**On the relation between short-range forces and the concept of neutrality in Hidrino and Widom-Larsen theories**

Jacques Chauveheid

**Abstract**

A unified theory

**[1]**details the relation between the strong-nuclear force and nuclear reactions, nuclear fusion included. The asymmetry between electron and nucleon sustains the primary concept of electrical neutrality through a weak nuclear force mechanism.__Key-words:__

*strong-nuclear force, weak interaction, two-body problem, Newton's third principle, neutron stars.*

__I. INTRODUCTION__**A. Preliminary remarks**

If quantum mechanics can
provide quantitative expressions of forces in conformity with the work of Erhenfest and the principle
of correspondence

**[2]**, recognized quantitative expressions for nuclear and weak forces do not currently exist**[3].**In addition, the four basic forces do not depend on temperature, since measured in vacuum between particles.
In one of his books

**[4]**, Abraham Pais recalled a comment by Rutherford during the 1914-1919 period**:**"the Coulomb forces dominate if v (speed of alpha particles) is sufficiently small", evidencing by these words the velocity-dependence of the strong-nuclear force. However, since Rutherford did not apparently refer to temperature, optimal conditions for nuclear fusion do not necessarily arise in disordered configurations characterized by extremely high temperatures, such as those encountered in stars like the sun. Even compared with galaxy formation, hot fusion in many stars seems the slowest and most inefficient physical phenomenon in the universe, because the sun's ten billion year lifetime has an order of magnitude similar to the age of the universe, this circumstance having been highly beneficial for the life on earth.
Although not based on equations, Rutherford’s conclusion constitutes the
essence of the “cold” approach to nuclear fusion and reactions starting from
moderate energy levels, instead of extreme temperatures hardly controlling with
precision the physical parameters ruling nuclear phenomena, besides the great
difficulty of chasing particle targets moving at incredibly high velocities. In
this view, a better theoretical understanding of these parameters will help
nuclear technologies.

**B. Theoretical antecedents**

Eddington mentioned the concept of asymmetric affine connection in
1921 and pointed out applications in microphysics, but he did not pursue this
idea

**[5]**. In 1922, Elie Cartan introduced geometric torsion, as the antisymmetric part of an asymmetric affine connection. In May 1929, Cartan wrote a letter to Einstein**[5]**in which he recommended the use of the differential formalism he developed, but Einstein did not follow Cartan's advice.
Between 1944 and 1950, J. Mariani published four papers dealing with
astrophysical magnetism

**[6]**and introduced an "ansatz" structurally similar to that used in the present theory. The German word "ansatz", used by Ernst Schmutzer (correspondence), refers to a supposed relationship between fields of distinct origin, for example geometric contrasting with physical. Einstein also used an ansatz when he identified gravitation with the 4-space metric, but he did not put it in the form of an equation, presumably because being trivial.
The organization of the paper is the following

**:**Section II details the Lagrangian formulation and the calculus of variations. Section III is about field equations and quantitative expressions of forces. Section IV introduces the short-range force between charged particles, first referred to as strong-nuclear between nucleons. Section V is on Yukawa and complexity. Section VI details the short-range forces in both systems electron-proton and electron-neutron, evidencing a weak nuclear mechanism in LENR technologies.
When not stated otherwise, mathematical conventions are those of reference

**[1]**.

__II. THE THEORY AND ITS METHOD__
Following Einstein's program, the field Lagrangian of this theory

**[1] is**essentially gravitational and electromagnetic, with five fields and a new constant. This Lagrangian retrieves Einstein's equation for gravitation, and Maxwell's linear electromagnetism in a first approximation. The torsion structure of this Lagrangian is part of an extended (4-dimensional) Einstein-Hilbert Lagrangian, whose full affine connection includes the torsion T schematically introduced by the ansatz T = FJ (without indices), where F is the electromagnetic field and J is the electric current density. Torsion produces three quadratic Lagrangian terms evidencing quadratic electromagnetic couplings subsequently describing short-range forces, besides magnetic moments coupled with electromagnetism.
A second Lagrangian, in simpler reduced form for motion, produces
the forces acting on massive matter, through coordinate variation. These forces
include short-range forces Interpreted as strong-nuclear and weak, besides spin
forces

