Graviweak Unification in the Visible and Invisible Universe and Inflation from the Higgs Field False Vacuum
aa r X i v : . [ h e p - ph ] J u l Graviweak Unification in the Visible andInvisible Universe and Inflation from theHiggs Field False Vacuum
C.R. Das ∗ and L.V. Laperashvili † Centre for Theoretical Particle Physics (CFTP), University of Lisbon,Avenida Rovisco Pais, 1 1049-001 Lisbon, PortugalandTheoretical Physics Division, Physical Research Laboratory,Navrangpura, Ahmedabad - 380 009, India The Institute of Theoretical and Experimental Physics,National Research Center “Kurchatov Institute”,Bolshaya Cheremushkinskaya, 25, 117218 Moscow, Russia
Abstract
In the present paper we develop the self-consistent
Spin (4 , SU (2) interactions in the assumption of theexistence of visible and invisible sectors of the Universe. It was shown that theconsequences of the multiple point principle predicting two degenerate vacua in theStandard Model (SM) suggest a theory of Inflation, in which the inflaton field σ starts trapped in a cold coherent state in the “false vacuum” of the Universe atthe value of the Higgs field’s VEV v ∼ GeV (in the visible world). Thenthe inflations of the two Higgs doublet fields, visible φ and mirror φ ′ , lead to theemergence of the SM vacua at the Electroweak scales with the Higgs boson VEVs v ≈
246 GeV and v ′ = ζv (with ζ ∼ Keywords: unification, gravity, mirror world, inflation,cosmological constant, dark energy
PACS: ∗ [email protected], [email protected] † [email protected] Introduction
In Ref. [1] a model of unification of gravity with the weak SU (2) gauge and Higgs fields wasconstructed, in accordance with Ref. [2]. Previously gravi-weak and gravi-electro-weakunified models were suggested in Ref. [3–5].In this investigation we imagine that at the early stage of the evolution of theUniverse the GUT-group was broken down to the direct product of gauge groups of theinternal symmetry U (4) and Spin(4,4)-group of the Graviweak unification.In the assumption that there exist visible and invisible (hidden) sectors of the Uni-verse, we presented the hidden world as a Mirror World (MW) with a broken MirrorParity (MP). In the present paper we give arguments that MW is not identical to thevisible Ordinary World (OW). We started with an extended g = spin (4 , L -invariantPlebanski action in the visible Universe, and with g = spin (4 , R -invariant Plebanskiaction in the MW. Then we have shown that the Graviweak symmetry breaking leadsto the following sub-algebras: ˜ g = s l (2 , C ) ( grav ) L ⊕ s u (2) L – in the ordinary world, and˜ g ′ = s l (2 , C ) ′ ( grav ) R ⊕ s u (2) ′ R – in the hidden world. These sub-algebras contain the self-dual left-handed gravity in the OW, and the anti-self-dual right-handed gravity in theMW. Finally, at low energies, we obtain a Standard Model (SM) group of symmetry andthe Einstein-Hilbert’s gravity. In this approach we have developed a model of Inflation, inwhich the inflaton σ , being a scalar SU (2)-triplet field, decays into the two Higgs SU (2)doublets of the SM: σ → φ † φ , and then the interaction between the ordinary and mirrorHiggs fields (induced by gravity) leads to the hybrid model of the Inflation.In Section 2 we considered the Plebanski’s theory of gravity, in which fundamentalfields are 2-forms, containing tetrads, spin connections and auxiliary fields. Then we haveused an extension of the Plebanski’s formalism of the 4-dimensional gravitational theory,and in Section 3 we constructed the action of the Graviweak unification model, describedby the overall unification parameter g uni . Section 4 is devoted to the Multiple Point Model(MPM), which allows the existence of several minima of the Higgs effective potential withthe same energy density (degenerate vacua). The MPM assumes the existence of the SMitself up to the scale ∼ GeV, and predicts that there exist two degenerate vacuainto the SM: the first one – at the Electroweak (EW) scale (with the VEV v ≃ v = v ∼ GeV).In Section 5 we consider the existence in the Nature of the Mirror World (MW) with abroken Mirror Parity (MP): the Higgs VEVs of the visible and invisible worlds are notequal, h φ i = v , h φ ′ i = v ′ and v = v ′ . The parameter characterizing the violation ofthe MP is ζ = v ′ /v ≫
1. We have used the result ζ ≃ φ and mirror Higgs boson φ ′ , which interact during Inflation via gravity. This interaction leads to the emergence ofthe SM vacua at the EW scales with the Higgs boson VEVs v ≈
