New Einstein-Hilbert type action with nonlinear SUSY and unity of - PowerPoint PPT Presentation

New Einstein-Hilbert type action with nonlinear SUSY and unity of nature Kazunari Shima and Motomu Tsuda (Saitama Institute of Technology) OUTLINE 1. Motivation 2. Nonlinear-supersymmetric general relativity theory( NLSUSYGR ) 3. Vacuum structure of NLSUSYGR : SUSYQED, SUSYYM 4. Cosmology and low energy particle physics of NLSUSYGR 5. Summary —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 1/63

1. Motivation @ The success of Two SMs, i.e. GR and GWS model. @ However, many unsolved fundamental problems in SMs: e.g., • Gravitational Force, • Space-time dimension four , • Three generations of quarks and leptons, • Chiral eigenstates, • Neutrino mass M ν • Dark Matter, Dark enegy; ρ D.E. ∼ ( M ν ) 4 ⇔ Λ (cosmological term)? • SUSY!?, Origin of SUSY breaking, etc. @ GR describes geometry of space-time. However, unpleasant differences between GR and SUGRA: ⇒ Geometry of Riemann space(Physical:[ x µ ], GL(4,R)) • General Relativity(GR) ⇐ While, ⇒ Geometry of superspace (Mathematical:[ x µ , θ α ], sPoicar´ • SUGRA ⇐ e ) = ⇒ New SUSY paradigm on specific physical space-time . —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 2/63

As for the particle spectrum based upon linear SUSY representation: @ Group theoretical Observation (Z.Phys.C18,25(1983),Euro.Phys.J.C7,341(1999)) : • Among all SO(N) sP, SM with just 3 generations emerges from a single irreducible rep. of only SO(10) sP. • 10 supercharges Q I , ( I = 1 , 2 , · · · . 10) are decomposed and assigned as follows: 10 SO (10) = 5 SU (5) + 5 ∗ SU (5) ⇔ 5 SU (5) GUT analogue multiplet of supercharges: 5 SU (5) = [ 3 ∗ c , 1 ew , ( e 3 ) : Q a ( a = 1 , 2 , 3) ] + [ 1 c , 2 ew , ( − e, 0) : Q m ( m = 4 , 5) ] . 3 , e 3 , e • Massless helicity states of gravity multiplet of SO(10) sP with CPT conjugation are specified by the helicity h = (2 − n 10! 2 ) and the dimension d [ n ] = n !(10 − n )! : | h > = Q I n Q I n − 1 · · · Q I 2 Q I 1 | 2 >, Q I n ( n = 0 , 1 , 2 , · · · , 10) : super charge 5 3 1 | h | 3 2 1 0 2 2 2 1 [0] 10 [1] 45 [2] 120 [3] 210 [4] d [ n ] 1 [10] 10 [9] 45 [8] 120 [7] 210 [6] 252 [5] 210 [4] —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 3/63

@ Spin 1 2 Dirac particles survivours after a tentative Higgs-like mechanism: SU (3) Q e SU (2) ⊗ U (1) ( ) ( ) ( ) ν e ν µ ν τ 0 e µ τ 1 − 1 ( E ) ( M ) − 2 5 / 3 ( ) ( ) a g 2 / 3   ( ) ( ) ( ) r u c t f m 3 − 1 / 3 ( ) i h   d s b   − 4 / 3 n o     4 / 3 P X 6 1 / 3 Q Y         − 2 / 3 R Z ( ) ( ) 0 N 1 N 2 8 − 1 E 1 E 2 @ One SM Higg-doublet survives in the low energy —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 4/63

• How to write down N=10 SUSY with gravity beyond N-G theorem in S-matrix ?! • We need (i) A certain degeneracy of space-time, (ii) General Relativity principle on physical SUSY space-time possesing space-time symmetries SO (1 , 3) , SL (2 , C ) , GL (4 , R ) . We show in this talk: • N=10 SUSY with gravity is obtained by the geometrical description of specific unstable physical (Riemann) space-time possesessing NLSUSY structure at each point . • The nonlinear(NL) SUSY invariant coupling of spin 1 2 fermion with spin 2 graviton circumvents the no-go theorem for SO(N > 8) Linear SUSY. ⇓ • New SUSY paragigm beyond the SMs indicating a certain gravitational composite structure for all particles and/or a fundamental fermionic structure. —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 5/63

