Classical SU ( 3 ) Gauge Field as a Dark Matter

The model of dark matter is presented where the dark matter is a classical gauge field. A spherical symmetric solution of Yang-Mills equation is obtained. The asymptotic behavior of the gauge fields and matter density is investigated. It is shown that the distribution of the matter density allows us interpret it as the dark matter. The fitting of a typical rotational curve with the rotational curve created by the spherical solution of SU(3) Yang-Mills equation is made


Introduction
One can give such definition for a dark matter [1]: "... dark matter is matter of unknown composition that does not emit or reflect enough electromagnetic radiation to be observed directly, but whose presence can be inferred from gravitational effects on visible matter."The nature of the dark matter of the Universe is one of the most challenging problems facing modern physics.Following to L. Smolin [2] there exist five great problems in the modern theoretical physics: Problem 1. Combine general relativity and quantum theory into a single theory that can claim to be the complete theory of nature.This is called the problem of quantum gravity.
Problem 2. Resolve the problems in the foundations of quantum mechanics, either by making sense of the theory as it stands or by inventing a new theory that does make sense.
Problem 3. Determine whether or not the various particles and forces can be unified in a theory that explains them all as manifestations of a single, fundamental entity.
Problem 4. Explain how the values of the free constants in the standard model of particle physics are chosen in nature.
Problem 5. Explain dark matter and dark energy.Or, if they don't exist, determine how and why gravity is modified on large scales.More generally, explain why the constants of the standard model of cosmology, including the dark energy, have the values they do.
The problem of the dark matter is the fifth one.Direct observational evidence for dark matter is found from a variety of sources:  On the scale of galactic halos, the observed flatness of the rotation curves of spiral galaxies is a clear indicator for dark matter. The measured orbital velocities of galaxies within galactic clusters have been found to be consistent with dark matter observations. In clusters of galaxies there is a hot intracluster gas.
Its temperature allows to measure gravitational potential of a cluster.These data are in agreement with measurements of galaxies speeds and show presence of dark matter. The direct evidence of dark matter has been obtained through the study of gravitational lenses.One of the strongest pieces of evidence for the existence of dark matter is following.Let us consider a rota- of stars in a galaxy.According to Newton law   M where r r is the mass at a given distance from the center of a galaxy; N G is the Newton gravitational constant.The rotational velocity, is measured [3,4] by observing 21 cm emission lines in HI regions (neutral hydrogen) beyond the point where most of the light in the galaxy ceases.Schematically a typical rotation curves of spiral galaxies is shown in Figure 1 (for details, see Refs.[5,6]). 1 r .This is not the case, as the rotation curves appear to be flat, i.e., outside the core of the galaxy.This implies that nt   M r r  beyond the point where the light stops.This fact is the evidence of the existence of dark matter.The detailed review for the experimental evidence of dark matter can be found in [7].
Another approach for the resolution of the dark matter problem is based on a modification of Newton's law or of general relativity, have been proposed to explain the behavior of the galactic rotation curves: (a) a modified gravitational potential [10,11]; (b) the Poisson equation for the gravitational potential is replaced by another equation [12][13][14][15][16][17]; (c) alternative theoretical models to explain the galactic rotation curves have been elaborated by Mannheim [17,18], Moffat and Sokolov [19] and Roberts [20]; (d) The idea that dark matter is a result of the bulk effects in brane world cosmological models was considered in [21,22].
All above mentioned approaches to the resolution dark matter problem are based on the assumption that the dark matter is one or another kind of quantum particles.The problem for this approach is that the most of these particles are hypothetical ones: till now these particles were not observed in the nature and in spite of general enthusiasm we do not have any confidence that these particles Here we propose will be discovered.the idea that the dark matter is a classical gauge field.This approach is based on the fact that in the consequence of the nonlinearity of Yang-Mills equations there exist a spherically symmetric solution without sources (color charge).The matter density in such solution is   that radically differs from the dist on matt nsity for Coulomb solution.Thus the proposed idea is that some galaxies are immersed in a cloud of a classical gauge field.The SU(3) classical gauge field does not interact with ordinary matter because ordinary matter is colorless.Thus one can suppose that SU(3) gauge field can be invisible matter in galaxies.The problem for such consideration is why the gauge field does not fill all Universe?The probable answer is that, in certain circumstances, the gauge field goes from a classical phase into a quantum phase.Probably such transition takes place at some distance from the center of the galaxy.
Let us note that in ributi er de Refs. [23][24][25][26][27] the similar approach for a dark energy is considered.In Ref. [23] it is shown that the Born-Infeld quantum condensate can play the role of dark energy in the present-day universe.In Ref. [24], it is demonstrated that both inflation and the latetime acceleration of the universe can be realized in modified Maxwell-  F R and Yang-Mills-  F R gravities.In Refs.[25][26][27] supposed that the energy is a condensate of Yang-Mills gauge field where the effective Lagrangian density of the YM field is calculated up to 1-loop order [28,29].With the logarithmic dependence on the field strength, the effective Lagrangian has a form similar to the Coleman-Weinberg scalar effective potential [30].
], it is dark

