Quantum vacuum, dark matter, dark energy and spontaneous supersymmetry breaking

We study the vacuum condensate characterizing many physical phenomena. We show that such a condensate may leads to non-trivial components of the dark energy and of the dark matter and may induces the spontaneous supersymmetry breaking, in a supersymmetric context. In particular, we consider the condensate induced by thermal states, fields in curved space-time and mixed particles.


I. INTRODUCTION
According to recent experimental data, the universe consists of 68% of dark energy [1]- [5] and about of 27% of dark matter. The rest is ordinary matter. The dark energy and the dark matter problem have been analyzed in different ways [6]- [31], however, their solution represents still a very big challenge.
Another object of study which has had a huge impact on contemporary physics is the Supersymmetry (SUSY) [32]- [37]. SUSY is a symmetry of nature that relates any boson to a fermion (called superpartner) with the same mass and internal quantum numbers, and viceversa. However, there is no evidence for the existence of the superpartners. Therefore, SUSY must be a broken symmetry, allowing for superparticles to be heavier than the corresponding Standard Model particles, or it must be ruled out as a fundamental symmetry. Intensive study has been devoted to the analysis of the possibility of SUSY breaking.
Here we report on recent results [52] according to which the vacuum condensate characterizing many phenomena [38]- [51], such as Hawking effect and fields in curved space, can explain the origin of dark energy and matter components [52] and can describe the spontaneous SUSY breaking [53]- [56]. In fact, all the phenomena inducing condensates have a non-zero vacuum energy which cannot be removed by use of the normal ordering procedure. The origin of the non trivial vacuum energy is due to the fact that the physical vacuum of such systems is a condensate of couples of particles and antiparticles which generate a positive value of the zero point energy. Such an energy can contribute to the dark sector of the universe and, in a supersymmetric context, induces the spontaneous SUSY breaking [53].
In particular, we analyze thermal states, fields in curved space and mixed particles and we show that dark matter components can be originated by the thermal vac- * e-mail address: capolupo@sa.infn.it uum of the hot plasma present at the center of a galaxy cluster (intracluster medium), by vacuum fluctuations of fields in curved space [57] and by the flavor neutrino vacuum [52]. Moreover, we show that dark energy contributes are given by vacuum condensates induced by the axion-photon mixing and by superpartners of mixed neutrinos [52].
We then consider the free Wees-Zumino model as a supersymmetric field theory, and we show that the presence of nonvanishing vacuum energy at the Lagrangian level implies that SUSY is spontaneously broken by the condensates. Next experiments using atomic systems characterized by vacuum condensate, could test our conjecture.
In Sec.II, we introduce the Bogoliubov transformations in QFT. In Sec.III, we compute the energy density and pressure of vacuum condensates induced by a generic Bogoliubov transformation for boson and fermion fields. In Secs.IV, V and VI, we present the contribution given to the energy of the universe by thermal states, with reference to the Hawking and Unruh effects, by fields in curved space and by particle mixing phenomena, respectively. The SUSY breaking induced by vacuum condensate is presented in Sec.VII and Sec.VIII is devoted to the conclusions.
A Bogoliubov transformation for bosons (similar discussion hold for fermions) assumes the form with a k (t) = a k e −iω k t , annihilators, such that a k |0 B = 0 and ω k = √ k 2 + m 2 . The coefficients satisfy the condi- and similar for fermions. The parameter ξ depends on the system one considers.
which induces an energy momentum tensor different from zero for |0(ξ, t) λ .

III. ENERGY-MOMENTUM TENSOR OF VACUUM CONDENSATE
In order to derive the state equation of the vacuum condensates |0(ξ, t) λ , (λ = B, F ) and their contributions to the energy density, one computes the expectation value of the free energy momentum tensor densities T µν (x) for real scalar fields and for Majorana fields on |0(ξ, t) λ , where, : ... :, denotes the normal ordering with respect to the original vacuum |0 λ . Since the off-diagonal components of Ξ λ µν (x) are zero, Ξ λ i,j (x) = 0, for i = j, the condensates behave as a perfect fluid and one can define their energy density and pressure as [52] respectively. For bosons, one has [52] and, in the particular case of the isotropy of the mo- , the energy density, the pressure and the state equation are respectively. For fermions, the energy density and the pressure are [52] and explicitly one has We also note that, being In the following, we denote with Θ(−ξ, x) = J −1 (−ξ, t)Θ(x)J(−ξ, t) the operators transformed by J(−ξ, t). All the equations above presented hold for many systems. The explicit form of the Bogoliubov coefficients specifies the particular system.

