established by Bronstein in 1936.
3. QUANTITIES SIMILAR TO ABJ–
ANOMALY
We consider the interactions of the photons and the
gluons with the quasigraviton in the one–loop
approximation.
In the Weinberg–Salam model there are
γ γ ϕ ϕγ γγ ϕ ϕϕγγγγ ,,,,,,, −−−−− WWWWWLLQQ –
verteces of the photon interactions. In Fig. 2 Feynman
diagrams are presented for the Gγ γ –interaction, where
G is the quasigraviton.
For each diagram with the virtual +W must be
diagram with the contribution of the scalar +ϕ field. So
besides diagrams 1B and 2B there are six diagrams
with one ϕ and two W , six diagrams with two ϕ and
one W , as well as two diagrams with three ϕ .
The amplitudes corresponding to the diagrams
2121 ,,, BBAA and others include the quadratically
divergent integrals. For the interactions of ϕγ +,, W
we use the Feynman rules of refs. [4]. The quadratically
81
divergent part of the amplitudes for the diagrams in
Fig. 2 іs given by
⋅−=→ )()()(
3
2)( 2
*
1
2 qGkkeGT quadr µ ννµ εεγγ (16)
∫
∑∑
++
++
−+
24
4
22
)2(
5
3
4
3
1
3
24
p
pdgg
ggN
W
i
L
i
Qc ii
πϕ
Fig. 2. Feynman diagrams for photon–
quasigraviton interaction
where )(),( 2
*
1 kk νµ εε are the polarization vectors of the
initial and the final photons, respectively, )(qGµ ν is the
tensor for the quasigraviton, 12 kkq −= , e is the value
of the electric charge of the electron and the proton; cN
is the number of the quark colours, In iQg)(16 and
iLg are coupling constants of the quasigralitational
interactions of the quarks and the leptons for i–th
generation, respectively. In the additive quark model
may be derived
protonQ gg =1 . (17)
If for each generation to assume
ii LQ gg = , (18)
then between the particles and antiparticles of the i–th
generation will be the long–range attractive force of the
form (12), (14). The coupling constants ϕg , iQg and
iLg , wg are determined by the Lagrangians (5)–(7) (
wvLQF ggggg ii === ; ).
Now we study the interaction of the gluon with the
quasigraviton. As in QCD the gluons interact with the
quarks, gluons g and the Faddev–Popov ghosts the one–
loop approximation is determined by the diagrams in
Fig. 3.
Fig. 3. Feynman diagrams for gluon–quasigraviton
interactions ggG →
For the quadratically divergent part of the ggG →
–interaction amplitude we derive
∫
∑
⊗
⊗
++⊗
−=→
,
)2(
4
15
3
4
)()()()(
24
4
2
2
*
1
p
pd
ggg
gqGkkggGT
pi Fgluon
i
Q
ab
QСD
ba
quadr
π
δ
εε µ ννµ
(19)
where )( 1ka
µε and )( 2
* kb
νε are the polarization vectors
of the initial and the final gluons, respectively, a and b
82
are the colour indexes; QCDg is the QCD coupling
constant. The interaction coupling constants of the
Faddeev–Popov ghosts Fpg and the gluon gluong with
the quasigraviton are determined by the Lagrangians
(5)–(7) at Fpgg =ϕ and eluonv gg = , respectively.
To eliminate the quadratic divergences in the
γγ →G and ggG → –amplitudes we use (18) and put
05
3
32 =++∑ ϕggg w
i
Qi , (20.a)
0
4
15
3
2 =++∑ pi Fgluon
i
Q ggg . (20.b)
By analogy with the photon–gluon and photon–
ghost interactions we assume that the quasigravitons do
not interact immediately with the gluons and the ghosts
(i.e. 0== Fpgluon gg ). Then we derive the next
relations from (18), (20):
0∑∑ ==
i
L
i
Q ii gg , (21)
wgg 5−=ϕ . (22)
In consequence of the relation (21) for the particles
and the antiparticles of the second (or the third )
generations must repulse the particles (and antiparticles)
of the first interaction.
4. RELATED TOPICS
4.1. Black holes. To satisfy (21) assume that the
particles of the third generation repulse the particles of
the first and the second generations. Consider big body
(e.g. the star), which consist of the first generation
particles. As consequence of the attractive gravitational
force the body dimension can be reduced. According to
the Pauli principle the particle energies will increase. At
some moment of the time the transitions into the
second-generation particles by means of the electroweak
interactions became favourable. The second-generation
particles will stable at enough quantity of the first
generation particles. In further the body reduction the
second-generation particles will transit into the third
generation particles. The last particles will be pushed
from the body. Such a way the body dimension cannot
be reduced to very small dimension. The third
generation particles will decay outside the body and the
body mass will decrease Therefore the black holes
ought not to appear.
