Physical parameters of the electroweak crossover

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.


Introduction
Baryon and lepton numbers are classically conserved quantities, but the chiral nature of weak interactions gives rise to the anomalous violation of baryon and lepton number currents at the quantum level. In practice, however, the processes violating B-and L-numbers are suppressed below a temperature scale of T c ∼ 100 GeV, thus making B and L effectively conserved in the present Universe. The critical temperature corresponds to the electroweak scale, where it has been suggested [1] that baryogenesis might have taken place. In the Standard Model, baryon number is violated by the Adler-Bell-Jackiw anomaly where j μ5 is the axial vector current, e the gauge coupling, and F i j the gauge field strength tensor. Eq. 1 expresses the fact that, in gauge theories, gauge invariance implies axial vector current non-conservation. However, apart from B-violation, any successful model of baryogenesis has also to fulfill the other two necessary Sakharov's conditions of C and CP violation as well as departure from equilibrium. These are not naturally satisfied by the Standard Model as-is, but a minimal extension is required. At temperatures above the electroweak scale, the rate of the sphaleron transitions is unsuppressed and has been accurately measured using effective theories on the lattice. At temperatures substantially below the electroweak scale, the Higgs field expectation value is large and the sphaleron rate is strongly suppressed.
The work presented in these proceedings is based on our previous work [3], where we use an effective electroweak theory on the lattice with multicanonical and real-time simulation methods to calculate the sphaleron rate through the electroweak crossover with Higgs mass of 125 GeV.

Theory
The electroweak theory possess a set of infinite nontrivial vacua (Fig. 1), each labeled by a Chern-Simons number The Chern-Simons current j μ CS is in turn related through the axial anomaly to the baryon-and lepton-number currents where the U(1) part of the theory is omitted. Transitions between vacua are possible by surmounting the potential barrier through sphaleron transitions. The sphaleron rate is strongly suppressed at low temperatures, where the potential barrier is high. At temperatures above the EWPT, though, transitions among vacua are made possible because of the availability of thermal energy. Each transition changes n CS by one unit and the baryon number by n g = 3

Theory on the lattice
We use large-scale lattice simulations and compute the sphaleron rate, the Higgs field expectation value and the critical temperature of the electroweak phase transition in the Standard Model.
The thermodynamics of the 4-dimensional electroweak theory is studied in 3 dimensions through dimensional reduction [5], a perturbative technique giving the correspondence between 4D and 3D parameters. The result is a SU(2) effective theory with the Higgs field φ and gauge field A μ (F i j ) and 3D effective parameters g 2 3 , λ 3 and m 2 3 . The time evolution of this effective SU(2) Higgs model is governed by Langevin dynamics [6]. The latter, however, is very slow on the lattice and can be substituted by any other dissipative procedure, heat bath in our case. One heat-bath sweep through the lattice corresponds to the real-time step Δt = a 2 σ el /4 [7], where a is the lattice spacing and σ el is the non-abelian color conductivity, the current response to infrared external fields.

Methods
In the symmetric phase we make use of canonical Monte Carlo simulations and approach the broken phase. At very low temperatures, the rate is highly suppressed and canonical methods do not work anymore.
Here, the computation is performed with multicanonical methods [8,9], which make use of a weight function that compensates the low-temperature suppression in the baryon violation rate. The obtained sphaleron rate is

The sphaleron rate
The measured sphaleron rate is shown in Fig. 6 with a shaded error band.
The freeze-out temperature T * is solved from the crossing of Γ and the Hubble rate, shown with the almost horizontal line. The sphaleron rate in the symmetric phase (T > T c ) is Γ/T 4 = (18 ± 3) α 5 W , and in the broken phase between 130 GeV < T < T c can be parametrized as log(Γ/T 4 ) = (0.83 ± 0.01) T/GeV − (147.7 ± 1.9).
The freeze-out temperature in the early Universe, where the Hubble rate wins over the baryon number violation rate, is T * = (131.7 ± 2.3) GeV.

Conclusions
The discovery of the Higgs particle of mass 125-126 GeV enables us to fully determine the properties of the symmetry breaking at high temperatures. Using lattice simulations of a 3D effective theory, we have located the crossover range at T c = (159 ± 1) GeV, determined the baryon number violation rate both above and well below the crossover point, and calculated the baryon freeze-out temperature in the early Universe, It is the first time that this fundamental parameter of the Standard Model has been so extensively and precisely studied in the full temperature range of the electroweak crossover. The obtained accuracy is the greatest achievable in the circumstance of exponential suppression, which occurs in the broken phase. Significant improvement in this direction is not around the corner any time soon.  Figure 3: Higgs field expectation value φ 2 as a function of temperature. and compared with the perturbative result in [12]. We notice a perfect match between the canonical and multicanonical results and a smooth transition from the symmetric to the broken phase.   [10]. Pure gauge refers to the rate in hot SU(2) gauge theory [11].