Guarcs in the Inside Hadronic Four-Dimensional Euclidean Space with Real Time

Non-perturbative effects are of great importance for the theory of space inside hadron. Supposing a sequence of QCD problems are concentrated in the branch of occurrences that can be described through the transition from the Minkowski space M(xM0,xM1,xM2,xM3) (M-space) into the Euclidean space inside hadron via the analytical extension of the time axis onto the lower semi plane xEi0=ixM0. In this case we get the Euclidean space with the imaginary time Eim(xEi0,xE1xE2,xE3,) (Eim is space), and XEi=XM is automatically VEi=iVM and 0 ≤ ||VEi|| ≤ 1. The use of such a space has brought to great results: the QCD valuum models, lattice calculations, string theory and so on. However, e.g. QCD in lattice can now be used only for the description of a limited class of hadronic elements of the matrix. There is no common and self-congruent description of the QCD vacuum heretofore, as well as confinement occurrence and a spontaneous disturbance of the chiral invariance. In the common case the rotation group of the Euclidean space in the plane (xE0, XE) presupposes that 0 ≤ ||VE|| ≤ ∞, while Eim0 ≤ ||VEi|| ≤ 1. At the same time Eim is not even a subspace of the Euclidean space, because it is not closed in respect of the operation of composition of vectors. Thus, an infinite velocity causing non-local (instantaneous) interactions and contained in some NQCD models lies outside the frames of Eim space. Non-local quark non-perturbative vacuum condensate plays a crucial role while creating realistic hadrons models [1]. At the same time the space correlation functions look like the curve of decreasing exponent [2] whose negative parameters include the distance of z=x-y while xEi0=const. In correspondence with [3,4] physics of non-locality starts to be seen at the distance of 0,2fm λ ≈ . The correlation length λ determines the spatial declining of bound gauge-invariant bilocal correlator of field gradient.


Introduction
Non-perturbative effects are of great importance for the theory of space inside hadron. Supposing a sequence of QCD problems are concentrated in the branch of occurrences that can be described through the transition from the Minkowski space M(x M0 ,x M1 ,x M2 ,x M3 ) (M-space) into the Euclidean space inside hadron via the analytical extension of the time axis onto the lower semi plane x Ei0 =ix M0 . In this case we get the Euclidean space with the imaginary time E im (x Ei0, x E1 x E2 ,x E3 ,) (E im is space), and X Ei= X M is automatically V Ei =iV M and 0 ≤ ||V Ei || ≤ 1.
The use of such a space has brought to great results: the QCD valuum models, lattice calculations, string theory and so on. However, e.g. QCD in lattice can now be used only for the description of a limited class of hadronic elements of the matrix. There is no common and self-congruent description of the QCD vacuum heretofore, as well as confinement occurrence and a spontaneous disturbance of the chiral invariance. In the common case the rotation group of the Euclidean space in the plane (x E0, X E ) presupposes that 0 ≤ ||V E || ≤ ∞, while E im 0 ≤ ||V Ei || ≤ 1. At the same time E im is not even a subspace of the Euclidean space, because it is not closed in respect of the operation of composition of vectors. Thus, an infinite velocity causing non-local (instantaneous) interactions and contained in some NQCD models lies outside the frames of E im -space. Non-local quark non-perturbative vacuum condensate plays a crucial role while creating realistic hadrons models [1]. At the same time the space correlation functions look like the curve of decreasing exponent [2] whose negative parameters include the distance of z=x-y while x Ei0= const.
In correspondence with [3,4] physics of non-locality starts to be seen at the distance of 0, 2fm λ ≈ . The correlation length λ determines the spatial declining of bound gauge-invariant bilocal correlator of field gradient.
In other studies, a minimal Gauss model, offered in [5], is used for condensates in a non-perturbative vacuum. The parameter of non-locality λ characterizes an average square of quarks' impulse in the QCD vacuum. Its estimations by means of QCD in lattice have shown the following range of probable values: 2 2 0.45 0.1ГэВ λ = ± q [6,7]. E im -is homomorphic in respect of the M-space and non-local, in other words, the instantaneous interactions even at some low λ value contradict with S principles.
It gives a reason to consider that the use of merely a part of fourdimensional Euclidean space volume in the models with E im does not allow using its potential to the full extent. The article expounds the first steps in the research of the inside hadronic four-dimensional Euclidean space with real time model E(x E0, x E1, x E2, x E3 ) (Е-space), where 0 ≤ ||V E || ≤ ∞, and its aim is to show the expedience of the studies in the Е-space as s probable prospective direction of sub-hadronic physics development. The article contains researches of the E-space properties in protons and it is presupposed that the obtained correlations have a common nature and can cover all the hadrons. Moreover it has been considered been considered that the models in the Е-space will not be an alternative for the theoretical developments in E im , but will extend their possibilities. The following requirement is the basic condition enabling this model to exist: Requirement 1: Space-time relations and regularities in the Е-space model mapped into the М -space must correspond to the principles of SR and be Lorentz-covariant.

