The equation of state and the sphaleron rate in the Standard Model Kari Rummukainen University of Helsinki and Helsinki Institute of Physics D'Onofrio, Rummukainen, Tranberg arXiv:1404.3565 D'Onofrio, Rummukainen, arXiv:1508.07161 Big Bang and the little bangs, CERN 8/2016 K. Rummukainen (Helsinki) SM equation of state 1 / 25 The \Bump" The \bump" has ceased to be (. or is it just resting?) the Standard Model rules (still) K. Rummukainen (Helsinki) SM equation of state 2 / 25 Phase transitions in the Standard Model No phase transitions in the Standard Model (at µ = 0) I QCD and EW \phase transitions" are cross-overs Many BSM models have a first order EW phase transition 7! I EW baryogenesis (Kainulainen) I Gravitational waves (Weir) K. Rummukainen (Helsinki) SM equation of state 3 / 25 In this talk: Precision results of EW: I Pseudocritical temperature I Equation of state - \softness", width of the cross-over I Sphaleron rate Why study these? I it's the Standard. I Background to cosmological applications, e.g. leptogenesis I Precision properties of physics K. Rummukainen (Helsinki) SM equation of state 4 / 25 Phase diagram of the electroweak SM After lots of activity on and off the lattice: > ! No phase transition at all, smooth \cross-over" for mHiggs∼72 GeV 130 symmetric phase 120 [Kajantie,Laine,K.R.,Shaposhnikov,Tsypin 95{98] 110 2nd order endpoint /GeV c T 100 see also 1st order transition [Csikor,Fodor, Heitger] 90 [G¨urtler,Ilgenfritz,Schiller,Strecha] broken Higgs phase 80 50 60 70 80 90 mH/GeV K. Rummukainen (Helsinki) SM equation of state 5 / 25 Overall EOS [Laine, Schr¨oder2006] 0.4 12 10 0.3 8 4 6 p / T w m = 150 GeV 0.2 2 4 H c m = 200 GeV s H 1/3 2 0.1 0 1 2 3 4 5 6 1 2 3 4 5 6 10 10 10 10 10 10 10 10 10 10 10 10 T / MeV T / MeV Perturbation theory + Lattice QCD + Hadron RG Here EW transition featureless K. Rummukainen (Helsinki) SM equation of state 6 / 25 Equation of state and the pseudocritical temperature K. Rummukainen (Helsinki) SM equation of state 7 / 25 3d effective theory Tool for perturbative and lattice computations Modes p > g 2T are perturbative (at weak coupling): can be integrated out in stages: 1. p>∼T : fermions, non-zero Matsubara frequencies ! 3d theory (dimensional reduction) 1/ T 2. Electric modes p ∼ gT 2 Obtain a \magnetic theory" for modes p<∼g T . Fully contains the non-perturbative thermal physics. K. Rummukainen (Helsinki) SM equation of state 8 / 25 3d effective theory Physics at T = 0 4D SM action: - Fix physics perturbatively £ (αS (MW ), GF , MH , MW , MZ , Mtop) £ £ £ Integration over scales T and gT (at 2{loop level) £ £ £ 3D continuum effective theory 2{loop countertermsB B in lattice perturbation theory BN 3D lattice theory K. Rummukainen (Helsinki) SM equation of state 9 / 25 0.394 3d effective Lagrangian: 0.3935 /T 2 3 0.393 g 1 1 0.3925 L = F aF a + B B + 0.3 4 ij ij 4 ij ij 0 y 2 y y 2 (Di φ) Di φ + m3φ φ + λ3(φ φ) y -0.3 -0.6 SU(2) + U(1) gauge + Higgs 0.295 Parameters: x 0.29 2 2 0.285 g3 ∼ gW T 2 0.3105 x ≡ λ3=g3 2 4 z 0.31 y ≡ m3=g3 02 2 0.3095 z ≡ g3 =g3 140 150 160 170 T/GeV K. Rummukainen (Helsinki) SM equation of state 10 / 25 Calculating the EOS Developed by Laine and Meyer 2015 Higgs mass parameter ν and T are the only dimensionful parameters For dimensional reasons: p(T ) p µ¯ ν2(¯µ) p (¯µ, ν2(¯µ); g 2(¯µ)) = R ; ; g 2(¯µ) − 0R T 4 T 4 T T 2 T 4 µ¯: MS scale, p0R: zero-temperature pressure g 2(¯µ): all dimensionless couplings in the SM Interaction measure: d p(T ) ∆ = T dT T 4 @(p =T 4) 2ν2[Z hφyφi ] 4p = − R − m 4d R + 0R @ ln(¯µ/T ) T 4 T 4 ≡ ∆1 + ∆2 + ∆3 hφyφi Applying 3d effective theory:∆ = ∆A + ∆B 3D 2 2 2 T K. Rummukainen (Helsinki) SM equation of state 11 / 25 Calculating the EOS y A B hφ φi3D Thus, ∆ = ∆1 + ∆3 + ∆2 + δ2 T 6 All ∆i 's computed perturbatively, uncertainty O(g ) [Laine, Meyer] y On the other hand, hφ φi3D can be perturbatively computed in the symmetric phase and in the broken phase 2 4 Expansions fail near m =g3 ≡ y ∼ 0 { i.e. just at the cross-over ) Non-perturbative evaluation of hφyφi needed ) Lattice simulations Using thermodynamic identities, other quantities can be obtained: p, e, heat capacity CV , speed of sound cs ... Laine and Meyer did the analysis, but using poor quality lattice data (provided by me et al.). ) Revisit with new simulations K. Rummukainen (Helsinki) SM equation