On microscopic structure of the QCD vacuum
OOn microscopic structure of the QCD vacuum
D.G. Pak,
1, 2, 3
Bum-Hoon Lee,
2, 4
Youngman Kim, Takuya Tsukioka, and P.M. Zhang Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Sogang University, Seoul 121-742, Korea Chern Institute of Mathematics, Nankai University, Tianjin 300071, China Asia Pacific Center of Theoretical Physics, Pohang, 790-330, Korea Rare Isotope Science Project, Institute for Basic Science, Daejeon 305-811, Korea School of Education, Bukkyo University, Kyoto 603-8301, Japan
We propose a new class of regular stationary axially symmetric solutions in a pure QCD whichcorrespond to monopole-antimonopole pairs at macroscopic scale. The solutions represent vacuumfield configurations which are locally stable against quantum gluon fluctuations in any small space-time vicinity. This implies that the monopole-antimonopole pair can serve as a structural elementin microscopic description of QCD vacuum formation through the monopole pair condensation.
PACS numbers: 11.15.-q, 14.20.Dh, 12.38.-t, 12.20.-mKeywords: monopole, confinement, vacuum
I. INTRODUCTION
One of most attractive mechanisms of confinement isbased on idea that the vacuum in quantum chromody-namics (QCD) vacuum represents a dual superconductorwhich is formed due to condensation of color magneticmonopoles [1–4]. Realization of this idea in the frame-work of a rigorous theory has not been possible due to twoprincipal difficulties: the first one, the color monopolesas physical particles have not been found in experiment,and so far there is no strict theoretical evidence of theirexistence in the real QCD (except unphysical singularmonopole solutions). The second obstacle represents along-standing problem of vacuum stability since late 70swhen it was shown that Savvidy-Nielsen-Olesen vacuumis unstable [5, 6]. There have been numerous attempts toresolve this issue. Unfortunately, most of vacuum modelslack the local vacuum stability at microscopic space-timescale. Besides, the known approaches to vacuum prob-lem are based on assumption that vacuum field is static.Such an assumption is not consistent with the quantummechanical principle which implies that elementary vac-uum fields (vortices, monopoles etc.) should vibrate atthe microscopic level [7, 8].In the present Letter we explore a novel approach fora microscopic description of the QCD vacuum based onstationary vacuum field configurations. There are sev-eral no-go theorems which impose strict limitations onthe existence of finite energy static and stationary solu-tions in a pure Yang-Mills theory [9–12]. To overcomethe restrictions imposed by these no-go theorems, it wasproposed in past that classical stationary non-solitonicinifnite energy wave solutions with a finite energy densitymay correspond to quasi-particles or to vacuum states inthe quantum theory [13]. Recently an example of a regu-lar stationary spherically symmetric solution with a finiteenergy density in a pure QCD has been proposed [14].The solution represents a system of interacting static Wu-Yang monopole and time dependent off-dagonal gluonfield. It has been proved that such a solution is stable against quantum gluon fluctuations [15]. We generalizethese results to the case of axially-symmetric fields andconstruct a new class of regular axially symmetric pe-riodic wave type solutions with a finite energy densityin a pure SU (2) and SU (3) QCD. The solutions possessan intrinsic mass parameter which determines a micro-scopic space-time scale. The averaging over the time pe-riod implies that the proposed solutions can be treatedas non-topological non-Abelian monopole-antimonopolepairs. The most important result is that our solutionsare locally stable under quantum gluon fluctuations inany small vicinity of each space point. The solutionscan be served as structural elements of the QCD vacuumformed through the monopole pair condensation in anal-ogy with the Cooper electron pair condensation in theordinary superconductor. II. AXIALLY-SYMMETRIC STATIONARYSOLUTIONS IN SU (2) QCD
