Nonlinear SUSY General Relativity Theory and Significances
aa r X i v : . [ h e p - t h ] D ec SIT-LP-11/12December, 2011
Nonlinear SUSY General Relativity Theoryand Significances ∗ Kazunari Shima † and Motomu Tsuda ‡ Laboratory of Physics, Saitama Institute of TechnologyFukaya, Saitama 369-0293, Japan
Abstract
We show some consequences of the nonliear supersymmetric general rela-tivity (NLSUSYGR) theory on particle physics, cosmology and their relations.They may give new insights into the SUSY breaking mechanism, dark energy,dark matter and the low enegy superpartner particles which are compatiblewith the recent LHC data. ∗ Based on the talk given by K. Shima at the 7th International Conference
Quantum Theoryand Symmetries , 07-13, August, 2011, Czech Technical University, Prague, Czech Republic † e-mail: [email protected] ‡ e-mail: [email protected] Introduction
Supersymmetry (SUSY) [1] and its spontaneous breakdown are profound notions es-sentially related to the space-time symmetry. Therefore it is natural to study themin the framework including the particle physics and the cosmology (gravitation) aswell. SO ( N ) super-Poincar´e (sP) symmetry may give a natural framework. We havefound by the group theoretical arguments that among all SO ( N ) sP groups only SO (10) sP can accomodate in the low energy the standard model (SM) with justthree generations of quarks and leptons in the single irreducible representation [2],where we have adopted the decomposition 10 SO (10) = 5 SU (5) + 5 ∗ SU (5) correspondingto SO (10) ⊃ SU (5) and assigned to 5 SU (5) the same quantum numbers as those of 5of SU (5) GUT [3]. Therefore it is an interesting problem to construct N = 10 SUSYtheory in curved space-time. For this purpose we must overcome the so called no-gotheorem of the S -matrix arguments in the local field theory for N > SO (1 , and SL (2 , C ) suggested bynonlinear supersymmetry (NLSUSY). Extending the geometric arguments of thegeneral relativity (GR) on ordinary Riemann space-time to new space-time we con-struct Einstein-Hilbert type action (NLSUSYGR theory) which is invariant underNLSUSY transformation. We discuss in this article some basic ideas and some con-sequences for the low energy particle physics and the cosmology of NLSUSY GR.We find that the self-contained phase transition of space-time plays crucial roles inour scenario. As the physical and the simplest case we consider N = 2 case explic-itly and show in the true vacuum of flat space-time that the lepton sector of SMwith U (1) gauge symmetry emerges as the composites of the fundamental Nambu-Goldstone(NG) fermion, i.e. the true vacuum of N = 2 NLSUSY GR is achieved bythe compositeness of particles of the SM. Nonlinear supersymmetric general relativity (NLSUSY GR) theory [4] is based uponthe general relativity (GR) principle and the nonlinear (NL) representation [5] ofSUSY [6]. In NLSUSY GR, four dimensional new space-time [4], as an ultimate2hape of nature, is introduced, where tangent flat space-time has the NLSUSYstructure, i.e. tangent space-time of four dimensional space-time manifold is speci-fied by not only the SO (1 ,
3) Minkowski coodinates x a but also SL (2 , C ) Grassmancoordinates ψ iα ( i = 1 , , · · · , N ) for N -NLSUSY. The Grassmann coordinates innew space-time are regarded as coset space coordinates of superGL (4 ,R ) GL (4 ,R ) , which canbe recasted as the NG-fermions ( superon ) for NLSUSY associated with the spon-taneous breakdown of super- GL (4 , R ) ( sGL (4 , R )) down to GL (4 , R ). By extend-ing the geometric arguments of Einstein GR in ordinary Riemann space-time tonew space-time, we obtain the fundamental action (NLSUSY GR theory) of theEinstein-Hilbert (EH) form [4]; L NLSUSYGR ( w ) = − c πG | w |{ Ω( w ) + Λ } , (1)where G is the Newton gravitational constant, Λ is a ( small ) cosmological constantindicating the NLSUSY structure in Riemann flat e aµ → δ aµ space-time, Ω( w ) is theunified Ricci scalar curvature of new space-time computed in terms of the unifiedvierbein w aµ (and the inverse w µA ) defined by w aµ = e aµ + t aµ ( ψ ) , t aµ ( ψ ) = κ i ( ¯ ψ i γ a ∂ µ ψ i − ∂ µ ¯ ψ i γ a ψ i ) , (2)and | w | = det w aµ . In Eq.