The Equation of State and the Sphaleron Rate in the Standard Model

Home , Sphaleron

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 † 2 † † 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 hφ†φi ] 4p = − R − m 4d R + 0R ∂ ln(¯µ/T ) T 4 T 4

≡ ∆1 + ∆2 + ∆3

hφ†φi Applying 3d eﬀective theory:∆ = ∆A + ∆B 3D 2 2 2 T

K. Rummukainen (Helsinki) SM equation of state 11 / 25 Calculating the EOS

† A B hφ φi3D Thus, ∆ = ∆1 + ∆3 + ∆2 + δ2 T 6 All ∆i ’s computed perturbatively, uncertainty O(g ) [Laine, Meyer] † 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 hφ†φi 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 hφ†φi susceptibility 4

χφ†φ: 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 hφ†φi 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→∞ Vt

5 4 In the symmetric phase, Γ ∝ αW T [Arnold, Son, Yaffe 97] 5 4 or rather Γ ∝ αW log(1/αW )T [Bödeker 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), ∃ continuum limit, very simple to use I Same “action” and cont. limit as in 3D thermo simulation ∃ 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. Rummukainen (Helsinki) SM equation of state 22 / 25 Multicanonical evolution

Deep in the broken phase the barrier between vacua is large: ⇒ Rate becomes very small

I Too small to be measured with std. method

I But still physically relevant! Rate can me measured with multicanonical methods, which enables to overcome potential barriers in simulations For broken phase sphaleron rate, methood developed by [Moore 99]

K. Rummukainen (Helsinki) SM equation of state 23 / 25 Result: sphaleron rate

Symmetric phase: Γ -10 = (8.0 ± 1.3) × 10−7 ≈ (18 ± 3)α5 pure gauge T 4 W -15

Broken phase: parametrized as -20 Γ T log 4 = (0.83 ± 0.01) − (147.7 ± 1.9) T GeV 4 -25 Γ/Τ

Errors dominated by systematics log -30 standard multicanonical fit perturbative Cosmology: Hubble cooling T˙ = −HT -35 Freeze-out temperature T∗ from Γ(T∗) -40 log[αH(T)/T] 3 = αH(T∗) with α ≈ 0.1015 [Burnier et T∗ al 05] -45 130 140 150 160 170 Baryon number freeze-out T / GeV

T∗ = 131.7 ± 2.3 GeV

K. Rummukainen (Helsinki) SM equation of state 24 / 25 Conclusions

Standard Model equation of state solved to 1% level

Pseudocritical temperature Tc = 159.6 ± 0.1 ± 1.5 GeV Cross-over weak but well deﬁned Width of the cross-over region 2-3 GeV

Baryon number freeze-out T∗ = 131.7 ± 2.3 GeV 4 5 Symmetric phase sphaleron rateΓ /T = (18 ± 3)αW Broken phase rate can be parametrized as log Γ/T 4 = (0.83 ± 0.01)T /GeV − (147.7 ± 1.9) . . . can be fed in to e.g. some leptogenesis scenarios

K. Rummukainen (Helsinki) SM equation of state 25 / 25