The 1-loop effective potential for the Standard Model in curved spacetime
Tommi Markkanen, Sami Nurmi, Arttu Rajantie, Stephen Stopyra
IIMPERIAL/TP/2018/TM/02
Prepared for submission to JHEP
The 1-loop effective potential for the Standard Modelin curved spacetime
Tommi Markkanen a Sami Nurmi b Arttu Rajantie c Stephen Stopyra d a,c,d Department of Physics, Imperial College London, SW7 2AZ, UK b Department of Physics, University of Jyva¨skyl¨a, P.O. Box 35, FI-40014 University of Jyva¨skyl¨a,Finland
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
The renormalisation group improved Standard Model effective potential inan arbitrary curved spacetime is computed to one loop order in perturbation theory. Theloop corrections are computed in the ultraviolet limit, which makes them independent ofthe 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 Higgs-curvature coupling, curvature contributions to the loop diagrams, and through the curva-ture dependence of the renormalisation scale. Together, these lead to significant curvaturedependence, which needs to be taken into account in cosmological applications, which isdemonstrated with the example of vacuum stability in de Sitter space. a r X i v : . [ h e p - ph ] J un ontents β -functions 20 Ever since the seminal work [1] the quantum corrected or effective potential has beenamongst the principal tools of quantum field theory. The effective potential in curvedspacetime can have a number of important cosmological impacts. A key example is theanalysis of vacuum stability in the early universe. The Standard Model (SM) of particlephysics predicts a metastable electroweak vacuum [2–11]. Its survival over inflation andreheating 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 spectatorfields such as the SM Higgs. In SM extensions, radiatively generated curvature couplingscan also produce primordial dark matter [41]. Smallness of curvature induced mass terms isalso a key condition required in the curvaton scenario [42] where massless spectator scalarssource the primordial perturbation. – 1 –rom a more fundamental point of view, in a quantum field theory setting the in-clusion of gravity in the form of background curvature leads to interesting an importantmodifications: the renormalization group (RG) running scale generically is influenced bythe curvature leading to curvature induced running . The importance of this effect wasfirst discovered in [19] and has since been shown to give rise to significant consequences invarious set-ups [30, 40, 43, 44]. Another crucial feature resulting from background gravityis the generation of new gravity-dependent operators, most famously of the non-minimalcoupling 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 thecurvature of the background.Making generic statements about the behaviour of a spectator field in curved spacetimeis unfortunately hindered by the calculational complexity of the problem: deriving thecomplete effective potential in an arbitrary curved spacetime is in general quite involvedand obtaining explicit results requires one to specify the set-up, including making a choicefor 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 fieldtheory principles, the ultraviolet (UV) behaviour of a theory must be state independent inorder to have unique divergent parts in the counter terms that are required for renderingthe theory finite. Furthermore, since techniques are available with which to extract the UVcontribution to the effective potential in a general curved spacetime, deriving the completeset of operators generated by the quantum corrections as well as investigating the RGrunning of constants can be performed without choosing a specific form of the backgroundmetric or the quantum state [56].In this work we calculate the UV contribution to the effective potential for the SMHiggs, in an arbitrary curved spacetime including all degrees of freedom contained in theSM to 1-loop order. We furthermore derive the complete set of β -functions with whichwe perform renormalization group improvement of the result. We will throughout work inthe approximation where the SM Higgs is a subdominant spectator while neglecting themetric fluctuations, which has been shown to be a very good approximation [57]. Recentlysimilar 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. Wehowever 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 quantumcorrections in curved space. Our calculation is based on the well-known Heat Kerneltechnique [74–80], which is essentially a gradient expansion, and we will in particular makeuse of the resummed form presented in [81, 82].We will also implement our result in the specific case of the de Sitter background andrevisit the analysis of electroweak vacuum stability during inflation. Requiring that theelectroweak vacuum survives inflation, we compute the lower bound for the non-minimalcoupling as function of the SM Higgs and top quark masses. As a new result, we showthat negative values of the non-minimal coupling are tightly constrained from below evenif the inflationary scale is well below the instability scale µ inst where λ ( µ inst ) = 0. Thissets a non-trivial lower bound on the non-minimal coupling even for low top mass values– 2 –or which µ inst is larger than the maximal inflationary scale allowed by the non-detectionof primordial gravitational waves.Our sign conventions for the metric and curvature tensors are ( − , − , − ) in the classi-fication of [83]. The derivation of the effective potential for a scalar field in an arbitrary curved spacetimefor theories containing scalar, fermion and gauge fields will be addressed in section 3 andimplemented for the full SM in de Sitter space in section 4. But first for illustrativepurposes we will show the necessary steps by using the self-interacting scalar field as a toymodel. Although simple, this model will exhibit all the qualitative features that arise inmore complicated theories when background curvature is not neglected in the derivationof the effective potential. We will also discuss renormalization group (RG) improvement incurved spacetime in this context. A point worth emphasizing is that we are only interestedin behaviour at the very high ultraviolet (UV) limit. This stems from the fact only the UVis relevant when discussing the radiative generation of operators not present at tree-leveland relatedly determining the RG running and the β -functions. For this reason we canmake use a large momentum approximation throughout, which will simplify the derivationconsiderably.The action for some generic massive, non-minimally coupled and self-interacting scalarfield χ reads S m = (cid:90) d x √− g (cid:20) ∂ µ χ ∂ µ χ − m χ − ξ Rχ − λ χ (cid:21) , (2.1)where R is the scalar curvature. The subscripts ”0” indicate bare or unrenormalizedparameters. In curved spacetime proper renormalization requires one also to introduce apurely 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 turnout to be important for the effective potential. Specifically, the gravitational action reads S g = − (cid:90) d x √− g (cid:20) V Λ , − κ R + α , R + α , R µν R µν + α , R µνδη R µνδη (cid:21) , (2.2)where κ = (16 πG ) − and V Λ , = (8 πG ) − Λ . Since we assume an unbounded spacethe terms (cid:3) χ and (cid:3) R are not present in the action as they may be removed by partialintegration.For a scalar field in the 1-loop approximation the effective potential can be studiedwithout 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 These lead to the traditional parametrization of the Einstein equation with R µν − Rg µν + g µν Λ = − πG T µν ; 2 √− g δS m δg µν = T µν . (2.3) – 3 –xtremely convenient as will become apparent in the following two sections, but for a modelcontaining a single scalar field the derivation can be completed by simply making use ofthe equations of motion.The derivation we are about to present is somewhat simpler than the traditional onefound in the seminal work [1] and standard textbooks [85, 86], mainly because it does notrely on an infinite summation of one-particle-irreducible Feynman diagrams and hence theoften non-trivial concept of symmetry factors never comes up. But more importantly forour 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 thequantized field as ˆ χ → (cid:104) ˆ χ (cid:105) + ˆ χ ≡ χ + ˆ χ , (2.4)to 1-loop order the equation of motion for the mean field χ and the fluctuation ˆ χ can bederived by first expanding the action to quadratic order S m = − (cid:90) d x √− g (cid:20) − ∂ µ χ ∂ µ χ + m χ + ξ Rχ + 2 λ χ (cid:21) − (cid:90) d n x √− g ˆ χ (cid:20) (cid:3) + ξ R + M ( χ ) (cid:21) ˆ χ + · · · , (2.5)where we have defined the flat space effective mass M ( χ ) = m + 3 λ χ . (2.6)The above leads to two coupled equations, one for the mean field and one for the fluctuation (cid:20) (cid:3) + m + ξ R + λ χ (cid:21) χ + 3 λ χ (cid:104) ˆ χ (cid:105) = 0 , (2.7) (cid:20) (cid:3) + ξ R + M ( χ ) (cid:21) ˆ χ = 0 , (2.8)Note that to this order of truncation the diagrams containing an odd number of externallegs drop out.The counter terms are obtained by defining the renormalized field with the wave func-tion renormalization factor Z [85] χ = √ Zχ , (2.9)and similarly setting Z = 1 + δZ , Zm = m + δm and Z λ = λ + δλ .For a constant