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Running inflation in the Standard Model The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation De Simone, Andrea, Mark P. Hertzberg, and Frank Wilczek. “Running Inflation in the Standard Model.” Physics Letters B 678, no. 1 (July 2009): 1–8. As Published http://dx.doi.org/10.1016/j.physletb.2009.05.054 Publisher Elsevier Version Author's final manuscript Accessed Thu May 26 17:50:05 EDT 2016 Citable Link http://hdl.handle.net/1721.1/102218 Terms of Use Creative Commons Attribution-Noncommercial-NoDerivatives Detailed Terms http://creativecommons.org/licenses/by-nc-nd/4.0/ Running Inflation in the Standard Model Andrea De Simone1 , Mark P. Hertzberg2 , and Frank Wilczek3 arXiv:0812.4946v3 [hep-ph] 9 Jul 2009 Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Abstract An interacting scalar field with largish coupling to curvature can support a distinctive inflationary universe scenario. Previously this has been discussed for the Standard Model Higgs field, treated classically or in a leading log approximation. Here we investigate the quantum theory using renormalization group methods. In this model the running of both the effective Planck mass and the couplings is important. The cosmological predictions are consistent with existing WMAP5 data, with 0.967 . ns . 0.98 (for Ne = 60) and negligible gravity waves. We find a relationship between the spectral index and the Higgs mass that is sharply varying for mh ∼ 120 − 135 GeV (depending on the top mass); in the future, that relationship could be tested against data from PLANCK and LHC. We also comment briefly on how similar dynamics might arise in more general settings, and discuss our assumptions from the effective field theory point of view. Key words: Higgs Boson, Cosmological Inflation, Standard Model PACS: 14.80.Bn, 98.80.Cq 1. Introduction The hypothesis that there was a period in the early history of the universe during which a local Lorentz invariant energy density – i.e., an effective cosmological term – dominated the equation of state, causing exponential expansion, explains several otherwise puzzling features of the present universe (flatness, isotropy, homogeneity) [1, 2, 3, 4]. It also suggests a mechanism whereby primordial density fluctuations arise through intrinsic fluctuations of quantum fields, leading to qualitative and semi-quantitative predictions that are consistent with recent observations. However the physics behind inflation remains mysterious. What, specifically, is the source of the energy density? Ideas ranging from fields associated with supersymmetry, string moduli, ghosts, branes, and others abound [5, 6, 7, 8, 9, 10]. One (or more) of them might be correct, but all are highly speculative, and none is obviously compelling. Alternatively, we can look for inflationary dynamics based on degrees of freedom already present in the Standard Model. We can also attempt to maintain the guiding philosophy of the 1 Electronic address: andreads at mit.edu address: mphertz at mit.edu 3 Electronic address: wilczek at mit.edu 2 Electronic Preprint submitted to Physics Letters B July 9, 2009 Standard Model, including gravity, to allow only local interactions which are gauge invariant and have mass dimension ≤ 4. Within this very restrictive framework, there remains the possibility to include the non-minimal gravitational coupling ξH † HR. Here H is the Higgs field, R is the Ricci scalar, and ξ is a dimensionless coupling constant, whose value is unknown and largely unconstrained by experiment.4 Indeed renormalization of the divergences arising in a self-interacting scalar theory in curved spacetime requires a term of this form [11]. The Higgs sector is then described, classically, by the Lagrangian Lh = −|∂H|2 + µ2 H † H − λ(H † H)2 + ξH † H R (1) where λ is the Higgs self coupling and µ is the Higgs mass parameter. It has been known for some time that such minimal classical Lagrangians can support inflation driven by an interesting interplay between the quartic term and the non-minimal coupling term [12, 13, 14, 15]. For ease of reference, we will call this general set-up “running inflation”; the name seems appropriate, since evolution of the effective Planck mass and the effective scalar mass is central to the dynamics.5 This quasi-renormalizable set-up allows use of renormalization group methods, as will be illustrated here. By quasi-renormalizable, we mean that the theory is renormalizable when gravity is treated classically; in particular, we ignore quantum corrections from graviton exchange (see Appendix B). In the investigation of (non-gravitational) quantum effects, it is appropriate to focus specifically on the Standard Model, for two reasons. First, because (as we will see) it illustrates important qualitative issues in a very concrete, familiar setting. Second, because it – or something close to it – might actually contain the degrees of freedom relevant to real-world inflation, in which case the specific predictions we derive could help describe reality. Recently, the idea that the Standard