Black Holes-Hedgehogs and Strings as Defects of the Universal Vacua
In the present paper, assuming the Multiple Point Principle (MPP) as a new law of Nature, we considered the existence of the two degenerate vacua of the Universe: a) the first Electroweak (EW) vacuum at GeV – “true vacuum”, and b) the second Planck scale “false vacuum” at GeV. In these vacua we investigated different topological defects. The main aim of this paper is an investigation of the black-hole-hedgehogs configurations as defects of the false vacuum. In the framework of the gravity, described by the Gravi-Weak unification model, we considered a black-hole solution, which corresponds to a “hedgehog” – global monopole, that has been “swallowed” by the black-hole with mass core GeV and radius GeV. Considering the results of the hedgehog lattice theory in the framework of the Yang-Mills gauge-invariant theory with hedgehogs in the Wilson loops, we have used the critical value of temperature for the hedgehogs confinement phase ( GeV). This result gave us the possibility to conclude that the SM shows a new physics (with contributions of the -triplet Higgs bosons) at the scale TeV. Theory predicts the stability of the EW-vacuum and the accuracy of the MPP.
Keywords: black holes, hedgehogs, topological defects, multiple point principle, effective potential, cosmological constant, degenerate vacua
PACS: 04.50.Kd, 98.80.Cq, 12.10.-g, 95.35.+d, 95.36.+x
The present article is devoted to studying of topological defects of the universal vacua.
During the expansion after the Planck era, the early Universe underwent a series of phase transitions as a result of which there were arisen such vacuum topological defects (widely discussed in literature) as monopoles or hedgehogs (point defects), strings (line defects), bubbles and domain walls (sheet defects). These topological defects appeared due to the breakdown of local or global gauge symmetries.
This paper is essentially based on the discovery that a cosmological constant of our Universe is extremely small, almost zero [1, 2, 3]. We considered a Multiple Point Principle (MPP) first suggested by D.L. Bennett and H.B. Nielsen , which predicts the existence in Nature of several degenerate vacua with very small energy density (cosmological constants).
The model developed in this article confirms the existence of the two degenerate vacua of the Universe: The first (“true”) Electroweak (EW) vacuum with VEV GeV, and the second (“false”) Planck scale vacuum with VEV GeV.
The main idea of this paper is the investigation of hedgehog’s configurations [5, 6] as defects of the false vacuum. We have shown that at superhigh (Planck scale) energies the black-holes-hedgehogs are responsible for the creation of the false vacuum of the Universe. In the framework of the gravity, we have obtained a solution for a global monopole, which is a black-hole-hedgehog at the Planck scale. Here we have used the gravity predicted by the Gravi-Weak unification model previously developed by authors in papers [7, 8, 9, 10, 11].
Using the results of Refs.[12, 13] obtained for the Yang-Mills theory of the gauge-invariant hedgehog-like structures in the Wilson loops, we have considered the lattice theory giving the critical value of temperature for the hedgehogs confinement phase. Considering the hedgehog lattice theory, we have concluded that hedgehogs can exist only at the energy scale GeV. Triplet Higgs fields (with ), which are responsible for the formation of hedgehogs, can show a new physics at the scale TeV.
