Understanding the origin of quark confinement in hadrons remains one of the most challenging problems in modern physics. Recently, the pressure distribution inside the proton was measured via deeply virtual Compton scattering. Surprisingly, strong repulsive pressure up to 10^{35} pascals, the highest so far measured in our universe, was obtained near the center of the proton up to 0.6 fm, combined with strong binding energy at larger distances. We show here that this profile can be derived semiquantitatively without any adjustable parameters using the rotating lepton model of composite particles (RLM), *i.e.* a proton structure comprising a ring of three gravitationally attracting rotating ultrarelativistic quarks. The RLM synthesizes Newton’s gravitational law, Einstein’s special relativity, and the de Broglie’s wavelength expression, thereby conforming with quantum mechanics, and also yields a simple analytical formula for the proton radius and for the maximum measured pressure which are in excellent agreement with the experimental values.

Keywords:
Hadron structure;
Proton structure;
Internal pressure distribution;
Rotating Lepton Model;
Relativistic neutrinos;
Quarks;
Gravitational mass

The proton consists of fundamental particles called quarks and gluons. Gluons carry the force that binds quarks together. Quarks are always confined in the composite particles in which they are located. The origin of quark confinement is still a subject of intensive study in modern physics. Recently, for the first time, the pressure distribution inside the proton was measured via deeply virtual Compton scattering [1, 2, 3, 4]. Strong repulsive pressure up to 10^{35} pascals, the highest so far measured in our universe, was obtained near the center of the proton up to 0.6 fm, combined with strong binding energy at larger distances. This recent pioneering experimental [1, 4] and theoretical [1, 2, 3] work on the deeply virtual Compton scattering has opened a new area of research on the fundamental gravitational properties of protons, neutrons and nuclei and has underlined that gravity plays an important, perhaps dominant, role inside protons and other hadrons [1–3].

The dominant role of gravity inside hadrons has also emerged in recent years from the rotating lepton model of composite particles (RLM) [5, 6], in which gravity causes confinement of highly energetic neutrinos in bound rotational states and thus leads to formation of quarks, hadrons and bosons [5–8]. This model follows exactly the steps of the Bohr treatment of the H atom and contains no adjustable parameters (Figure 1). It synthesizes Newton’s gravitational law, Einstein’s special relativity [9], and de Broglie’s wavelength expression, thereby conforming with quantum mechanics. The model does not require any new theory. Furthermore, it fits extremely well with current experimental evidence for the masses and other properties of hadrons. The simple structure of the RLM for the proton is also suggested by the earlier proposed [10] bagel-shaped proton geometry (Figure 1). Without any adjustable parameters, the RLM predicts the masses of hadrons [5, 6], but also of bosons [7, 8] with an accuracy of one percent [5, 6, 7, 8, 11, 12].

Two important suggestions have emerged from the RLM analysis [5, 6]: First, quarks and hadrons consist primarily of rotating neutrinos, and their mass is due to the kinetic energy of the rotating neutrinos. Second, the Strong Force can be viewed as the relativistic gravitational force. Here the RLM is used to describe quantitatively the measured pressure distribution in protons and to derive simple analytical formulae for the proton radius and for the maximum measured pressure of 10^{35} pascals. We also show that a proton structure comprising a ring of three rotating ultrarelativistic quarks with radius 0.63 fm describes the proton’s measured pressure profile semi-quantitatively without any adjustable parameters.

Similar to the Bohr model of the H atom, but utilizing gravitational instead of electrostatic forces, the RLM for the proton comprises only two equations, *i.e.*

(1)
F
=
γ
m
o
v
2
/
r
e
=
G
m
g
2
3
r
e
2

where *r _{e}* is the radius of the circle defined by the centers of the three rotating particles, which form an equilateral triangle,

(2)
γ
m
o
vr
e
=
n
ℏ

which is the historical basis of quantum mechanics and introduces quantization via the integer number *n*. Accounting for the equivalence principle, *m _{g}* equals the inertial mass,

(3)
m
g
=
m
i
=
γ
3
m
o
.

Solution of equations (1), (2) and (3) for *n* = 1, the ground state, gives

(4)
γ
q
=
3
1
/
12
(
m
P
l
/
m
o
)
1
/
3
;
m
p
=
3
γ
q
m
o
=
3
m
q
=
3
13
/
12
(
m
P
l
m
o
2
)
1
/
3

where

Setting *m _{p}*=938.272 MeV/c

(5)
r
e
=
ƛ
q
=
ℏ
/
γ
q
m
o
c
=
ℏ
/
m
q
c
=
3
ℏ
/
m
p
c
=
0.63
f
m
,

in very good agreement with the experimental value [1, 16]. It is worth noting that the first equation (4), together with (3), dictate that the gravitational mass,
*i.e.*

(6)
m
g
=
γ
q
3
m
o
=
3
1
/
4
m
P
l
.