**[7]**not calculated so far. A third quantum Lagrangian is also part of the theory but does not include yet the specific quadratic field couplings relative to the short-range forces, which might lead (?) to strong-nuclear and weak forces expressed in terms of probability packets**[2]**.
Due to the local character of physical forces, implied by the
presence of the current J and the non-vanishing of torsion in massive matter,
the first and second Lagrangians only define field equations and forces inside
massive matter, here synonymous to electrically charged particles such as
electrons and quarks. This is so because forces act on massive matter at its
precise location, not close to it in vacuum if one accepts that massive particles
are not singularities of fields in vacuum. Nevertheless, this formalism is
easily extended to the usual description of forces in vacuum, together with
cosmology

**[1]**.
Somewhat summarizing the nuclear problem, a coupling of torsion to
massive matter, via electrodynamics, evidences the quadratic structure of
torsion in an enlarged Einstein-Maxwell theory implying squared
electromagnetism covering short-range forces such as nuclear and weak.

About equations,
attempting to extract all interactions from a system of field equations is not
applicable in this theory, because one derives field equations by varying
separately all fields representing variable entities in a field Lagrangian.
From a second Lagrangian for motion, one obtains a unique equation of motion,
containing all interactions, by varying the 4-coordinates of one massive
(charged) particle. There is therefore no reason to confuse both processes,
apart from their common field Lagrangian. Moreover, the same Lagrangian field
couplings were introduced in a third quantum Lagrangian that includes
Schrödinger's field, which produces a wave equation retrieving Schrödinger's
equation in a non-relativistic weak field approximation. This operation leads
to energy definition through Hamiltonian wave-solutions of Schrödinger's
time-dependent equation (of evolution)

**[1]**.
In conclusion, these
three Lagrangians describe the fields inside matter and the forces acting on
this massive matter, all massive particles being electrically charged.
Moreover, these forces are naturally extended in vacuum. In relation with this,
physics appears quite far from a unique theoretical model, besides the Standard
Model and a long list covering classical mechanics, thermodynamics, large
number of quantum approaches, etc...

__III. THE EQUATIONS__
In general relativity,
the trajectories of matter are geodesics. This is the consequence of the
variational postulate δS = 0, whose action S is defined by the line-integral
implying the gravitational force

**[8]**in a curved 4-space in general relativity**:**
S =

**∫**- mcds =**∫**p_{k}dx^{k}, (1a)
where p

_{k}is the 4-momentum defined by
p

_{k}≡ mcdx_{k}/ds ; (ds^{2}≡ - dx_{k}dx^{k}). (1b)
For other interactions,
gravity is switched off

**[5]**and the formalism of special relativity is used. Such interactions are derivable from the action**[8]**
S =

**∫**(p_{k}+ eN_{k})dx^{k}, (2)
where e is the electric
charge and N

_{k}is the enlarged vector potential constructed from the Lagrangian densities containing the electric current density. An ensuing equation of motion, encompassing all interactions besides gravity, is derived from (2) by the sole variation of coordinates. One verifies
δ(ds

^{2}) = 2ds(δds) = - 2dx_{k}δdx^{k}. (3)
Eq. (3) implies

(δp

_{k})dx^{k}≡ 0, (4)
obtained by replacing p

_{k}with its expression (1b). For the variation of (2), one uses δdx^{k}= dδx^{k}and δN_{k}= ∂_{i}N_{k}δx^{i}, integrating by parts according to the known procedure**[8]**and finds
δS =

**│**(p_{k}+ eN_{k})δx^{k}**│**-**∫**[dp_{k}- e(∂_{k}N_{i}- ∂_{i}N_{k})dx^{i}]δx^{k}(5)
(from points A fo B).

In accordance with usual
limit conditions, the null-variation of S (δS = 0) implies the equation of
motion

F

_{k}≡ dp_{k}/dt = e(∂_{k }N_{n}- ∂_{n}N_{k})dx^{n}/dt, (6)
where F

_{k}is the generalization of the Newtonian force in special relativity. Since dt is not an invariant, F_{k}is not a vector.
Since (2) avoids the
summation symbol for various particles, the equation of motion (6) is
restricted to a system formed by different fields and only one particle, at the
approximation that one moving body does not affect the fields. The aim is to
delimit the general problem of interactions in the simplest case, being aware
that this technique is apparently limited to a 2-body problem. This method is
now applied to the field Lagrangian £