Model Higgs field, non-minimally coupled to gravity, can lead to inflation was proposed in Ref. [17]. Those authors argued that the radiative corrections to the potential are negligible and hence the inflationary parameters can be computed using the classical Lagrangian. They found that the cosmological predictions are in good agreement with cosmological data, independent of the Standard Model parameters, such as λ. On the other hand the authors of Ref. [18] criticized their approach, suggesting that the quantum corrections to the potential can be very important. They concluded that a Higgs lighter than 230 GeV cannot serve as the inflaton, because the predicted spectral index is ruled out by WMAP5 data [19]. Ref. [18] only incorporated quantum corrections at leading log order, extrapolated from low energies. Here, in contrast, we will compute the full renormalization group improved effective action at 2-loops. We conclude that running inflation based upon a Standard Model Higgs makes predictions that are consistent with current cosmological data, and leads to firm predictions for the PLANCK satellite and the LHC. Our main result is a correlation between the spectral index and the Higgs mass, see Fig. 1. This correlation is absent in the classical theory. The origin of the correlation lies in the interactions of the Standard Model, which dictate the form of the effective action. In Section 2 we review inflation with non-minimally coupled scalars. In Section 3 we investigate the classical theory of the Higgs non-minimally coupled to gravity. In Section 4 we describe our method for obtaining the quantum corrected effective action. We compute all the inflation4 For 5 The ξ = −1/6 the Higgs is conformally coupled to gravity. term “running inflation” was used in a different context in [16]. 2 0.990 mt=17 3 GeV V 0.975 71 Ge V e 69 G 0.980 m t=1 m t=1 Spectral index ns 0.985 0.970 0.965 122 124 126 128 130 132 Higgs mass mh HGeVL 134 136 Figure 1: The spectral index ns as a function of the Higgs mass mh for a range of light Higgs masses. The 3 curves correspond to 3 different values of the top mass: mt = 169 GeV (red curve), mt = 171 GeV (blue curve), and mt = 173 GeV (orange curve). The solid curves are for αs (mZ ) = 0.1176, while for mt = 171 GeV (blue curve) we have also indicated the 2-sigma spread in αs (mZ ) = 0.1176 ± 0.0020, where the dotted (dot-dashed) curve corresponds to smaller (larger) αs . The horizontal dashed green curve, with ns ' 0.968, is the classical result. The yellow rectangle indicates the expected accuracy of PLANCK in measuring ns (∆ns ≈ 0.004) and the LHC in measuring mh (∆mh ≈ 0.2 GeV). In this plot we have set Ne = 60. ary observables numerically and present results in Section 5. Finally, we review our results and discuss their significance in Section 6. 2. Non-Minimal Inflation Here we briefly review the recipe to compute inflationary observables, which will be used in the later sections, and the latest observational constraints. Consider a real scalar field φ non-minimally coupled to gravity via the Ricci scalar R. The class of effective actions we consider is Z √ 1 1 S = d4 x −g m2Pl f (φ)R − k(φ)(∂φ)2 − V (φ) , (2) 2 2 where we allow for a general coefficient of the Ricci scalar f (φ), general coefficient of kinetic energy k(φ), and general potential V (φ). Here mPl ' 2.43 × 1018 GeV is the reduced Planck mass; we are effectively assuming that the field φ is stabilized at the end of inflation with f (φ0 ) ≈ 1, as will be the case for the Standard Model Higgs. The cosmology of this theory is most easily studied by performing a conformal transformation to the so-called “Einstein frame” where the gravity sector is canonical 21 m2Pl RE . This is achieved E by defining the Einstein metric as gµν = f (φ)gµν . The corresponding Einstein frame potential is VE (φ) = 3 V (φ) . f (φ)2 (3) Furthermore, the kinetic energy in the Einstein frame can be made canonical with respect to a new field σ, defined through the equation dσ dφ 2 ≡ k(φ) 3 2 f 0 (φ)2 , + m f (φ) 2 Pl f (φ)2 (4) (the second term here comes from transforming the Ricci scalar). In this frame, the action takes the canonical form Z 1 2 1 4 √ 2 S = d x −gE m RE − (∂E σ) − VE (σ(φ)) , (5) 2 Pl 2 which is amenable to straightforward analysis. The inflationary dynamics and cosmological predictions is determined by the shape of the potential VE . In the usual way, we introduce the first and second slow-roll parameters, which control the first and second derivatives of the potential, respectively. Using the chain rule, these are 2 −2 dσ 1 2 VE0 mPl , (6) (φ) = 2 VE dφ " −2 −3 2 # VE00 dσ VE0 dσ d σ 2 η(φ) = mPl − , (7) VE dφ VE dφ dφ2 where a prime denotes a derivative with respect to φ. Similarly, the third slow-roll parameter ζ is related to the third derivative of the potential as ζ 2 = m4Pl (d3 VE /dσ 3 )(dVE /dσ)/VE2 . The number of e-foldings of slow-roll inflation is given by an integral over φ: Z φ 1 dσ dφ̃ q Ne (φ) = √ , (8) 2 mPl φend (φ̃) dφ̃ where φend is the value of the field at the end of inflation, defined by ' 1. The number of