246 GeV and v ′ = ζ v (with ζ ∼ v = v ∼ GeV. Section 7 containsConclusions.
General Theory of Relativity (GTR) was formulated by Einstein as dynamics of themetrics g µν . Later, Plebanski [6] and other authors (see for example [7,8]) presented GTRin the self-dual approach, in which fundamental variables are 1-forms of connections A IJ In this paper the superscript ’prime’ denotes the M- or hidden H-world. e I : A IJ = A IJµ dx µ , e I = e Iµ dx µ . (1)Also 1-form A = A IJ γ IJ is used, in which generators γ IJ are products of generators ofthe Clifford algebra Cl (1 , γ IJ = γ I γ J . Indices I, J = 0 , , , η IJ = diag(1 , − , − , − g µν . In this case connection belongs tothe local Lorentz group SO (1 , Spin (1 , SU ( N ) or SO ( N ) gauge and Higgs fields (see [2]), thegauge algebra is g = spin ( p, q ), and we have I, J = 1 , ...p + q . In our model of unificationof gravity with the weak SU (2) interactions we consider a group of symmetry with the Liealgebra spin (4 , I, J run over all 8 × I, J = 1 , .., , B IJ = e I ∧ e J = 12 e Iµ e Jν dx µ ∧ dx ν , F IJ = 12 F IJµν dx µ ∧ dx ν , where F IJµν = ∂ µ A IJν − ∂ ν A IJµ +[ A µ , A ν ] IJ , which determines the Riemann-Cartan curvature: R κλµν = e Iκ e Jλ F IJµν . Also 2-forms of B and F are considered : B = 12 B IJ γ IJ , F = 12 F IJ γ IJ , F = dA + 12 [ A, A ] . (2)The well-known in literature Plebanski’s BF -theory is submitted by the following gravi-tational action with nonzero cosmological constant Λ: I ( GR ) = 1 κ Z ǫ IJKL (cid:18) B IJ ∧ F KL + Λ4 B IJ ∧ B KL (cid:19) , (3)where κ = 8 πG N , G N is the Newton’s gravitational constant, and M red.P l = 1 / √ πG N .Considering the dual tensors: F ∗ µν ≡ √− g ǫ ρσµν F ρσ , A ⋆IJ = 12 ǫ IJKL A KL , we can determine self-dual (+) and anti-self-dual (-) components of the tensor A IJ : A ( ± ) IJ = (cid:0) P ± A (cid:1) IJ = 12 (cid:0) A IJ ± iA ⋆ IJ (cid:1) . (4)Two projectors on the spaces of the so-called self- and anti-self-dual tensors P ± = 12 (cid:0) δ IJKL ± iǫ IJKL (cid:1) carry out the following homomorphism: so (1 ,
3) = sl (2 , C ) R ⊕ sl (2 , C ) L . (5)As a result of Eq. (5), non-zero components of connections are only A ( ± ) i = A ( ± )0 i . Insteadof (anti-)self-duality, the terms of left-handed (+) and right-handed (-) components areused. Plebanski [6] and other authors [7, 8] suggested to consider a gravitational action inthe (visible) world as a left-handed sl (2 , C ) ( grav ) L - invariant action, which contains self-dualfields F = F (+) i and Σ = Σ (+) i (i=1,2,3): I ( grav ) (Σ , A, ψ ) = 1 κ Z h Σ i ∧ F i + (cid:0) Ψ − (cid:1) ij Σ i ∧ Σ j i . (6)3ere Σ i = 2 B i , and Ψ ij are auxiliary fields, defining a gauge, which provides equivalenceof Eq. (6) to the Einstein-Hilbert gravitational action: I ( EG ) = 1 κ Z d x ( R − Λ) , (7)where R is a scalar curvature, and Λ is the Einstein cosmological constant. On a way of unification of the gravitational and weak interactions we considered anextended g = spin (4 , I ( A, B,
Φ) = 1 g uni Z M (cid:28) BF + B Φ B + 13 B ΦΦΦ B (cid:29) , (8)where h ... i means a wedge product, g uni is an unification parameter, and Φ IJKL are aux-iliary fields.Having considered the equations of motion, obtained by means of the action (8),and having chosen a possible class of solutions, we can present the following action forthe Graviweak unification (see details in Refs. [1, 2]): I ( A, Φ) = 18 g uni Z M h Φ F F i , (9)where h Φ F F i = d x ǫ µνρσ Φ µν ϕχIJ KL F ϕχIJ F ρσKL , (10)and Φ µν ρσabcd = ( e fµ )( e gν ) ǫ fgkl ( e ρk )( e σl ) δ abcd . (11)A spontaneous symmetry breaking of our new action that produces the dynamics ofgravity, weak SU (2) gauge and Higgs fields, leads to the conservation of the followingsub-algebra: ˜ g = s l (2 , C ) ( grav ) L ⊕ s u (2) L . Considering indices a, b ∈ { , , , } as corresponding to I, J = 1 , , ,