mean field χ the renormalized quantum corrected equation of motion(2.7) reduces to finding the minimum of the effective potential, which to 1-loop order canbe written as V (cid:48) eff ( χ ) = (cid:20) ξR + m + λχ (cid:21) χ + 3 λχ (cid:104) ˆ χ (cid:105) − δV (cid:48) ( χ ) = 0 , (2.10)where δV ( χ ) contains the counter terms for which from now on we use the unifying notation δc i . The effective potential straightforwardly follows from integration V eff ( χ ) ≡ V (0) ( χ ) + V (1) ( χ ) + · · · = (cid:90) χ V (cid:48) eff ( ˜ χ ) d ˜ χ , (2.11)– 4 –here the superscripts ”(0)” and ”(1)” denote the tree-level and 1-loop pieces, respectively.For a scalar field, to 1-loop order finding a solution in the UV approximation forthe quantum field in terms of modes is relatively simple even when the curvature of thebackground is included in the discussion. For completeness however we first present thederivation in flat space. As usual, in flat space the solutions to (2.8) to 1-loop order can be expressed as a modeexpansionˆ χ = (cid:90) d k (cid:112) (2 π ) e i k · x (cid:104) ˆ a k f k ( t ) + ˆ a †− k f ∗ k ( t ) (cid:105) ; f k ( t ) = e − iωt √ ω ; ω ≡ k + M ( χ ) , (2.12)where [ˆ a k , ˆ a † k (cid:48) ] = δ (3) ( k − k (cid:48) ) , [ˆ a k , ˆ a k (cid:48) ] = [ˆ a † k , ˆ a † k (cid:48) ] = 0 and k is the momentum with k ≡ | k | .The effective mass M ( χ ) is found from (2.6). It is then trivial to use the mode solutionand write V (cid:48) eff ( χ ) = m χ + λχ + 3 λχ (cid:90) d k π ) (cid:112) k + M ( χ ) − δV (cid:48) ( χ ) , (2.13)which by performing the integral over χ as in (2.11) and using the standard formulae fordimensional regularization [85] gives the 1-loop effective potential V eff ( χ ) = 12 m χ + λ χ + M ( χ )64 π (cid:20) log (cid:18) M ( χ ) µ (cid:19) −
32 + (cid:110) − (cid:15) − log(4 π ) + γ e (cid:111) + O ( (cid:15) ) (cid:21) − (cid:20) δV Λ + 12 δm χ + δλ χ (cid:21) , (2.14)where the divergences are expressed in terms of n = 4 − (cid:15) and we have introduced theusual renormalization scale µ . In the MS subtraction scheme, which we will from now onuse throughout, the divergent pole at n →
4, the log(4 π ) and the Euler constant in thewavy brackets would be removed by a proper choice of the renormalization counter terms, δV Λ , δm and δλ . Note that even in flat space a divergence ∝ m is generated and strictlyspeaking the cosmological constant counter term δV Λ introduced by the gravitational action(2.2) is required. In curved spacetime we can define a properly normalized ansatz for the modes by first re-stricting our background to a homogeneous and isotropic one described via the Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) metric given in cosmic time as ds = dt − a d x , (2.15)then rescaling the field as in the previous section and finally writingˆ χ = (cid:90) d k (cid:112) (2 πa ) e i k · x (cid:104) ˆ a k f k ( t ) + ˆ a †− k f ∗ k ( t ) (cid:105) ; f k ( t ) = e − i (cid:82) t W dt (cid:48) √ W , (2.16)– 5 –hich after inserting into the equation of motion for the fluctuation (2.8) gives a relationfor
W W = k a + M ( χ ) + ¨ aa
32 (4 ξ −
1) + (cid:18) ˙ aa (cid:19) (cid:0) ξ − (cid:1) + 3 ˙ W W − ¨ W W . (2.17)Importantly, in practice the ansatz (2.16) provides useful solutions only as a high momen-tum expansion. This is also the reason why it and the results that follow resemble verymuch the flat space results of the previous subsection: when probing the very high UVthe global structure of spacetime is not visible as locally any smoothly curved manifold isnearly flat.We will solve for W from (2.17) iteratively as an expansion in terms of large k/a . Thefirst few orders may be written as W = (cid:114) k a + M ( χ ) + ( ξ − / R + O ( k/a ) − , (2.18)which contain all terms leading to divergences in four dimensions and where R is againthe scalar curvature. It is now straightforward to calculate the 1-loop contribution to thevariance, which can again be calculated with standard dimensional regularization (cid:104) ˆ χ (cid:105) = µ (cid:15) (cid:90) d n − k (2 πa ) n − (cid:112) ( k/a ) + M ( χ ) + ( ξ − / R = M ( χ ) + ( ξ − / R π (cid:20) log (cid:18) M ( χ ) + ( ξ − / Rµ (cid:19) − − (cid:15) − log(4 π ) + γ e (cid:21) . (2.19)Like in the previous section by using (2.10) and (2.11) and choosing the appropriate counterterms we can write the 1-loop correction to the renormalized curved spacetime effectivepotential in a form very similar to the flat space result in (2.14) V (1) ( χ ) = (cid:0) M ( χ ) + ( ξ − / R (cid:1) π (cid:20) log (cid:18) | M ( χ ) + ( ξ − / R | µ (cid:19) − (cid:21) + O ( R ) . (2.20)A few comments are now in order. The notation O ( R ) indicates an inherent ambiguityin the derivation in terms of operators that are purely gravitational at tree-level: anycontribution ∝ R log, R µν R µν log or R µνδη R µνδη log results in a finite contribution for V (cid:48) eff ( χ ) and will thus be invisible to a derivation including only the divergent terms inthe effective equation of motion. We have also neglected any possible imaginary part ofthe effective potential by using an absolute value in the logarithm. It is well-known fromflat space that integration over the infrared modes may give rise to a complex result forthe 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 UVexpansion. Furthermore, we have left in the same non-logarithmic finite pieces that aregenerated in the flat space MS prescription (cf. eg. (2.14)).As (2.20) clearly shows, the O ( R )-type terms couple to the scalar field and are thusrelevant for the effective potential. Next we will briefly present their derivation for theself-interacting scalar field model. – 6 – .2.1 via Heat Kernel techniques The derivation of the previous section via an UV expansion for the 1-loop approximationis 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 itbecomes apparent that more sophisticated (and formal) technology is needed, namely theHeat Kernel techniques to be discussed in section 3. This is also useful for obtaining allthe O ( R ) terms in (2.20).Functional determinants are widely used in quantum field theory in flat space and werefer the reader to [85] for more discussion for their use in traditional particle physics. Inthis regard, we can express the 1-loop quantum correction from (2.5) via a ’tracelog’ (cid:90) d x √− g V (1) ( χ ) = − i (cid:104) (cid:3) + M ( χ ) + ξRχ (cid:105) , (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 simplyapply the results of section 3, specifically subsection (3.1) to (2.21) in order to write V eff ( χ ) = 12 m χ + ξ Rχ + λ χ + V Λ − κR + α R + α R µν R µν + α R µνδη R µνδη + (cid:0) M ( χ ) + ( ξ − / R (cid:1) π (cid:20) log (cid:18) | M ( χ ) + ( ξ − / R | µ (cid:19) − (cid:21) + (cid:0) R µνδη R µνδη − R µν R µν (cid:1) π (cid:20) log (cid:18) | M ( χ ) + ( ξ − / R | µ (cid:19) (cid:21) . (2.22)The operators that are generated via radiative corrections in curved spacetime can be seenfrom the 1-loop correction in (2.22), which is why they needed to be present already attree-level in (2.1–2.2) and are a part of the complete V eff ( χ ). Furthermore, as all operatorscouple to the renormalization scale µ they cannot be made to vanish for all scales which isfelt in the dynamics of the scalar field due to the χ -dependence of the logarithms in (2.22). Here we perform the RG analysis of the self-interacting scalar field model (2.1–2.2) anddiscuss RG improvement in curved space. Early work on RG improving the effectivepotential in flat space may be found in [88–91]. Studies in curved spacetime include [60–62, 65–73], see also the textbook [92].The Callan-Symanzik equation is first and foremost an expression of renormalizationscale invariance: in principle the renormalization scale µ is an arbitrary choice and physicalquantities should not depend on it. For the effective potential this translates as demanding dV eff ( χ ) dµ = 0 , (2.23)where we emphasize due to the coupling between χ and all the gravitational operators in(2.2) the above includes all operators in the original action as visible in (2.22).