Ver. 1.50 2001 ( )
1 4 2 Effective Theory 5 2.1 Effective theory... 5 2.2 massless 2-flavor QCD... 5 2.3..................... 9 2.4 Standard model... 10 2.5... 11 2.6... 13 3 Supersymmetry 15 3.1 Supersymmetry... 15 3.2 Superspace....................... 16 3.3 Chiral superfield................... 17 3.4... 19 3.5..................... 20 3.6..................... 21 3.7... 25 4 Inflationary Vacua 29 4.1 Slow-roll inflation................... 29 4.2 Supersymmetric inflation... 30 4.3 Supergravity corrections... 32 4.4 Quantum fluctuations... 33 4.5..................... 35 4.6 Dynamical inflation... 38 4.7 Inflation... 40 4.8 Dilaton... 42 5 Higher Dimensions 47 5.1..................... 47 5.2...................... 48 5.3 Effective theory... 52 6 Cosmological Constant 54 6.1...................... 54 6.2...................... 55 6.3 Adjustment mechanism... 57 2
6.4 Changing gravity................... 58 6.4.1 Traceless gravity... 59 6.4.2 Quantum theory... 61 6.4.3 3-form fields... 64 6.4.4 Extra dimensions... 65 6.5 Kaluza-Klein reduction... 71 6.6......................... 73 6.7 Brane world...................... 75 6.8 Spacetime inflation... 77 6.9 Quintessence...................... 79 7 82 3
1 20 ( ) framework standard model standard model Standard model effective theory 4
2 Effective Theory effective theory ( 2.6 ) 2.1 Effective theory effective theory effective theory partition function : Z[J] = Dϕ e i(s[ϕ] +J ϕ). (2.1) ϕ source J N Z[J] ϕ χ : Z[J] = Dχ e is eff [χ,j]. (2.2) ϕ χ action S eff effective theory integrating out 1 2.2 massless 2-flavor QCD effective theory massless 2-flavor QCD 5
ϕ gauge (gluon) 2-flavor up quark down quark QCD massless 2-flavor QCD pion effective theory massless QCD dimensional transmutation Gauge-invariant hadron mass massive integrate out massless pion symmetry symmetry massless QCD symmetry (2.3) SU(2) L SU(2) R U(1) B. SU(2) F quark chiral symmetry baryon number symmetry pion SU(2) L SU(2) R SU(2) F flavor symmetry 3 π a (a =1, 2, 3) χ SU(2) L SU(2) R SO(4) SU(2) F SO(3) SO(3) 3 chiral symmetry massless symmetry massive mass mass matrix : ( (ū d) m u d ). (2.4) m constant chiral symmetry ( dynamical ) 6
m symmetry m symmetry m symmetry massless symmetry pion action symmetry SO(4) pion SO(3) 3 SO(4) SO(4) 4 ϕ i (i =1,, 4) SO(4)-invariant constraint : ϕ 2 i = Λ 2. (2.5) SO(4) symmetry Λ QCD dynamical scale dimensional transmutation QCD scale Λ µ : Λ µe 8π 2 bg 2 (µ). (2.6) b β g running gauge coupling ϕ i 4 invariant Lagrangian : L eff = V (ϕ 2 i ) + 1 2 µϕ i µ ϕ i +. (2.7) effective Lagrangian ϕ 2 i =Λ 2 π a : L eff = 1 µ π a µ π a +. (2.8) 2 1 + π2 b Λ 2 (2.5) constraint ϕ 4 π a : π a ϕ a 1 + ϕ2 b 4Λ 2. (2.9) ϕ 2 b 3 (2.7) V ϕ2 i Λ2 dynamics 7
exact effective theory [ ] ϕ 2 i =Λ 2 [ ] π a SO(3) 3 SO(4) symmetry nonlinear SO(4)-invariant action SO(4) 4 linear ϕ i linear invariant action SO(4) symmetry SO(4) symmetry ϕ 2 i =Λ2 Lagrangian 3 nonlinear SO(4)-invariant action effective theory QCD strong dynamics pion pion interaction π a Lagrangian action massless massless 8
massless pole singular singular pion nonsingular Lagrangian implicit nontrivial gauge theory effective theory QCD nontrivial effective theory effective theory (effective) 2.3 partition function exact effective theory nontrivial picture exact chiral Lagrangian chiral perturbation theory Λ QCD Λ cutoff Cutoff regularization cutoff Λ integrate out 9
Λ cutoff effective theory effective theory higher-dimensional term derivative expansion chiral perturbation theory ( ) higher-dimensional term effective theory exact 2.4 Standard model standard model Standard model effective theory standard model higher-dimensional term effective theory standard model standard model effective theory standard Einstein chiral perturbation theory standard model 10
derivative expansion effective theory background effective theory ( ) effective theory framework framework standard model 2.5 QED Yang-Mills Einstein motivation effective theory Weinberg-Salam model weak boson massive 4-Fermi Weinberg-Salam model fermion 4-Fermi higher-dimensional term 4-Fermi higher-dimensional term Weinberg-Salam model higher-dimensional term Weinberg-Salam model 11
weak boson 4-Fermi higher-dimensional term asymptotic freedom (Weinberg-Salam model asymptotically free ) 4-Fermi 1 1 gauge boson Higgs 1 derivative expansion gauge Higgs 12
2 : 2 asymptotically free [ ] Effective theory? [ ] effective theory relevant operator, irrelevant operator relevant operator dominant operator effective theory 2.6 effective theory standard model 13
