Elementary Particles in the Early Universe

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N. 3 fQatmCrmdnmc r ae iden- taken are Chromodynamics Quantum of (3) Abstract 1 fsm fisprmtr enables parameters its of some of eal en t non-evident its means nerally fEetoekMdl The Model. Electroweak of t som ipegopt a to group simple a nsforms nl ofimdb discover- by confirmed ently scnetdwt system with connected is r mlctdpyia system physical omplicated sne.Tedeveloped The esented. rcin.Sadr Model Standard eractions. vstehigh-temperature the ives rcino t symmetry its of traction l(M oss fElec- of consist (SM) el h esaeconstructed are dels eaino ru con- group of peration neatos swl as well as interactions, up neat tlooks It interact. t SU group fteelementary the of se ytepow- the by ished grneadare and ng-range ce oemasses, lose icles h Standard The . ret contraction. to erse vrefo the from iverse SU (3) tatosof ntractions 2 fElec- of (2) SU × SU (2; (2) j ) × × U U (1) (1) generalization. Quantum groups [5], which are simultaneously non-commutative and non- cocommutative Hopf algebras, present a good example of similar generalization since pre- viously Hopf algebras with only one of these properties were known. But when the contraction of some mathematical or physical structure is performed one can reconstruct the initial structure by the deformation in the narrow sense moving back along the contraction way. 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. In this paper we investigate the high-temperature limit of Standard Model generated by contraction of the gauge groups SU(2) and SU(3). Similar very high temperatures can exist in the early Universe after inflation and reheating on the first stages of the Hot Big Bang [6]. At these times the elementary particles demonstrate rather unusual properties. It appears that the SM Lagrangian falls to a number of terms which are distinguished by the powers of the contraction parameter ǫ 0. As far as the temperature in the hot Universe is connected → with its age, then moving forward in time, i.e. back to high-temperature contraction, we conclude that after the Universe creation elementary particles and their interactions pass a number of stages in their evolution from infinite temperature state up to SM state. These stages of quark-gluon plasma formation and color symmetries restoration are distinguished by the powers of contraction parameter and consequently by times of its creations. From the contraction of Standard Model we can classify the stages in time as earlier-later, but we can not determine their absolute date. To estimate the absolute date we use additional assumptions.

2 High-Temperature Lagrangian of EWM

It was shown in papers [2, 3, 4], that in zero energy limit the gauge group SU(2; j) and the fundamental representation space C2(j) are transformed as follows

jz α jβ jz jν ju z (j)= 1′ = 1 = u(j)z(j), z(j)= l ,..., l ,..., ′ z jβ¯ α¯ z e d 2′ ! − ! 2 ! l ! l ! 2 2 2 det u(j)= α + j β = 1, u(j)u†(j) = 1, (1) | | | | so for j 0 the first components of the lepton and quark doublets become infinitely small → in comparison to their second components. On the contrary, when energy (temperature) increases the first components of the doublets become greater than their second ones. In the infinite temperature limit the second components of the lepton and quark doublets will be infinitely small as compared to their first components. To describe this limit we introduce [7, 8] new contraction parameter ǫ and new consistent rescaling of the group SU(2; ǫ) and the space C2(ǫ) in the form

z α ǫβ z z (ǫ)= 1′ = 1 = u(ǫ)z(ǫ), ′ ǫz ǫβ¯ α¯ ǫz 2′ ! − ! 2 ! 2 2 2 det u(ǫ)= α + ǫ β = 1, u(ǫ)u†(ǫ) = 1. (2) | | | |

2 Hermite form 2 2 2 z†(ǫ)z(ǫ)= z + ǫ z (3) | 1| | 2| remain invariant under contraction ǫ 0. → In the contraction scheme (2) the standard boson ﬁelds and left lepton and quark ﬁelds are transformed as follows

W ± ǫW ±,Z Z , A A . µ → µ µ → µ µ → µ e ǫe , d ǫd , ν ν , u u . (4) l → l l → l l → l l → l The next reason for inequality of the first and second doublet components is the special mechanism of spontaneous symmetry breaking, which is used to generate mass of vector bosons and other elementary particles of the model. In this mechanism one of Lagrangian 0 ground states φvac = is taken as vacuum of the model and then small field v ! excitations v + χ(x) with respect to this vacuum are regarded. So Higgs boson field χ and constant v are multiplied by ǫ. As far as masses of all particles are proportionate to v we obtain the following transformation rule

