Beyond the SM, Supersymmetry

Transcription

1 Beyond the SM, 1/ 44 Beyond the SM, A. B. Lahanas University of Athens Nuclear and Particle Physics Section Athens - Greece

2 Beyond the SM, 2/ 44 Outline 1 Introduction 2 Beyond the SM Grand Unified Theories The Gauge Hierarchy Problem (GHP) 3 Introducing SUSY

3 Beyond the SM, 2/ 44 Outline 1 Introduction 2 Beyond the SM Grand Unified Theories The Gauge Hierarchy Problem (GHP) 3 Introducing SUSY

4 Beyond the SM, 2/ 44 Outline 1 Introduction 2 Beyond the SM Grand Unified Theories The Gauge Hierarchy Problem (GHP) 3 Introducing SUSY

5 The Standard Model

6 Beyond the SM, 3/ 44 Introduction The Standard Model The Standard Model Unifies Electromagnetic and Weak forces. It has been tested in the laboratory and theory agrees with experiment to an uprecentedent accuarcy. As a mathematical structure is a non-abelian gauge theory based on the group SU(2) U(1) which is broken spontaneously to the U em (1) by the Higgs fields. The carriers of EW forces are the photon ( massless ) and three masive spin - 1 bosons, Z, W ± with masses m z = GeV and m W = GeV.

7 Beyond the SM, 3/ 44 Introduction The Standard Model The Standard Model Unifies Electromagnetic and Weak forces. It has been tested in the laboratory and theory agrees with experiment to an uprecentedent accuarcy. As a mathematical structure is a non-abelian gauge theory based on the group SU(2) U(1) which is broken spontaneously to the U em (1) by the Higgs fields. The carriers of EW forces are the photon ( massless ) and three masive spin - 1 bosons, Z, W ± with masses m z = GeV and m W = GeV.

8 Beyond the SM, 3/ 44 Introduction The Standard Model The Standard Model Unifies Electromagnetic and Weak forces. It has been tested in the laboratory and theory agrees with experiment to an uprecentedent accuarcy. As a mathematical structure is a non-abelian gauge theory based on the group SU(2) U(1) which is broken spontaneously to the U em (1) by the Higgs fields. The carriers of EW forces are the photon ( massless ) and three masive spin - 1 bosons, Z, W ± with masses m z = GeV and m W = GeV.

9 Beyond the SM, 4/ 44 Introduction The force carriers :

10 Beyond the SM, 4/ 44 Introduction The matter :

11 Beyond the SM, 5/ 44 Introduction Its non-abelian structure has been tested in particle processes. Z exchange contributes to W-paired production e e + W W + through Z W W +, characteristic of a non - Abelian coupling!

12 Beyond the SM, 5/ 44 Introduction Its non-abelian structure has been tested in particle processes. Z exchange contributes to W-paired production e e + W W + through Z W W +, characteristic of a non - Abelian coupling!

13 Beyond the SM, 6/ 44 Introduction

14 Beyond the SM, 7/ 44 Introduction The LEP2 last run in 2000 searched for a Higgs for energies up to E CM 207 GeV in the associated production SM Higgs was not discovered and only limits on its mass have been derived, from experiment and theory m H E CM M Z = GeV Non - SM model Higgs masses, like in have lower mass bounds!

15 Beyond the SM, 8/ 44 Introduction Open Questions Although SM is in agreement with experimental observation there are still open questions

16 Beyond the SM, 8/ 44 Introduction Open Questions Too many parameters G F, M Z, α em, α s, θ QCD, 12-fermion masses, ν - mixing parameters and masses, Higgs-boson mass. Why three generations? Why quarks mix? Substructute of Leptons and Quarks? Higgs sector? Strong and EW forces Unify? Gravity how does it fit? Why the EW scale is sixteen orders of magnitude smaller than the Planck scale?...

17 Beyond the SM, 8/ 44 Introduction Open Questions Need move Beyond the SM physics!

18 Beyond the SM, 9/ 44 Beyond the SM Grand Unified Theories GUT Unification Attempts to Unify Electroweak and QCD forces in a simple gauge group encompassing SU c (3) SU(2) U(1) took place between A.D.. Groups SU(5), SO(10), E 7... In all these schemes proton decays Other decay channels are possible, p π + ν...

19 Beyond the SM, 10/ 44 Beyond the SM Grand Unified Theories The unification scale turns out to be M GUT GeV and proton s lifetime τ p 1 α GUT M GUT 4 m p yrs! Current proton decay experiments yield τ p > yrs ruling out these unification schemes. M GUT However if the unification scale M GUT is larger, like in supersymmetric unification, this obstacle is evaded!

20 Beyond the SM, 10/ 44 Beyond the SM Grand Unified Theories The unification scale turns out to be M GUT GeV and proton s lifetime τ p 1 α GUT M GUT 4 m p yrs! Current proton decay experiments yield τ p > yrs ruling out these unification schemes. M GUT However if the unification scale M GUT is larger, like in supersymmetric unification, this obstacle is evaded!