Effective theory standard model field content coupling consistency unique configuration space ( ) effective theory background configuration background configuration effective theory ( 2.1 ) 14
3 Supersymmetry supersymmetric supersymmetric effective theory ( 3.7 ) 3.1 Supersymmetry Supersymmetry supersymmetry 2 Hamiltonian H = ω(a a + b b) (3.1) a b symmetry a b b a Q : Q = ω(a b + b a). (3.2) convention ω [H, Q] =0 (3.3) Q symmetry a b bosonic fermionic Q bosonic symmetry bosonic fermionic Q fermionic {Q, Q} =2H (3.4) fermionic symmetry supersymmetry 4 fermionic Q Q Hamiltonian 4 P µ {Q α,q β } = 2Γ µ αβp µ (3.5) 15
Γ µ αβ α, β =1, 2, 3, 4 µ Q 4 Hamiltonian Q H Q 2 H supersymmetry Q Hamiltonian supersymmetry supercharge 3.2 Superspace supersymmetry superspace supersymmetry P µ ( ) a µ {x µ } x µ = x µ +a µ ϕ(x µ ) ϕ (x µ ) ϕ(x µ ) ϕ (x µ )=ϕ(x µ ). (3.6) action ϕ Lagrangian : S = d 4 x L(ϕ(x µ )). (3.7) supersymmetry Supersymmetry [P µ,p ν ]=0 Q 4 {Q α,q β } =0 16
P µ {x µ } P µ =0 superspace Q α Grassmann ξ α superspace {θ α } θ α = θ α + ξ α Superspace superfield Φ(θ α ) Φ(θ α ) Φ (θ α )=Φ(θ α ). (3.8) supersymmetric Lagrangian L = d 4 θ F(Φ(θ α )) (3.9) action superspace dθ 1 f(θ 1 )=f 1 ; f(θ 1 )=f 0 + θ 1 f 1 (3.10) Berezin 3.3 Chiral superfield superfield Φ(θ α ) 4 Grassmann supercharge 4 θ α 2 θ (2 ) θ θ superfield Φ(θ, θ ) θ supersymmetry Φ(θ, θ ) chiral superfield Φ(θ) Grassmann odd θ 2 θθθ =0 Φ(θ) Φ(θ) =φ + θψ + θ 2 F. (3.11) 17
φ = Φ(0) Φ(θ) complex field φ complex scalar field ψ 2 chiral fermion F complex scalar field φ ψ 4 1 3 φ = Φ(0) Φ 1 2 θ 1 F 2 2 chiral superfield Φ(θ) supersymmetric Lagrangian 2 d 2 θd 2 θ K(Φ, Φ ) (Kähler potential ) Lagrangian : L K = d 2 θd 2 θ K(Φ, Φ ). (3.12) Chiral superfield θ Φ W (Φ) (superpotential ) d 2 θ : L W = d 2 θw(φ) + h.c. (3.13) Lagrangian Lagrangian Kähler potential K(Φ, Φ )=ΦΦ (3.14) superpotential W (Φ) 3 W (Φ) = Λ 2 Φ+ 1 2 mφ2 + 1 3 λφ3. (3.15) potential Berezin fermion ψ =0 Φ =φ+θ 2 F : L K = FF, L W = F W(φ) φ +h.c. (3.16) superpotential W (Φ) θ Berezin W (φ) kinetic term F 2 F ( kinetic term ) F : V = W(φ) φ 2 = Λ 2 + mφ + λφ 2 2. (3.17) 18
superpotential supersymmetry φ 4 supersymmetric interaction fermion interaction Supersymmetry φ V supersymmetric supersymmetry φ ( V ) supersymmetry φ V m = λ =0 Λ 2 0 V > 0 supersymmetry 3.4 Supersymmetric Lagrangian supersymmetry interaction massless massless chiral symmetry Supersymmetry fermion interaction fermion supersymmetry complex scalar field phase rotation φ exp(iζ)φ m 2 φ 2 supersymmetry chiral superfield superpotential Φ Φ d 2 θ supersymmetric standard model Higgs ( ) supersymmetry ( ) R supersymmetry R phase rotation Φ superfield Φ component φ ψ scalar fermion 19
phase chiral superfield φ ψ θ : Φ(θ) Φ (θ) =e iqζ Φ(e iζ θ). (3.18) R (U(1) R ) R R θ superpotential : W (Φ (θ)) = e 2iζ W (Φ(e iζ θ)). (3.19) superpotential R supersymmetry Φ R charge q =2 W (Φ) = Λ 2 Φ Φ 2 Φ 3 R symmetry superpotential 3.5 Supersymmetry symmetry supersymmetry supersymmetry supercharge linear fermion boson supersymmetry ( ) Supercharge H = Q 2 Q gauge constraint Q constraint gauge supersymmetry Gauge symmetry supersymmetry superpotential W (Φ) = Λ 2 Φ Φ linear constant positive supersymmetry interaction ( inflation ) 20
Φ chiral superfield X(θ) Y (θ) : W (Φ,X,Y) = Φ(Λ 2 X 2 )+MXY. (3.20) Λ M 1 constant Φ superpotential φ x y : V = Λ 2 x 2 2 + Mx 2 + 2φx My 2. (3.21) X(0) = x Y (0) = y φ Λ 2 x 2 2 y Mx 2 x V supersymmetry 3.6 supersymmetry supersymmetry supersymmetry Planck scale supersymmetry effective theory cutoff Planck scale supersymmetry Planck scale physics supersymmetry effective theory supersymmetry Planck scale physics supersymmetry dimensional transmutation dynamical scale NonSUSY QCD quark chiral symmetry supersymmetry 21
SU(2) doublet 4 4 chiral superfield Q α i (θ) α =1, 2 gauge gauge theory (Supersymmetric gauge theory gauge theory supersymmetric matter chiral superfield ) QCD dynamical scale condensation gaugeinvariant Gauge-invariant Φ ij Q α i Q β j ɛ αβ SU(2) ɛ αβ i j 1 4 6 6 gauge-invariant 6 Q i chiral superfield 4 4 SU(4) chiral symmetry SO(6) 6 Φ ij condensation SU(4) SO(6) chiral symmetry condensation ( doublet 4 doublet 2 6 gauge theory global anomaly 4 condense 4 ) [ ] Gauge-invariant Φ ij [ ] Gauge theory gauge invariant operator Gauge-dependent gauge order parameter gauge-invariant Φ ij chiral Lagrangian massless supersymmetric QCD Φ ij pion gauge- 22