– 7 –he requirement in (2.23) leads to the well-known Callan-Symanzik equations for theeffective potential in terms of the β -functions and the anomalous dimension γ [89] (cid:26) µ ∂∂µ + β c i ∂∂c i − γχ ∂∂χ (cid:27) V eff ( χ ) = 0 , β c i ≡ µ ∂c i ∂µ , γ ≡ µ ∂ log √ Z∂µ , (2.24)where the c i stands for all the parameters of the action with summation over the repeatedindex i assumed. Z is the wave function renormalization introduced in (2.9) defining therenormalized field. It should be kept in mind that the wave function renormalizationhas a dependence on the renormalization scale µ and so does then the renormalized field, χ ( µ ) = Z ( µ ) − / χ .To 1-loop order, the above can be written as (cid:26) β c i ∂∂c i − γχ ∂∂χ (cid:27) V (0) ( χ ) = − µ ∂∂µ V (1) ( χ ) . (2.25)In general one needs the anomalous dimension γ as an input before the equation maybe solved, which requires its determination by means other than the effective potential,for example from a direct evaluation of the 2-point function. This poses no additionalcomplications in curved spacetime over the usual flat space case, since the γ in a curvedspacetime derivation must be identical to the flat space result and can be taken fromstandard literature. This is because the gravitational couplings such a ξ and the α ’s ineq. (2.2) must only couple to operators that vanish in the flat space limit. If this were notthe case and γ did contain e.g. a contribution from α it would indicate that the size of agravitational operator could also affect the running of all parameters in flat space, whichis not tenable from a purely physical point of view.Given γ , one may solve for the β -functions and the running constants and finally usethem to improve the limit of applicability of the effective potential, with the end resultin principle should be independent of µ . As one can show however, this is not the casefor any perturbative result, but rather there is always some residual µ -dependence left,which is an artefact of our inability to solve the effective potential exactly. In [89] it wasfirst proposed that in order to minimize the error from the neglected higher order termsthe scale µ can be chosen such that the logarithms in the loop correction remain small.Choosing a particular form for µ is allowed since the complete result must be independentof µ as demanded by (2.24).As an example we first discuss RG improvement in the 1-loop approximation in flatspace for the simple scalar field model in (2.1). The β -functions to 1-loop order are easy toderive by using (2.14) and (2.24), and noting that to 1-loop order the anomalous dimensionvanishes, γ = O ( λ ), for scalar field with only a quartic interaction term, which results in β λ = 9 λ π ; β m = 3 m λ π . (2.26)It is straightforward to solve the above λ ( µ ) = λ ( µ )1 − λ ( µ )8 π log( µ/µ ) ; m ( µ ) = m ( µ ) (cid:2) − λ ( µ )8 π log( µ/µ ) (cid:3) / , (2.27)– 8 –here the scale µ fixes the physical input values of the parameters which in principle areprovided by the appropriate measurements. Using the above we may easily write down theeffective potential with running constants V eff ( χ ( µ )) = 12 m ( µ ) χ ( µ ) + λ ( µ )4 χ ( µ )+ (cid:0) m ( µ ) + 3 λ ( µ ) χ ( µ ) (cid:1) π (cid:20) log (cid:18) m ( µ ) + 3 λ ( µ ) χ ( µ ) µ (cid:19) − (cid:21) , (2.28)where for clarity we have have denoted all µ -dependence explicitly and neglected the run-ning vacuum energy V Λ ( µ ).The scale µ is usually chosen to match the energy scale of the process one is consideringbecause then the logarithms appearing in the loop corrections are generally small, andtherefore the loop expansion can be expected to converge faster.However, in the case of the effective potential, the characteristic energy scale generallydepends on the field value. Therefore it becomes natural to make the scale µ depend onthe field value χ . More precisely, we define a suitable function µ ∗ ( χ ), chosen in such away that the loop corrections are small. This leads to the renormalisation group improved (RGI) effective potential, V RGI ( χ ) = 12 m ( µ ∗ ) Z ( µ ) Z ( µ ∗ ) χ + λ ( µ ∗ )4 Z ( µ ) Z ( µ ∗ ) χ + 164 π (cid:18) m ( µ ∗ ) + 3 λ ( µ ∗ ) Z ( µ ) Z ( µ ∗ ) χ (cid:19) (cid:20) log (cid:18) m ( µ ∗ ) + 3 λ ( µ ∗ ) Z ( µ ) Z ( µ ∗ ) χ µ ∗ (cid:19) − (cid:21) , (2.29)where µ ∗ = µ ∗ ( χ ), by χ we denote the field defined with fixed renormalisation scale µ ,i.e., χ = χ ( µ ), and the field renormalisation factors are Z / ( µ ) Z / ( µ ) = exp (cid:18) − (cid:90) log (cid:16) µµ (cid:17) γ ( t )d t (cid:19) . (2.30)Here we point out that we reserve the word ’improved’ for the result where µ -independenceis exploited in order to optimize the convergence of the perturbative expansion i.e. for (2.29)but not for (2.28) in contrast to some other works.In simple cases, when all particle masses are proportional to the field χ , it is commonto choose µ = χ [93]. However, in situations which involve other energy scales (for examplethe spacetime curvature in our case) this does not necessarily work, and one needs a generalpresciption for determining µ for each field value. A natural choice is to set µ in such away that the one loop correction vanishes, as was recently advocated in [94]. In the caseof Eq. (2.28), this means choosing µ = µ ∗ , where µ ∗ is given by solving the equation µ ∗ = e − / (cid:20) m ( µ ∗ ) + 3 λ ( µ ∗ ) Z ( µ ) Z ( µ ∗ ) χ (cid:21) . (2.31)Even in this simple theory this equation cannot be solved analytically, but numerically itis straightforward. With the scale choice (2.31), the RGI potential can be written simply– 9 –s V RGI ( χ ) = 12 m ( µ ∗ ) Z ( µ ) Z ( µ ∗ ) χ + λ ( µ ∗ )4 Z ( µ ) Z ( µ ∗ ) χ . (2.32)We can now repeat the process of RG improvement for the case of an arbitrary curvedspacetime. Note that at no point in our derivation have we used an expansion assumingthe background curvature to be a small correction on top of the flat space result. This isimportant as the case of significant background curvature is precisely the relevant one formany applications. The only assumption is that the quantized matter field is energeticallysub-dominant, which means that it can be treated as a spectator field on a curved classicalbackground [57].In addition to (2.26) in curved spacetime β -functions related to the gravitational terms(2.2) emerge, which are easy to derive by using (2.24) for the curved spacetime effectivepotential (2.22), which results in β ξ = 6 λ ( ξ − / π ; β V Λ = m / π ; β κ = − m ( ξ − / π ; β α = ( ξ − / / π ; β α = − / π ; β α = 1 / π . (2.33)The above could easily be solved and used to improve the perturbative result (2.22) verysimilarly to the flat space case in (2.28).From a calculational point of view RG improvement in curved spacetime proceeds asa very natural extension of the usual steps made in flat space. There are however veryimportant qualitative differences that arise in the presence of background curvature. Forexample, in the simple scalar field case in flat space we optimized the expansion in theimproved potential (2.28) by making the logarithms small with the choice (2.31). Dueto the R -dependence in the logarithm in (2.22) the analogous choice in curved spacetimewould in be µ ∗ = e − / (cid:12)(cid:12)(cid:12)(cid:12) m ( µ ∗ ) + 3 λ ( µ ∗ ) Z ( µ ) Z ( µ ∗ ) χ + (cid:18) ξ ( µ ∗ ) − (cid:19) R (cid:12)(cid:12)(cid:12)(cid:12) . (2.34)This is a generic feature that arises in curved spacetimes: whenever we optimize the ex-pansion in the improved potential we unavoidably introduce a curvature dependence inthe running scale i.e. it leads to curvature induced running of the parameters. In thecase of high background curvature, R (cid:29) m ( µ ) + 3 λ ( µ ) χ , the dominant contribution tothe running of parameters comes from the scalar curvature, an effect which obviously iscompletely missed when using a flat space approximation for the potential. In the contextof a potential electroweak vacuum instability in the early Universe this effect was firstdiscovered in [19].Another important feature with qualitatively significant consequences is the generationof gravitational operators that couple to the scalar field, which we have already mentionedon several occasion. In addition to the well-known non-minimal coupling ξ leading to atree-level coupling between curvature and the scalar field, the O ( R )-type operators visiblein (2.22) introduce potentially significant modifications in the effective potential. As seenfrom (2.33) all these operators are generically radiatively induced.– 10 –utting all this together, we can write the final expression for the RGI effective po-tential, including the gravitational terms, as: V RGI ( χ ) = 12 (cid:0) m ( µ ∗ ( χ )) + ξ ( µ ∗ ( χ )) R (cid:1) Z ( µ ) Z ( µ ∗ ( χ )) χ + Z ( µ ) Z ( µ ∗ ( χ )) λ ( µ ∗ ( χ ))4 χ + V Λ ( µ ∗ ( χ )) − κ ( µ ∗ ( χ )) R + α ( µ ∗ ( χ )) R + α ( µ ∗ ( χ )) R µν R µν + α ( µ ∗ ( χ )) R µνδη R µνδη + M ( χ )64 π (cid:20) log (cid:18) |M ( χ ) | µ ∗ ( χ ) (cid:19) − (cid:21) + (cid:0) R µνδη R µνδη − R µν R µν (cid:1) π (cid:20) log (cid:18) |M ( χ ) | µ ∗ ( χ ) (cid:19)(cid:21) , (2.35)where: M ( χ ) = m ( µ ∗ ( χ )) + 3 λ ( µ ∗ ( χ )) Z ( µ ) Z ( µ ∗ ( χ )) χ + (cid:18) ξ ( µ ∗ ( χ )) − (cid:19) R, (2.36) χ is the field renormalized at µ , and µ ∗ ( χ ) is the chosen scale for the renormalizationgroup improvement, as a function of χ .Now we can proceed to discuss the derivation of the curved spacetime effective potentialfor more involved theories containing fermions and gauge fields in addition to scalars.Despite a substantial increase in calculational work qualitatively the main modificationsone encounters are the same as in the simple scalar field case just discussed. In this section we present the steps for the derivation of an effective potential containingscalar, fermion and gauge fields with the proper gauge-fixing terms at the ultraviolet limit.Although the main application we have in mind is the specific case of the SM on a ho-mogeneous and isotropic spacetime relevant for cosmological applications, the results arepresented in a general form applicable for arbitrary field content and an arbitrary back-ground metric. We also note that many of the slightly formal results of this section becomeclearer when they are implemented in practice, which we will do in section 4 for the SM inde Sitter space.Following the standard discussion found in [85] first we briefly review the use of func-tional determinants in the calculation