21 Beyond the SM, 11/ 44 Beyond the SM The Gauge Hierarchy Problem (GHP) The presence of a scalar Higgs field poses a severe problem if a theory at a highier scale, Λ, couples to the SM M W = g v 2 = 80.4 GeV = v = 247 GeV M W Higgs mass is of the order of the EW scale, m H 2 = 2 λ v 2 = m H M W Higgs mass is subject to radiative corrections and its value changes! δm 2 H Corrections are δm H 2 = k Λ2, with k = O(g 2,...)

22 Beyond the SM, 12/ 44 Beyond the SM The Gauge Hierarchy Problem (GHP) Corrected Higgs mass m H 2 = 2 λ v 2 + k Λ 2 SM corrections pose no problem, Λ M W and m H stays O(M W ) In the presence of a highier scale theory, Λ M W, Gravity is one example, m H Λ. Higgs mass is driven to high values, not M W! To avoid Higgs mass from becoming large a fine - tuning of k is necessary Gravity = k M2 W M P GUTs = k M2 W M GUT Couplings should be tuned to many decimal places, Unnatural! If the high scale theory is supersymmetric this problem is evaded!

23 Beyond the SM, 13/ 44 Beyond the SM The Gauge Hierarchy Problem (GHP) In : Particles go in pairs! To each Fermion F there is a Boson B with same mass and same couplings. Only their spins differ by half a unit Their contributions to Higgs mass exactly cancel! When SUSY is broken softly couplings same but masses differ m B 2 m F 2 M S 2

24 Beyond the SM, 14/ 44 Beyond the SM The Gauge Hierarchy Problem (GHP) Leading corrections, Λ 2, cancel, next to leading ln Λ δm H 2 = k ( m B 2 m F 2 ) ln Λ k M S 2 ( ln Λ ) corrections are not large due to ln Λ To keep corrections of the order of the EW scale = M S O(1 TeV ) Important for collider searches, supersymmetric particles have masses in the TeV range, supersymmetry is likely to be discovered at the LHC! controls the UV corrections more efficiently than an ordinary field theory!

25 Beyond the SM, 15/ 44 Beyond the SM The Gauge Hierarchy Problem (GHP) The cosmological constant problem Vacuum energy of zero point fluctuations in gravity Vacuum energy T mn = Λ g µν M P 4 g µν E vac M P 4 Observations point to a much lower value E vac ( 10 3 ev ) MP 4! In broken leading corrections cancel between fermions and bosons ( ) 2 Evac S MS 2 M 2 MS P = M P 4 = M P 4! M P Much improvement but still far from M 4 P

26 Beyond the SM, 16/ 44 Introducing SUSY Why Historical note! 1970, ( SUSY ) was invoked to explain the masslessness of neutrino. ( Volkov, Akulov - Wess, Zumino ) Goldstone modes of SB theories with fermionic generators are massless fermions ( Goldstinos ). These could be the neutrinos! 1981, it was called back as resolution of the Gauge Hierarchy Problem in GUTs.

27 Beyond the SM, 16/ 44 Introducing SUSY Why With SM promoted to a SUSY model : SM gauge couplings unify at a scale M GUT GeV. The top quark mass drives the Higgs potential mass parameter to µ 2 < 0 in a natural way and the hierarchy M W M GUT is understood. Supersymmetric GUT models predict larger unification scales ( better chance to reconcile proton lifetime with experimental data ) The quartic Higgs self-couplings are not arbitrary, λ g 2. Higgs masses are bounded m H < 135 GeV Rich phenomenology: Predicts more Higgses, and sparticles with masses in the TeV scale, likely to be discovered at LHC. Predicts WIMP candidates for DM. Gauged SUSY is a Supergravity Theory believed to be the low energy manifestation of String Theories!

28 Beyond the SM, 17/ 44 Introducing SUSY SUSY in QM Bosonic oscillator H B = x 2 + p 2 ( = ω a a + 1 ) 2 2 a x + i p = [ a, a ] = 1 States n B have energies E n = ω (n ), n = 0, 1,... Fermionic oscillator H F ( = ω b b 1 ) 2 { b, b } = 1, b 2 = b 2 = 0 Two states 0 F, 1 F with energies E 0 = ω/2, E 1 = + ω/2

29 Beyond the SM, 17/ 44 Introducing SUSY SUSY in QM Bosonic oscillator H B = x 2 + p 2 ( = ω a a + 1 ) 2 2 a x + i p = [ a, a ] = 1 States n B have energies E n = ω (n ), n = 0, 1,... Fermionic oscillator H F ( = ω b b 1 ) 2 { b, b } = 1, b 2 = b 2 = 0 Two states 0 F, 1 F with energies E 0 = ω/2, E 1 = + ω/2