Inside Hadronic Euclidean Frames of Reference
In the Е-space no frames of reference, which are microscopic in reality, can be physically implemented. To determine the spatial coordinates the laboratory frame of reference LFR with the coordinates (x M0, x M1, x M 2, x M3 ) has been used, where hadron rests, and dx E= dx M . The own time of particles in the LFR is admitted to be the temporal coordinate x E0, Thus the Е-space is "subsidiary" towards the М-space. Definition 1: Inner hadronic four-dimensional Euclidean Frame of Reference (x E0, x E1 , x E2 , x E3 ) EFR, is a system, where the space coordinates are indexed by the coordinates of (x M1, x M 2, x M3 ) LFR and the own time of the particles is equal to the own time of the particles in the LFR Where V Mi is the velocity of the i-number particle in the LFR. The transition to the other IFR is carried out by means of Lorenz transformation. The Е -space of the real particles corresponds to the М-space upper closed cone From (3.4) we can make a conclusion that there is a 4-vector of energy-momentum in the EFR, and its invariant is equal to the invariant of the corresponding relativistic 4-vector and it is one more symmetry between the LFR and EFR. Translating the 4-vector of the particle in LFR through (2.3), we obtain These values stay  -invariant.
Formula (3.3) testifies to an unusual behavior in the E-space of the kinetic energy: it diminishes when the speed grows. The next unit will demonstrate that it is so because of the energy-mass exchange between quarks and gluons.

Mechanics of quark in the proton
Here we use the model where quarks are considered electrically neutral particles, and we admit that in the center of a proton there is a hypothetical source creating central-symmetrical attractive potential V(r) of strong interactions. It is considered that this simplified model will provide the possibility to determine some peculiarities of quarks motion in the proton.
The E-invariant Lagrange function of the quark in the potential V(r) where m q -constituent mass of the quark. Е-invariance of this function is ensured by dx E = dx M potential V(r) will be identical for each proton in LFR.
On the analogy with (3.3) the energy of the system "quarkpotential V(r) » E V =E Eq -V(r) =const. (3.6) If a particle is under the influence of power parallel to the velocity, that it will change the momentum as follows: ( ) (3.7) The alteration of the energy ( ) (3.8) (3.9) From the (3.9b) and (3.6) it follows that E V =0. The zero-value of V E is a result of the fact that gluons have not been taken into account.
To ensure the constant proton mass, the alteration of the quark kinetic motion must be compensated by the relevant alteration of gluons energy -mass. Taking gluons into account.
The preserved energy of "quark -gluon -potential V(r) system makes This equation has a solution, if v G =v q . Then in the potential V(r) And the energy From Esq. (3.3) and (3.13) we can draw a conclusion, that the energy -mass of the quark translates into the energy -mass of the gluon, and their sum makes equals m q . At the same time P EG =P Eq and gluons are moving along with quarks creating valon. As a result, the constituent mass of quarks includes zero rest mass. This determines the dynamical relation of constituent and current quarks' masses. The quark mass diminishes as it approaches to the centre of a proton. This corresponds to the existing idea that quark has a minimum mass under a big transferred to it q 2 momentum. There are some scientific studies devoted to the NQCD, in which gluons are described as possessors of dynamical energy -mass [8]. Contains an approximate solution of Dyson-Schwinger equation, where a propagator of non-perturbative gluon is regulated by the dynamical generated mass of a gluon. The usage of this propagator gives an opportunity to calculate sections of pp-scattering and achieve a good concord of calculations with experimental data for an effective gluon mass of 370 MeV [9], this value corresponds to m q in the nucleon. The fact that gluon has peculiarities of a massive particle is confirmed by calculations in lattice [10,11]. In the papers [12,13] different non-zero masses of gluons have also been studied. Let us examine the quark motion in the linearly increasing potential V(r) =cr. The zero orbital moment of a proton along with experimental studies of the charge distribution in proton means that the quark is vibrating along the diameter towards the center of a proton. Let us presuppose that the quark vibrates under the power of |F z |=constant along the z axis which has a null in of the center a proton. Basing n the eq.
Where clz / = z q a F m is a "classic" acceleration? Then The dependence of the quark energy on the radius is From the (3.6) we can draw a conclusion that V(r max =r p ) =m q , where r p is the radius of a proton and r . Under the condition that in the coordinates (x E0 , z) quark makes a circumference with a radius The digits before the root take turns depending on the Thus, a vibrating quark makes two half circumferences with z>0 and z<0, moved at 1 clz 2a − . Figure 1 shows the graph of the quark oscillations. The calculation involves the rms radius of the proton r p =0.84fm.
Here we can show how the formula (3.6) is functioning. Under z=0 and V=0 the speed makes v Ezq =∞ and E Eq =0 (points A, C, E). Under a and v Ezq =0 as well as E Eq =m q , as well as V=m q (points B, D). And therefore E Eq -V=0.
This brings up a question: how do the oscillations of quarks provide total zero momentum in the motionless proton while they are oscillations? Under multi-particle interactions, a symmetric disposition of particles corresponds to the minimum of energy and therefore a proton possesses a spherical symmetry and that means that 3 quarks make diametric oscillations creating a space angle π and their impulses are getting balanced. This supposition correlates with analytical studies described in [14]; according to them effective fields in baryons has a Y-shaped configuration of quarks' plane making an equilateral triangle. This conclusion has also been confirmed by calculations in lattice [15].

