Mirror symmetry of quantum Yang–Mills vacua and cosmological implications
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Abstract
We find an argument related to the existence of a \(\mathbb {Z}_2\)symmetry for the renormalization group flow derived from the bare Yang–Mills Lagrangian, and show that the cancellation of the vacuum energy may arise motivated both from the renormalization group flow solutions and the effective Yang–Mills action. In the framework of the effective Savvidy’s action, two Mirror minima are allowed, with exactly equal and hold opposite sign energy densities. At the cosmological level, we explore the stability of the electric and magnetic attractor solutions, both within and beyond the perturbation theory, and find that thanks to these latter the cancellation between the electric and the magnetic vacua components is achieved at macroscopic space and time separations. This implies the disappearance of the conformal anomaly in the classical limit of an effective Yang–Mills theory. In this picture, the tunneling probability from the Mirror vacua to the other vacua is exponentially suppressed in the quantum nonthermal state – similarly to what happens for electroweak instantonic tunneling solutions. Specifically, we show that, in a dynamical Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological background, the Nielsen–Olsen argument – on the instability of uniform chromoelectric and chromomagnetic Mirror vacua – is subtly violated. The chromomagnetic and chromoelectric uniform vacua are unstable only at asymptotic times, but at those times the attractor to a zero energy density is already reached. The two vacua can safely decay into one anisotropic vacuum that has zero energydensity inside the Fermi confinement volume scale. We also discover a new surprising pattern of solitonic and antisolitonic spacelike solutions, which are sourced by the Yang–Mills dynamics coupled to the Einstein’s equations in FLRW. We dub such nonperturbative configurations, which are directly related to dynamical cancellation mechanism of the vacuum energy, as chronons, or \(\chi \)solutions.
1 Introduction
The ground state of quantum Yang–Mills (YM) theories plays a crucial role in both particle physics and cosmology. In particular the gluon condensate, a strongly coupled system in quantum chromo dynamics (QCD), largely determines nontrivial properties of the topological QCD vacuum. In a nonperturbative mechanism, this is responsible e.g. for the color confinement effects and the hadron mass generation – for a comprehensive review on the topological QCD vacuum, see e.g. Ref. [1] and references therein. Currently, the cosmological constant (CC) scenario, with the vacuum equation of state \(w\equiv p/\epsilon =1\), is preferred to the Dark Energy (DE) paradigm to unveil the latetime acceleration epoch, as supported by a wealth of data from the Supernovae type IA [2] and Cosmic Microwave Background [3] observations. Despite of many DE/CC models existing in the literature, there is not a compelling resolution of the CC problem, i.e. why the CC term is so small compared to the other scales of Nature, and why it is positive.
From the Quantum Field Theory (QFT) viewpoint, the ground state energy density of the Universe should account for a bulk of various contributions from existing quantum fields, at energy scales ranging from the Quantum Gravity (Planck) scale, \(M_\mathrm{PL}\simeq 1.2\cdot 10^{19}\ \hbox {GeV}\), down to the QCD confinement scale, \(\Lambda _\mathrm{QCD}\simeq 0.1\ \hbox {GeV}\). Even the relatively wellknown vacuum subsystems of the Standard Model (SM), such as the Higgs and the quarkgluon condensates, exceed by far the observed CC. This is often considered as a severe problem [4, 5] – for a recent review on this topic, see e.g. Ref. [6] and references therein. For confined QCD, with SU(3) color gauge symmetry, there is a rather unique contribution to the ground state energy of the Universe that emerges from the nonperturbative quantumtopological fluctuations of the quark and gluon fields [1, 7, 8, 9], namely, \(\epsilon ^\mathrm{QCD}_\mathrm{top} \simeq (5\pm 1)\times 10^{9}\;\text {MeV}^4\). Given the fact that the CC term observed in astrophysical measurements is very small (and positive), \(\epsilon _\mathrm{CC}\simeq 3\times 10^{35}\,\mathrm{MeV}^4 ,\) to the first approximation one must exclude the negativevalued topological vacuum contribution with an accuracy of a few tens of decimal digits.