4, and indices m, n as corresponding to indices
I, J = 5 , , ,
8, we can present a spontaneous violation of theGraviweak unification symmetry in terms of the 2-forms: A = 12 ω + 14 E + A W , where ω = ω ab γ ab is a gravitational spin-connection, which corresponds to the sub-algebra sl (2 , C ) ( grav ) L . The connection E = E am γ am corresponds to the non-diagonal componentsof the matrix A IJ , described by the following way (see [2]): E = eϕ = e aµ γ a ϕ m γ m dx µ . Theconnection A W = A mn γ mn gives: A W = A iW τ i , which corresponds to the sub-algebra su (2) L of the weak interaction. Here τ i are the Pauli matrices with i = 1 , , ϕ m = ( ϕ, ϕ i ), we can consider a symmetrybreakdown of the Gravi-Weak Unification, leading to the following OW-action [1]: I ( OW ) ( e, ϕ, A, A W ) = 38 g uni Z M d x | e | (cid:18) | ϕ | R − | ϕ | + 116 R abcd R abcd − D a ϕ † D a ϕ − F iW ab F iW ab (cid:19) . (12)4n Eq. (12) we have the Riemann scalar curvature R ; | ϕ | = ϕ † ϕ is a squared scalar field,which from the beginning is not the Higgs field of the Standard Model; D ϕ = dϕ +[ A W , ϕ ]is a covariant derivative of the scalar field, and F W = dA W + [ A W , A W ] is a curvatureof the gauge field A W . The third member of the action (12) is a topological term in theGauss-Bone theory of gravity (see for example [9, 10]).Lagrangian in the action (12) leads to the nonzero vacuum expectation value (VEV)of the scalar field: v = h ϕ i = ϕ , which corresponds to a local minimum of the effective po-tential V eff ( ϕ ) at v = R /
3, where R > G N is defined by the expres-sion: 8 πG N = ( M ( red. ) P l ) − = 64 g uni v , (13)a bare cosmological constant is equal toΛ = 34 v , (14)and g W = 8 g uni / . (15)The coupling constant g W is a bare coupling constant of the weak interaction, which alsocoincides with a value of the constant g = g W at the Planck scale. Considering therunning α − ( µ ), where α = g / π , we can carry out an extrapolation of this rate to thePlanck scale, what leads to the following estimation [11, 12]: α ( M P l ) ∼ / , (16)and then the overall GWU parameter is: g uni ∼ . . The radiative corrections to the effective Higgs potential, considered in Refs. [13,14], bringto the emergence of the second minimum of the effective Higgs potential at the Planckscale. It was shown that in the 2-loop approximation of the effective Higgs potential,experimental values of all running coupling constants in the SM predict an existenceof the second minimum of this potential located near the Planck scale, at the value v = ϕ min ∼ M P l .In general, a quantum field theory allows an existence of several minima of theeffective potential, which is a function of a scalar field. If all vacua, corresponding to theseminima, are degenerate, having zero cosmological constants, then we can speak about theexistence of a multiple critical point (MCP) at the phase diagram of theory consideredfor the investigation (see Refs. [16, 17]). In Ref. [16] Bennett and Nielsen suggested theMultiple Point Model (MPM) of the Universe, which contains simply the SM itself up tothe scale ∼ GeV. In Ref. [18] the MPM was applied (by the consideration of the twodegenerate vacua in the SM) for the prediction of the top-quark and Higgs boson masses,which gave: M t = 173 ± , M H = 135 ± . (17)Later, the prediction for the mass of the Higgs boson was improved by the calculation ofthe two-loop radiative corrections to the effective Higgs potential [13,14]. The predictions:125 GeV . M H .