**[1]**defined by
£/√g ≡ (1/2K)[R + T

^{a,bc}(T_{a,bc}+ ΦJ_{(a}F_{bc)})] + A_{i}J^{i}
- (1/4μ

_{o})F_{ik}F^{ik}+^{ }αΦJ_{i}J^{i}, (7)
where g

_{ab}is the symmetric metric tensor with g ≡ - det(g_{ab}). T_{a,bc}is the torsion tensor and parentheses around three indices mean their cyclic permutation. J_{k}is the electric current density, A_{k}is the 4-vector potential, F_{ik}is the electromagnetic tensor. R is Riemann's scalar, K is Einstein's constant of gravitation and μ_{o}is the magnetic permeability of vacuum. Φ is a scalar field and α is a (new) constant. Furthermore, agreeing with Poincaré's definition of science as a system of relations**[9]**, only the relation between torsion and physical fields is meaningful regarding the relation between geometry and physics.
In line with the next
equations (16) to (18), the line-integral (2) for motion will arise as a
4-volume integration of the Lagrangian terms, which in (7) include the electric
current density J

^{i }containing the 4-velocity c(dx^{i}/ds) for coordinate variation. For motion, one therefore discards the terms not containing J^{i}**.**
In special relativity
(gravitation switched off

**[5]**), this procedure thus leads to the second reduced Lagrangian scalar L for motion, given by
L ≡ (1/2K)T

^{a,bc}ΦJ_{(a}F_{bc)}+ A_{i}J^{i}+ αΦJ_{i}J^{i}, (8)
where

T

_{a,bc}= - (Φ/2)J_{(a}F_{bc) }(9)
is the equation-definition
for torsion

**[1].**One details
J

_{k}≡ ρ_{o}u_{k}= ρ_{o}c(dx_{k}/ds) = = ρ(dx_{k}/dt) (10)
(u

_{k}is the 4-velocity), where ρ_{o }is the rest electric charge density and ρ is this charge density in the referential of the observer, so that (10) implies
ρ = ρ

_{o }/ (1 - v^{2}/c^{2})^{1/2}. (11)
Using (9), one puts (8)
in the form

L = N

_{i}J^{i}+ αΦJ_{i}J^{i}, (12)
with

N

_{a}≡ A_{a}- (3Φ^{2 }/ 4K)J_{(a}F_{ik)}F^{ik}, (13)
due to the substitution
of torsion by the right member of (9) and the identity

J

_{(a}F_{bc).}J^{(a}F^{bc)}_{ }= 3J_{a}F_{bc }J^{(a}F^{bc)}. (14)
One first shows that the
second "nuclear" term including J

_{i}J^{i }in the right member of (12) produces the line-integral (1a) after 4-volume integration, according to the mass condition**[1]**
m = αΦρ

_{o}e, (15)
in which the Φ-field
plays a key-role in next equations (16) to (18). Φ originally came from
solutions of Einstein's equation for gravitation in a static, spherically
symmetric space-time inside matter

**[1]**. Such solutions would not exist if αΦ were a constant. The Φ-field was introduced as a variable physical quantity, uniting in matter the mass m with its corresponding electric charge e, both quantities varying in relation with their common radius.
Using (15) and the
notation u

_{k}≡ J_{k}/ρ_{o}for relativist 4-velocity, one writes**∫**dt(αΦJ

_{k}J

^{k})d

^{3}x =

**∫**dt(αΦρ

_{o}u

_{k}ρ)(dx

^{k}/dt)d

^{3}x =

**∫**(m/e)u

_{k}ρdx

^{k}d

^{3}x

= (1/e)

**∫**p_{k}ρd^{3}xdx^{k}=**∫**p_{k}dx^{k }, (16)
where Φρ

_{o}becomes a constant at the particle level, in relation with m and e becoming the respective mass and electric charge of a particle. N_{i}J^{i}in (12) then gives**∫**dt(N

_{k}J

^{k})d

^{3}x =

**∫**dt(N

_{k}ρdx

^{k}/dt)d

^{3}x =

**∫**eN

_{k}dx

^{k}, (17)

and gets

S ≡

**∫**(L.d^{3}x)dt =**∫**(p_{k}+ eN_{k})dx^{k}. (18)
Field equations and equations of
motion are distinct objects. From the first field Lagrangian £, one gets field
equations by varying the fields. From the second matter Lagrangian L,
simplified for motion, one derives an equation of motion by varying the
coordinates representing the location of matter.