invariant effective Lagrangian symmetry supersymmetric QCD asymptotically free dynamical scale condensation condense Φ ij Φ ij chiral symmetry SO(6) 6 gauge interaction dynamics condense SO(6) symmetry chiral Lagrangian QCD pion Supersymmetric supersymmetry effective theory superpotential gauge interaction dynamics 6 Φ ij SO(6) condense superpotential Φ ij 6 Φ Φ a (a =1,, 5) gauge singlet chiral superfield Z Z a 6 a Φ a a 1 5 superpotential W (Q α i,za )=Z a Φ a superpotential Z a (chiral superfield Z a scalar field Z a (0) ) Φ a Φ a =0 6 Φ, Φ a dynamical scale condense superpotential Φ dynamical scale condense superpotential supersymmetric gauge theory Φ =Λ 2 dynamical scale [ ] Φ condence [ ] gauge SU(2) doublet 4 Q i (θ) (i =1,, 4) superpotential supersymmetric gauge theory dynamics condensation QCD chiral symmetry breaking 23
condensation gauge-invariant (Q i Q j ) order parameter Doublet 4 flavor symmetry SU(4) SO(6). (3.22) Gauge-invariant (Q i Q j ) flavor symmetry 6 1 5 (Q i Q j ) (Φ, Φ a ); a =1,, 5. (3.23) (Φ, Φ a ) condence SO(6) symmetry symmetry SO(6)-invariant : Φ 2 +Φ 2 a =Λ 4. (3.24) 6 dynamical scale Λ 2 (3.24) condence (Q i Q j ) ɛ ijkl (Q i Q j )(Q k Q l ) (Φ, Φ a ) condensation (Q i Q j ) condense superpotential chiral superfield Q i superpotential massless supersymmetric QCD superpotential W = Z a Φ a. (3.25) superpotential supersymmetric gauge theory superpotential Z a (0) V = Φ a 2 +. (3.26) Supersymmetry Φ a (3.24) : Φ =Λ 2. (3.27) 24
gauge theory supersymmetry dynamics Planck scale Gauge theory dynamics ( inflation ) superpotential W (Φ α i,z a,z)=z a Φ a + λzφ (3.28) superpotential λzφ Φ symmetry superpotential λ Φ a Φ ( )condense Φ Λ 2 condence Q i W (Z) =λλ 2 Z supersymmetry effective effective theory dynamical supersymmetry breaking supersymmetry V = W(φ) φ 2 (3.29) 3.7 effective theory supersymmetry effective theory Supersymmetry 25
supersymmetry configuration space ( ) supersymmetry Supersymmetry standard model Higgs supersymmetry standard model inflaton inflation supersymmetric physics inflationary vacua supersymmetry gauge symmetry ψ 3 Gauge symmetry( 3 ) 1 26
2 QED quark lepton strong interaction week interaction 3 gauge symmetry deep inelastic scattering gauge symmetry J/ψ ( anomaly cancellation ) 4 standard model ( effective theory standard model effective theory ) gauge symmetry supersymmetry ( 3 ) supersymmetry supersymmetry gauge gauge supersymmetry 1 supergravity Supergravity QED standard model standard model supersymmetry supersymmetric 2 minimal (MSSM) quark lepton dark matter strong interacton weak interacton inflation dynamics Dark matter R Inflation inflaton supersymmetry supersymmetry dynamics framework 3 dynamical supersymmetry breaking effective theory dynamical effective theory dynamical breaking supersymmetry ( supersymmetric inflation ) 27
( ) Higgs supersymmetry( inflation) ( 3 Higgs supersymmetry standard model ) supersymmetry inflaton Higgs supersymmetry 3 standard model 4 supersymmetry supersymmetry supersymmetry ( 3.1 ) 28
4 Inflationary Vacua supersymmetry ( 4.8 ) 4.1 Slow-roll inflation Supersymmetry inflation inflaton ϕ 2.4 10 18 GeV (reduced Planck scale) 1 effective theory cutoff cutoff scale V (ϕ) V (ϕ) < 1 (4.1) reduced Planck scale inflation( ) slow-roll inflation inflation supersymmetry (ϕ >0,λ>0): V (ϕ) = λϕ n. (4.2) n effective theory ( ) ϕ n : ϕ > n. (4.3) reduced Planck scale n 29
effective theory V inflation effective theory 0 ϕ n V 1 λ 1 (λ 1) effective theory ϕ cutoff scale cutoff scale effective theory V ( ) effective theory n 1 ϕ large-field type inflation model building 1 field ϕ 1 small-field type inflation n =0 V constant : V (ϕ) = λ. (4.4) ϕ small field λ effective theory effective theory slow-roll inflation 4.2 Supersymmetric inflation small-field type supersymmetry φ chiral superfield v 2 superpotential : W = v 2 φ. (4.5) φ (Kähler potential effective theory 30
) v 4 constant : V v 4. (4.6) supersymmetry ( ) inflation inflation Friedmann phase Slow-roll inflation φ v 2 φ φ n : W = v 2 φ φ n. (4.7) W φ V v 2 nφ n 1 2 (4.8) φ ( ) φ χ : W = φ(v 2 χ n ), (4.9) V v 2 χ n 2 + nφχ n 1 2. (4.10) v 2 φ χ n v 2 condense (4.5 ) inflation supersymmetry 31
4.3 Supergravity corrections superpotntial rigid supersymmetry inflation supergravity supergravity : { ( ) } V = e K 2 1 K F 2 3 W 2. (4.11) φ φ F superpotential : F = W φ + K W. (4.12) φ K Kähler potential dθ 4 kinetic term tree-level φ 2 effective theory : K = φ 2 +. (4.13) Supergravity negative 3 W 2 Rigid supersymmetry Hamiltonian Q 2 V ( Hamiltonian ) negative supersymmetry positive cosmological constant 3 W 2 supersymmetry supersymmetry effective supersymmetry gravity effective theory 32