the 1-loop effective potential in flat space. Thegenerating functional defined via a path integral in flat space reads Z [ J ] = (cid:90) D ϕ e iS [ ϕ ]+ i (cid:82) d x Jϕ , (3.1)where ϕ is some generic scalar field with possibly non-trivial group structure and J is theusual source term.By performing a Legendre transformation asΓ[ ϕ ] = − i log Z [ J ] − (cid:90) d x J ( x ) ϕ ( x ) , (3.2)– 11 –ne obtains the effective action Γ[ ϕ ] and, for a constant field, ϕ the effective potential asΓ[ ϕ ] ≡ Γ (0) [ ϕ ] + Γ (1) [ ϕ ] + · · · ≡ (cid:90) d x L eff ϕ = const. = − V eff ( ϕ ) (cid:90) d x . (3.3)The standard approach in the 1-loop approximation, which contains only terms quadraticin fluctuations, comes by using the path integral generalization of the formulae (cid:90) e − x i A ij x j dx ∝ (cid:112) det A ij , (cid:90) e − x i A ij x ∗ j dxdx ∗ ∝ A ij , (cid:90) e − ¯ c i A ij c j d ¯ cdc ∝ det A ij ; { c i , c j } = 0 , (3.4)to write the 1-loop contribution to the effective potential as a sum of ’tracelogs’Γ (1) [ ϕ ] = i (cid:88) k n k log det D k = i (cid:88) k n k Tr log D k . (3.5)with the pre-factors determined by the field content and the group structure of the theoryin question.The above summarizes the standard approach for the derivation of effective action to 1-loop order, which can also be generalized for the case of a curved spacetime. Symbolicallynothing changes for the tracelogs, but in curved spacetime finding explicit expressionsfor them becomes non-trivial due replacement of ∂ µ with the ∇ µ -operator containing thespacetime connection and in the integral over spacetime indices the inclusion of the measure √− g .Following the steps and the conventions of [84] we can express any effective actiongiven in terms of a tracelog as an integral over a fictitious time parameter τ often calledproper time,Γ (1) [ ϕ ] = i D = − i (cid:20)(cid:90) ∞ dττ e − iτD (cid:21) = − i (cid:90) d x √− g (cid:90) ∞ dττ tr [ K ( τ ; x, x )] , (3.6)where D is yet an unknown matrix, ”Tr” denotes a trace over all indices, ”tr” a trace withthe spacetime coordinates excluded and K ( τ ; x, y ) ≡ e − iτD , (3.7)is the heat kernel . Note that no assumptions of the underlying theory have been madewhen writing (3.6) and (3.7), except perhaps that the integral is well-defined.The efficiency of heat kernel formalism lies in the existence of approximation schemesallowing one to perform the proper time integral in (3.6) and hence obtain a solution forthe effective action: if the matrix D can be expressed as D = (cid:3) + X , (3.8)where X is an arbitrary matrix in spin and group degrees of freedom, a small τ expansion ofthe heat kernel becomes analytically tractable. It will turn out that to 1-loop order scalars,– 12 –ermions and gauge fields all can be manipulated to have the form (3.8) even in the presenceof background curvature. Note that the (cid:3) -operator contains the metric connection (andpossibly the spin connection) via the ∇ µ -operator, as is described in section 3.8 of [56] forfields with any spin.The small τ expansion of the heat kernel corresponds to the UV or local limit andhas been studied and implemented in the physical context by a number of people over theyears. Notable early work on heat kernel techniques may be found in [74–79] and see [80]for a review.In [81, 82] a form of the heat kernel that sums all scalar curvature contributions in thesmall proper time approximation was provided. This is obtained via the ansatz K ( τ ; x, x ) = i (4 πiτ ) n/ exp (cid:20) − iτ (cid:16) X − R (cid:17)(cid:21) Ω( τ ) , (3.9)where we have analytically continued the dimensions to n and whereΩ( τ ) = ∞ (cid:88) k =0 a k ( iτ ) k ; a = 1 , a = 0 , (3.10)with a = − R µν R µν + 1180 R µνρσ R µνρσ − (cid:3) R + 16 (cid:3) X + 112 W αβ W αβ . (3.11)The W αβ is defined as W αβ Ψ = (cid:2) ∇ α , ∇ β (cid:3) Ψ , (3.12)and depends on spacetime structure of the particular field contribution Ψ to the effectivepotential i.e. it will be different for scalars, fermions and gauge fields as will be illustratedin sections 3.1 – 3.3. It is precisely the form (3.9) that turns out to correspond to theelementary UV derivation of section 2.We can now write down the effective action (3.6) by using eqs. (3.9) – (3.11) andperforming the integral over proper time in n spacetime dimensions,Γ (1) [ ϕ ] = 12(4 π ) n/ (cid:90) d n x √− g tr (cid:34)(cid:18) M µ (cid:19) n − ∞ (cid:88) k =0 (cid:0) M (cid:1) − k a k Γ (cid:16) k − n (cid:17)(cid:35) = − (cid:90) d n x √− g π tr (cid:20) M (cid:18) log |M | ˜ µ − (cid:19) + 2 a log |M | ˜ µ (cid:21) + . . . (3.13)where we have only included the leading logarithmic terms, denoted the curved spacetimeeffective mass as M ≡ X − R , (3.14)and absorbed the divergences in the renormalization scale aslog ˜ µ = log µ + 24 − n − γ E + log(4 π ) . (3.15)– 13 –ote that all group and spacetime indices are present implicitly. In flat space in the MSrenormalization scheme the last three terms on the RHS of (3.15) are subtracted by thecounter terms after the full expression is expanded in the limit n → µ → log µ . We will also make use of this replacement although in curvedspace non-logarithmic finite pieces containing curvature dependence are generated as theresult of the interplay between the pole ∝ (4 − n ) − and n -dependent terms in R, R µν and R µνρσ . These terms we absorb in ξ , κ , V Λ and the α ’s in the tree level action (2.1) and(2.2) .In what follows we use the formula (3.13) to compute the one-loop contributions to theeffective potential from scalar, fermion and gauge fields. For the scalar case we can easilysee 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 asthe 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 (cid:3) -type terms in (3.11) as theywill not give rise to divergences or µ -dependence when µ is a constant as in MS, whichfollows from the assumption of an unbounded Universe and partial integration. 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) parametrizingthe 1-loop contribution to the effective action via an effective mass parameter now denotedas m s with a non-minimal coupling givesΓ (1) s [ ϕ ] = i (cid:2) (cid:3) + m s + ξR (cid:3) . (3.16)The effective mass in (3.14) then becomes simply M s = m s + (cid:18) ξ − (cid:19) R , (3.17)where ”s” stands for scalar and similarly the relevant higher order curvature terms are a ,s = − R µν R µν + 1180 R µνρσ R µνρσ , (3.18)where we used that fact that for a scalar φ one has (cid:2) ∇ µ , ∇ ν ] φ = 0 ⇒ W µν = 0 . (3.19)Here we further emphasize that all scalars have the same a -contribution. From the SM Lagrangian (4.11) one sees that a typical fermion contribution needs somework before (3.13) can be used, since it is not in the form (3.8), butΓ (1) f [ ϕ ] = − i Tr log (cid:2) i ∇ µ γ µ − m f (cid:3) , (3.20) For the conformal anomaly this is of course not possible, however this contribution does not couple tothe Higgs and hence in the subsequent discussion can be ignored. – 14 –here much like for scalars, the subscript ” f ” stands for fermion. Using very similar stepsas in flat space i.e. the fact that log det = Tr log and thatdet [ i ∇ µ γ µ − m f ] = [det( i ∇ µ γ µ − m f ) det( γ γ )( i ∇ µ γ µ − m f )] / = [det( − i ∇ µ γ µ − m f )( i ∇ µ γ µ − m f )] / , (3.21)with the help of the relation ( γ µ ∇ µ ) = (cid:3) + R/ (1) f [ ϕ ] = − i (cid:2) (cid:3) + m f + R/ (cid:3) , (3.22)and the fermionic effective mass (3.14) in curved spacetime M f = m f + R . (3.23)Unlike for the scalar case the W µν term from (3.11) gives a no-zero contribution due to thespin connection of the fermion. By using [82] (cid:2) ∇ µ , ∇ ν ] ψ = − R µναβ γ α γ β ψ = W µν ψ , (3.24)for Dirac fermions ψ in curved spacetime and familiar identities from trace technology forthe combination of four γ -matrices we can writetr { a ,f } = tr { Group } (cid:20) − R µν R µν − R µνρσ R µνρσ (cid:21) . (3.25)In the above we have performed the trace over Dirac indices, but for completeness left inthe trivial trace over any group indices, which for example for the SM quarks simply givesan overall factor of 3 from the three different colors and a factor of 1 for leptons. The gauge contributions to the effective potential are the most non-trivial to calculate dueto the explicit dependence on R µν and ∇ µ ∇ ν as is visible from (4.11) for the SM .The main difficulty comes from gauge fixing. We use the the so-called R ξ or background(’t Hooft) gauges [85] to fix the gauge, which in order not to create confusion with the non-minimal coupling we parametrize with ζ . Generically, gauge fields give rise to contributionsof the form Γ (1) g [ ϕ ] = i (cid:20) (cid:3) g µν + (cid:18) ζ − (cid:19) ∇ µ ∇ ν + m g g µν + R µν (cid:21) . (3.26)The above can be simplified by first splitting the vector into scalar and orthogonal compo-nents as A µ = A µ ⊥ + ∇ µ A ; ∇ µ A µ ⊥ = 0 , (3.27)with which and the help of standard commutator formula (cid:2) ∇ µ , ∇ ν (cid:3) A ρ = − R ραµν A α , (3.28) For a similar derivation we refer the reader to section 7.9 of [84]. – 15 –e can write (cid:20) (cid:3) g νµ + (cid:18) ζ − (cid:19) ∇ ν ∇ µ + m g g νµ + R νµ (cid:21) A µ = (cid:2) (cid:3) g νµ + m g g νµ + R νµ (cid:3) A µ ⊥ + ζ − (cid:2) (cid:3) g νµ + ζm g g νµ + R νµ (cid:3) ∇ µ A . (3.29)From this it