PHYSICAL REVIEW D 97, 116012 (2018) Decay rate of electroweak vacuum in the standard model and beyond So Chigusa,1 Takeo Moroi,1 and Yutaro Shoji2 1Department of Physics, University of Tokyo, Tokyo 113-0033, Japan 2Institute for Cosmic Ray Research, The University of Tokyo, Kashiwa 277-8582, Japan (Received 26 March 2018; published 14 June 2018) We perform a precise calculation of the decay rate of the electroweak vacuum in the standard model as well as in models beyond the standard model. We use a recently developed technique to calculate the decay rate of a false vacuum, which provides a gauge invariant calculation of the decay rate at the one- loop level. We give a prescription to take into account the zero modes in association with translational, dilatational, and gauge symmetries. We calculate the decay rate per unit volume, γ, by using an analytic formula. The decay rate of the electroweak vacuum in the standard model is estimated to be 3 þ40þ184þ144þ2 log10γ × Gyr Gpc ¼ −582−45−329−218−1 , where the first, second, third, and fourth errors are due to the uncertainties of the Higgs mass, the top quark mass, the strong coupling constant and the choice of the renormalization scale, respectively. The analytic formula of the decay rate, as well as its fitting formula given in this paper, is also applicable to models that exhibit a classical scale invariance at a high energy scale. As an example, we consider extra fermions that couple to the standard model Higgs boson, and discuss their effects on the decay rate of the electroweak vacuum. DOI: 10.1103/PhysRevD.97.116012 I. INTRODUCTION vacuum can be evaluated through a rate of bubble nucle- ation in unit volume and unit time as formulated in [26,27]. In the standard model (SM) of particle physics, it has The rate is expressed in the form of been known that the Higgs quartic coupling may become negative at a high scale through quantum corrections, so γ ¼ Ae−B; ð1:1Þ that the Higgs potential develops a deeper vacuum. The detailed shape of the Higgs potential depends on the Higgs where B is the action of a so-called bounce solution, and and the top masses; with the recently observed Higgs mass prefactor A is quantum corrections having mass dimension of ∼125 GeV, it has been known that the electroweak (EW) 4. The bounce solution is an Oð4Þ symmetric solution of vacuum is not absolutely stable if the SM is valid up to ∼1010 1 the Euclidean equations of motion, connecting the two GeV or higher. In such a case, the EW vacuum can vacua. Although the dominant suppression of the decay rate decay into the deeper vacuum through tunneling in quan- comes from B, the prefactor A is also important. This is tum field theory. The lifetime of the EW vacuum has been because of large quantum corrections from the top quarks one of the important topics in particle physics and and the gauge bosons. Thus, it is essential to calculate both cosmology. A and B to determine the decay rate precisely. In the SM, The decay rate of the EW vacuum has been discussed for there are infinite bounce solutions owing to (i) the classical a long time. The calculation of the decay rate at the one- scale invariance at a high energy scale, (ii) the global loop level first appeared in [15] and was also discussed in ð2Þ ð1Þ ð1Þ – symmetries corresponding to SU L × U Y=U EM,as other literature [16 25]. However, there are subtleties in the well as (iii) the translational invariance. For the calculation treatment of zero modes related to the gauge symmetry of the prefactor A, a proper procedure to take account of the breaking which make it difficult to perform a precise and effects of the zero modes related to (i) and (ii) were not well reliable calculation of the decay rate. The lifetime of a understood until recently. In addition, the previous calcu- lations of A were not performed in a gauge-invariant way, 1For the absolute stability of the EW vacuum in the SM, which made the gauge invariance of the result unclear. see [1–14]. Recently, a prescription for the treatments of the gauge zero modes was developed [28,29], based on which a Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. complete calculation of the decay rate of the EW vacuum Further distribution of this work must maintain attribution to became possible. The calculation has been performed by the author(s) and the published article’s title, journal citation, the present authors in a recent publication [25] and also by and DOI. Funded by SCOAP3. [24]. The purpose