supergravity φ superpotential n =5 : W = v 2 φ λ 5 φ5. (4.14) φ 5 5 Kähler potential φ 2 φ 4 : K = φ 2 + κ 4 φ 4 +. (4.15) reduced Planck scale 1 cutoff effective theory φ 4 cutoff scale κ 1 supergravity φ slow-roll inflation : ( V v 2 λ ) 2 4 ϕ4 κ 2 v4 ϕ 2. (4.16) ϕ chiral superfield φ real : ϕ 2Reφ. (4.17) Kähler potential (4.16) 2 ϕ 2 v 4 inflation κ κ Kähler potential reduced Planck scale Planck scale physics effective inflation dynamics cutoff 4.4 Quantum fluctuations slow-roll parameter ɛ η slow-roll condition ɛ 1 ( ) V 2, η V 2 V V, (4.18) 33
slow-roll parameter 1 ɛ, η < 1, (4.19) slow-roll condition 1 slow-roll inflation inflation inflation inflation Inflation inflaton inflation inflation inflation ( ) 2.7K 10 5 K 10 5 K slow-roll inflation inflaton 34
slow-roll inflation : V 1 4 /ɛ 1 4 0.027 6.7 10 16 GeV. (4.20) horizon ϕ v 10 2 κe 30κ (4.21) index spectral index : n s 1 6ɛ + 2η 1 2κ. (4.22) n s 1 n s 1 slow-roll parameter κ 0.1 n s 1 slow-roll parameter ɛ slow-roll inflation 1 (4.20) V inflation energy scale GUT scale effective theory Planck scale inflation inflation sector picture 4.5 inflation : W = Λ 2 Z(λ λ φ 2 ) (4.23) v 2 Z(1 gφ 2 ). (4.24) 35
Z φ chiral superfield λ, λ g coupling Λ 2 ( ) reduced Planck scale 1 Λ 2 1 v 2 coupling superpotential Kähler potential : K = Z 2 + φ 2 + k 1 Z 2 φ 2 k 2 4 Z 4 +. (4.25) Z φ mixing term 4 4 k 1 k 2 1 coupling superpotential Kähler potential supergravity : V = e K (K AB F A F B 3 W 2 ). (4.26) F superpotential : F A W φ A + K φ A W. (4.27) (4.26) K AB Kähler potential 2 2 K, (4.28) φ A φ B field 1 supergravity W W : W φ A = W = 0 = V φ A = V = 0. (4.29) F gravity ridid supersymmetry F V supergravity F supersymmetry order parameter rigid supersymmetry 36
(4.24) superpotential superpotential : W Z = v2 (1 gφ 2 ), (4.30) W = 2v 2 gzφ. (4.31) φ Z φ condense Z superpotential : Z = 0, (4.32) φ = ± 1. g (4.33) superpotential Kähler potential supergravity : V v 4 1 gφ 2 2 + (1 k 1 )v 4 φ 2 + k 2 v 4 Z 2 (4.34) v 4 κ 2 v4 ϕ 2. (4.35) κ 2g + k 1 1, (4.36) ϕ 2Reφ. (4.37) Z Z inflation (hybrid inflation) Z φ inflation v 4 ϕ 2 Kähler potential coupling coupling 1 1 ϕ inflaton slow-roll inflation supersymmetry 37
4.6 Dynamical inflation supersymmetric inflaton v 2 effective theory Einstein gravity inflation effective theory cutoff inflation inflation dynamical inflation supersymmetry supersymmetry breaking inflation inflation small-field type : 4 4 1 ϕ ( ) ( 4 2 ) 38
big bang v 4 1 dimensional transmutation [ ] Supersymmetric inflation model superpartner [ ] supersymmetry inflation inflaton : 5 ( 5 1 ) inflation inflation 2 big bang ( 5 3 ) supersymmetry inflaton inflation scale GUT scale 4 supersymmetry weak scale superpartner supersymmetry inflaton inflatino (inflaton superpartner) inflation inflation 39
4.7 Inflation supersymmetric inflation dynamical inflation Slow-roll inflation inflaton ϕ ϕ cosmological constant inflation inflation Hubble parameter H : ṙ = Hr. (4.38) Hubble radius 1 1 = Hr H, (4.39) r H = H 1. (4.40) 1 r H Einstein Hubble parameter Einstein : H 2 = V = v4 3 3, (4.41) H = v2. 3 (4.42) inflation Hubble radius 1/v 2 v inflation inverse inflation ϕ inflation 40
inflationary phase Einstein singularity Planck scale Einstein gravity t E 1. (4.43) Planck Planck energy inflaton inflaton : E 1 2 ( ϕ)2 + V (ϕ). (4.44) ϕ E Planck energy inflation V E energy scale chaotic ϕ slow-roll inflation ϕ inflation primary inflation GUT scale inflation Planck scale inflation primary inflation inflation primordial inflation primary inflation large-field inflation Planck scale field inflation primary inflation primordial inflaton primordial inflation primordial inflation ( ) inflation supersymmetric inflation [D.H. Lyth and A. Riotto, arxiv:hep-ph/9807278 ] 41
4.8 Dilaton dimensional transmutation coupling gauge coupling variable coupling gauge coupling dilaton Φ chiral superfield dilaton Φ gauge field kinetic term coupling ( real part) gauge coupling 1 : Φ = 1 g 2. (4.45) e 8π2 Φ = e 8π 2 g 2 (4.46) dilaton dilaton dilaton : 6 42
dilaton gauge coupling free interaction interaction runaway type free Free consistency effective theory dilaton toy model dilaton moduli field 6 1 effective theory runaway type effective theory( 6 2 ) gauge coupling interaction 7 43
chaotic dilaton dynamical inflation dilaton sector 7 Φ dilaton ϕ inflaton inflaton 7 1 dilaton inflaton dynamical scale runaway type coupling dynamical scale inflaton dilaton gauge coupling dynamical scale gauge coupling ( 7 2 ) inflaton chaotic initial condition dilaton ( ) ( geometric classical patch ) inflaton dilaton Free theory free 7 2 inflation chaotic initial condition fluctuation ( ) dilaton coupling superpotential ( ) dynamical inflation supersymmmetry inflation (inflationary vacuum selection) chaotic initial condition inflation effective theory dilaton gauge coupling 44
