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Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Higgs ξ-inflation for the 125–126 GeV Higgs (Based on arXiv:1306.6931) Kyle Allison University of Oxford Seminar, University of Sussex November 4, 2013 Conclusions Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Outline ¶ Inflation from a particle physics perspective · Higgs ξ-inflation: Tree level ¸ Radiative corrections ¹ Conclusions Radiative corrections Conclusions Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Part I Inflation from a particle physics perspective Conclusions Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions What is inflation? I A (supposed) period of accelerating expansion of the early universe I Before the radiation-dominated era, the scalar potential dominated the energy density of the universe ,→ space grows exponentially Proposed to explain I . Flatness problem . Horizon problem −→ I Produces a nearly scale-invariant spectrum of density fluctuations Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Achieving inflation I Let scalar field φ be the “inflaton” V (φ) Inflation φ(~x , t) = φ(t) + δφ(~x , t) I Need a slowly rolling field for inflation φ̇2 V (φ), I φ̈ 3H φ̇ Inflation ends In terms of the slow roll parameters 00 M2 V 0 2 V 2 ≡ Pl , η ≡ MPl , 2 V V Reheating φ(t) inflation occurs so long as < 1, I |η| < 1 ⇐⇒ Inflation After inflation, φ oscillates about the minimum and its kinetic energy is converted into SM particles ←− Reheating Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Matching observations I The amount of inflation described by N = number of times space expands by a factor of e (e-folds) Z tf N= ti I 1 Hdt = MPl Z φi φf dφ √ 2 Precise constraints come from measuring primordial density fluctuations via the cosmic microwave background (CMB) δφ(x) δgµν δTµν δT δρ Quantum fluctuations exist at all scales during inflation and produce a nearly scale-invariant spectrum Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Matching observations I The predicted temperature anisotropies from inflation are δT 2 16π V (φ` ) field value when scale ` leaves = , φ = ` 4 (φ ) horizon during inflation T ` 45MPl ` I Measurement of δT /T for ` ' 3000 Mpc gives V (φ∗ ) field value N∗ ' 50–60 e-folds 4 ' 5.6 × 10−7 MPl , φ∗ = before end of inflation (φ∗ ) I Two other parameters are particularly useful for constraining inflationary models ns = 1 − 6 + 2η r = 16 Spectral index, deviation from scale-invariance Tensor-to-scalar ratio, size of gµν perturbations As above, ns (φ) and r (φ) are evaluated at φ∗ Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Matching observations I Many inflationary models have been constructed (Planck collaboration, 2013) Conclusions Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Candidates for the inflaton I I Identity of inflaton still unknown; often assumed to be a new weakly-coupled scalar field, but . . . . . . we have just discovered our first fundamental scalar field Can the Higgs boson be the inflaton? . . . . . h4 chaotic inflation (Linde, 1983) ←− experimentally disfavoured 7 Quasi-flat SM potential (Isidori et al., 2008) ←− too few e-folds 7 False vacuum inflation (Masina & Notari, 2012) ←− needs second scalar 7 New Higgs inflation (Germani & Kehagias, 2010) ←− new scale M < MPl ? Higgs ξ-inflation (Bezrukov & Shaposhnikov, 2008) ←− unitarity issues ? Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Part II Higgs ξ-inflation: Tree level Conclusions Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Model definition I I I Higgs ξ-inflation is based on a non-minimal coupling of the Higgs doublet to the Ricci scalar M2 L = − Pl R − ξH † HR + LSM 2 This is the only local, gauge-invariant interaction with dimension ≤ 4 For computing tree-level predictions, use unitary gauge H = √12 h0 ,→ Jordan frame action Z 2 MPl λ 4 ξh2 4 √ 2 SJ = d x −g − 1 + 2 R + (∂µ h) − h 2 4 MPl Slow-roll calculations require a minimal gravity sector ,→ Conformal transformation ξh2 gµν → g̃µν = Ω2 gµν , Ω2 = 1 + 2 MPl Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Model definition I The resulting Einstein frame action is Z SE = d 4x 2 Ω2 = 1 + ξh2 /MPl 2 p M2 1 Ω2 + 6ξ 2 h2 /MPl λh4 2 −g̃ − Pl R̃ + (∂ h) − µ 2 2 Ω4 4Ω4 I Define a new scalar field χ with canonical kinetic term s 2 Ω2 + 6ξ 2 h2 /MPl dχ = dh Ω4 I The Einstein frame action is then Z 2 p MPl 1 λ[h(χ)]4 4 2 SE = d x −g̃ − R̃ + (∂µ χ) − U(χ) , U(χ) = 2 2 4Ω4 √ U(χ) flattens for h(χ) & MPl / ξ ←− Inflationary region I Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Model predictions I Can perform slow-roll calculations with U(χ) 2 0 2 MPl 4M 4 U = ' 2 Pl4 2 U 3ξ h 4 00 4MPl ξh2 2 U η = MPl ' 2 4 1− 2 U 3ξ h MPl I √ Inflation ends at hend ' 1.07MPl / ξ when ' 1 I √ Working backwards, N = 59 e-folds produced at h∗ ' 9.14MPl / ξ I The spectral index and tensor-to-scalar ratio for Higgs ξ-inflation are ns (χ∗ ) ' 0.967, (Bezrukov & Shaposhnikov, 2008) r (χ∗ ) ' 0.0031 Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Model predictions I Can perform slow-roll calculations with U(χ) 2 0 2 MPl 4M 4 U = ' 2 Pl4 2 U 3ξ h 4 00 4MPl ξh2 2 U η = MPl ' 2 4 1− 2 U 3ξ h MPl I √ Inflation ends at hend ' 1.07MPl / ξ when ' 1 I √ Working backwards, N = 59 e-folds produced at h∗ ' 9.14MPl / ξ I The spectral index and tensor-to-scalar ratio for Higgs ξ-inflation are ns (χ∗ ) ' 0.967, (Planck collaboration, 2013) r (χ∗ ) ' 0.0031 Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Perturbative unitarity breakdown I Generating the observed primordial density fluctuations requires √ U(χ∗ ) 4 ' 5.6 × 10−7 MPl =⇒ ξ ' 48000 λ ' 17000 (χ∗ ) I Such a large value ξ ∼ 104 creates a problem: √ (1) Perturbative unitarity breaks down at MPl /ξ MPl / ξ p 2 + ξh2 ξ MPl ξ 2 2 ξh R −→ 2 ĥ2 ĝ ' ĥ ĝ for h ' 0 2 2 2 MPl MPl + ξh + 6ξ h (2) New physics entering at Λ = MPl /ξ is naively expected to affect the potential in an uncontrollable way (3) Self-consistency of Higgs ξ-inflation is questionable I Proponents argue that the scale of perturbative unitarity breakdown depends on h (it is larger during inflation) and so does not spoil the inflationary predictions ,→ Assumes scale of new physics is background field-dependent Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Perturbative unitarity breakdown I For Mh ' 125–126 GeV, λ can run to very small values near the Planck scale (where inflation occurs) I How does this affect the predictions for Higgs ξ-inflation? We expect √ ξ ' 48000 λ 17000 Can this address the perturbative unitarity breakdown problem? I Actually, Mt must be about 2–3σ below its central value for Higgs ξ-inflation to be possible (Degrassi et al., 2012) . Glass half empty: Model disfavoured at 2–3σ . Glass half full: Special region within 2–3σ of measurements I Need to go beyond the tree level Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Part III Radiative corrections Conclusions Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Renormalization group equations I The most important higher-loop effect is the running of λ I Need the RG equations for Higgs ξ-inflation 2 p ∂LE dχ Jordan πh = g̃ 0ν ∂˜ν h −−−−→ = −g̃ dh ∂ ḣ Ω 2√ −g dχ dh Canonical commutation relation gives [h, πh ] = i~δ 3 (~x − ~y ) where s(h) = I 2 1+ξh2 /MPl 2 1+(1+6ξ)ξh2 /MPl =⇒ ' The physical Higgs propagators are suppressed during inflation, but not those of the NambuGoldstone bosons 1 1+6ξ [h, ḣ] = s(h)i~δ 3 (~x − ~y ) for large h s t h t̄ h 2 ḣ Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Renormalization group equations I Two slightly different ways of dealing with the suppressed Higgs