In Section 2 we reviewed the Multiple Point Principle (MPP) suggested by D.L. Bennett and H.B. Nielsen . In the assumption of the existence of the two degenerate vacua (Electroweak vacuum at GeV, and Planck scale one at GeV), Froggatt and Nielsen  obtained the first prediction of the top-quark and Higgs boson masses, which was further improved by several authors in the next approximations. Section 3 is devoted to the general properties of topological defects of the universal vacua. We considered topological defects in the “false vacuum”, which is presented as a spherical bubble spontaneously produced in the de Sitter-like universe. The space-time inside the bubble, which we refer to as a “true vacuum”, has the geometry of an open Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. Section 4 is devoted to the Gravi-Weak unification (GWU) model [7, 8, 9, 10, 11] as an example of the gravity. Subsection 4.1 considers the existence of the de Sitter solutions in the Planck phase. Subsection 4.2 is devoted to calculations of parameters of the GWU-model, where we predicted the Planck scale false vacuum VEV equal to GeV. In Section 5 we have investigated the hedgehog’s configurations as defects of the false vacuum. We obtained a solution for a black-hole in the framework of the gravity, which corresponds to a global monopole “swallowed” by a black-hole. The metric around of the global monopole was considered in Subsection 5.1. The mass , radius and “horizon radius” of the black-hole-hedgehog were estimated in Subsection 5.2. Section 6 is devoted to the lattice-like structure of the false vacuum which is described by a non-differentiable space-time: by a foam of black-holes, having lattice-like structure, in which sites are black-holes with “hedgehog” monopoles inside them. This manifold is described by a non-commutative geometry predicted an almost zero cosmological constant. The phase transition from the “false vacuum” to the “true vacuum” was considered in Section 7, where it was shown that the Electroweak spontaneous breakdown of symmetry created new topological defects of EW vacuum: the Abrikosov-Nielsen-Olesen closed magnetic vortices (“ANO strings”) of the Abelian Higgs model and Sidharth’s Compton phase objects. Then the “true vacuum” (EW-vacuum) again presents the non-differentiable manifold with non-commutative geometry, and again has an almost zero cosmological constant. Here we estimated the black-hole-hedgehog’s mass and radius: GeV and GeV near the second vacuum . In Subsection 7.1 we emphasize that due to the energy conservation law, the vacuum density before the phase transition is equal to the vacuum density after the phase transition, and we have
Therefore, we confirmed the Multiple Point Principle: we have two degenerate vacua and with an almost zero vacuum energy (cosmological constants). This means that our EW-vacuum, in which we live, is stable. The Planck scale vacuum cannot be negative: , these potentials are equal exactly. In Section 8 hedgehogs in Wilson loops of the Yang-Mills theory, and phase transitions in this theory were investigated using the results of Refs. [12, 13]. Their lattice results gave the critical value of the temperature for the hedgehog’s confinement phase: , and this result gives the value of critical temperature GeV. In Section 9 we show that the hedgehog’s confinement happens at energy TeV, which is a threshold energy of the production of a pair of the -triplet Higgs bosons. In Section 10 we reviewed the problem of the vacuum stability in the Standard Model. In Section 11 we show that hedgehogs can contribute at energy scale GeV. Therefore, a triplet Higgs field provides a new physics at the scale TeV. In this Section 11 we predict an exact stability of the EW-vacuum and the accuracy of the MPP.
2 Degenerate vacua of the Universe
This paper is based on the new law of Nature named the Multiple Point Principle (MPP) which was suggested by D.L. Bennett and H.B. Nielsen in Ref.. The MPP means: There exist in Nature several degenerate vacua with very small energy density, or cosmological constants.
Vacuum energy density of our Universe is the Dark Energy (DE). It is related with cosmological constant by the following way:
where is the reduced Planck mass: GeV. At present, cosmological measurements give:
which means a tiny value of the cosmological constant:
This tiny value of was first predicted by B.G. Sidharth in 1997 year [1, 2]. In the 1998 year S. Perlmutter, B. Schmidt and A. Riess  were awarded the Nobel Prize for the discovery of the accelerating expansion of the Universe.
Having an extremely small cosmological constant of our Universe, Bennett, Froggatt and Nielsen [4, 14, 15, 16] assumed to consider only zero, or almost zero, cosmological constants for all vacua existing in Nature.
The MPP theory was developed in a lot of papers by H.B. Nielsen and his collaborators (see for example, Refs. [4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] and recent Refs. [32, 33, 34, 35] by other authors).
Restricted to pure Standard Model (SM) we assumed the existence of only three vacua:
Present Electroweak vacuum, “true vacuum”, in which we live.
It has vacuum expectation value (VEV) of the Higgs field equal to:
High Higgs field vacuum, “false vacuum” – Planck scale vacuum, which has the following VEV:
From experimental results for these three vacua, cosmological constants – minima of the Higgs effective potentials – are not exactly equal to zero. Nevertheless, they are extremely small. By this reason, Bennett, Froggatt and Nielsen assumed to consider zero cosmological constants as a good approximation. Then according to the MPP, we have a model of pure SM being fine-tuned in such a way that these three vacua proposed have just zero energy density.