Classical Gauge Theory ve a short introduction
In this section we would like to gi to SU(N) Yang-Mills gauge theory.The corresponding Lagrangian is where , , 1,2, ,   is the field strength; B A  is the SU(N) gauge potentia stan l; g is the coupling cont; BCD f are the structure constants for the SU(N) gauge group.The corresponding Yang-Mills field equations are 0.
Here we will consider N eq re non For our goals we will consider practically the same equations as for monopole but with different boundary conditions.Strictly speaking the solutions of Equations ( 4)- (11) for almost all boundary conditions are singular (the energy is infinite) and only for some special choice of boundary conditions we have regular solution (monopole solution with finite energy).In this work we use solution with infinite energy but we assume that at some distance from the origin the classical phase undergoes a transition to a quantum phase.

Ansatz for the Gauge Potential
The classical SU(3) Yang-Mills gauge field B A  we choose in the following form [31]  x form and by definition where  are the Gell-Mann matrices.

uations
The corresponding Yang-Mills Equations (3) with the

Yang-Mills Eq
potential ( 4)- (11) and here we introduce the dimensionless radius 0 x r r  , r 0 is an arbitrary constant.The asymptotic behavior of the functions     , v x w x by x  1 are [34]       Using the asymptotic behavior of the gauge potential (15) (16) the asymptotic behavior of the energy density is.

Numerical Investigation
In this subsection we present the typical numerical sothe consequence of lution of Equation's ( 13) and ( 14).In the occurrence of the factor 2  x in the front of left hide side of Equations ( 13) and ( 14) we have to start the numerical calculations not from 0 x  but from from = 1 x   .To do that we search an approximate solution near to the origin in te form After the substitution into Yang-Mills Equations ( 13) and ( 14), we have

The Comparison wi Curve of Spiral Galaxies
The idea presented in this work is that in a galaxy there n ordinary visible matt ich ectromagnetic waves).The vie expe tars in the galaxy out-th a Universal Rotation exists a er (barionic matter which glows) and an invisible matter (classic gauge field wh does not interact with el sible matter is immersed into a cloud of the classical gauge field.The main goal of this work is to compare th rimental rotational curve for s side light core with the rotational curve created by the distribution of the classical gauge field.
In Ref. [35] a Universal Rotation Curve of spiral galaxies is offered that describes any rotation curve at any radius with a very small cosmic variance 1.97 0.72 0.44 log 0.78 1.97 0.72 0.44 log , 0.78 is the optical radius and where is the dimensionle ass of the color fields is the dimensionless coupling constant with the experimental rotation curve for the dark matter (29).
Far away from the center the dimensionless energy density    is given from Section 2.3.The details of fitting 0 defined near to the center of galaxy re according to Equation (32) the dif-    is maximal.Thus the asymptotic e rotation curve for the domain filled with the SU(3) gauge field is  15) and ( 16).Small values of 0 r (34)-( 36) mean that we can use the asymptotical behavior (15 practically in the whole region of radius r .

Cut-Off the Region Filled with Classical Gauge Field
The gauge field distribution considered here has an ina the he value.portant to note that the gauge field in ) and (16) finite energy in the consequence of the asymptotic beh vior (18) of the energy density (18).Consequently it is necessary to have a mechanism to cut-off the distribution of the classical gauge fields in the space.In our opinion it can happen in the following way.At some distance from the origin the gauge field undergoes transition from the classical state to quantum one and t quantum field tends very fast to its vacuum expectation It is very im the vacuum state must be described in a nonperturbative rves (33) are manner.
The physical reason why such transition takes place is following.As we see from Equations ( 15) and ( 16) and Figure 2 asymptoticly the gauge potentials are oscillating functions with increasing frequency.Far from the origin the frequency is so big that it is necessary quantum fluctuations take into account.In this way the transition from the classical state to quantum one takes place.
To estimate a transition radius we follow to the Heisenberg uncertainty principle For the ansatz ( 10) and ( 11) Let us to introduce the physical com nt of the pone To an accurac a num al factor the fluctuations of the SU(3) color electric field are y of eric For the ansatz ( 10) and ( 11) The physical components of the g 1,3,4,6,8 Now we assume that th of t ponent The volume V  where supposed quantum fluctuations

