_{1}

^{*}

When the electrically charged elementary particles of “normal” matter like protons and electrons mutually interact, their masses interact gravitationally and charge electrically. There is no interaction between the mass of one and charge of other particle. In this paper, we describe a prediction of the existence of a “pseudo-charge” of the same size as the common elementary electric charge, which seems to be possessed by neutrinos. If the prediction is relevant to the reality, the pseudo-charge interacts with the mass of normal particle in the interaction between this particle and neutrino. Consequently, the cross-section in a collision between neutrino and particle of normal matter is many orders of magnitude lower than that in the mutual collisions of normal-matter particles. However, the pseudo-charge of one neutrino interacts with the pseudo-charge of other neutrino in a mutual interaction of neutrinos and, consequently, their collisional cross-section is predicted to be again relatively large, essentially the same as that in, e.g., electron-electron collisions. We propose an experimental verification of the possible existence of neutrinic pseudo-charge with the help of two mutually crossing neutrino beams.

Albert Einstein devoted many years of his scientific activities to a search for an unified description of four well-known interactions in the universe. From his era, this effort has continued till the present.

In the course of the unification, we achieved a certain progress, which was published in our two previous papers [

The concept of the interaction described by the solution also explains why the orientation of gravity must be the same as the orientation of primary electric force between the charges of opposite polarity, why the magnitude of the charge of all electrically charged elementary particles is the same, and why there is inertia linked to the mass of particles, but not to their electric charge.

The orientation of gravity with respect to the orientation of electric force between two charged particles was, basically, derived from the equation

where

The quadrate

where

with

In these equations,

If we establish operators

then Equation (1) ca be re-written to the matrix form

where

Auxiliary parameter

Because we need only three matrices of the set of four to decompose Equation (1) and, moreover, there are a lot of combinations of the sequential order of these matrices, there is a plenty of possibilities for the decomposition, whereby each of these possibilities implies a specific kind of interaction between two considered particles. To describe the interaction between, in the given context, “normal” particles, as electrons or protons, one specific combination of all was used and this case is reminded in Section 2. Assuming the series of the com- binations with

Of the combinations listed in Section 3, we choose one interesting interaction and suppose that this interaction can be assigned to some hypothetical particles. The interesting feauture of the interaction is that the electric charge of particles possessing the relevant property does not interact with the electric charge of normal particles, but with their mass. So, we further refer to this special charge as “pseudo charge”, this interaction as to the “mixed interaction” and corresponding hypothetical particles as the “mixed-interaction particles” (M-IPs, here- inafter). We analyze some further basic properties of them in Section 4. Because these properties appear to be, at least qualitatively, identical to those, which are known for neutrinos, we identify the M-IPs to these tiny, weird particles and derive a consequence relevant to the dark matter in the universe.

An experimenal verification of the pseudo charge of neutrinos, if exists, is proposed in Section 5. We suppose that the experiment could be performed already at the present, using the technical facilities of the most advanced physical laboratories (particle accelerators) in the world.

The Maxwell equations comprise a dependence on time. To eliminate this quantity in course to find a solution, the most obvious (if not unique) way is to assume the solution of the vectors of electric intensity and magnetic induction in the forms

respectively. In these forms,

In our new concept of the unified interaction, we do not regard the time solutions for

Let us now consider an electrically charged particle, which generates the electric field in the surrounding space. The time dependences (9) and (10) are, however, valid wherever in the space. In every point, the vectors are harmically changing and, hence, they obviously represent the description of a spherical wave centred on the particle. There exists a possibility that this wave can deliver an impulse to another particle and, thus, accelerate it. However, the impulse must change its size and (what is important, in the given context) its orientation. Hence, the force action corresponding to such an interaction mechanism would result only in an oscillation of influenced object. No permanet action in a single direction, which is characteristic for the electrostatic force and gravity, could occur in this concept.

