Available online at www.sciencedirect.com Nuclear and Particle Physics Proceedings 273–275 (2016) 2363–2365 www.elsevier.com/locate/nppp Physical parameters of the electroweak crossover Michela D’Onofrio Department of Physics and Helsinki Institute of Physics, PL 64 (Gustaf H¨allstr¨ominkatu 2), FI-00014 University of Helsinki, Finland Kari Rummukainen Department of Physics and Helsinki Institute of Physics, PL 64 (Gustaf H¨allstr¨ominkatu 2), FI-00014 University of Helsinki, Finland Anders Tranberg Faculty of Science and Technology, University of Stavanger, N-4036 Stavanger, Norway Abstract We use large-scale lattice simulations to compute the rate of baryon-number violating processes, the sphaleron rate, the Higgs field expectation value, and the critical temperature of the electroweak phase transition in the Standard Model. Keywords: baryogenesis, sphaleron rate, baryon number, electroweak crossover, lattice simulations, standard model 1. Introduction necessary Sakharov’s conditions of C and CP violation as well as departure from equilibrium. These are not Baryon and lepton numbers are classically conserved naturally satisfied by the Standard Model as-is, but a quantities, but the chiral nature of weak interactions minimal extension is required. gives rise to the anomalous violation of baryon and lep- ton number currents at the quantum level. In practice, Ε however, the processes violating B- and L-numbers are Εsph suppressed below a temperature scale of Tc ∼ 100 GeV, ff N thus making B and L e ectively conserved in the present −2 −1 0 1 2 3 CS Universe. The critical temperature corresponds to the electroweak scale, where it has been suggested [1] that Figure 1: Vacuum structure of the pure-gauge electroweak theory. The baryogenesis might have taken place. In the Standard energy of gauge field configurations as a function of Chern-Simons number [2]. Model, baryon number is violated by the Adler-Bell- Jackiw anomaly At temperatures above the electroweak scale, the rate 2 of the sphaleron transitions is unsuppressed and has μ5 e αβμν ∂μ j = − FαβFμν, (1) been accurately measured using effective theories on the π2 16 lattice. At temperatures substantially below the elec- where jμ5 is the axial vector current, e the gauge cou- troweak scale, the Higgs field expectation value is large pling, and Fij the gauge field strength tensor. Eq. 1 ex- and the sphaleron rate is strongly suppressed. presses the fact that, in gauge theories, gauge invariance The work presented in these proceedings is based on implies axial vector current non-conservation. our previous work [3], where we use an effective elec- However, apart from B-violation, any successful troweak theory on the lattice with multicanonical and model of baryogenesis has also to fulfill the other two real-time simulation methods to calculate the sphaleron http://dx.doi.org/10.1016/j.nuclphysbps.2015.09.392 2405-6014/© 2015 Elsevier B.V. All rights reserved. 2364 M. D’Onofrio et al. / Nuclear and Particle Physics Proceedings 273–275 (2016) 2363–2365 rate through the electroweak crossover with Higgs mass Each transition changes nCS by one unit and the of 125 GeV. baryon number by ng = 3 B(t f ) − B(ti) = ng [nCS (t f ) − nCS (ti)]. 2. Theory The electroweak theory possess a set of infinite non- 3. Theory on the lattice trivial vacua (Fig. 1), each labeled by a Chern-Simons number We use large-scale lattice simulations and compute the sphaleron rate, the Higgs field expectation value and ≡ 3 0 nCS d xjCS the critical temperature of the electroweak phase transi- tion in the Standard Model. g2 g = − 3 ijk + . The thermodynamics of the 4-dimensional elec- π d x Tr AiA jk i AiA jAk 64 3 troweak theory is studied in 3 dimensions through di- μ mensional reduction [5], a perturbative technique giv- The Chern-Simons current jCS is in turn related through the axial anomaly to the baryon- and lepton-number cur- ing the correspondence between 4D and 3D parameters. ff