30 Beyond the SM, 17/ 44 Introducing SUSY SUSY in QM Bosonic oscillator H B = x 2 + p 2 ( = ω a a + 1 ) 2 2 a x + i p = [ a, a ] = 1 States n B have energies E n = ω (n ), n = 0, 1,... Fermionic oscillator H F ( = ω b b 1 ) 2 { b, b } = 1, b 2 = b 2 = 0 Two states 0 F, 1 F with energies E 0 = ω/2, E 1 = + ω/2

31 Beyond the SM, 18/ 44 Introducing SUSY The combined Hamiltonian H = H B + H F States n, s = n s with energies E n,s = ω ( n + s ) States labelled by Quantum numbers : s = 0, 1, n = 0, 1, 2... s declares the fermionic content : s = 0 no-fermion, s = 1 one fermion Except the vacuum E 0,0 the energy spectrum is degenerate! m, 0, m 1, 1 have same energy E n,s = ω m Is there a symmetry behind? Q = ω a b, Q = ω a b and H close a graded Lie algebra! {Q, Q } = 2 H [ Q, H ] = [ Q, H ] = 0 Q 2 = Q 2 = 0

32 Beyond the SM, 18/ 44 Introducing SUSY The combined Hamiltonian H = H B + H F States n, s = n s with energies E n,s = ω ( n + s ) States labelled by Quantum numbers : s = 0, 1, n = 0, 1, 2... s declares the fermionic content : s = 0 no-fermion, s = 1 one fermion Except the vacuum E 0,0 the energy spectrum is degenerate! m, 0, m 1, 1 have same energy E n,s = ω m Is there a symmetry behind? Q = ω a b, Q = ω a b and H close a graded Lie algebra! {Q, Q } = 2 H [ Q, H ] = [ Q, H ] = 0 Q 2 = Q 2 = 0

33 Beyond the SM, 19/ 44 Introducing SUSY For systems based on this algebra: Q, Q generate Supersymmetric ( SUSY ) transformations and they transform fermionic states to bosonic and v.v. The Hamiltonian H commutes with Q, Q, respects SUSY. Degeneracy is a consequence of this symmetry! The Hamiltonian has energies E 0, H = 1 2 ( Q Q + Q Q ) If the symmetry breaks spontaneously, the vacuum state vac is not invariant Q vac, Q vac = 0 The vacuum energy is strictly positive E vac = Q vac 2 + Q vac 2 > 0 and sets the scale of spontaneous breaking of!

34 Beyond the SM, 19/ 44 Introducing SUSY For systems based on this algebra: Q, Q generate Supersymmetric ( SUSY ) transformations and they transform fermionic states to bosonic and v.v. The Hamiltonian H commutes with Q, Q, respects SUSY. Degeneracy is a consequence of this symmetry! The Hamiltonian has energies E 0, H = 1 2 ( Q Q + Q Q ) If the symmetry breaks spontaneously, the vacuum state vac is not invariant Q vac, Q vac = 0 The vacuum energy is strictly positive E vac = Q vac 2 + Q vac 2 > 0 and sets the scale of spontaneous breaking of!

35 Beyond the SM, 20/ 44 Introducing SUSY Physical System? Electron moving on a plane under the influence of a constant magnetic field B B plane H 0 = 1 ( 2 ( π x 2 + π 2 y ) = ω B a a + 1 ) 2 ω B = e B m e c Larmor frequency Electron carries spin - 1/2, couples to B B B H s = µ B ω B = g s 2 With the gyromagnetic ratio g s = 2 ( H s = ω B b b 1 ) 2 σ z 2 with = b = ( ) ( ) 0 1 and b = 0 0

36 Beyond the SM, 20/ 44 Introducing SUSY Physical System? Electron moving on a plane under the influence of a constant magnetic field B B plane H 0 = 1 ( 2 ( π x 2 + π 2 y ) = ω B a a + 1 ) 2 ω B = e B m e c Larmor frequency Electron carries spin - 1/2, couples to B B B H s = µ B ω B = g s 2 With the gyromagnetic ratio g s = 2 ( H s = ω B b b 1 ) 2 σ z 2 with = b = ( ) ( ) 0 1 and b = 0 0

37 Beyond the SM, 20/ 44 Introducing SUSY Physical System? Electron moving on a plane under the influence of a constant magnetic field B B plane H 0 = 1 ( 2 ( π x 2 + π 2 y ) = ω B a a + 1 ) 2 ω B = e B m e c Larmor frequency Electron carries spin - 1/2, couples to B B B H s = µ B ω B = g s 2 With the gyromagnetic ratio g s = 2 ( H s = ω B b b 1 ) 2 σ z 2 with = b = ( ) ( ) 0 1 and b = 0 0

38 Beyond the SM, 21/ 44 Introducing SUSY The total Hamiltonian H tot = H 0 + H s is like the supersymmetric Hamiltonian studied before! The energy spectrum ( E n,σ = ω B n σ ) σ = +1, spin, 2 σ = 1, spin Two energy eigenstates for each Landau level E m = ω B m, m = 1, 2,... m,, m 1, One state 0, to the vacuum energy, E 0 = 0, not degenerate! Action of Q, Q : Q m, = m 1, Q m, = m + 1, flips spin up flips spin down QED corrections induce g s 2 and break SUSY since the Hamiltonian is not supersymmetric!

