The 1-loop effective potential for the Standard Model in curved spacetime
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ The 1-loop effective potential for the Standard Model in curved spacetime © Authors, 2018 Published version Markkanen, Tommi; Nurmi, Sami; Rajantie, Arttu; Stopyra, Stephen Markkanen, T., Nurmi, S., Rajantie, A., & Stopyra, S. (2018). The 1-loop effective potential for the Standard Model in curved spacetime. Journal of High Energy Physics, 2018(6), Article 40. https://doi.org/10.1007/JHEP06(2018)040 2018
JHEP06(2018)040 Published for SISSA by Springer Received:April 27, 2018 Accepted:June 3, 2018 Published:June 7, 2018 The 1-loop effective potential for the Standard Model in curved spacetime Tommi Markkanen,aSami Nurmi,bArttu Rajantieaand Stephen Stopyraa aDepartment of Physics, Imperial College London, London SW7 2AZ, U.K. bDepartment of Physics, University of Jyva¨skyl¨a, P.O. Box 35, FI-40014, Finland E-mail: [email protected],[email protected], [email protected],[email protected] Abstract: The renormalisation group improved Standard Model effective potential in an arbitrary curved spacetime is computed to one loop order in perturbation theory. The loop corrections are computed in the ultraviolet limit, which makes them independent of the choice of the vacuum state and allows the derivation of the complete set of β-functions. The potential depends on the spacetime curvature through the direct non-minimal Higgscurvature coupling, curvature contributions to the loop diagrams, and through the curvature dependence of the renormalisation scale. Together, these lead to significant curvature dependence, which needs to be taken into account in cosmological applications, which is demonstrated with the example of vacuum stability in de Sitter space. Keywords: Classical Theories of Gravity, Field Theories in Higher Dimensions, Field Theories in Lower Dimensions ArXiv ePrint: 1804.02020 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP06(2018)040
JHEP06(2018)040 Contents 1 Introduction 1 2 Effective potential for a self-interacting scalar field 3 2.1 1-loop in flat space 5 2.2 1-loop in curved spacetime 5 2.2.1 Via Heat Kernel techniques 7 2.3 RG improvement in the presence of curvature 7 3 Effective potential via the Heat Kernel 11 3.1 Scalar 14 3.2 Fermion 14 3.3 Gauge 15 4 The standard model 17 4.1 The β-functions 20 5 A case study: the SM in de Sitter space 22 5.1 Scale choice 24 5.2 Potential 25 5.3 Gauge dependence 27 5.4 Implications for vacuum stability 28 6 Conclusions 32 1 Introduction Ever since the seminal work [1] the quantum corrected or effective potential has been amongst the principal tools of quantum field theory. The effective potential in curved spacetime can have a number of important cosmological impacts. A key example is the analysis of vacuum stability in the early universe. The Standard Model (SM) of particle physics predicts a metastable electroweak vacuum [2–11]. Its survival over inflation and reheating is a non-trivial consistency requirement both for the SM and its extensions [12– 39]. The stability conditions crucially depend on curved spacetime contributions [12,19,40] in the effective potential which affect the behaviour of energetically subdominant spectator fields such as the SM Higgs. In SM extensions, radiatively generated curvature couplings can also produce primordial dark matter [41]. Smallness of curvature induced mass terms is also a key condition required in the curvaton scenario [42] where massless spectator scalars source the primordial perturbation. – 1 –
JHEP06(2018)040 From a more fundamental point of view, in a quantum field theory setting the inclusion of gravity in the form of background curvature leads to interesting an important modifications: the renormalization group (RG) running scale generically is influenced by the curvature leading to curvature induced running. The importance of this effect was first discovered in [19] and has since been shown to give rise to significant consequences in various set-ups [30,40,43,44]. Another crucial feature resulting from background gravity is the generation of new gravity-dependent operators, most famously of the non-minimal coupling between scalar fields and the scalar curvature of space, as already discussed in [45–47]. These profound features are not visible in an approximation that neglects the curvature of the background. Making generic statements about the behaviour of a spectator field in curved spacetime is unfortunately hindered by the calculational complexity of the problem: deriving the complete effective potential in an arbitrary curved spacetime is in general quite involved and obtaining explicit results requires one to specify the set-up, including making a choice for the background and the quantum state of interest. For examples of such calculations, see [48–55]. There are however some aspects that are universal. According to standard field theory principles, the ultraviolet (UV) behaviour of a theory must be state independent in order to have unique divergent parts in the counter terms that are required for rendering the theory finite. Furthermore, since techniques are available with which to extract the UV contribution to the effective potential in a general curved spacetime, deriving the complete set of operators generated by the quantum corrections as well as investigating the RG running of constants can be performed without choosing a specific form of the background metric or the quantum state [56]. In this work we calculate the UV contribution to the effective potential for the SM Higgs, in an arbitrary curved spacetime including all degrees of freedom contained in the SM to 1-loop order. We furthermore derive the complete set of β-functions with which we perform renormalization group improvement of the result. We will throughout work in the approximation where the SM Higgs is a subdominant spectator while neglecting the metric fluctuations, which has been shown to