If the effective potential has three degenerate minima, then the following requirements are satisfied:
Here we assume that:
Assuming the existence of the two degenerate vacua in the SM:
a) the first Electroweak vacuum at GeV, and
b) the second Planck scale vacuum at GeV,
Froggatt and Nielsen predicted in Ref.  the top-quark and Higgs boson masses:
In Fig. 1 it is shown the existence of the second (non-standard) minimum of the effective Higgs potential in the pure SM at the Planck scale.
3 Topological defects of the universal vacua
Topological structures in fields are as important as the fields themselves. The presence of defects determines special features of the vacuum.
It is well known that in the early Universe topological defects may be created in the vacuum during the vacuum phase transitions. The early Universe underwent a series of phase transitions, each one spontaneously breaking some symmetry in particle physics and giving rise to topological defects of some kind, which can play an essential role throughout the subsequent evolution of the Universe.
In the context of the General Relativity, Barriola and Vilenkin (see Ref. ) studied the gravitational effects of a global monopole as a spherically symmetric topological defect. The authors found, that the gravitational effect of the global monopole is repulsive in nature. Thus, one may expect that the global monopole and cosmological constants are connected through their common manifestation as the origin of repulsive gravity. Moreover, both cosmological constant and vacuum expectation value (VEV) are connected while the VEV is connected to the topological defects. All these points lead us to a simple conjecture: There must be a common connection among the cosmological constant, topological defects and the vacuum expectation values (VEVs).
Different phase transitions have resulted during the expansion of the early Universe after the Planck era. They produced the formation of the various kind of topological defects: point defects (monopoles, hedgehogs, etc.); line defects (strings, vortices), and sheet defects (for example, domain walls). The topology of the vacuum manifold dictates the nature of these topological defects, appearing due to the breakdown of local or global gauge symmetries.
In the present paper, we shall discuss another potentially observable manifestation of topological defects. It has been shown in Ref.  that topological defects, like spherical domain walls and circular loops of cosmic string, can be spontaneously produced in a de Sitter-like universe. The initial radii of walls and strings are close to the de Sitter horizon, which corresponds to the Universe radius:
In the present paper we study the evolution of the two bubbles: one having a “false vacuum”, and the other one having a “true vacuum”. The bubble, which we shall refer to as the false vacuum, to be a de Sitter space with a constant expansion rate . It is convenient to use flat de Sitter coordinates to describe the background of the inflating false vacuum:
The space-time inside the bubble, which we shall refer to as a true vacuum, has the geometry of an open Friedmann-Lemaitre-Robertson-Walker (FLRW) universe (see for example review ):
where is a scale factor with cosmic time . In the true vacuum we have a constant expansion rate , which has the meaning of the slow-roll inflation rate inside the bubble at early stage of its evolution.
4 Gravi-Weak unification and hedgehogs as defects of the false vacuum
In the paper  (using the ideas of Refs. and ) we have considered a -group of the gravi-weak unification which is spontaneously broken into the . Such a model was constructed in agreement with experimental and astrophysical results. We assumed that after the Bing Bang there existed a Theory of the Everything (TOE) which rapidly was broken down to the direct product of the following gauge groups:
And below the see-saw scale ( GeV) we have the SM group of symmetry:
The action of the Gravi-Weak unification (obtained in Ref. ) was given by the following expression:
where is a parameter of the graviweak unification, parameters (with ) are “bare” coupling constants of the higher derivative gravity, is the Riemann curvature scalar, is the Ricci tensor, is a squared triplet Higgs field, where (with ) is an isovector scalar belonging to the adjoint representation of the gauge group of symmetry. In Eq.(15):
is a covariant derivative, and
is a curvature of the gauge field of the Yang-Mills theory. The coupling constant is a “bare” coupling constant of the weak interaction.
In a general case of the gravity, the action contains matter fields and can be presented by the following expression:
where corresponds to the part of the action associated with matter fields, fermions and Higgs fields.