moduli ( ) ( ) moduli dynamical inflation Gauge theory ( ) 4 asymptotic freedom 5 asymptotically free CFT inflationary dynamics dynamical inflation inflation 5 inflation 4 dynamical inflation 4 inflation inflationary 4 supersymmetry N Supersymmetry 4 supersymmmetry (N =1) N =2 4 supersymmetry dynamical scale Supersymmetry N 1 SUSY [ ] effective theory SUSY model? [ ] MSSM MSSM running coupling 45
flavor ( 4.1 ) 46
5 Higher Dimensions ( 5.3 ) 5.1 effective theory framework effective theory vacuum selection supersymmetric inflation inflationary vacuum dynamical inflation gauge coupling dilaton moduli 3 Minkowski 3 4 Minkowski MODERN KALUZA-KLEIN THEORIES Nordström On the Posibility of Unification of the Electromagnetism and Gravitation ( interaction ) scalar potential Minkowski ( ) 5 5 electromagnetic theory 5 Abstract It is shown that a unified treatment of the electromagnetic and gravitational fields is possible if one views the four-dimensional space time as a surface in a five-dimensional world. subspace 4 brane world Brane world 4 picture Nordström 47
Kaluza-Klein Einstein gravity 5 5 4 Kaluza-Klein 5.2 ( Newton ) crossover (string theory) string ( point theory ) conformal field theory picture Hilbert string picture worldsheet perturbation M Theory : type IIA string M theory 10D N=2 11D SUGRA. S 1 48
11 supergravity supersymmetry maximal S 1 10 N =2 supersymmetry Kaluza-Klein reduction : ( ) g MN A M. φ A N 11 metric 10 metric g MN A M φ (10D N = 2 supergravity) effective theory type IIA string evidence ( ) coupling dilaton coupling 11 11 supergravity paralell 11 string picture M theory type IIA M theory M theory S 1 Type IIA string dilaton strong coupling string/m theory( ) effective theory standard model standard model GUT model standard model 11 supergravity full M theory 11 supergravity unique fundamental unique standard model standard model 49
effective theory standard model full ( ) unique Hamiltonian observable type IIA,B (II) heterotic SO(32),E 8 E 8 (H) type I (I) M theory (M) : 8 9 II M 10 N =2 supersymmetry (supercharge 32 ) I H 50
N =1 supersymmetry (supercharge 16 ) 8 supersymmetry Supersymmetry heterotic M theory M theory 9 2 orbifold S 1 /Z 2 Z 2 S 1 9 1 11 orbifold R 10 S 1 /Z 2 S 1 /Z 2 (Z 2 ) 10 2 11 (bulk) : 10 effective theory bulk 11 supergravity supersymmetry 10 N =2 Z 2 (Z 2 ) supersymmetry 10 N =1 10 N = 1 supergravity field content anomalous anomaly cancellation field content string twisted sector string perturbation effective theory anomaly cancellation 51
E 8 E 8 E 8 10 E 8 E 8 heterotic string M theory/s 1 type IIA string M theory/(s 1 /Z 2 ) E 8 E 8 heterotic string effective theory (duality) heterotic M theory heterotic string Heterotic M theory S 1 /Z 2 R 10 10 boundary brane world picture heterotic string 10 effective 11 picture effective theory Brane world effective theory brane Nordström heterotic M theory toy model realistic effective theory brane world 5.3 Effective theory effective theory α (gauge ) i : ϕ i α( x, t) ϕ i α, x(t). (5.1) gauge 0 1 gauge gauge gauge nontrivial 4 3 3 52
R 3 nontrivial ϕ i α( x, t) i α x content (ϕ i α, x (t)) field content field content brane world effective theory brane world cosmological constant ( 5.1 ) 53
6 Cosmological Constant naturalness ( 6.5 ) 6.1 Effective theory framework naturalness cosmological constant supersymmetry supersymmetry weak scale superpartner weak scale effective theory supersymmetry Einstein : G µν = T µν = Λg µν. (6.1) Einstein : G µν = R µν 1 2 g µνr (6.2) stress-energy tensor T µν Bianchi identity cosmological constant Λ background geometry background (6.1) traceless part trace part 2 : G µν 1 4 g µνg =0, G = R = 4Λ. (6.3) traceless part cosmological constant trace part cosmological constant background 54
background 0 naive effective theory weak scale contribution : Λ m W 4. (6.4) effective theory framework effective theory framework cosmological constant background Effective theory cutoff background nontrivial Poincaré invariance background effective theory framework flat background effective theory background effective theory framework effective theory effective theory picture 6.2 Cosmological constant Weinberg [S. Weinberg, Rev. Mod. Phys. 61 (1989) 1] : 55
