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Home , False vacuum decay, Inflation (cosmology)

False as a unstable state Krzysztof URBANOWSKI

Institute of , University of Zielona G´ora, Szafrana 4a, PL 65–516 Zielona G´ora, Poland

e–mail: [email protected] 1 Introduction Starting from the asymptotic expression (4) for a(t) and using (6) after some algebra one finds for considered. The standard calculations shows that the same volume factors t ≫ T that should appear in both early and late decay rate estimations (see Krauss and Dent [8, 14] ). i i The problem of false vacuum decay became famous after the publication of pioneer papers by h (t) ≃ E +(− ) c + (− )2 c + ..., (8) This means that the calculations of cross–over time T can be applied to survival probabilities per M t→∞ min t 1 t 2 Coleman and his colleagues [1, 2, 3]. The instability of a physical system in a state which is not an unit volume. For the same reasons within the quantum field the quantity EM (t) can be where c = c∗, i =1, 2,...; (coefficients c depend on ω(E)). This last relation means that absolute minimum of its density, and which is separated from the minimum by an effective i i i replaced by the energy per unit volume ρM (t)= EM (t)/V because these volume factors V appear c c 2 1 in the numerator and denominator of the formula (6) for hM (t). This conclusion seems to hold potential barrier was discussed there. It was shown, in those papers, that even if the state of EM (t) ≃ Emin − ..., γM (t) ≃ 2 + ..., (for t ≫ T ). (9) t2 t false false the early is too cold to activate a ”thermal” transition (via thermal fluctuations) to the when considering the energy E0 (t) of the system in false vacuum state |0i because Universe lowest energy (i.e. ”true vacuum”) state, a quantum decay from the false vacuum to the true These properties take place for all unstable states which survived up to times t ≫ T . is assumed to be homogeneous and isotropic at suitably large scales. So at such scales to a suffi- Note that from (9) it follows that lim E (t)= E and lim γ (t)=0. false 0 false vacuum may still be possible through a barrier penetration via macroscopic quantum tunneling. t→∞ M min t→∞ M ciently good accuracy we can extract properties of the ρ0 = EM /V = E0 /V For the density ω(E) of the form (3) (i. e. for a(t) having the asymptotic form given by (4)) we false false Not long ago, the decay of the false vacuum state in a cosmological context has attracted interest, of the system in the false vacuum state |0i from properties of the energy E0 (t) of the sys- have false false false especially in view of its possible relevance in the process of tunneling among the many vacuum (1) tem in this state defining ρ0 (t) as ρ0 (t) = E0 (t)/V . This means that in the case of a η (Emin) states of the string landscape (a set of vacua in the low energy approximation of ). c1 = λ +1, c2 =(λ + 1) . (10) meta–stable (unstable or decaying, false) vacuum the following important property of κ(t) holds: η(Emin) In many models the scalar field potential driving inflation has a multiple, low–energy minima or false ”false vacuua”. Then the absolute minimum of the energy density is the ”true vacuum”. ρ0 (t) − ρbare For the most general form (3) of the density ω(E) (i. e. for a(t) having the asymptotic form given κ(t) ≡ false , Recently the problem of the instability the false vacuum state triggered much discussion in the by (4) ) we have ρ0 − ρbare context of the discovery of the Higgs–like resonance at 125 — 126 GeV (see, eg., [4, 5, 6, 7]). (1) η (Emin) where ρ = E /V is the energy density of the true (bare) vacuum. From the last equations In the recent analysis [5] assuming the validity of the up to Planckian it c1 = λ +1, c2 =(λ + 1) . (11) bare min η(Emin) the following relation follows was shown that a Higgs mass m < 126 GeV implies that the electroweak vacuum is a metastable h The energy densities ω(E) considered in quantum mechanics and in quantum field theory can be false false state. This means that a discussion of Higgs vacuum stability must be considered in a cosmological 1 described by ω(E) of the form (3), eg. quantum field theory models correspond with λ = . ρ0 (t) − ρbare =(ρ0 − ρbare) κ(t). framework, especially when analyzing inflationary processes or the process of tunneling among the 2 A general form of many vacuum states of the string landscape. Thus, because for tT is presented in panels B of Figs (1) whereas for t ≫ T we have false vacuum regions can survive up to the times much later than times when the exponential decay — (3). These results were obtained for ω(E)= ωBW (E). The crossover time T , that is the time law holds. In the mentioned paper by Krauss and Dent the attention was focused on the possible region where fluctuations of PM (t) and EM (t) take place depend on the value of the parameter ~2 0 0 false false s =(E − E )/Γ in the model considered : The smaller s the shorter T . ρ (t) − ρbare =(ρ − ρbare) κ(t) ≃± d2 , (t ≫ T ), (16) behavior of the unstable false vacuum at very late times, where deviations from the exponential 0 M min M 0 0 0 t2 decay law become to be dominat. where d = d∗. The units ~ =1= c will be used in the next formulas. Analogous relations (with The aim of this presentation is to analyze properties of the false vacuum state as an unstable state, 2 2 the same κ(t)) take place for Λ(t)= 8πG ρ(t), or Λ(t)=8πGρ(t) in ~ = c =1 units: the form of the decay law from the canonical decay times t up to asymptotically late times and to c2 discuss the late time behavior of the energy of the false vacuum states. Λ(t) − Λbare = (Λ0 − Λbare) κ(t), (17)