e-foldings must be matched to the appropriate normalization of the data set and the cosmic history, with a typical value being Ne ' 60; we return to this point in Section 5. The amplitude of density perturbations in k-space is specified by the power spectrum: Ps (k) = ∆2R k k∗ ns (k)−1 , where ∆2R is the amplitude at some “pivot point” k ∗ , predicted by inflation to be VE 2 ∆R = , 24π 2 m4Pl ∗ (9) (10) k and measured by WMAP5 to be ∆2R = (2.445 ± 0.096) × 10−9 at k ∗ = 0.002 Mpc−1 [19]. The corresponding spectral index ns , running of the spectral index α ≡ dns /d ln k, and tensor to 4 scalar ratio r, are given to good approximation by ns = 1 − 6 + 2η , (11) α = −242 + 16η − 2ζ 2 , (12) r = (13) 16. The combined WMAP5 plus baryon-acoustic-oscillations (BAO) and supernovae (SN) data considerably constrain ns and r. Assuming negligible α, as will be the case for running inflation, the constraints are: 0.93 < ns < 0.99 and r < 0.22 (at 95% confidence level). 3. Classical Analysis Without essential loss we can rotate the Higgs doublet so that it takes the form H T = √ (1/ 2)(0, v + φ). Only the real field φ will play a role in our analysis. Specializing to gauge invariant, dimension ≤ 4 operators, without higher derivatives, the functions f (φ), k(φ), and V (φ) must take the form f (φ) = 1 + ξφ2 , m2Pl k(φ) = 1, V (φ) = λ 2 (φ − v 2 )2 , 4 (14) where v ' 246.2 GeV is the vacuum expectation value for the Higgs field, setting the electroweak scale. The self coupling λ is in one-to-one correspondence with the Higgs mass, namely m2h = 2λv 2 . Current experimental bounds on the Higgs mass (and hence λ) are as follows: 114.4 GeV < mh . 182 GeV, 0.11 < λ . 0.27, (15) where the lower bound comes from direct searches and the upper bound comes from a global fit to precision electroweak data (95% CL) [20]. In this theory, inflation takes place at energies many orders of magnitude above the electroweak scale (φ2 ≫ v 2 ). Hence, during inflation the potential is well approximated by the quartic potential: V (φ) = λ4 φ4 , and this form of the classical potential will be sufficient throughout this Letter. The corresponding potential in the Einstein frame is then VE (φ) = (1 λ 4 4φ ξφ2 2 +m 2 ) Pl , (16) √ which approaches a constant V0 ≡ λm4Pl /4ξ 2 at large field values φ mPl / ξ (we assume ξ > 0). This fact allows slow-roll inflation to take place [12, 14, 17]. It is notable that through this mechanism slow-roll inflation emerges unusually “naturally”. √ It is useful to define the dimensionless quantity ψ ≡ ξ φ/mPl which controls the cosmological evolution: inflationary stage (ψ 1), the end of inflation (ψ ∼ 1), and the low-energy regime (ψ 1). Indeed the potential VE plotted in Fig. 2 displays the familiar quartic behavior for small ψ values, but asymptotes to a constant for large ψ. 5 Einstein frame potential VE V0 1.0 0.8 0.99 0.6 0.98 0.97 0.4 0.96 0.95 0.2 0.94 6 0.0 0 2 7 4 8 9 6 Higgs Ψ = 10 8 11 12 10 12 Ξ ΦmPl Figure 2: The potential √ in the Einstein frame VE , normalized to a reference value V0 ≡ λm4Pl /4ξ 2 , as a function of the Higgs field ψ = ξ φ/mPl . The dashed green curve is the classical case (independent of Higgs mass), the solid blue (red) curve is the quantum case with Higgs mass mh = 126.5 GeV (mh = 128 GeV). We have set mt = 171 GeV and αs (mZ ) = 0.1176 for this plot. The inset focusses on the slow-roll inflationary regime. Using eqs. (4), (6), and (7), the slow-roll parameters are readily computed. The exact results are not very transparent. They simplify for large ξ, which is the case of physical interest: 4 4 1 16 3 2 ' , η ' − 1 − , ζ ' 1 − . (17) 3ψ 4 3ψ 2 ψ2 9ψ 4 ψ2 We see that at large ψ (during slow-roll inflation) η is dominant, and will primarily control the predictions for the spectral index. The number of e-foldings is computed from eq. (8) giving 3 1 + ψ2 2 Ne ' ψ 2 − ψend − ln , (18) 2 4 1 + ψend where ψend ' (4/3)1/4 is the value of ψ at the end of inflation ( ' 1). Eqs. (17) and (18) provide a parametric description of (Ne ), η(Ne ), and ζ(Ne ), thus determining ns , α, and r as a function √ of Ne , i.e., we can trade the unknown value of the Higgs field during inflation φ(= mPl ψ/ ξ) for the number of e-foldings Ne . For Ne = 60 we find the following results for the spectral index, the running of the spectral index, and the tensor to scalar ratio: ns ' 0.968, α ' −5.2 × 10−4 , r ≈ 3.0 × 10−3 . (19) We see that α and r are rather small. This will remain qualitatively true in the quantum theory, but the corrections to ns are quite important, as we explore in detail in the √ next section. Finally, using eq. (10) and expanding to leading order in 1/ψ ∼ 1/ Ne , the amplitude of density fluctuations is found to be λ N2 ∆2R ' 2 e 2 . (20) ξ 72π 6 Since this must be O(10−9 ), it is impossible to satisfy for λ = O(0.1) and ξ = O(0.1) (which might be considered “natural” values). One possibility is that λ is extremely small, but that is incompatible with experimental bounds on the Higgs mass, see eq. (15), and is not stable under renormalization. Instead, following [17], we assume ξ = O(104 ) in order to obtain the correct amplitude of density fluctuations with λ = O(0.1). The need to dial a parameter to large or small values, so that ∆2R is consistent with observations, is a common feature to all known inflation models. It will also apply in the quantum theory. 