On microscopic structure of the QCD vacuum

We propose a new class of regular stationary axially symmetric solutions in a pure QCD which correspond to monopole-antimonopole pairs at macroscopic scale. The solutions represent vacuum field 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 element in microscopic description of QCD vacuum formation through the monopole pair condensation.


I. INTRODUCTION
One of most attractive mechanisms of confinement is based on idea that QCD vacuum represents a dual superconductor which is formed due to condensation of color magnetic monopoles [1][2][3][4][5]. Such a mechanism was confirmed in lattice studies [6][7][8][9][10], however, realization of this idea in the framework of a rigorous theory has not been possible due to two principal difficulties: the first one, monopoles as physical particles have not been found in experiment, and so far there is no strict theoretical evidence for their existence in the realistic theories of fundamental interactions. All known monopole solutions are either singular or need for their construction additional new hypothetical particles like Higgs scalar fields. The second obstacle represents a long-standing problem of vacuum instability since late 70s when it was shown that the Savvidy-Nielsen-Olesen vacuum is unstable [11,12]. There were numerous attempts to resolve this issue, the most advanced vacuum model is known as a "spaghetti" vacuum [13][14][15]. Unfortunately, most of QCD vacuum constructions lack the local vacuum stability at microscopic scale.
In the present Letter we explore the idea that stationary monopole like solutions can provide a stable vacuum condensation in QCD. An idea that stationary nonsolitonic solutions may correspond to (quasi-) particles was sounded long time ago [16][17][18]. It was supposed as well that vortex strings forming the "spaghetti" vacuum should be vibrating at small scale due to quantum mechanical consideration [15]. The stationary non-linear solutions in the pure QCD possess intriguing features like the intrinsic mass scale parameter, zero spin and vanishing total color magnetic charge. Besides, the origin of color confinement is intimately related to gauge invariance of the vacuum [4]. So that a Weyl symmetric classical monopole solution with a total vanished color magnetic charge seems more preferable field configuration to form a stable vacuum condensate [19][20][21].
We start with a standard Lagrangian of a pure SU (N ) Yang-Mills theory and respective equations of motion where (a = 1, 2, .., N ) denotes the color index, and (µ, ν = r, θ, ϕ, t) are the world indices in the spherical coordinates. In the case of SU (2) QCD one can generalize the static axially-symmetric Dashen-Hasslacher-Neveu (DHN) ansatz [22] by adding a temporal component of the gauge potential The ansatz leads to a system of five partial differential equations (PDE) which is still invariant under residual U (1) gauge transformations [23-25] where λ(r, θ, t) is an arbitrary gauge function. One can fix the residual symmetry by passing to a singular gauge where the Abelian potential K 3 describes static monopoles immersed in the field of dynamical offdiagonal gluons [26]. In the axially-symmetric case the singular gauge is not suitable for numeric solving since the reduced system of PDEs includes three second order hyperbolic type equations and two constraints containing first and second order space derivatives. So we choose a Lorenz gauge by introducing the gauge fixing terms With this all five equations for K i become well-defined hyperbolic differential equations which can be solved by using standard numeric recipes. To solve the equations arXiv:1703.09635v1 [hep-th] 28 Mar 2017 we impose boundary conditions which are consistent with local solutions near the boundaries and with finite energy density condition where C 4 is an arbitrary number. In the asymptotic region, r ∞, the equations admits the following basis solutions in the leading order where the mass scale parameter M appears due to the presence of scaling invariance of the equations, (r → M r, t → M t), and it defines a class of conformally equivalent solutions. In further numeric calculations we set M = 1 without loss of generality. We use a method applied in solving equations for the static sphaleron solution [24,25]. First we decompose the functions K i (r, θ, t) in Fourier series in a consistent manner with the local and asymptotic solutions Substituting the series decomposition truncated at a finite order n = N into the classical action one can perform integration over the time period. Variation of the simplified classical action provide final Euler equations for the coefficient functionsK [24,25]. The obtained equations forK (n) i (r, θ) represent a well-defined system of two-dimensional elliptic PDEs which is suitable for numeric solving. We solve the equations starting from the leading order approximation with the functions K (1) i (r, θ) and trace convergence of the full solution with subsequent solving the system of PDEs up to order n = 6. The obtained numeric solution up to order n = 6 implies that even order functionsK vanish identically. The solution for the monopole-antimonopole pair in the leading and subleading order is presented in Fig. 1. The energy density is finite at the origin and has maximums located at the origin and along the torus center line. In the asymptotic region the energy density decreases along the radial direction as 1 r 2 . In the case of SU (3) QCD we look for an essentially SU (3) stationary monopole like solutions which do not correspond to SU (2) embedded solutions. We introduce the following axially-symmetric ansatz which includes non-vanishing gauge potentials corresponding to all three 4 ; (f) the two-dimensional energy density plot.