propagators (1) Insert a factor s for each off-shell Higgs in the RG equations of the SM (De Simone, Hertzberg & Wilczek, 2009) (2) View the effect as a suppression of the Higgs coupling to SM fields. For ξ 1, use the RG equations from the chiral electroweak theory (Bezrukov & Shaposhnikov, 2009) I Have tried to reconcile the differences, but have been unable to reproduce the results from method (2) using Feynman diagrams I For this analysis, use the two-loop RG equations derived using method (1) with leading three-loop corrections to λ, yt and γ I Note the running of ξ is given by 1 βm2 , βξ = ξ + 6 m2 ξ(MPl /ξ0 ) = ξ0 Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Effective potential I The SM effective potential must also be modified by the suppressed Higgs propagators ←− result seems to be frame-dependent Renormalization prescription II (Jordan frame) I Perturb the tree-level SM potential about the background value, compute the masses of the perturbations, then transform to the Einstein frame Mh2 = 3sλh2 , MG2 = λh2 , 2 MW = g 2 h2 , 4 MZ2 = (g 2 + g 02 )h2 , ... 4 The higher-loop corrections take the usual Coleman-Weinberg form with these modified particle masses 4 Mh λh4 1 Mh2 3 3MG4 MG2 3 + ln − + ln − + . . . U(χ) = 4Ω4 16π 2 4Ω4 µ2 2 4Ω4 µ2 2 I Choose the renormalization scale µ = h to minimize the log terms Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Effective potential I The Einstein frame prescription is similar but we perform the conformal transformation before computing the particle masses Renormalization prescription I (Einstein frame) I Transform the tree-level SM potential to the Einstein frame, perturb it about the background value, and compute the masses of the perturbations ξh2 2 4 2 1 − 2 λh 3sλh g 2 h2 MPl 2 , MG2 = λh , MW U0 = =⇒ Mh2 = = ,... 2 4 4 4 ξh 4Ω Ω Ω 4Ω2 1+ 2 MPl 4 1 Mh2 3 3MG4 MG2 3 λh4 Mh U(χ) = + ln 2 − + ln 2 − + ... 4Ω4 16π 2 4 µ 2 4 µ 2 I There is an additional suppression of Mh2 and MG2 in this prescription I Choose the renormalization scale µ = h/Ω to minimize the log terms Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Effective potential I The additional suppression of Mh2 and MG2 makes little numerical difference to the effective potential since the contributions of these masses are already small (λ 1) I The most important difference between the Einstein and Jordan frame renormalization prescriptions is the functional dependence µ(h) ( √ h 2 2 Prescription I (Einstein frame) 1+ξh /MPl µ= h Prescription II (Jordan frame) Can have a large impact on the effective potential during inflation! I For this analysis, consider both prescriptions and use the two-loop effective potential with the appropriately modified particle masses I Moreover, define the effective Higgs self-coupling λeff (µ) through U(χ) ≡ λeff (µ)h4 4Ω4 Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Two-loop inflationary predictions Use the RG equations to run the initial values of Mh , Mt , αs , . . . up to the inflationary scale and use the two-loop effective potential I To explore λeff (µ) 1, replace Mt with λmin eff ≡ min {λeff (µ)} ,→ fine-tuning 0.02 For a fixed Mh , λmin and α , M in GeV s eff I Minimum effective Higgs quartic coupling Λmin eff I h 0.01 (1) Choose ξ0 . Adjust Mt to give the desired λmin eff 0.00 (2) Determine U/ at N = 59 e-folds before the end of Α HM L=0.1184 -0.01 Ξ =1000 inflation (3) Repeat the steps above -0.02 170 171 172 173 until the correct U/ Top quark mass M in GeV normalization is obtained (4) Compute the predictions for ns and r s Z 0 t I 124 125 126 127 The numerical results are presented as a function of λmin eff 174 175 Inflation from a particle physics perspective Higgs ξ-inflation: Tree level I I Mh in GeV s -5 -4.5 -4 -3.5 Z -3 -2.5 -2 min eff 0.971 0.970 0.969 0.968 0.967 