4. Quantum Analysis We now consider how quantum corrections modify the classical results of the previous section. In order to do so, we need to compute the effective action that takes into account the effects of particles of the Standard Model interacting with the Higgs boson through quantum loops. The frame we calculate in is the original “Jordan” frame which defines the theory. The quantum theory modifies all three functions f (φ), k(φ), V (φ) from the classical expressions in eq. (14). The quantum corrections to the classical kinetic sector k(φ) = 1 arise from wave-function renormalization, and are approximately ξ–independent. It is simple to check that at large ξ the second term in eq. (4) scales as ξ 0 ∼ 1, while the contribution from the k(φ)/f (φ) term scales as 1/ξ. Hence corrections to k(φ) occur with a factor 1/ξ, in addition to suppression by loop factors and couplings. The quantum corrections to the classical gravity sector f (φ) = 1 + ξφ2 /m2Pl are more subtle. Let us start by considering the case of a (classical) background gravitational field. In this case the conformal anomaly induces a 1-loop β-function for ξ given by [21] 3 2 1 02 6ξ + 1 2 2λ + yt − g − g . (21) βξ = (4π)2 4 4 The term proportional to λ, coming from Higgs running in a loop (see Fig. 3(a)), is potentially important during inflation. We will return to this point soon when we include the (classical) back reaction of gravity, and argue that in fact this contribution is negligible. The remaining terms arise from external leg corrections and cancel against wave-function renormalization to good approximation. Hence corrections to f (φ) are ignorable also. Finally we turn to the computation of the potential sector V (φ). Let us begin with the flat space analysis. The RG improved potential for the Higgs in the Standard Model is (see e.g. Ref. [22] for a review) 1 V (φ) = λ(t(φ)) G(t(φ))4 φ4 , (22) 4 (φ ≫ v) where t(φ) = ln(φ/µ), and µ is the normalization point; taken to be µ = mt in this Rt Letter. Here λ(t) encodes the running of λ, while G(t) = exp(− 0 dt0 γ(t0 )/(1 + γ(t0 ))), where γ is the anomalous dimension of the Higgs field, encodes wave-function renormalization. The running of λ is governed by the renormalization group equation: dλ/dt = βλ /(1 + γ). At 1-loop it is " # 1 9 4 3 2 02 3 04 2 4 2 2 02 βλ = 24λ − 6 yt + g + g g + g + λ(12yt − 9g − 3g ) . (23) (4π)2 8 4 8 7 (a) (b) (c) (d) (e) (f) Figure 3: Some representative Feynman diagrams. Top row: renormalization of the conformal coupling ξ with Higgs in loop (a), and renormalization of top quark’s Yukawa coupling with gauge boson (b) and Higgs (c) across vertex. Bottom row: renormalization of quartic coupling λ with Higgs (d), top quark (e), and gauge boson (f) in loop. At low energies, the two most important terms here are the self coupling 24λ2 (see Fig. 3(d)), which tries to drive λ to large positive values, and the top quark −6 yt4 (see Fig. 3(e)), which tries to drive λ towards zero. This is summarized in Fig. 4. This leads to a delicate interplay between the Higgs mass and the top mass. For mh mt , the 24λ2 term dominates and λ will eventually hit a Landau pole at high energies. For mh mt , the −6 yt4 dominates and λ will go negative which is a sign of vacuum instability. The “Goldilocks” window for the Higgs mass, where the theory is both perturbative and stable up to very high energies is also the regime in which the quantum corrections are relatively small, allowing for slow-roll inflation. At high energies, the contribution from gauge bosons (see Fig. 3(f)) are important and increase λ. In the recent work of Barvinsky et al. [18] the top quark’s Yukawa coupling was approximated √ by the tree level value: yt = 2 mt /v for all energy scales. This provides a significant negative contribution to βλ , forcing λ to negative values and vacuum instability in large regions of parameter space. Instead it is essential to include the running of the top Yukawa coupling in the analysis: yt 9 2 17 02 9 2 2 βyt = y − 8g − g − g , (24) s (4π)2 2 t 4 12 which is negative due to the large negative contribution from the strong coupling −8gs2 yt (see Fig. 3(b)). Hence yt runs to smaller values at high energies; see Fig. 4. In our work, we have included the complete running of the 5 couplings: λ, yt , gs , g, and g 0 to 2 loops, to ensure accurate results.6 The β-functions are summarized in Appendix A. Furthermore, we have adopted the pole mass matching scheme for the Higgs and top masses, 6 The 3-loop running is unknown for the Standard Model, but would need to be abnormally large to have an effect. 