Metastable Dark Energy ∗ Ricardo G

Physics Letters B 764 (2017) 271–276

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Physics Letters B

www.elsevier.com/locate/physletb

Metastable dark energy ∗ Ricardo G. Landim , Elcio Abdalla

Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05314-970 São Paulo, São Paulo, Brazil a r t i c l e i n f o a b s t r a c t

Article history: We build a model of metastable dark energy, in which the observed vacuum energy is the value of the Received 3 November 2016 scalar potential at the false vacuum. The scalar potential is given by a sum of even self-interactions up Received in revised form 22 November 2016 to order six. The deviation from the Minkowski vacuum is due to a term suppressed by the Planck scale. Accepted 22 November 2016 The decay time of the metastable vacuum can easily accommodate a mean life time compatible with the Available online 25 November 2016 age of the universe. The metastable dark energy is also embedded into a model with SU(2) symmetry. Editor: M. Trodden R The dark energy doublet and the dark matter doublet naturally interact with each other. Athree-body Keywords: decay of the dark energy particle into (cold and warm) dark matter can be as long as large fraction of Dark energy the age of the universe, if the mediator is massive enough, the lower bound being at intermediate energy Dark matter level some orders below the grand uniﬁcation scale. Such a decay shows a different form of interaction Particle physics between dark matter and dark energy, and the model opens a new window to investigate the dark sector from the point-of-view of particle physics. © 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction retical origin of this constant is still an open question, with several attempts but with no definitive answer yet. At the present age, around ninety five percent of the universe There is a wide range of alternatives to the cosmological con- corresponds to two kinds of energy whose nature is largely un- stant, which includes canonical and non-canonical scalar fields known. The first one, named dark energy, is believed to be respon- [8–19], vector fields [20–27], holographic dark energy [28–35], sible for the current accelerated expansion of the universe [1,2] modifications of gravity and different kinds of cosmological fluids and is dominant at present time (∼ 68%) [3]. In addition to the [5–7]. baryonic matter (5%), the remaining 27% of the energy content of In addition, the two components of the dark sector may inter- the universe is a form of matter that interacts, in principle, only act with each other [36,37] (see [38] for a recent review), since gravitationally, known as dark matter. The simplest dark energy their densities are comparable and the interaction can eventually candidate is the cosmological constant, whose equation of state is alleviate the coincidence problem [39,40]. Phenomenological mod- in agreement with the Planck results [3]. els have been widely explored in the literature [37,41–45,7,31–34, This attempt, however, suffers from the so-called cosmological 46–50]. On the other hand, field theory models that aim a consis- constant problem, ahuge discrepancy of 120 orders of magnitude tent description of the dark energy/dark matter interaction are still between the theoretical (though rather speculative) prediction and few [51–53,19]. the observed data [4]. Such a huge disparity motivates physicists Here we propose a model of metastable dark energy, in which to look into more sophisticated models. This can be done either the dark energy is a scalar field with a potential given by the sum looking for a deeper understanding of where the cosmological con- of even self-interactions up to order six. The parameters of the stant comes from, if one wants to derive it from first principles, model can be adjusted in such a way that the difference between or considering other possibilities for accelerated expansion, such the energy of the true vacuum and the energy of the false one is as modifications of general relativity (GR), additional matter fields − the observed vacuum energy (10 47 GeV4). Other models of false and so on (see [5–7] and references therein). Moreover, the theo- vacuum decay were proposed in [54,55,52] with different poten- tials. Adifferent mechanism of metastable dark energy (although with same name) is presented in [56]. Furthermore, adark SU(2) * Corresponding author. R E-mail addresses: [email protected] (R.G. Landim), [email protected] model is presented, where the dark energy doublet and the dark (E. Abdalla). matter doublet naturally interact with each other. Such an inter- http://dx.doi.org/10.1016/j.physletb.2016.11.044 0370-2693/© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 272 R.G. Landim, E. Abdalla / Physics Letters B 764 (2017) 271–276 action opens a new window to investigate the dark sector from the point-of-view of particle physics. Models with SU(2)R symmetry are well-known in the literature as extensions of the standard model introducing the so-called left–right symmetric models [57–61]. Recently, dark matter has also been taken into account [62–69]. However, there is no similar effort to insert dark energy in a model of particle physics. We begin to attack this issue in this paper, with the dark SU(2)R model. The remainder of this paper is structured as follows. In Sect. 2 we present a model of metastable dark energy. It is embedded into a dark SU(2)R model in Sect. 3 and we summarize our results in Sect. 4. We use natural units (h¯ = c = 1) throughout the text.