be a very good approximation [57]. Recently similar calculations, primarily in the context of the SM vacuum instability during inflation, have been performed in [19,27,58,59] and see [43,60–73] for related earlier studies. We however emphasize that the current work is the first one to present the complete result i.e. it includes all degrees of freedom of the SM along with all operators generated by quantum corrections in curved space. Our calculation is based on the well-known Heat Kernel technique [74– 80], which is essentially a gradient expansion, and we will in particular make use of the resummed form presented in [81,82]. We will also implement our result in the specific case of the de Sitter background and revisit the analysis of electroweak vacuum stability during inflation. Requiring that the electroweak vacuum survives inflation, we compute the lower bound for the non-minimal coupling as function of the SM Higgs and top quark masses. As a new result, we show that negative values of the non-minimal coupling are tightly constrained from below even if the inflationary scale is well below the instability scale µinst where λ(µinst) = 0. This sets a non-trivial lower bound on the non-minimal coupling even for low top mass values – 2 –
JHEP06(2018)040 for which µinst is larger than the maximal inflationary scale allowed by the non-detection of primordial gravitational waves. Our sign conventions for the metric and curvature tensors are (−,−,−) in the classification of [83]. 2 Effective potential for a self-interacting scalar field The derivation of the effective potential for a scalar field in an arbitrary curved spacetime for theories containing scalar, fermion and gauge fields will be addressed in section 3and implemented for the full SM in de Sitter space in section 4. But first for illustrative purposes we will show the necessary steps by using the self-interacting scalar field as a toy model. Although simple, this model will exhibit all the qualitative features that arise in more complicated theories when background curvature is not neglected in the derivation of the effective potential. We will also discuss renormalization group (RG) improvement in curved spacetime in this context. A point worth emphasizing is that we are only interested in behaviour at the very high ultraviolet (UV) limit. This stems from the fact only the UV is relevant when discussing the radiative generation of operators not present at tree-level and relatedly determining the RG running and the β-functions. For this reason we can make use a large momentum approximation throughout, which will simplify the derivation considerably. The action for some generic massive, non-minimally coupled and self-interacting scalar field χreads Sm=Zd4x√−g1 2∂µχ0∂µχ0−1 2m2 0χ2 0−ξ0 2Rχ2 0−λ0 4χ4 0,(2.1) where Ris the scalar curvature. The subscripts “0” indicate bare or unrenormalized parameters. In curved spacetime proper renormalization requires one also to introduce a purely gravitational part to the action as such operators are radiatively generated [56,84]. As we will show, the running of these at tree-level purely gravitational operators will turn out to be important for the effective potential. Specifically, the gravitational action reads Sg=−Zd4x√−gVΛ,0−κ0R+α1,0R2+α2,0RµνRµν +α3,0RµνδηRµνδη,(2.2) where1κ0= (16πG0)−1and VΛ,0= (8πG0)−1Λ0. Since we assume an unbounded space the terms χ2and Rare not present in the action as they may be removed by partial integration. For a scalar field in the 1-loop approximation the effective potential can be studied without the need of more sophisticated approaches, namely the Heat Kernel technology [74– 80]. For more complicated theories however, the Heat Kernel approach does prove to be 1These lead to the traditional parametrization of the Einstein equation with Rµν −1 2Rgµν +gµν Λ0=−8πG0Tµν ;2 √−g δSm δgµν =Tµν .(2.3) – 3 –
JHEP06(2018)040 extremely convenient as will become apparent in the following two sections, but for a model containing a single scalar field the derivation can be completed by simply making use of the equations of motion. The derivation we are about to present is somewhat simpler than the traditional one found in the seminal work [1] and standard textbooks [85,86], mainly because it does not rely on an infinite summation of one-particle-irreducible Feynman diagrams and hence the often non-trivial concept of symmetry factors never comes up. But more importantly for our purposes, the derivation can very easily be generalized to curved spaces. In order to derive the quantum corrected or effective equations of motion we shift the quantized field as ˆχ0→ hˆχ0i+ ˆχ0≡χ0+ ˆχ0,(2.4) to 1-loop order the equation of motion for the mean field χand the fluctuation ˆχcan be derived by first expanding the action to quadratic order Sm=−1 2Zd4x√−g−∂µχ0∂µχ0+m2 0χ2 0+ξ0Rχ2 0+ 2λ0 4χ4 0 −1 2Zdnx√−gˆχ0+ξ0R+M2(χ0)ˆχ0+··· ,(2.5) where we have defined the flat space effective mass M2(χ) = m2 0+ 3λ0χ2 0.(2.6) The above leads to two coupled equations, one for the mean field and one for the fluctuation h+m2 0+ξ0R+λ0χ2 0iχ0+ 3λ0χ0hˆχ2 0i= 0 ,(2.7) h+ξ0R+M2(χ0)iˆχ0= 0,(2.8) Note that to this order of truncation the diagrams containing an odd number of external legs drop out. The counter terms are obtained by defining the renormalized field with the wave function renormalization factor Z[85] χ0=√Zχ , (2.9) and similarly setting Z= 1 + δZ,Zm2 0=m2+δm2and Z2λ0=λ+δλ. For a constant mean field χthe renormalized quantum corrected equation of motion (2.7) reduces to finding the minimum of the effective potential, which to 1-loop order can be written as V0 eff(χ) = hξR +m2+λχ2iχ+ 3λχhˆχ2i−δV 0(χ) = 0 ,(2.10) where δV (χ) contains the counter terms for which from now on we use the unifying notation δci. The effective potential straightforwardly follows from integration Veff(χ)≡V(0)(χ) + V(1)(χ) + ··· =Zχ V0 eff(˜χ)d˜χ , (2.11) where the superscripts ”(0)” and ”(1)” denote the tree-level and 1-loop pieces, respectively. – 4 –