supersymmetry, supergravity, superstrings; anthoropic considerations; adjustment mechanisms; changing gravity; quantum cosmology. supersymmetry etc. SUSY effective theory anthropic consideration anthropic principle ( ) Anthropic principle correlation function anthropic principle Cosmological constant cosmological constant negative positive inflation ( ) cosmological constant cosmological constant logical ( 6.9 ) quantum cosmology cosmological constant adjustment mechanism changing gravity effective theory 56
6.3 Adjustment mechanism effective theory adjustment mechanism Adjustment mechanism Cosmological constant background ϕ : S ϕ = d 4 x ( g U(ϕ)R + 1 ) 2 µϕ µ ϕ V (ϕ). (6.5) V (ϕ) U(ϕ) : U (ϕ)r = V (ϕ). (6.6) background ϕ : µ ϕ =0. (6.7) U (ϕ) 0, V (ϕ) 0 (6.8) ϕ R (adjustment) backgound Minkowski : g µν η µν. (6.9) cosmological constant V fine tuning V (ϕ) statement U(ϕ) metric φ Weyl 1 V field redefinition statement 57
standard model action contribute : S = S ϕ + S standard. (6.10) Einstein Einstein background trace part : U(ϕ)R =4Λ. (6.11) ϕ R 0 : R 0. (6.12) (Λ fine tune ) U(ϕ) : U(ϕ). (6.13) U(ϕ) background value Planck scale interaction decouple decouple Λ background flat 6.4 Changing gravity effective theory changing gravity Changing gravity : 1. traceless gravity, 2. 3-form fields, 3. extra dimensions. ( ) 58
6.4.1 Traceless gravity traceless gravity ( ) Einstein Einstein cosmological constant static cosmological constant static cosmological constant motivation (Einstein ) Einstein Einstein traceless part : G µν 1 4 g µνg = T µν 1 4 g µνt. (6.14) Einstein cosmological constant trace part traceless part traceless gravity trace part Einstein Einstein gravity Einstein gravity gauge theory redundant G µν, T µν D µ G µν = D µ T µν =0 (6.15) traceless gravity D µ : ν G = ν T. (6.16) G = T + const. (6.17) Einstein trace part Einstein cosmological constant traceless gravity 59
traceless gravity Einstein gravity cosmological constant Einstein cosmological constant Lagrangian traceless gravity traceless gravity flat Einstein cosmological constant flat Hamilton : q =, (6.18) ṗ = (6.19) ( q,p ) 1 ( ) : q(t) =q(q 0,p 0,t), (6.20) p(t) =p(q 0,p 0,t). (6.21) q 0,p 0 (6.18),(6.19) : q 0 = q 0 (q, p, t), (6.22) p 0 = p 0 (q, p, t). (6.23) q 0,p 0 q(t),p(t) q 0 (t),p 0 (t) q 0 =0, (6.24) ṗ 0 =0 (6.25) q 0,p 0 cosmological constant 60
given traceless gravity effective theory 6.4.2 Quantum theory Lagrange (6.14) Lagrangian Einstein gravity concept Einstein gravity spin 2 massless 2 : g µν = g νµ. (6.26) Lorentz covariance Lorentz metric negative norm gauge theory Gauge symmetry diffeomorphism: ε µ, Weyl Weyl transrormation: λ Einstein garvity Weyl gauge δ λ g µν = λ(x)g µν. (6.27) φ δ λ φ =0 (6.28) 61
topological gravity ɛ µ λ gauge symmetry Einstein gravity diffeomorphism gauge symmetry gauge symmetry ϕ diffeomorphism-invariant : ϕ 2 = d 4 x g ϕ 2. (6.29) ϕ Dϕ diffeomorphism diffeomorphism diffeomorphism invariance Einstein gravity diffeomorphism gauge symmetry Weyl diffeomorphism Weyl (6.29) Weyl g µν Weyl diffeomorphism invariance Weyl invariance Weyl anomaly 5 gauge symmetry 4 diffeomorphism 4 1 3 Lorentz volume-preserving diffeomorphism : µ ε µ =0. (6.30) volume-preserving g µν : δ ε det g µν =0. (6.31) 1 1 Weyl volume-preserving diffeomorphism gauge symmetry 2 gauge symmetry 62
: ϕ 2 = d 4 x ϕ 2. (6.32) Einstein gravity (6.29) g Weyl diffeomorphism g volume-preserving diffeomorphism gauge symmetry 4 gauge symmetry g µν massless spin 2 2 Weyl volume-preserving diffeomorphism gauge symmetry Weyl invariance 2 g µν Weyl : g µν g 1/4 g µν. (6.33) g µν Weyl : det g µν = 1. (6.34) Einstein gravity Lagragian : L = U(ϕ)R + 1 2 gµν µ ϕ ν ϕ V (ϕ). (6.35) Weyl invariance Volume-preserving diffeomorphism Einstein gravity g g =1 (6.35) g µν Einstein gravity cosmological constant V (ϕ) constant Lagragian constant : δl δg µν = g 1/4 ( δl δg µν 1 4 gµν g ρσ δl δg ρσ ) =0. (6.36) traceless (6.14) (6.33) Einstein 1 gauge traceless Einstein 63
Einstein recover cosmological constant background ( flat ) Einstein gravity cosmological constant Einstein gravity cosmological constant Lagragian flat background flat cosmological constant background background 6.4.3 3-form fields traceless gravity Einstein gravity Einstein gravity 3-form field : F µνρσ = [µ A νρσ]. (6.37) 3 gauge field strength Einstein gravity action : 1 S = d 4 x gf µνρσ F µνρσ. (6.38) : µ ( gf }{{ µνρσ } )=0. (6.39) ϕɛ µνρσ 4 4 field strength 1 1 [R. Bousso and J. Polchinski, arxiv:hep-th/0004134 ] 64