Consequently the relativistic gravitational force and potential energy computed from eq (1) is

(7)
F
=
G
m
g
2
3
r
e
2
=
G
(
ℏ
c
/
G
)
r
e
2
=
ℏ
c
r
e
2
;
U
=
−
ℏ
c
r
e

which are the values anticipated for the strong force between two quarks [16]. These values are a factor of *α*^{−1} = 137.035 stronger than the Coulombic attraction and potential energy of an *e*^{+}*e*^{−} pair at the same distance.

The rotating quark ring structure of the proton according to the RLM allows for the quantitative description of the potential energy and pressure distribution inside a proton and their comparison with those measured in the pioneering reference [1]. We first show in Figure 1, bottom, that the rotating quark ring of the RLM creates two types of attractive strong relativistic gravitational forces. First, those pointing inward and confining the three neutrinos to their rotational orbit at

Thus, considering the distance of the quarks A and B fixed, and quark C at a variable location r on the perpendicular bisector of AB (Figure 2, bottom) one obtains, as shown in the methods section, that the total gravitational force, *F*, exerted by quarks A and B to quark C in the *x* direction is

(8)
F
r
=
−
2
ℏ
c
r
+
λ
−
q
/
2
r
2
+
r
λ
−
q
+
λ
−
q
2
3
/
2

which, in dimensionless form, is given by

(9)
F
(
z
)
=
−
ℏ
c
(
ƛ
q
/
2
)
2
[
2
(
z
+
1
)
[
(
z
+
1
)
2
+
3
]
3
/
2
]

where

(10)
z
=
r
/
λ
−
q
/
2

Equation (8) is plotted in Figure 3, as *F*(*r*)/4*π* vs *r* and *z*, since this quantity is equal to *r*^{2}*p*(*r*) (expressed in 10^{−2} GeV/fm) which has been used in the *y* axis of ref. [1] as shown in Figures 2 and 3. In this way, direct comparison of equation (8) with reference [1] is possible. The shaded part of Figure 3 is the one compared in Figure 2 with the experimental data of [1]. There is remarkable semiquantitative agreement between reference [1] and equations (8) and (9).

It is also worth noting that the force expressions (8) and (9) describe in a simple way the two key characteristics of the Strong Force, *i.e.* asymptotic freedom (*F* = 0 at *z*+1 = 0, *i.e.* at
*i.e. dF*/*dr* > 0 up to the force minimum at
*r*=0.071 fm, in excellent agreement with the experimental results [1], as shown in Figures 2 and 3.

Utilizing *F* = −*dU*/*dr*, where *U* is the potential energy of quark C due to its interaction with A and B one obtains

(11)
U
(
z
)
=
−
2
3
m
p
c
2
[
2
[
(
z
+
1
)
2
+
3
]
1
/
2
]

which is plotted in Figure 4. The minimum *U* value occurs at *z* = −1, *i.e.* at

(12)
U
m
i
n
=
−
4
3
3
m
p
c
2
=
−
0.723
G
e
V

while the corresponding value at the rotational center *z* = 0 and *r* = 0 fm is

(13)
U
o
=
−
2
(
m
p
c
2
/
3
)
=
−
0.626
G
e
V

for quark C and

(14)
U
p
=
3
U
o
=
−
2
m
p
c
2
=
−
2
(
938.3
G
e
V
)

for the entire proton, as shown in Figure 4. This is consistent with the virial theorem, *i.e.*

(15)
U
p
=
−
2
T
p

as shown in Figure 4, since, according to the RLM, the kinetic energy, *T _{p}*, of the relativistic neutrinos is the rest energy of the proton.

Equations (8) to (14) show that, while the potential energies *U _{min}* and

(16)
F
≈
4
ℏ
c
λ
−
q
2
=
4
m
p
c
2
/
3
2
ℏ
c
=
3.16
⋅
10
5
N
=
2
G
e
V
/
f
m

and the pressure, *p*, computed from *p* = *F*/4*πr*2 is of the order of

(17)
p
≈
4
ℏ
c
π
λ
−
q
4
=
4
π
(
m
p
c
2
/
3
)
4
(
ℏ
c
)
3
=
2.56
⋅
10
35
P
a

in very good agreement with reference [1]. Indeed, as shown in Figures 2, 3 and 4, these values provide a good measure of the interquark forces and pressure inside the proton, in very good agreement with the experimental values measured in [1].

In summary, there is at least semiquantitative agreement between the RLM and the recent pioneering experimental results for force and pressure in the proton [1]. This agreement provides strong support for the simple rotating quark model and demonstrates the excellent quality of the experimental data of references [1] and [4] and of their theoretical treatment [1]. It also underlines the importance of gravitational forces inside the proton and suggests that, as previously proposed [5], the Strong Force can be viewed as relativistic gravity. This suggests that the simple methodology used here to model the proton structure and to compute the mass and basic properties of protons can be also applied to other hadrons as well.