39 Beyond the SM, 21/ 44 Introducing SUSY The total Hamiltonian H tot = H 0 + H s is like the supersymmetric Hamiltonian studied before! The energy spectrum ( E n,σ = ω B n σ ) σ = +1, spin, 2 σ = 1, spin Two energy eigenstates for each Landau level E m = ω B m, m = 1, 2,... m,, m 1, One state 0, to the vacuum energy, E 0 = 0, not degenerate! Action of Q, Q : Q m, = m 1, Q m, = m + 1, flips spin up flips spin down QED corrections induce g s 2 and break SUSY since the Hamiltonian is not supersymmetric!

40 Beyond the SM, 21/ 44 Introducing SUSY The total Hamiltonian H tot = H 0 + H s is like the supersymmetric Hamiltonian studied before! The energy spectrum ( E n,σ = ω B n σ ) σ = +1, spin, 2 σ = 1, spin Two energy eigenstates for each Landau level E m = ω B m, m = 1, 2,... m,, m 1, One state 0, to the vacuum energy, E 0 = 0, not degenerate! Action of Q, Q : Q m, = m 1, Q m, = m + 1, flips spin up flips spin down QED corrections induce g s 2 and break SUSY since the Hamiltonian is not supersymmetric!

41 Beyond the SM, 21/ 44 Introducing SUSY The total Hamiltonian H tot = H 0 + H s is like the supersymmetric Hamiltonian studied before! The energy spectrum ( E n,σ = ω B n σ ) σ = +1, spin, 2 σ = 1, spin Two energy eigenstates for each Landau level E m = ω B m, m = 1, 2,... m,, m 1, One state 0, to the vacuum energy, E 0 = 0, not degenerate! Action of Q, Q : Q m, = m 1, Q m, = m + 1, flips spin up flips spin down QED corrections induce g s 2 and break SUSY since the Hamiltonian is not supersymmetric!

42 Beyond the SM, 22/ 44 In Particle Theory SUSY may play a fundamental role! A Femion, Boson symmetry fused with the Poincare symmetry, if Poincare is promoted to a graded Lie algebra.! The Algebra : Poincare symmetry : [ P m, P n ] = 0 [ P m, M rs ] = i ( η mr P s ( r s ) ) [ M mn, M rs ] = i ( η ms M nr + η nr M ms ( r s ) ) Spinor - representation ψ has two Weyl components ψ α ( ) ψ1 ψ = ψ 2 ψ 2 Its antispinor ψ has two Weyl components ψ α ψ = ( ψ2 ψ 1 ) ψ α

43 Beyond the SM, 22/ 44 In Particle Theory SUSY may play a fundamental role! A Femion, Boson symmetry fused with the Poincare symmetry, if Poincare is promoted to a graded Lie algebra.! The Algebra : Poincare symmetry : [ P m, P n ] = 0 [ P m, M rs ] = i ( η mr P s ( r s ) ) [ M mn, M rs ] = i ( η ms M nr + η nr M ms ( r s ) ) Spinor - representation ψ has two Weyl components ψ α ( ) ψ1 ψ = ψ 2 ψ 2 Its antispinor ψ has two Weyl components ψ α ψ = ( ψ2 ψ 1 ) ψ α

44 Beyond the SM, 23/ 44 The simplest supe-poincare algebra includes two spinorial charges Q, Q commuting with P m { Q α, Q α } = 2 ( σ m ) α α Pm [ Q, P m ] = [ Q, P ] m = 0 N = 1 SUSY Highier Supersymmeries have more spinorial charges, N > 1! Q, Q generate SUSY transformations and transform a bosonic state to a fermionic ( and v.v. ) Q B = F, Q F = B ( idem Q ) The Hamiltonian is positive definite operator, and energies are E 0. H = 1 4 ( Q 1 Q 1 + Q 1 Q 1 + Q 2 Q 2 + Q 2 Q 2 ) Exact SUSY : The vacuum state is invariant Q vac = Q vac = 0 and the vacuum energy E vac = 0! SB SUSY : The vacuum state is non-invariant Q vac, Q vac = 0 and the vacuum energy E vac > 0!

45 Beyond the SM, 23/ 44 The simplest supe-poincare algebra includes two spinorial charges Q, Q commuting with P m { Q α, Q α } = 2 ( σ m ) α α Pm [ Q, P m ] = [ Q, P ] m = 0 N = 1 SUSY Highier Supersymmeries have more spinorial charges, N > 1! Q, Q generate SUSY transformations and transform a bosonic state to a fermionic ( and v.v. ) Q B = F, Q F = B ( idem Q ) The Hamiltonian is positive definite operator, and energies are E 0. H = 1 4 ( Q 1 Q 1 + Q 1 Q 1 + Q 2 Q 2 + Q 2 Q 2 ) Exact SUSY : The vacuum state is invariant Q vac = Q vac = 0 and the vacuum energy E vac = 0! SB SUSY : The vacuum state is non-invariant Q vac, Q vac = 0 and the vacuum energy E vac > 0!