143 GeV in Ref. [13], and 129 ± M H ≈
126 GeV observed at the LHC.5he authors of Ref. [15] have shown that the most interesting aspect of the measured valueof M H is its near-criticality. They have thoroughly studied the condition of near-criticalityin terms of the SM parameters at the high (Planck) scale. They extrapolated the SMparameters up to large energies with full 3-loop NNLO RGE precision. All these resultsmean that the radiative corrections to the Higgs effective potential lead to the value ofthe Higgs mass existing in the Nature.Having substituted in Eq.(13) the values of g uni ≃ . G N = 1 / π ( M red.P l ) ,where M red.P l ≈ . · GeV, it is easy to obtain the VEV’s value v , which in this caseis located near the Planck scale: v = v ≈ . · GeV . (18)Such a result takes place, if the Universe at the early stage stayed in the ”false vacuum”,in which the VEV of the Higgs field is huge: v = v ∼ GeV . The exit from this statecould be carried out only by means of the existence of the second scalar field. In thepresent paper we assume that the second scalar field, participating into the Inflation, isthe mirror Higgs field, which arises from the interaction between the Higgs fields of thevisible and invisible sectors of the Universe.
As it was noted at the beginning of this paper, we assumed the parallel existence in theNature of the visible (OW) and invisible (MW) (mirror) worlds.Such a hypothesis was suggested in Refs. [19, 20].The Mirror World (MW) is a mirror copy of the Ordinary World (OW) and containsthe same particles and types of interactions as our visible world, but with the oppositechirality. Lee and Yang [19] were first to suggest such a duplication of the worlds, whichrestores the left-right symmetry of the Nature. The term “Mirror Matter” was introducedby Kobzarev, Okun and Pomeranchuk [20], who first suggested to consider MW as ahidden (invisible) sector of the Universe, which interacts with the ordinary (visible) worldonly via gravity, or another (presumably scalar) very weak interaction.In the present paper we consider the hidden sector of the Universe as a Mirror World(MW) with broken Mirror Parity (MP) [21–25]. If the ordinary and mirror worlds areidentical, then O- and M-particles should have the same cosmological densities. But thisis immediately in conflict with recent astrophysical measurements [43–45]. Astrophysicaland cosmological observations have revealed the existence of the Dark Matter (DM),which constitutes about 25% of the total energy density of the Universe. This is fivetimes larger than all the visible matter, Ω DM : Ω M ≃ G SM of the Standard Model was enlarged to G SM × G ′ SM ′ ,where G SM stands for the observable SM, while G ′ SM ′ is its mirror gauge counterpart.Here O(M)- particles are singlets of the group G SM ( G SM ′ ).It was assumed that the VEVs of the Higgs doublets φ and φ ′ are not equal: h φ i = v, h φ ′ i = v ′ , and v = v ′ . The parameter characterizing the violation of the MP is ζ = v ′ /v ≫
1. Astrophysicalestimates give: ζ > , ζ ∼
100 (see references in [32, 33]).6he action I ( MW ) in the mirror world is represented by the same integral (12), inwhich we have to make the replacement of all OW-fields by their mirror counterparts: e, φ, A, A W , R → e ′ , φ ′ , A ′ , A ′ W , R ′ . Then: G ′ N = ζ G N , Λ ′ = ζ Λ , M ′ red.P l = ζ M red.P l . (19)However, g ′ W = g W : it is supposed that at the early stage of evolution of the Universe,when the GUT takes place, mirror parity is unbroken, what gives g ′ uni = g uni . It is well-known that the hidden (invisible) sector of