χ ǫχ, v ǫv, m ǫm , p = χ, W, Z, e, u, d. (5) → → p → p After transformations (4), (5) the EWM boson Lagrangian [9, 10] can be represented in the form

1 2 1 2 2 3 + 4 L (ǫ)= + ǫ L + ǫ gW W −χ + ǫ L , (6) B −4Zµν − 4Fµν B,2 µ µ B,4 where

2 2 2 + 1 2 2 3 λ 4 g + + 2 g + 2 L = m W W − m χ λvχ χ + W W − W −W + W W −χ , B,4 W µ µ − 2 χ − − 4 4 µ ν − µ ν 4 µ ν (7) 2 1 2 1 2 2 1 + gmz 2 g 2 2 LB,2 = (∂µχ) + mZ (Zµ) µν µν− + (Zµ) χ + 2 (Zµ) χ 2 2 − 2W W 2 cos θW 8 cos θW − + + 2ig W W − W −W sin θ + cos θ − µ ν − µ ν Fµν W Zµν W − i + + i + + e A W − − W + eA W − − W −2 µ Wµν ν −Wµν ν 2 ν Wµν µ −Wµν µ − i + + g cos θ Z W − − W −2 W µ Wµν ν −Wµν ν − h 2 2 2 + + e + 2 Z W − − W W + W − (A ) − ν Wµν µ −Wµν µ − 4 µ µ ν − i + + + 2 2 2 2 W W + W −W − A A + W + W − (A ) − µ ν µ ν µ ν ν ν µ − h i o 2 2 2 g + 2 cos θ W + W − (Z ) − 4 W µ µ ν − + + + 2 2 2 2 W W + W −W − Z Z + W + W − (Z ) − µ ν µ ν µ ν ν ν µ − h i o 3 + + 1 + + eg cos θ W W −A Z +W W −A Z W W − + W W − (A Z + A Z ) . (8) − W µ µ ν ν ν ν µ µ− 2 µ ν ν µ µ ν ν µ The lepton Lagrangian in terms of electron and neutrino ﬁelds takes the form

2 LL(ǫ)= LL,0 + ǫ LL,2 = g = νl†iτ˜µ∂µνl + er†iτµ∂µer + g′ sin θwer†τµZµer g′ cos θwer†τµAµer + νl†τ˜µZµνl+ − 2 cos θw

2 g cos 2θw +ǫ el†iτ˜µ∂µel me(er†el + el†er)+ el†τ˜µZµel − 2 cos θw − g + ee†τ˜µAµel + ν†τ˜µW el + e†τ˜µW −νl . (9) − l √2 l µ l µ The quark Lagrangian in terms of u- and d-quarks ﬁelds can be written as

2 L (ǫ)= L ǫm (u†u + u†u )+ ǫ L , (10) Q Q,0 − u r l l r Q,2 where L = d†iτ ∂ d + u†iτ˜ ∂ u + u†iτ ∂ u Q,0 r µ µ r l µ µ l r µ µ r− 1 1 2e g′ cos θ d†τ A d + g′ sin θ d†τ Z d + u†τ˜ A u + −3 w r µ µ r 3 w r µ µ r 3 l µ µ l g 1 2 2 2 2 + sin θw ul†τ˜µZµul + g′ cos θwur†τµAµur g′ sin θwur†τµZµur, (11) cos θw 2 − 3 3 − 3 e L = d†iτ˜ ∂ d m (d†d + d†d ) d†τ˜ A d Q,2 l µ µ l − d r l l r − 3 l µ µ l− g 1 2 2 g + sin θw dl†τ˜µZµdl + ul†τ˜µWµ dl + dl†τ˜µWµ−ul . (12) − cos θw 2 − 3 √2 h i The complete Lagrangian of the modiﬁed model is given by the sum L(ǫ) = LB(ǫ)+ LL(ǫ)+ LQ(ǫ) and can be written in the form