46 Beyond the SM, 23/ 44 The simplest supe-poincare algebra includes two spinorial charges Q, Q commuting with P m { Q α, Q α } = 2 ( σ m ) α α Pm [ Q, P m ] = [ Q, P ] m = 0 N = 1 SUSY Highier Supersymmeries have more spinorial charges, N > 1! Q, Q generate SUSY transformations and transform a bosonic state to a fermionic ( and v.v. ) Q B = F, Q F = B ( idem Q ) The Hamiltonian is positive definite operator, and energies are E 0. H = 1 4 ( Q 1 Q 1 + Q 1 Q 1 + Q 2 Q 2 + Q 2 Q 2 ) Exact SUSY : The vacuum state is invariant Q vac = Q vac = 0 and the vacuum energy E vac = 0! SB SUSY : The vacuum state is non-invariant Q vac, Q vac = 0 and the vacuum energy E vac > 0!

47 Beyond the SM, 23/ 44 The simplest supe-poincare algebra includes two spinorial charges Q, Q commuting with P m { Q α, Q α } = 2 ( σ m ) α α Pm [ Q, P m ] = [ Q, P ] m = 0 N = 1 SUSY Highier Supersymmeries have more spinorial charges, N > 1! Q, Q generate SUSY transformations and transform a bosonic state to a fermionic ( and v.v. ) Q B = F, Q F = B ( idem Q ) The Hamiltonian is positive definite operator, and energies are E 0. H = 1 4 ( Q 1 Q 1 + Q 1 Q 1 + Q 2 Q 2 + Q 2 Q 2 ) Exact SUSY : The vacuum state is invariant Q vac = Q vac = 0 and the vacuum energy E vac = 0! SB SUSY : The vacuum state is non-invariant Q vac, Q vac = 0 and the vacuum energy E vac > 0!

48 Beyond the SM, 24/ 44 Fundamental SUSY representations : The basic representations, or multiplets, for model building Multiplet name Particle content Spin content Chiral φ ψ 0 1/2 Vector A µ λ 1 1/2 Gravity g µν ψ µ 2 3/2 Every multiplet consists of a boson and a L - handed Weyl fermion with spin differing by 1/2. They are connected by SUSY transformations SUSY : φ ψ, A µ λ, g µν ψ µ The recipe to construct SUSY Lagrangians well - known In flat space-time, if N chiral - multiplets are involved (φ i, ψ i ) i L = µ φ i 2 + i ψi σ µ µ ψ i W φ i ( 2 ) W ψ i ψ j + h.c. φ i φ j

49 Beyond the SM, 25/ 44 W is the superpotential W (φ) = λ ijk φ i φ j φ k + m ij φ i φ j Bosonic and Fermionic mass eigenstates have equal number of d.o.f and same mass as a Manifestation of m B = m F The scalar potential is positive definite : V = W φ i 2 0 At the vacuum state φ i the minimum V 0 of the potential : Exact SUSY : V 0 = 0 and SB SUSY : V 0 0 and m B = m F m B m F Only broken, spontaneously or other, can be realized in nature to lift mass degeneracy!

50 Beyond the SM, 26/ 44 To build supersymmetric gauge theories need include vector-multiplets carrying gauge quantum numbers ( A (a) µ, λ (a) ) with a = gauge index.

51 Beyond the SM, 26/ 44 SUSY vertices are related!

52 Beyond the SM, 26/ 44 SUSY vertices are related! Essential for cancellations among graphs! δm H 2 = 0 if SUSY is exact! δm 2 H δm H 2 δmh 2 = m A 2 µ mλ 2 if SUSY is broken spontaneously or softly

53 Beyond the SM, 27/ 44 The minimal The minimal SUSY extension of the SM is the MSSM, To every SM particle its SUSY partner is introduced and the d.o.f. are doubled! Rich phenomenology : Predicts new particles ( sparticles ), partners of the known SM particles. Needs two Higgs doublets ( unlike one needed in SM! ), and five Higgses remain after SSB!

54 Beyond the SM, 27/ 44 The minimal The minimal SUSY extension of the SM is the MSSM, To every SM particle its SUSY partner is introduced and the d.o.f. are doubled! Rich phenomenology : Predicts new particles ( sparticles ), partners of the known SM particles. Needs two Higgs doublets ( unlike one needed in SM! ), and five Higgses remain after SSB!