the Universe interacts with theordinary (visible) world only via gravity, or another very weak interaction (see for example[20, 30, 34]). In particular, the authors of Ref. [35] assumed, that along with gravitationalinteraction there also exists the interaction between the initial Higgs fields of both OW-and MW-worlds: V int = α h ( ϕ † ϕ )( ϕ ′† ϕ ′ ) , (20)which begin to interact during the Inflation via gravitational interactions. The existenceof the second Higgs field ϕ ′ could be the cause of the hybrid inflation (see the model ofHybrid inflation by A. Linde [36]), bringing the Universe out of the “false vacuum” withthe VEV v ∼ GeV. This circumstance provided the subsequent transition to thevacuum with the Higgs VEV v existing at the Electroweak (EW) scale. Here v ≈ I ( OW ) ( e, ϕ, A, A W ) = 364 g uni Z M d x | e | (cid:18) | ϕ | R − | ϕ | + ... (cid:19) . (21)Considering the background value R ≃ R , we can find a minimum of the potential: V eff ( ϕ ) ∼ − | ϕ | R + 34 | ϕ | (22)at ϕ = < ϕ > = v . Here v = R /
3. Then according to (13), we obtain: I ( OW ) ( e, ϕ, A, A W ) = Z M d x √− g (cid:18) M redP l v (cid:19) (cid:18) | ϕ | R − | ϕ | + ... (cid:19) . (23)In the action (23) the Lagrangian includes the non-minimal coupling with gravity [37–39].We see that the field ϕ is not stuck at ϕ anymore, but it can be represented as ϕ = ϕ − σ = v − σ, (24)where the scalar field σ is an inflaton. Here we see that in the minimum, when ϕ = v ,the inflaton field is zero ( σ = 0) and then it increases with falling of the field ϕ .Considering the expansion of the Lagrangian around the background value R ≃ R in powers of the small value σ/v , and leaving only the first-power terms, we can presentthe gravitational part of the action as: I ( grav OW ) = Z M d x √− g (cid:0) M redP l (cid:1) (cid:18) R | − σ/v | − Λ | − σ/v | + ... (cid:19) . (25)7ere Λ = v = R /
4. Using the last relations, we obtain: I ( grav OW ) = Z M d x √− g (cid:0) M redP l (cid:1) (cid:18) Λ − m | σ | + ... (cid:19) , (26)where m = 6 and m is the bare mass of the inflaton in units M red.P l = 1.In the Einstein-Hilbert action the vacuum energy is: ρ vac = (cid:0) M redP l (cid:1) Λ . (27)In our case (26) the vacuum energy density is negative: ρ = − (cid:0) M redP l (cid:1) Λ . (28)However, assuming the existence of the discrete space-time of the Universe at the Planckscale and using the prediction of the non-commutativity suggested by B.G. Sidharth[40, 41], we obtain that the gravitational part of the GWU action has the vacuum energydensity equal to zero or almost zero.Indeed, the total cosmological constant and the total vacuum density of the Universecontain also the vacuum fluctuations of fermions and other SM boson fields:Λ ≡ Λ eff = Λ ZMD − Λ − Λ ( NC ) s + Λ ( NC ) f , (29)where Λ ZMD is zero modes degrees of freedom of all fields existing in the Universe, andΛ ( NC ) s,f are boson and fermion contributions of the non-commutative geometry of the dis-crete spacetime at the Planck scale. If according to the theory by B.G. Sidharth [40, 41],we have: ρ (0) vac = (cid:0) M redP l (cid:1) Λ (0) = (cid:0) M redP l (cid:1) (Λ ZMD − Λ − Λ ( NC ) s ) ≈ , (30)then Eq. (26) contains the cosmological constant Λ (0) ≈
0. In Eqs. (29) and (30) thebosonic (scalar) contribution of the non-commutativity is: ρ ( NC )( scalar ) = m s (in units : ~ = c = 1) , (31)which is given by the mass m s of the primordial scalar field ϕ . Then the discrete spacetimeat the very small distances is a lattice (or has a lattice-like structure) with a parameter a = λ s = 1 /m s . This is a scalar length: a = λ s ∼ − GeV − , which coincides with the Planck length: λ P l = 1 /M P l ≈ − GeV − . The assumption:Λ (0) = Λ