8 0.044 0.042 0.040 0.038 1.5 0.036 0.034 0.032 0.030 30 32 34 36 Commutator function s Slow-Roll Inflation 1.0 Top yt 0.5 Light Higgs Λ 0.0 0 5 15 20 t = ln HΦmtL 10 25 30 35 Figure 4: This plot summarizes some of the most important effects of the renormalization group flow. The red curve shows the running of the quartic coupling λ(t)/λ(0) for a light Higgs mh = 126.5 GeV. The dotted purple curve is the top running yt (t)/yt (0) and the dot-dashed cyan curve is the commutator function s(t), with ξ = 2.3 × 103 and µ = mt . The right-hand region is the slow-roll inflationary regime; here λ rises (and so ns does too), as highlighted by the inset. given in the Appendix of [23]. For the sake of brevity, we do not reproduce the pole matching details here. We now consider the effective potential V (φ) including the effect of the non-minimal coupling to gravity ξφ2 R. The calculation is difficult to perform exactly. However, we can obtain approximate results for large ξ fairly simply. Following [12], one can heuristically identify a nonstandard commutator for φ as follows. From eqs. (4) and (5) we see that when the gravity sector 2 is canonical, the kinetic sector is non-canonical − 12 (∂E φ)2 (dσ/dφ) . On a spatial hypersurface, the canonical momentum corresponding to φ is 2 2 √ ∂L √ dσ dσ µν µν = −gE (gE nµ ∂ν φ) π= = −g (g nµ ∂ν φ) f (φ) , (25) dφ dφ ∂ φ̇ where nµ is a unit timelike vector. Imposing standard commutation relations for φ and π, we learn that [φ(x), φ̇(y)] = i ~ s(φ) δ (3) (x − y), with s(φ) = f −1 (φ) dσ dφ −2 1+ = ξφ2 m2Pl 2 ξφ 1 + (6ξ + 1) m 2 . (26) Pl For φ mPl /ξ (the √ low energy regime) we recover the ordinary value of the commutator s = 1, while for φ mPl / ξ (the inflationary regime) we see a suppression in the commutator by a factor of s = 1/(6ξ + 1). So in the inflationary regime with ξ 1, quantum loops involving the Higgs field are heavily suppressed. To summarize, our prescription for the renormalization group improved effective potential in the presence of non-minimal coupling is to assign one factor of s(φ) = s(µ et ) for every off-shell Higgs that runs in a quantum loop. This factor is plotted as the dot-dashed cyan curve in Fig. 4. 9 In eq. (23), for example, this prescription means the replacement 24 λ2 → 24 s2 λ2 , as that term arises from two Higgs off-shell propagators, while all other terms are untouched since they only involve other fields in loops (see Appendix A for more details). This provides an important modification to the high energy running of couplings, and explains why the running of ξ from the diagram of Fig. 3(a) is suppressed. Apart from this modification, the RG improved analysis is as standard, as summarized in eq. (22). We have checked our prescription against detailed analytical calculations of the effective action of non-minimally coupled scalars in the literature (e.g., see [24, 25]) and have found excellent agreement. We assume that quantum corrections from graviton exchange are small, see Appendix B. 5. Results and Predictions After numerically solving the set of 5 coupled renormalization group differential equations of Appendix A for the couplings: λ, yt , gs , g, and g 0 , we have obtained the effective potential V (φ) in the full quantum theory, as a function of input parameters, such as the Higgs mass. Some representative potentials in the Einstein frame are given in Fig. (2). The inset clearly exhibits variation of the effective potential VE (φ) with Higgs mass, which was absent in classical case. As we lower the Higgs mass, approaching the instability, the magnitude of the first derivative is raised and the that of the second is lowered (see the blue and red curves). This leads to modifications to the cosmological parameters. Following the recipe we outlined earlier in Section 2, we are able to efficiently compute the spectral index in the RG improved theory using Mathematica. Recall that in the classical theory, ns is independent of the parameters of the Standard Model, and its value was found to be ns ' 0.968 (for Ne = 60). In the quantum theory, we find that ns depends on several of the Standard Model parameters, in particular on the Higgs and top masses, see Fig. 1. As the top mass is varied through its experimentally allowed range (169 GeV . mt . 173 GeV) the spectral index varies noticeably. In particular, as we lower the Higgs mass towards vacuum instability, the spectral index increases substantially. To achieve successful inflation with ns < 0.99, we require αs (mZ ) − 0.1176 mt − 171 GeV − 1.4 GeV ± δ, (27) mh > 125.7 GeV + 3.8 GeV 2 GeV 0.0020 where δ ∼ 2 GeV indicates theoretical uncertainty from higher order corrections (such as 3-loop). This bound almost coincides with that from absolute stability presented in Ref. [23]. Note that near the boundary λ is small, so the corresponding ξ to obtain the observed ∆2R is reduced from its classical value ξ ∼ 104 by an order of magnitude or so to ξ ∼ 103 . Let us now trace the chain of logic behind the rise in ns . For a light Higgs, βλ is dominated by the top and gauge boson contributions. For a heavy top, the top contribution is dominant at low energies, causing βλ to be negative and thus driving λ to low values as the energy is increased. At the same time, the top Yukawa coupling runs, with dominant contributions coming from gauge fields and Higgs