55 Beyond the SM, 28/ 44 The breaking of SUSY No - mass degeneracy among fermions and bosons is observed in nature! must be broken to lift up masses of SUSY - partners of the known particles ( sparticles ). 1 Spontaneous breaking of SUSY is an elegant mechanism. A Goldstone fermion ( Goldstino ) appears due to the fermionic character of the SUSY generators. This is not any of the neutrinos (? ) 2 If broken explicitly should preserve its good properties in the UV regime. Absence of quadratic divergences to protect Higgses from getting large masses = Softly broken Supersymmetries Softly broken Supersymmetries occur in a variety of scenarios. In SUPERGRAVITY it can be realized through gravitational interactions ( Gravity mediated breaking )

56 Beyond the SM, 28/ 44 The breaking of SUSY Other known types of breaking : Gauge mediation. Anomaly mediation. Breaking by anomalous U(1) s. Extra Dimension mechanisms...

57 Beyond the SM, 28/ 44 The breaking of SUSY

58 Beyond the SM, 29/ 44 N = 1 Supergravity ( SUGRA ) The Poincare group can be gauged. Gauge fields for P m, M mn = the vierbein em e µ m, spin-connection ω µ mn gct transformations are gauge transformations if the curvarure R µν m = D µ eν n D ν eµ m = 0 Then spin-connection is not independent ω µ mn = ω µ mn ( e ) The vierbein is covariantly constant D µ e ν m = 0 Then gct connection Γ k µ ν is the Christoffle connection. The Einstein - Hilbert Lagrangian L EH = 1 2 k(4) 2 e R can be viewed as a Poincare gauge invariant Lagrangian!

59 Beyond the SM, 29/ 44 N = 1 Supergravity ( SUGRA ) The Poincare group can be gauged. Gauge fields for P m, M mn = the vierbein em e µ m, spin-connection ω µ mn gct transformations are gauge transformations if the curvarure R µν m = D µ eν n D ν eµ m = 0 Then spin-connection is not independent ω µ mn = ω µ mn ( e ) The vierbein is covariantly constant D µ e ν m = 0 Then gct connection Γ k µ ν is the Christoffle connection. The Einstein - Hilbert Lagrangian L EH = 1 2 k(4) 2 e R can be viewed as a Poincare gauge invariant Lagrangian!

60 Beyond the SM, 29/ 44 N = 1 Supergravity ( SUGRA ) The Poincare group can be gauged. Gauge fields for P m, M mn = the vierbein em e µ m, spin-connection ω µ mn gct transformations are gauge transformations if the curvarure R µν m = D µ eν n D ν eµ m = 0 Then spin-connection is not independent ω µ mn = ω µ mn ( e ) The vierbein is covariantly constant D µ e ν m = 0 Then gct connection Γ k µ ν is the Christoffle connection. The Einstein - Hilbert Lagrangian L EH = 1 2 k(4) 2 e R can be viewed as a Poincare gauge invariant Lagrangian!

61 Beyond the SM, 29/ 44 N = 1 Supergravity ( SUGRA ) The Poincare group can be gauged. Gauge fields for P m, M mn = the vierbein em e µ m, spin-connection ω µ mn gct transformations are gauge transformations if the curvarure R µν m = D µ eν n D ν eµ m = 0 Then spin-connection is not independent ω µ mn = ω µ mn ( e ) The vierbein is covariantly constant D µ e ν m = 0 Then gct connection Γ k µ ν is the Christoffle connection. The Einstein - Hilbert Lagrangian L EH = 1 2 k(4) 2 e R can be viewed as a Poincare gauge invariant Lagrangian!

62 Beyond the SM, 30/ 44 Global SUSY can be gauged ( = Local SUSY or Supergravity )! In addition to e µ m, ω µ mn need gauge fields for the SUSY generators Q, Q = ψ µ, ψµ µ both vector and fermion fields! They are combined to form a 4-component massless Majorana fermion Ψ µ carrying spin - 3 2, the superpartner of the graviton, the Gravitino. The graviton ( spin - 2) and the gravitino ( spin ) constitute the graviton multiplet. The invariant Lagrangian undel local SUSY transformations L susy EH = 1 2 k (4) 2 e R 1 µ 2 ɛµνρσ Ψµ γ 5 γ ν D ρ Ψ σ Describes the gravitational interactions of only the graviton and the gravitino. D µ = µ i 2 ωmn µ σ mn the spin covariant derivative.

63 Beyond the SM, 30/ 44 Global SUSY can be gauged ( = Local SUSY or Supergravity )! In addition to e µ m, ω µ mn need gauge fields for the SUSY generators Q, Q = ψ µ, ψµ µ both vector and fermion fields! They are combined to form a 4-component massless Majorana fermion Ψ µ carrying spin - 3 2, the superpartner of the graviton, the Gravitino. The graviton ( spin - 2) and the gravitino ( spin ) constitute the graviton multiplet. The invariant Lagrangian undel local SUSY transformations L susy EH = 1 2 k (4) 2 e R 1 µ 2 ɛµνρσ Ψµ γ 5 γ ν D ρ Ψ σ Describes the gravitational interactions of only the graviton and the gravitino. D µ = µ i 2 ωmn µ σ mn the spin covariant derivative.