2 Unstable states in short or, Λ(t) = Λbare + (Λ0 − Λbare) κ(t). (18) If |Mi is an initial unstable state then the survival probability, P(t), equals P(t)= |a(t)|2, where a(t) is the survival amplitude, a(t)= hM|M; ti, and a(0) = 1, and, |M; ti = exp[−itH] |Mi, H is the total Hamiltonian of the system under considerations. (The units ~ = c = 1 are One may expect that Λ0 equals to the calculated within quantum field used in this presentation). The spectrum, σ(H), of H is assumed to be bounded from below, theory. From (18) it is sen that for t −∞. min min Λ(t) ≃ Λ , for (t 0, b > 0 and 0 < q < 1. This means that the decay law P(t) of unstable Λ(t) ≃ Λbare + Λ0 κ(t). (21) states decaying in the vacuum can not be described by an exponential function of time t if time t is suitably long, t → ∞, and that for these lengths of time P(t) tends to zero as t → ∞ more slowly Note that for t ≫ T there should be (see (16)) than any exponential function of t. The analysis of the models of the decay processes shows that 2 0 0 d2 α P(t) ≃ exp[−Γ M t], (where ΓM is the decay rate of the state |Mi), to an very high accuracy at Λ0 κ(t) ≃ 8πG 2 ≡± 2 , for (t ≫ T ). (22) 0 t t the canonical decay times t: From t suitably later than the initial instant t0 up to t ≫ τ =1/ΓM , (τ is a lifetime), and smaller than t = T , where T is the crossover time and it denotes the time t for which the non–exponential deviations of a(t) begin to dominate. 5 Final Remarks In general, in the case of quasi–stationary (metastable) states it is convenient to express a(t) in the following form: a(t)= ac(t)+ anon(t), where ac(t) is the exponential (canonical) part of a(t), that Parametrization following from quantum theoretical treatment of metastable vacuum states can 0 i 0 0 is ac(t)= N exp[−it(EM − 2 ΓM )], (EM is the energy of the system in the state |Mi measured explain why the with the time–dependent cosmological constant Λ(t) are worth con- at the canonical decay times, N is the normalization constant), and anon(t) is the non–exponential sidering and may help to explain the cosmological constant problem [16, 17]. The time dependence part of a(t)). For times t ∼ τ: |ac(t)|≫|anon(t)|. The crossover time T can be found by solving α2 of Λ of the type Λ(t) = Λbare + t2 was assumed eg. in [18] but there was no any explanation the following equation, what physics suggests such a choice of the form of Λ. Earlier analogous form of Λ was obtained in |a (t)| 2 = |a (t)| 2. (2) c non [19], where the invariance under scale transformations of the generalized Einstein equations was The amplitude anon(t) exhibits inverse power–law behavior at the late time region: t ≫ T . In- studied. Such a time dependence of Λ was postulated also in [20] as the result of the analysis deed, the integral representation (1) of a(t) means that a(t) is the Fourier transform of the energy of the hypothesis. The cosmological model with time dependent Λ of the above distribution function ω(E). Using this fact we can find asymptotic form of a(t) for t → ∞. Figure 2:The case s0 = 50. Panel A — Axes: y = P(t), The logarithmic scale), x = t/τ. postulated form was studied also in [21] and in much more recent papers. Results are rigorous. If to assume that Panel B, Axes: y = κ(t), x = t/τ. The horizontal red dashed line denotes y = κ(t)=1, The nice feature and maybe even the advantage of the formalism presented in Sect. 4 is that in the 0 case of the universe with metastable (false) vacuum if one realized that the decay of this unstable ω(E)=(E − E )λ η(E) ∈ L (−∞, ∞), (3) that is EM (t)= EM . min 1 vacuum state is the quantum decay process then it emerges automatically that there have to exist def (k) d the true of the system that is the true (or bare) vacuum with the minimal energy, (where 0 ≤ λ < 1), and η(Emin) = η0 > 0, and η (E)= dE η(E), (k =0, 1,...,n), exist and def (k) E > −∞, of the system corresponding to him and equivalently, ρ = E /V , or Λ . they are continuous in [E , ∞), and limits lim η(k)(E) = η exist, and min bare min bare min E→Emin+ 0 What is more, in this case such Λ ≡ Λ(t) emerges that at suitable late times it has the form de- lim (E − E )λ η(k)(E)=0 for all above mentioned k, then one finds that E→∞ min scribed by relations (21), (22). In such a case the function κ(t) given by the relation (12) describes i λ+1 time dependence for all times t of the energy density ρM (t) or the cosmological ”constant” ΛM (t) a(t) ∼ (−1) e−iEmint − Γ(λ + 1) η (4) and it general form is presented in panels B in Figs (1) — (3). Note that results presented in t 0 t→∞ h  Sections 2 — 4 are rigorous. The formalism mentioned was applied in [14, 15], where cosmological λ+2 α2 i (1) models with Λ(t) = Λbare ± t2 were studied: The most promising result is reported in [15] where + λ − Γ(λ + 2) η0 + ... = anon(t).  t i using the parametrization following from the mentioned quantum theoretical analysis of the decay process of the unstable vacuum state an attempt was made to explain the small today’s value of From (4) it is seen that asymptotically late time behavior of the survival amplitude a(t) depends the cosmological constant Λ. This shows that formalism and the approach described in this paper rather weakly on a specific form of the energy density ω(E). The same concerns a decay curves and in [14, 15] is promising and can help to solve the cosmological constant and other cosmological P(t) = |a(t)|2. A typical form of a decay curve, that is the dependence on time t of P(t) when problems and it needs further studies, especially it to take into account the LHC result concerning t varies from t = t0 = 0 up to t > 30τ is presented in panels A od Figs (1) — (3). Re- the mass of the and cosmological consequences of this result. sults presented in these Figures were obtained for the Breit–Wigner energy distribution function, Acknowledgments: The work was supported in part by the NCN grant No DEC- 0 N ΓM 2013/09/B/ST2/03455. ω(E) ≡ ωBW = 2π Θ(E − Emin) 0 2 0 2, which corresponds with λ =0 in (3). (E−EM ) +(ΓM /2)