JHEP06(2018)040 For a scalar field, to 1-loop order finding a solution in the UV approximation for the quantum field in terms of modes is relatively simple even when the curvature of the background is included in the discussion. For completeness however we first present the derivation in flat space. 2.1 1-loop in flat space As usual, in flat space the solutions to (2.8) to 1-loop order can be expressed as a mode expansion ˆχ=Zd3k p(2π)3eik·xhˆakfk(t) + ˆa† −kf∗ k(t)i;fk(t) = e−iωt √2ω;ω2≡k2+M2(χ),(2.12) where [ˆak,ˆa† k0] = δ(3)(k−k0),[ˆak,ˆak0] = [ˆa† k,ˆa† k0] = 0 and kis the momentum with k≡ |k|. The effective mass M2(χ) is found from (2.6). It is then trivial to use the mode solution and write V0 eff(χ) = m2χ+λχ3+ 3λχ Zd3k 2(2π)3 1 pk2+M2(χ)−δV 0(χ),(2.13) which by performing the integral over χas in (2.11) and using the standard formulae for dimensional regularization [85] gives the 1-loop effective potential Veff(χ) = 1 2m2χ2+λ 4χ4+M4(χ) 64π2log M2(χ) µ2−3 2+−2 −log(4π)+ γe+O() −δVΛ+1 2δm2χ2+δλ 4χ4,(2.14) where the divergences are expressed in terms of n= 4 −and we have introduced the usual renormalization scale µ. In the MS subtraction scheme, which we will from now on use throughout, the divergent pole at n→4, the log(4π) and the Euler constant in the wavy brackets would be removed by a proper choice of the renormalization counter terms, δVΛ, δm2and δλ. Note that even in flat space a divergence ∝m4is generated and strictly speaking the cosmological constant counter term δVΛintroduced by the gravitational action (2.2) is required. 2.2 1-loop in curved spacetime In curved spacetime we can define a properly normalized ansatz for the modes by first restricting our background to a homogeneous and isotropic one described via the Friedmann- Lemaˆıtre-Robertson-Walker (FLRW) metric given in cosmic time as ds2=dt2−a2dx2,(2.15) then rescaling the field as in the previous section and finally writing ˆχ=Zd3k p(2πa)3eik·xhˆakfk(t) + ˆa† −kf∗ k(t)i;fk(t) = e−iRtWdt0 √2W,(2.16) – 5 –
JHEP06(2018)040 which after inserting into the equation of motion for the fluctuation (2.8) gives a relation for W W2=k2 a2+M2(χ) + ¨a a 3 2(4ξ−1) + ˙a a23 48ξ−1+3˙ W2 4W2−¨ W 2W.(2.17) Importantly, in practice the ansatz (2.16) provides useful solutions only as a high momentum expansion. This is also the reason why it and the results that follow resemble very much the flat space results of the previous subsection: when probing the very high UV the global structure of spacetime is not visible as locally any smoothly curved manifold is nearly flat. We will solve for Wfrom (2.17) iteratively as an expansion in terms of large k/a. The first few orders may be written as W=rk2 a2+M2(χ)+(ξ−1/6) R+O(k/a)−2,(2.18) which contain all terms leading to divergences in four dimensions and where Ris again the scalar curvature. It is now straightforward to calculate the 1-loop contribution to the variance, which can again be calculated with standard dimensional regularization hˆχ2i=µ 2Zdn−1k (2πa)n−1 1 p(k/a)2+M2(χ)+(ξ−1/6) R =M2(χ)+(ξ−1/6) R 16π2log M2(χ)+(ξ−1/6) R µ2−1−2 −log(4π) + γe. (2.19) Like in the previous section by using (2.10) and (2.11) and choosing the appropriate counter terms we can write the 1-loop correction to the renormalized curved spacetime effective potential in a form very similar to the flat space result in (2.14) V(1)(χ) = M2(χ)+(ξ−1/6) R2 64π2log |M2(χ)+(ξ−1/6) R| µ2−3 2+O(R2).(2.20) A few comments are now in order. The notation O(R2) indicates an inherent ambiguity in the derivation in terms of operators that are purely gravitational at tree-level: any contribution ∝R2log, RµνRµν log or RµνδηRµνδη log results in a finite contribution for V0 eff(χ) and will thus be invisible to a derivation including only the divergent terms in the effective equation of motion. We have also neglected any possible imaginary part of the effective potential by using an absolute value in the logarithm. It is well-known from flat space that integration over the infrared modes may give rise to a complex result for the effective potential which is usually taken to indicate a finite lifetime of the state [87], however this effect is not correctly represented in an approach that is based on an UV expansion. Furthermore, we have left in the same non-logarithmic finite pieces that are generated in the flat space MS prescription (cf. eg. (2.14)). As (2.20) clearly shows, the O(R2)-type terms couple to the scalar field and are thus relevant for the effective potential. Next we will briefly present their derivation for the self-interacting scalar field model. – 6 –
JHEP06(2018)040 2.2.1 Via Heat Kernel techniques The derivation of the previous section via an UV expansion for the 1-loop approximation is to illustrate the modifications that arise when background curvature is not neglected. For deriving the effective potential for a theory including also fermions and gauge fields it becomes apparent that more sophisticated (and formal) technology is needed, namely the Heat Kernel techniques to be discussed in section 3. This is also useful for obtaining all the O(R2) terms in (2.20). Functional determinants are widely used in quantum field theory in flat space and we refer the reader to [85] for more discussion for their use in traditional particle physics. In this regard, we can express the 1-loop quantum correction from (2.5) via a ‘tracelog’ Zd4x√−g V (1)(χ) = −i 2Tr log h+M2(χ) + ξRχ2i,(2.21) as is well-known. This approach can also be generalized to the case of a curved spacetime. The detailed derivation and formulae may be found in section 3, but here we will simply apply the results of section 3, specifically subsection (3.1) to (2.21) in order to write Veff(χ) = 1 2m2χ2+ξ 2Rχ2+λ 4χ4+VΛ−κR +α1R2+α2RµνRµν +α3RµνδηRµνδη +M2(χ)+(ξ−1/6) R2 64π2log |M2(χ)+(ξ−1/6) R| µ2−3 2 + 1 90 RµνδηRµνδη −RµνRµν 64π2log |M2(χ)+(ξ−1/6) R| µ2.