ɛ µνρσ Levi-Civita g constant : gϕ = c. (6.40) c 3-form gauge constant 1 (6.39),(6.40) : S = d 4 x g c ɛ αβγδ g αµ g βν g γρ g δσ ɛ µνρσ c = d 4 x gc 2. (6.41) g g c 2 cosmological constant Einstein gravity cosmological constant changing gravity 6.4.4 Extra dimensions extra dimension ( Weinberg review [V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B125 (1983) 139] ) toy model changing gravity Einstein gravity traceless gravity 3 6 action : S 6 = d 6 x g( 1 2 R Λ 6). (6.42) Lagrangian parameter cosmological constant 6 Λ 6 tune 6 background 6 effective 4 65
4 (6.42) background mertric warped compactification : ds 2 = σ(r)ḡ µν (x)dx µ dx ν dr 2 ρ(r)dθ 2. (6.43) 4 ḡ µν (x) 4 metric 2 (extra dimensions) (r, θ) θ 0 2π background Background θ σ(r) ρ(r) r dr 2 factor r rescaling 1 convention (6.43) effective 4 background (6.42) 6 Einstein 4 metric ḡ µν 4 parametrization 4 Einstein ḡ µν 4 Einstein 4 cosmological constant 6 Einstein (6.43) r ( θ) constant effective cosmological constant σ ρ 3 σ 2 σ + 3 σ ρ 4 σ ρ 1 ρ 2 4 ρ + 1 2 3 σ 2 2 σ + σ 2 σ ρ ρ ρ ρ = Λ 6 + Λ 4 σ, (6.44) = Λ 6 + 2Λ 4 σ, (6.45) 2 σ σ + 1 σ 2 = Λ 2 σ 2 6 + 2Λ 4 σ. (6.46) Λ 4 σ ρ Einstein gauge theory 66
Bianchi identity (6.44)-(6.46) z : σ z 4 5. (6.47) ρ z : ρ = C 2 z 2 z 6 5. (6.48) C z : V z = V z. (6.49) V (z) = 5 16 Λ 6z 2 25 24 Λ 4z 6 5 (6.50) z(r) r V Einstein Newton z coordinate singularity smooth geometory factor ρ ( conical ) condition : boundary condition z ρ(0) = 0, (6.51) ( ρ) (0) = 1. (6.52) z (0) = 0, (6.53) z (0) = C, (6.54) z(0) = 1. (6.55) σ z convention (6.43) x (6.55) (6.53),(6.54) smoothness condition 67
(6.50) Λ 6 given parameter Λ 4 effective 4 cosmological constant Λ 4 Λ 4 =0 Λ 6 > 0 11 boundary condition 11 12 12 13 ( 13 ) 12 θ 0 2π parametrize dθ 2 68
13 ρ(r) z =0 extra dimension effective 4 cosmological constant 2 1 1 5 extra volume finite effective theory volume ( ) effective 4 noncompact extra 6 metric (determinant ) degenerate metric 4 metric ḡ µν (x) singularity 4 effective cosmological constant variable Einstein gravity physics cosmological constant toy model 6 cosmological constant pure gravity realistic 4 standard model matter 3-brane standard model field localize 3-brane : S = S 6 + d 4 x g 4 λ. (6.56) r=0 69
3-brane background constant energy density λ warped compactification σ ρ (6.44) : 3 σ 2 σ +...+ 1 ρ 2 ρ = Λ 6 + Λ 4 σ λ 2πɛ Θ(ɛ r); (6.57) ρ ɛ +0. (6.58) 3-brane source (6.57) 3 Θ 3-brane source r =0 singularity regularize step function pure gravity brane 12 brane tension 14 : 14 boundary condition : z (ɛ) = 0, (6.59) z (ɛ) = C(1 λ ), 2π (6.60) z(ɛ) = 1. (6.61) pure gravity backgound brane tension background pure gravity 4 effective cosmological constant changing gravity 70
6.5 Kaluza-Klein reduction effective theory heterotic M theory brane world effective theory Kaluza-Klein reduction 5 4 5 R 4 S 1 /Z 2 5 extra dimension action : S = dyd 4 x g 5 (R 5 Λ 5 ) + d 4 x g 4 L 1 + d 4 x g 4 L 2. (6.62) y=0 extra dimension y boundary 3-brane 1 5 (bulk) pure gravity 2 3 3-brane matter action g 5 g 4 5 4 metric R 5 Λ 5 5 cosmological constant y =0 brane effective 4 brane l brane 5 5 4 effective theory regular 5 metric parametrize : ( ) g 4 (g 5 )=. (6.63) 1 g 4 zero mode Kaluza-Klein mode : y=l g 4 (x, y) =ḡ(x)a(y)+ n ḡ n (x)a n (y). (6.64) : a(0) = 1, a(l) =a l. (6.65) (6.62) action 4 schematic effective theory 71
: S eff V + d 4 x ḡ( R Λ) + d 4 x ḡl 1 d 4 x ḡa l4 L 2 + O(m 1 KK ); (6.66) V = l 0 dy a 4 a 1 = M Planck 2. (6.67) bulk gravity zero-mode 4 y (6.67) bulk effective 4 Einstein action 4 Planck scale (6.62) action 5 fundamental scale 1 (6.66) 2 brane a(0) 1 convention ḡ metric 3 brane a l warp factor metric Kaluza-Klein mode mass m KK (6.67) volume fundamental scale large-volume extra dimension 5 fundamental scale 4 Planck scale (6.67) volume extra dimension l warp factor a fundamental scale GUT scale weak scale ( energy scale ) 4 fundamental scale volume Witten heterotic M theory 15(a) MSSM gauge interaction coupling constant α 1,2,3 (GUT scale) dimensionless coupling constant α G Gauge unification point fundamental scale (string scale) GUT scale Planck scale bulk volume factor Coupling bulk 15(c) coupling GUT scale 5 brane gauge interaction 5 bulk 15(b) [J. Polchinski, arxiv:hep-th/9812104] 72
15 6.6 Adjustment mechanism background Einstein fine tuning Planck decouple 73
flat background changing gravity cosmological constant (Einstein gravity input parameter ) dynamical variable (integration constant) flat background flat background flat effective theory Adjustment mechanism flat background changing gravity background flat traceless gravity Einstein gravity traceless gravity adjustment mechanism action : ( S 4 = d 4 x U(ϕ)R + 1 ) 2 µϕ µ ϕ V (ϕ). (6.68) R traceless gravity Weyl invariant : U (ϕ)r = V (ϕ). (6.69) background : µ ϕ =0. (6.70) constant background V (ϕ) flat background : U (φ) 0, V (φ) 0 R 0. (6.71) R Einstein gravity Einstein trace part decoupling unrealistic traceless gravity trace part R decouple background fine tuning V (ϕ) 74