4.2. Red shift. Compare the light emission from big
bodies consisting of the particles of the first, second ,
and third generations (now (21) is not obligatory). In
consequence of the small electron mass the light, the
ultraviolet, and X-rays are radiated mainly by the
electrons. At large the second and the third generation
particle concentration the radiation of the electrons is
suppresed as: (i) a lot of electrons transited into muons;
(ii) The electrons do not radiate to obey the Pauli
principle. Therefore with the growth of the second and
the third generation particle concentrations the
intensities of the light, the ultraviolet, and X-rays
radiations will decrease. Besides, in consequence of the
increasing of the gravitation field strength the light
frequency will be reduced. Therefore at the reduction of
the light radiation intensity the shift to the red side will
increase. Note that such dependence is explained
usually by the Hubble law. At very small the electron
concentration the body can stay invisible. Possible such
bodies can be observed in the investigations of high-
energy the γ -quanta and the neutrinos.
4.3. Gravitational waves. The lengths λ of the
predicted gravitational waves are rather large: from 3⋅
103 m to 14103 ⋅ m. These waves ought to have the
higher modes. We can expect that the ratio of the
intensities of the high mode (with the l -wave length) to
the ground mode equals nl 2)/( λ , where n is the
order of the differential equation for the gravitational
waves )2( ≥n . Therefore the receiving antenna of the
dimension l about one meter has an insufficient
sensitivity to detect the gravitational waves, as such
antenna can record effectively only high modes, which
intensities decrease (to 1210− for 310=λ m). To observe
Gravitational waves of 63 1010 −=λ m we propose to
use Earth as the detector. For this the apparatus to
record the oscillations or the dislocations must be placed
in some points of Earth. This apparatus can be based on
the refraction of the laser light. The gravitational waves
can be observed by the time delay corresponding to the
passage of the wave through different points with the
light velocity.
4.4. Motion of Sun in Galaxy and Earth climate.
Propose that Sun movs in the orbit at the period about
27 Mil. years and pass through the Galaxy spiral. At the
passage through the Galaxy spiral Sun can loss some
planets and join another planets. In the Galaxy spiral the
planet orbits, the planet angular momenta, and the
magnetic fields are changed. Therefore in the spiral the
Earth climate, the direction of the axis of the rotation,
and the angular velocity, the strength of the magnetic
field, the places of North and South magnetic poles of
Earth can be changed. In particular, possibly 65 Mil.
years ago Earth transited to higher orbit and the Earth
climate become more cold. The hypothesis on Sun
motion in Galaxy may be considered as true if the times
of the mass death of animals coincide with the times of
the strong changes of the Earth magnetic field.
REFERENCES
1. A.I. Akhiezer, S.V Peletminskij. Fields and
main interactions. Kiev: “Naukova Dumka”, 1986,
552 p. (in Russian).
2. V. De Alfaro, S. Fubinis, G. Furlan,
G. Rossettin. Currents in hadron physics.
Amsterdam –London: “North–Holland Publishing
Company”, 1973, 724 p.
3. A.I. Akhiezer, V.B. Berestetskij. Quantum
electrodynamics. M:”Nauka”, 1981, 623 p. (in
Russian.)
4. A.I Akhiezer, S.V Peletminskij. Theory of
main interactions. Kiev: “Naukova Dumka”, 1993,
570 p. (in Russian).
83
|
| id | nasplib_isofts_kiev_ua-123456789-79429 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-30T22:59:03Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kulish, Yu.V. 2015-04-01T19:37:46Z 2015-04-01T19:37:46Z 2001 Interactions of elementary particles similar to gravitational one / Yu.V. Kulish // Вопросы атомной науки и техники. — 2001. — № 6. — С. 80-83. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 04.25.Dm, 02.70.Bf, 04.04.70.Bw https://nasplib.isofts.kiev.ua/handle/123456789/79429 It is shown that in the non-relativistic limit the Newtonian gravity law can be derived at the equality of the coupling constants for the electron, proton, neutron interactions with the spin-2 massless particle (quasigraviton). The interactions of the photons and the gluons with quasigraviton in one-loop approximation are studied at the contributions of quarks, leptons, W, and the scalar fields. From the absence of the quadratic divergences it is derived that the sums of the coupling constants of the quasigravitational interactions of all the quarks and all the leptons vanish. Therefore the black holes do not appear. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Quantum field theory Interactions of elementary particles similar to gravitational one Взаимодействия элементарных частиц подобны гравитационным Article published earlier |
| spellingShingle | Interactions of elementary particles similar to gravitational one Kulish, Yu.V. Quantum field theory |
| title | Interactions of elementary particles similar to gravitational one |
| title_alt | Взаимодействия элементарных частиц подобны гравитационным |
| title_full | Interactions of elementary particles similar to gravitational one |
| title_fullStr | Interactions of elementary particles similar to gravitational one |
| title_full_unstemmed | Interactions of elementary particles similar to gravitational one |
| title_short | Interactions of elementary particles similar to gravitational one |
| title_sort | interactions of elementary particles similar to gravitational one |
| topic | Quantum field theory |
| topic_facet | Quantum field theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79429 |
| work_keys_str_mv | AT kulishyuv interactionsofelementaryparticlessimilartogravitationalone AT kulishyuv vzaimodeistviâélementarnyhčasticpodobnygravitacionnym |