IV. VACUUM CONTRIBUTIONS OF THERMAL STATES, HAWKING AND UNRUH EFFECTS
In the context of the Thermo Field Dynamics (TFD) [43]- [45], the physical vacuum of systems at non-zero temperature is the thermal vacuum state |0(ξ(β)) λ , where β ≡ 1/(k B T ), k B is the Boltzmann constant and λ = B, F . The state |0(ξ(β)) λ is obtained by means of a Bogoliubov transformation similar to the ones presented in Section I, (for details see [45]) and the ther- k χ k , (χ = a for bosons and α for fermions) is the number operator [44].
The thermal Bogoliubov coefficients are given by U T k = e βω k e βω k ±1 and V T k = 1 e βω k ±1 , with − for bosons and + for fermions, and ω k = √ k 2 + m 2 . Such coefficients, used in Eqs. (8), (9) and (13), (14) give the contributions of the thermal vacuum states to the energy and pressure. In particular, for temperatures of order of the cosmic microwave radiation, i.e. T = 2.72K, one find that photons and particles with masses of order of (10 −3 − 10 −4 )eV contribute to the energy radiation with ρ ∼ 10 −51 GeV 4 and state equations, w = 1/3 [64]. On the other hand, non-relativistic particles give negligible contributions. Moreover, the thermal vacuum of the hot plasma filling the center of galaxy clusters, which has temperatures of order of (10 ÷ 100) × 10 6 K, has an energy density of (10 −48 −10 −47 )GeV 4 and a state equation w = 0.01. Such values of ρ and w are in agreement with the ones of the dark matter.
The thermal states can describe also the Unruh and of the Hawking effects; however, both of the phenomena do not contribute to the energy of the universe, since the temperatures are very low [64].

V. VACUUM CONTRIBUTION OF FIELDS IN CURVED BACKGROUND
Fields in curved background are also characterized by condensed vacuum and Bogoliubov transformations [48]. For such fields, the energy density and pressure depend on the particular metric considered. Here one considers the spatially flat Friedmann Robertson-Walker metric ds 2 = dt 2 − a 2 (t)dx 2 = a 2 (η)(dη 2 − dx 2 ) , where a is the scale factor, t is the comoving time, η is the conformal time, η(t) = The energy density and pressure are expressed as [65,66] where K is the cut-off on the momenta, φ k are mode functions and φ ′ k denotes the derivative of φ k with respect to the conformal time η. Assuming at late time the cutoff on the momenta much smaller than the comoving mass of the field, K ≪ ma and setting m ≫ H, for an arbitrary Robertson-Walker metric in infrared regime, one has [57] The state equation is w curv ≃ 0, which coincides with the one of the dark matter. Numerical values compatible with the ones of dark matter are found when mK 3 a 3 ∼ 10 −45 GeV 4 .