ZMD − Λ − Λ ( NC ) s ≈ ρ DE = (cid:0) M redP l (cid:1) Λ ( NC ) f , (33)considering the non-commutative contribution of light primordial neutrinos as a dominantcontribution to ρ DE , which coincides with astrophysical measurements [43–45]: ρ DE ≈ (2 . × − eV) . (34)8eturning to the Inflation model, we rewrite the action (26) as: I ( grav OW ) = − Z M d x √− g (cid:0) M redP l (cid:1) (cid:18) Λ + m | σ | + ... (cid:19) , (35)where the positive cosmological constant Λ is not zero, but is very small.Taking into account the interaction of the ordinary and mirror scalar bosons ϕ and ϕ ′ , given by equation analogous to Eq. (20) [35], we obtain: I ( grav ) = Z M d x √− g "(cid:18) M redP l v (cid:19) (cid:18) | ϕ | R − | ϕ | − α h | ϕ | | ϕ ′ | + ... (cid:19) + ... + Z M d x √− g M ′ redP l v ′ ! (cid:18) | ϕ ′ | R ′ − | ϕ ′ | − α h | ϕ | | ϕ ′ | + ... (cid:19) + ... . (36)Considering the Planck scale Higgs potential, corresponding to the action (36), we have: V ( ϕ, ϕ ′ ) ≃ (cid:18) M redP l v (cid:19) (cid:18) − | ϕ | R + 34 | ϕ | + α h | ϕ | | ϕ ′ | (cid:19) + M ′ redP l v ′ ! (cid:18) − | ϕ ′ | R ′ + 34 | ϕ ′ | + α h | ϕ | | ϕ ′ | (cid:19) . (37)According to (13), we have: (cid:18) M redP l v (cid:19) = M ′ redP l v ′ ! , (38)and the local minima at ϕ = v and ϕ ′ = v ′ are given by the following conditions: ∂V ( ϕ, ϕ ′ ) ∂ | ϕ | (cid:12)(cid:12) | ϕ | = v = (cid:18) M redP l v (cid:19) (cid:18) − R + 32 v + 2 α h | ϕ ′ | (cid:19) = 0 , (39) ∂V ( ϕ, ϕ ′ ) ∂ | ϕ ′ | (cid:12)(cid:12) | ϕ ′ | = v ′ = (cid:18) M redP l v (cid:19) (cid:18) − R ′ + 32 v ′ + 2 α h | ϕ | (cid:19) = 0 , (40)which give the following solutions: v ≃ R − α h | ϕ ′ | , (41) v ′ ≃ R ′ − α h | ϕ | , (42)and V ( ϕ = v, ϕ ′ = v ′ ) = − (cid:2) ( M redP l ) R + ( M ′ redP l ) R ′ (cid:3) = − ( M redP l ) Λ − ( M ′ redP l ) Λ ′ . (43)Finally, according to (13), we obtain: V ( ϕ = v, ϕ ′ = v ′ ) == − (1 + ζ )( M redP l ) Λ , (44)what gives the negative vacuum energy density.9owever, the cosmological constant is not given by Eq. (44). According to the ideasof non-commutativity given by B.G. Sidharth in Refs. [40–42], it must be replaced by thecosmological constant Λ, which is related with the potential V ( ϕ = v, ϕ ′ = v ′ ) and withthe Dark Energy density (34) by the following way: V ( ϕ = v, ϕ ′ = v ′ ) = ( M redP l ) Λ = ρ DE , (45)where Λ = Λ eff + Λ ′ eff , Λ eff is given by Eq. (29), and Λ ′ eff is its mirror counterpart.Then using the notation: ϕ = v − σ and ϕ ′ = v ′ − σ ′ , (46)and neglecting the terms containing α h as very small, it is not difficult to see that thepotential near the Planck scale is: V ( ϕ, ϕ ′ ) = ( M redP l ) (Λ + m | σ | + m ′ | σ ′ | + ... ) , (47)where m ≃ m ′ ≃ ζ (compare with (34)).The local minimum of the potential (47) at ϕ = v , when σ = 0, and ϕ ′ = v ′ ( σ ′ = 0) gives: V ( v, ϕ ′ ) = ( M redP l ) (Λ + m ′ | σ ′ | + ... ) . (48)The last equation (48) shows that the potential V ( v ) grows with growth of σ ′ , i.e. withfalling of the field ϕ ′ . It means that a barrier of potential grows and at some value σ ′ = σ ′ | in potential begins its inflationary falling. Here it is necessary to comment thatthe position of the minimum also is displaced towards smaller ϕ (bigger σ ), according tothe formula (41).Our next step is an assumption that during the inflation σ decays into the two Higgsdoublets of the SM: σ → φ † + φ. (49)As a result, we have: σ = a d | φ | , (50)where φ is the Higgs doublet field of