Fig. 1. Scalar potential (1) with arbitrary parameters and values. The difference 2. A model of metastable dark energy between the true vacuum at ϕ± ≈±1.2and the false vacuum at ϕ0 = 0is − ∼ 10 47 GeV4. The current stage of accelerated expansion of the universe will be described by a canonical scalar field ϕ at a local minimum ϕ0 As usual in quantum field theory it is expected that the pa- − of its potential V (ϕ), while the true minimum of V (ϕ) is at ϕ± = rameter λ is smaller than one, thus, if we assume λ ∼ 10 1, the − ϕ. The energy of the true vacuum is below the zero energy of the Eq. (2) gives ∼ 10 47 GeV4 for m ∼ O(MeV). Bigger values of λ false vacuum, so that this difference is interpreted as the observed imply smaller values of m. Therefore, the cosmological constant is − value of the vacuum energy (10 47 GeV4). determined by the mass parameter and the coupling of the quartic We assume that by some mechanism the scalar potential is interaction. ϕ6 positive definite (as e.g. in supersymmetric models) and the true The potential (1) with the term is shown in Fig. 1. M2 vacuum lies at zero energy. As we will see below this value is pl adjusted by the mass of the scalar field and the coefficient of the 2.1. Decay rate quartic and sixth-order interaction. The rate at which the false vacuum decays into the true vacuum state will be calculated. The process of barrier penetration in which the metastable false The computation of the decay rate is based on the semi- vacuum decays into the stable true vacuum is similar to the old in- classical theory presented in [70]. The energy of the false vacuum = flationary scenario and it occurs through the formation of bubbles state at which ϕ 0is given by [71] ⎡ ⎤ of true vacuum in a false vacuum background. After the barrier penetration the bubbles grow at the speed of light and eventually 1 ⎣ ⎦ E0 =− lim ln exp (−S E [ϕ; T ]) dϕ( x,t) , (3) collide with other bubbles until all space is in the lowest energy T →∞ T state. The energy release in the process can produce new particles x,t and a Yukawa interaction g ¯ can account for the production ϕψψ where S E [ϕ; T ] is the Euclidean action, of a fermionic field which can be the pressureless fermionic dark + T matter. However, as we will see, the vacuum time decay is of the 2 2 order of the age of the universe, so another dominant process for 3 1 ∂ϕ 1 2 S E = d x dt + (∇ϕ) + V (ϕ) . (4) the production of cold dark matter should be invoked in order to 2 ∂t 2 recapture the standard cosmology. − T 2 If one considers a scalar field ϕ with the even self-interactions up to order six, one gets The imaginary part of E0 gives the decay rate and all the fields ϕ( x, t) integrated in Eq. (3) satisfy the boundary conditions 2 2 m 2 λ 4 λ 6 V (ϕ) = ϕ − ϕ + ϕ , (1) ϕ( x, +T /2) = ϕ( x, −T /2) = 0 . (5) 2 4 32m2 where m and λ are positive free parameters of the theory and the The action (4) is stationary under variation of the fields that coefficient of the ϕ6 interaction is chosen in such a way that the satisfy the equations potential (1) is a perfect square. This choice will be useful to cal- 2 δS E ∂ ϕ culate the false vacuum decay rate. =− −∇2ϕ + V (ϕ) = 0(6) The potential (1) has extrema at ϕ = 0, ϕ± =±√2m and ϕ = δϕ ∂t2 0 λ 1 √ϕ± , but it is zero in all of the minima (ϕ0 and ϕ±). In order to and are subject to the boundary conditions (5). In order to get 3 have a cosmological constant, the potential should deviate slightly the solution of Eq. (6) we make an ansatz that the field ϕ( x, t) is from the perfect square (1). Once the coupling present in GR is the invariant under rotations around x0, t0 in four dimensions, which Planck mass M pl it is natural to expect that the deviation from the in turn is valid for large T [72]. The ansatz is − Minkowski vacuum is due to a term proportional to M 2. Thus we pl ( x,t) = ( ) with ≡ ( x − x )2 + (t − t )2 . (7) ϕ6 ϕ ϕ ρ ρ 0 0 assume that the potential (1) has a small deviation given by . M2 pl In terms of the Eq. (7), the field equation (6) becomes Although the value of the scalar field at the minimum point ϕ± also changes, the change is very small and we can consider that 2 d ϕ + 3 dϕ = the scalar field at the true vacuum is still ± √2m . The difference V (ϕ). (8) λ dρ2 ρ dρ between the true vacuum and the false one is The above equation of motion is analogous to that of a particle 6 − ≈ 64m at position ϕ moving in a time ρ, under the influence of a poten- V (ϕ0) V (ϕ±) . (2) dϕ 3 2 tial −V (ϕ) and a viscous force − 3 . This particle travels from λ M pl ρ dρ R.G. Landim, E. Abdalla / Physics Letters B 764 (2017) 271–276 273