VI. VACUUM CONTRIBUTIONS OF PARTICLE MIXING
The particle mixing concerns neutrinos and quarks in fermion sector, axions, kaons, B 0 , D 0 , and η−η ′ systems, in boson sector. In the case of mixing between two fields, it is expressed as where, θ is the mixing angle, ϕ i (θ, x) are the mixed fields and ϕ i (x) are the free fields, with i = 1, 2.
-Boson mixing -The energy density and pressure of the vacuum condensate induced by mixed bosons are respectively. One can see that the kinetic and gradient terms of the mixed vacuum are equal to zero [52] 0| : Then, Eqs. (25) and (26) reduce to and the state equation coincides with the ones of the cosmological constant, w B mix = −1. By setting ∆m 2 = |m 2 2 − m 2 1 |, one has where K is the cut-off on the momenta.
-In the case of axion-photon mixing, for magnetic field strength B ∈ [10 6 −10 17 ]G, axion mass m a ≃ 2×10 −2 eV , sin 2 a θ ∼ 10 −2 and a Planck scale cut-off, K ∼ 10 19 GeV , one obtain a value of the energy density ρ axion mix = 2.3 × 10 −47 GeV 4 , which is of the same order of the estimated upper bound on the dark energy.
-In the case of superpartners of the neutrinos, considering masses m 1 = 10 −3 eV and m 2 = 9 × 10 −3 eV , such that ∆m 2 = 8 × 10 −5 eV 2 and assuming sin 2 θ = 0.3, one obtains, ρ B mix = 7 × 10 −47 GeV 4 for a cut-off on the momenta K = 10eV . Smaller values of the mixing angle lead to values of ρ B mix which are compatible with the estimated value of the dark energy also in the case in which the cut-off is K = 10 19 GeV , indeed ρ B mix depends linearly by sin 2 θ [52].
-Fermion mixing -The energy density and pressure are where ψ i (−θ, x) are the flavor neutrino fields or the quark fields. Being 0| : The state equation is then w F mix = 0, which is the one of the dark matter. The the energy density is By considering masses of order of 10 −3 eV , such that ∆m 2 ≃ 8 × 10 −5 eV 2 and a cut-off on the momenta K = m 1 + m 2 , one obtains ρ F mix = 4 × 10 −47 GeV 4 , which is in agreement with the estimated upper bound of the dark matter. For K of order of the Plank scale one has ρ F mix ∼ ×10 −46 GeV 4 . Notice that the quark confinement inside the hadrons should inhibit the gravitational interaction of the quark vacuum condensate. Thus the quark condensate should not affect the dark matter of the universe.

VII. SUSY BREAKING AND VACUUM CONDENSATE
We show that vacuum condensate provides a new mechanism of spontaneous SUSY breaking. We start by a situation in which SUSY is preserved at the lagrangian level and study the effects of vacuum condensation. We consider a Bogoliubov transformation acting simultaneously and with the same parameters on the bosonic and on the fermionic degrees of freedom in order not to break SUSY explicitly. Since, in any field theory which has manifest supersymmetry at the lagrangian level, a nonzero vacuum energy implies the spontaneous SUSY breaking [34], then the vacuum condensate (which is characterized by nontrivial energy) breaks SUSY spontaneously.
The effects of a Bogoliubov transformation are analyzed in the Wess-Zumino model described by the Lagrangian [67] where ψ is a Majorana spinor field, S is a scalar field and P is a pseudoscalar field. This Lagrangian is invariant under supersymmetry transformations [67].
respectively. Then we have the final result which is different from zero and positive. Notice that, Eq.(45), holds for disparate physical phenomena. As remarked above, the explicit form of the Bogoliubov coefficients V ψ k and V B k specifies the particular system.
Laser cooling experiments could allow to test the mechanism of SUSY breaking here presented. Indeed, the Wess-Zumino model in 2 + 1 dimensions can be obtained by a mixture of cold atoms-molecules trapped in two dimensional optical lattices [68]. In this case SUSY is preserved at zero temperature and is broken at T = 0. Then, a signature of SUSY breaking in such a system can be probed by the detection of the constant background noise due to the nonzero energy of the thermal vacuum, given by (45). In the thermal case, V B k = 1 e βω k −1 and V ψ k = 1 e βω k +1 , where β is the inverse temperature (in units such that k B = 1). Considering a two-dimensional optical lattice and a tuning of the parameters such that the effective mass of the emerging fields is zero, the vacuum energy density, due to Eq. (45), and representing the background noise, is given by < H > = 14 π ζ(3) T 3 .

VIII. CONCLUSIONS
The vacuum condensates characterizing many systems can contribute to the dark matter and to the dark energy. Dark matter contributes derive by thermal vacuum of intercluster medium, by the vacuum of fields in curved space and by the neutrino flavor vacuum. Dark energy contribute are given by axion-photon mixing. We have also shown that, in a supersymmetric field theory, vacuum condensates may leads to spontaneous SUSY breaking.