follows that the gauge tracelog splits into two separate pieces. The first one isalmost of the form required by the hear kernel results, were it not the orthogonality con-straint ” ⊥ ”. Its effect can be deduced by studying the unconstrained eigenvalue equation (cid:2) (cid:3) g µν + m g g µν + R µν (cid:3) A ν = λA µ (cid:12)(cid:12) · ∇ µ ⇔ (cid:2) (cid:3) + m g (cid:3) ∇ ν A ν = λ ∇ ν A ν . (3.30)These eigenvalues have to be removed from the constrained ones resulting from A µ ⊥ , givingus symbolically the relationeigenvalues (cid:110)(cid:2) (cid:3) g µν + m g g µν + R µν (cid:3) A ν ⊥ (cid:111) = eigenvalues (cid:110)(cid:2) (cid:3) g µν + m g g µν + R µν (cid:3) A ν (cid:111) − eigenvalues (cid:110) (cid:2) (cid:3) + m g (cid:3) ∇ ν A ν (cid:111) , (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 = 1all terms resulting from the ∇ µ ∇ ν piece in (3.26) must vanish, which completely fixes theremaining contribution allowing us to write for the gauge fieldΓ (1) g [ ϕ ] = i (cid:2) (cid:3) g µν + m g g µν + R µν (cid:3) − i (cid:2) (cid:3) + m g (cid:3) + i (cid:2) (cid:3) + ζm g (cid:3) , (3.32)where only the first operator is a matrix in terms of spacetime indices and the other twoare scalars.As one may see from (3.32) a gauge field will result in 3 separate pieces with theeffective masses from (3.14) (cid:0) M g (cid:1) µν = m g g µν + R µν − g µν R ; M g,s = m g − R M g,ζs = ζm g − R . (3.33)For gauge fields we can again use (3.28) (cid:2) ∇ µ , ∇ ν (cid:3) A α = − R αµν β A β = ( W µν ) αβ A β ⇒ ( W µν ) δα ( W µν ) βδ = R δµνα R µν βδ , (3.34)for obtaining an expression for the a -contribution for the first term on the right hand sideof (3.32) ( a ,g ) βα = − g βα R µν R µν + g βα R µνρσ R µνρσ + 112 R δµνα R µν βδ , (3.35)with ” g ” for gauge and the remaining two scalar pieces trivially give rise to two contribu-tions as in (3.18). – 16 – The Standard Model
By making use of the results we derived in the previous section we present the steps forcalculating the effective potential for the SM Higgs to 1-loop order in curved spacetime. Wewill show the explicit derivation in a set-up including only the Higgs doublet, the massivevector bosons W ± and Z and the top quark, as from this result a generalization whichincludes the complete particle content of the SM is straightforward.We start with the Lagrangian L SM = L YM + L F + L Φ + L GF + L GH + · · · (4.1)with L YM = − (cid:0) F aµν (cid:1) −
14 ( F µν ) + · · · ; a = 1 , , L Φ = ( D µ Φ) † ( D µ Φ) + m Φ † Φ − ξR Φ † Φ − λ (Φ † Φ) ; (4.3) L F = ¯ Q L iγ µ D µ Q L + ¯ t R iγ µ D µ t R + (cid:0) − y t ¯ Q L ( iσ )Φ ∗ t R + h . c . (cid:1) + · · · ; (4.4)where Φ and Q L are the Higgs and the left-handed top/bottom doublets, respectivelyΦ = 1 √ (cid:32) − i ( χ − iχ ) ϕ + ( h + iχ ) (cid:33) ; Q L = (cid:32) tb (cid:33) L , (4.5)and where ϕ is the vacuum expectation value and finally the χ i ’s are the would be Goldstonebosons. Since our calculation is performed in curved space the covariant derivative, inaddition to the gauge connection, contains a metric dependence via the covariant ∇ µ D µ = ∇ µ − igτ a A aµ − ig (cid:48) Y A µ ; τ a = σ a / . (4.6)Also note that the γ -matrices satisfy the curved space generalization of the usual relation (cid:8) γ µ , γ ν (cid:9) = g µν , (4.7)which as shown in section 3.2 plays a role in the contributions from the fermions. Thelast two terms in (4.1), L GF and L GH , are the gauge fixing and ghost contributions. Asdiscussed in section (3.3) we will use the background gauge where the gauge parametersare left unspecified. With these choices the gauge fixing Lagrangian can then be writtenas L GF = − G ; G j = 1 (cid:112) ζ j (cid:0) ∇ µ A jµ − ζ j E ja χ a (cid:1) ; j = 1 , , , a = 1 , , , (4.8)with the definitions E ia ≡ g g
00 0 g − g (cid:48) ; A µ ≡ A µ . (4.9)– 17 –inally, since we are fixing gauge in a non-Abelian gauge theory we must also introduce aghost term. The ghost Lagrangian can be written from L GH = ¯ c i (cid:18) δG i δα j (cid:19) c j , (4.10)where G i is the gauge fixing function from (4.8), the c ’s are the ghost fields and the α i theparameters of the gauge transformations.Now we may write the Lagrangian to quadratic order in fluctuations; it requires somealgebra but is a straightforward exercise. At this stage the only complication arising fromhaving a curved background is the fact that covariant derivatives for the gauge fields donot commute as given in (3.28). Taking this into account one gets L SM = m ϕ − λ ϕ − h (cid:2) (cid:3) + m h + ξR (cid:3) h + ¯ t [ i ∇ µ γ µ − m t ] t + W + µ (cid:20) (cid:3) g µν + (cid:18) ζ W − (cid:19) ∇ µ ∇ ν + m W g µν + R µν (cid:21) W − ν + 12 Z µ (cid:20) (cid:3) g µν + (cid:18) ζ Z − (cid:19) ∇ µ ∇ ν + m Z g µν + R µν (cid:21) Z ν − (cid:88) a =1 , χ a (cid:2) (cid:3) + ζ W m W + m χ + ξR (cid:3) χ a − χ (cid:2) (cid:3) + ζ Z m Z + m χ + ξR (cid:3) χ − (cid:88) a =1 , ¯ c a (cid:2) (cid:3) + ζ W m W (cid:3) c a − ¯ c (cid:2) (cid:3) + ζ Z m Z (cid:3) c + · · · , (4.11)where we have chosen separate gauge fixing parameters for the W ± and Z contributionsand defined the mass parameters m h = − m + 3 λϕ , m t = y t ϕ , m W = g ϕ ,m Z = g + ( g (cid:48) ) ϕ , m χ = − m + λϕ . (4.12)For an intermediate result we use the steps shown in (3.1), (3.2) and (3.3) of section3 to write (4.11) in terms of tracelogs that are calculable with the heat kernel technology.Explicitly this gives V (1)SM ( ϕ ) = − i (cid:2) (cid:3) + m h + ξR (cid:3) + i (cid:2) (cid:3) + m t + R/ (cid:3) − i Tr log (cid:2) (cid:3) g µν + m W g µν + R µν (cid:3) + i Tr log (cid:2) (cid:3) + m W (cid:3) − i (cid:2) (cid:3) g µν + m Z g µν + R µν (cid:3) + i (cid:2) (cid:3) + m Z (cid:3) − i Tr log (cid:2) (cid:3) + ζ W m W + m χ + ξR (cid:3) − i (cid:2) (cid:3) + ζ Z m Z + m χ + ξR (cid:3) + i Tr log (cid:2) (cid:3) + ζ W m W (cid:3) + i (cid:2) (cid:3) + ζ Z m Z (cid:3) + · · · , (4.13) with the usual mass eigenstates W ± µ = 1 √ (cid:0) A µ ∓ iA µ (cid:1) ; Z µ = 1 (cid:112) g + ( g (cid:48) ) (cid:0) gA µ − g (cid:48) A µ (cid:1) , – 18 –here the definitions for the mass parameters can be found in (4.12). Note that the ζ -dependent mass terms given by (3.33) for the W ± µ and Z µ fields have been canceled by theghost 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 = (cid:88) σ = scalars n σ π (cid:20) M σ (cid:18) log |M σ | µ − (cid:19) + 2 a ,s log |M σ | µ (cid:21) , (4.15)which specifically for the Lagrangian in (4.1) contains the expressions h ; n h = +1 , M h = m h + (cid:18) ξ − (cid:19) R ,W ± ; n W,s = − , M W,s = m W − R ,Z ; n Z,s = − , M Z,s = m Z − R ,χ , χ ; n χ,W = +2 , M χ,W = ζ W m W + m χ − R ,χ ; n χ,Z = +1 , M χ,Z = ζ Z m Z + m χ − R ,c , c ; n c,W = − , M c,W = ζ W m W − R ,c ; n c,Z = − , M c,Z = ζ Z m Z − R , (4.16)with the a ,s given by (3.18).The contribution from the top quark is straightforward to express via (3.13) with thehelp of section 3.2 V (1)SM ( ϕ ) fermion,t = − π (cid:20) { Group }M t (cid:18) log |M t | µ − (cid:19) + 2 tr { a ,f } log |M t | µ (cid:21) , (4.17)where we have explicitly calculated the trace over Dirac indices, the effective mass can befound in (3.23), tr { a ,f } is given by (3.25) and due to color in the SM tr { Group } = 3 forthe top quark (tr { Group } = 1 for leptons).Finally we can address the contributions coming from gauge fields that have non-trivial structure in terms of spacetime/Lorentz indices. Unsurprisingly, this is the most– 19 –omplicated 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 + · · · = 264 π (cid:20) ( M W ) βα ( M W ) νβ (cid:18) log | ( M W ) αν | ˜ µ − g αν (cid:19) + 2( a ,g ) βα log | ( M W ) αβ | ˜ µ (cid:21) + 164 π (cid:20) ( M Z ) βα ( M Z ) νβ (cid:18) log | ( M Z ) αν | ˜ µ − g αν (cid:19) + 2( a ,g ) βα log | ( M Z ) αβ | ˜ µ (cid:21) + · · · , (4.18)where ( M Z ) µν and ( M W ) µν can be found from (3.33) and ( a ,g ) βα from (3.35). Thereason we have left in the divergent renormalization scales (3.15) in (4.18) is that since thetrace depends on the dimensions of spacetime, we must not choose n = 4 before explicitlycalculating it. Due to the presence of the logarithm with Lorentz indices in general theabove is a fairly non-trivial expression. However, when one limits to the case with onlydiagonal elements in R µν such as FLRW the sums can be explicitly performed. For examplefor the piece coming from Z µ in the FLRW case we can then write V (1)SM ( ϕ ) gauge,Z = 164 π (cid:20) ( M Z ) ( M Z ) (cid:18) log | ( M Z ) | µ − (cid:19) + 2( a ,g ) log | ( M Z ) | µ (cid:21) + 364 π (cid:20) ( M Z ) ii ( M Z ) ii (cid:18) log | ( M Z ) ii | µ − (cid:19) + 2( a ,g ) ii log | ( M Z ) ii | µ (cid:21) , (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 prescriptionis such that the result coincides with the standard parametrization in the flat space limit(see e.g. [93]) . The remaining W ± ν piece 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 resultincluding 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 straightforwardas is adding degrees of freedom beyond the SM. For an explicit example, see section 5. β -functions By following the procedures we introduced for the simple scalar field model in section 2.3we can now derive all the β -functions of the SM in curved spacetime. This includes thewell-known