of state 12 / 25 Higgs field expectation value 0.6 Red dots: 3D lattice 0.5 (continuum limit) Green bands: 0.4 I Broken phase: 2-loop Coleman-Weinberg [Kajantie et 0.3 >/T φ + al 95] φ < I Symmetric phase: 3-loop 0.2 [Laine and Meyer 2015] Agreement remarkably good 0.1 away from the cross-over P.T. does not converge near the 0 cross-over 140 145 150 155 160 165 170 T/GeV K. Rummukainen (Helsinki) SM equation of state 13 / 25 Higgs susceptibility & pseudocritical temperature 10 β=6 8 β=9 β=16 φ 6 + Define pseudocritical T : maximum φ χ location of the hφyφi susceptibility 4 χφyφ: 2 Tc = 159:6 ± 0:1 ± 1:5 GeV 0 140 145 150 155 160 165 170 1st error: lattice errors T/GeV 2nd error: estimate of the 10 systematic uncertainty in 3d action 8 φ + φ χ 6 Right: continuum extrapolation of χ 4 2 156 157 158 159 160 161 162 T/GeV K. Rummukainen (Helsinki) SM equation of state 14 / 25 Interaction measure 0.7 Convert hφyφi to interaction measure ∆ = ( − 3p)=T 4 4 Band: lattice errors + variation 0.65 µ¯ = (0:5 ::: 2)πT . - 3p)/T ε ( Non-trivial feature between broken phase (low T ) and symmetric phase 0.6 (high T ) behaviour Width of the cross-over region ∼ 2 ::: 3 GeV 140 145 150 155 160 165 170 T/GeV K. Rummukainen (Helsinki) SM equation of state 15 / 25 Pressure and energy density 34.6 Pressure: 4 34.4 e/T Z T 0 p(T ) p(T0) 0 ∆(T ) 4 − 4 = dT 0 1% T T T 34.2 0 T0 4 Use reference temperature , 3p/T 4 34 T0 = 140 GeV, e/T 4 pert. pressure p(T0)=T0 = 11:173 33.8 [Laine, Schr¨oder2006] 4 3p/T Energy density e = ∆ + 3p 33.6 p(T0) has ∼ 1% uncertainty ! 140 145 150 155 160 165 170 uncertainty for e and p T/GeV K. Rummukainen (Helsinki) SM equation of state 16 / 25 More thermodynamics 0.334 142 1/3 0.332 141 0.33 140 2 3 c s /T V 0.328 C 139 0.326 w 138 0.324 137 140 145 150 155 160 165 170 140 145 150 155 160 165 170 T/GeV T/GeV 0 Heat capacity CV = e (T ) 2 0 0 Speed of sound: cs = p =e EOS parameter w = p=e Cross-over well defined, but very soft! K. Rummukainen (Helsinki) SM equation of state 17 / 25 Masses 3 2 Higgs and W screening masses Effective cos θWeinberg 0,8 1.1 m =9 H βG m =16 H βG m 3 β =9 0,6 W G 1 m 3 =16 W βG 0,4 γ 0.9 A m/T 0,2 0.8 cos2 θW 0 0.7 140 145 150 155 160 165 170 140 145 150 155 160 165 170 T/GeV T/GeV K. Rummukainen (Helsinki) SM equation of state 18 / 25 Sphaleron rate K. Rummukainen (Helsinki) SM equation of state 19 / 25 Background Anomaly: baryon number B and gauge topology are connected: Z t Z 3 µν ~ ∆B = ∆L = 3∆NCS = 2 dt dVF Fµν 32π 0 Baryogenesis Rate in thermal equilibrium: h(∆N (t))2i Γ = lim CS V ;t!1 Vt 5 4 In the symmetric phase, Γ / αW T [Arnold, Son, Yaffe 97] 5 4 or rather Γ / αW log(1/αW )T [B¨odeker 98] In the broken phase the rate is exponentially suppressed Turning off of the rate is important for some baryogenesis scenarios K. Rummukainen (Helsinki) SM equation of state 20 / 25 Calculation of the sphaleron rate Non-perturbative ) real-time lattice simulations Several methods: Classical EOM [Ambjorn, Krasnitz 95; + many] I UV modes (HTL) make result lattice spacing depednent [Arnold 97] Classical + HTL effective theories [Moore, Hu, Muller 97; Bodeker, Moore, K.R 99] I More control over UV modes, no continuum limit I Used also in studies of plasma instabilities in HIC B¨odeker method: [B¨odeker 98], heat bath version [Moore 98] 2 I Fully dissipative evolution of soft (g T ) modes X Exact to leading log order log(1=g 2), 9 continuum limit, very simple to use I Same \action" and cont. limit as in 3D thermo simulation 9 lot of lattice results in pure gauge, few in broken EW phase. Here: physical Higgs mass K. Rummukainen (Helsinki) SM equation of state 21 / 25 Evolution of NCS Symmetric T = 152GeV, broken T = 145GeV, deeply broken T = 140GeV (with mH = 113GeV) [D'Onofrio et al 12] 12 1 60 10 0.8 8 0.6 CS CS CS 40 0.4 6 0.2 0 4 20 -0.2 2 -0.4 Chern-Simons number N Chern-Simons number N Chern-Simons number N -0.6 0 0 -0.8 -2 -1 -20 5 5 5 5 5 0.0 5.0×104 1.0×105 1.5×105 2.0×105 0 1×10 2×10 3×10 4×10 5×10 0 1×105 2×105 3×105 4×105 5×105 Measurements Measurements Measurements 1 0 10 1.4 1.2 0.8 -1 10 1 0.6 0.8 -2 10 0.6 0.4 Normalized distribution Normalized distribution 0.4 normalized distribution (log) -3 10 0.2 0.2 -4 0 0 10 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Chern-Simons number N Chern-Simons number N CS CS Chern-Simons number NCS K.