(2.22) The operators that are generated via radiative corrections in curved spacetime can be seen from the 1-loop correction in (2.22), which is why they needed to be present already at treelevel in (2.1)–(2.2) and are a part of the complete Veff (χ). Furthermore, as all operators couple to the renormalization scale µthey cannot be made to vanish for all scales which is felt in the dynamics of the scalar field due to the χ-dependence of the logarithms in (2.22). 2.3 RG improvement in the presence of curvature Here we perform the RG analysis of the self-interacting scalar field model (2.1)–(2.2) and discuss RG improvement in curved space. Early work on RG improving the effective potential in flat space may be found in [88–91]. Studies in curved spacetime include [60– 62,65–72], see also the textbook [92]. The Callan-Symanzik equation is first and foremost an expression of renormalization scale invariance: in principle the renormalization scale µis an arbitrary choice and physical quantities should not depend on it. For the effective potential this translates as demanding dVeff(χ) dµ = 0 ,(2.23) where we emphasize due to the coupling between χand all the gravitational operators in (2.2) the above includes all operators in the original action as visible in (2.22). – 7 –
JHEP06(2018)040 Note that all group and spacetime indices are present implicitly. In flat space in the MS renormalization scheme the last three terms on the r.h.s. of (3.15) are subtracted by the counter terms after the full expression is expanded in the limit n→4 resulting in the replacement log ˜µ2→log µ2. We will also make use of this replacement although in curved space non-logarithmic finite pieces containing curvature dependence are generated as the result of the interplay between the pole ∝(4 −n)−1and n-dependent terms in R, Rµν and Rµνρσ. These terms we absorb in ξ,κ,VΛand the α’s in the tree level action (2.1) and (2.2).2 In what follows we use the formula (3.13) to compute the one-loop contributions to the effective potential from scalar, fermion and gauge fields. For the scalar case we can easily see that the first term in the expansion (3.13) corresponds with the elementary derivation (2.20) in section (2.2). Like in (2.20) we have only included the real part of the result as the infrared modes potentially giving a complex result are not included in the local (UV) expansion (3.9) with (3.11). Finally, we choose to drop the -type terms in (3.11) as they will not give rise to divergences or µ-dependence when µis a constant as in MS, which follows from the assumption of an unbounded Universe and partial integration. 3.1 Scalar Since scalar fields will always result in a tracelog given via an operator of the form (3.8), one may directly implement the expression in (3.13). Much like in (2.21) parametrizing the 1-loop contribution to the effective action via an effective mass parameter now denoted as mswith a non-minimal coupling gives Γ(1) s[ϕ] = i 2Tr log +m2 s+ξR.(3.16) The effective mass in (3.14) then becomes simply M2 s=m2 s+ξ−1 6R , (3.17) where “s” stands for scalar and similarly the relevant higher order curvature terms are a2,s =−1 180RµνRµν +1 180RµνρσRµνρσ ,(3.18) where we used that fact that for a scalar φone has ∇µ,∇ν]φ= 0 ⇒Wµν = 0 .(3.19) Here we further emphasize that all scalars have the same a2-contribution. 3.2 Fermion From the SM Lagrangian (4.11) one sees that a typical fermion contribution needs some work before (3.13) can be used, since it is not in the form (3.8), but Γ(1) f[ϕ] = −iTr log i∇µγµ−mf,(3.20) 2For the conformal anomaly this is of course not possible, however this contribution does not couple to the Higgs and hence in the subsequent discussion can be ignored. – 14 –
JHEP06(2018)040 where much like for scalars, the subscript “f” stands for fermion. Using very similar steps as in flat space i.e. the fact that log det = Tr log and that det [i∇µγµ−mf] = [det(i∇µγµ−mf) det(γ5γ5)(i∇µγµ−mf)]1/2 = [det(−i∇µγµ−mf)(i∇µγµ−mf)]1/2,(3.21) with the help of the relation (γµ∇µ)2=+R/4 leads to Γ(1) f[ϕ] = −i 2Tr log +m2 f+R/4,(3.22) and the fermionic effective mass (3.14) in curved spacetime M2 f=m2 f+R 12 .(3.23) Unlike for the scalar case the Wµν term from (3.11) gives a no-zero contribution due to the spin connection of the fermion. By using [82] ∇µ,∇ν]ψ=−1 4Rµναβγαγβψ=Wµνψ , (3.24) for Dirac fermions ψin curved spacetime and familiar identities from trace technology for the combination of four γ-matrices we can write tr{a2,f }= tr{ 1 Group}−1 45RµνRµν −7 360RµνρσRµνρσ.(3.25) In the above we have performed the trace over Dirac indices, but for completeness left in the trivial trace over any group indices, which for example for the SM quarks simply gives an overall factor of 3 from the three different colors and a factor of 1 for leptons. 3.3 Gauge The gauge contributions to the effective potential are the most non-trivial to calculate due to the explicit dependence on Rµν and ∇µ∇νas is visible from (4.11) for the SM.3 The main difficulty comes from gauge fixing. We use the so-called Rξor background (’t Hooft) gauges [85] to fix the gauge, which in order not to create confusion with the nonminimal coupling we parametrize with ζ. Generically, gauge fields give rise to contributions of the form Γ(1) g[ϕ] = i 2Tr log gµν +1 ζ−1∇µ∇ν+m2 ggµν +Rµν.(3.26) The above can be simplified by first splitting the vector into scalar and orthogonal components as Aµ=Aµ ⊥+∇µA;∇µAµ ⊥= 0 ,(3.27) with which and the help of standard commutator formula ∇µ,∇νAρ=−RραµνAα,(3.28) 3For a similar derivation we refer the reader to section 7.9 of [84]. – 15 –