We start with a standard Lagrangian of a pure SU ( N )Yang-Mills theory and corresponding equations of motion L = − F aµν F aµν , ( D µ F µν ) a = 0 , (1)where ( a = 1 , , .., N ) denote the color indices, and( µ, ν = r, θ, ϕ, t ) are the space-time indices in the spheri-cal coordinates. Let us first consider stationary solutionsin a pure SU (2) QCD. One can generalize a static axiallysymmetric Dashen-Hasslacher-Neveu (DHN) ansatz [16]as follows A r = K ( r, θ, t ) , A θ = K ( r, θ, t ) , A ϕ = K ( r, θ, t ) ,A ϕ = K ( r, θ, t ) , A t = K ( r, θ, t ) . (2)The ansatz leads to a system of five differential equationswhich are invariant under residual U (1) gauge transfor- a r X i v : . [ h e p - t h ] D ec mations with a gauge parameter λ ( r, θ, t ) [17–19] K (cid:48) = K ∂ r λ, K (cid:48) = K + ∂ θ λ, K (cid:48) = K + ∂ t λ,K (cid:48) = K cos λ + K sin λ,K (cid:48) = K cos λ − K sin λ. (3)To fix the residual symmetry we choose a Lorenz gaugeby introducing the gauge fixing terms L SU (2) g.f. = −
12 ( ∂ r K + 1 r ∂ θ K − ∂ t K ) . (4)We apply a method used in solving equations for thesphaleron solution [18, 19]. First, we decompose the func-tions K i in the Fourier series K i =1 − ( r, θ, t ) = δ i C + (cid:88) n =1 , ,... ˜ K ( n ) i ( nM r, θ ) cos( nM t ) ,K ( r, θ, t ) = (cid:88) n =1 , ,... ˜ K ( n )5 ( nM r, θ ) sin( nM t ) , (5)where C is an arbitrary number, M is a mass scale pa-rameter which defines a class of conformally equivalentfield configurations due to the presence of scaling invari-ance in a pure QCD, ( r → M r, t → M t ). The structureof the series expansion (5) implies that all time aver-aged color electric components of the field strength F aµν vanish identically. Substituting the series decompositiontruncated at a finite order n f into the classical action,and performing integration over the time period, one re-sults in a reduced action which implies Euler equationsfor the coefficient functions ˜ K ( n ≤ n f ) i ( r, θ ). The obtainedreduced equations for the modes ˜ K ( n ≤ n f ) i ( r, θ ) representa well-defined system of elliptic partial differential equa-tions which can be solved by applying standard numericrecipes.To solve the equations for ˜ K ( n ≤ n f ) i ( r, θ ) we impose thefollowing Dirichlet boundary conditions˜ K ( n ) i ( r, θ ) | r =0 = 0 , ˜ K ( n ) i ( r, θ ) | θ =0 ,π = 0 . (6)Asymptotic behavior at ( r → ∞ ) of the functions ˜ K ( n ) i is determined by the equations of motion as follows˜ K ( n )1 (cid:39) a ( n )1 ( θ ) sin( nM r ) r , ˜ K ( n )5 (cid:39) a ( n )5 ( θ ) cos( nM r ) r , ˜ K ( n )2 , , (cid:39) a ( n )2 , , ( θ ) cos( nM r ) , (7)where a ( n ) i ( θ ) are periodic angle functions. We choosethe lowest angle modes for a ( n ) i ( θ ) consistent with thefinite energy density condition, in particular, in the lead-ing order one has a (1) i (cid:54) =2 ( θ ) = c i sin θ , a (1)2 ( θ ) = c sin(2 θ ).With this one can solve numerically the equations up tothe sixth order of series decomposition in the numericdomain (0 ≤ r ≤ L, ≤ θ ≤ π ). The obtained nu-meric solution implies that even order coefficient func-tions ˜ K ( n =2 k )1 , , , and odd order functions ˜ K ( n =2 k − vanish (a) (b)(c) (d)(e) (f) FIG. 1: Solution profile functions in the leading and sub-leading order: (a) ˜ K (1)1 ; (b) ˜ K (1)2 ; (c) ˜ K (1)3 ; (d) ˜ K (1)5 ; (e)˜ K (2)4 ; (f) the energy density plot ( C = 2 , g = 1 , M = 1). identically. A typical solution in the leading and sublead-ing order approximation is presented in Fig. 1.One has fast convergence for the obtained numeric so-lution. The solution profile functions of the third andfourth order, ˜ K ( n =3 , i , provide corrections less than 2%by a norm with respect to the solution obtained in theleading and subleading order. The fifth and sixth ordercorrections are less than 0 . θ = π/
2. For largevalues of the radius of the sphere enclosing the numericdomain, the energy density decreases as 1 R .In the leading order one has non-vanishing time av-eraged color magnetic fields (cid:104) F θϕ (cid:105) t (cid:39) ˜ K (1)2 ˜ K (1)3 and (cid:104) F rϕ (cid:105) t (cid:39) ˜ K (1)1 ˜ K (1)3 , which create magnetic fluxes cor-responding to a pair of non-topological monopole andantimonopole located at the origin. It is unexpectedthat a regular monopole-antimonopole pair solution inthe limit of zero distance between the monopoles doesexist in the real QCD. A brief overview of the structureof non-Abelian monopole-antimonopole fields and detailsof the numeric solution are presented in [20]. III. AXIALLY-SYMMETRIC STATIONARYSOLUTIONS IN SU (3) QCD