running in a loop, with the gauge √ fields slightly dominant causing yt to decrease with energy.7 At very high energies φ mPl / ξ (the inflationary regime), the Higgs running in the loop is highly suppressed, causing yt to jump to even lower values. Hence the 7 Note that the 2-loop term −108yt gs4 /(4π)4 in βyt (see eq. (34)) speeds up the running compared to 1-loop. 10 top contribution to the running of λ becomes subdominant, the gauge boson contributions now dominate and λ rises, as seen in Fig. 4 (inset). Since λ is concave up, this increases η and hence the spectral index. In Fig. 1 and in all plots we have chosen the reference value N0 = 60. For Ne close to N0 , we can Taylor expand ns to linear order: ns (Ne ) = ns (N0 ) + dns (Ne − N0 ) + . . . . dNe (28) Now, the spectral index is in fact a function of all the parameters, including Ne and ξ: ns = ns (Ne , ξ, . . .). As in the classical theory, we have fixed ξ such √ that the amplitude of density fluctuations is in agreement with observations (requiring ξ ∼ 104 λ). In this way, we can think of ξ = ξ(Ne ), so from the chain rule dns ∂ns ∂ns dξ = + . dNe ∂Ne ∂ξ dNe (29) The first term is precisely the (negative) of the running of the spectral index α = dns /d ln k, while the second term is found to be very small numerically. Hence to a good approximation we can write ns (Ne ) ≈ ns (N0 ) − α(N0 )(Ne − N0 ). (30) We plot the running of the spectral index α(N0 = 60) in Fig. 5 (left). We see that α ≈ −5 × 10−4 (as in the classical case), with some variation for low Higgs masses as we approach the instability. However, this is still far too small to be detected by PLANCK, which is expected to be only sensitive to α = O(10−2 ) [26]. Hence the main usefulness of Fig. 5 (left) is that it should be used in accompaniment with Fig. 1 and eq. (30) to infer the value of ns for different values Ne (as long as Ne does not vary too far from N0 = 60). The actual number of e-foldings of inflation is related to the wavenumber of interest k, the energy density during inflation VE , the energy density at the end of inflation Vend , and the energy density at the end of reheating ρreh [6] Ne ' 62 − ln k VE 1 VE Vend 1 1 + ln ln + ln − . a0 H0 4 (1016 GeV)4 4 Vend 12 ρreh (31) Since the Higgs is strongly coupled to Standard Model fields, reheating is expected to occur automatically. As Ne has only a weak dependence on ρreh , the details of reheating are rather inconsequential to our mass bounds, but may be calculable [27]. According to [28], Treh ∼ 1013.5 GeV giving Ne ' 59 for the classical theory. In our case, we must take into account the variation in the scale of inflation due to the quantum corrections. In Fig. 5 (right) we plot r versus the Higgs mass. Since we have fixed ξ such that the amplitude of density fluctuations is at the observed value, the energy density of inflation VE is simply proportional to r. Using eqs. (10) and (13), we have 4 r 3 . (32) VE = π 2 m4Pl ∆2R r ≈ 7.8 × 1015 GeV 2 0.003 Since r changes by a factor of order 2, as we vary the Higgs mass, then VE changes by the same amount. From eq. (31), rescaling VE and Vend by a factor of p, say, the number of e-foldings is shifted by ∆Ne = 61 log p, which is ≈ 0.1 for p = 2. Hence the variation in Ne with the Higgs mass is very small. 11 -5.2 mt = 173 GeV G eV 17 1 t= m -5.5 -5.6 4.5 3 GeV -5.4 5.0 mt=17 mt =169 GeV V 71 Ge -5.3 GeV Tensor to scalar ratio r H103 L 5.5 m t=1 169 m t= Running of spectral index Α H´ 104 L 6.0 4.0 3.5 3.0 122 124 126 128 130 132 Higgs mass mh HGeVL 134 136 122 124 126 128 130 132 Higgs mass mh HGeVL 134 136 Figure 5: The running of the spectral index α (×104 ) (left panel) and the tensor to scalar ratio r (×103 ) (right panel) as a function of the Higgs mass mh . The 3 solid curves correspond to 3 different values of the top mass: mt = 169 GeV (red curve), mt = 171 GeV (blue curve), and mt = 173 GeV (orange curve). The horizontal dashed green curve, with α ' −5.2 × 10−4 and r ' 3.0 × 10−3 , is the classical result. We have set αs = 0.1176 and Ne = 60 in this plot. 6. Discussion A number of papers have discussed bounds on the Higgs mass coming from demanding stability of the vacuum, e.g., see [22, 23, 29, 30]. Cosmological constraints only require metastability on the lifetime of the universe, which places the constraint mh & 105 GeV [23]. However, if we further demand that the Higgs drive inflation, we find that heavier Higgs are required: mh & 126 GeV (depending on the top mass, see eq. (27)), which essentially coincides with the bounds from absolute stability. Furthermore, by demanding that the theory remains perturbative to high energies (mh . 