NLSUSY Nambu-Goldstone Fermion Composite Model of Nature - PowerPoint PPT Presentation

NLSUSY Nambu-Goldstone Fermion Composite Model of Nature —Nonlinear-Supersymmetric General Relativity Theory— Kazunari Shima (Saitama Institute of Technology) OUTLINE 1. Motivation 2. Nonlinear-Supersymmetric General Relativity Theory( NLSUSYGR ) 3. Phase Transition( Big Decay ) to Riemann Space-time and Matter 4. SMs of Cosmology and Low Energy Particle Physics from NLSUSYGR 5. Summary —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 1/60

1. Motivation @ How to unify Two SMs for space-time and matter, i.e. GRT and GWS model are confirmed. @ SUSY may be an essential notion beyond SMs, → MSSM, SUSYGUT, SUGRA • SUSY stabilizes the low mass Higgs particle!? @ Many unsolved basic problems in SMs: • Origin of SUSY breaking, • Proton decay, • Three generations of quarks and leptons, • ν oscillations, • Dark Matter, Dark enegy density; ρ D ∼ ( M ν ) 4 ⇔ Λ (cosmological term) @ SUSY constitutes space-time symmetry and describes geometry of space-time. @Geometry and symmetry of specific space-time ⇒ Geometry of superspace (Mathematical:[ x µ , θ α ], sPoicar´ • SUGRA ⇐ e ) While, ⇒ Geometry of Riemann space(Physical:[ x µ ] , GL(4,R)) • General Relativity(GRT) ⇐ = ⇒ New SUSY paradigm on particular physical space-time . —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 2/60

@ SUSY and its spontaneous breakdown are profound notions essentially related to the space-time symmetry, therefore, to be studied in particle physics, cosmology(gravitation) and their relations. = ⇒ SO(N) superPoincar´ e(sP) symmetry gives a natural framework. @ We found group theoretically (Z.P,1983.E.P.J.,1999) : • SM with just three generations equipped with ν R emerges from one irrep representation of SO(10) sP with the decomposition 10 = 5 + 5 ∗ corresponding to SO (10) ⊃ SU (5) , where 5 SU (5) GUT quantum numbers are assigned to 5 . • Proton is stable due to the selection rule despite SU (5) , 1 provided all particles are regarded as composites of fundamental spin 2 objects 5 = 5 SU (5) GUT (Superon Quintet Model)( SQM, spin 1 2 ). SO(N > 8) Linear(L) SUSY = ⇒ NO-GO theorem in S-matrix ! SUSY indicates gravitational compositeness of matter before BB? —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 3/60

SU (3) Q e SU (2) ⊗ U (1) � � � � � � ν 1 ν 2 ν 3 0 l 1 l 2 l 3 1 − 1 E − 2 � � � � � � 2 u 1 u 2 u 3 3 − 1 d 1 d 2 d 3 � � 3 h 3 − 4 o 3 4     P X 3 1 6 Q Y     3     − 2 R Z 3 � � � � 0 N 1 N 2 8 − 1 E 1 E 2 —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 4/60

@ A way to field theoretical breakthrough: We show in this talk: The nonlinear(NL) SUSY invariant coupling of spin 1 • 2 fermion with spin 2 graviton is crucial to circumvent the no-go theorem of S-matrix arguments for SO(N > 8) Linear SUSY. • This is attributed to the geometrical description of particular (empty) unstable space-time unifying: 1 the fundamental object(spin NLSUSY) and the background space-time 2 manifold(general relativity) . • There may be a certain composite (SQM) structure and/or a fundamental fermionic structure beyond the SM. —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 5/60

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 Super − P oincare . P oincare • ψ is quantized canonically in compatible with SUSY algebra. —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 6/60

2. Nonlinear-Supersymmetric General Relativity (NLSUSYGR) 2.1. New Space-time as Ultimate Shape of Nature We consider the following new (unstable) 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 ψ α the coordinates of the the coset space superGL (4 ,R ) turning to the NLSUSY NG GL (4 ,R ) fermion (called superon hereafter) and x a are attached at every curved space-time point. —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 7/60

• 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 spacetime ( 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. —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 8/60

2.2. Nonlinear-Supersymmetric General Relativity (NLSUSYGR) We have found that geometrical arguments of Einstein general relativity(EGR) can be extended to new (unstable) space-time : • Unified vierbein of new space-time: 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 ) (Note: The first and the second indices of t represent those of γ -matrix and the covariant derivative, respectively.) • N -extended NLSUSYGR action of EH-type in new (empty) space-time : —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 9/60

N -extended NLSUSY GR action: c 4 L NLSUSY GR ( 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 unified vierbein of new space-time, • e aµ ( x ) : the ordinary vierbein for the local SO(1,3) of EGR, • t aµ ( ψ ( x )) : the mimic vierbein for the local SL(2,C) composed of the stress-energy- momentum of NG fermion ψ ( x ) I (called superons ), • Ω( w ) : the unified Ricci 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 ) w νa ( x ) : unified metric tensors of new space-time. • G : the Newton gravitational constant. • Λ : cosmological constant in new space-time indicating NLSUSY of tangent space. —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 10/60

• No-go theorem for has been circumvented in a sense that SO(N > 8) SUSY with the non-trivial gravitational interaction has been constructed by using NLSUSY, i.e. the vacuum degeneracy. • Note that SO (1 , D − 1) ∼ = SL ( d, C ) , i.e. = 2( d 2 − 1) D ( D − 1) 2 holds only for D = 4 , d = 2 . NLSUSYGR scenario predicts 4 dimensional space-time . —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 11/60

• 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 . • Also Λ > 0 in the action is now fixed uniuely to give the correct sign to the kinetic term of ψ ( x ) and indicates (i) the positive potential minimum V P.E. ( w ) = Λ > 0 for w aµ ( x ) and (ii) the negative dark energy density interpretation for Λ ( → Sec.4). —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 12/60

2.3. Symmetries of NLSUSY GR(N-extended action) • NLSUSY GR action is invariant at least under the following space-time symmetries which is homomorphic to sP: [new NLSUSY] ⊗ [local GL(4 , R)] ⊗ [local Lorentz] ⊗ [local spinor translation] (4) and • the following internal symmetries for N-extended NLSUSY GR ( with N-superons ψ I ( I = 1 , 2 , ..N ) ) : [global SO(N)] ⊗ [local U(1) N ] ⊗ [chiral] . (5) —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 13/60

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) Because (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. —SCGT14Mini/KMI, Nagoya/05-07/03/2014/K.Shima — 14/60