(2), e aµ is the ordinary vierbein of GR for the local SO (3 , t aµ ( ψ ) is the mimic vierbein for the local SL (2 , C ) ψ i ( x ) which is recastedas the stress-energy-momentum tensor of the NG fermion ψ i ( x ) created by the sub-sequent spontaneous breakdown of SUSY. κ with the dimemsion (mass) − is anarbitrary constant of NLSUSY representing the fundamental volume ( κ ) of four di-mensional flat space-time which subsequently reinterpreted as the superon-vacuumcoupling constant. Note that e aµ and t aµ ( ψ ) contribute equally to the curvature ofspace-time Ω( w ), which realizes the Mach’s principle in ultimate space-time.The N -NLSUSY GR action (1) possesses the following space-time and internalsymmetries homomorphic to SO ( N ) ( SO (10)) sP symmetry [7], i.e. L NLSUSYGR ( w )is invariant under[new NLSUSY] ⊗ [local GL(4,R)] ⊗ [local Lorentz] ⊗ [local spinor translation] ⊗ [global SO(N)] ⊗ [local U(1) N ] ⊗ [Chiral].Note that the no-go theorem is overcome (circumvented) in a sense that the nontivial N -extended SUSY gravity theory with N >
Particle astrophysics of NLSUSY GR
New empty (except the constant energy density Λ) space-time described by the( matter free ) EH-type NLSUSY GR action equipped with the positive potentialminimum V P.E. = Λ > V P.E. = 0. It decays(called
Big Decay ) spontaneously to ordinary Riemann space-time with the NGfermion ( superon matter) described by the ordinary EH action with the cosmologicalterm, the NLSUSY action for the N NG fermions superon and their gravitationalinteractions, which is called SGM action for superon-graviton(SG) model space-time.By expanding (1) around e aµ the SGM action is given as follows; L SGM ( e, ψ ) = − c πG e | w VA |{ R ( e ) + Λ + T ( e, ψ ) } (3)where R ( e ) is the Ricci scalar curvature of ordinary EH action, T ( e, ψ ) repre-sents highly nonlinear gravitational interaction terms of ψ i , and | w VA | = det w ab =det( δ ab + t ab ) is the NLSUSY invariant four dimensional volume [5]. We can easily seethat the cosmological term in L NLSUSYGR ( w ) of Eq.(1) (i.e. the constant energy den-sity of ultimate space-time) mediated to the second term in SGM action (3) reducesto the NLSUSY action [5], L NLSUSY ( ψ ) = − κ | w VA | in Riemann-flat e aµ ( x ) → δ µa space-time. Therefore, the arbitrary constant κ of NLSUSY should be fixed to κ − = c Λ8 πG . (4)The Big Decay is the phase transition sGL (4 ,R ) GL (4 ,R ) of space-time. It produces the masslessNG fermion, the ordinary Riemann space-time and a fundamental mass scale ofthe spontaneous breakdown of SUSY (SBS) depending on the Λ and G throughthe relation (4). We will show that the effect of Big Decay is mediated to the(low energy) particle physics in (asymptotic) Riemann-flat space-time. Note that L SGM ( e, ψ ) (3) (massless superon-graviton model) preserves V P.E. = Λ >
0, whichleaves the problem of identifying the true vacuum V P.E. = 0 for the massless fermion-graviton world.The nonlinear model sometimes can be related (converted) to the (equivalent)linear theory which is tractable. NLSUSY is also the case and the consequentlinearized theory possesses the true vacuum V P.E. = 0, where the (massless) particlespectrum is determined by the space-time sP symmetry and all particles and forcesbecome composites of ψ i ( x ). We will investigate the low energy physics of NLSUSY4R through the NL/L SUSY relation. To see the (low energy) particle physicscontent in (asymptotic) Riemann-flat space-time we focus on N = 2 SUSY in twodimensional space-time for simplicity, for in the SGM scenario N = 2 case gives theminimal and realistic N = 2 LSUSY QED model [8]. By performing the systematicarguments we can find the NL/L SUSY relation between the N = 2 NLSUSY modeland N = 2 LSUSY QED theory in Riemann-flat space-time [9, 10]; L N =2SGM ( e, ψ ) −→ L N =2NLSUSY ( ψ ) = L N =2LSUSYQED ( V , Φ ) + [tot . der . terms] . (5)In the relation (5), the N = 2 NLSUSY action L N =2NLSUSY ( ψ ) for the two (Majo-rana) NG-fermions superon ψ i ( i = 1 ,