Recently, in Ref. [21] by some of the authors it was shown that a possible elimination of the QCD contribution to the cosmological constant could be achieved by means of the existence of an additional “Mirror QCD” sector whose (nonperturbative) vacuum energydensity contributes with an opposite sign to the conventional QCD trace anomaly. Disregarding the Anthropic Principle, the main issue of this approach is the need for a significant fine tuning between the usual QCD and Mirror QCD vacua parameters which would be a problem for getting a naturally small CC term. Within this paper we show that even in the framework of standard QFT it is possible to recover as a result the cancellation of SU(2) Yang–Mills (YM) contributions to the vacuum energy within the same theory. This achievement holds a certain generality, since SU(2) subgroups of SU(N) YM theories can always be picked out, being the ones that must be accounted for the cosmological applications. The vacua compensation mechanism will be analyzed for effective YM theories, in both the perturbative and the nonperturbative cases, and then applied to address the QCD electric and magnetic condensates. Our approach is based on the Savvidy vacuum model [10, 11, 12, 13], as an effective method describing the ground state dynamics in quantum YM field theories at long distances. Interestingly enough, the Savvidy vacuum model has received a further support from another approach based on the analysis of the gluon condensation within the framework of the Functional RG (FRG) [14, 15, 16].
As the main result of this work, we find the stability conditions of the considered Savvidy vacuum solutions for the gaugeinvariant homogeneous gluon condensate, and obtain analytic expressions for the density, the pressure and the scale factor in the nonstationary Friedmann–Lemaître–Robertson–Walker (FLRW) Universe filled with the gluon condensate, which fluctuates near the minimum of the effective Lagrangian.
2 Effective YM theory and the mirror symmetry
We may start showing how to recover the effective action of SU(N) YM theories, following the seminal Ref. [10] recently followed by Refs. [17, 18, 19, 20, 21]. We then generalize these findings for a nonstationary FLRW background of expanding Universe.
Through the rest of the paper we will work only with spatially averaged quantities, thus from now on we remove the \(\langle \dots \rangle \), for simplicity. Our approach must be thought as a chromodynamical mean field theory, in analogy to many condensed matter models.^{1} In the minimum of the effective Lagrangian, the spatiallyhomogeneous CE and CM condensates correspond to positive and negativevalued energy densities, respectively. In a nonstationary background of expanding Universe, these condensates yield stable deSitter (dS) and antideSitter (AdS) attractor solutions with positive and negative cosmological constants, respectively.
3 Mirror symmetry
The Perturbation Theory can be applied to the effective action in the limit of large mean fields, i.e. \(\mathcal {J}\rightarrow \infty \), away from the nonperturbative ground state. We now comment on the oneloop results obtained by Savvidy for SU(N) YM theories, and then focus on a different strategy to account for allloops corrections, based on the FRG approach. The latter has been developed in its cosmological applications in Ref. [15], accounting for the SU(2) gauge symmetry.
In Fig. 1, we show the effective SU(2) YM theory Lagrangian dependence on \(\mathcal {J}/\lambda ^4\) corresponding to one particular branch of the RG equation (4) with \(\mathcal {J}>0\) (CE configuration). As anticipated, there is a single minimum in the nonperturbative domain \(0<\mathcal {J}^*<\lambda ^4\), hence, identified with the CE condensate. The Mirror CM condensate solution can then be obtained by means of \(\mathbb {Z}_2\) transformations (6), and it corresponds to the conventional oneloop result for the trace anomaly in SU(N) YM gluodynamics (known e.g. from lattice QCD simulations). Notably, applying the \(\mathbb {Z}_2\) transformations to the physical scale of the CE configuration (17) one gets a smaller physical scale of the CM configuration, such that the corresponding CM condensate \(\mathcal {J}^*<0\) appears in the perturbative \(\mathcal {J}^*>\lambda ^4\) domain, with a positive \(\bar{g}_{1}^2(\mathcal {J}^*) > 0\).