Models of E-Invariant Quantized Motion of Massive Particles
A peculiarity of inside hadronic E-space is that its size in the three- dimensional space is comparable to the Compton quark wave length and the maximum value of quark kinetic energy makes m q . According to the quantum mechanics the minimum quark energy in the limited space must excess its mass. This is also applicable for oscillators' energy in the quantum field theory. Thus the wave equations cannot be applied in our case, including probability interpretation. Though the quarks' behavior in hadrons has a casual nature and the definite metric of the E-space enables to precede straight forward to the probability characteristics.

Free scalar particle
The model is oriented towards the inside hadronic space, in which a particle cannot be free, so this part is of a methodic character.
Let us introduce the probabilistic space indexed by Е-elements and defined by three quantities (Ω,∑µ) where Ω is a multitude of eve, ∑ σ is algebra of Ω subsets and µ is a positive measure normalized, and µ(Ω) ≤1. If X E is the real random variable and X Å ∈ Ω , then the distribution of X E is the probabilistic measure on Ω µ =P(X E0 <X E <X E ) Definition 2: The state of the particle is described by the function x x x describe a scalar particle then its Lagrangian will equal Ò From this it follows that equations for each phase of the i-cycle are to be solutions of (4.1) Equation (4.4a) has the following solution As attractive potential V(r) ≥ 0 equally affects the particle as well as the antiparticle, according to (3.6) For several cycles, segments 0 E x form the simplest stream with no aftereffect. For the free particle in the translation phase the displacement vector of the particle E x and vector Å k are co-directed and (4.4b) has the following solution x is the unit vector. Probability density , i.e. probability density is also positive and the mathematical expectation of the particle displacement in the translation phase is As has been assumed ,τ E and χ E are the components of E-vector and with (2.2) where τ Ml is the average cycle duration in LRF. E-invariant is a value which equals quantum of action .
On the grounds of (4.10) we consider the cycle duration in LRF x M0l to be the two-dimensional random vector with random coordinates x E0 and x E , distributed by the exponential law. Then x M0l is also distributed is also relativistically covariant, and equals to relativistic Lagrangian accurate to a coefficient and changes from  to 0.
Thus the motion of the particle in E-space is discreet, consists of alternating translation and localization phases and the resultant action for every cycle equals a quantum of motion. The averaged graph of free -particle motion in  is a random step function with the average step length τ E and the average step height E χ .
The average duration of free -particle cycle in LRF quantizes time