Using the metric formalism, we obtain the following field equations:
, is the gravitational constant, and is the energy-momentum tensor derived from the matter action .
4.1 The existence of the de Sitter solutions at the early time of the Universe
It is well-known that at the early time of the Universe an acceleration era is described by the de Sitter solutions (see for example [52, 53]). The investigation of the problem that de Sitter solutions exist in the case of the action (15) was considered by authors of Ref.. Our model  is a special case of the more general model , and we can assume that the Universe is inherently de Sitter. Then the 4-spacetime is a hyperboloid in a 5-dimensional Minkowski space under the constraint:
where is a radius of a curvature of the de Sitter space, or simply “the de Sitter radius”. The Hubble expansion of the Universe is then viewed as a process that approaches the asymptotic limit of a pure space which is de Sitter in nature, evidenced that the cosmological constant describes the dark energy (DE) substance, which has become dominant in the Universe at later times:
where is the dark energy density and the critical density is:
where is the Hubble constant:
Identifying the Einstein tensor as
we see that the only nontrivial component that satisfies this equation is a constant for the Ricci scalar:
As it was shown in Ref., the nontrivial vacuum solution to the action (15) is de Sitter spacetime with a non-vanishing Higgs vacuum expectation value (VEV) of the triplet Higgs scalar field : . The standard Higgs potential in Eq.(15) has an extremum at (with ), corresponding to a de Sitter spacetime background solution:
which implies vanishing curvature:
solving the field equations , and strictly minimizing the action (15).
Based on this picture, the origin of the cosmological constant (and DE) is associated with the inherent spacetime geometry, and not with vacuum energy of particles (we consider their contributions later). We note that as a fundamental constant under the de Sitter symmetry, is not a subject to quantum corrections. Local dynamics exist as fluctuations with respect to this cosmological background. In general, the de Sitter space may be inherently unstable. The quantum instability of the de Sitter space was investigated by various authors. Abbott and Deser  have shown that de Sitter space is stable under a restricted class of classical gravitational perturbations. So any instability of the de Sitter space may likely have a quantum origin. Ref. demonstrated through the expectation value of the energy-momentum tensor for a system with a quantum field in a de Sitter background space that in general, it contains a term that is proportional to the metric tensor and grows in time. As a result, the curvature of the spacetime would decrease and the de Sitter space tends to decay into the flat space (see Ref.). The decay time of this process is of the order of the de Sitter radius:
Since the age of our universe is smaller than , we are still observing the accelerating expansion in action.
4.2 Parameters of the Gravi-Weak unification model
Assuming that at the first stage of the evolution (before the inflation), the Universe had the de Sitter spacetime – maximally symmetric Lorentzian manifold with a constant and positive background scalar curvature – we have obtained the following relations from the action (15):
1) The vacuum expectation value – the VEV of “the false vacuum” – is given by the de Sitter scalar curvature :
2) At the Planck scale the squared coupling constant of the weak interaction is:
leads to the following GW-action:
Now considering the VEV of the false vacuum as , we have:
The Einstein-Hilbert action of general relativity with the Einstein’s cosmological constant is given by the following expression:
3) The comparison of the Lagrangian with the Lagrangian given by Eq.(34) near the false vacuum leads to the following relations for the Newton’s gravitational constant and reduced Planck mass:
5 The solution for the black-holes-hedgehogs
A global monopole is described by the part of the Lagrangian given by the action (34), which contains the -triplet Higgs field , VEV of the second vacuum and cosmological constant :
Here we have:
and in Eq.(42) we have the compensation of the Einstein’s cosmological term. Then
where the Higgs potential is:
This potential has a minimum at , in which it vanishes:
where with , and are some structural functions. This solution is pointing radially. Here is parallel to – the unit vector in the radial, and we have a “hedgehog” solution of Refs. [5, 6]. The terminology “hedgehog” was first suggested by Alexander Polyakov in Ref. .