In this theory of
motion, massive charged particles are not point-like. However, the framework of
motion relative to such particles has no interest in a variable charge density
inside matter. One therefore interprets
ρ

_{o}as the average value of rest charge density.

__IV. THE SHORT-RANGE FORCE AT THE ELECTRO-NUCLEAR APPROXIMATION__
To evidence short-range
forces, one discards the Lorentz force engendered by A

_{a}in the right member of (13), whose second term will produce an attractive force in 1/r^{5}, of intensity proportional to the square of non-relativistic momentum multiplied by the particle volume at rest (see further). This short-range force is distance-dependent**[10, 11]**, in opposition to spin forces of greater local character.
One now calculates the
components f

_{a}of this short-range force in the approximation of electromagnetism reduced to its electric components, due to the non-relativistic neglect of the magnetic field**[2]**. Accordingly, this short-range force of electro-nuclear character reads
f

_{a}≡ dp_{a}/dt = e(∂_{a}Q_{b}- ∂_{b}Q_{a})dx^{b}/dt, (19)
with

Q

_{a}= - (3Φ^{2 }/ 4K)J_{(a}F_{ik)}F^{ik}, (20)
from (13), thus without
A

_{a}implying the Lorentz force. Using (15), Eq. (20) then becomes
Q

_{a}= - (3m^{2 }/ 4Kα^{2}e^{2}ρ_{o}^{2})J_{(a}F_{ik)}F^{ik}. (21)
Due to the existence of
quarks, the simplest interaction between nucleons is a 6-body problem. However,
one will treat the system nucleon-nucleon as a 2-body problem in a first
approximation.

Rectangular coordinates
characterize the referential Oxy where a first static proton is located at the
origin O. A second proton moves above the x-axis, its velocity

**v**being parallel to the y-axis. Using x^{o}= ct in the approximation of point-like protons, the components E_{x}and E_{y}of the electric field produced by the proton at rest are
E

_{x}= cF_{xo}= ex/r^{3}; E_{y}= cF_{yo}= ey/r^{3}(22)
(r

^{2}≡ x^{2}+ y^{2}; v_{y}≡ v), at the approximation of Maxwell's electric field in vacuum.
The electric current
density reads

J

_{y}= ρ_{o}v_{y }/ Γ**;**J_{x}= 0, (23)
where Γ ≡ (1 - v

^{2}/c^{2})^{1/2}. (24)
One calculates

J

_{(x}F_{ik)}F^{ik}= 2J_{y}F_{xo}F_{yo}**;**J_{(y}F_{ik)}F^{ik }= - 2J_{y}(F_{xo})^{2}, (25)
and finds

Q

_{x}= (-3m^{2}v / 2Kα^{2}ρ_{o}Γc^{2})(xy / r^{6}), (26a)
Q

_{y}= (3m^{2}v / 2Kα^{2}ρ_{o}Γc^{2})(x^{2 }/ r^{6}), (26b)
Q

_{o}= 0. (27).
The components of f

_{a}then read
f

_{x}= e(∂_{x}Q_{y}- ∂_{y}Q_{x})v, (28a)
f

_{y}= 0. (28b)
Writing x/r = sinδ,
where δ is here the angle between the straight line defined by the two proton
centers and the velocity of the moving proton, Eq, (28a) gives

f

_{x}= (-9m^{2}v^{2}Vsinδ**/**2Kα^{2}c^{2}Γ)r^{-5}(29)
(f

_{y}= 0) where V ≡ e/ρ_{o}is the proton volume, defined before in the approximation of a constant rest electric charge density ρ_{o.}
In the more general case of two
particles with respective charges e

_{1}and e_{2}, the particle 1 being at rest, (29) goes over into
f

_{x}= -[9(m_{2})^{2}(v_{2})^{2}V_{2}sinδ / 2Kα^{2}c^{2}Γ ](e_{1}**/**e_{2})^{2}r^{-5}, (30)
where m