2) is written in d = 2 as follows; L N =2NLSUSY ( ψ )= − κ | w VA | = − κ (cid:26) t aa + 12! ( t aa t bb − t ab t ba ) (cid:27) = − κ (cid:26) − iκ ¯ ψ i ∂ψ i − κ ( ¯ ψ i ∂ψ i ¯ ψ j ∂ψ j − ¯ ψ i γ a ∂ b ψ i ¯ ψ j γ b ∂ a ψ j ) (cid:27) , (6)where κ is a constant with the dimension (mass) − , which satisfies the relation (4).On the other hand, in Eq.(5), the N = 2 LSUSY QED action L N =2SUSYQED ( V , Φ )is constructed from a N = 2 minimal off-shell vector supermultiplet and a N = 2 off-shell scalar supermultiplet denoted V and Φ respectively. Indeed, the most general L N =2SUSYQED ( V , Φ ) in d = 2 with a Fayet-Iliopoulos (FI) D term and Yukawainteractions, is given in the explicit component form as follows for the massless case; L N =2SUSYQED ( V , Φ ) = −
14 ( F ab ) + i λ i ∂λ i + 12 ( ∂ a A ) + 12 ( ∂ a φ ) + 12 D − ξκ D + i χ ∂χ + 12 ( ∂ a B i ) + i ν ∂ν + 12 ( F i ) + f ( A ¯ λ i λ i + ǫ ij φ ¯ λ i γ λ j − A D + φ D + ǫ ab AφF ab )+ e (cid:26) iv a ¯ χγ a ν − ǫ ij v a B i ∂ a B j + ¯ λ i χB i + ǫ ij ¯ λ i νB j − D ( B i ) + 12 A ( ¯ χχ + ¯ νν ) − φ ¯ χγ ν (cid:27) + 12 e ( v a − A − φ )( B i ) . (7)where ( v a , λ i , A, φ, D ) ( F ab = ∂ a v b − ∂ b v a ) are the staffs of the minimal off-shell vectorsupermultiplet V representing v a for a U (1) vector field, λ i for doublet (Majorana)fermions, A for a scalar field in addition to φ for another scalar field and D for anauxiliary scalar field, while ( χ , B i , ν , F i ) are the staffs of the (minimal) off-shell5calar supermultiplet Φ representing ( χ, ν ) for two (Majorana) fermions, B i fordoublet scalar fields and F i for auxiliary scalar fields. Also ξ in the FI D term is anarbitrary dimensionless parameter turning to a magnitude of SUSY breaking mass,and f and e are Yukawa and gauge coupling constants with the dimension (mass) (in d = 2), respectively. The N = 2 LSUSY QED action (7) can be rewritten in thefamiliar manifestly covariant form by using the superfield formulation (for furtherdetails see Ref.[10]).In the relation (equivalence) of the two theories (5), all component fields of( V , Φ ) of the N = 2 LSUSY QED action (7) are expressed uniquely as compositesof the NG fermions ψ i . We call them SUSY invariant relations , which terminate at O (( ψ i ) ) in d = 2 and N = 2,( V , Φ ) ∼ ( ξ, ξ i ) κ n − ( ψ j ) n | w VA | + · · · ( n = 0 , , , (8)where ξ i is arbitrary demensionless ( SO (2)) overall parameters in the SUSY invari-ant relations for Φ and ( ψ j ) = ¯ ψ j ψ j , ǫ jk ¯ ψ j γ ψ k , ǫ jk ¯ ψ j γ a ψ k , etc. For example, someof SUSY invariant relations for V in the Wess-Zumino gauge are v a = − i ξκǫ ij ¯ ψ i γ a ψ j | w | ,λ i = ξ h ψ i | w | − i κ ∂ a { γ a ψ i ¯ ψ j ψ j | w |} i A = ξκ ¯ ψ i ψ i | w | , · · · which are promissing features for the SGM scenario. The explicit form [9] of theSUSY invariant relations (8) are obtained systematically in the superfield formulation(for example, see Refs.[10, 11, 12]). The familiar LSUSY transformations amongthe component fields of the LSUSY supermultiplets ( V , Φ ) are reproduced amongthe composite LSUSY supermultiplets by taking the NLSUSY transformations ofthe constituents ψ i . We just mention that four-NG fermion self-interaction terms(i.e. the condensation of ψ i ) appearing only in the auxiliary fields F i of the scalarsupermultiplet Φ are crucial for the local U (1) gauge symmetry of LSUSY theoryin SGM scenario [9, 10]. The relation (5) are shown explicitly (and systematically)by substituting Eq.