190 GeV), we establish a correlation between both stability and triviality bounds, and inflation. More precisely, we have established a mapping between the renormalization group flow and the cosmological spectral index. Over a substantial range of parameter space the classical value ns ' 0.968 (for Ne = 60) emerges as a good approximation, but there are corrections. Given a detailed microphysical theory, such as the Standard Model, we can explicitly calculate such corrections, as summarized in Fig. 1. This plot displays a sharp rise in the spectral index towards 0.98, or so, as we approach vacuum instability for a light Higgs. It is likely that the Standard Model is only the low-energy limit of a more complete theory, accommodating the facts that do not find explanation within the Standard Model, such as neutrino masses, dark matter, baryon asymmetry, etc. Our methodology is still applicable, so long as we can control the relevant β-functions. In principle some quite different scalar field, not connected to the Standard Model Higgs, could drive running inflation. The central requirement is a large coefficient for the φ2 R term. It is possible that such a coefficient could emerge as some sort of Clebsch-Gordan coefficient, or from the coherent addition of several smaller terms (involving more basic scalars φj ). It is also possible to consider, in the same spirit, the dimension 3 interaction φR, which arises for generic scalar fields, though not of course for the Standard Model Higgs. Furthermore, as discussed in 12 Appendix B and Refs. [35, 36], the inclusion of higher dimension operators may significantly affect the predictions of the original theory and even spoil its validity (as is the case in many inflationary models). These possibilities, and their possible embedding in unified field theory or string theory, deserve further investigation. If the Higgs boson exists and it is in the mass range considered in this Letter, the LHC will discover it and will determine the Higgs mass with a precision of about 0.1% [31], which means an uncertainty ∆mh ≈ 0.2 GeV. In order to extract accurate correlations between the inflationary observables and the Higgs mass it is crucial to improve the precision with which we know the other parameters of the Standard Model, in particular the top quark mass and the strong coupling. The current value of the top mass from direct observation of events is mt = (171.2 ± 2.1) GeV [20]. In the near future, the LHC will improve the determination of the top mass, but relatively large systematic uncertainties will prevent a top mass determination to better than 1 GeV; more conservatively, the top mass will be determined at LHC with an error ∆mt = 1 ÷ 2 GeV. Looking further ahead, the ILC is expected to be able to measure the top mass to ∼ 100 MeV. So together with the measured Higgs mass from the LHC and improved precision on the strong coupling, as well as calculating higher order effects and reheating details, running inflation in the Standard Model will predict a rather precise value for the spectral index. We would like to thank M. Amin, F. D’ Eramo, A. Guth, and C. Santana for useful discussions, and A. Riotto for comments on the manuscript. We thank F. L. Bezrukov, A. Magnin, and M. Shaposhnikov for correspondence. The work of ADS is supported in part by the INFN “Bruno Rossi” Fellowship. The work of ADS, MPH, and FW is supported in part by the U.S. Department of Energy (DoE) under contract No. DE-FG02-05ER41360. Note Added Our Letter appeared simultaneously on the arXiv with Ref. [32], which also studied the quantum corrections to inflation driven by the Standard Model Higgs. The central conclusion of both papers is that the classical analysis provides a good approximation over a wide range of parameters, but that quantum corrections are calculable and can be quantitatively significant. For a top mass of mt = 171.2 GeV, Ref. [32] found that in order to have successful inflation the Higgs mass is constrained to be in the range: 136.7 GeV < mh < 184.5 GeV, and the spectral index decreases from its classical value as mh approaches the lower boundary. In this note we briefly discuss the similarities and differences between their analysis and ours.8 In our analysis, we computed the full RG improved effective potential. We did this including (i) 2-loop beta functions, (ii) the effect of curvature in the RG equations (through the function s), (iii) wave-function renormalization, and (iv) accurate specification of the initial conditions through proper pole matching. On the other hand, [32] did not compute the full effective potential or include any of the items (i)–(iv).9 Instead Ref. [32] approximated the potential at leading log order with couplings evaluated at an inflationary scale after running them at 1-loop (this is one step beyond [18] where couplings were not run). 8 The discussion here refers to version 1 of [32]. wave-function renormalization was not included in [32], external leg corrections in the running of λ were included. However these two effects roughly cancel against one another. 