an initial value ϕi and ρ = 0 and reaches ϕ = 0at ρ →∞. The The decay time is obtained inverting the above expression, Euclidean action (4) for the rotation invariant solution becomes 12 1/4 −25 143 m −4 ∞ tdecay ≈ 10 exp 10 λ s . (18) 2 GeV 2 3 1 ∂ϕ S E = 2π ρ dρ + V (ϕ) . (9) 2 ∂ρ The expression for the decay time gives the lowest value of the 0 mass parameter m for which (18) has at least the age of the uni- 17 The metastable vacuum decay into the true vacuum is seen as verse (10 s). Therefore the mass parameter should be the formation of bubbles of true vacuum surrounded by the false −12 dϕ m ≥ 10 GeV , (19) vacuum outside. The friction term dρ is different from zero only − at the bubble wall, since the ﬁeld is at rest inside and outside. The for λ ∼ 10 1. Thus, it is in agreement with the values for m at decay rate per volume of the false vacuum, in the semi-classical which the scalar potential describes the observed vacuum energy, approach, is of order as discussed in the last section. The mass of the scalar ﬁeld can − be smaller if the coupling λ is also smaller than 10 1. The de- −4 cay rate (17) is strongly suppressed for larger values of m. The ≈ M exp(−S E ), (10) V bubble radius given in Eq. (13) for the mass parameter (19) is R ≥ 0.03 cm. where M is some mass scale. When S E is large the barrier penetration is suppressed and the mass scale is not important. This is Notice that the axion would still be a possibility, although it the case when the energy of the true vacuum is slightly below the arises in a quite different context. We can also consider the grav- energy of the false vacuum, by an amount , considered here as itational effect in the computation of the decay rate. In this case − M2 small as ∼ 10 47 GeV4. On the other hand, the potential V (ϕ) is ¯ pl R R the new action S has the Einstein–Hilbert term 2 , where not small between ϕ0 and ϕ±. is the Ricci scalar. The relation between the new action S¯ and the We will use the so-called ‘thin wall approximation’, in which ϕ old one S E can be deduced using the thin wall approximation and is taken to be inside of a four-dimensional sphere of large radius R. it gives [73] For a thin wall we can consider ρ ≈ R in this region and since R S E is large we can neglect the viscous term, which is proportional to S¯ = , (20) 2 3/R at the wall. The action (9) in this approximation is + R 2 1 2 2 π 4 2 3 S E − R + 2π R S1 , (11) where S E and R are given by Eqs.√ (14) and (13), respectively, in 2 3M the absence of gravity, and = √ pl is the value of the bubble where S is a surface tension, given by 1 radius when it is equal to the Schwarzschild radius associated with ϕ+ the energy released by the conversion of false vacuum to true one. √ √ − For ∼ 10 47 GeV4 we get ∼ 1027 cm, thus the gravitational S = 2 dϕ V , (12) 1 correction R/ is very small. Larger values of m give larger R, ϕ0 implying that the gravitational effect should be taken into account. Even so, the decay rate is still highly suppressed. for small . The action (11) is stationary at the radius