How well the oneloop approximation reproduces the allloops vacuum state, given by the nonperturbative groundstate solutions in Eq. (5)? We can answer this question focusing on the case of SU(2), which is also relevant for cosmology, in the framework of FRG [14, 15, 16]. As is illustrated explicitly by two curves in Fig. 1 for the CE branch, the oneloop and the allloops CE solutions approach the zero of the effective action at exactly the same values of \(\mathcal {J}=0\) and \(\mathcal {J}=\lambda ^4\). The solutions also exhibit minima that, although do not coincide, are very close to each other: at one loop, \({\mathcal {J}^*}/{\lambda ^4} = \frac{1}{e} \simeq 0.3679 , \) and \({\mathcal {L}^*_\mathrm{eff}}/{\lambda ^4} = \pm {b}/(192\pi ^2 e) \simeq \pm \, 2.135\cdot 10^{3}\); at all loops \({\mathcal {J}^*}/{\lambda ^4} \simeq 0.3693 , \) and \({\mathcal {L}^*_\mathrm{eff}}/{\lambda ^4} = \pm \, 2.163\cdot 10^{3} .\) Remarkably, the CE groundstate solutions for oneloop and allloops cases differ only at a permille level. By means of the Mirror symmetry, the same applies for the CM configuration as well.
It is worth emphasizing that is not reductive to focus on SU(2) YM theory. For any SU(N) gauge group, the cosmological instantiation will be provided by the SU(2) subgroups, for which an isomorphism between indices of the adjoint representation and spatial indices may be recovered. On the other hand, the calculation of the supertrace would be technically very difficult to be achieved. Because of the lack of any physical advantage, we can skip this point without any loss of generality and physical insight.
4 Homogeneous YM condensates
To address the characteristic time scales that are required for this mechanism to take place, let us consider a deviation from the exact partial solution, which describes the evolution of U(t), and study numerically the general solution of the equations of motion – see Appendix B. We first choose the subset of the initial conditions satisfying \(Q_0\equiv Q(t=t_0)>1\), and then discuss the results of the numerical analysis qualitatively. For this choice of the initial conditions, Fig. 2 (left) illustrates the physical time evolution of the total energy density (in dimensionless units) of the homogeneous gluon condensate \(U=U(t)\), namely \(T^0_0 (t) \equiv \bar{\epsilon } + T^{0,\mathrm{U}}_0(t)\). In Appendix B we show the explicit expression of \(T^{0,\mathrm U}_0\) and \(\bar{\epsilon }\), respectively, as functionals of U(t). In Fig. 2 (middle) we display the corresponding result for the trace of the total gluon EMT \(T^\mu _\mu (t)\equiv 4\bar{\epsilon } + T^{\mu ,\mathrm U}_\mu (t)\) in dimensionless units, and the corresponding solution for the logarithm of the scale factor is given in Fig. 2 (right).
The period of the \(T^\mu _\mu (t)\) oscillations is practically time independent, which can also be proven analytically, while a small residual timedependence appears due to a possibly large deviation from \(Q=1\). Here we used \(\xi \simeq 4\) (following Ref. [17]) while a change of \(\xi \) would only affect the asymptotic values of \(T^0_0(t)\) and \(T^\mu _\mu (t)\) at large t. Although the amplitude of the condensate U(t) possesses quasiperiodic singularities, as is seen in Fig. 3, the evolution of its energy density \(T^{0,\mathrm{U}}_0(t)\) (see Fig. 2, left), as well as of the pressure (or \(T^{\mu ,\mathrm{U}}_\mu (t)\), see Fig. 2, middle), remain continuous in time. One immediately notices that the general solution asymptotically reaches a fix branch. This happens after a number of oscillations of the function Q(t), whose amplitude approaches unity at large physical times t, i.e. \(Q(t\rightarrow \infty )\Rightarrow 1\), for any initial conditions satisfying \(Q_0>0\). During such a relaxation regime, the total energy density of the QCD vacuum (composed of the conventional QCD trace anomaly term – the CM condensate – and the considered CE homogeneous condensate) continuously decreases and eventually vanishes in the asymptotic limit \(t\gg t_0\). Note, this regime is accompanied by a decelerating expansion of the Universe.