JHEP06(2018)040 we can write gνµ +1 ζ−1∇ν∇µ+m2 ggνµ +RνµAµ =gνµ +m2 ggνµ +RνµAµ ⊥+ζ−1gνµ +ζm2 ggνµ +Rνµ∇µA . (3.29) From this it follows that the gauge tracelog splits into two separate pieces. The first one is almost of the form required by the hear kernel results, were it not the orthogonality constraint “⊥”. Its effect can be deduced by studying the unconstrained eigenvalue equation gµν +m2 ggµν +RµνAν=λAµ·∇µ ⇔+m2 g∇νAν=λ∇νAν.(3.30) These eigenvalues have to be removed from the constrained ones resulting from Aµ ⊥, giving us symbolically the relation eigenvaluesngµν +m2 ggµν +RµνAν ⊥o = eigenvaluesngµν +m2 ggµν +RµνAνo−eigenvaluesn+m2 g∇νAνo,(3.31) where the last piece is simply a scalar contribution. The second term in (3.29) we can evaluate by invoking consistency: at the limit ζ= 1 all terms resulting from the ∇µ∇νpiece in (3.26) must vanish, which completely fixes the remaining contribution allowing us to write for the gauge field Γ(1) g[ϕ] = i 2Tr log gµν +m2 ggµν +Rµν−i 2Tr log +m2 g+i 2Tr log +ζm2 g,(3.32) where only the first operator is a matrix in terms of spacetime indices and the other two are scalars. As one may see from (3.32) a gauge field will result in 3 separate pieces with the effective masses from (3.14) M2 gµν =m2 ggµν +Rµν −gµν 6R;M2 g,s =m2 g−R 6;M2 g,ζs =ζm2 g−R 6.(3.33) For gauge fields we can again use (3.28) ∇µ,∇νAα=−Rα µν βAβ= (Wµν)αβAβ⇒(Wµν)δ α(Wµν)β δ=Rδ µνα Rµν β δ,(3.34) for obtaining an expression for the a2-contribution for the first term on the right hand side of (3.32) (a2,g)β α=−gβ α 180RµνRµν +gβ α 180RµνρσRµνρσ +1 12Rδ µνα Rµν β δ,(3.35) with “g” for gauge and the remaining two scalar pieces trivially give rise to two contributions as in (3.18). – 16 –
JHEP06(2018)040 4 The standard model By making use of the results we derived in the previous section we present the steps for calculating the effective potential for the SM Higgs to 1-loop order in curved spacetime. We will show the explicit derivation in a set-up including only the Higgs doublet, the massive vector bosons W±and Z0and the top quark, as from this result a generalization which includes the complete particle content of the SM is straightforward. We start with the Lagrangian LSM =LYM +LF+LΦ+LGF +LGH +··· (4.1) with LYM =−1 4Fa µν2−1 4(Fµν)2+··· ;a= 1,2,3 (4.2) LΦ= (DµΦ)†(DµΦ) + m2Φ†Φ−ξRΦ†Φ−λ(Φ†Φ)2; (4.3) LF=¯ QLiγµDµQL+¯ tRiγµDµtR+−yt¯ QL(iσ2)Φ∗tR+ h.c.+··· ; (4.4) where Φ and QLare the Higgs and the left-handed top/bottom doublets, respectively Φ = 1 √2 −i(χ1−iχ2) ϕ+ (h+iχ3)!;QL= t b!L ,(4.5) and where ϕis the vacuum expectation value and finally the χi’s are the would be Goldstone bosons. Since our calculation is performed in curved space the covariant derivative, in addition to the gauge connection, contains a metric dependence via the covariant ∇µ Dµ=∇µ−igτaAa µ−ig0Y Aµ;τa=σa/2.(4.6) Also note that the γ-matrices satisfy the curved space generalization of the usual relation γµ, γν=gµν ,(4.7) which as shown in section 3.2 plays a role in the contributions from the fermions. The last two terms in (4.1), LGF and LGH, are the gauge fixing and ghost contributions. As discussed in section (3.3) we will use the background gauge where the gauge parameters are left unspecified. With these choices the gauge fixing Lagrangian can then be written as LGF =−1 2G2;Gj=1 pζj∇µAj µ−ζjEjaχa;j= 1,2,3,4; a= 1,2,3, (4.8) with the definitions Eia ≡ g0 0 0g0 0 0 g 0 0 −g0 ;A4 µ≡Aµ.(4.9) – 17 –
JHEP06(2018)040 Finally, since we are fixing gauge in a non-Abelian gauge theory we must also introduce a ghost term. The ghost Lagrangian can be written from LGH = ¯ciδGi δαjcj,(4.10) where Giis the gauge fixing function from (4.8), the c’s are the ghost fields and the αithe parameters of the gauge transformations. Now we may write the Lagrangian to quadratic order in fluctuations; it requires some algebra but is a straightforward exercise. At this stage the only complication arising from having a curved background is the fact that covariant derivatives for the gauge fields do not commute as given in (3.28). Taking this into account one gets4 LSM =m2 2ϕ2−λ 4ϕ4−1 2h+m2 h+ξRh+¯ t[i∇µγµ−mt]t +W+ µgµν +1 ζW−1∇µ∇ν+m2 Wgµν +RµνW− ν +1 2Z0 µgµν +1 ζZ−1∇µ∇ν+m2 Zgµν +RµνZ0 ν −1 2X a=1,2 χa+ζWm2 W+m2 χ+ξRχa−1 2χ3+ζZm2 Z+m2 χ+ξRχ3 −X a=1,2 ¯ca+ζWm2 Wca−¯c3+ζZm2 Zc3+··· ,(4.11) where we have chosen separate gauge fixing parameters for the W±and Z0contributions and defined the mass parameters m2 h=−m2+ 3λϕ2, m2 t=y2 t 2ϕ2, m2 W=g2 4ϕ2, m2 Z=g2+ (g0)2 4ϕ2, m2 χ=−m2+λϕ2.(4.12) For an intermediate result we use the steps shown in (3.1), (3.2) and (3.3) of section 3 to write (4.11) in terms of tracelogs that are calculable with the heat kernel technology. Explicitly this gives V(1) SM (ϕ) = −i 2Tr log +m2 h+ξR+i 2Tr log +m2 t+R/4 −iTr log gµν +m2 Wgµν +Rµν+iTr log +m2 W −i 2Tr log gµν +m2 Zgµν +Rµν+i 2Tr log +m2 Z −iTr log +ζWm2 W+m2 χ+ξR−i 2Tr log +ζZm2 Z+m2 χ+ξR +iTr log +ζWm2 W+i 2Tr log +ζZm2 Z+··· ,(4.13) 4with the usual mass eigenstates W± µ=1 √2A1 µ∓iA2 µ;Z0 µ=1 pg2+ (g0)2gA3 µ−g0Aµ, – 18 –
JHEP06(2018)040 where the definitions for the mass parameters can be found in (4.12). Note that the ζdependent mass terms given by (3.33) for the W± µand Z0 µfield s have been canceled by the ghost contribution. It proves convenient to split (4.13) into scalar, fermion and gauge contributions as V(1) SM (ϕ) = V(1) SM (ϕ)scalar +V(1) SM (ϕ)fermion +V(1) SM (ϕ)gauge .(4.14) Collecting all the scalar pieces and using (3.13) along with section 3.1 one gets V(1) SM (ϕ)scalar =X σ=scalars nσ 64π2M4 σlog |M2 σ| µ2−3 2+ 2a2,s log |M2 σ| µ2,(4.15) which specifically for the Lagrangian in (4.1) contains the expressions h;nh= +1 ,M2 h=m2 h+ξ−1 6R , W±;nW,s =−2,M2 W,s =m2 W−R 6, Z0;nZ,s =−1,M2 Z,s =m2 Z−R 6, χ1, χ2;nχ,W = +2 ,M2 χ,W =ζWm2 W+m2 χ−R 6, χ3;nχ,Z = +1 ,M2 χ,Z =ζZm2 Z+m2 χ−R 6, c1, c2;nc,W =−2,M2 c,W =ζWm2 W−R 6, c3;nc,Z =−1,M2 c,Z =ζZm2 Z−R 6,(4.16) with the a2,s given by (3.18). The contribution from the top quark is straightforward to express via (3.13) with the help of section 3.2 V(1) SM (ϕ)fermion,t =−1 64π24 tr{ 1 Group}M4 tlog |M2 t| µ2−3 2+ 2 tr{a2,f }log |M2 t| µ2, (4.17) where we have explicitly calculated the trace over Dirac indices, the effective mass can be found in (3.23), tr{a2,f }is given by (3.25) and due to color in the SM tr{ 1 Group}= 3 for the top quark (tr{ 1 Group}= 1 for leptons). Finally we can address the contributions coming from gauge fields that have nontrivial structure in terms of spacetime/Lorentz indices. Unsurprisingly, this is the most complicated piece, which we can evaluate with the help of section 3.3: V(1) SM (ϕ)gauge ≡V(1) SM (ϕ)gauge,W +V(1) SM (ϕ)gauge,Z +··· =2 64π2(M2 W)β α(M2 W)ν βlog |(M2 W)α ν| ˜µ2−gα ν 3 2+ 2(a2,g)β αlog |(M2 W)α β| ˜µ2 – 19 –