2 3 4 L(ǫ)= L + ǫL1 + ǫ L2 + ǫ L3 + ǫ L4. (13) ∞ The contraction parameter is monotonous function ǫ(T ) of the temperature with the prop- erty ǫ(T ) 0 for T . Very high energies can exist in the early Universe just after its → →∞ creation. It is well known that to gain a better understanding of a physical system it is useful to investigate its properties for limiting values of physical parameters. It follows from the decomposition (13) that there are five stages in evolution of the Electroweak Model after the creation of the Universe which are distinguished by the powers of the contraction parameter ǫ. This offers an opportunity for construction of intermediate limit models. One can take the Lagrangian L for the initial limit system, then add L1 and obtain the ∞ second limit model with the Lagrangian 1 = L + L1. After that one can add L2 and L ∞ obtain the third limit model 2 = L + L1 + L2. The last limit model has the Lagrangian L ∞ 3 = L + L1 + L2 + L3. L ∞ In the infinite temperature limit (ǫ = 0) Lagrangian (13) is equal to

1 2 1 2 L = µν µν + ν†iτ˜µ∂µνl + u†iτ˜µ∂µul+ ∞ −4Z − 4F l l

4 int + er†iτµ∂µer + dr†iτµ∂µdr + ur†iτµ∂µur + L (Aµ,Zµ), (14) ∞ where int g 2e L (Aµ,Zµ)= νl†τ˜µZµνl + ul†τ˜µAµul+ ∞ 2 cos θw 3

g 1 2 2 +g′ sin θ e†τ Z e + sin θ u†τ˜ Z u g′ cos θ e†τ A e w r µ µ r cos θ 2 3 w l µ µ l w r µ µ r w − − − 1 1 2 2 g′ cos θ d†τ A d + g′ sin θ d†τ Z d + g′ cos θ u†τ A u g′ sin θ u†τ Z u . − 3 w r µ µ r 3 w r µ µ r 3 w r µ µ r − 3 w r µ µ r (15) We can conclude that the limit model includes only massless particles: photons Aµ and neutral bosons Zµ, left quarks ul and neutrinos νl, right electrons er and quarks ur, dr. This phenomenon has simple physical explanation: the temperature is so high, that particle mass becomes negligible quantity as compared to its kinetic energy. The electroweak interactions become long-range because they are mediated by the massless neutral Z- bosons and photons. Let us note that Wµ±-boson fields, which correspond to the translation subgroup of Euclid group E(2), are absent in the limit Lagrangian L (14). ∞ Similar high energies can exist in early Universe after inflation and reheating on the first stages of the Hot Big Bang [6, 11] in pre-electroweak epoch. However more interesting is the Universe evolution and the limit Lagrangian L can be considered as a good approx- ∞ imation near the Big Bang just as the nonrelativistic mechanics is a good approximation of the relativistic one at low velocities. int From the explicit form of the interaction part L (Aµ,Zµ) it follows that there are no interactions between particles of different kind, for∞ example neutrinos interact only with each other by neutral currents. All other particles are charged and interact with particles of the same sort by massless Zµ-bosons and photons. Particles of different kind do not interact. It looks like some stratification of the Electroweak Model with only one sort of particles in each stratum. From contraction of the Electroweak Model we can classify events in time as earlier- later, but we can not determine their absolute time without additional assumptions. Al- ready at the level of classical gauge fields we can conclude that the u-quark first restores its mass in the evolution of the Universe. Indeed the mass term of u-quark in the Lagrangian (13) L = m (u u + u†u ) is proportional to the first power ǫ, whereas the mass terms 1 − u r† l l r of Z-boson, electron and d-quark are multiplied by the second power of the contraction parameter 2 1 2 2 ǫ m (Z ) + m (e†e + e†e )+ m (d†d + d†d ) . (16) 2 Z µ e r l l r d r l l r At the same time massless Higgs boson χ and charged W -boson are appeared. They restore their masses after all other particles of the Electroweak Model because their mass terms are multiplied by ǫ4. The electroweak interactions between elementary particles are restored mainly in the epoch which corresponds to the second order of the contraction parameter. There is one + 3 term in Lagrangian (6) L3 = gWµ Wµ−χ proportionate to ǫ . The final reconstruction of the electroweak interactions takes place at the last stage ( ǫ4) together with restoration ≈ of mass of all particles. Two other generations of leptons and quarks are developed in a similar way: for infinite energy there are only massless right µ- and τ-muons, left µ- and τ-neutrinos, as well as

5 massless left and right quarks cl, cr,sr,tl,tr, br. c- and t-quarks ﬁrst acquire their mass and after that µ-, τ-muons, s-, b-quarks become massive.