A brief review of NLSUSY: • Take flat space-time specified by x a and ψ α . • Consider one form ω a = dx a + κ 2 2 i ( ¯ ψγ a dψ − d ¯ ψγ a ψ ) , κ is an arbitrary constant with the dimension l +2 . • δω a = 0 under δx a = iκ 2 2 (¯ ζγ a ψ − ¯ ψγ a ζ ) and δψ = ζ with a global spinor parameter ζ . • An invariant acction( ∼ invariant volume) is obtained: ω 0 ∧ ω 1 ∧ ω 2 ∧ ω 3 = S = − 1 d 4 xL V A , ∫ ∫ 2 κ 2 L V A is N=1 Volkov-Akulov model of NLSUSY given by L VA = − 1 2 κ 2 | w V A | = − 1 1 + t aa + 1 [ 2 ( t aa t bb − t ab t ba ) + · · · ] , 2 κ 2 | w V A | = det w ab = det( δ a b + t ab ) , t ab = − iκ 2 ( ¯ ψγ a ∂ b ψ − ¯ ψγ a ∂ b ψ ) , which is invariant under N=1 NLSUSY transformation: κ ζ − iκ (¯ ζγ a ψ − ¯ δ ζ ψ = 1 ζγ a ψ ) ∂ a ψ . ← → NG fermioon for SB SUSY • ψ is NG fermion (the coset space coordinate) of superP oincare . P oincare • ψ is quantized canonically in compatible with SUSY algebra. —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 6/63

2. Nonlinear-Supersymmetric General Relativity (NLSUSYGR) 2.1. New Space-time as Ultimate Shape of Nature We consider new (unstable) physical space-time inspired by nonlinear(NL) SUSY: The tangent space of new space-time is specified by SL(2,C) Grassmann coordinates ψ α for NLSUSY besides the ordinary SO(1,3) Minkowski coordinates x a , i.e., the coordinate ψ α of the the coset space superGL (4 ,R ) turning to the NLSUSY NG fermion GL (4 ,R ) (called superon hereafter) are attached at every curved space-time point besides x a . —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 7/63

• Ultimate shape of nature ⇐ ⇒ (empy) unstable space-time: w aµ : unified vierbein { x a , ψ i α } { x µ } Λ w aµ − → δ a µ New space-time ( Locally homomorphic non-compact groups SO(1,3) and SL(2,C) for space-time symmetry are analogous to compact groups SO(3) and SU(2) for gauge symmetry of ’t Hooft-Polyakov monopole, though SL(2,C) is realized nonlinearly. ) • Note that SO (1 , 3) ∼ = SL (2 , C ) is crucial for NLSUSYGR scenario. 4 dimensional space-time is singled out. —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 8/63

2.2. Nonlinear-Supersymmetric General Relativity (NLSUSYGR) We have found that geometrical arguments of Einstein general relativity(GR) can be extended to new (unstable) space-time . • Unified vierbein w aµ ( x ) ( ulvierbein ) of new space-time: (Note: Grassmann d.o.f. induces the imaginary part of w aµ ( x ) .) w aµ ( x ) = e aµ + t aµ ( ψ ) , w µa ( x ) = e µa − t µa + t µρ t ρa − t µσ t σρ t ρa + t µκ t κσ t σρ t ρa , w aµ ( x ) w µb ( x ) = δ ab t aµ ( ψ ) = κ 2 ψ I γ a ∂ µ ψ I − ∂ µ ¯ 2 i ( ¯ ψ I γ a ψ I ) , ( I = 1 , 2 , .., N ) (By conventions the first index A and the second index B of t AB represent those of γ -matrix and the derivative, respectively.) • N -extended NLSUSYGR action of Eienstein-Hilbert(EH)-type for new space-time. = ⇒ —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 9/63