JHEP06(2018)040 +1 64π2(M2 Z)β α(M2 Z)ν βlog |(M2 Z)α ν| ˜µ2−gα ν 3 2+ 2(a2,g)β αlog |(M2 Z)α β| ˜µ2 +··· ,(4.18) where (M2 Z)µν and (M2 W)µν can be found from (3.33) and (a2,g)β αfrom (3.35). The reason we have left in the divergent renormalization scales (3.15) in (4.18) is that since the trace depends on the dimensions of spacetime, we must not choose n= 4 before explicitly calculating it. Due to the presence of the logarithm with Lorentz indices in general the above is a fairly non-trivial expression. However, when one limits to the case with only diagonal elements in Rµν such as FLRW the sums can be explicitly performed. For example for the piece coming from Z0 µin the FLRW case we can then write V(1) SM (ϕ)gauge,Z =1 64π2(M2 Z)0 0(M2 Z)0 0log |(M2 Z)0 0| µ2−3 2+ 2(a2,g)0 0log |(M2 Z)0 0| µ2 +3 64π2(M2 Z)i i(M2 Z)i ilog |(M2 Z)i i| µ2−5 6+2(a2,g)i ilog |(M2 Z)i i| µ2, (4.19) where very importantly there is no sum over the repeated spatial indices denoted with “i”. As we did for scalars and fermions, also for the gauge fields our renormalization prescription is such that the result coincides with the standard parametrization in the flat space limit (see e.g. [93]).5The remaining W± νpiece may be obtained in a similar fashion. With (4.15), (4.17) and (4.18) we have shown the calculation for the full 1-loop result including all contributions contained in the Lagrangian given in (4.1). As is apparent from (4.15), (4.17) and (4.18) the generalization to include the complete SM is straightforward as is adding degrees of freedom beyond the SM. For an explicit example, see section 5. 4.1 The β-functions By following the procedures we introduced for the simple scalar field model in section 2.3 we can now derive all the β-functions of the SM in curved spacetime. This includes the well-known β-functions that can be calculated in flat space and can be found from standard references, see for example [6,7,89,93], and the ones connected to the dynamics of a curved spacetime. The βs that are relevant in curved spacetime are defined by the operators in the purely gravitational part of the action (2.2) and the non-minimal term proportional to ξ. With the help of formulae from section 3we can use the 1-loop approximation to the Callan-Symanzik equation (2.25) and the SM anomalous dimension [95] γ=1 16π2Y2−9g2 4−3(g0)2 4−ζW g2 2−ζZ 1 4g2+ (g0)2,(4.20) to write the gravitational β-functions 16π2βξ=ξ−1 612λ+ 2Y2−3(g0)2 2−9g2 2(4.21) 5The factor of 5/6 in the second line of (4.19) is the result of the interplay between the pole ∝(4 −n)−1 and the (n−1)-contribution from the trace. – 20 –
JHEP06(2018)040 16π2βVΛ= 2m4(4.22) 16π2βκ= 4m2ξ−1 6,(4.23) 16π2βα1= 2ξ2−2ξ 3−277 144 ,(4.24) 16π2βα2=571 90 ,(4.25) 16π2βα3=−293 720 ,(4.26) with Y2≡3(y2 u+y2 c+y2 t) + 3(y2 d+y2 s+y2 b)+(y2 e+y2 µ+y2 τ), Y4≡3(y4 u+y4 c+y4 t) + 3(y4 d+y4 s+y4 b)+(y4 e+y4 µ+y4 τ).(4.27) We emphasize that (4.21)–(4.26) include the contributions from the entire SM and are exhaustive in terms of the generated operators.6 The other one-loop SM β-functions are given by, for example [7,89]. Since there are curvature dependent loop corrections for all the fermions in the theory, it is not necessarily correct to ignore the light fermions, so we include the running of all the Yukawa couplings. These can be found in [96,97] for example, 16π2βyt=yt3 2(y2 t−y2 b) + Y2−17 12(g0)2+9 4g2+ 8g2 3,(4.28) 16π2βyb=yb3 2(y2 b−y2 t) + Y2−5 12(g0)2+9 4g2+ 8g2 3,(4.29) 16π2βyl=yl3 2y2 l+Y2−45 12(g0)2+9 4g2,(4.30) 16π2βλ= 24λ2−3λ(g0)2+ 3g2+3 41 2(g0)4+ (g0)2g2+3 2g4+ 4Y2λ−2Y4,(4.31) 16π2βm2=m212λ−3 2(g0)2−9 2g2+ 2Y2,(4.32) 16π2βg0=41 6(g0)3,16π2βg=−19 6g3,16π2βg3=−7g4 3,(4.33) where βylis the lepton beta function for l=e, µ, τ. For reference, the beta functions are defined as βX≡µ(dX/dµ), g0is the U(1) coupling, gis the SU(2) coupling, and g3 the SU(3) coupling. Beta functions for the other generations of fermions can be obtained from eqs. (4.28) and (4.29) by substituting yt→yu, ycand yb→yd, ys, leaving the gauge couplings and Y2the same. In priciple also the gauge parameters ζZand ζWwould run according to their respective β-functions. Their running is however not relevant for a one loop calculation: the ζ’s only enter in the loop correcting and do not couple to other one loop β-functions making the running a two loop effect and with no loss of generality one may treat them as constant, which will be our choice. 6The particle content of the SM may be found from tables 1and 2. – 21 –
JHEP06(2018)040 Ψi nidin0 iM2 i 1 2 3/2−34/15 m2 W+H2 W±2 6 5/6−34/5m2 W+H2 3−2 3/2 4/15 m2 W−2H2 4 1 3/2−17/15 m2 Z+H2 Z05 3 5/6−17/5m2 Z+H2 6−1 3/2 2/15 m2 Z−2H2 q 7 −12 −12 3/2 38/5m2 q+H2 l13 −15 −4 3/2 38/15 m2 l+H2 h16 1 3/2−2/15 m2 h+ 12(ξ−1/6)H2 χW17 2 3/2−4/15 m2 χ+ζWm2 W+ 12(ξ−1/6)H2 χZ18 1 3/2−2/15 m2 χ+ζZm2 Z+ 12(ξ−1/6)H2 cW19 −2 3/2 4/15 ζWm2 W−2H2 cZ20 −1 3/2 2/15 ζZm2 Z−2H2 Table 1. Contributions to the effective potential (5.2) with tree-level couplings to the Higgs. Ψ stands for W±,Z0, the 6 quarks q, the 3 charged leptons l, the Higgs h, the Goldstone bosons χW and χZand the ghosts cWand cZ. The masses are defined as in (4.12). For the boundary conditions of the running couplings at the EW scale t= 0, we use the precise matching relationships between pole masses and MS parameters found in [7], supplemented with one-loop results for the remaining fermions in the theory [98]. Unless otherwise stated, we used top quark and Higgs boson pole masses of Mt= 173.34 GeV and Mh= 125.15 GeV respectively, with other pole masses found in [99]. 5 A case study: the SM in de Sitter space The general results derived above using the heat kernel method hold for an arbitrary curved spacetime. As a specific example, we apply them in the de Sitter space where R= 12H2, RµνRµν = 36H4, RµνδηRµνδη = 24H4,(5.1) and the Hubble rate His constant. Substituting these into the expressions derived in sections 3 and 4, we find that the 1-loop contribution to the effective potential of the SM Higgs in the MS scheme, including the complete set of quarks and leptons as well as the photon and the gluons, is given by V(1) SM (ϕ) = 1 64π2 31 X i=1 niM4 ilog |M2 i| µ2−di+n0 iH4log |M2 i| µ2,(5.2) where the inputs can be read from tables 1and 2. They are split as the degrees of freedom that directly couple to the Higgs in table 1and degrees of freedom that do not in table 2. Recall that our computation gives the UV limit of the effective potential. A generic expression for M2 iis thus a function of the form M2 i(ϕcl, µ) = κi(µ)Z(Mt) Z(µ)ϕ2 cl− κ0 i(µ) + θi(µ)H2, where the coupling-dependent κi, κ0 i, θican be read from tables 1and 2, – 22 –