9 Though 13 The lower bound on the Higgs mass we find in eq. (27) is about 11 GeV lower than that found in Ref. [32] (mh > 136.7 GeV). This numerical discrepancy is due to several of the above simplifications, but the dominant difference comes from (i) inclusion or not of 2-loop effects (importantly, the −108yt gs4 /(4π)4 term in βyt , see eq. (34)), and a second significant difference comes from (iv) pole matching. Higher order effects (such as 3-loop) and uncertainty in the strong coupling αs (mZ ) also modify the bound, as we summarized in eq. (27). A precise upper bound on the Higgs mass (mh < 184.5 GeV) is stated in [32]. The basis of this is the famous “triviality bound”, see e.g., Ref. [33], which has little to do with inflation. The theory ultimately requires a cutoff, and exactly how low a cutoff one feels comfortable with (or equivalently, how large a value of λ one regards as acceptable) is arguable. We feel that our stated semi-quantitative bound mh . 190 GeV adequately represents the situation. In [32] ns decreases as mh approaches its minimum value, while we find that ns increases (see Fig. 1). This behavior depends critically on the value of yt during inflation, as compared to the value of the gauge couplings. If yt is small, then ns increases, and vice versa. Ref. [32] overestimated yt during inflation and hence obtained the opposite behavior. This is primarily due to ignoring items (i) and (ii) above. Ignoring (i) misses the 2-loop term −108yt gs4 /(4π)4 in βyt , and ignoring (ii) maintains the 1-loop term 29 yt3 /(4π)2 in βyt during inflation. Finally, [32] computes quantum corrections with both a field-independent cutoff (as we use) and a field-dependent cutoff in the original “Jordan” frame. Either procedure defines a possible model, but the field-independent cutoff is more in the spirit of the motivating arguments, based on dimension ≤ 4 effective Lagrangians. A. 2-Loop RG Equations In this appendix we list the RG equations for the couplings λ, yt , g 0 , g, gs at energies above mt at 2-loop [34]. In each case, we write dλ/dt = βλ /(1 + γ), etc., where t = ln φ/µ. Also, we insert one factor of the commutator function s(µ et ) (see eq. (26)) for each off-shell Higgs propagator.10 For the Higgs quartic coupling we have 1 3 4 2 02 2 2 2 4 2 02 2 βλ = 2g + g + g 24s λ − 6y + + −9g − 3g + 12y λ t t (4π)2 8 " 02 1 8g 1 6 4 02 2 04 06 6 4 2 + 915g − 289g g − 559g g − 379g + 30sy − y + 32g + 3sλ t t s (4π)4 48 3 73 39 629 04 + λ − g 4 + g 2 g 02 + sg + 108s2 g 2 λ + 36s2 g 02 λ − 312s4 λ2 8 4 24 # 9 21 19 45 85 + yt2 − g 4 + g 2 g 02 − g 04 + λ g 2 + g 02 + 80gs2 − 144s2 λ . (33) 4 2 4 2 6 10 We have carefully extracted out all Higgs propagators contributions at 1-loop order by the appropriate insertion of factors of s. For the 2-loop contributions we have only inserted s for the obvious terms. The complete set of insertions are tedious and provide negligible corrections. 14 For the top Yukawa coupling we have " yt 9 2 17 02 9 2 yt 23 3 1187 04 2 βyt = − g − g − 8gs + syt + − g 4 − g 2 g 02 + g + 9g 2 gs2 (4π)2 4 12 2 (4π)4 4 4 216 # 19 02 2 225 131 + g gs − 108gs4 + g2 + g 02 + 36gs2 syt2 + 6 −2s2 yt4 − 2s3 yt2 λ + s2 λ2 (34) . 9 16 16 For the gauge couplings gi = {g 0 , g, gs } we have βgi = 3 X 1 1 g 3 bi + g3 Bij gj2 − sdti yt2 , (4π)2 i (4π)4 i j=1 (35) with b = ((40 + s)/6, −(20 − s)/6, −7), 199/18 9/2 44/3 35/6 12 , B = 3/2 11/6 9/2 −26 dt = (17/6, 3/2, 2). (36) Finally, the anomalous dimension of the Higgs field is 2 1 3g 02 9g 2 γ = − + − 3y t (4π)2 4 4 1 271 4 9 2 02 431 04 5 9 2 17 02 27 4 2 2 3 2 − g − g g − sg − g + g + 8g y + sy − 6s λ (37) . s t (4π)4 32 16 96 2 4 12 4 t B. Remarks on Running Inflation as an EFT The Lagrangian analyzed in this Letter is not renormalizable in the conventional sense, nor is it “technically natural” from the point of view of effective field theory. In this appendix we remark on the validity of such a theory at high energies (for related discussions see [35, 36]) and elaborate on the spirit of our calculations. The novelty of running inflation is to introduce the non-minimal coupling ξφ2 R into the low energy Lagrangian, which is allowed by all known symmetries of the Standard Model and gravity. This term is dimension 4 in the same sense that the kinetic term g µν ∂µ φ∂ν φ is also. However, if we expand around flat space gµν = ηµν + hµν /mPl then the new term is dimension 5 at leading order, plus an infinite tower of corrections ξφ2 R ∼ ξφ2 h/mPl + . . . , which is connected to the non-renormalizability of gravity in 4 dimensions. This suggests that non-minimally coupled theories becomes strongly interacting at scales Λ ∼ mPl /ξ. This can be compared to minimally coupled theories with Λ ∼ mPl . Without any protecting symmetry, we cannot forbid infinite towers of corrections to the dimension 4 effective Lagrangian L4 , including those of the form X φ n X φ n 2 4 + ξφ R bn + ..., L = L4 + λφ an Λ Λ n>0 n>0 15 which applies to non-minimal models (with Λ ∼ mPl /ξ) and minimal models (with Λ ∼ mPl ). The values of the higher order Wilson coefficients an , bn cannot be determined without knowledge of the behavior of gravity at energy scales above Λ, since these terms arise from graviton exchange. If we take a naive estimate an , bn = O(1), then the required flatness of the inflationary potential is jeopardized. This applies both to running inflation and to many minimal inflation models, such as m2 φ2 chaotic inflation, since in both cases φ > Λ during inflation. 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