64 Beyond the SM, 31/ 44 To build realistic models need add matter fields and gauge bosons. Promote chiral and vector multiplets to gravity supermultiplets! The general N = 1 Supergravity Lagrangian, in D = 4, has been derived 1982, Cremmer, Ferrara, Girardello, van Proyen Expressed in terms of three arbitrary functions depending on the scalar fields involved φ, φ W (φ) Superpotential K( φ, φ ) Kahler potential f ab (φ) Gauge kinetic function N=1, D=4 Supergravity may be conceived as effective superstring theories valid at E M P, well below the string scale E M string, upon compactifying the extra 6 dimensions on particular manifolds.

65 Beyond the SM, 31/ 44 To build realistic models need add matter fields and gauge bosons. Promote chiral and vector multiplets to gravity supermultiplets! The general N = 1 Supergravity Lagrangian, in D = 4, has been derived 1982, Cremmer, Ferrara, Girardello, van Proyen Expressed in terms of three arbitrary functions depending on the scalar fields involved φ, φ W (φ) Superpotential K( φ, φ ) Kahler potential f ab (φ) Gauge kinetic function N=1, D=4 Supergravity may be conceived as effective superstring theories valid at E M P, well below the string scale E M string, upon compactifying the extra 6 dimensions on particular manifolds. W (φ), K( φ, φ ), f ab (φ) will be known once physics beyond the Planck scale is better known!

66 Beyond the SM, 31/ 44 Supergravities in highier dimensions ( D > 4 ) seem to play a key role in String theory and Branes. N = 1, D = 11 is utilized for a unified view of certain types of String Theories....

67 Beyond the SM, 32/ 44 Gravity mediated breaking of N=1 SUGRA The features : N = 1 local is spontaneously broken at a scale M s by some hidden ( invisible at low energies ) fields. The emerging Goldstino is absorbed by the gravitino, superhiggs effect, which gets a mass m 3/2 M2 s M P The induced breaking is felt through gravitational interactions by the visible fields and the theory at energies E < M P is a softly broken global M s L = L SUSY + L soft breaking at low energies in encoded within L soft!

68 Beyond the SM, 32/ 44 Gravity mediated breaking of N=1 SUGRA The features : N = 1 local is spontaneously broken at a scale M s by some hidden ( invisible at low energies ) fields. The emerging Goldstino is absorbed by the gravitino, superhiggs effect, which gets a mass m 3/2 M2 s M P The induced breaking is felt through gravitational interactions by the visible fields and the theory at energies E < M P is a softly broken global M s L = L SUSY + L soft breaking at low energies in encoded within L soft!

69 Beyond the SM, 33/ 44 The soft Lagrangian L soft The generic form of the soft breaking part : L soft = µ 2 i j φ i φ j ( mi 2 j φ i φ j + A i j k φ i φ j φ k + M a 2 λ a λ a + h.c. ) Sources of global breaking : Gaugino masses Scalar masses Mixing masses Trilinear couplings M a m i j µ i j A i j k M P Their values at M P depend on the details of the local breaking and are, in general, of order m 3/2! Their values at EW energies are set by solving the RGEs from Planck M P to M W

70 Beyond the SM, 34/ 44 Magnitude of local breaking scale? M s GeV = m 3/2 100 GeV O(1) TeV Mass splittings expected between particles and sparticles, m B 2 m F 2 O(1) TeV as demanded for the resolution of the GHP! Particularly appealing scenario : Soft parameters are universal at the unification scale m i j m 0 M a m 1/2 A i j k A 0 The most economic model, msugra, realized in particular models with canonical forms for K, f ab which is supported phenomenologically ( absence of FCNC ). All predictions depend on parameters m 0, m 1/2, A 0

71 Beyond the SM, 34/ 44 Particular models of gravity mediated SUSY breaking are more predictive: Polonyi : m 0 = m 3/2, A 0 = (3 3) m 3/2, m 1/2 = O(m 3/2 ) Dilaton-dominated : m 0 = m 3/2, m 1/2 = A 0 = 3 m 3/2 No - scale : m 1/2, m 0, A 0 m 3/2 m 3/2 determined by radiative corrections! Non-universal models compatible with phenomenological requirements do also exist!

72 Beyond the SM, 35/ 44 MSSM The minimal Supersymmetric SM ( MSSM ): SM world SUSY world Quarks and Leptons 3 families Squarks and Sleptons 3 families ( ) ( ) ul Quarks : u d L c d L c ũl Squarks : ũ L d c d L L c L ( ) ( ) νl Leptons : e e L c νl Sleptons : ẽ L ẽ L c L e L Gauge bosons U(1) : B µ SU(2) : W 1, 2, 3 Gauginos Bµ µ W µ 1, 2, 3 W µ G µ 1, 2..., 8 g 1, 2..., 8 SU(3) : G 1, 2..., 8 ẽ L Bino µ Winos Gluinos