The same quantities can be also studied for initial conditions of opposite sign, i.e. for \(0<Q_0<1\). Numerical results for the latter regime are reported in Appendix B. In this case the general solution asymptotically approaches the dS regime as well, in full analogy with the \(Q_0>1\) case. A qualitatively similar situation is realized for \(Q_0<0\). The dS (for \(Q_0>0\)) and AdS (for \(Q_0<0\)) solutions, therefore, appear as two attractor (or tracker) solutions of the EinsteinYM system, providing a dynamical mechanism for the elimination of the gluon vacuum component of the ground state energy of the Universe. This happens asymptotically, at macroscopic spacetime scales and for arbitrary initial conditions and parameters of the model.
5 Conclusions and remarks
We found an argument for the cancellation of the vacuum energy of QCD in the infrared limit that is related to the existence of an emergent \(\mathbb {Z}_2\) Mirror symmetry of the RG flow – derived from the bare Lagrangian. We showed that the cancellation of the vacuum energy is motivated both from the RG flow solutions and the effective action. We then commented on the relevance of vacuum fluctuations constituting chromoelectric vacuum solutions, which we argue counterbalance the conventional chromomagnetic contributions. We presented at this purpose an estimate of these latter contributions, and argued about the dynamical cancellation of the vacuum energy that they induce. We insisted that this is not in disagreement with lattice QCD. Our arguments are indeed dynamical, pertaining nonequilibrium configurations, and restricted to the infrared limit, to which we referred for the disappearance of the QCD vacuum beyond the confinement lengthscale.
We claim that both the chromomagnetic and chromoelectric solutions coexist in the infrared limit of QCD, while the topological effect is zero according to Ref. [12]. Their attractor nature unavoidably leads to their compensation, and consequently induces a gross suppression of the net QCD vacuum contribution in the ground state of the Universe, naturally and without any fine tuning. While the quantumtopological effect may not be entirely excluded, the observed attractor nature of the QCD vacuum components may indicate similar selftuning properties of the heterogeneous QCD vacuum, and point to a universal mechanism of the mutual net cancellation of all its components in the infrared limit of the theory, as required by cosmological observations.

he appearance of negative values of \(\bar{g}^2\) for the CE configuration in our analysis highlights a nonperturbative nature of the CE condensate and should not worry the reader about the appearance of ghosts. First of all, only gaugeinvariant quantities were deployed in this analysis, including the operator \(\mathcal {J}\), and thus both \(\beta (\mathcal {J})\) and \(\bar{g}^2(\mathcal {J})\). Nonetheless, the local loss of Lorentz invariance prevents from extending the ghost theorem to a case, like the one accounted for here, in which violations of the Lorentz symmetry – in particular, of the boost invariance for the considered spatially homogeneous CE and CM configurations – appear locally. For instance, in the confined phase of YM theories the presence of CM vortices leads to a groundstate that is not spatially isotropic. This subtly allows to avoid the ghost theorem, which is formulated under the Lorentz invariance assumption.