We are looking for essentially SU (3) field configura-tions which do not reduce to embedded SU (2) solutions.We start with a more general Lagrangian which includesadditional gauge fixing terms L gen = L − (cid:88) a =2 , , α a ∂ r A a + 1 r ∂ θ A a − ∂ t A a ) , (8)where α a are arbitrary real numbers. An axially sym-metric ansatz contains the following non-vanishing com-ponents of the gauge potential corresponding to three I, U, V -type SU (2) subgroups of SU (3) A = K , A = K , A = K , A = K ,A = Q , A = Q , A = Q , A = Q ,A = S , A = S , A = S , A = S ,A = K , A = K , (9)where the fields K i , Q i , S i depend on space-time coordi-nates ( r, θ, t ). The ansatz is consistent with the Eulerequations obtained from the Lagrangian L gen and leadsto a system of fourteen partial differential equations forthe field variables K i , Q i , S i . Note, that due to the in-troduced gauge fixing terms the equations for K i , Q i , S i do not admit any residual symmetry and represent well-defined second order hyperbolic differential equations.One can simplify further the system of equations for K i , Q i , S i applying the following reduction ansatz Q , , = − S , , = − K , , ,Q = (cid:0) −
12 + √ (cid:1) K ,S = (cid:0) − − √ (cid:1) K ,K , = − √ K , (10)with setting the parameters α = α = α . Withoutloss of generality we choose α a = 1. Direct substitutionof the reduction ansatz into the equations for K i , Q i , S i leads to only five linearly independent differential equa-tions which contain four second order hyperbolic equa-tions for the fields K , , , ( r, θ, t ) and one quadratic con-straint with first order partial derivatives r ∂ t K − r ∂ r K − ∂ θ K + 2 r ( ∂ t K − ∂ r K )+ cot θ ( ∂ r K − ∂ θ K ) + 92 csc θK K = 0 , (11) r ∂ t K − r ∂ r K − ∂ θ K + r cot θ ( ∂ t K − ∂ r K ) − cot θ∂ θ K + 92 csc θK K = 0 , (12) r ∂ t K − r ∂ r K − ∂ θ K + cot θ∂ θ K +3 r ( K − K ) K + 3 K K = 0 , (13) r ∂ t K − r ∂ r K − ∂ θ K + 2 r ( ∂ t K − ∂ r K )+ cot θ ( ∂ t K − ∂ θ K ) + 92 csc θK K = 0 , (14)2 r ( K ∂ t K − K ∂ r K ) + K (cot θK − ∂ θ K )+ K ( − ∂ θ K + r ( ∂ t K − ∂ r K )) = 0 . (15)The system of equations (11-15) is not suitable for nu-meric solving due to the presence of the constraint withderivatives and lack of explicit functions defining theboundary conditions on two-dimensional surfaces clos-ing the three-dimensional numeric domain. To overcomethis problem we employ the same method as in the caseof SU (2) QCD using the Fourier series decompositionfor the fields K i , Q i , S i as in (5) (the Abelian potential K has a decomposition similar to one for K ). Afterintegration over the time period in the classical actionone can derive the Euler equations for the field modes˜ K ( n ) i , ˜ Q ( n ) i , ˜ S ( n ) i depending on two coordinates ( r, θ ). Weapply a reduction ansatz to the obtained Euler equationsfor ˜ K ( n ) i , ˜ Q ( n ) i , ˜ S ( n ) i ˜ Q ( n )1 , , = − ˜ S ( n )1 , , = − ˜ K ( n )1 , , , ˜ Q ( n )4 = (cid:0) −
12 + √ (cid:1) ˜ K ( n )4 , ˜ S ( n )4 = (cid:0) − − √ (cid:1) ˜ K ( n )4 , ˜ K ( n )3 , = − √
32 ˜ K ( n )4 , (16)where ( n = 1 , , , ... ), and we impose a condition thatall even modes ˜ K (2 k ) i , ˜ Q (2 k ) i , ˜ S (2 k ) i vanish. Such a condi-tion resolves the constraint (15) and reduces the spaceof general solutions to the subspace of solutions with adefinite parity under the reflection θ → π − θ . It is re-markable that the ansatz (16) reduces the total numberof equations to four independent second order hyperbolicequations in each order of the Fourier series decomposi-tion (5) without any additional constraints.Note that the Abelian potentials ˜ K , are equal to eachother as it takes place in the case of Abelian Weyl sym-metric homogeneous magnetic fields providing an abso-lute minimum of the quantum effective potential [21, 22].We demonstrate existence of another type of monopolepair solution which is different from the SU (2) stationarymonopole pair considered above. To find such a solutionwe apply asymptotic conditions (7) for the functions ˜ K ( n ) i containing even angle