73 Beyond the SM, 36/ 44 The Higgs sector in the MSSM is different from that of the SM! In the MSSM two scalar Higgs doublets H 1, H 2 are needed to give masses to down ( T 3 = 1/2 ) and up ( T 3 = +1/2 ) fermions! Higgs scalars ( ) H 0 1 ( H + 2 ) Higgsinos ( ) H 0 1 ( H + 2 ) H 1 = H 1, H 2 = H 2 0 H 1 = H 1, H2 2 = H 2 0 The new fermionic d.o.f. in the MSSM do not introduce gauge anomalies. Both H 1, H 2 develop vev s v cos β 0 H 1 = 2, H 2 = v sin β 0 2 M W = g v determines v but not the relative size of the vev s 2 tan β = H 0 2 / H 0 1

74 Beyond the SM, 36/ 44 The Higgs sector in the MSSM is different from that of the SM! In the MSSM two scalar Higgs doublets H 1, H 2 are needed to give masses to down ( T 3 = 1/2 ) and up ( T 3 = +1/2 ) fermions! Higgs scalars ( ) H 0 1 ( H + 2 ) Higgsinos ( ) H 0 1 ( H + 2 ) H 1 = H 1, H 2 = H 2 0 H 1 = H 1, H2 2 = H 2 0 The new fermionic d.o.f. in the MSSM do not introduce gauge anomalies. Both H 1, H 2 develop vev s v cos β 0 H 1 = 2, H 2 = v sin β 0 2 M W = g v determines v but not the relative size of the vev s 2 tan β = H 0 2 / H 0 1

75 Beyond the SM, 37/ 44 After EW breaking states mix! Mass eigenstates are not the Gauge eigenstates!

76 Beyond the SM, 38/ 44 Names Spin Mass eigenstates Gauge Eigenstates Squarks 0 c 1 c 2 s 1 s 2 in 3 colors t 1 t 2 b1 1 b2 2 ũ 1 ũ 2 d1 1 d2 2 ũ L ũ c ũ ũ c L L c c d L c d L c 1 c 2 s 1 s 2 c L c L c s L s L c t 1 t 2 b 1 b 2 t L t c b L b c L ẽ 1, ẽ 2 ν e ẽ L, ẽ c Sleptons 0 µ 1, µ 2 ν µ µ L, µ c L L τ 1, τ 2 ν τ τ L, τ c Charged Higgses 0 H ± H 2 + H 1 Neutral Higgses 0 H 0 h 0 A H 0 Neutralinos 1/2 χ 0 1 χ 0 2 χ 0 3 χ 0 4 Majorana Charginos 1/2 χ + 1 χ + 2 Dirac L L ν L e ν µ L ν L τ H 1 0 H 2 0 H 2 0 B W 0 H 1 0 W ± H 2 + Gluinos 1/2 g g g g g g in 8 colors Majorana H 1

77 Beyond the SM, 39/ 44 The Higgs sector of MSSM There are 5 Higgs fields surviving EW SSB in the MSSM. 2 charged and 3 neutral! Phenomenology different than in the SM. H 0 1 H0 2 Two are CP - even, H 0, h 0 mixtures of H 1 0 H0 2 ( ) ( ) ( H 0 cos α sin α H 0 1 = sin α cos α h 0 H 2 0 ) The lightest mode h 0 has a mass < 135 GeV! Not like the SM Higgs, its couplings are different The SM LEP bound on the Higgs mass, GeV does not apply!

78 Beyond the SM, 40/ 44 Associated Higgs production at LEP last run at a C.M. energy 207 GeV did not discover Higgs! This assumes a SM Higgs coupling! In the MSSM σ SUSY = sin 2 (α β) σ SM The bound weakens m h > 91 GeV! A discovery of a light Higgs below GeV, or other Higgs states could be a signal for

79 Beyond the SM, 41/ 44 In msugra the arbitrary parameters are m 0, m 1/2, A 0, tan β In addition the sign of the parameter µ, which mixes the Higgs multiplets, µ H 1 H 2, is a parameter. Mass spectrum and interaction vertices depend on these!

80 Beyond the SM, 42/ 44 Schematic sample of sparticle mass spectrum in msugra model

81 Beyond the SM, 43/ 44 SUSY in Cosmology SUSY predicts a compelling Dark Matter candidates! The lightest neutralino qualifies as a WIMP Dark Matter candidate It interacts weakly Its mass is in the range 100 GeV few TeV If SUSY is the low energy manifestation of Supergravity the gravitino may also be the DM candidate...

82 Beyond the SM, 43/ 44 SUSY in Cosmology SUSY predicts a compelling Dark Matter candidates! The lightest neutralino qualifies as a WIMP Dark Matter candidate It interacts weakly Its mass is in the range 100 GeV few TeV If SUSY is the low energy manifestation of Supergravity the gravitino may also be the DM candidate...

83 Beyond the SM, 44/ 44 SUSY at LHC Models of low energy predict new particles ( SUSY particles or sparticles ) with masses in the TeV scale. is within LHC reach! Absence of sparticle discovery so far puts limits on their masses and constrains the parameters of any model! WMAP data, astrophysical data, and other DM searches constrain SUSY theories more than accelerator experiments ( at the present stage )....