_{2}, v_{2}, V_{2}, e_{2}are the respective mass, velocity, volume at rest (V_{2 }≡ e_{2}/ρ_{o}), and electric charge of the moving particle, e_{1}being the charge of the particle at rest, r being the distance between them. Eqs. (28) and (30) determine the short-range force exerted by particle 1 on particle 2, which is perpendicular to the velocity of particle 2. The eventual quark structure also implies that (e_{1})^{2 }and (e_{2})^{2}are sums of squared quark charges in the case of nucleons. Accordingly the expression "summed charge squared", worth 1 for a proton and 2/3 for a neutron, figures in ref**[12]**. Moreover, Eq. (30) applies to all massive particles, here built on electricity (Mie's idea, see below).
These sums of squared
quark charges also relate to the equation e = Cr, unnumbered formula between
Eqs. (45) and (46) in ref.

**[1]**where e is the charge, C is a constant and r is the radius of a fundamental charged particle such as a quark, which presents another relation with the words "neutron mean squared intrinsic charge radius" in ref.**[13]**. The 1/ r^{5}dependence of the strong interaction came out in 1926 after unsuccessful attempts with 1/ r^{2}and 1/ r^{4}**[4, 7]**, reference in which A. Pais recalls the discovery of a non-central component of the nuclear force, discovered by Schwinger and Bethe in 1939**[4]**. This non-central character was confirmed in the forties**[4]**. Attractive forces are not necessarily central and the range of the strong-nuclear force is infinite, however its intensity rapidly decreases with distance, reason why this force is referred to as short-range.
There is more on the
subject of fundamentals, briefly recalled now. As dynamics is an essential
feature of physical theories

**[2]**, Mariani's ansatz toward field unification centered on the motion of massive matter, but the problem of motion has little to do with a field theory in vacuum,where particles are singularities of the fields (see above). One thus sees that the problem of motion essentially resides inside massive matter, which implies interior solutions of field equations in matter**[1]**. In 1912, Gustav Mie introduced this idea of matter constituted by fields. Hermann Weyl detailed Mie's theory in his book*Space-Time-Matter*(Dover, NY 1952, p. 206), in which Weyl reproduced Mie's words when writing**:***matter is "purely" electrical in nature*. Einstein and Leopold Infeld retook this idea of matter constituted by fields in*The Evolution of Physics*(Simon & Schuster, NY 1938, p. 242).
The present theory in matter reproduces field theories in
torsion-free situations, Equations of motion for gravitation and
electromagnetism (Lorentz's force) are also retrieved. Furthermore, one retrieves quantum mechanics
for the hydrogen atom from identical field couplings, including Dirac's
magnetic dipole and spin-orbit energies, by introducing additional constants
such as the electron charge and mass, besides Planck's constant

**[1]**. On the other hand, the quantum treatment of strong-weak couplings has been done regarding magnetic moments**[1]**, but the procedure has to be applied to short-range forces, strong-nuclear and weak, because quantum theory is over-imposed on the classical structure.

__V. YUKAWA AND COMPLEXITY__
Since perpendicular to
velocity according to Eqs. (28), the short-range force defined by Eq. (30) does
not produce energy (see below). Moreover, this force is firmly non-central,
which contrasts with central forces oriented along straight lines connecting
two particle centers

**[11]**. A complex and chaotic kinematics, together with a rather unexpected dynamics characterize therefore strong-nuclear forces manifesting a tendency toward unpredictability and instability**[4]**, besides radioactivity opposing nuclear stability.
Apart from this
stability issue, nuclear forces are velocity-dependent and complex

**[4]**, so that Lev Landau's suggested to limit the study of the strong-nuclear force to binary nuclear interactions ("two by two"**[7]**). This was probably due to the Yukawa distance and the short-range character of strong-nuclear forces, to what A. Pais added "*the nuclear 2-body problem is just too complicated*"**[4]**. However, Pais' words mean that the complexity resides in the nuclear problem, not necessarily in theories describing nuclear phenomena.
About this issue of
complexity, the present theory may also look complicated, example of its
non-linear version of Maxwell's theory, whose approximation for the electric
field in Eqs. (22) may be responsible for the vanishing work (energy) produced
by the strong-nuclear force, according to Eqs. (28). From another standpoint,
perpendicularity does not materialize exactly in the real world because
3-dimensional orthogonality is not an invariant in 4-dimensional relativity
theory. In contrast, the attractive character of forces is an invariant
according to a definite arrow of time.