3 QCD with contracted gauge group

Strong interactions of quarks are described by the QCD. Like the Electroweak Model QCD is a gauge theory based on the local color degrees of freedom [12]. The QCD gauge group is SU(3), acting in three dimensional complex space C3 of color quark states. The SU(3) gauge bosons are called gluons. There are eight gluons in total, which are the force carrier of the theory between quarks. The QCD Lagrangian is taken in the form

8 i µ j 1 α µν α = q¯ (iγ )(Dµ)ijq Fµν F , (17) L q − 4 X αX=1 where Dµq are covariant derivatives of quark ﬁelds q = u, d, s, c, b, t

q q λα 1 R D q = ∂ ig Aα q, q = q q C , (18) µ µ − s 2 µ  2  ≡  G  ∈ 3 q3 qB         a a a gS is the strong coupling constant, t = λ /2 are generators of SU(3), λ are Gell-Mann matrices in the form

1 i 1 · · · − · · · λ1 = 1 , λ2 = i , λ3 = 1 ,  · ·   · ·   · − ·   · · ·   · · ·   · · ·        1 i · · · · − · · · λ4 = , λ5 = , λ6 = 1 ,  · · ·   · · ·   · ·  1 i 1  · ·   · ·   · ·        1 · · · 1 · · λ7 = i , λ8 = 1 , (19)  · · −  √  · ·  i 3 2  · ·   · · −  gluon stress tensor     F α = ∂ Aα ∂ Aα + g f αβγAβAγ , (20) µν µ ν − ν µ s µ ν with the nonzero antisymmetric on all indices structure constant of the gauge group: 1 f 123 = 1, f 147 = f 246 = f 257 = f 345 = , 2

1 √3 f 156 = f 367 = , f 458 = f 678 = , (21) −2 2 where [tα,tβ] = if αβγ tγ, α, β, γ = 1,..., 8. Mass terms m q¯iq are not included as far − q i as they are present in the electroweak Lagrangian.

6 The choice of Gell-Mann matrices in the form (19) ﬁx the basis in SU(3). This enable us to write out the covariant derivatives (18) in the explicit form

3 1 8 1 2 4 5 Aµ + Aµ Aµ iAµ Aµ iAµ g √3 − − s A1 + iA2 1 A8 A3 A6 iA7 Dµ = I∂µ i  µ µ √3 µ µ µ µ  = − 2 4 5 6 − 7 −2 8  Aµ + iAµ Aµ + iAµ √ Aµ   − 3    ARR ARG ARB g µ µ µ = I∂ i s AGR AGG AGB , (22) µ − 2  µ µ µ  ABR ABG ABB  µ µ µ  where   1 1 2 ARR = A8 + A3 , AGG = A8 A3 , ABB = A8 , µ √3 µ µ µ √3 µ − µ µ −√3 µ RR GG BB GR 1 2 ¯RG Aµ + Aµ + Aµ = 0, Aµ = Aµ + iAµ = Aµ , BR 4 5 ¯RB BG 6 7 ¯GB Aµ = Aµ + iAµ = Aµ , Aµ = Aµ + iAµ = Aµ . (23) Let us note, that in QCD the special mechanism of spontaneous symmetry breaking is absent, therefore gluons are massless particles. The contracted special unitary group SU(3; κ) is deﬁned by the action

q1′ u11 κ1u12 κ1κ2u13 q1 q′(κ)=  κ1q2′  =  κ1u21 u22 κ2u23   κ1q2  = κ1κ2q′ κ1κ2u31 κ2u32 u33 κ1κ2q3  3            = U(κ)q(κ), det U(κ) = 1,U(κ)U †(κ) = 1 (24) on the complex space C3(κ) in such a way that the hermitian form