3S1 3. A dark SU(2)R model R , (13)

As an example of how the metastable dark energy can be em- and at the stationary point the action (11) becomes bedded into a dark sector model we restrict our attention to a model with SU(2) symmetry. Both dark energy and dark matter 27π 2 S4 R 1 are doublets under SU(2) and singlets under any other symme- S E . (14) R 2 3 try. Presumably, the dark sector interacts with the standard model 1 particles only through gravity. After the spontaneous symmetry Using the potential (1) into Eq. (12) we obtain + − breaking by the dark Higgs ﬁeld φ, the gauge bosons Wd , Wd = = m3 and Zd acquire the same mass given by mW mZ gv/2, where S1 = , (15) v is the VEV of the dark Higgs. The dark SU(2) model contains a λ R dark matter candidate ψ, adark neutrino νd (which can be much which in turn gives the Euclidean action at the stationary point in lighter than ψ), and the dark energy doublet ϕ, which contains + the thin wall approximation (11) ϕ0 and ϕ , the latter being the heaviest particle. After symmetry + breaking ϕ0 and ϕ have different masses and both have a po- 2 12 27π m (ϕ†ϕ)3 tential given by Eq. (1) plus the deviation . The interaction S E . (16) M2 2λ4 3 pl between the ﬁelds are given by the Lagrangian Substituting the action (16) into the decay rate (10) with ∼ − 10 47 GeV4 and the mass scale being M ∼ 1GeVfor simplicity, we L = + +μ + − −μ + 0 0μ int g Wdμ JdW Wdμ JdW Zdμ JdZ , (21) have where the currents are m 12 − ≈ exp −10143 4 GeV4 (17) λ . + 1 V GeV μ ¯ μ 0 μ ¯ + ¯ + μ 0 J = √ [νdRγ ψR + i(ϕ ∂ ϕ − ϕ ∂ ϕ )] , (22) dW 2 −μ 1 + + ϕ6 = √ [ ¯ μ + μ ¯ 0 − ¯ 0 μ ] 1 The term is very small and can be ignored. JdW ψR γ νdR i(ϕ ∂ ϕ ϕ ∂ ϕ ) , (23) M2 pl 2 274 R.G. Landim, E. Abdalla / Physics Letters B 764 (2017) 271–276

+ 0 Fig. 2. Feynman diagram for the decay ϕ → ϕ + ψ + νd .

