Higgs inflation: dark matter, detectability and unitarity

Rose Lerner University of Helsinki and Helsinki Institute of Physics In collaboration with John McDonald (Lancaster University)

0909.0520 (Phys. Rev. D 80 (2009) 123507)

0912.5463 (JCAP 1004:015,2010)

1005.2978 (Phys. Rev. D82 (2010) 103525)

1103.xxxx (to appear) Outline

1. Higgs inflation: basic ideas

2. S inflation and observational predictions

3. Unitarity and the scale ?

4. A new unitarity-conserving model

5. Conclusions Higgs inflation  [Early] D. S. Salopek, J. R. Bond and J. M. Bardeen, Phys. Rev. D 40, 1753 (1989)  F. L. Bezrukov and M. Shaposhnikov, Phys. Lett. B 659 (2008) 703, arXiv:0710.3755 [hep-th] and arXiv:0904.1537 [hep-ph]  F. Bezrukov, D. Gorbunov and M. Shaposhnikov, arXiv:0812.3622 [hep-ph]  F. L. Bezrukov, A. Magnin and M. Shaposhnikov, arXiv:0812.4950 [hep-ph]  A. O. Barvinsky, A. Y. Kamenshchik and A. A. Starobinsky, JCAP 0811 (2008) 021 [arXiv:0809.2104 [hep-ph]]  A. O. Barvinsky, A. Y. Kamenshchik, C. Kiefer, A. A. Starobinsky and C. Steinwachs, arXiv:0904.1698 [hep-ph]  A.O. Barvinsky, A.Yu. Kamenshchik, C. Kiefer, A.A. Starobinsky, C.F. Steinwachs, arXiv:0910.1041 [hep-ph]  A.O. Barvinsky, A.Y. Kamenshchik, C. Kiefer, C. F. Steinwachs, Phys.Rev.D81:043530,2010., arXiv:0911.1408 [hep-th]  A. De Simone, M. P. Hertzberg and F. Wilczek, arXiv:0812.4946 [hep-ph]  [Reheating] J. Garcia-Bellido, D. G. Figueroa and J. Rubio, arXiv:0812.4624 [hep-ph]  S. C. Park and S. Yamaguchi, JCAP 0808 (2008) 009 [arXiv:0801.1722 [hep-ph]]  [With DM] T. E. Clark, B. Liu, S. T. Love and T. terVeldhuis, arXiv:0906.5595 [hep-ph]  [Naturalness] J. L. F. Barbon and J. R. Espinosa, Phys. Rev.D79 (2009) 081302, arXiv:0903.0355 [hep-ph].  [Naturalness] C. P. Burgess, H. M. Lee, and M. Trott, JHEP 09 (2009) 103, arXiv:0902.4465 [hep-ph].  [Naturalness] M. P. Hertzberg, arXiv:1002.2995  [Naturalness] C. P. Burgess, H. M. Lee, and M. Trott, JHEP 07 (2010) 007, arXiv:1002.2730 [hep-ph]  [SUSY] M. B. Einhorn and D. R. T. Jones, arXiv:0912.2718 [hep-ph]  [SUSY] S. Ferrara, R. Kallosh, A. Linde, A. Marrani, and A. Van Proeyen, Phys. Rev. D82 (2010) 045003, arXiv:1004.0712  [SUSY] S. Ferrara, R. Kallosh, A. Linde, A. Marrani, and A. Van Proeyen, arXiv:1008.2942 [hep-th]  AND MORE!! Jordan frame action Standard model Higgs doublet

required by renormalisation in curved space-time [Bezrukov & Shaposhnikov (2008)] Inflation  If ξ ~ 0 then require λ to be ~10-13 not Higgs boson!  With λ ~ 0.1, effective Planck mass proportional to h2 other masses proportional to h2 → approximately scale invariant action Curvature perturbation spectrum requires ξ ~104  Consideration of radiative corrections essential… Radiative corrections  Renormalisation group improved effective action:

 All couplings run:

 suppression of scalar propagator due to scalar-graviton mixing: In the Einstein frame...

Slow roll inflation easily formulated with minimal gravitational term Canonical normalisation  unitary gauge:

 canonically normalise with

 potential:

“asymptotic shift symmetry of action” Inflation

Figure: Bezrukov Tree-level results:

 Radiative corrections incorporated in full calculation Reheating in Higgs inflation  Calculation is in principle possible, as all couplings to the SM are known... but extremely difficult to calculate  Bezrukov, Gorbunov & Shaposhnikov [JCAP 0906 (2009) 029]  mechanism is parametric resonance to gauge bosons (ignores backreaction) 13 14  very approximate calculation; TR ~ 10 – 10 GeV  Garcia-Bellido, Figueroa & Rubio [Phys. Rev. D79 (2009) 063531]  mechanism is combined preheating (perturbative decay and non- perturbative decay via parametric resonance) and backreaction is important  a more careful calculation, but conclude that further numerical work needed S-inflation  Add a WIMP dark matter candidate to Higgs inflation  S is a stable, real gauge singlet scalar which is thermal relic cold dark matter.