Within a
non-relativistic approximation (Γ= 1), one makes (e

_{1})^{2}= (e_{2})^{2}and takes the absolute value of the strong-nuclear force between two protons or two neutrons from Eq. (30), equaling its right member to mv^{2}/(R/2) for motion around the center of mass and assuming a circular motion. The factor v^{2}then simplifies in both members, which gives
R

^{4}≈ 9mV / 4Kα^{2}c^{2}, (31)
(sinδ = 1).

The binding of two
protons, or two neutrons, consequently implies their fixed separation defining
the Yukawa distance R figuring in Eq. (31) for a common particle volume (see
further). R is currently valued at 1.4 fermis and allows the calculation of the
constant α according to a nucleon radius of 0.7 fermis (approximation).

__VI. THE WEAK NUCLEAR FORCE IN THE SYSTEM ELECTRON-NUCLEON__**A. Electrons and nucleons**

In ref.

**[7]**, Landau wrote**:**"...quantum mechanics occupies a very original position in the range of physical theories; it contains classical mechanics as a limiting case and at the same time needs this limit to be founded" (translation). In agreement with Bohr's correspondence, one then sees the importance of a classical theory of motion and forces, before a quantum treatment evidencing probability packets**[2]**, which would bring new elements, besides reassuring the calculation of the important tunnel effect within an appropriate framework.
In this classical
approach, the strong-nuclear force does not produce energy to realize nuclear
fusion in a first approximation, This situation looks deceiving but leads to an
apparently positive conclusion. Nuclear fusion, based on the unique role of the
strong-nuclear force, still does not present a high degree of probability after
many years of hot fusion experiments. Other phenomena might therefore play a
significant role, if theoretically validated. In addition, LENR experiments
already produce excess heat, which establishes the experimental foundation for
future developments, starting from simple configurations, to improve in a
second stage.

Before neutron formation
with emission of a neutrino, the electron capture by a proton is a bound system
electron-proton constituting a two-body system respecting Newton's principle of
equality between action and reaction, in an energy conservative bound system
not interfering with energy-momentum conservation of fields around it. The
phenomenon of orbital electron capture from
atomic levels K, L, M exists in neutron stars

**[14]**, its classical description implies the presence of two short-range forces, detailed now without writing the electrostatic force between proton and electron, because this force cancels out in both members of next Eq. (33) expressing equality of action and reaction for motion around the center of mass.
Quark charges in a proton are 2/3, 2/3, -1/3, whose sum of squared
charges = 1 is the same as the squared charge of the electron, which implies (e

_{1})^{2}= (e_{2})^{2 }in Eq.(30). Within a non-relativistic approximation (Γ =1), one writes an equation whose left member is the intensity of the short-range force exerted by a proton on an electron, the right member expressing the converse. Simplifying both members by
9 / 2Kα

^{2}c^{2}r^{5}, (32)
yields

(m

_{e})^{2}(v_{e})^{2}V_{e}= (m_{p})^{2}(v_{p})^{2}V_{p,}(33)
with sinδ =
1 for both forces in opposition, in relation with round orbits of particle
centers (analogy with Bohr's 1913 model). The square root of both members
produces

m

_{e}v_{e}= m_{p}v_{p}(V_{p }/_{ }V_{e})^{1/2}. (34)
In Eq. (30),
the attractive short-range force is proportional to the squared
non-relativistic momentum of a particle, now retrieved in the left member of
Eq. (33), whose square root gives m

_{e}v_{e}in the left member of Eq. (34). The time-derivative of both members then leads to
[m(dv/dt)]

_{e}= [m(dv/dt)]_{p}(V_{p }/_{ }V_{e})^{1/2}. (35)_{ }
whose left member is the force dp/dt figuring in Eq. (19), acting
now on the electron. For consistency, therefore not only in relation with
Newton's third principle, one writes

[m(dv/dt)]

_{e}= [m(dv/dt)]_{p}, (36)
so that Eq. (35) reduces to

V

_{p}= V_{e,}(37)
the
electron and the proton have therefore the same volume in this theory.