Mh in GeV 124 125 126 127 Αs HMZ L=0.1184 0.966 10-5 10-4.5 10-4 10-3.5 10-3 10-2.5 10-2 Minimum effective Higgs quartic coupling Λmin eff Tensor-to-scalar ratio r I 5000 Conclusions 124 Non-minimal coupling can be as 2000 125 −4.4 126 small as ξ ∼ 400 for λmin ∼ 10 eff 1000 127 ,→ still too large to address the 500 Α HM L=0.1184 perturbative unitarity problem 200 min −4.4 10 10 10 10 10 10 10 Need larger ξ for λeff < 10 Minimum effective Higgs quartic coupling Λ Eventually the effective potential develops a second minimum that spoils Higgs ξ-inflation Predictions for ns and r remain within the 1σ region Spectral index ns I Non-minimal coupling Ξ0 Results for prescription I Radiative corrections 0.0035 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005 Mh in GeV 124 125 126 127 Αs HMZ L=0.1184 0.0000 10-5 10-4.5 10-4 10-3.5 10-3 10-2.5 10-2 Minimum effective Higgs quartic coupling Λmin eff Inflation from a particle physics perspective Higgs ξ-inflation: Tree level I I Non-minimal coupling Ξ0 Results for prescription II Radiative corrections 5500 5000 4500 4000 3500 Mh in GeV Two regions for prescription II: 124 125 large and small λmin regions 126 eff 127 3000 min The large λeff region behaves 2500 similarly to prescription I but the Α HM L=0.1184 2000 10 10 10 10 running of λeff (µ) more important Minimum effective Higgs quartic coupling Λ Can only have ξ as small as about ξ ∼ 2000–4000 before the potential develops a second minimum Predictions for ns and r show more variation, still within 1σ s -3.5 0.98 Tensor-to-scalar ratio r I -3 Z -2.5 -2 min eff Spectral index ns I Conclusions 0.97 0.96 Mh in GeV 124 125 126 127 0.95 0.94 0.93 0.92 10-3.5 Αs HMZ L=0.1184 10 -3 -2.5 10 Minimum effective Higgs quartic coupling -2 10 Λmin eff 0.004 0.003 0.002 Mh in GeV 124 125 126 127 Αs HMZ L=0.1184 0.001 0.000 10-3.5 -3 10 10-2.5 10-2 Minimum effective Higgs quartic coupling Λmin eff Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Results for prescription II I The small λmin eff region is very different from the other results 0.35 0.35 Tensor-to-scalar ratio Tensor-to-scalar ratio r r 0.30 0.30 0.25 0.25 0.20 0.20 0.15 0.15 0.10 0.10 H10-4.10 , 58L H10-4.15 , 75L H10-4.10 , 58L -4.10 H10-4.15H10 , 75L , 76L H10-4.10 , 76L H10-4.05 , 78L H10-4.05 , 78L -4.0 H10 , 80L H10-4.0 , 80L H10-3.95 , 84L Planck+WP+BAO: LCDM+r H10-3.95 , 84L Planck+WP+BAO: LCDM+r Planck+WP+BAO: LCDM+r+Neff Planck+WP+BAO: LCDM+r+Neff Planck+WP+BAO: LCDM+r+YHe Planck+WP+BAO: LCDM+r+YHe H10-4.05 , 63L H10-4.05 , 63L H10-3.9 , 90L H10-3.9 , 90L Mh in GeV Mh in GeV 124 124 124.5 124.5 H10-3.875 , 96L H10-3.875 , 96L 0.05 0.05 0.00 0.00 I 0.94 0.94 0.96 0.96 0.98 0.98 Spectral index ns Spectral index ns 1.00 1.00 1.02 1.02 Allows ξ ∼ 90 at 2–3σ with an observable level of r , but there is still a perturbative unitarity problem Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Part IV Conclusions Radiative corrections Conclusions Inflation from a particle physics perspective Higgs ξ-inflation: Tree level Radiative corrections Conclusions Conclusions I Higgs ξ-inflation is one of the few remaining inflationary models that does not require scalar fields in addition to those in the SM I The breakdown of perturbative unitarity at MPl /ξ (below the scale of inflation) has long been a potential problem for this model I We have investigated whether the recently measured Higgs mass, for which λeff (µ) 1 near the Planck scale, can address this problem µ Prescription I Prescription II I h 2 1+ξh2 /MPl √ h λmin eff ξ ns r −4.6 & 400 0.97 . 0.003 −3.3 & 2000 ∼ 90 0.96–0.97 0.97–1.00 . 0.003 0.15–0.25 & 10 & 10 ∼ 10−3.9 The perturbative unitarity problem remains but small λmin eff allows a new region of Higgs ξ-inflation with observable tensor-to-scalar ratio Thank you for your attention!