It is commonly retained that the Savvidy’s CE vacuum cannot be stable, since it is uniformly distributed in space (and has positive energy density). This statement is based on the old famous proof of Nielsen and Olsen (N.O.) in Ref. [38]. Mutatis Mutandis, the same instability argument would imply the destabilization of the Mirror CM vacuum. However, the N.O. proof is not in contradiction with our results: we consider the evolution of the YM condensates in a dynamical spacetime, whilst the N.O. argument is formulated on a rigid Minkowski spacetime. This is enough to ensure that results reported in this analysis cannot be excluded on the basis of the N.O. proof. The N.O. instability is expected only after a relaxation timescale of \(\tau \simeq 50\div 100 \, \Lambda _\mathrm{QCD}^{1}\) or similar – see e.g. Fig. 2, where the dynamical evolution of the energy–momentum tensor of the condensate as a function of the cosmological time is displayed. The Savvidy’s equations of motion for both the CE and CM condensates, at system with the Einstein’s equations, provide a nonequilibrium fields system that was analyzed for the first time only within this work. Intriguingly, even starting from an initial nonzero energy–density, the evolution of the CE and CM YM condensates trigger a mutual screening, flowing towards a zeroenergy density attractor.

In principle, after the cosmological relaxation time \(\tau \simeq 50\div 100 \, \Lambda _\mathrm{QCD}^{1}\), the Mirror Savvidy’s vacuum states can decay into more complicated anisotropic vacuum states, in accordance with the N.O. argument. Nonetheless, the decay from a CE Savvidy’s vacuum into a CM vacuum state – either a CM Savvidy’s vacuum or an anisotropic (averaged dominated) CM vacuum state – has to be exponentially suppressed. Indeed, since the CM and CE states are always separated by an energy barrier proportional to \(\Lambda _\mathrm{QCD}\), at present times in the Universe, the tunnelling probability from the positiveenergy CE vacuum to the CM minimal (negative) energy state is roughly \(\Gamma \sim e^{\Lambda _\mathrm{QCD}/T_{0}}\simeq 10^{4 \times 10^{11}}\, \). This is not a surprising result, since a similar suppression of the tunnelling at present times also happens in the case of electroweak instantons (sphalerons). This argument allows the final anisotropic vacuum state to retain an energy density (averaged in the \(\sim \Lambda ^{3}\) confinement volume) that is vanishing.

The coexistence in the “Fermi world” of both the Mirror vacua can be sustained by nonperturbative and interpolating configurations. This is suggested by the well known case of a scalar field theory with a double wells potential: a kinklike profile can interpolate among two minima \(v\) and v (with v the VEV scale), which are contained in domain walls. It is very well known, for a scalar field related to the Higgs mechanism of a YM theory, that the change of the kink profile corresponds to the presence of monopole or vortex solutions that are localized inside the domain walls. Within the context of the YM theories, there is a strong evidence, sustained by numerical simulations, that the confinement phenomena are related by the formation of a network of ’t Hooft monopoles or, alternatively, chromovortices – see e.g. Refs. [34, 35, 36, 37]. Both the scenarios highly suggest that these nonperturbative solutions may interconnect the CE and CM vacuum energies. In our case, the kink scalar profile may correspond to the effective \(\mathcal {J}\)kink field. In other words, the cancellation mechanism proposed in our paper seems to naturally marry the confinement pictures that arise from numerical simulations. A fascinating picture to be envisaged consists into a sort of “ferromagnetic confinement” inside the Fermi world. In ferromagnetism, the ferromaterial show several domain regions with certain different magnetic orientations. In a similar way, after the YM dimensional transmutation, several different domains, containing YM monopoles of vortices, interconnect the CE and CM vacua stabilised by the domain walls.