of this paper is to give a more complete 2470-0010=2018=97(11)=116012(34) 116012-1 Published by the American Physical Society SO CHIGUSA, TAKEO MOROI, and YUTARO SHOJI PHYS. REV. D 97, 116012 (2018) and detailed discussion about the calculation of the decay the quartic coupling constant negative at a high scale. In the rate. In [25], we have numerically evaluated the functional SM, the instability occurs when the Higgs amplitude determinants of fluctuation operators, which are necessary becomes much larger than the EW scale. Since the typical for the calculation of the decay rate. Here, we perform the field value for the bounce configuration is around that scale, calculation analytically; a part of the analytic results was we can neglect the quadratic term in the Higgs potential. first given in [24]. The effects of the zero modes and the In this section, we use a toy model with Uð1Þ gauge modes with the lowest angular momentum are carefully symmetry to derive relevant formulas. The calculation of taken into account, on which previous works had some the decay rate of the EW vacuum is almost parallel to that in confusion. We give fitting formulas of the functional the case with Uð1Þ gauge symmetry; the application to the determinants based on analytic results, which are useful SM case will be explained in the next section. for the numerical calculation of the decay rate. We also provide a C þþpackage to study the electroweak vacuum A. Setup stability (ELVAS), which is available at [30]. In this paper, we discuss the calculation of the decay rate Let us first summarize the setup of our analysis. We of the EW vacuum in detail. We first present a detailed study the decay rate of a false vacuum whose instability is formulation of the calculation of the decay rate at the one- due to an RG running of the quartic coupling constant of a loop level. We derive a complete set of analytic formulas scalar field, Φ. We assume that Φ is charged under the Uð1Þ that can be used for any models that exhibit classical scale gauge symmetry (with charge þ1); the kinetic term invariance at a high energy scale like the SM. Then, as one includes of the important applications, we calculate the decay rate of L ∋ ½ð∂ − ÞΦ†ð∂ − ÞΦ ð Þ the EW vacuum in the SM. We find that the lifetime of the kin μ igAμ μ igAμ ; 2:1 EW vacuum is much longer than the age of the Universe. There, we see that one-loop corrections from the top quark where Aμ is the gauge field and g is the gauge coupling and the gauge bosons are very large although there is an constant, while we consider the following scalar potential: accidental cancellation. It shows the importance of A for the evaluation of a decay rate. We also evaluate the decay VðΦÞ¼λðΦ†ΦÞ2: ð2:2Þ rates of the EW vacuum for models with extra fermions that couple to the Higgs field. In such models, the EW vacuum The quartic coupling, λ, depends on the renormalization tends to be destabilized compared with that of the SM since scale, μ, and is assumed to become negative at a high scale the quartic coupling of the Higgs field is strongly driven to due to the RG effect. As we have mentioned before, we a negative value. (For discussion about the stability of the neglect the quadratic term assuming that λ becomes negative EW vacuum in models with extra particles, see [22,31– at a much higher scale. In this setup, the scalar potential has 54].) We consider three models that contain, in addition to scale invariance at the classical level. In the application to the the SM particles, (i) vectorlike fermions having the same case of the SM, Φ corresponds to the Higgs doublet and λ SM charges as the left-handed quark and the right-handed corresponds to the Higgs quartic coupling constant. down quark, (ii) vectorlike fermions with the same SM Hereafter, we perform a detailed study of the effects of charges as left-handed lepton and right-handed electron, the fields coupled to Φ on the decay rate of the false and (iii) a right-handed neutrino. We give constraints on vacuum. We consider a Lagrangian that contains the their couplings and masses, requiring that the lifetime of the following interaction terms: EW vacuum be long enough. L ∋ κσ2jΦj2 þð Φψ¯ ψ þ Þþ ðΦÞ ð Þ This paper is organized as follows. In Sec. II,we int y L R H:c: V ; 2:3 summarize the formulation for the decay rate at the one- loop level, where we provide an analytic formula for each where σ is a real scalar field, and ψ L and ψ R are chiral field that couples to the Higgs boson.