modes a ( n )2 , ( θ ) and odd angle modes a ( n )1 , ( θ ). Imposing vanishing Dirichlet boundary condi-tions at the origin and along the boundaries ( θ = 0 , π ), (a) (b)(c) (d)(e) (f) FIG. 2: Profile functions for the monopole pair solution inthe leading order: (a) ˜ K (1)1 ; (b) ˜ K (1)2 ; (c) ˜ K (1)4 ; (d) ˜ K (1)5 ; (e)the energy density plot; (f) the energy density contour plotin Cartesian coordinates ( g = 1 , M = 1). we solve the equations for ˜ K ( n ) i in the fifth order approxi-mation. The obtained solution profile functions ˜ K (1) i andenergy density plots are presented in Fig. 2.Consider the magnetic flux structure of the obtainedsolution when the observation time is much larger thanthe period of time oscillations. The time averaged radialcomponents of the Abelian field strength in the leadingorder include only non-linear terms (cid:104) F θϕ (cid:105) t = −
34 ˜ K (1)2 ˜ K (1)4 , (cid:104) F θϕ (cid:105) t = + 34 ˜ K (1)2 ˜ K (1)4 , (17)where the Abelian potential ˜ K (1)4 contains the angle de-pendent factor sin θ , and ˜ K (1)2 is an even function withrespect to the reflection symmetry θ → π − θ . Themagnetic fields ˜ F , θϕ create opposite magnetic fluxesthrough a sphere of radius R with a center at the originwhich correspond to non-topological antimonopole andmonopole located at one point. Careful numeric analysisshows that stationary solutions corresponding to a lowestenergy in a chosen finite numeric domain are classified byonly two parameters, the conformal parameter M and an amplitude c of the oscillating Abelian potential ˜ K (1)4 . Adetailed structure of the numeric solution up to the fifthorder series decomposition is given in [20]. IV. MICROSCOPIC QUANTUM STABILITY OFTHE STATIONARY SOLUTIONS
The Savvidy QCD vacuum based on the classical ho-mogeneous color magnetic field is unstable due to thepresence of an imaginary part of the effective action [5, 6].Usually one expects that introducing time dependentcolor fields as vacuum makes worse the vacuum stabilitysince the color electric field leads to an imaginary partof the effective action as well [23]. Surprisingly, it hasbeen found that non-linear plane wave solutions makethe problem of vacuum stability more soft, in a sense,that an equation for the unstable modes is very similarto the equation for an electron in the periodic poten-tial [14]. This gives a hint that one can find a properstationary periodic wave type solution which providesa stable vacuum. Indeed, recently it has been provedthat a stationary spherically symmetric generalized Wu-Yang monopole solution leads to a stable vacuum [15].We will prove the quantum stability of the stationaryaxially-symmetric solutions under small quantum gluonfluctuations in the case of SU (2) and SU (3) QCD.To verify the stability of the solutions it is suitableto apply the quantum effective action formalism. Weconsider one-loop quantum effective action expressed interms of functional operators [24] S = −
12 Tr ln[ K abµν ] + Tr ln[ M ab FP ] ,K abµν = − δ ab δ µν ∂ t − δ µν ( D ρ D ρ ) ab − f acb F cµν , (18) M ab FP = − ( D ρ D ρ ) ab , where the Wick rotation t → − it has been performedto provide the causal structure of the operators, and D µ , F cµν are defined with a classical background field B aµ .The operators K abµν and M ab FP correspond to gluon andFaddeev-Popov ghosts. Note that the expression (18) isvalid for arbitrary background field and does not dependon a chosen gauge for the background field due to useof a gauge covariant background formalism [25]. Effec-tive action describes the vacuum-vacuum amplitude, andthe presence of an imaginary part of the action impliesvacuum instability. Therefore, if the operator K abµν is notpositively defined then an unstable mode will appear asan eigenfunction corresponding to a negative eigenvalueof the following “Schr¨odinger” equation K abµ Ψ bν = λ Ψ aµ , (19)where the “wave functions” Ψ aµ ( t, r, θ, ϕ ) describe gluonfluctuations. Note that the ghost operator M ab FP is posi-tively defined and does not produce instability [6]. Thepotential in the operator K abµ does not depend on theasimuthal angle. Due to this one can separate a cor-responding angle dependent part from the function Ψ aµ and solve the