Nonlinear-SUSY General Relativity Theory(NLSUSYGR) -Unification of - PowerPoint PPT Presentation

Nonlinear-SUSY General Relativity Theory(NLSUSYGR) -Unification of space-time and matter- Kazunari Shima Saitama Institute of Technology OUTLINE 1. New view of SUSY 2. Nonlinear-supersymmetric general relativity theory( NLSUSYGR ) 3. Linearization of NLSUSY and vacuum of NLSUSYGR : SUSYQED 4. Cosmology and low energy particle physics of NLSUSYGR 5. Nonlinear vector-spinor SUSYGR(3/2NLSUSYGR) 6. Summary —Strings and Fields 2017/07-11/08/YITP, Kyoto — 1/ ??

1. New view of NL and L SUSY @ The success of Two SMs, i.e. GR and GWS model. However, many unsolved fundamental problems in SMs: e.g., • Unification of two SMs. • Space-time dimension four , • Three generations of quarks and leptons, • Tiny Neutrino mass M ν , proton decay and GUT • Dark Matter, Dark enegy; ρ D.E. ∼ ( M ν ) 4 ⇔ Λ (cosmological term)? = ⇒ SUGRA!?, Origin of SUSY breaking, · · · etc. @ GR describes geometry of space-time. However, unpleasant differences between GR andSUGRA: ⇒ Geometry of Riemann space-time(Physical:[ x µ ], GL(4,R)) • GR ⇐ ⇒ Geometry of superspace (Mathematical:[ x µ , θ α ], sPoicar´ • SUGRA ⇐ e ) = ⇒ New SUSY paradigm on specific physical space-time!. —Strings and Fields 2017/07-11/08/YITP, Kyoto — 2/ ??

The three-generations structure based upon linear(L) SUSY representation: @Among all SO(N) sP, SM with just 3 generations emerges from one irreducible rep. of only SO(10) sP. • 10 supercharges Q I , ( I = 1 , 2 , · · · . 10) are embedded as follows: 10 SO (10) = 5 SU (5) + 5 ∗ SU (5) 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 ⇔ 5 SU (5) GUT are [ Q a : ¯ d -type, Q m :Lepton-type] supercharges, • Massless helicity states of gravity multiplet of SO(10) sP with CPT conjugation 10! are specified by the helicity h = (2 − n 2 ) and the dimension d [ n ] = n !(10 − n )! : | h > = Q n Q n − 1 · · · Q 2 Q 1 | 2 >, Q n ( n = 0 , 1 , 2 , · · · , 10) : supercharge 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] —Strings and Fields 2017/07-11/08/YITP, Kyoto — 3/ ??

@ Spin 1 2 (Dirac) state survivours after superHiggs ( SU(2): preliminary) SU (3) Q e SU (2) ⊗ U (1) ( ) ( ) ( ) ν e ν µ ν τ 0 e µ τ 1 − 1 ( E ) − 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 Higgs doublet state survives in h = 0 state. —Strings and Fields 2017/07-11/08/YITP, Kyoto — 4/ ??

• How to construct N=10 SUSY with gravity beyond No-Go theorem in S-matrix ? • To circumvent the No-Go theorem we consider a certain degeneracy of space- time. We show in this talk: • N=10 SUSY with gravity is obtained by the geometric description of General Relativity principle on specific unstable physical (Riemann) space-time whose tangent space possesses NLSUSY structure. ⇓ • A new SUSY paradigm beyond SMs is proposed, which indicates: a gravitational compositeness or a fundamental fermionic internal structure of all particles. —Strings and Fields 2017/07-11/08/YITP, Kyoto — 5/ ??

A quick 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 Nambu-Goldstone(NG) fermion (the coset space coordinate) of superP oincare . P oincare • ψ is quantized canonically in compatible with SUSY algebra. —Strings and Fields 2017/07-11/08/YITP, Kyoto — 6/ ??

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 Minkowski coordinates x a for SO(1,3), i.e., the coordinate ψ α of the the coset space superGL (4 ,R ) turning to the NLSUSY NG GL (4 ,R ) fermion (called superon hereafter) are attached at every curved space-time point besides x a . —Strings and Fields 2017/07-11/08/YITP, Kyoto — 7/ ??

• 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. —Strings and Fields 2017/07-11/08/YITP, Kyoto — 8/ ??

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. = ⇒ —Strings and Fields 2017/07-11/08/YITP, Kyoto — 9/ ??

N -extended NLSUSY GR action: ( Phys.Lett.B501,237(2001), 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. —Strings and Fields 2017/07-11/08/YITP, Kyoto — 10/ ??

• NLSUSYGR scenario fixes the arbitrary constatnt κ 2 to κ 2 = ( c 4 Λ 16 πG ) − 1 , with the dimension ( length ) 4 ∼ ( enegy ) − 4 . • Λ > 0 in the action L NLSUSYGR allows negative dark energy density interpretation of Λ in the Einstein equation. → Sec.4. • No-go theorem for N > 8 SUGRA has been circumvented by using NLSUSY, i.e. by the vacuum(flat space) degeneracy. • Note that SO (1 , D − 1) ∼ = 2( d 2 − 1) holds = SL ( d, C ) , i.e. D ( D − 1) 2 for only D = 4 , d = 2 . NLSUSYGR scenario predicts 4 dimensional space-time. —Strings and Fields 2017/07-11/08/YITP, Kyoto — 11/ ??

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) —Strings and Fields 2017/07-11/08/YITP, Kyoto — 12/ ??

For example: • 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) 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 (??) 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. —Strings and Fields 2017/07-11/08/YITP, Kyoto — 13/ ??