Free spinor particle
Spinor function x x also should be a solution to the Dirac equation in E and describe two phases of motion. To derive the Dirac equation model in E we need to take into account that The sense of such E-space partition is in the fact that in it the rotation is only possible in and consequently only bispinors have effect. Let us factorize (4.2) where l 0 Equation for the translation phase is and in compliance with (4.8) the solution is When movement is along axis х 3 ( ) ( ) All the features of the quantum theory of the scalar particle are valid for spinors as well. But in the latter case we have a new detail of helicity. In E, the helicity of massive fermions is only observed in the translation phase, and it is a "good" quantum number, whereas in M the helicity of massive fermions with a nonzero mass can't be a quantum number characterizing the particle, since it can be inverted by appropriate Lorentz transformations. Nevertheless, in nature, there exist left and right fermions that are quite different particles and this is seen in E.

Neutral spin or particle in the strong potential
If the particle is affected by the attractive potential which in the general case equals V(x 0 ,x E ), then (4.13) will take the form x works then in the localization phase (4.21) The solution to this equation is (4.22) The average duration of the localization phase is The equation for the translation form will take the form The average extent of the translation phase is With the quantized motion for V(r)=cr (3.9) takes the following form (4.27) Equation (4.27b) proves that while the translation phase is on when x E0 =const, there are instant nonlocal interactions in E. However, when mapped in M they take place with speed с.

Application of the Model of Quantized Motion of Quarcs to Determine Some Properties of Quarks in Protons Quantized motion of quarks
The calculation of the quantized motion of quarks has been done on the basis of the IVC (Figure 1) on the assumption that the quark moves along the axis z which passes though the centre of the proton, parameters of motion V τ i and V χ zi being of average value. The following data are used in the calculation: root-mean-square radius of the proton r p =0.84 fm and the quark localization phase starts. The acceleration portion starts with the translation phase at point B when v Ez =0 and the end of the translation phase coordinate x E0 has become more than 0.84fm.
The quark deceleration in the second half-period of oscillation starts also with the localization phase at point C for v Ez =∞ and z<0 and the calculation is done in the way similar to the first half-period. Here the following peculiarity is disclosed: the coordinates of the beginning of the second oscillation (0.08fm 0.32fm) are close to the accepted coordinates of the beginning of the first oscillation (0.0, 0.3fm).

Charge distribution in the proton
Central-symmetric motion of quarks (Section 3.1) makes it possible to confine to the calculation of the charge distribution for one quark considering that its charge equals the charge of a proton. The calculation is done on the assumption that V(r) =cr and the charge distribution is defined by the probability of the quark being at a given point of radius r=|Z| and this probability must be determined from the Ml i E i i . As the calculations show that the second oscillation practically repeats the first oscillation the parameters of the first oscillation are accepted as the calculation basis. The calculation of the charge density has been done under the condition that the charge is located in the spherical layer with a unit thickness which has radius r. After the approximation by the exponential function the equation for the charge density calculation is obtained For the comparison the experimental data for the electric formfactor of the proton have been used which are usually described by dipole approximation 2 2 (1 / 0.71) − = + G q [16] for the preset square of 4-momentum q 2 . This dependence gives the experimental value of charge density Graph j c (r) systematically exceeds j e (r). It is connected with the fact that definitional domain j c (r) equals 0<r<0.85 fm and the box under j c (r) equals 1 ≈ . Definitional domain j c (r) equals 0<r<∞ and the box under this curve on the section 0<r<0.9fm equals 0.6.

Conclusion
It is stated that in the E-space model, Radius-vector rotations group does not mix temporal and spatial coordinates;-kinetic energy diminishes when the speed grows. This determines the existence of constituent and current quarks and describes the dynamic relation of their masses; -to describe quantum movement in the E-space, wave equations cannot be applied. The application of the random function theory has shown that the quarks' movement consists of localization and translation phases;-helicity of massive fermions can be observed only during translation phase and is a "good" quantum number;an infinite velocity and non-local interactions connected with it while mapping in the M-space does not upset the RS-principles: the maximum interaction transmission velocity and the maintenance of causality principle; -the proton charge calculation result plausibly agrees with the experimental data; the four-dimensional values in the E-space are the 4-vector with scalar invariants which have analogies in the M-space;-the E-invariant models have a transparent physical content and are no alternative for the existing QCD methods, but expand their possibilities.