the Standard Model. The Higgs field φ also interactsdirectly with field φ ′ , according to the interaction (20) given by Ref. [35]. It has a timeevolution and modifies the shape of the barrier so that at some value φ ′ E can roll down thefield ϕ . This possibility, which we consider in our paper, is given by the so-called HybridInflation scenarios [36]. Here we assume that the field φ begins the inflation at the value φ | in ≃ H .Using the relations given by GWU, we obtain near the local “false vacuum” thefollowing gravitational potential: V ( φ, φ ′ ) ≃ Λ + λ | φ | + λ ′ | φ ′ | + a h | φ | | φ ′ | , (51)where λ = 12 a d and λ ′ = 12( a ′ d ) are self-couplings of the Higgs doublet fields φ and φ ′ ,respectively.Returning to the problem of the Inflation, we see that the action of the GWU theoryhas to be written near the Planck scale as: I grav ≃ − Z M d x √− g (cid:0) M redP l (cid:1) (cid:18) Λ + λ | φ | + λ ′ | φ ′ | + a h | φ | | φ ′ | + ... (cid:19) , (52)10here the cosmological constant Λ is almost zero (has an extremely tiny value).The next step is to see the evolution of the Inflation in our model, based on theGWU with two Higgs fields φ and mirror φ ′ .In the present investigation we considered only the result of such an Inflation, whichcorresponds to the assumption of the MPP, that cosmological constant is zero (or almostzero) at both vacua: at the ”first vacuum” with VEV v = 246 GeV and at the ”secondvacuum” with VEV v = v ∼ GeV. If so, we have the following conditions of theMPP (see section 4): V eff ( φ min ) = V eff ( φ min ) = 0 , (53) ∂V eff ∂ | φ | (cid:12)(cid:12) φ = φ min = ∂V eff ∂ | φ | (cid:12)(cid:12) φ = φ min = 0 . (54)Considering the total Universe as two worlds, ordinary OW and mirror MW, we presentthe following expression for the low energy total effective Higgs potential (which is farfrom the Planck scale): V eff = − µ | φ | + 14 λ ( φ ) | φ | − µ ′ | φ ′ | + 14 λ ′ ( φ ′ ) | φ ′ | + 14 α h ( φ, φ ′ ) | φ | | φ ′ | , (55)where α ( φ, φ ′ ) is a coupling constant of the interaction of the ordinary Higgs field φ withmirror Higgs field φ ′ .According to the MPP, at the critical point of the phase diagram of our theory,corresponding to the ”second vacuum”, we have: µ = µ ′ = 0 , λ ( φ ) ≃ , λ ′ ( φ ′ ) ≃ , (56)and then α h ( φ , φ ′ ) ≃ , if V criteff ( v ) ≃ . (57)At the critical point, corresponding to the first EW vacuum v = 246 GeV, we also have V criteff ( v ) ≃
0, according to the MPP prediction of the existence of the almost degeneratevacua in the Universe.Then we can present the full scalar Higgs potential by the following expression: V eff ( φ, φ ′ ) = 14 (cid:0) λ ( | φ | − v ) + λ ′ ( | φ ′ | − v ′ ) + α h ( φ, φ ′ )( | φ ′ | − v ′ ) | φ | (cid:1) , (58)where we have shifted the interaction term: V int = 14 α h ( φ, φ ′ ) | φ | | φ ′ | (59)in such a way that the interaction term vanishes, when φ ′ = φ ′ = v ′ , recovering theusual Standard Model.At the end of the Inflation we have: φ ′ = φ ′ E , and the first vacuum value of V eff isgiven by: V eff ( v , φ ′ E ) = 14 (cid:0) λ ′ ( | φ ′ E | − v ′ ) + α h ( v , φ ′ E )( | φ ′ E | − v ′ ) v (cid:1) = 0 , (60)and ∂V eff ∂ | φ | (cid:12)(cid:12)(cid:12)(cid:12) φ = v φ ′ = φ ′ E (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (61)This means that the end of the Inflation occurs at the value: φ ′ E = v ′ = ζ v , (62)11hich coincides with the VEV < φ ′ > of the field φ ′ at the first vacuum in the mirrorworld MW. Thus, V eff ( φ, φ ′ E ) = 14 λ ( | φ | − v ) , (63)which means the Standard Model with the first vacuum, having the VEV v ≈