The field equations for in the flat metric reduces to a single equation for :
where is the core radius of the hedgehog. In the flat space the hedgehog’s core has the following size:
The function grows linearly when and exponentially approaches unity as soon as . Barriola and Vilenkin  took outside the core which is an approximation to the exact solution. As a result, the functions and are constrained by the following conditions:
5.1 The metric around of the global monopole
The most general static metric around of the global monopole is a metric with spherical symmetry:
For this metric the Ricci tensor has the following non-vanishing components:
The energy-momentum tensor of the monopole is given by
Considering the approximation used by Barriola and Vilenkin in Ref., we obtain an approximate solution for monopole-hedgehog taking out the core of the hedgehog (see also Refs.[57, 58, 59, 60, 61]). In this case scalar curvature is constant and Eq.(20) comes down to the Einstein’s equation:
where the energy-momentum tensor is given by the following approximation:
and then asymptotically (when ) we have:
From Eq.(56) we obtain a general relation for the function :
In the limit we obtain:
5.2 The mass, radius and horizon radius of the black-hole-hedgehog
Eq.(61) suggests the following equation for the hedgehog mass :
The function was estimated in Ref. at :
and we obtain an approximate value of the hedgehog mass:
There is a repulsive gravitational potential due to this negative mass. A freely moving particle near the core of the black-hole experiences an outward proper acceleration:
We have obtained a global monopole with a huge mass (66), which has a property of the hedgehog. This is a black-hole solution, which corresponds to a global monopole-hedgehog that has been “swallowed” by a black-hole. Indeed, we have obtained the metric result by M. Barriola et.al.  like:
A black hole has a horizon. A horizon radius is found by solving the equation :
and we have a solution:
According to Eq. (37), , and we obtain the black-hole-hedgehog with a horizon radius:
We see that the horizon radius is more than the hedgehog radius :
and our concept that “a black hole contains the hedgehog” is justified.
6 Lattice-like structure of the false vacuum and non-commutativity
Now we see, that at the Planck scale the false vacuum of the Universe is described by a non-differentiable space-time: by a foam of black-holes, having lattice-like structure, in which sites are black-holes with “hedgehog” monopoles inside them. This manifold is described by a non-commutative geometry [1, 2, 62, 63, 64, 65, 66, 67, 68].
That a cosmological constant is given by a tiny value:
where is the Hubble rate in the early Universe:
That a Dark Energy density is very small:
That a very small DE-density provides an accelerating expansion of our Universe after the Big Bang.
Sidharth proceeded from the following points of view  : Modern Quantum Gravity  (Loop Quantum Gravity, etc.,) deal with a non-differentiable space-time manifold. In such an approach, there exists a minimal space-time cut off , which leads to the non-commutative geometry, a feature shared by the Fuzzy Space-Time also.
If the space-time is fuzzy, non-differentiable, then it has to be described by a non-commutative geometry with the coordinates obeying the following commutation relations:
Eq. (75) is true for any minimal cut off .
Previously the following commutation relation was considered by H.S. Snyder :
which shows that effectively 4-momentum is replaced by
Then the energy-momentum formula becomes as:
In such a theory the usual energy momentum dispersion relations are modified [66, 67]. In the above equations, stands for a minimal (fundamental) length, which could be the Planck length , or for more generally – Compton wavelength .
Writing Eq. (79) as
where is the usual (old) expression for energy, and is the new additional term in modification. can be easily verified as – for boson fields, and – for fermion fields with masses , respectively. These formulas help to identify the DE density, what was first realized by B.G. Sidharth in Ref. .
DE density is a density of the quantum vacuum energy of the Universe. Quantum vacuum, described by Zero Point Fields (ZPF) contributions, is the lowest state of any Quantum Field Theory (QFT), and due to the Heisenberg’s principle has an infinite value, which is renormalizable.
As it was pointed out in Refs. [71, 63], the quantum vacuum of the Universe can be a source of the cosmic repulsion. However, a difficulty in this approach has been that the value of the cosmological constant turns out to be huge , far beyond the value which is observed by astrophysical measurements. This phenomenon has been called “the cosmological constant problem” .