R V /U U 0 fundamental scale V 0 tune Cosmological constant tuning fine tuning adjustment mechanism decouple statement Weyl rescaling statement definite 6.7 Brane world traceless gravity Weyl invariance brane world setup Cosmological constant variable 3-form field : 16 Brane world 16 5 bulk 4 effective 4 cosmological 75
constant variable setup y =0 3-form field A µνρ sector 4 effective theory cosmological constant variable adjustment mechanism brane scalar ϕ Kaluza-Klein reduction bulk brane l action y = l brane action brane adjusting scalar ϕ S l = d 4 x ( g U(ϕ)R + 1 ) 2 µϕ µ ϕ V (ϕ) (6.72) y=l g bulk induced metric g ϕ Weyl V ϕ bulk induced metric V statement setup y = l y =0 y = l sector Kaluza-Klein reduction effective theory y = l brane induced metric metric ḡ µν y = l warp factor a l : g µν =ḡ µν a l. (6.73) action S l = d 4 x ḡ y=l ( Ū( ϕ) R + 1 ) 2 µ ϕ µ ϕ V ( ϕ) (6.74) contribution ϕ ϕ a l, (6.75) Ū( ϕ) U(ϕ)a l, (6.76) V ( ϕ) 2 V (ϕ)a l (6.77) 76
y =0 warp factor 1 convention ḡ y = 0 brane quark lepton metric coupling U(ϕ) V (ϕ) Ū( ϕ) V ( ϕ) Kaluza-Klein reduction warp factor rescaling ((6.66) 3 ) U V rescale U R rescale rescale 4 effective theory action : Ū ( ϕ) R = V ( ϕ). (6.78) background ϕ : µ ϕ =0. (6.79) 4 R U V U (ϕ)a l R = V (ϕ)a l 2. (6.80) R a l U V 5 fundamental scale U V fine tuning order one U 0 V order one V (ϕ) U (ϕ) 1 (6.81) R a l (6.82) a l warp factor 4 adjustment mechanism fine tuning warp factor tuning 6.8 Spacetime inflation warp factor 77
inflation i) spatial inflation: 4 : ds 4 2 = dt 2 a(t) d x 2. (6.83) 3 scale factor( ) a(t) cosmological constant Λ : a(t) e t Λ. (6.84) de Sitter slice 3 4 5 ii) spacetime inflation: 5 warped compactification : ds 5 2 = a(y)ḡ µν dx µ dx ν dy 2. (6.85) a(y) warp factor 5 bulk cosmological constant Λ analogous : a(y) e y Λ. (6.86) 3 4 background 5 extra dimension evolution cosmological constant negative anti-de Sitter slice spatial inflation evolution timelike spacelike extra spacelike evolution extra dimension l a l e l Λ (6.87) l warp factor exponential exponential warp factor large volume Randall Sundrum e +y Λ bulk warp factor volume Planck scale ((6.67) ) y = l brane warp factor ((6.66) 3 ) 78
warp factor l background flat effective theory configuration 16 inflation spatial inflation evolution 3 flatness cosmological constant background 4 spacetime flat flatness extra evolution 4 flat background evolution inflation flat fundamental scale 3 (effective theory ) flat effective theory spacetime inflation brane fundamental scale background 3-brane (effective theory ) 3-brane background flat effective theory adjustment mechanism effective theory effective theory SUSY flat background flat SUSY flat background moduli point effective theory 6.9 Quintessence supersymmetric background cosmological constant warp factor brane l depend extra 79
supernova cosmological parameter cosmological constant critical density cosmilogical constant quintessence photon neutrino baryonic matter dark matter 5 : 1. photons, 2. neutrinos, 3. baryonic matter, 4. dark matter, 5. quintessence. quintessence ( late inflation adjusting scalar ) dark energy spacetime inflation background curvature setup brane positive negative supernova positive [ ] Dark energy dark matter density [ ] coincidence ( ) effective theory dark energy cosmological constant dark matter order anthropic principle cosmological constant anthropic principle coincidence 80
coincidence cosmological constant ( ) spacetime inflation extra dimension standard model ( ) ( SUSY breaking ) dark energy coincidence energy density effective theory cosmological constant framework effective theory ( 6.1 ) 81
7 : 17 X ( ) Y X 4 i) ( ) (20 ) higher dimension index external ii) string theory ( ) higher dimension string iii) supersymmetry ( ) fermionic string X Grassmann Grassmann odd extra dimension 82
Y i) effective theory ( ) effective ii) vacuum structure ( ) background vacuum dynamics framework i) effective theory framework brane world scenario string theory heterotic M theory duality SUSY dynamical scale generation ii) vacuum structure spacetime inflatioary background string theory ( 8) SUSY dynamical inflation selection crossover topics crossover (higher dimension, string, SUSY, effective theory, vacuum structure) crossover Z Y effective theory vacuum selection effective theory 83
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