Indeed the use of RLM has been explored recently to compute the masses of several hadrons [5, 6], but also bosons [7, 8], with an accuracy of one percent [5, 6, 7, 8, 11, 12], using similar simple rotating lepton structures, the components of which are chosen on the basis of the decay products of the corresponding composite particles. It thus appears that the present methodology may be of broader usefulness for the study of the structure and mass of composite particles.

Figure 5 shows the model geometry. Particles A, B and C, of gravitational mass *m _{g}* each, lie on a circle of center M and radius R. The distance of points A and B is fixed and equal to

One notes in Figure 5 that

(18)
cos
φ
=
(
ƛ
q
/
2
)
+
r
(
A
C
)

and

(19)
cos
φ
=
(
A
C
)
2
R

Consequently

(20)
λ
−
q
/
2
+
r
2
R
=
A
C
2
=
4
R
2
cos
2
φ

Therefore

(21)
λ
−
q
/
2
+
r
=
2
R
cos
2
φ

When C reaches C^{*}, then _{}*φ* = 30*∘*, thus cos^{2}

We denote *F _{A}* and

(22)
F
A
=
F
B
=
−
G
m
g
2
(
A
C
)
2

and we consider the centripetal force, *F*, acting on particle C. It is

(23)
F
=
2
F
A
cos
φ
=
−
2
G
m
g
2
(
A
C
)
2
cos
φ

Noting from (18) that

we can write equation (23) as

(24)
F
=
−
2
G
m
g
2
[
r
+
(
ƛ
q
/
2
)
]
(
A
C
)
3

From Pythagoras’s theorem it follows

(25)
A
C
2
=
A
D
2
+
r
+
λ
−
q
/
2
2
=
3
λ
−
q
2
/
4
+
r
2
+
λ
−
q
r
+
λ
−
q
2
/
4

thus

(26)
A
C
2
=
λ
−
q
2
+
λ
−
q
r
+
r
2

From (24) and (26) it follows

(27)
F
=
−
2
G
m
g
2
r
+
λ
−
q
/
2
r
2
+
r
λ
−
q
+
λ
−
q
2
3
/
2

For *m _{g}* =

(28)
F
=
−
2
G
ℏ
c
/
G
r
+
λ
−
q
/
2
r
2
+
r
λ
−
q
+
λ
−
q
2
3
/
2
=
−
2
ℏ
c
r
+
λ
−
q
/
2
r
2
+
r
λ
−
q
+
λ
−
q
2
3
/
2

which is equation (8). Introducing

The force component in

(29)
F
z
=
−
4
ℏ
c
λ
−
q
2
2
z
+
1
z
+
1
2
+
3
3
/
2

and its minimum occurs when

(30)
2
[
(
z
+
1
)
2
+
3
]
3
/
2
−
2
(
z
+
1
)
3
2
[
(
z
+
1
)
2
+
3
]
1
/
2
2
(
z
+
1
)
=
0

which leads to

(31)
2
[
(
z
+
1
)
2
+
3
]
3
/
2
=
6
(
z
+
1
)
2
[
(
z
+
1
)
2
+
3
]
1
/
2

and to

(32)
2
[
(
z
+
1
)
2
+
3
]
=
6
(
z
+
1
)
2

Defining

(33)
(
z
+
1
)
2
=
y

it follows from (32) that

(34)
2
(
y
+
3
)
=
6
y
⇒
y
=
3
2

Consequently it follows

(35)
(
z
+
1
)
2
=
3
2
z
+
1
=
3
2

which implies that the minimum occurs at

(36)
z
=
3
/
2
−
1
=
0.225

consequently

(37)
r
=
(
ƛ
q
/
2
)
z
=
0.071
f
m

and the maximum at

(38)
z
=
−
1
−
3
/
2
=
−
2.22

therefore at

(39)
r
=
(
ƛ
q
/
2
)
z
=
−
0.70
f
m

The recently measured for the first time internal pressure profile in a proton [1] is described semiquantitatively by the Rotating Lepton Model (RLM) of composite particles. There is excellent agreement between model and experiment regarding the radius (0.63 fm) and the maximum pressure (~2.5·10^{35} Pa).

This agreement provides very strong support for the validity of the rotating lepton model (RLM) and its great advantage over the standard model (SM) which neglects gravitational forces and does not allow for any such pressure computations except perhaps via inclusion of more adjustable parameters.

The RLM also explains that the strong repulsive pressure reported in [1] near the center of the proton is in reality due to the strong gravitational attraction by the rotating neutrino ring, which attracts matter equally strongly inside and outside its radius.

We are very thankful to Professor Donna Greenberg from Harvard University for numerous helpful discussions and style corrections.

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© 2019 C. G. Vayenas *et al*., published by De Gruyter

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