The permanent single-direction action occurs, in our concept, assuming the validity of old idea of screening one object by another, which was suggested by Le Sage in 1782. In more detail, he introduced the idea of fluid fulfilled with particles, which deliver an impulse to the objects randomly from all directions. A screening of one object by another disturbs the spherical symmetry of impulse delivered to a given object and, thus, the complete compensation of impulse delivered to this object. Since there is delivered a lower impulse from the direction of screening, the particle is accelerated in this direction due to the excess of impulse delivered from the opposite direction. However, while Le Sage applied his idea to the macroscopic bodies, we apply it to the “cores” of ele-

mentary particles. The size of the cores is gauged to be about

Planck length and

Our interaction mechanism is based on concept of permanently irradiated and absorbed wave by the considered test particle. When the wave is irradiated, it delivers a reaction impulse to the particle. If the returning wave impacts the particle, it simply delivers to it the carried impulse. Because of the perfect spherical symmetry of the outcoming and incoming waves, the impulse in a given direction is exactly compensated by the equal-size but oppositely oriented impulse from the opposite direction. Thus, the net impulse per time unit, i.e. the force, is zero.

The test particle is accelerated when the symmetry is disturbed. In analogy with the Le Sage’s concept, it can happen by screening of the impacting wave by another particle(s) or when the test particle is forced to accelerate and the symmetry is then broken due to the blue shift and red shift of the parts of wave in the direction of acceleration and in opposite direction, respectively. The first reason leads to the force laws and the second one to the law of inertia.

In our concept of interaction, every elementary particle is regarded as a source of waving. The amplitude of generated spherical wave increases with the decreasing radial distance from the particle, down to a certain critical distance, where the existence of wave reaches so high measure that the wave is able to absorb the wave of other particle. We assume that the absorption is proportional to the wave amplitude, which corresponds, in the case of static interaction, to the amplitude of the electric intensity, in the critical distance. Thus, the absorption ability, i.e. the effective absorption cross-section in fact, can be derived from the electric intensity in the critical distance. Since the intensity has both real-valued and imaginary-valued components, there are obviously the absorbing agents in both the real sub-space and in the imaginary sub-space. Each of these agents has its specific, real-valued or imaginary-valued, cross-section.

In physics, the quantities acquiring the imaginary values are regarded as unobservable and, usually, are ignored in the interval, where their values has non-zero imaginary component. In our concept, the imaginary part of the wave generated by the test particle in the position of an absorbing particle is absorbed by the agent in the imaginary sub-space. Mathematically, the corresponding force due to the absorption is proportional to the product of imaginary part of the electric intensity and imaginary cross-section. This product of two imaginary values is however real-valued and, therefore, observable in the real world. In this way also the imaginary parts of physical quantities appear to be observable.

In our earlier papers [

Function

where

In Equation (14), the r-dependent part of the first term of

which is formally identical to Equation (1) and, thus, it can be decomposed and solved in the same manner as the latter.

Let us denote the Pauli matrices as

We remind the properties of these matrices:

To obtain the solution for the normal-matter particles interacting in the common way, we put

where

We denoted

The total energy of particle with the rest angular frequency

In the hydrogen atom as well as in the macroscopic regime of unified interaction, energy

for two oppositely charged elementary particles.

Subsequently, the relations (5) giving the terms of wave vector

where we utilized de Broglie relation

As we mentioned above, the wave gains the ability of the effective absorption of other wave in some very small radial distance from the particle, which we denote by

In the distance

Part

elementary particles (as seen in relation (28)), we neglect

The argument of functions sine and cosine, kr, in relations (19) and (20) equals the ratio

Using all expressions for the distance

These relations characterize the absorption ability of the charged particle with rest mass

In distance r, the total surface area of the spherical wave is

with the plus for the positive and minus for the negative charge in the last relation. It appeared that the components

The variants of force corresponding to

While product

uct

We can see that product

In contrast to B-variant, the amplitude of the A-variant of intensity,

To describe the common, well-known interaction between the paticles of normal matter in Section 2, we considered one specific combination of matrices

If

Solving this matrix equation, we obtain two variants,

It appears that the A-variant is the same for all studied combinations and equals

for

in the case of variant A.