where the fields K i , Q i , S i depend on (r, θ, t). We employ the same method as in the case of SU (2) QCD and solve the reduced equations for the field modesK up to the fifth order of series decomposition (7). In general, the obtained system of reduced equations describes a wide class of regular stationary axially-symmetric solutions. To find a subclass of Weyl symmetric solutions one can simplify further the system of reduced equations applying the following ansatz where n = 1, 3, 5 and the even modesK vanish. The ansatz is consistent with the system of equations for the field modes and reduces the number of the equations in each order (n = 1, 3, 5) to four independent PDEs for the basic modesK (n) 1,2,4,5 . Note that Abelian field strengths corresponding to potentialsK 3,8 are equal to each other as it takes place in the case of Abelian Weyl symmetric homogeneous magnetic fields [19,20]. To solve the equations we set the following boundary conditions which are consistent with the local solutions at the origin r = 0 and satisfy the finite energy density conditioñ where the initial functions c (n) 1 (θ) are arbitrary functions. We use a non-linear stationary solver based on the Newton iterative method which finds a solution starting with proper chosen initial profile functions. With this one obtains a numeric solution which is presented in Fig. 2. One can observe that SU (2) and SU (3) solutions for the Abelian gauge potentials K 3 , K 4 in Figs. 2c, 3c have the same profiles in the asymptotic region where both of them contain the angle dependent factor sin 2 θ. The presence of this factor implies that a corresponding radial component of the color magnetic field, H θϕ = ∂ θ K 3 , creates magnetic fluxes through the upper and lower semispheres with opposite signs, and a total magnetic flux vanishes identically. Such a behavior is an inherent feature of the non-Abelian theory which admits a solution for a pair of monopole and antimonopole located at one point [27], there is such an analog in the Abelian gauge theory. There is only one known solution in the realistic theories, the sphaleron, which can be treated as a monopole-antimonopole pair [28,29].
Let us now consider the stability of the vacuum condensates made of monopole pair fields. To verify whether a given classical solution can provide a stable quantum vacuum condensate (in one-loop approximation) it is suitable to apply the quantum effective action formalism. A one-loop effective action is expressed in terms of functional operator logarithms as follows [30,31]   B a µ . The operators K ab µν and M ab FP correspond to gluon and Faddeev-Popov ghosts. Note that expression (11) is valid for arbitrary background field and does not depend on a chosen gauge for the background fiedl due to use of a gauge covariant background formalism [30,31]. Effective action describes the vacuum-vacuum amplitude, and the presence of an imaginary part of the action implies instability of the vacuum condensate. So that, if the operator K ab µν is not positively defined then an unstable mode will appear as an eigenfunction corresponding to a negative eigenvalue of the "Schroedinger" type equation where the "wave functions" Ψ a µ (t, r, θ, ϕ) describe gluon fluctuations. Note that the ghost operator is positively defined and does not produce instability [12]. Substituting interpolation functions for the stationary monopole solutions in the leading order as a background field one can solve the eigenvalue equation (12). In the case of SU (2) QCD the full eigenvalue spectrum is divided into four sub-spectra corresponding to four factorized systems of equations: (I) Ψ 2 4 , (II) Ψ 1 4 , Ψ 3 4 , (III) Ψ 2 The lowest eigenvalue is reached by a solution to type II equations, the corresponding eigenfunctions are plotted in Fig. 3. Other type equa- the vacuum condensate is confirmed numerically for solutions with asymptotic amplitudes for the Abelian field K 3 (or K 4 ) in the interval (0 ≤ b i ≤ 2). For large amplitude values of the background solution negative eigenvalues appear in the spectrum. Remind, that the spherically symmetric monopole solution provides a stable vacuum condensate with asymptotic amplitude values less than a critical one, a cr 0.56 [26]. So that the monopoleantimonopole solutions provide better stability of corresponding vacuum condensates.
To trace the origin of stability of the vacuum condensate let us consider the classical Yang-Mills Lagrangian written in terms of Weyl symmetric fields [20] where G p µν are Abelian field strength corresponding to the gauge potentials A 3,8 µ , the complex fields W p µ represent off-diagonal gluons, V [(W ) 4 ] is a quartic potential, and the index p counts the Weyl symmetric combinations of the gauge potentials. The second and third terms in the Lagrangian includes cubic interaction terms corresponding to anomaly magnetic moment interaction which is precisely the source of the Nielsen-Olesen vacuum instability [12]. Direct substitution of the spherically symmetric SU (2) monopole ansatz given in [26] and axiallysymmetric SU (2) DHN ansatz into the Lagrangian implies that such cubic terms remain non-vanishing. It is surprising, in the case of SU (3) QCD despite each cubic contribution in (13) corresponding to I, U, V Weyl combinations does not vanish, their total sum vanishes identically. This reveals the origin of color confinement in QCD, namely, the existence of SU (3) Weyl symmetric classical monopole solution and absence of anomaly magnetic moment interaction terms in the Lagrangian provide stability and gauge invariance of the vacuum monopole condensate. This immediately implies the color confinement in QCD [4]. The absence of a stable monopole condensate in the electroweak theory based on the gauge group SU (2) × U (1) leads to gauge noninvariant vacuum and, as a consequence, to spontaneous symmetry breaking irrespective of presence of the Higgs potential which is introduced in a pure phenomenological way.
We conclude, the embedded SU (2) stationary monopole solution is not symmetric under the Weyl group transformation, and it represents a saddle point in the space of SU (3) axially-symmetric stationary solutions. Contrary to this the Weyl symmetric SU (3) monopole solution realizes a deepest minimum of the effective potential (at least on the space of axiallysymmetric field configurations) and provides a stable vacuum monopole pair condensate. Such a solution can serve as a structural element in further construction of the microscopic theory of the QCD vacuum. This strongly supports the QCD vacuum model as a dual color superconductor as it was conjectured in the seminal papers [2][3][4][5]. A rich structure of non-linear stationary solutions in QCD, which includes non-linear plane waves, spherical and axially-symmetric monopole like solutions, suggests a novel way to describe non-perturbative phenomena in hadron physics, in particular, in description of glueball spectrum [32,33]. These issues will be considered in a separate paper.