(8) into the LSUSY QED action (7) [9, 10].Now we briefly show the (physical) true vacuum structure of N = 2 LSUSY QEDaction (7) related (equivalent) to the N = 2 NLSUSY action (6) [13]. The vacuumis determined by the minimum of the potential V P.E. ( A, φ, B i , D ) in the action (7).The potential is given by using the equation of motion for the auxiliary field D as V P.E. ( A, φ, B i ) = 12 f ( A − φ + e f ( B i ) + ξf κ ) + 12 e ( A + φ )( B i ) ≥ , (9)6he configurations of fields corresponding to true vacua V P.E. ( A, φ, B i ) = 0 in( A, φ, B i )-space in the potential (9) are classified according to the signatures ofthe parameters e, f, ξ, κ .By adopting the simple parametrization ( ρ, θ, φ, ω ) for the vacuum configurationof ( A, φ, B i )-space and by expanding the fields ( A, φ, B i ) around the vacua, e.g. A = − ( k + ρ ) cos θ cos ϕ cosh ω,φ = ( k + ρ ) sinh ω, ˜ B = ( k + ρ ) sin θ cosh ω, ˜ B = ( k + ρ ) cos θ sin ϕ cosh ω.A = φ = 0 or B i = 0 , we obtain the particle (mass) spectra of the linearized theory N = 2 LSUSY QED.We have found one of the vacua V P.E. ( A, φ, B i ) = 0 describes N = 2 LSUSY QEDcontainingone charged Dirac fermion ( ψ Dc ∼ χ + iν ),one neutral (Dirac) fermion ( λ D ∼ λ − iλ ),one massless vector (a photon) ( v a ),one charged scalar ( φ c ∼ θ + iϕ ),one neutral complex scalar ( φ ∼ ρ (+ iω )),with masses m φ = m λ D = 4 f k = − ξfκ , m ψ Dc = m φ c = e k = − ξe κf , m v a = 0,which are the composites of NG-fermions superon and the vacuum breaks SUSYalone spontaneously (The local U (1) is not broken. For further detailes, see [13, 14]).For large N case we can anticipate the large (broken) SU ( N ) LSUSY with differentlarge mass scales.Remarkably these arguments show that the true vacuum of ((asymptotic) Riemann-flat space-time of) L N =2SGM ( e, ψ ) is achieved by the compositeness (eigenstates) offields of the supermultiplets of global N = 2 LSUSY QED. This phenomena maybe regarded as the relativistic second order phase transition of massless superon-graviton system, which is dictated by the symmetry of space-time (analogous to thesuperconducting states achieved by the Cooper pair). Here we should notice thatR-parity (for N ≥
2) may not be a good quantum number in the true vacuum ofSGM scenario as seen from the particle spectra (without superpartners) mentionedabove. These situations are very fabourable in constructing the consistent modelwith the recent LHC data which exclude the TeV scale SUSY breaking. As for thecosmological significances of N = 2 SUSY QED in the SGM scenario, the (physi-cal) vacuum for the above model explains (predicts) simply the observed mysterious7numerical) relation between the (dark) energy density of the universe ρ D ( ∼ c Λ8 πG )and the neutrino mass m ν , ρ obs D ∼ (10 − GeV ) ∼ ( m ν ) ∼ Λ G ( ∼ g sv2 ) , (10)provided − ξf ∼ O (1) and λ D is identified with the neutrino, which gives a newinsight into the origin of (small) mass [13, 15] and produce the mass hielarchy bythe factor ef ( ∼ O (cid:16) m e m ν (cid:17) in case of ψ Dc as electron! ).Furthermore, the neutral scalar field φ ( ∼ ρ ) with mass ∼ O ( m ν ) of the radialmode in the vacuum configuration may be a candidate of the dark matter , for N = 2LSUSY QED structure and the radial mode in the vacuum are preserved in therealistic large N SUSY GUT model. (Note that ω in the model is a NG bosonand disappears provided the corresponding local gauge symmetry is introduced asin the standard model.) These arguments show the potential of the SGM scenariowhich gives unified understandings for particle physics and cosmology. The no-gotheorem for N >
N > massive theory with SSB.As for the magnitude of the bare coupling constant, by taking the more general auxiliary-field structure for the general off-shell vector supermultiplet ( v a , λ i , A, φ, D,M ij , Λ , C ) [16] and establishing the NL/L SUSY relation we have shown that themagnitude of the bare (dimensionless) gauge coupling constant e (i.e. the fine struc-ture constant α = e π ) is expressed (determined) in terms of vacuum values ofauxiliary-fields [16]: e = ln( ξ i ξ − )4 ξ C , (11)where e is the bare gauge coupling constant, ξ , ξ i and ξ C are the vacuum-value scaleparameters (the relative magnitudes in the NL/L SUSY relation) of auxiliary-fieldsof the general off-shell supermultiplet in d = 2. This mechanism is natural and veryfavourable for SGM scenario as a theory of everything . We have proposed a new paradigm for describing the unity of nature, where theultimate shape of nature is new unstable ( V P.E >