β -functions that can be calculated in flat space and can be found from standardreferences, see for example [6, 7, 89, 93], and the ones connected to the dynamics of a curvedspacetime. The β s that are relevant in curved spacetime are defined by the operators inthe purely gravitational part of the action (2.2) and the non-minimal term proportional to ξ . The factor of 5 / ∝ (4 − n ) − and the ( n − – 20 –ith the help of formulae from section 3 we can use the 1-loop approximation to theCallan-Symanzik equation (2.25) and the SM anomalous dimension [95] γ = 116 π (cid:20) Y − g − g (cid:48) ) − ζ W g − ζ Z (cid:0) g + ( g (cid:48) ) (cid:1) (cid:21) , (4.20)to write the gravitational β -functions16 π β ξ = (cid:18) ξ − (cid:19)(cid:20) λ + 2 Y − g (cid:48) ) − g (cid:21) (4.21)16 π β V Λ = 2 m (4.22)16 π β κ = 4 m (cid:18) ξ − (cid:19) , (4.23)16 π β α = 2 ξ − ξ − , (4.24)16 π β α = 57190 , (4.25)16 π β α = − , (4.26)with Y ≡ y u + y c + y t ) + 3( y d + y s + y b ) + ( y e + y µ + y τ ) ,Y ≡ y u + y c + y t ) + 3( y d + y s + y b ) + ( y e + y µ + y τ ) . (4.27)We emphasize that (4.21 – 4.26) include the contributions from the entire SM and areexhaustive in terms of the generated operators. The other one-loop SM β -functions are given by, for example [7, 89]. Since there arecurvature dependent loop corrections for all the fermions in the theory, it is not necessarilycorrect to ignore the light fermions, so we include the running of all the Yukawa couplings.These can be found in [96, 97] for example,16 π β y t = y t (cid:20)
32 ( y t − y b ) + Y − (cid:18) g (cid:48) ) + 94 g + 8 g (cid:19)(cid:21) , (4.28)16 π β y b = y b (cid:20)
32 ( y b − y t ) + Y − (cid:18)
512 ( g (cid:48) ) + 94 g + 8 g (cid:19)(cid:21) , (4.29)16 π β y l = y l (cid:20) y l + Y − (cid:18) g (cid:48) ) + 94 g (cid:19)(cid:21) , (4.30)16 π β λ = 24 λ − λ (cid:0) ( g (cid:48) ) + 3 g (cid:1) + 34 (cid:18)
12 ( g (cid:48) ) + ( g (cid:48) ) g + 32 g (cid:19) + 4 Y λ − Y , (4.31)16 π β m = m (cid:20) λ −
32 ( g (cid:48) ) − g + 2 Y (cid:21) , (4.32)16 π β g (cid:48) = 416 ( g (cid:48) ) , π β g = − g , π β g = − g , (4.33) The particle content of the SM may be found from tables 1 and 2. – 21 –here β y l is the lepton beta function for l = e, µ, τ . For reference, the beta functionsare defined as β X ≡ µ (d X/ d µ ), g (cid:48) is the U (1) coupling, g is the SU(2) coupling, and g the SU (3) coupling. Beta functions for the other generations of fermions can be obtainedfrom eqs. (4.28) and (4.29) by substituting y t → y u , y c and y b → y d , y s , leaving the gaugecouplings and Y the same.In priciple also the gauge parameters ζ Z and ζ W would run according to their respective β -functions. Their running is however not relevant for a one loop calculation: the ζ ’s onlyenter in the loop correcting and do not couple to other one loop β -functions making therunning a two loop effect and with no loss of generality one may treat them as constant,which will be our choice.For the boundary conditions of the running couplings at the EW scale t = 0, we usethe 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]. Unlessotherwise stated, we used top quark and Higgs boson pole masses of M t = 173 .
34 GeV and M h = 125 .
15 GeV respectively, with other pole masses found in [99].
The general results derived above using the heat kernel method hold for an arbitrary curvedspacetime. As a specific example, we apply them in the de Sitter space where R = 12 H , R µν R µν = 36 H , R µνδη R µνδη = 24 H , (5.1)and the Hubble rate H is constant.Substituting these into the expressions derived in Sections 3 and 4, we find that the1-loop contribution to the effective potential of the SM Higgs in the MS scheme, includingthe complete set of quarks and leptons as well as the photon and the gluons, is given by V (1)SM ( ϕ ) = 164 π (cid:88) i =1 (cid:26) n i M i (cid:20) log (cid:18) |M i | µ (cid:19) − d i (cid:21) + n (cid:48) i H log (cid:18) |M i | µ (cid:19) (cid:27) , (5.2)where the inputs can be read from tables 1 and 2. They are split as the degrees of freedomthat directly couple to the Higgs in table 1 and degrees of freedom that do not in table 2.Recall that our computation gives the UV limit of the effective potential.– 22 – able 1 . Contributions to the effective potential (5.2) with tree-level couplings to the Higgs. Ψstands for W ± , Z , the 6 quarks q, the 3 charged leptons l , the Higgs h , the Goldstone bosons χ W and χ Z and the ghosts c W and c Z . The masses are defined as in (4.12). Ψ i n i d i n (cid:48) i M i / − / m W + H W ± / − / m W + H − / / m W − H / − / m Z + H Z / − / m Z + H − / / m Z − H q 7 − −
12 3 / / m q + H l − − / / m l + H h
16 1 3 / − / m h + 12( ξ − / H χ W
17 2 3 / − / m χ + ζ W m W + 12( ξ − / H χ Z
18 1 3 / − / m χ + ζ Z m Z + 12( ξ − / H c W − / / ζ W m W − H c Z − / / ζ Z m Z − H A generic expression for M i is thus a function of the form M i ( ϕ cl , µ ) = κ i ( µ ) Z ( M t ) Z ( µ ) ϕ − κ (cid:48) i ( µ ) + θ i ( µ ) H , where the coupling-dependent κ i , κ (cid:48) i , θ i can be read from tables 1 and 2,using field-dependent masses for all the Standard Model particles. The one-loop effectivepotential with running couplings is then given by V effSM ( ϕ ( µ )) = − m ( µ ) ϕ ( µ ) + ξ ( µ )2 Rϕ ( µ ) + λ ( µ )4 ϕ ( µ ) + V Λ ( µ ) − κ ( µ ) H + α ( µ ) H + 164 π (cid:88) i =1 (cid:26) n i M i ( µ ) (cid:20) log (cid:18) |M i ( µ ) | µ (cid:19) − d i (cid:21) + n (cid:48) i H log (cid:18) |M i ( µ ) | µ (cid:19) (cid:27) . (5.3)The different gravitational terms α R , α R µν R µν , α R µνδη R µνδη in (2.2) have combinedto a single term α ( µ ) H due to the de Sitter relations (5.1). The β function of the coupling α is determined by using eqs. (4.24 – 4.26) which yield β α = 116 π (cid:20) ξ − ξ − (cid:21) . (5.4)All other β functions are directly given in section 4.1.– 23 – able 2 . Contributions to the effective potential (5.2) that do not to couple to the Higgs attree-level. Ψ stands for the photon γ , the 8 gluons g , the 3 neutrinos ν and the ghosts c γ and c g . Ψ i n i d i n (cid:48) i M i
21 1 3 / − / H γ
22 3 5 / − / H − / / − H
24 8 3 / − / H g
25 24 5 / − / H − / / − H ν − − / / H c γ − / / − H c g − / / − H The renormalization group improved potential is found by numerically solving for thefull set of β functions and substituting the results into (2.2). We use a modification of amethod recently employed by two of the authors in [29, 100]. Briefly, this method consistsof computing the running of the couplings at a set of discrete points, and using these toconstruct a C continuous interpolating piecewise polynomial to describe the running inlogarithmic space. This results in function g i ( µ ) for the couplings, and together with ascale choice µ ( ϕ ), gives a numerical expression for the potential. The final expression isthen: V RGISM ( ϕ cl ) = 12 (cid:20) − m ( µ ∗ ( ϕ cl )) + ξ ( µ ∗ ( ϕ cl )) R (cid:21) Z ( M t ) Z ( µ ∗ ( ϕ cl )) ϕ + λ ( µ ∗ ( ϕ cl ))4 Z ( M t ) Z ( µ ∗ ( ϕ cl )) ϕ + V Λ ( µ ∗ ( ϕ cl )) − κ ( µ ∗ ( ϕ cl )) H + α ( µ ∗ ( ϕ cl )) H + 164 π (cid:88) i =1 (cid:26) n i M i ( ϕ cl ) (cid:20) log (cid:18) |M i ( ϕ cl ) | µ ∗ ( ϕ cl ) (cid:19) − d i (cid:21) + n (cid:48) i H log (cid:18) |M i ( ϕ cl ) | µ ∗ ( ϕ cl ) (cid:19) (cid:27) , (5.5)where M i ( ϕ cl ) = M i ( ϕ cl , µ ∗ ( ϕ cl )) is the relevant mass-term defined at scale µ ∗ ( ϕ cl ), ϕ cl = ϕ ( M t ) is the field evaluated at the electroweak scale, and α = 144 α + 36 α + 24 α . The renormalisation group improvement procedure discussed in section 2.3 consists ofchoosing the renormalisation scale in such a way that the loop correction vanishes. Applying– 24 –he same method to the full SM case amounts to setting µ = µ ∗ given by the condition (cid:88) i =1 (cid:20) n i M i ( µ ∗ , ϕ ) (cid:18) log (cid:18) |M i ( µ ∗ , ϕ ) | µ ∗ (cid:19) − d i (cid:19) + n (cid:48) i H log (cid:18) |M i ( µ ∗ , ϕ ) | µ ∗ (cid:19)(cid:21) = 0 (5.6)However, unlike in the simple scalar theory of section 2.3, this equation may nowhave several solutions as it contains multiple terms with different mass scales M i . Thisis illustrated in fig. 1 which shows two solutions of eq. (5.6). The individual logarithms M i log( M i /µ )) are not necessarily small for all solutions. Therefore, eq. (5.6) alone isnot enough to ensure convergence of the one-loop effective potential which is accurate upto log squared corrections, such as O (( y t / π ) M t log ( M t /µ )). Fig. 1 also shows therange where the dominant log contributions are small for each solution. Outside this rangethe logs grow large and using µ ∗ given by eq. (5.6) does not give a reliable result. This isdemonstrated in fig. 2 which shows the effective potentials computed for the two differentsolutions of eq. (5.6). ϕ cl / GeV µ ∗ ( ϕ c l ) / G e V ϕ cl / GeV µ ∗ ( ϕ c l ) / G e V Solution 1Solution 2 µ ∗ = ϕ cl µ ∗ = aϕ + bR Figure 1 . Two solutions of Eq. (5.6) for a range of ϕ cl , and two different values of ξ EW and H . A‘best fit’ choice as in Eq. 5.7 is also plotted, along-side the µ = ϕ choice. Left: ξ EW = ξ ( M t ) = 1 / H = 10 GeV, a = 0 . b = 0 . ξ EW = ξ ( M t ) = − H = 10 GeV, a = 0 . b = 0 . W, Z, ϕ, t ,all satisfy (cid:12)(cid:12)(cid:12) log M µ (cid:12)(cid:12)(cid:12) <
5, small dots where this is not true.