JHEP06(2018)040 Ψi nidin0 iM2 i 21 1 3/2−17/15 H2 γ22 3 5/6−17/5H2 23 −1 3/2 2/15 −2H2 24 8 3/2−136/15 H2 g25 24 5/6−136/5H2 26 −8 3/2 16/15 −2H2 ν27 −29 −2 3/2 19/15 H2 cγ30 −1 3/2 2/15 −2H2 cg31 −8 3/2 16/15 −2H2 Table 2. Contributions to the effective potential (5.2) that do not to couple to the Higgs at tree-level. Ψ stands for the photon γ, the 8 gluons g, the 3 neutrinos νand the ghosts cγand cg. using field-dependent masses for all the Standard Model particles. The one-loop effective potential with running couplings is then given by Veff SM(ϕ(µ)) = −1 2m2(µ)ϕ2(µ)+ ξ(µ) 2Rϕ2(µ)+ λ(µ) 4ϕ4(µ)+ VΛ(µ)−12κ(µ)H2+α(µ)H4 +1 64π2 31 X i=1 niM4 i(µ)log |M2 i(µ)| µ2−di+n0 iH4log |M2 i(µ)| µ2.(5.3) The different gravitational terms α1R2,α2RµνRµν,α3RµνδηRµνδη in (2.2) have combined to a single term α(µ)H4due to the de Sitter relations (5.1). The βfunction of the coupling αis determined by using eqs. (4.24)–(4.26) which yield βα=1 16π2288ξ2−96ξ−1751 30 .(5.4) All other βfunctions are directly given in section 4.1. The renormalization group improved potential is found by numerically solving for the full set of βfunctions and substituting the results into (2.2). We use a modification of a method recently employed by two of the authors in [29,100]. Briefly, this method consists of computing the running of the couplings at a set of discrete points, and using these to construct a C1continuous interpolating piecewise polynomial to describe the running in logarithmic space. This results in function gi(µ) for the couplings, and together with a scale choice µ(ϕ), gives a numerical expression for the potential. The final expression is then: VRGI SM (ϕcl) = 1 2−m2(µ∗(ϕcl)) + ξ(µ∗(ϕcl))RZ(Mt) Z(µ∗(ϕcl))ϕ2 cl +λ(µ∗(ϕcl)) 4 Z2(Mt) Z2(µ∗(ϕcl))ϕ4 cl+VΛ(µ∗(ϕcl))−12κ(µ∗(ϕcl))H2+α(µ∗(ϕcl))H4 +1 64π2 31 X i=1 niM4 i(ϕcl)log |M2 i(ϕcl)| µ2 ∗(ϕcl)−di+n0 iH4log |M2 i(ϕcl)| µ2 ∗(ϕcl), (5.5) where M2 i(ϕcl) = Mi(ϕcl, µ∗(ϕcl)) is the relevant mass-term defined at scale µ∗(ϕcl), ϕcl = ϕ(Mt) is the field evaluated at the electroweak scale, and α= 144α1+ 36α2+ 24α3. – 23 –
JHEP06(2018)040 instanton in the limit where Ris fixed - the so called ‘fixed background approximation’. This is given by BHM =8π2∆Vξ(ϕHM) 3H4,(5.15) where His the Hubble rate, R= 12H2, and ∆Vξ(ϕ) = V0(ϕ) + ξ 2ϕ2R−V0(ϕfv)−ξ 2ϕ2 fv.(5.16) This means that BHM is proportional to the difference in Height between the top of the barrier and the false vacuum, with the potential evaluated in the Jordan frame and including the 1 2ξϕ2Rterm as if it were part of the potential. Stability during inflation requires that the probability of decay is sufficiently low that the expected number of separate causal regions in which a bubble was nucleated in our past light-cone, is fewer than 1. The survival of a single causal region that decayed to the false vacuum could potentially destabilize the universe if it were to then continue expanding after inflation ended.8If Ne-folds of inflation are visible, there are approximately e3N such causal regions, and so the number that decayed during inflation is ndecayed =e3Np(N, 1),(5.17) where p(N, 1) is the probability of a single causal region decaying after Ne-folds of inflation. Note that a bubble forming during inflation always expands to fill the causal region (1 Hubble volume) that it fills [26], but can expand no further because the expansion of spacetime out-paces the bubble wall, that can only move at causal velocities. The probability per unit time that a bubble forms within a Hubble volume during inflation is given by γ=VHubblecH4e−BHM =4πc 3He−BHM .(5.18) Thus, the probability of decaying between e-folds N1and N2is p(N2, N1) = 4πc 3Zt2 t1 dtHe−BHM =kZN2 N1 dN Ne−BHM .(5.19) Where dN N=Hdt, and Nis the number of e-folds. kis an unknown O(1) factor. For constant Hubble rate, which we will assume here for simplicity, this means ndecayed =klog(N) exp 3N−8π2∆Vξ(ϕHM) 3H4.(5.20) The condition ndecayed <1 then translates to H < A∆Vξ(ϕHM)1 4,(5.21) 8Though it is unclear precisely what would happen to such a bubble that started expanding during inflation, only for inflation to end, as opposed to a bubble forming in flat space - here we will assume that it expands and envelops the whole spacetime. – 30 –