A global monopole is a heavy object formed as a result of the gauge-symmetry breaking during the phase transition of the isoscalar triplet system. The black-holes-hedgehogs are similar to elementary particles, because a major part of their energy is concentrated in a small region near the monopole core. Assuming that the Planck scale false vacuum is described by a non- differentiable space-time having lattice-like structure, where sites of the lattice are black-holes with “hedgehog” monopoles inside them, we describe this manifold by a non-commutative geometry with a minimal length . Using the non-commutative theory of the discrete space-time, B.G. Sidharth predicted in Refs. [2, 63] a tiny value of the cosmological constant: GeV as a result of the compensation of ZPF contributions by non-commutative contributions of the hedgehog lattice.
7 The phase transition from the “false vacuum” to the “true vacuum”
In the Guendelman-Rabinowitz theory  of the universal vacua, the authors investigated the evolution of the two phases:
one being the “false vacuum” (Planck scale vacuum), and
the other – the “true vacuum” (EW-scale vacuum).
By cosmological theory, the Universe exists in the Planck scale phase for extremely short time. By this reason, the Planck scale phase was called “the false vacuum”. The presence of hedgehogs as defects is responsible for the destabilization of the false vacuum. The decay of the false vacuum is accompanied by the decay of the black-holes-hedgehogs. These configurations are unstable, and at some finite cosmic temperature which is called the critical temperature , a system exhibits a spontaneous symmetry breaking, and we observe a phase transition from the bubble with the false vacuum to the bubble with the true vacuum. After the phase transition, the Universe begins its evolution toward the low energy Electroweak (EW) phase. Here the Universe underwent the inflation, which led to the phase having the VEV GeV. This is a “true” vacuum, in which we live.
Guendelman and Rabinowitz  also allowed a possibility to consider an arbitrary domain wall between these two phases. During the inflation, domain wall annihilates, producing gravitational waves and a lot of the SM particles, having masses.
The Electroweak spontaneous breakdown of symmetry leads to the creation of the topological defects in the EW vacuum. They are the Abrikosov-Nielsen-Olesen closed magnetic vortices (“ANO strings”) of the Abelian Higgs model [73, 74], and Sidharth’s Compton phase objects [75, 76]. Then the electroweak vacuum again presents the non-differentiable manifold, and again we have to consider the non-commutative geometry.
Kirzhnits  and Linde  were first who considered the analogy between the Higgs mechanism and superconductivity, and argued that the SM (SU(2)-doublet) Higgs field condensate GeV disappears at high temperatures, leading to the symmetry restoration. As a result, at high temperatures all fermions and bosons are massless. These conclusions were confirmed, and the critical temperature was estimated (see review by A. Linde ).
At the early stage, the Universe was very hot, but then it began to cool down. Black-holes-monopoles (as bubbles of the vapour in the boiling water) began to disappear. The temperature dependent part of the energy density died away. In that case, only the vacuum energy will survive. Since this is a constant, the Universe expands exponentially, and an exponentially expanding Universe leads to the inflation (see review ). While the Universe was expanding exponentially, so it was cooling exponentially. This scenario was called supercooling in the false vacuum. When the temperature reached the critical value , the Higgs mechanism of the SM created a new condensate , and the vacuum became similar to superconductor, in which the topological defects are magnetic vortices. The energy of black-holes is released as particles, which were created during the radiation era of the Universe, and all these particles (quarks, leptons, vector bosons) acquired their masses through the Yukawa coupling mechanism . Therefore, they acquired the Compton wavelength, . Then according to the Sidharth’s theory of the cosmological constant, in the EW-vacuum we again have lattice-like structures formed by bosons and fermions, and the lattice parameters “” are equal to the Compton wavelengths: .
As it was shown in Ref., the Planck scale vacuum energy density (with the VEV ) is equal to:
and the EW-vacuum gives:
In the above equations “NC” means the “non-commutativity” and “ZPF” means “zero point fields”.
