Finite temperature symmetry (non)restoration Thermal mass restores symmetry (eg: magnets) Λ2 divergence at T = 0 −→ T2 correction at T = 0 So, naturalness =⇒ symmetry non-restoration? Not true for boson-fermion cancellations How about same-spin partners? “Symmetry non-restoration at high-temperatures, in Little-Higgs models” – Phys.Rev.D72 (2005) 043520

FTFT for EWPT is hard “It should be emphasized, however, that the application of perturbation theory to EWPT physics is fraught with uncertainties as well as the potential for ambiguities.” – New J.Phys. 14 (2012) 125003 Perturbation theory is not expected to be well behaved IR divergences, resummations; high-temperature approximations Lattice methods are the best option (3d EFT)

FTFT for EWPT is hard, but. . . “At the same time, perturbation theory is the only feasible approach for surveying a broad range of BSM scenarios [...] One can then perform Monte Carlo computations that focus on a more limited range of parameters within a given model.” – New J.Phys. 14 (2012) 125003

Our goal Investigate ﬁnite temperature behaviour Simple model UV completable (up to naturalness issues) Same-spin partners Phenomenologically motivated Does a theory with same spin partners have symmetry non-restoration at high temperature?

A ﬂagship problem Quadratic divergent contributions to Higgs mass 3×(−4)×(y2 t /2)Λ2 16π2 3×3×(g2/2)Λ2 4×16π2 × 2 (6×(λ/4)+3×2(λ/4)) 16π2 × 2 1% tuning 4% tuning 6% tuning So, plausibly Λ not too large (eg: 5TeV above) Bottom up: New physics around the corner. . . LHC?! There are also other reasons to expect new physics at TeV scale

Things haven’t quite gone to plan No real BSM hints at the LHC yet But.. how?! LHC was supposed to generically produce anything and everything coloured, up to few 100 GeV! (eg: top partners)

Things haven’t quite gone to plan No real BSM hints at the LHC yet But.. how?! LHC was supposed to generically produce anything and everything coloured, up to few 100 GeV! (eg: top partners) Could we reconcile with naturalness, at least at one-loop?

Our caricature model Toy model motivated by the same idea. Twin image of SM (A←→B) Focus on the top sector (forget all other fermions) Forget the gauge bosons for now (Set gi = 0)

Ingredients: SU(4)/SU(3) NLSM (2) Parametrize NLSM in terms of NGBs H = HA HB = if h √ h† h sin √ h† h f if h √ h † h cos √ h† h f where SM Higgs is h = h1 h2 (SU(2)A doublet) SU(4) symmetry prevents Λ2 contributions to m2 h from the Goldstones!

Ingredients: SU(4)/SU(3) NLSM (3) Exploit SU(2)A SU(2)B H SU(2)A×SU(2)B − − − − − − − − − − → | HA | | HB | = f sin v f f cos v f ≈ v f − v2 2f HB HA vEW v f

Ingredients: Top sector (1) [SU(3) × SU(2)]A × [SU(3) × SU(2)]B global symmetry SM quarks: QA (3A , 2A ) and UA (3A , 1A ) Twin partners: QB (3B , 2B ) and UB (3B , 1B ) Only up-type quark interacts with vev Lup-type yukawa = y ¯ QA( HA)UA + ¯ QB( HB)UB + h.c. Yukawas break SU(4) but have twin symmetry (A ↔ B)

Twin mechanism HA H† A HB H† B h h y y h h ×yf − y 2f + h h × yf − y 2f Discrete Z2 symmetry causes quadratic terms to mimic SU(4) symmetry (which is otherwise broken) − 3y2Λ2 8π2 H† A HA + H† B HB = − 3y2 8π2 Λ2|H|2

The eﬀective potential, at one-loop Eﬀective (free) energy after integrating out ﬂuctuations treating ﬂuctuations as free modes (at one-loop level) Zero-point energy cost VCW = i (−1)Fi ni d3k 1 2 k2 + m2

The eﬀective potential, at one-loop Eﬀective (free) energy after integrating out ﬂuctuations treating ﬂuctuations as free modes (at one-loop level) Zero-point energy cost VCW = i (−1)Fi ni d3k 1 2 k2 + m2 For studying phase, can drop ﬁeld-independent terms

The eﬀective potential, at one-loop (2) VCW ≡ V T=0 1 = dofs: i (−1)Fi ni 64π2 m4 i log m2 i µ2 ∗ − 3 2 m2 tA = y2f 2 sin2 v f and m2 tB = y2f 2 cos2 v f Goldstones cannot conspire to give a potential for one amongst themselves m2 h ∼ 3y2 8π2 m2 tB log Λ2 m2 tB ∼ f π 2 =⇒ f ∼ 500GeV

µH† A HA for asymmetric vacuum HB HA vEW f HB HA vEW f Need to push up m2 h while keeping down vEW Use µ and increase f Small µ technically natural Soft breaking, needn’t consider one-loop corrections

Twin-Higgs model at zero temperature: Upshot Quadratic divergences cancelled between same-spin partners No dofs charged under both SM and twin Very weak constraints from experiments

Twin-Higgs model at zero temperature: Upshot Quadratic divergences cancelled between same-spin partners No dofs charged under both SM and twin Very weak constraints from experiments What happens when we heat the theory?

Matsubara formalism Compactify and Wick rotate time direction t → −iτ , τ ∈ S1 R=β periodic BCs for bosons ωB n = 2πT n anti-periodic BCs for fermions ωF n = 2πT n + 1 2 Corrected two-point function; Replace d4k → ωn d3k

Finite temperature eﬀective potential Veﬀ = Vtree + V T=0 1 + ∆V T 1 ∆V T 1 = i T4 2π2 niJb/f m2 i T2 Let’s look at Jb/f and high temperature approximations

Resumming the daisies ∼ λT2 IR divergences at high temperature; Expansion in λT2 m2 Resum: m2 thermal = m2 T=0 + ΠT for zero modes (3d EFT) Use this eﬀective mass in the m3T term Expansion parameter improves to λT meﬀ (3d perturbativity)

LσM completion H = HA HB = 1 √ 2 FA FB Fluctuating degrees of freedom m2 R = −m2 + 3λ (F2 A + F2 B )2 7 × m2 G = −m2 + λ (F2 A + F2 B )2 m2 tA = y2F2 A 2 , m2 tB = y2F2 B 2

LσM completion H = HA HB = 1 √ 2 FA FB Fluctuating degrees of freedom m2 R = −m2 + 3λ (F2 A + F2 B )2 7 × m2 G = −m2 + λ (F2 A + F2 B )2 m2 tA = y2F2 A 2 , m2 tB = y2F2 B 2 Plug into formula for 1-loop corrections, to get Veﬀ

At high temperatures Push to high temperature =⇒ m2T2 approximation good Symmetry restoration due to thermal mass Since µ2 for A sector, we expect EW symmetry to be restored before radial VEV shrinks to origin Lack perturbative control close to phase transition At high temperature, restored phase with minimum at origin λT meﬀ ∼ √ λ

Summary Goal: High temperature symmetry (non?)restoration, in theory with same-spin partners Considered example of Twin-Higgs inspired model Symmetry is restored in NLSM If UV completed as LσM then radial vev shrinks to zero

Stepping back and taking stock Imperfect arguments; working intuition; spirit of no-go Heuristic thermodynamic argument F = E − TS Broken symmetry at high temperatures, if possible, might require some surprising mechanism, or diﬀerent conditions Any unnaturalness gets recycled into symmetry restoration SUSY breaks at T > 0 and leads to symmetry restoration