JHEP06(2018)040 Figure 8. Stability analysis for Mt= 173.34GeV, Mh= 125.15GeV. The red region has on average more than one bubble nucleation event within the observable universe during inflation, while the green region has less than one such event. µinst is defined as the renormalization scale (in flat space) at which λ(µinst) = 0. where A=8π2 3(log k+ log log N+ 3N) 1 4 = 0.617 ±0.004,(5.22) for N= 60 e-folds and the uncertainty given by assuming 10−2< k < 102, illustrating the weak dependence on k. We apply this condition to the potential computed in this paper, using the solution of eq. (5.6) with smallest logs at the top of the barrier, for a range of ξ and H- the results are plotted in figure 8. We have also checked that this stability analysis is not affected by using a different scale choice, such as µ2=aϕ2 cl +bR: this produced virtually identical results to figure 8. The condition (5.21) is of the same form but slightly stronger than the bound used in [19], 8π2V(φ) 3H4>1 which corresponds to A= (8π2/3)1/4≃2.26. As discussed in refs. [25, 26,30,100], the bound could be improved further by accounting for possibility for the field to flow back across the barrier due to evolution during inflation and by including the possible impact of CdL solutions. The effect of changing the top mass at constant His shown in figure 9. Together these plots illustrate that even for Hubble rates somewhat below the instability scale (defined by λ(µinst) = 0), negative ξcan quickly destabilize the potential. Note also that because of the running of ξ,ξEW = 0 is qualitatively equivalent to having ξ < 0, since the nonminimal coupling runs to negative values at higher energies, and if the optimal scale choice is µ2∼Rfor small ϕ, this means ξ < 0 for the whole range of the potential for any non-zero Hubble rate. – 31 –
JHEP06(2018)040 171 171.5 172 172.5 173 173.5 174 174.5 175 175.5 176 Mt/GeV -104 -103 -102 -101 -100 0 100 ξEW H = 1.0579µinst H = 0.10579µinst H = 0.010579µinst ξEW = 0 ξEW = 1/6 Figure 9. Plot of the boundary between stability and instability for different Hubble rates, and top masses. Note that µinst is defined by λ(µinst) = 0 using mt= 173.34 GeV for comparison. Note that since the results in this paper use only 1-loop running, low values of mtcan be unstable for values that would lead to an absolutely stable vacuum at 3-loops. 6 Conclusions In this work we derived the renormalization group improved effective potential in curved spacetime for the SM Higgs including the complete SM particle content to one loop order in perturbation theory. Our calculation included the UV limit of the loop corrections and thus contains the universal contribution that must be shared by all quantum states possessing the coinciding UV divergent behavior. We also presented the complete set of β-functions for the SM to one loop order, including all operators that are generated in curved spacetime. As an application we investigated the behavior of the SM Higgs in de Sitter space in the context of electroweak vacuum instability. Our use of the UV expansion means that the effective potential does not include infrared contributions, which can be large in the presence of light scalar fields. Locally the infrared contributions can always be absorbed into rescaling of background quantities and therefore do not affect our ultraviolet results. The global effects of these infrared modes can be studied by using the stochastic inflation approach [111] with the effective ultraviolet potential computed here as the input. Broadly speaking our results highlight two important, and often overlooked, aspects that arise whenever quantum fields are investigated in situations for which the curvature of spacetime is non-negligible: the first is that the renormalization group running sees the energy scale set by the curvature of the background, a mechanism which we called curvature induced running. For cosmologically interesting cases where the field is a light spectator with respect to the Hubble rate, the curvature can give the dominant contribution to the renormalization group running. The second aspect is the generation of operators – 32 –
JHEP06(2018)040 invisible in flat space. In addition to the well-known non-minimal coupling there are 5 other operators (see (2.2)) generated via loops in curved spacetime. For the SM in curved spacetime the β-functions imply the generation of all such operators resulting in important modifications, as is apparent from the results of section 4.1. There is no compelling reason to assume similar contributions not to arise for theories beyond the SM, which can be studied by straightforward generalizations of our results. The application to de Sitter space shows clearly the impact of making use of an effective potential calculated in curved spacetime. The standard procedure of optimizing the convergence of the loop expansion by an appropriate renormalization scale choice is made more complicated by the additional scale introduced by curvature. As we showed, even in the simple case of de Sitter space finding a physically motivated scale choice is non-trivial. Using the effective potential, we demonstrated for the SM that negative values of the non-minimal coupling are tightly constrained from below by the requirement of vacuum stability during inflation. Importantly, this is true even for inflationary scales well below the scale of instability and hence for low top mass values for which the instability occurs above the maximal inflationary scale allowed by the non-detection of a primordial tensor spectrum. Acknowledgments The authors thank Jos´e Espinosa and Hardi Veerm¨ae for useful comments on the manuscript. TM and AR are supported by the STFC grant ST/P000762/1, and SS by the Imperial College President’s PhD Scholarship. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. References [1] S.R. Coleman and E.J. Weinberg, Radiative corrections as the origin of spontaneous symmetry breaking,Phys. Rev. D 7 (1973) 1888 [INSPIRE]. [2] S. Chigusa et al., Decay rate of electroweak vacuum in the standard model and beyond, arXiv:1803.03902 [INSPIRE]. [3] S. Chigusa et al., State-of-the-art calculation of the decay rate of electroweak vacuum in the standard model,Phys. Rev. Lett. 119 (2017) 211801 [arXiv:1707.09301] [INSPIRE]. [4] A. Andreassen, W. Frost and M.D. Schwartz, Scale invariant instantons and the complete lifetime of the standard model,Phys. Rev. D 97 (2018) 056006 [arXiv:1707.08124] [INSPIRE]. [5] A.V. Bednyakov, B.A. Kniehl, A.F. Pikelner and O.L. Veretin, Stability of the electroweak vacuum: gauge independence and advanced precision,Phys. Rev. Lett. 115 (2015) 201802 [arXiv:1507.08833] [INSPIRE]. [6] G. Degrassi et al., Higgs mass and vacuum stability in the standard model at NNLO,JHEP 08 (2012) 098 [arXiv:1205.6497] [INSPIRE]. – 33 –
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