2 2 2 2 2 q†(κ)q(κ)= q + κ q + κ q (25) | 1| 1 | 2| 2 | 3| remains invariant, when the contraction parameters tend to zero: κ , κ 0. Transition 1 2 → from the classical group SU(3) and space C3 to the group SU(3; κ) and space C3(κ) is given by the substitution

q q , q κ q , q κ κ q , 1 → 1 2 → 1 2 3 → 1 2 3 AGR κ AGR, ABG κ ABG, ABR κ κ ABR, (26) µ → 1 µ µ → 2 µ µ → 1 2 µ RR GG BB and diagonal gauge ﬁelds Aµ , Aµ , Aµ remain unchanged. Substituting (26) in (17), we obtain the quark part of Lagrangian in the form

µ gs 2 µ RR q(κ)= iq¯1γ ∂µq1 + q1 γ Aµ + L q ( 2 | | X g +κ2 iq¯ γµ∂ q + s q 2 γµAGG + q q¯ γµAGR +q ¯ q γµA¯GR + 1 2 µ 2 2 2 µ 1 2 µ 1 2 µ | | 2 2 µ gs 2 µ BB µ BR µ ¯BR +κ1κ2 iq¯3γ ∂µq3 + q3 γ Aµ + q1q¯3γ Aµ +q ¯1q3γ Aµ + 2 | | 7 µ BG µ ¯BG 2 (2) 2 2 (4) + q2q¯3γ Aµ +q ¯2q3γ Aµ = Lq∞ + κ1Lq + κ1κ2Lq . (27) ) Let us introduce the notations

∂Ak ∂ Ak ∂ Ak , [k,m] Ak Am AmAk, (28) ≡ µ ν − ν µ ≡ µ ν − µ ν then the gluon tensor has the following components g F 1 = κ ∂A1 + s 2[2, 3] + κ2 ([4, 7] [5, 6]) , µν 1 2 2 − g F 2 = κ ∂A2 + s 2[1, 3] + κ2 ([4, 6] + [5, 7]) , µν 1 2 − 2 g F 3 = ∂A3 + s κ22[1, 2] κ2[6, 7] + κ2κ2[4, 5] , µν 2 1 − 2 1 2 g F 4 = κ κ ∂A4 s [1, 7] + [2, 6] + [3, 5] √3[5, 8] , µν 1 2 − 2 − g F 5 = κ κ ∂A5 + s [1, 6] [2, 7] + [3, 4] √3[4, 8] , µν 1 2 2 − − g F 6 = κ ∂A6 + s κ2 ([2, 4] [1, 5]) + [3, 7] + √3[7, 8] , µν 2 2 1 − g F 7 = κ ∂A7 + s κ2 ([1, 4] + [2, 5]) [3, 6] √3[6, 8] , µν 2 2 1 − − g √3 F 8 = ∂A8 + s κ2 κ2[4, 5] + [6, 7] . (29) µν 2 2 1 The gluon part of Lagrangian is as follows 1 (κ)= F α F µν α = Lgl −4 µν

1 2 2 2 2 2 = H3 + H8 + κ1 F1 + F2 + 2H3F3 + −4( +κ2 G2 + G2 + 2H G 2√3H G + κ4F 2 + κ44G2+ 2 6 7 3 3 − 8 3 1 3 2 3 2 2 2 2 +κ1κ2 P4 + P5 + 2 (F1G1 + F2G2 + F3G3 + F6G6 + F7G7+ h +√3H P + κ2κ4 G2 + G2 4G P + 8 3 1 2 1 2 − 3 3 i 4 2 2 2 4 4 2 + κ1κ2 F6 + F7 + 2F3P3 + κ1κ24P3 . (30) ) where F = ∂A1 + g [2, 3], F = ∂A2 g [1, 3], 1 s 2 − s g g G = s ([4, 7] [5, 6]) , G = s ([4, 6] + [5, 7]) , 1 2 − 2 2