As already explained in Section 2, the force action of an acting particle on a test particle depends on the intensity of the test particle and, therefore, on its amplitude, actually given by relation (35) in the A-variant. And, the force action depends on the absorption rate, the factor of which can be derived applying relation (35) to the acting particle in the distance

Then Equation (35) in

For variant B, i.e. for

Let us now describe the possible kinds of macroscopic-type interaction with the help of corresponding force, which is the product of the intensity amplitude (relation (35) in the case of A-variant or the form given in the fourth column of

No. | solution for | solution in | ||||
---|---|---|---|---|---|---|

1. | ||||||

2. | ||||||

3. | ||||||

4. | ||||||

5. | ||||||

6. |

Within variant A, there is only the force, which is proportional to

Concerning variant B, it appears that this part of force corresponds to the common primary electric force or another, newly predicted, kinds of force. Among the found combinations, the primary-electric interaction between two particles of normal matter is described by forms, which are listed as No. 4 in

The solutions of B-variant Nos. 3 and 5 in

Because of two signs in front of both above-given forms, each force, on the hypothetical as well as normal- matter particle, can be as repulsive as attractive. If we consider this kind of force between a proton as the test particle and the hypothetical particle with the rest mass again equal to the rest mass of proton,

As stated in Section 2, the inertia force occurs due to an acceleration of particle causing a blue shift and red shift of emitted and absorbed wave. Hence, the force depends on the angular frequency of the wave. The intensity linked to the solution of B-variant No. 3 and 5 is reciprocally proportional the square root of mass and, hence, the angular frequency of the wave associated with the particle. Thus, we could expect a dependence of inertia force of this particle on the B-variant of solution for the radial part of the intensity. However, the corresponding integral to calculate the inertia force equals exactly zero, therefore the inertia force is, also here, generated only by the mass-linked term of A-variant (see [

The solution of B-variant No. 2 in

Finally, there are the solutions of B-variant Nos. 1 and 6, which are practically identical in respect to the macroscopic-interaction properties, in

In this section, let us consider the solution of B-variant No. 6 (

In other words, while the force between two particles of normal matter is the product of (i) real-valued mass

We note, there is also the common gravitational interaction between the M-IP and normal-matter particle. Its magnitude is, however, expected to be many orders of magnitude lower than the magnitude of the force described above. (The ratio of the magnitudes of both forces cannot be specified numerically, since we do not know the size of the rest mass of M-IP.)

We see that the M-IPs are predicted to possess an electric charge, which however does not interact with the charge of normal-matter particles, but with the mass of the latter. This peculiar property was the reason of why we established, in Section 1, referring to this charge as “psedo-charge” to distinguish it from the common electric charge.

Because the solution of B-variant No. 6 contains both signs, plus and minus, the force between the M-IP and particle of normal matter can be as repulsive as attractive. So the force between two M-IPs. In contrast to normal, primary-electric force, the orientation of force between two M-IPs with pseudo-charges of the same (opposite) polarities is attractive (repulsive), however.

As stated in Section 2, the inertia occurs due to asymmetry of the spherical wave caused by an accelerated motion. In the direction of the acceleration (in the opposite direction), the wave is blue-shifted (red-shifted) and, therefore, its frequency increases (decreases) and, thus, the carried impulse is enlarged (reduced). The shift of the wavelength can, however, appear only if the formula describing the wave, which corresponds to the amplitude of the electric-field intensity, contains the frequency or angular frequency.

One can see that while the solution of A-variant is proportional to mass

Let us now estimate the collisional cross-section, when the M-IPs collide with the electrically charged elementary particles of normal matter. At the collision, the kinetic energy of incident M-IPs is

where

Z_{i} and Z_{t} are the ionization degrees of the incident and target particles, respectively. Since we consider charged elemenary particles,

The size of the pseudo-charge is the same as the size of elementary electric charge. Hence, factor

One can immediately reveal that this cross-section is extremely small. For example, it is about the factor of

Because neutrinos have an extremely small (inertial) mass, if any finite, their effects on the particles of normal matter are negligible regardless they possess or do not possess the pseudo-charge. Even if these particles have the pseudo-charge, they remain hardly detectable. An exception, which can be utilized in an experimental detection of pseudo-charge, is discussed in Section 5.

Of course, the consequence of the pseudo-charge, if exists, on the structure of the universe would be significant. Neutrinos usually have a large kinetic energy per unit mass, therefore they are deliberated from various bounds with the other particles, among themselves, or with the macroscopic objects. The exception can, however, be expected in the case of compact objects, like those residing in the centers of galaxies and quasars. The extremly strong gravity of these objects can obviously trap the neutrinos tending to pass through them or in their close vicinity. If we do a thinking experiment in which we want to move with such the trapped neutrinos, their motion is possible, due to the strong bounding, only if we move the whole object. So, the bound neutrinos acquire the inertia of the compact object and they then behave, with this “borrowed” inertia, as the particles with the mass equal to

The interstellar and intergalactic space, as well as the space among the clusters of galaxies is likely fulfilled with a “neutrino pseudo-plasma”. (We identify the pseudo-plasma to a gas consisting of ionized particles possessing the pseudo-charge.) We assume the plasma instead some bound neutrinic structures because of the large kinetic energy of neutrinos per unit mass. The large energy means that the neutrino “soup” is hot and, hence, in the form of the pseudo-plasma.