0) space-time described by the8LSUSY GR action L NLSUSYGR ( w ) in the form of the free EH action for empty space-time with the constant energy density. Big Decay of new space-time L NLSUSYGR ( w )creates ordinary Riemann space-time with massless spin- superon described by theSGM action L SGM ( e, ψ ) with ( V P.E >
0) and ignites Big Bang of space-time andmatter accompanying the dark energy (cosmological constant). Interestingly onRiemann-flat tangent space (in the local frame), the familiar renormalizable LSUSYtheory emerges on the true vacuum ( V P.E = 0) of SGM action L SGM ( e, ψ ) as msslesscomposite-eigenstates of superon. We can anticipate in the true vacuum the largergauge symmetries and the consequent different mass scales for the NL/L SUSYrelation for the larger N . We have seen that the physics before/of the Big Bangmay play crucial roles for understanding unsolved problems of the universe and theparticle physics.In fact, we have shown explicitly that N = 2 LSUSY QED theory as the realistic U (1) gauge theory emerges in the physical field configurations on the true vacuum of N = 2 NLSUSY theory on Minkowski tangent space-time, which gives new insightsinto the origin of mass and the cosmological problems. The cosmological implicationsof the composite SGM scenario seems promissing but deserve further studies.Remarkably the physical particle states of N = 2 LSUSY as a whole look thesimilar structure to the lepton sector of ordinary SM with the local U (1) and theimplicit global SU (2) [8] disregarding the R-parity, i.e. without the trivial super-partner. Such SUSY breaking mechanism may allow the SUSY model constructionwithout introucing apriori the superpartners, which is compatible with the recentLHC data excluding the low (TeV) mass superpartners. (Note that the scalar mode ω is a NG boson and disappears provided the corresponding local gauge symmetryis introduced.) We anticipate that the physical consequences obtained in d = 2 holdin d = 4 as well, for the both have the similar structures of on-shell helicity statesof N = 2 supermultiplet though scalar fields and off-shell (auxiliary field) structuresare modified (extended). However, the similar investigations in d = 4 are urgent forthe realistic model building based upon SUSY.The extension to large N , especially to N = 5 is important for superon quintethypothesis of SGM scenario with N = 10 = 5 + 5 ∗ for equipping the SU (5) GUTstructure [3] and to N = 4 may shed new light on the mahematical structures ofthe anomaly free non-trivial d = 4 field theory. ( N = 10 SGM predicts double-charge heavy lepton state E and new one neutral singlet massive vector state[2]). Further investigations on the spontaneous symmetry breaking for N ≥ T c ) superconductivity or superfluidity and SGM scenario. The structure (symmetry)of the bulk (space-time) determines the resulting spectra.Linearizing SGM action L SGM ( e, ψ ) on curved space-time, which elucidates thetopological structure of space-time [17], is a challenge. The corresponding NL/LSUSY relation will give the supergravity (SUGRA) [18, 19] analogue with the vac-uum which breaks SUSY spontaneously.Locally homomorphic non-compact groups SO (1 ,
3) and SL (2 , C ) for space-timedegrees of freedom are analogues of compact groups SO (3) and SU (2) for gaugedegrees of freedom of ’t Hooft-Polyakov monopole. They are special, because theyare unique homomorphic groups among SO (1 , D −
1) and SL ( d, C ), i.e. D ( D − d −
1) (12)holds for only D = 4 , d = 2 . (13)NLSUSYGR/SGM scenario predicts four dimensional space-time. ( D = d = 1 istrivial.)Finally we just mention that NLSUSY GR and the subsequent SGM scenario forthe spin- NG fermion [7, 20] is in the same scope.Our discussion shows that considering the physics before/of the Big Bang maybe significant for cosmology and the (low energy) particle physics as well, whereSUSY and its spontaneous breakdown may play crucial roles and leave evidenceswhich can be tested in the low energy.The authors would like to thank Professor T. Okano, Department of Mathematicsof SIT for giving the proof of Eq. (13). 10 eferences [1] J. Wess and J. Bagger,
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