8 g g H = ∂A3, F = g [1, 2], G = s [6, 7], P = s [4, 5], 3 3 s 3 − 2 3 2 g P = ∂A4 s [1, 7] + [2, 6] + [3, 5] √3[5, 8] , 4 − 2 − g P = ∂A5 + s [1, 6] [2, 7] + [3, 4] √3[4, 8] , 5 2 − − g g G = ∂A6 + s [3, 7] + √3[7, 8] , F = s ([2, 4] [1, 5]) , 6 2 6 2 − g g G = ∂A7 s [3, 6] + √3[6, 8] , F = s ([1, 4] + [2, 5]) , 7 − 2 7 2 8 H8 = ∂A . (31) In the framework of Cayley-Klein scheme [2] the gauge group SU(3; κ) has two one- parameter contractions κ 0, κ = 1 and κ 0, κ = 1, as well as one two-parameter 1 → 2 2 → 1 contraction κ , κ 0. We consider the following contraction: κ = κ = κ = ǫ 0, 1 2 → 1 2 → which corresponds to the inﬁnite temperature limit of QCD. The quark part of Lagrangian (27) is represented as a sum of terms proportional to zero, the second and forth powers of contraction parameter ǫ 2 (2) 4 (4) (ǫ)= L∞ + ǫ L + ǫ L (32) Lq q q q and gluon part is represented as a sum

2 (2) 4 (4) 6 (6) 8 (8) (ǫ)= L∞ + ǫ L + ǫ L + ǫ L + ǫ L . (33) Lgl gl gl gl gl gl In the infinite temperature limit κ = ǫ 0 the most parts of gluon tensor components → are equal to zero and the expressions for two nonzero components are simplified 1 √3 F 3 = ∂ A3 ∂ A3 = F RR F GG , F 8 = ∂ A8 ∂ A8 = F RR + F GG , µν µ ν − ν µ 2 µν − µν µν µ ν − ν µ 2 µν µν (34) so we can write out the QCD Lagrangian in this limit explicitly L∞ = Lq∞ + Lgl∞ = L∞ µ gs 2 µ RR 1 RR 2 1 GG 2 1 RR GG = iq¯Rγ ∂µqR + qR γ Aµ Fµν Fµν Fµν Fµν . (35) q 2 | | − 4 − 4 − 4 X From we conclude that only the dynamic terms for the first color component of massless L∞ quarks survive under infinite temperature, which means that the quarks are monochromatic, and the terms also survive, which describe the interactions of these components with R-gluons. Besides R-gluons there are also G-gluons, which do not interact with the quarks. Similarly to the Electroweak Model starting with (ǫ) (32) and (ǫ) (33) one can Lq Lgl construct a number of intermediate models for QCD, which describe the gradual restoration of color degrees of freedom for the quarks and the gluon interactions in the Universe evolution. It follows from Lagrangian (ǫ)= (ǫ)+ (ǫ), that the total reconstruction of the L Lq Lgl quark color degrees of freedom will take place after the restoration of all quark mass ( ǫ2) ≈ at the same time with the reestablishment of all electroweak interactions ( ǫ4). Complete ≈ color interactions start to work later because some of them are proportionate to the eighth power ǫ8.

9 4 Estimation of boundary values

As it was mentioned the contraction of gauge group of QCD gives an opportunity to order in time different stages of its development, but does not make it possible to bear their absolute date. Let us try to estimate this date with the help of additional assumptions. The equality of the contraction parameters for QCD and the EWM is one of these assumptions. Then we use the fact that the electroweak epoch starts at the temperature T4 = 13 100 GeV (1 GeV = 10 K) and the QCD epoch begins at T8 = 0, 2 GeV . In other words we assume that complete reconstruction of the EWM, whose Lagrangian has minimal terms proportionate to ǫ4, and QCD, whose Lagrangian has minimal terms proportionate to ǫ8, take place at these temperatures. Let us denote by ∆ cutoff level for ǫk, k = 1, 2, 4, 6, 8, i.e. for ǫk < ∆ all the terms proportionate to ǫk are negligible quantities in Lagrangian. At last we suppose that the contraction parameter inversely depends on temperature A ǫ(T )= , (36) T where A is constant. As far as the minimal terms in the QCD Lagrangian are proportional to ǫ8 and QCD 8 8 8 is completely reconstructed at T8 = 0, 2 GeV , we have the equation ǫ (T8)= A T8− =∆ 1/8 1/8 and obtain A = T8∆ = 0, 2∆ GeV . The minimal terms in the EWM Lagrangian 4 4 are proportional to ǫ and it is reconstructed at T4 = 100 GeV , so we have ǫ (T4) = 4 4 1/4 1/8 1/4 1/8 A T4− = ∆, i.e. T4 = A∆− = T8∆ ∆− = T8∆− and we obtain the cutoff level 1 8 2 8 22 ∆ = (T8T4− ) = (0, 2 10− ) 10− , which is consistent with the typical energies of the · ≈ k k k Standard Model. From the equation ǫ (Tk)= A Tk− = ∆ we obtain