[R.L. & McDonald (2009); see also Clark, Liu, Love & terVeldhuis (2009)] Reheating in S-inflation  S is stable so cannot decay... but reheating can still occur. Following B+S, we find reheating is similar to Higgs inflation:  Higgs bosons produced non-adiabatically at zero crossings of inflaton oscillation  Initially, Higgs bosons decay between zero crossings (negligible energy transfer)  Decreased amplitude of inflaton oscillation eventually prevents Higgs decay  A parametric resonance develops and energy rapidly transferred from inflaton to Higgs – which then annihilate to relativistic particles Reheating  S-inflation:

 Higgs inflation:

[Bezrukov & Shaposhnikov, 2008]  So (tree-level) spectral index apparently reasonably well determined: 0.965 < n < 0.967

However...  These are extremely rough estimates  More careful work by Garcia-Bellido, Figueroa & Rubio gave an inconclusive answer Spectral index predictions

Real singlet S Tensor to scalar ratio predictions

Real singlet S Origin of different predictions?  n dominated by tree-level η:

 But determined by ε, through (fixed N):

 Larger ε → larger → larger n  Dominant radiative corrections to ε are of different sign for S-inflation and Higgs inflation .  this is because running of ξ is negligible for S-inflation and very important for Higgs inflation. BUT...

Does the theory become unnatural at the scale ? Naturalness and unitarity

∑(probabilities) ≠ 1

Theory Calculation incomplete incomplete

Don’t use Add new terms perturbative methods

Which calculations Fine tuning? require strong coupling? Scales in the theory Two possible problems...

Non-canonical kinetic term Non-polynomial potential

Should see unitarity violation in both Jordan frame and in Einstein frame (equivalence theorem: scattering matrix is invariant under non-linear local field redefinitions) Einstein frame kinetic sector  Unitarity violation at a scale is determined by a tree- level calculation of scattering of non-identical scalars - via graviton exchange (Jordan frame) or - via non-renormalisable interaction (Einstein frame):

 Therefore, new terms such as should be added to the Lagrangian, requiring fine-tuning and thus spoiling the naturalness of the theory  [Pure singlet case does not violate unitarity in kinetic sector (as was once suggested) because a cancellation between s, t and u diagrams occurs – see Hertzberg 2010.] Einstein frame potential sector

Einstein frame:

 naive expansion → unitarity violation at  but expansion only valid for  and we know potential is tending to non-interacting at large field values → perhaps expansion is wrong? Potential sector is different  potential is flat at large field values  scale of unitarity violation depends on the background field  expanding potential around gives (canonically normalised)

 so violates unitarity at energy

can be arbitrarily large

 kinetic unitarity violation independent of background v  → potential sector has a different character Strong coupling

 The theory is not perturbative at the scale Λ = Mp/ξ [Han and Willenbrok 2005]  Specifically, the imaginary part of the one-loop partial wave is half of the tree-level partial wave at scale Λ  ... but the calculation of unitarity violation is a tree-level calculation!  Also, shown by H+W that a theory can violate unitarity at tree level but NOT when summed to all orders – perhaps this is happening here?  Alternative is strong coupling regime at scale Λ.  Implies that can’t calculate scattering amplitudes BUT inflation analysis unchanged. Scale-dependent cut-off Bezrukov and Shaposhnikov 2010  Calculate power-counting cut-off (lower limit) in singlet model  expanding around background field and metric: now: Λ ~ Mp/ξ 2 reheating: Λ ~ ξφ /Mp inflation: Λ ~ ξ1/2φ  Similar in both Jordan and Einstein frames

 BUT the lowest cut-off determines the new terms we need to add to the action!  (If strong coupling instead, then ok) Asymptotic safety Atkins and Calmet 2010  Weinberg’s asymptotically safe gravity: gravitational coupling → non-trivial UV fixed point  Planck mass runs  non-minimal coupling is ξ zero at fixed point

 so Mp increases and ξ decreases → Λ = Mp/ξ increases  such a fixed point may exist for the theory  if so, then no new degrees of freedom are required for a consistent theory Does the theory become unnatural at the scale ?

Maybe! But needs further (probably non- perturbative) analysis to find out. Using derivative couplings of the Higgs to the Einstein tensor Linearising Higgs kinetic term with an additional field

If “yes”, how can the theory be unitarised?

Adding terms to the theory to cancel the problem terms Derivative couplings Germani & Kehagias 2010  Idea: derivative coupling of Higgs to Einstein tensor

 Problem: interaction term [Atkins & Calmet 2011] gives (from expanding around and )  but WMAP results give , i.e. unitarity problems! (original paper only checked that ) Linearising kinetic term Giudice & Lee 2010  Idea: add real field σ with constraint σ2 = Λ 2 + φ 2  Non-minimal coupling of Higgs negligible  Tree-level parameters n,r identical to original model  Radiative corrections depend on unknown σ couplings

 Issues: if ξ negligible, is this really Higgs inflation? Minimal modifications necessary to conserve unitarity R.L. & McDonald 2010 Einstein frame action:

Jordan frame action: Observational predictions

 Unitarity-conserving model: ~ 0.975

 Original model: ~ 0.966 for N = 60  Radiative corrections not suppressed for Higgs case → need singlet or SUSY version

Wait for results of Planck! Conclusions  Large non-minimal coupling to gravity → scalar with O(1) coupling can be the inflaton  Spectral index ~ 0.966 (N=60)  Higgs-inflation: n < 0.966  S-inflation: n > 0.966  (Probably) distinguishable by means of Planck, direct detection DM experiments and LHC.  Theory may be strongly coupled at scale  OR new physics necessary – a minimal implementation gives n ~ 1 – 3/(2N) ~ 0.975 (N=60)

0909.0520, 0912.5463, 1005.2978, and 1103.xxxx