We could accompany eq. (5.6) with the additional condition that the individual loga-rithms M i log( M i /µ )) remain small. However, there are two problems with this proce-dure. First, it is not always possible to find any solution µ ∗ for which the logs are smallwhile eq. (5.6) is satisfied exactly. Second, even when such a solution exists, it may notgive a continuous potential over the whole range of ϕ cl .– 25 –o obtain a continuous potential, one has to relax the requirement that the one loopcorrection vanishes completely, and instead choose the renormalization scale in a continuousway so that the loop correction remains small albeit non-zero. A simple scale choice thatapproximatively achieves this is µ = aϕ + bR, (5.7)where ϕ cl = ϕ ( M t ) . (5.8)There are several ways to determine the constants a, b , for example as the best fit valuesthat interpolate between the different solutions of eq. (5.6), or chosen to minimise the sizeof the logarithm terms in the potential. A similar choice without the numerical fitting wasused in [19, 40]. The scale choice (5.7) is dominated by the curvature R for small ϕ andby the field for large ϕ and can therefore minimize loop terms of type log( M /µ ) where M is a sum of R and ϕ dependent entries. This works well if the effective potential doesnot depend too strongly on the scale µ across the relevant range. As an example, we plotin figure 3 the effective potential V ( ϕ, µ ) for different µ at constant ϕ , H = 10 GeV and ξ = . If the scale is well chosen, then it should lie in a relatively flat region of the resultingcurve. It is noticed that the renormalization scale can drastically change V ( ϕ, µ ) if ϕ isclose to the barrier. This is perhaps unsurprising, as the potential rapidly changes frompositive to negative values there. ϕ cl / GeV -10 -10 -10 -10 -10 [ V ( ϕ c l ) − V ( ) ] / G e V Solution 1Solution 2
Figure 2 . RGI Potentials associated to solution 1 and solution 2 for ξ EW = − , H = 10 GeV,from fig. 1. Note that solution 1 is completely unstable, while solution 2 possesses a barrier. Thescale µ inst is the renormalization scale at which λ ( µ inst ) = 0. – 26 – µ / GeV -10 -10 -10 -10 -10 -10 -10 V ( ϕ c l , µ ) − V ( , µ ) µ i n s t ϕ cl = 2 . × GeV ϕ cl = 6 . × GeV ϕ cl = 8 . × GeVSolution 1, t (6 . × GeV)Solution 2, t (6 . × GeV)
Figure 3 . Renormalisation scale dependence of the approximated effective potential for H =10 GeV and ξ EW = 1 /
6. For most values of ϕ cl , the potential does not vary considerably. Theexception is in the vicinity of the barrier, where changing µ can rapidly shift the potential frompositive to negative values, since it moves around the zero of the potential. To consider a wide range of possible scenarios, we plot in figures 4 to 6 the potential as afunction of the “classical” field ϕ cl . We also plot the potential in units of the instability scale µ inst , defined as the renormalisation scale at which λ ( µ inst ) = 0, rather than in units of GeV.The reason for this is that the instability scale given by the one-loop calculation ( µ inst =9 . × GeV) is quite different from the more accurate three-loop value ( µ inst = 6 . × GeV) (computed using 3 loop running of the Standard Model beta functions, see forexample [101]), and this is reflected in the scale of the corresponding potentials. Expressingthe potential in units of µ inst should therefore allow a more meaningful comparison andmore reliable physical conclusions.In figs. 4, 5, 6, we show the effective potential for a range of different Hubble rates, H ,and non-minimal couplings ξ EW = ξ ( µ = M t ) (as evaluated at the electroweak scale). Thescale µ is chosen to be Eq. (5.7), in order to give a continuous potential for all ϕ . Thereare various ways of picking the coefficients a and b ; in the case of the potentials plotted infigs. 4, 5, 6, we choose a and b such that a weighted sum of the squares of the logs, S = (cid:80) i M i log( M i /µ ) (cid:80) i M i , (5.9)is minimized at both a small scale ( ϕ cl = M t ) and a large scale ( ϕ cl = M P ).– 27 – -4 -2 ϕ cl / µ inst -0.3-0.2-0.100.10.20.3 [ V ( ϕ c l ) − V ( ) ] / µ i n s t H = 1 . × − µ inst H = 0 . µ inst H = 0 . µ inst H = 0 . µ inst H = 0 . µ inst H = 0 . µ inst H = µ inst H = 1 . × µ inst Figure 4 . Example RGI potentials for ξ EW = −
1, in units of µ inst defined by λ ( µ inst ) = 0. -4 -2 ϕ cl / µ inst -0.3-0.2-0.100.10.20.3 [ V ( ϕ c l ) − V ( ) ] / µ i n s t H = 1 . × − µ inst H = 0 . µ inst H = 0 . µ inst H = 0 . µ inst H = µ inst H = 102 . µ inst Figure 5 . Example RGI potentials for ξ EW = 0, in units of µ inst defined by λ ( µ inst ) = 0. Notice that for negative ξ , sufficiently large H erases the barrier altogether. For positive ξ , the opposite occurs: larger H raises the height of the barrier. The case of ξ = 0 at theelectroweak scale is effectively the same as a negative ξ scenario, because the coupling runs– 28 – -5 ϕ cl / µ inst -10 -10 -10 [ V ( ϕ c l ) − V ( ) ] / µ i n s t H = 0 . µ inst H = 0 . µ inst H = µ inst H = 13 . µ inst H = 102 . µ inst Figure 6 . Example RGI potentials for ξ EW = 1 /
6, in units of µ inst defined by λ ( µ inst ) = 0. to negative values at scales above the electroweak scale, and the potential is evaluated at µ = bR for small ϕ cl (thus, unless R = 0, the relevant scale is never the electroweak scalefor any ϕ cl , and so ξ < The gauge dependence of the instability scale for the SM in flat space was recently inves-tigated in [95, 102–104].It is well known that the stationary points of the effective potential for a gauge theoryare independent of the gauge, even if the potential is not. This is described by the Nielsenidentity for the effective potential [105] ∂V SMeff ( ϕ ) ∂ζ = C ( ϕ, ζ ) ∂V SMeff ( ϕ ) ∂ϕ . (5.10)From the above it trivially follows that at extremal points the effective potential is gaugeindependent. In flat space this fact ensures that vacuum decay rates are gauge independentquantities [106] and the same can be seen for the Hawking-Moss instanton in de Sitter space,since it’s action depends only on the height of the barrier (see section 5.4).The Nielsen identity, however, only applies to the RGI effective potential up to theorder of truncation, and therefore it is important to check how much its extremal pointsdepend on the gauge choice. In fig. 7 we plot the potential for different choices of thegauge parameter. This shows that across a reasonably large parameter space for the gaugeparameters ζ W , ζ Z , the height of the potential barrier stays roughly the same (outside thisparameter range, a numerical instability related to choosing the scale that sets the sum of– 29 – ϕ cl / GeV -10 -10 -10 -10 -10 V ( ϕ c l ) / G e V ζ Z = ζ W = -100 ζ Z = ζ W = -68.4211 ζ Z = ζ W = -36.8421 ζ Z = ζ W = -5.2632 ζ Z = ζ W = 26.3158 ζ Z = ζ W = 57.8947 ζ Z = ζ W = 89.4737 Figure 7 . Gauge dependence of the potential for H = 10 GeV and ξ EW = 0. The extremal shouldbe gauge-independent and our one-loop approximation satisfies this condition relatively well. Y-axisis scaled as y = sign(x) log(1 + | x | ), which is logarithmic for large positive and negative values - thiscauses the sharp apparent change across zero. the quantum corrections was encountered). Fig. 7, computed with the ‘physical’ (smallestlog) solution, also shows that unlike the extrema, the scale at which the potential turnsover to negative values is sensitive to the gauge parameter, due in most part to the gaugedependence of the anomalous dimension. Stability of the vacuum during inflation depends on two factors: the nucleation of truevacuum bubbles, and the expansion of space-time. This issue has been considered, forexample, by [25, 26, 30]. Here, we will present a simplified analysis based on the Hawking-Moss instanton. This neglects some potentially important effects: see [25] for a discussionof this. Note also that below a certain critical Hubble rate, Coleman de Luccia (CdL)bounces are expected to dominate [107, 108].The decay rate predicted by Hawking-Moss instantons is of the formΓ = cH exp( − B HM ) , (5.11)where c is an O (1) constant, and the prefactor is proportional to H on dimensionalgrounds . The decay exponent, B HM , is given by[109], including non-minimal coupling, A proper analysis would involve computing the full functional determinant[107]; the precise pre-factor,however, will not matter much due to the exponential dependence on B HM – 30 –y B HM = 24 π M (cid:34) V ( ϕ fv ) (cid:18) − ξϕ M (cid:19) − V ( ϕ HM ) (cid:18) − ξϕ M (cid:19) (cid:35) . (5.12)Here ϕ fv is the false vacuum field value, ϕ HM the top of the barrier, and V represents theeffective potential without the ξϕ R non-minimal coupling term. This result includes theeffect of gravitational backreaction of the Hawking-Moss solution, and can be seen mosteasily in the Einstein frame, where the potential takes the form[110]˜ V ( ˜ ϕ ) = V ( ϕ ( ˜ ϕ )) (cid:16) − ξϕ ( ˜ ϕ ) M (cid:17) . (5.13)˜ V and ˜ ϕ are the canonical Einstein frame potential and field, respectively. Thus in theEinstein frame, Eq. (5.12) is just the usual Hawking-Moss formula. Generically, the actionof an O (4) symmetric bounce solution is given by S = 2 π (cid:90) r max d ra ( r ) (cid:20)