N -extended NLSUSY GR action: ( Phys.Lett.B501,237(2001), Phys.Lett.B507,260(2001) .) c 4 L NLSUSYGR ( w ) = − 16 πG | w |{ Ω( w ) + Λ } , (1) | w | = det w a µ = det( e a µ + t a µ ( ψ )) , (2) µ ( ψ ) = κ 2 ψ I γ a ∂ µ ψ I − ∂ µ ¯ 2 i ( ¯ t a ψ I γ a ψ I ) , ( I = 1 , 2 , .., N ) (3) • w aµ ( x )(= e aµ + t aµ ( ψ )) : the vierbein of new space-time(ulvierbein) • e aµ ( x ) : the ordinary vierbein for the local SO(1,3) d.o.f.of GR, • t aµ ( ψ ( x )) : the mimic vierbein for the local SL(2,C) d.o.f. composed of the stress-energy-momentum of NG fermion ψ ( x ) I (called superons), • Ω( w ) : the scalar curvature of new space-time in terms of w aµ , • s µν ≡ w aµ η ab w bν , s µν ( x ) ≡ w µa ( x ) η ab w νa ( x ) : metric tensors of new space-time. • G : the Newton gravitational constant. • Λ : cosmological term in new space-time indicating NLSUSY of tangent space. —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 10/63

• Remarkably NLSUSYGR scenario fixes the arbitrary constatnt κ 2 to κ 2 = ( c 4 Λ 8 πG ) − 1 , with the dimension ( length ) 4 ∼ ( enegy ) − 4 . • The sign Λ > 0 in the action L NLSUSYGR is now fixed uniuely, (i) which gives the correct sign to the kinetic term of ψ ( x ) in the energy momentum tensor and (ii)allows the negative dark energy density interpretation of Λ in the Einstein equation. ( → Sec.4). • No-go theorem for N > 8 with gravity has been circumvented by using NLSUSY, i.e. by the vacuum(flat space) degeneracy. = 2( d 2 − 1) holds • Note that SO (1 , D − 1) ∼ = SL ( d, C ) , i.e. D ( D − 1) 2 for only D = 4 , d = 2 . NLSUSYGR scenario predicts 4 dimensional space-time . —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 11/63

2.3. Symmetries of NLSUSY GR(N-extended action) • Space-time symmetries ( ∼ sP ) : [new NLSUSY] ⊗ [local GL(4 , R)] ⊗ [local Lorentz] (4) • Internal symmetries for N-extended NLSUSY GR (N-superons ψ I ( I = 1 , 2 , ..N ) ): [global SO(N)] ⊗ [local U(1) N ] ⊗ [chiral] . (5) —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 12/63

Examples: • Invariance under the new NLSUSY transformation ; δ ζ ψ I = 1 κζ I − iκ ¯ µ = iκ ¯ ζ J γ ρ ψ J ∂ ρ ψ I , δ ζ e a ζ J γ ρ ψ J ∂ [ µ e a ρ ] . (6) (6) induce GL(4,R) transformations on w aµ and the unified metric s µν δ ζ w a µ = ξ ν ∂ ν w a µ + ∂ µ ξ ν w a δ ζ s µν = ξ κ ∂ κ s µν + ∂ µ ξ κ s κν + ∂ ν ξ κ s µκ , ν , (7) where ζ is a constant spinor parameter, ∂ [ ρ e aµ ] = ∂ ρ e aµ − ∂ µ e aρ and ξ ρ = − iκ ¯ ζ I γ ρ ψ I . Commutators of two new NLSUSY transformations (6) on ψ I and e aµ close to GL(4,R), [ δ ζ 1 , δ ζ 2 ] ψ I = Ξ µ ∂ µ ψ I , [ δ ζ 1 , δ ζ 2 ] e a µ = Ξ ρ ∂ ρ e a µ + e a ρ ∂ µ Ξ ρ , (8) where Ξ µ = 2 i ¯ ζ I 1 γ µ ζ I 2 − ξ ρ 1 ξ σ 2 e aµ ∂ [ ρ e aσ ] . q.e.d. —HotGRG/9-15/08/2015/Quy Nhon, Vietnam — 13/63