rents The result is a SU(2) e ective theory with the Higgs field φ and gauge field Aμ (Fij) 2 μ μ g αβμν a a ∂μ( j + j ) = ng AαβAμν , (2) B L 16π2 = 1 a a + φ † φ + 2φ†φ + λ φ†φ 2, L FijFij (Di ) (Di ) m3 3( ) (4) ∂ μ = ∂ μ , 4 μ jB ng μ jCS (3) ff 2 λ 2 and 3D e ective parameters g3, 3 and m3. The time where the U(1) part of the theory is omitted. Transitions evolution of this effective SU(2) Higgs model is gov- between vacua are possible by surmounting the poten- erned by Langevin dynamics [6]. The latter, however, sphaleron transitions tial barrier through . The sphaleron is very slow on the lattice and can be substituted by any rate is strongly suppressed at low temperatures, where other dissipative procedure, heat bath in our case. One the potential barrier is high. At temperatures above the heat-bath sweep through the lattice corresponds to the EWPT, though, transitions among vacua are made pos- 2 real-time step Δt = a σel/4[7], where a is the lattice sible because of the availability of thermal energy. spacing and σel is the non-abelian color conductivity, the current response to infrared external fields. The Standard Model 130 4. Methods symmetric phase 120 In the symmetric phase we make use of canoni- perturbation theory cal Monte Carlo simulations and approach the broken phase. At very low temperatures, the rate is highly sup- 110 2nd order pressed and canonical methods do not work anymore. endpoint Here, the computation is performed with multicanon- /GeV c ical methods [8, 9], which make use of a weight func- T 100 tion that compensates the low-temperature suppression in the baryon violation rate. The obtained sphaleron rate is 90 Higgs phase (n (t) − n (0))2 Γ ≡ lim CS CS . (5) t→∞ Vt 80 50 60 70 80 90 5. The sphaleron rate mH/GeV The measured sphaleron rate is shown in Fig. 6 Figure 2: The order of the electroweak phase transition shown in terms of Higgs mass [4]. The phase transition is first order up to with a shaded error band. The freeze-out tem- Γ mH ∼ 70 GeV, where it becomes second order. For higher values perature T∗ is solved from the crossing of and of the Higgs mass, the transition is a crossover. the Hubble rate, shown with the almost horizontal M. D’Onofrio et al. / Nuclear and Particle Physics Proceedings 273–275 (2016) 2363–2365 2365 line. The sphaleron rate in the symmetric phase -10 pure gauge (T > Tc)is Γ/ 4 = ± α5 , T (18 3) W -15 and in the broken phase between 130 GeV < T < Tc can be parametrized as -20 log(Γ/T 4) = (0.83 ± 0.01) T/GeV − (147.7 ± 1.9). 4 -25 The freeze-out temperature in the early Universe, where Γ/Τ log -30 standard the Hubble rate wins over the baryon number violation multicanonical rate, is T∗ = (131.7 ± 2.3) GeV. fit perturbative -35 6. Conclusions α -40 log[ H(T)/T] The discovery of the Higgs particle of mass 125– 126 GeV enables us to fully determine the properties of -45 the symmetry breaking at high temperatures. Using lat- 130 140 150 160 170 tice simulations of a 3D effective theory, we have lo- T / GeV Figure 4: The sphaleron rate as a function of temperature at the cated the crossover range at Tc = (159 ± 1) GeV, de- crossover. Both canonical and multicanonical results are shown and termined the baryon number violation rate both above are in good agreement. The perturbative result is from [10]. Pure and well below the crossover point, and calculated the gauge refers to the rate in hot SU(2) gauge theory [11]. baryon freeze-out temperature in the early Universe, T∗ = (131.7 ± 2.3) GeV. It is the first time that this fundamental parameter of These results represent intrinsic properties of the the Standard Model has been so extensively and pre- Minimal Standard Model, as well as provide input for cisely studied in the full temperature range of the elec- leptogenesis calculations, in particular for models with troweak crossover. The obtained accuracy is the greatest electroweak-scale leptons. 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