Can a pNGB Higgs lead to symmetry Can a pNGB Higgs lead to symmetry non-restoration? non-restoration? (w/ Can Kilic) (w/ Can Kilic) Sivaramakrishnan Swaminathan Sivaramakrishnan Swaminathan arXiv:1508.05121 arXiv:1508.05121 20 October 2015 (Many-body theory) 20 October 2015 (Many-body theory)
Motivations Motivations What happens to electroweak symmetry at high temperatures? Baryogenesis (How was matter created?) Heuristic thermodynamic argument hints at symmetry restoration Counter examples (Rochelle salts, �nite chemical potential, etc.) F = E − TS
The idea (a la Landau-Ginzburg) The idea (a la Landau-Ginzburg) Let the order parameter be a �eld Free-energy cost of homogeneous �eld con�gurations Include quantum effects ("integrate out" all �uctuations) Think "bubbles" ( ) Position of minimum corresponds to particle density in the condensate F = −β log Z
Integrating out the �uctuations Integrating out the �uctuations Cost of zero-point energies of �eld �uctuations around a putative background �eld value: harmonic oscillator for each mode VCW (H) VCW = STr ∫ [ ] d3k⃗ 1 2 + k2 m2 − − − − − − − √ = STr [ (H) + (log ( ) − )] m2 Λ2 16π2 (H) m4 64π2 (H) m2 Λ2 3 2 At one-loop, the cost of one-�uctuation depends only on the background, and is independent of other �uctuations "Integrate out �uctuations" : up to what scale?
At �nite temperature At �nite temperature For each �uctuation: Modify spectral function by adding thermal occupations (ϕ, T) = (ϕ) + Δ (ϕ, T) Veff, 1-loop VCW V T 1 Δ = ( ) V T 1 ∑ i T 4 2π2 ni Jb/f m2 i T 2 ( ) Jb xi ( ) Jf xi = dt log [1 − ] ∫ ∞ 0 t2 e− + xi t2 √ = dt log [1 + ] ∫ ∞ 0 t2 e− + xi t2 √
High temperature expansion High temperature expansion (x) Jb − (x) Jf = = − π4 45 − 7π4 360 + x π2 12 + x π2 24 − πx 3 2 6 − log ( ) x2 32 x ab + log ( ) x2 32 x af + … + … Note the correspondence with the zero temperature contribution bosons: fermions: − Λ2 16π2 Λ2 16π2 ⟶ T 2 12 ⟶ . T 2 24 Both fermions and bosons contribute to with the same sign! T 2
The hierarchy problem The hierarchy problem Mass term: Self energy of background �eld, at zero-external momentum Mass term, and hence VEV depends sensitively on the UV-cutoff! Supersymmetry models try to exploit the opposite sign between fermion (quark) and scalar ("squark") corrections above.
SU(4)/SU(3) SU(4)/SU(3) NLSM (2) NLSM (2) H [ ] = ≈ [ ] − → − − − − − − − − − − − − − − − − − Exploit [ ] SU(2) A SU(2) B |⟨ ⟩| HA |⟨ ⟩| HB ⎡ ⎣ ⎢ f sin ( ) v f f cos ( ) v f ⎤ ⎦ ⎥ v f − v2 2f What happens to ? 3 : eaten by gauge bosons 3 : eaten by gauge bosons 1 : heavy radial �uctuation 1 : "magnitude" of the SM Higgs H R SU(2) A R SU(2) B R R
A caricature model A caricature model Twin image of SM (A B) Forget the gauge bosons for now (Set ) Focus on the top quark sector (forget all other fermions) ⟷ = 0 gi Top sector Top sector = y [ (ϵ ) + (ϵ ) ] + h.c. Lup-type yukawa Q ¯ A HA UA Q ¯ B HB UB Yukawas break but have twin symmetry SU(4) (A ↔ B)
Insights Insights The leading dependence cancels away ( just like ) Subleading temperature dependence ( ) restores symmetry Those corrections are unavoidable, since terms are what give the Higgs a �nite mass ( ), in such models T T 2 Λ2 log T log Λ 125GeV
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