Assuming by example that hedgehogs form a hypercubic lattice with lattice parameter , we have the negative energy density of such a lattice equal to:
If this energy density of the hedgehogs lattice compensates the Einstein’s vacuum energy (45), we have the following equation:
Using the estimation (38), we obtain:
Therefore hedgehogs have a huge mass of order of the Planck mass. Eq.(66) predicts a radius of the hedgehog’s core:
7.1 Stability of the EW vacuum
Here we emphasize that due to the energy conservation law, the vacuum density before the phase transition (for ) is equal to the vacuum density after the phase transition (for ), therefore we have:
The analogous link between the Planck scale phase and EW phase was considered in the paper . It was shown that the vacuum energy density (DE) is described by the different contributions to the Planck and EW scale phases. This difference is a result of the phase transition. However, the vacuum energy densities (DE) of both vacua are equal, and we have a link between gravitation and electromagnetism via the Dark Energy. According to the last equation (88), we see that if (at the Planck scale) is almost zero, then (at EW scale) also is almost zero, and we have a triumph of the Multiple Point Principle: we have two degenerate vacua with almost zero vacuum energy density. Almost zero cosmological constants are equal:
where is a cosmological constant for -vacuum with VEV (here ).
Now we see that we have obtained a very important result: our vacuum, in which we live, is stable. The Planck scale vacuum cannot be negative: exactly.
8 Hedgehogs in the Wilson loops and the phase transition in the Yang-Mills theory
The authors of Ref.  investigated the gauge-invariant hedgehog-like structures in the Wilson loops of the Yang-Mills theory. In this model the triplet Higgs field vanishes at the centre of the monopole : , and has a generic hedgehog structure in the spatial vicinity of this monopole.
In the Yang-Mills theory, a hedgehog structure can be entirely defined in terms of Wilson-loop variables . In general, we consider an untraced Wilson loop, beginning and ending at the point on the closed loop :
To improve the analogy with the triplet Higgs field , we subtract the singlet part from :
This is a traceless adjoint operator similar to the field . Associating the triplet part of Wilson loop with the triplet Higgs field , we notice the following property: As the Higgs field vanishes in all points , belonging to the monopole trajectory, similarly vanishes on the hedgehog loop :
In conventional superconductivity , Abrikosov vortices are singularities in the superconducting condensate (i.e., in the Cooper-pair field). Abrikosov vortices are “two-dimensional hedgehogs” (see Ref.). In the core of the Abrikosov’s vortices, the superconductivity is broken, and the normal state is restored. As temperature increases, the condensate weakens, and nucleation of the vortices due to thermal fluctuations strengthens. Thus, the higher the temperature is, the density of the (thermal) vortices should be larger. It can be expected in the YM theory that the density of hedgehog loops is also sensitive to the phase transition.
The order parameter of the phase transition is the vacuum expectation value (trace) of the Polyakov line:
Here is a temperature and VEV is . Functional , called the thermal Wilson line, is a basic variable in an effective theory, which describes the properties of the finite-temperature phase transition of the system. In the confinement phase, the expectation value of the Polyakov line is zero: , indicating that the free energy of a single quark becomes infinite when . In the deconfinement phase, the Polyakov line has a non-zero expectation value: , and the quarks are no longer confined.
Considering lattice model of the Yang-Mills theory, Belavin, Chernodub and Kozlov showed numerically that the density of hedgehogs structures in the thermal Wilson-Polyakov lines is very sensitive to the finite-temperature phase transition. The hedgehog line density behaves like an order parameter: the density is almost independent of the temperature in the confinement phase and changes substantially as the system enters the deconfinement phase. These authors obtained a very important result: , which shows that the critical temperature , corresponding to the hedgehogs confinement, is smaller than the Planck scale value.
Then the critical temperature is:
9 Threshold energy of the -triplet Higgs bosons
Eq.(92) also gives the critical value of the coupling constant of the Yang-Mills theory:
The renormalization group equation (RGE) for (see for example  and references there) is given by the following expression:
where , and GeV is the top quark mass.
Usually RGE is a function of : . Then
For -gauge theory and , and we obtain the following RGE equation :
Then we can calculate using the following result:
This result means that the hedgehog’s confinement happens at energy of 10 TeV, which is a threshold energy of the production of a pair of the -triplet Higgs bosons :