1 = = 0μ = [¯ μ − ¯ μ + + μ ¯ + − ¯ + μ + Fig. 3. Differential decay rate d (29) as a function of s12 for M 1000 GeV, m1 J νdRγ νdR ψR γ ψR i(ϕ ∂ ϕ ϕ ∂ ϕ ) = = ∼ −27 −2 dZ 2 1MeV, m2 100 GeV, m3 0and Gd 10 GeV . − i(ϕ0∂μϕ¯ 0 − ϕ¯ 0∂μϕ0)] . (24) The decay rate can be evaluated from [74] The currents above are very similar to the ones in the elec- 1 1 2 troweak theory. The main differences are that there is no hyper- d = |M| ds12ds23 , (29) + (2 )3 32M3 charge due to U (1)Y and there is a new doublet, given by ϕ π and 0. ϕ where for a given value of s12, the range of s23 is determined by Among the interactions shown in Eq. (21), it is of interest to its values when p 2 is parallel or antiparallel to p 3 + → 0 + + calculate the decay rate due to the process ϕ ϕ ψ νd. The three-body decay leads to a cold dark matter particle whose 2 = ∗ + ∗ 2 − ∗2 − 2 − ∗2 − 2 mass can be accommodated to give the correct relic abundance, to (s23)max (E2 E3) E2 m2 E3 m3 , (30) a dark neutrino which is a hot/warm dark matter particle, and to 2 0 a scalar field ϕ . Similar to the weak interactions, we assume that ∗ ∗ 2 ∗2 2 ∗2 2 (s23)min = (E + E ) − E − m + E − m . (31) the energy involved in the decay is much lower than the mass 2 3 2 2 3 3 of the gauge fields, thus the propagator of W is proportional to √ ∗ 2 2 ∗ 2 2 The energies E = (s −m +m )/(2 s ) and E = (M − s − g /m and the currents interact at a point. We can also define √ 2 12 1 2 12 3 12 W 2 m3)/(2 s12) are the energies of particles 2 and 3 in the s12 rest 2 frame [74]. The invariant s , in turn, has the limits g Gd 12 ≡ √ , (25) 8m2 2 2 W 2 (s12)max = (M − m3) ,(s12)min = (m1 + m2) . (32) where Gd is the dark coupling. With the limits for s12 (32) and for s23 (30)–(31) and with the The Feynman diagram for the decay is shown in Fig. 2 and the amplitude squared (28) we can integrate Eq. (29) for different val- amplitude for the decay is ues of masses, in order to get the decay time tdec . The plot of d as a function of s12 is shown in Fig. 3 and 2 g μ 5 the decay rate is the area under the curve. For illustrative pur- M = (P + p1)μu(p3)γ (1 + γ )u(p2), (26) 2 poses, we set the mass of the particles as being M = 1000 GeV, 4mW m1 = 1MeV, m2 = 100 GeV and m3 = 0GeV. With these values of where the labels 1, 2 and 3are used, respectively, for the particles masses, the decay time is of the order of the age of the universe 0 − −2 − −2 ϕ , ψ and νd. The energy–momentum conservation implies that (1017 s) with G ∼ 10 27 GeV , while with G ∼ 10 26 GeV + d d = + + 15 P p1 p2 p3, where P is the four-momentum of the field ϕ the decay time is tdec ∼ 10 s. If g is for instance of the same ± and M will be its mass. order of the fine-structure constant, the gauge bosons W and Zd + → 0 + + d The averaged amplitude squared for the decay ϕ ϕ ψ have masses around 1011 GeV in order to the decay time to be νd is 1015 s. Such decay times are compatible with phenomenological models of interacting dark energy, where the coupling is propor- |M|2 = 2 [ + · ][ + · ] 16Gd 2 (P p1) p2 (P p1) p3 tional to the Hubble rate [49,38]. In addition, depending on the values of the free parameters, the mass of the gauge bosons can 2 − (P + p1) (p2 · p3 + m2m3) . (27) be of the same order of the grand unified theories scale.

Using the energy–momentum conservation and defining the in- 4. Conclusions 2 2 variants sij as sij ≡ (pi + p j) = (P − pk) , we can reorganize the amplitude squared. The three invariants are not independent, In this paper we presented a model of metastable dark energy, + + = 2 + 2 + 2 + 2 obeying s12 s23 s13 M m1 m2 m3 from their definitions in which the dark energy is a scalar field with a potential given and the energy–momentum conservation. With all these steps we by a sum of even self-interactions up to order six. The parameters eliminate s13 and get of the model can be adjusted in such a way that the difference between the energy of the true vacuum and the energy of the false − |M|2 = 2[− 2 − + 2 + 2 + 2 + 3 one is around 10 47 GeV4. The decay of the false vacuum to the 16Gd 2s12 2s12s23 2(M m1 m2 m3)s12 true one is highly suppressed, thus the metastable dark energy can + 2 (m2 m3) 2 2 2 2 explain the current accelerated expansion of the universe. We do + s23 − 2m2m3(M + m ) − 2m M 2 1 1 not need a very tiny mass for the scalar field (as it happens for + 2 some models of quintessence), in order to get the observed value 2 2 2 (m2 m3) − 2m (m + m ) − ] . (28) of the vacuum energy. 2 1 2 2 R.G. Landim, E. Abdalla / Physics Letters B 764 (2017) 271–276 275

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