One
repeats the same procedure, by adapting Eq. (33) to the bound system
electron-neutron, taking into account the sum of squared quark charges
worth 2/3 for the neutron, which gives

(m

_{e})^{2}(v_{e})^{2}V_{e}(2/3) = (m_{n})^{2}(v_{n})^{2}V_{n}(3/2). (38)
In
relation with Eq. (36), one writes

m

_{e}v_{e}= m_{n}v_{n}, (39)
for
equality of absolute values of momenta, so that Eq. (38) reduces to

V

_{n}= (4/9)V_{e}. (40)
implying V

_{n}= (4/9)V_{p}in relation with Eq. (37). This result would be unphysical if the proton and neutron volumes need to be equal for defining a unique Yukawa length in Eq. (31). However, the imperfect equivalence between (p-p) and (n-n) interactions is an established fact (“very similar” in ref.**[7]).**In this view, the 18 % difference for R in Eq. (31) might be physical, and not important enough to discard this type of capture, in spite of its apparent lack of observational data. All this seems good news for the possibility of electron capture by a neutron, part of the Widom-Larsen theory**[15]**which brings forth a neutral deuteron in conformity with Eq. (38).
The spherical symmetry, fundamental in Bohr's model of the hydrogen
atom and Schrödingers's equation in atomic theory, besides the statistical
interpretation of the wave-function, seems typical in relation with a classically
defined “weak nuclear” force mechanism in Eq. (33), referring to two distinct
forces of equal intensity in the system electron-proton. The weak force acts on
the electron with the same intensity as if the source were another electron,
instead of a proton, and the second
strong-nuclear force acts on the proton with the same intensity as if the
source were another proton, instead of an electron. Short-range forces work
this way because proportional to the squared mass of the particle acted upon,
multiplied by the factor (e

_{1}**/**e_{2})^{2}previously defined in relation with quarks.
Here,
the bound system electron-proton represents the electron capture through two
forces, strong-nuclear and weak, system literally characterizing a “weak
nuclear” mechanism of forces, according to which proton and electron react. In
relation with this, ref.

**[14]**indicates that “β-decay applies to all nuclear reactions implying neutrinos or anti-neutrinos” summarized by
p
+ e

**↔ n + ν**^{-}_{e}, (41)
adding
that “all these reactions are ruled by the weak nuclear force”, represented
here by two forces, strong-nuclear and weak of equal intensity, which are
Newton's inevitable action and reaction. The left member of (41) presents the
electron capture explaining the neutralization of the Coulomb
barrier (including eventual shielding during definite time intervals before
neutron formation ?). The Widom-Larsen theory includes this electron capture
producing a neutron, subsequently fusing with a nucleus since being essentially
acted upon by the strong-nuclear force between nucleons.

In
conclusion, the bound system electron-proton with two distinct forces, weak and
strong-nuclear of equal intensities, contrasts with the scenario of two
strong-nuclear forces representing action and reaction in the usual outline of
nuclear fusion between protons, as in the sun where the presence of electrons
in momentary bound systems electron-proton could play a role in the fusion of
hydrogen

**[16]**, but this is another story. Obviously, the differentiation between short-range forces in the weak nuclear scenario of electron capture by a proton**[14]**supports the low energy approach to nuclear fusion.**B. Numbers**

From Eq. (30), the coupling constant of the weak force
between two electrons is (m

_{e})^{2}, which constitutes a first number. According to the approximation m_{n}≈ m_{p}, the strong nuclear force between two nucleons is characterized by the coupling constant (m_{p})^{2}. Based on m_{p}= 1836 m_{e}, the coupling constant of the strong-nuclear force between nucleons is 3.37 x 10^{6}times greater than the coupling constant of the weak force between electrons, this the second number.
In the bound system electron-nucleon, the weak nuclear
mechanism includes the two coupling constants (m

_{p})^{2 }and (m_{e})^{2}because two distinct forces are present. Since the weak force acts on the electron and the strong-nuclear force acts on the nucleon, the mean value of these two coupling constants would be half the second number above, so roughly 1.69 x 10^{6}because (m_{e})^{2}is negligible in regard to (m_{p})^{2}. However, this third number is unrelated to relative intensities of these two forces in a bound system, where the equality between action and reaction implies the equality of both force intensities. In relation with Eq. (30), the non-central short-range force is velocity-dependent, so that relative force intensities are relative values of squared non-relativistic momenta, in opposition to relative values of coupling constants.

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