 In addition, the stabilization at asymptotically large times during the Universe’s expansion of the “false vacuum”, namely the positiveenergy CE vacuum state, allows to reach an energy density that corresponds to the average energy of the two local minima. Decay of a “false vacuum” characterized by a positive cosmological constant, to a “true vacuum” can be achieved thanks to the Coleman de Luccia (CdL) instantonic solutions, and is exponentially suppressed. According to Ref. [39], the CdL decay rate per unit time and unit volume can be exactly determined to bewhere A is a factor (irrelevant to this argument) derived from quantum corrections, \(\ell \) is the typical lengthscale of the system and \(\varkappa ^2 = 8\pi G\) is the Planck length squared. For the positive cosmological constant solution, the typical lengthscale corresponds to the dS radius, i.e. \(\ell \sim R_\mathrm{dS}\). It follows that the probability of decay from the “false” dS to the “true” AdS vacuum reads$$\begin{aligned} \Gamma _\mathrm{CdL} \sim A e^{48.33 \frac{\ell ^2 }{\varkappa ^2}} \ll \ell ^{4}, \end{aligned}$$and that is generically very small – see e.g. Ref. [40]. These generic conditions actually make the “false” and positiveenergy CE vacuum we dealt with in this analysis stable at all relevant cosmological time scales, also in full agreement with arguments given above.$$\begin{aligned} P_\mathrm{dec} \simeq R_\mathrm{dS}^4 \, \Gamma _\mathrm{CdL} \ll 1, \end{aligned}$$

We may consider the case in which the Universe is filled with a positiveenergy meanfield solution, which is such that the \(\mathbb {Z}_2\) Mirror symmetry is broken for initial conditions (in general, away from the ground state) chosen at initial time \(t=t_0\) in the early Universe. Consequently, the total YM field energy density is nonzero. This means that at \(t=t_0\) any domainwall has not formed yet for the chosen (\(\mathbb {Z}_2\)violating) initial condition selected. Nonetheless, domainwalls will form later in time, due to the attractor nature of the solutions recovered here, and thanks to the \(\mathbb {Z}_2\) symmetry that is restored globally at late times of the cosmological evolution. We then imagine the initial conditions such that the Universe expands, but that a generic mixed state is chosen that entails both positiveenergy (CE) and negativeenergy (CM) meanfields, with a total energy density that is positive, but not yet corresponding to asymptotic tracker solutions and are away from the ground state (i.e. the groundstate of the effective YM action is not yet reached). This situation may correspond to a universe filled with radiation, for instance, the gluon plasma. At \(t\sim t_0\), the CdL decay of the positiveenergy CE configurations happens quite efficiently, such that the CE meanfield in a vicinity the “false” vacuum starts to decay, populating the CM ground state, which is the “true” vacuum. Due to local inhomogeneities, such decays do not happen at every spacetime point with the same rate, but fluctuate from point to point, although carrying an averaged decay rate \(\Gamma (t)\) that is generically a function of the cosmological time. For the case of QCD, the typical range of the gluon field fluctuations in the gluon plasma is around an inverse of the proton mass, more precisely 0.3 fm, as known from particle phenomenology.
At some time \(t^* \gg t_0\), with \(t^* \ll \Gamma (t^*)^{1}\), the Universe quickly expands, and both the initially large CE meanfield and the initially small CM meanfield, which appears due to the CdL decay, reach their attractors (i.e. for each of the two configuration, the corresponding fields roll down to their respective minima of the effective Lagrangian), allowing for exactly opposite contributions to the cosmological constant. Nonetheless, this happens at different 3space points regions, which on average are separated by lengthscales of size \(\sim 0.3\ \hbox {fm}\). This process then entails the emergence of spatiallyseparated stabilised patches of CM and CE vacua – in other words, “pockets” of “false” and “true” vacua that correspond to dS and AdS attractor solutions necessarily emergent at \(t^{*}\gg t_0\). As we have elaborated in the point above, the patches of “false” vacuum with positive cosmological constant (dS solution) do stabilize and remain metastable with an extremely long lifetime, way above the age of the Universe. We emphasize that the attractor solutions may not be reached simultaneously at every 3space point, but that conversely at \(t=t^*\) some small patches of space, with size \(\sim 0.3 \, \mathrm{fm}\), may survive either in the CM condensate or in the CE condensate states, providing on average a vanishing contribution to the cosmological constant. In this sense, the \(\mathbb {Z}_2\) symmetry is restored on average at \(t^*\) when each of the patches in the YM system effectively reaches its cosmological attractor. At the same time, the presence of the \(\mathbb {Z}_2\) symmetry between the different patches of solutions implies the formation of domainwalls between them as was mentioned above. The CdL decay basically terminates, and the patches get “locked” forever. This picture fits well with all we know (also experimentally) about the magnetic domains and the spontaneous magnetization in ferromagnetic materials strongly favouring the corresponding picture of confinement and provides its dynamical realisation in real physical time.