  take place
The period of space oscillations by 0 r r  can defined in the follo way We suppose that the place where the SU(3) classical co me ntum fluctuations in the volume 4 V r r lor field beco s quantum one is defined as the place where the qua of the corresponding field becomes comparable with magnitude of these fields Substituting of Equations ( 40), ( 15)-( 16), ( 44), ( 45), ( 47) and (48) into Equation ( 37) we obtain where is the dimensionless coupling constant that is similar to the fine structure constant in quan-  .If we choose 1 1 g  and from Figure 2 we take 0.4 A we see that omparable that is c with 2 6.28   .Thus in this section we have shown that if the condition (49) is true then at some distance from the center the transition from the l phase to quantum one classica occurs.Unfortunately the rough estimation presented in a the r akes pl xact evaluation of the place where such transition happens it is necessary to have non-perturbative which are missing at the moment.

4.
es are not experimenteract-this section does not llow us to calculate adius where such transition t ace.For the e quantization methods

Invisibility of the Color Dark Matter
The main question in any dark matter model is its invisibility.For the model presented here the answer is very simple: the SU(3) color matter (dark matter in this context) is invisible because color gauge fields interact with color charged particles only.But at the moment in the nature we do not know any particles with SU(3) color charge.In principle such particles can be SU(3) monopoles but up to now the monopol tally registered.
For more details we write SU(3) Lagrangian in ing with matter where

Conclusion
In this work we have suggested the idea that the dark matter model is SU(3) gauge field.We have shown that in SU(3) Yang-Mills theory there exists a spherical symmetric distribution of the gauge potential with slow decreasing matter density.The asymptotic behavior of the nsity allow us to describe the rotational curve for the stars in elliptic galaxies.The fitting of the typical rotational curve gives us parameters w ment with the param solution of Yang-M erical

Appendix A: Heisenberg's Quantization of Strongly Interacting Fields Applying for Gauge Fields
In this section we would like to bring some arguments that nonperturbative quantized SU(3) gauge field tends very quickly to zero.To do so we use the Heisenberg approach [36] for a nonperturbative quantization of a nonlinear spinor field.In quantizing strongly interacting SU(3) gauge fields-via Heisenberg's non-perturbative method [37] where Q is a quantum state.However this equation will in it's turn contain other, higher order Green's functions.Repeating these steps leads to an infinite set of equations connecting Green's functions of ever increasing order.Let us note that absolutely similar idea is applied in turbulent hydrodynamics for correlation functions all orders.This construction, leading to an infinite set of coupled, differential equations, does not have an exact, analytical solution and so must be handled using some approximation.The basic approach in this case is to give some physically reasonable scheme for cutting off the infinite set of equations for the Green's functions.One can use Heisenberg's approach to reduce the initial SU(3) Lagrangian to an effective Lagrangian describing two interacting scalar fields (for details see Ref. [29]).Two scalar fields  and  appear in such approach.We assume that in the first approximation two points Green's functions are bilinear combinations of scalar fields  and  where ; the same for remaining indexes with some constants 1,2 1,2 .The assumptions (A3)-(A8) allows us to average the SU(3) Lagrangian and bring it to the form with the field equations  are constants.We suppose that this solution describes the non-perturbative quantized SU(3) gauge field after the transition from classical phase to quantum one occurs.We see that the non-perturbative quantized gauge field decreases very quickly (exponentially) after transition to the quantum phase and consequently the total mass becomes finite one.

Appendix B: Fitting of Rotational Curve of Gauge Field
For the fitting of the rotational curve (33) we use the data from the Universal Rotational Curve (27).The fitting equation is equation (33) in the form

Figure 1 .
Figure 1.Schematical rotation curve of spiral galaxies.If the bulk of the mass is associated with light, then beyond the point where most of the light stops,   M r

3 
case of the Yang-Mills uations.Equations (3) a linear generalizations of Maxwell equations.The spherally symmetric static solution in electrodynamics is Coulomb potential.Well known spherically symmetric static solution for the SU(2) Yang-Mills equations are famous monopole and instanton solutions.The monopole solution has finite energy and is a special solution for corresponding equa- ansatz (4)-(11) is the SU(3) generalization of ansatz for the SU(2) monopole[32,33].The SU(3) ansatz gives us a spherically symmetric energy distribution that is necessary to describe a spherical dark matter distribution in a galaxy.The coset com  are written in the matri

Figure 3 .Figure 3
Figure 2. The profiles of functions 

0 In Figure 4
the profiles of the dark matter rotational curves (29) (for different values 0.5,1.0,15L L  ) and fitting cu presented.The value of parameter 1one can use the asymptotical behavior ( fluctuation of the color electric potential ai A ; V  is the volume where the quannot any summation over repeating indexes.
It is easy to see that asymptoticly the solution has the formCopyright © 2013 SciRes.JMP one first replaces the classical fields by field operators B