246 GeV.
In the present paper we constructed a model of unification of gravity with the weak SU (2)gauge and Higgs fields. Imagining that at the early stage of the evolution the Universe wasdescribed by a GUT-group, we assumed that this Grand Unification group of symmetrywas quickly broken down to the direct product of the gauge groups of internal symmetryand Spin(4,4)-group of the Graviweak unification.Also we assumed the existence of visible and invisible (hidden) sectors of the Uni-verse. We have given arguments that modern astrophysical and cosmological measure-ments lead to a model of the Mirror World with a broken Mirror Parity (MP), in whichthe Higgs VEVs of the visible and invisible worlds are not equal: h φ i = v, h φ ′ i = v ′ and v = v ′ . We estimated a parameter characterizing the violation of the MP: ζ = v ′ /v ≫
1. We have used the result: ζ ∼
100 obtained by Z. Berezhiani and hiscollaborators. In this model, we showed that the action for gravitational and SU (2)Yang–Mills and Higgs fields, constructed in the ordinary world (OW), has a modifiedduplication for the hidden (mirror) sector of the Universe (MW).Considering the Graviweak symmetry breaking, we have obtained the following sub-algebras: ˜ g = s u (2) ( grav ) L ⊕ s u (2) L – in the ordinary world, and ˜ g ′ = s u (2) ′ ( grav ) R ⊕ s u (2) ′ R – in the hidden world. These sub-algebras contain the self-dual left-handed gravity in theOW, and the anti-self-dual right-handed gravity in the MW. We assumed, that finally atlow energies, we have a Standard Model and the Einstein-Hilbert’s gravity.We reviewed the Multiple Point Model (MPM) by D.L. Bennett and H.B.Nielsen,who assumed the existence of several minima of the Higgs effective potential with thesame energy density (degenerate vacua of the SM). In the assumption of zero cosmologicalconstants, MPM postulates that all the vacua, which might exist in the Nature (as minimaof the effective potential), should have zero, or approximately zero, cosmological constant.The prediction that there exist two vacua into the SM: the first one – at the Electroweakscale ( v ≃
246 GeV), and the second one – at the Planck scale ( v ∼ GeV), wasconfirmed by calculations in the 2-loop approximation of the Higgs effective potential.The prediction of the top-quark and Higgs masses was given in the assumption that thereexist two vacua into the SM.In the above-mentioned theory we have developed a model of Inflation. Accordingto this model, a singlet field σ , being an inflaton, starts trapped from the “false vacuum”of the Universe at the value of the Higgs field’s VEV v = v ∼ GeV. Then duringthe Inflation σ decays into the two Higgs doublets of the SM: σ → φ † φ . The interactionbetween the ordinary and mirror Higgs fields φ and φ ′ , induced by gravity, generates ahybrid model of the Inflation in the Universe. Such an interaction leads to the emergenceof the SM vacua at the Electroweak scales: with the Higgs boson VEVs v ≈
246 GeV –in the OW, and v ′ = ζ v – in the MW. L.V. Laperashvili greatly thanks the Niels Bohr Institute (Copenhagen, Denmark) andProf. H.B. Nielsen for hospitality, collaboration and financial support. L.V.L. also deeply12hanks the Department of Physics and University of Helsinki for hospitality and financialsupport, and Prof. M. Chaichian and Dr A. Tureanu for fruitful discussions and advises.C.R. Das acknowledges a scholarship from the Funda¸c˜ao para a Ciˆencia e a Tecnolo-gia (FCT, Portugal) (ref. SFRH/BPD/41091/2007), and greatly thanks the Departmentof Physics, Jyv¨askyl¨a University, in particular Prof. Jukka Maalampi (HOD) for hospital-ity and financial support. This work was partially supported by FCT through the projectsCERN/FP/123580/2011, PTDC/FIS-NUC/0548/2012 and CFTP-FCT Unit 777 (PEst-OE/FIS/UI0777/2013) which are partially funded through POCTI (FEDER). C.R. Dasalso sincerely thanks Physical Research Laboratory and Prof. Utpal Sarkar (Dean) forVisiting Scientist position.
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