High temperature limit of the Standard Model due to gauge groups contraction

The high temperature (high energy) limit of the Standard Model is developed with the help of contractions its gauge groups. The elementary particles evolution in the early Universe from Plank time up to several milliseconds is deduced from this limit theory. Particle properties at the infinite temperature look very unusual: all particles are massless, only neutral Z-bosons, u-quarks, neutrinos and photons are survived in this limit. The weak interactions become long-range and are mediated by neutral currents, quarks have only one color degree of freedom.


Introduction
The modern theory of elementary particles known as Standard Model consist of Electroweak Model, which unified electromagnetic and weak interactions, as well as Quantum Chromodynamics (QCD), describing their strong interactions. The Standard Model gives a good description of the experimental data and has been recently confirmed by the discover of Higgs boson at LHC. No new physics was observed in the experiments up to now. So today the Standard Model is the only theory of elementary particles and if we are interested in particle properties in the early Universe we need use its high temperature (high energy) limit.
Standard Model is a gauge theory with the direct product of a simple groups SU (3) × SU (2) × U (1) as a gauge group. The well known in physics operation of group contraction (or transition) [1] transforms initial group into some more simple limit group. For a symmetric physical system the contraction of its symmetry group means a transition to some limit state. We use this method in order to re-establish the evolution of elementary particles and their interactions in the early Universe. We are based on the modern knowledge of the particle world which is concentrated in Standard Model. For this we investigate the high-temperature (highenergy) limit of Standard Model generated by appropriate contractions of the gauge groups SU (2) and SU (3) [2][3][4].

High-temperature lagrangian of Electroweak Model
The contracted group SU (2; ) is obtained [3] by the consistent rescaling of the fundamental representation of SU (2) and the space C 2 as follows This transformation induces the following rescaling of the standard boson fields as well as the left lepton and quark fields: From the special mechanism of spontaneous symmetry breaking, which is used to generate mass of vector bosons and other elementary particles of the model, the following transformation rule for Higgs boson field χ and masses of all particles is obtained. After transformations (2), (3) the Lagrangian of the Electroweak Model can be represented in the form The explicit form of itermediate parts of Lagrangian L k was given in [3]. In particular the infinite temperature limit ( = 0) Lagrangian is equal to The contraction parameter is monotonous function (T ) of the temperature of a system under consideration with the property (T ) → 0 for T → ∞. Very high temperatures can exist in the early Universe just after its creation on the first stages of the Hot Big Bang [5] in pre-electroweak epoch.

Quantum Chromodynamics at high temperature
Like the Electroweak Model QCD is a gauge theory based on the local color degrees of freedom [6]. The QCD gauge group is SU (3), acting in three dimensional complex space of color quark states (3) gauge bosons (gluons) are the force carrier of the theory between the quarks. The QCD Lagrangian is taken in the form where D μ q are covariant derivatives of quark fields g S is the strong coupling constant, t a = λ a /2 are generators of SU (3), λ a are Gell-Mann matrices. Gluon stress tensor is introduced as usual The contracted special unitary group SU (3; κ) is defined by the action on the complex space C 3 (κ) in such a way that the hermitian form remains invariant, when the contraction parameters tend to zero: κ 1 , κ 2 → 0. We take these parameters identical to the contraction parameter of the Electroweak Model: κ 1 = κ 2 = κ = , so the limit κ = → 0 corresponds to the infinite temperature limit. Transition from the classical group SU (3) and space C 3 to the group SU (3; ) and space C 3 ( ) is given by the substitution and diagonal gauge fields A RR μ , A GG μ , A BB μ remain unchanged. Then from (6) we obtain the quark part of Lagrangian The gluon part of (6) is cumbersome, but can be represented as a sum of terms [4] In the infinite temperature limit the most parts of gluon tensor components are equal to zero, so we can write out the QCD Lagrangian L ∞ explicitly Let us introduce the notations The Standard Model passes in this limit through several stages, which are distinguished by the powers of contraction parameter, i.e. by powers of the temperature. Starting with L q ( ) (4), L q ( ) (12) and L gl ( ) (13) one can construct a number of intermediate models, namely, one model with explicit Lagrangian for each stage [4]. These models describe the restoration of particles properties in the Universe evolution.

Conclusion
The contraction of gauge group of the Standard Model gives an opportunity to order in time different stages of its development, but does not make it possible to bear their absolute dates. boundary values of the temperature in the early Universe (GeV): T 1 = 10 18 , T 2 = 10 7 , T 3 = 10 3 , T 4 = 10 2 , T 6 = 1, T 8 = 2 · 10 −1 . The obtained estimation for "infinity" temperature T 1 ≈ 10 18 GeV is comparable with Planck energy ≈ 10 19 GeV, where the gravitation effects are important. So the developed evolution of the elementary particles does not exceed the range of the problems described by electroweak and strong interactions. At the infinite temperature limit (T > 10 18 GeV) all particles including vector bosons lose their masses and electroweak interactions become long-range. Monochromatic massless quarks exchange by only one sort of R-gluons. It follows from the explicit form of Lagrangians L int ∞ (A μ , Z μ ) (5) and L ∞ (14) that only the particles of the same sort interact with each other. Particles of different sorts do not interact. It looks like some stratification of leptons and quarkgluon plasma with only one sort of particles in each stratum.
At the level of classical gauge fields it is already possible to give some conclusions on the stages of elementary particles mass appearance in the Universe evolution. In particular we can conclude that half of quarks (≈ , 10 18 GeV > T > 10 7 GeV) first restore they mass. Then Z-bosons, electrons and other quarks become massive (≈ 2 , 10 7 GeV > T > 10 3 GeV). Finally Higgs boson χ and charged W ± -bosons restore their masses (≈ 4 , T ≤ 10 2 GeV).