12 ˙ ϕ + V ( ϕ ) − M (cid:18) − ξϕ M (cid:19) R (cid:21) , (5.14)where a ( r ) is the scale factor in the Euclidean metric, d s = d r + a ( r )dΩ and r is a (Eu-clidean) radial co-ordinate. Using Eq. (5.14) we can compute B HM of the Hawking-Mossinstanton in the limit where R is fixed - the so called ‘fixed background approximation’.This is given by B HM = 8 π ∆ V ξ ( ϕ HM )3 H , (5.15)where H is the Hubble rate, R = 12 H , and∆ V ξ ( ϕ ) = V ( ϕ ) + ξ ϕ R − V ( ϕ fv ) − ξ ϕ . (5.16)This means that B HM is proportional to the difference in Height between the top of the bar-rier and the false vacuum, with the potential evaluated in the Jordan frame and including the ξϕ R term as if it were part of the potential.Stability during inflation requires that the probability of decay is sufficiently low thatthe expected number of separate causal regions in which a bubble was nucleated in our pastlight-cone, is fewer than 1. The survival of a single causal region that decayed to the falsevacuum could potentially destabilize the universe if it were to then continue expandingafter inflation ended. If N e-folds of inflation are visible, there are approximately e N such causal regions, and so the number that decayed during inflation is n decayed = e N p ( N, , (5.17)where p ( N,
1) is the probability of a single causal region decaying after N e-folds of inflation.Note that a bubble forming during inflation always expands to fill the causal region (1 Though it is unclear precisely what would happen to such a bubble that started expanding duringinflation, only for inflation to end, as opposed to a bubble forming in flat space - here we will assume thatit expands and envelops the whole spacetime. – 31 –ubble volume) that it fills[26], but can expand no further because the expansion of space-time out-paces the bubble wall, that can only move at causal velocities. The probabilityper unit time that a bubble forms within a Hubble volume during inflation is given by γ = V Hubble cH e − B HM = 4 πc He − B HM . (5.18)Thus, the probability of decaying between e-folds N and N is p ( N , N ) = 4 πc (cid:90) t t d tHe − B HM = k (cid:90) N N d NN e − B HM . (5.19)Where d NN = H d t , and N is the number of e-folds. k is an unknown O (1) factor. Forconstant Hubble rate, which we will assume here for simplicity, this means n decayed = k log( N ) exp (cid:18) N − π ∆ V ξ ( ϕ HM )3 H (cid:19) . (5.20)The condition n decayed < H < A ∆ V ξ ( ϕ HM ) , (5.21)where A = (cid:20) π k + log log N + 3 N ) (cid:21) = 0 . ± . , (5.22)for N = 60 e-folds and the uncertainty given by assuming 10 − < k < , illustrating theweak 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 analysisis not affected by using a different scale choice, such as µ = aϕ + bR : this producedvirtually identical results to figure 8.The condition (5.21) is of the same form but slightly strong than the bound used in[19], π V ( φ )3 H > A = (8 π / / (cid:39) .
26. As discussed in Refs. [25,26, 30, 100], the bound could be improved further by accounting for possibility for thefield to flow back across the barrier due to evolution during inflation and by including thepossible impact of CdL solutions.The effect of changing the top mass at constant H is shown in figure 9. Together theseplots illustrate that even for Hubble rates somewhat below the instability scale (defined by λ ( µ inst ) = 0), negative ξ can quickly destabilize the potential. Note also that because ofthe running of ξ , ξ EW = 0 is qualitatively equivalent to having ξ <
0, since the non-minimalcoupling runs to negative values at higher energies, and if the optimal scale choice is µ ∼ R for small ϕ , this means ξ < igure 8 . Stability analysis for M t = 173 . M h = 125 . µ inst is defined as the renormalization scale (in flat space)at which λ ( µ inst ) = 0.
171 171.5 172 172.5 173 173.5 174 174.5 175 175.5 176 M t / GeV -10 -10 -10 -10 -10 ξ E W H = 1.0579 µ inst H = 0.10579 µ inst H = 0.010579 µ inst ξ EW = 0 ξ EW = 1/6 Figure 9 . Plot of the boundary between stability and instability for different Hubble rates, andtop masses. Note that µ inst is defined by λ ( µ inst ) = 0 using m t = 173 .
34 GeV for comparison. Notethat since the results in this paper use only 1-loop running, low values of m t can be unstable forvalues that would lead to an absolutely stable vacuum at 3-loops. – 33 – Conclusions
In this work we derived the renormalization group improved effective potential in curvedspacetime for the SM Higgs including the complete SM particle content to one loop orderin perturbation theory. Our calculation included the UV limit of the loop correctionsand thus contains the universal contribution that must be shared by all quantum statespossessing 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 incurved spacetime. As an application we investigated the behavior of the SM Higgs in deSitter space in the context of electroweak vacuum instability.Our use of the UV expansion means that the effective potential does not includeinfrared contributions, which can be large in the presence of light scalar fields. Locally theinfrared contributions can always be absorbed into rescaling of background quantities andtherefore do not affect our ultraviolet results. The global effects of these infrared modescan be studied by using the stochastic inflation approach [111] with the effective ultravioletpotential computed here as the input.Broadly speaking our results highlight two important, and often overlooked, aspectsthat arise whenever quantum fields are investigated in situations for which the curvatureof spacetime is non-negligible: The first is that the renormalization group running seesthe energy scale set by the curvature of the background, a mechanism which we calledcurvature induced running. For cosmologically interesting cases where the field is a lightspectator with respect to the Hubble rate, the curvature can give the dominant contributionto the renormalization group running. The second aspect is the generation of operatorsinvisible in flat space. In addition to the well-known non-minimal coupling there are 5other operators (see (2.2)) generated via loops in curved spacetime. For the SM in curvedspacetime the β -functions imply the generation of all such operators resulting in importantmodifications, as is apparent from the results of section 4.1. There is no compelling reasonto assume similar contributions not to arise for theories beyond the SM, which can bestudied by straightforward generalizations of our results.The application to de Sitter space shows clearly the impact of making use of an ef-fective potential calculated in curved spacetime. The standard procedure of optimizingthe convergence of the loop expansion by an appropriate renormalization scale choice ismade 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 isnon-trivial. Using the effective potential, we demonstrated for the SM that negative val-ues of the non-minimal coupling are tightly constrained from below by the requirement ofvacuum stability during inflation. Importantly, this is true even for inflationary scales wellbelow the scale of instability and hence for low top mass values for which the instabilityoccurs above the maximal inflationary scale allowed by the non-detection of a primordialtensor spectrum. – 34 – cknowledgments 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 Col-lege President’s PhD Scholarship.
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