1/8 A T8∆ k−8 88−15k T = = = T8∆ 8k 10 4k GeV. (37) k ∆1/k ∆1/k ≈ Simple calculations give the following estimations for the boundary values of the temper- 18 7 3 2 ature in the early Universe (GeV ): T1 = 10 , T2 = 10 , T3 = 10 , T4 = 10 , T6 = 1, T = 2 10 1. The obtained estimation for ”inﬁnity” temperature T 1018 GeV is 8 · − 1 ≈ comparable with Planck energy 1019 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. p It should be noted that for the power function class ǫ(T )= BT − the estimations for temperature boundary values are very weakly dependent on power p. So for p = 10 we obtain practically the same Tk as for the simplest function (36) with p = 1.

Conclusion

We have investigated the high-temperature limit of the SM which was obtained from the ﬁrst principles of the gauge theory as contraction of its gauge group. It was shown that the mathematical contraction parameter is proportional to the temperature and its zero limit corresponds to the inﬁnite temperature limit of the Model. The SM passes in this limit through several stages, which are distinguished by the powers of contraction parameter, what gives the opportunity to classify them in time as earlier-later. To determine the absolute date of these stages the additional assumptions was used, namely: the inversely

10 dependence ǫ on the temperature (36) and the cutoff level ∆ for ǫk. Unknown parameters are determined with the help of the QCD and EWM typical energies. The exact expressions for the respective Lagrangians for any stage in the SM evolution are presented. On the base of decompositions (13), (32), (33) the intermediate models Lk for any temperature scale are constructed. It gives an opportunity to draw a conclusions on interactions and properties of elementary particles in each of considered epoch. The presence of the several intermediate models in the interval from Plank energy 1019 GeV up to the EWM typical energy 102 GeV instead of only one model automatically takes away so-called hierarchy problem of the SM [12]. At the infinite temperature limit (T > 1018 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. Besides R-gluons there are also G- gluons, which do not interact with quarks. It follows from the explicit form of Lagrangians int L (Aµ,Zµ) (15) and (35) that only the particles of the same sort interact with each ∞ L∞ other. Particles of different sorts do not interact. It looks like some stratification of leptons and quark-gluon 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 appearance of elementary particles mass on the different stages of the Universe evolution. In particular we can conclude that half of quarks ( ǫ, 1018 GeV > T > ≈ 107 GeV ) first restore they mass. Then Z-bosons, electrons and other quarks become massive ( ǫ2, 107 GeV > T > 103 GeV ). Finally Higgs boson χ and charged W -bosons ≈ ± restore their masses because their mass terms are multiplied by ǫ4 (T < 102 GeV ). In a similar way it is possible to describe the evolution of particle interactions. Self- action of Higss boson appears with its mass restoration. At the same epoch start interactions of four W ±-bosons, as well as of two Higgs and two W -bosons (7). The only one term in Lagrangian, which is proportional to the third power of ǫ, describes 3 interaction of Higss boson with charged W ±-bosons (T < 10 GeV ). The rest of the electroweak particle interactions are appeared in the second order of the contraction parameter (107 GeV > T > 103 GeV ). Some part of color interactions between quarks in Lagrangian (27) is proportional to ǫ2 (T < 107 GeV ) and the rest part is proportional to ǫ4 (T < 102 GeV ). Therefore the complete restoration of quark color degrees of freedom takes place after the appearance of quark masses ( ǫ2, T < 107 GeV ) (16) together ≈ with the restoration of all electroweak interactions ( ǫ4,T< 102 GeV ). Complete color ≈ 8 1 interactions start later because they are proportional to ǫ (T < 10− GeV ). The evolution of elementary particles and their interactions in the early Universe obtained with the help of contractions of gauge groups of the SM does not contradict the canonical one [6], according to which the QCD phase transitions take place later then the electroweak phase transitions. The developed evolution of the SM present the basis for a more detailed analysis of different phases in the formation of leptons and quark-gluon plasma. The author is thankful to V. V. Kuratov and V. I. Kostyakov for helpful discussions. The study is supported by Program of UrD RAS project N 15-16-1-3.

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