3 Instantaneous energy and instantaneous Figure 3:The case s0 = 50. The enlarged part of Fig. 2. Panel A — Axes: y = P(t), References The logarithmic scale), x = t/τ. Panel B, Axes: y = κ(t), x = t/τ. decay rate [1] S. Coleman, Phys. Rev. D 15, 2929 (1977). [2] C.G. Callan and S. Coleman, Phys. Rev. D 16, 1762 (1977). The amplitude a(t) contains information about the decay law P(t) of the state |Mi, that is about 4 Cosmological applications 0 0 [3] S. Coleman and F. de Lucia, Phys. Rev. D 21, 3305 (1980). the decay rate ΓM of this state, as well as the energy EM of the system in this state. This infor- mation can be extracted from a(t). Indeed if |Mi is an unstable (a quasi–stationary) state then Krauss and Dent in their paper [8] mentioned earlier made a hypothesis that some false vacuum [4] A. Kobakhidze, A. Spencer–Smith, Phys. Lett. B 722, 130,(2013). ∼ 0 i 0 false true a(t) = exp[−i(EM − ΓM ) t] ≡ ac(t) for t ∼ τ. So, there is regions do survive well up to the time T or later. Let |Mi = |0i , be a false, |0i – a true, [5] G. Degrassi, et al.,JHEP 1208, 098, (2012). 2 false vacuum states and E0 be the energy of a state corresponding to the false vacuum measured at 0 i 0 ∂ac(t) 1 true [6] J. Elias–Miro, et al., Phys. Lett. B 709, 222, (2012). EM − ΓM ≡ i , (t ∼ τ), (5) the canonical decay time and E0 be the energy of true vacuum (i.e. the true ground state of 2 ∂t ac(t) the system). As it is seen from the results presented in previous Section, the problem is that the [7] Wei Chao, et al., Phys. Rev. D 86, 113017, (2012). false in the case of quasi–stationary states. energy of those false vacuum regions which survived up to T and much later differs from E0 [8] L. M. Krauss, J. Dent, Phys. Rev. Lett., 100, 171301 (2008). The standard interpretation and understanding of the quantum theory and the related construc- [13]. [9] S. Winitzki, Phys. Rev. D 77, 063508 (2008). tion of our measuring devices are such that detecting the energy E0 and decay rate Γ 0 one is sure Now, if one assumes that Etrue ≡ E and Efalse = E0 . and takes into account results of the M M 0 min 0 M [10] L. A. Khalfin, Zh. Eksp. Teor. Fiz. 33, 1371 (1957); [Sov. Phys. JETP 6, 1053 (1958)]. that the amplitude a(t) has the canonical form ac(t) and thus that the relation (5) occurs. Taking previous Section (including those in Panels B of Figs (1 ) — (3) then one can conclude that the the above into account one can define the ”effective Hamiltonian”, hM , for the one–dimensional energy of the system in the false vacuum state has the following general properties: [11] K. Urbanowski, Eur. Phys. J. D, 54, (2009). subspace of states H|| spanned by the normalized vector |Mi as follows [11, 12] false true [12] K. Urbanowski, Cent. Eur. J. Phys. 7, 696 (2009). E0 (t)= E0 +∆E · κ(t), (13) def ∂a(t) 1 def i false true false [13] K. Urbanowski, Phys. Rev. Lett., 107, 209001 (2011). hM = i = EM (t) − γM (t). (6) where ∆E = E0 − E0 and κ(t) ≃ 1 for t ∼ τ0