Within the case of \(\mathcal {N}=1\) SYM, described by the VenezianoYankielowicz (VY) superpotential, the dimensional transmutation is related to the gaugino condensation [41]. An initial \(\mathbb {Z}_{N}\) symmetry, interpolating N different vacua of the VY Lagrangian, is spontaneously broken down to \(\mathbb {Z}_{2}\), due to the formation of domain walls. Since the VY and the Savvidy’s Lagrangians are very similar, one might suggest that, in a nonSUSY model, the initial Mirror \(\mathbb {Z}_{2}\) symmetry of the Savvidy’s model can be violated by other nonperturbative configurations. Although this latter may represent an open possibility, nonetheless, according to simulations, the fact that the vacuum is dominated by the formation of YM monopole condensates or chromovortices seems to point exactly in the opposite direction. Indeed, these latter nonperturbative configurations preserve the \(\mathbb {Z}_{2}\) Mirror symmetry of the vacua. Even more, such configurations may emerge due the Mirror symmetry. Even though their interactions may lead to a spontaneous symmetry breaking of the Mirror vacua, if we average on a confinement sphere with a finite radius \(\Lambda _\mathrm{YM}^{1}\), we expect that the averaged energy splitting among the Mirror vacua will still vanish. In this sense, the ground state of a nonSUSY YM theory may “sit close” to the correspondent SYM ones.

In Fig. 3 we have displayed an unexpected result, that the uniform parts of the CE and CM YM condensates form a periodic pattern of spikes in cosmological times. These latter show up as quantum groundstate solutions of the equations of motion for the CE and CM YM condensates, coupled to the Einstein’s equation in the FLRW background, where the effective Lagrangian used as a classical fieldtheoretical model. The solutions are localized in time instants, but are not localized in 3space, i.e. they cannot be interpreted as instantons. In other words, these solutions are spacelike gauge solitons that appear and decay with a periodic time series. We were tempted to dub these solitons chronons, or \(\chi \)solutions, as they likely remind Sbranes within the context of string theory [42]. Chronons and antichronons trigger the dynamical screening mechanism depicted in Fig. 2, dictating the dynamical flow to the net zeroenergy density cosmological attractor.

A further appealing possibility is that a nonperfect screening among the two CE and CM contributions may provide a source of dark energy, with the same dynamics analyzed in Refs. [16, 18, 30, 31], as an effect of a soft dynamical breaking of the Mirror symmetry (e.g. in the firstorder semiclassical gravity correction to the QCD ground state [18]), while a screening mechanism of the Planckian contribution to the vacuum energy may arise from virtual black holes [32].
Footnotes
 1.
For example, the GinzburgLandau model describes the evolution of spatially averaged observables in superconductive materials, which in turn are crystals with local impurities and anisotropies – see e.g. Ref. [33].
 2.
 3.
We thank H.P. Pavel for pointing out to us the ansatz (A11) and the simple rescaling of the coupling constant.
Notes
Acknowledgements
Useful discussions with O. Teryaev, S. Brodsky, M. Faber, D. Antonov and D. Sivers are gratefully acknowledged. We would like to thank also G. Veneziano for a detailed constructive criticism on this subject. A.M. wishes to acknowledge support by the Shanghai Municipality, through the Grant No. KBH1512299, and by Fudan University, through the Grant No. JJH1512105. R.P. is supported in part by the Swedish Research Council Grants, contract numbers 62120134287 and 201605996, by CONICYT Grants PIA ACT1406 and MEC80170112, as well as by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No 668679). The work has been performed in the framework of COST Action CA15213 “Theory of hot matter and relativistic heavyion collisions” (THOR). This work was supported in part by the Ministry of Education, Youth and Sports of the Czech Republic, project LT17018.
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