eigenvalue equation in a three-dimensionaldomain ( t, r, θ ). Substituting interpolation functions forthe stationary magnetic solutions in the leading order,one can solve the eigenvalue equation (19). In the caseof SU (2) stationary solution a full eigenvalue spectrumis divided into four sub-spectra corresponding to fourdecoupled systems of equations: (I) Ψ , (II) Ψ , Ψ ,(III) Ψ , Ψ , Ψ , Ψ , (IV) Ψ , Ψ , Ψ , Ψ , Ψ . The low-est eigenvalue is positive, and it is reached by a solutionsatisfying the system of equations (II), the correspond-ing eigenfunctions are plotted in Fig. 3 ( g = 1 , M = 1).Other systems of equations, (I,III,IV), have a similar (a) (b) FIG. 3: Non-vanishing eigenfunctions corresponding to thelowest eigenvalue λ = 0 . ; (b) Ψ . structure of the eigenvalue spectrum.In the case of SU (3) stationary monopole pair solutionthe equations in (19) are not factorized, and one has tosolve a full set of thirty two differential equations. Nu-merical results of solving the “Schr¨odinger” eigenvalueequation with the stationary monopole-antimonopolebackground field show that the eigenvalue spectrum ispositively defined. The obtained dependence of the low-est eigenvalue on the size of the chosen numeric domainfor large values of L confirms the positiveness of theeigenvalue spectrum, Fig. 4. This proves the quantumstability of stationary monopole-antimonopole pair so-lutions in QCD. The quantum stability of SU (2) and SU (3) stationary background fields has been checked forsolutions with amplitude values of the Abelian potential K ( r, θ, t ) in the interval (0 ≤ c ≤
2) and with conformalparameter values (0 ≤ M ≤ M, c unstable modes appears which destabilize the vacuum. V. WEYL SYMMETRY AND MICROSCOPICSTRUCTURE OF THE QCD VACUUM
Let us consider symmetry properties of essentially SU (3) stationary solutions. The classical Yang-Mills La-grangian can be rewritten in terms of Weyl symmetric (cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) L Λ FIG. 4: Dependence of the ground state eigenvalue λ ( L )on the size L of the numeric domain in the cases of SU (3)(red curve) and SU (2) (blue curve) stationary monopole-antimonopole pair solutions ( g = 1 , M = 1). fields [22] L = (cid:88) p =1 , , (cid:110) −
16 ( G | pµν ) − | D pµ W pν − D pν W pµ | − ig G pµν W ∗ pµ W pν (cid:111) − L (4) int [ W ] , (20)where G pµν are Abelian field strengths containing thegauge potentials A , µ , the complex fields W pµ representoff-diagonal gluons, and the index p counts the Weyl sym-metric gauge potentials. Note that the Lagrangian is notWeyl symmetric under permutation of I, U, V subgroups SU (2) since the quartic interaction term L (4) int [ W ] is notfactorized into a sum of separate parts corresponding to I, U, V sectors. It is remarkable that the Lagrangian onthe space of essentially SU (3) stationary solutions pos-sesses a high symmetric structure. First of all, substi-tuting the reduction ansatz (10) for general functions K i , Q i , S i into the Lagrangian, one can verify that L (4) int obtains an explicit Weyl symmetric form L (4) int [ W ] = 98 (cid:88) p =1 , , (cid:16) ( W ∗ pµ W pµ ) − ( W ∗ µ ) ( W ν ) (cid:17) (21)Secondly, each I, U, V sector in the Lagrangian (20)contains cubic interaction terms corresponding to theanomaly magnetic moment interaction which is preciselythe source of the Nielsen-Olesen vacuum instability [6]. Itis surprising, within the framework of the ansatz (10) onehas complete mutual cancellation of all cubic interactionterms. This implies that on the space of Weyl symmet-ric fields the classical action describes a generalized λφ theory.A simple consideration shows that our approach toQCD vacuum problem based on stationary Weyl sym-metric monopole pair solutions opens a new perspec-tive towards construction of a microscopic theory of vac-uum and vacuum phase transitions. First of all, notethat a system of separated stationary generalized Wu-Yang monopoles and antimonopoles can not be stabledue mutual attraction between the monopole and anti-monopole. In addition, despite on the quantum stabilityof a sinlge stationary spherically symmetric monopole,the solution is rather classically unstable with respectto small axially-symmetric field deformations [14]. Thisimplies that axially-symmetric solutions are more prefer-able as candidates for the vacuum. Another importantfeature of the monopole pair solution is that it representsa non-trivial essentially non-Abelian field configurationwhich describes a pair of monopole and antimonopole lo-cated at one point. This implies that monopole and an-timonopole, as well as two monopole-antimonopole pairswith opposite color orientations, can merge into a sta-ble state with a finite energy density in the limit ofzero distance between the monopole and antimonopole.In other words, the existence of a stable solution fora pair of monopole and antimonopole located at onepoint prevents from annihilation and disappearance ofthe monopoles. This is contrary to the case of Dirac andWu-Yang pair of monopole and antimonopole which an-nihilate when they meet each other.We expect that the QCD vacuum is formed due tocondensation of monopole-antimonopole pairs, and themicroscopic vacuum structure is characterized by fewparameters: the conformal parameter M , the amplitudeof oscillations c of the Abelian gauge potential K in theasymptotic region, and the concentration of monopolepairs at zero temperature. Numeric analysis shows thatwith increasing temperature the internal energy of eachmonopole pair increases and at some critical values of theparameters ( M > , c >
2) the monopole-antimonopolepair becomes unstable. Note that in the confinementphase the vacuum averaging value of the gluon fieldoperator (cid:104) | A aµ | (cid:105) vanishes since the size of the hadron ismuch larger than the characteristic length λ M = 2 π/M of the vacuum monopole field oscillations. To describedynamics of the vacuum structure at finite temperatureone should apply the Euclidean functional integralformalism with time integration in the finite interval(0 ≤ t ≤ β = 1 /kT ). It is clear that at high enoughtemperature the upper integration limit β will be lessthan λ M . This will lead to a non-vanishing vacuumaveraging value of the gluon field operator, (cid:104) | A aµ | (cid:105) , andtransition to the deconfinement phase with spontaneoussymmetry breaking where the gluon can be observed asa color object. VI. CONCLUSION
In conclusion, we have proposed a new class of regularaxially-symmetric stationary solutions in a pure SU (2)and SU (3) QCD. The solutions possess interesting fea-tures such as an intrinsic mass parameter, a vanishingclassical canonical spin density. Such properties serveas a heuristic argument to existence of a stable quan-tum vacuum condensate in the quantum theory. Af-ter time averaging over the period the solutions corre-spond to color magnetic field configurations which haveasymptotic behavior similar to one of the non-Abelianmonopole-antimonopole pair. A careful numeric analysisconfirms stability of the stationary solutions under smallgluon fluctuations within the framework of one-loop ef-fective action formalism. As it is known, in QCD thequantum dynamics leads to generation of the mass gap,or the so-called vacuum gluon condensate parameter. Sothat, the mass scale parameter M of the classical sta-tionary solutions is related to the finite mass gap param-eter and characterizes the microscopic scale of the vac-uum structure. The presence of such a parameter allowsto describe phase transitions in QCD. The most impor-tant step in construction of the full microscopic theory ofthe QCD vacuum is to study condensation of monopole-antimonopole pairs. This will be considered in a separatepaper. Acknowledgments
One of authors (DGP) thanks Prof. C.M. Bai forwarm hospitality during his staying in Chern Insti-tute of Mathematics and Dr. Ed. Tsoy for usefuldiscussions of numeric aspects. The work is sup-ported by: (YK) Rare Isotope Science Project of Inst.for Basic Sci. funded by Ministry of Science, ICTand Future Planning, and National Reserach Foun-dation of Korea, grant NRF-2013M7A1A1075764;(BHL) NRF-2014R1A2A1A01002306 and NRF-2017R1D1A1B03028310; (CP) Korea Ministry ofEducation, Science and Technology, Pohang city, andNRF-2016R1D1A1B03932371; (DGP) Korean Federa-tion of Science and Technology, Brain Pool Program,and grant OT-Φ2-10. [1] Y. Nambu, Phys. Rev.
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