Because of the neutrinos trapped inside the compact objects, the homogenity of the pseudo-plasma in the universe is obviously disturbed. In a smaller or larger region around every compact object, there is probably a relatively larger concentration of neutrinos with the positive sign of pseudo-charge (this pseudo-charge attracts the normal matter). On contrary, a relatively larger concentration of negative pseudo-charge neutrinos can be expected especially in the space among the galaxy clusters, if we believe that the total pseudo-charge of the universe is zero.

As we mentioned above, the dominant interaction between the particles of normal matter and neutrinos, when the latter are identified to the M-IPs, is that between the mass of former and pseudo-charge of the latter. In the interaction between two neutrinos/M-IPs, the dominant interaction must be, however, that between their pseudo-charges, since the product of pseudo-charge,

Since the magnitude of the elementary pseudo-cherge is the same as the magnitude of elementary electric charge, the magnitude of elementary pseudo-charge-with-pseudo-charge interaction is exactly equal to the magnitude of elementary charge-with-charge interaction. Hence, the collisional cross-section in the mutual collisions between neutrinos/M-IPs is again given by relation (38) with

Because of this “normal” size of the collisional cross-section, a significant fraction of neutrinos in a collimated beam should be effectively dispersed by another neutrino beam, which crosses the first beam. The measured flux of neutrinos should detectably change in dependence on whether the crossing beam is switched on or off. This is the principle to experimentally reveal the eventual existence of the pseudo-charge of neutrinos.

At the experiment, one has to be aware of the possibility that there are three types of neutrinos, electron, muon, and tau, and not all may possess the pseudo-charge. This situation can be analogous as that with, e.g., nucleons or pions, some of which are electrically neutral.

Our analysis of the potential kinds of interaction, based essentially on the Maxwellian electromagnetism, predicts an existence of elementary particles possessing a pseudo-charge with the size equal to that of common elementary electric charge, which however does not interact with the charge of normal particles, but with the mass of the latter.

In a collision of such a hypothetical particle with a particle of “normal” matter, as proton or electron, the collisional cross-section is many orders of magnitude smaller than that in, e.g., proton-proton collision. (At low incident energies, the cross-section in the hypothetical particle to proton collision is about 36 orders of magnitude smaller than the cross-section in the proton to proton collision.) Because of this property, we suggest to identify these hypothetical, pseudo-charge possessing, particles to neutrinos.

In the mutual collisions of the hypothetical particles/neutrinos, the pseudo-charge of incident particle is predicted to interact with the pseudo-charge of target particle. So, we have the pseudo-charge-with-pseudo-charge interaction, which is analogous to the charge-with-charge interaction in, e.g., proton-proton or electron-electron collisions. Hence, the neutrino-neutrino collisional cross-section, if neutrino actually possesses the pseudo-charge, is relatively large, essentially the same as that in, e.g., electron-electron collisions (assuming that the incident particles have the same energy, of course).

If the neutrino-neutrino collisional cross-section is as large as predicted (like that of “normal” charged particles in their mutual collisions), it can be verified in an experimental way, most probably using the current facilities, or facilities which can be constructed using the currently available technologies. Namely, we are able to measure a flux of neutrinos in a beam. If this beam is crossed by neutrinos emitted in other beam, the latter should significantly disperse the neutrinos in the former beam and a decrease of the flux would reveal the existence of the neutrinic pseudo-charge.

The work was supported, in part, by the VEGA―the Slovak Grant Agency for Science, grant No. 2/0031/14, and by the Slovak Research and Development Agency under the contract No. APVV-0158-11.

LubošNeslušan, (2015) Neutrinos as the Particles with Mixed Interaction and Prediction of Large Neutrino-Neutrino Collisional Cross-Section. Journal of Modern Physics,06,1756-1767. doi: 10.4236/jmp.2015.612177