The rudiments of Supersymmetry 1/ 53. Supersymmetry I. A. B. Lahanas. University of Athens Nuclear and Particle Physics Section Athens - Greece

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1 The rudiments of Supersymmetry 1/ 53 Supersymmetry I A. B. Lahanas University of Athens Nuclear and Particle Physics Section Athens - Greece

2 The rudiments of Supersymmetry 2/ 53 Outline 1 Introduction Motivation The Gauge Hierarchy Problem (GHP) 2 The SUSY fundamentals Introducing SUSY The Algebra of Supersymmetry Supermultiplets 3 Supersymmetric Field Theories SUSY Lagrangians Breaking global SUSY 4 Conclusions Summary of this Lecture

3 The rudiments of Supersymmetry 2/ 53 Outline 1 Introduction Motivation The Gauge Hierarchy Problem (GHP) 2 The SUSY fundamentals Introducing SUSY The Algebra of Supersymmetry Supermultiplets 3 Supersymmetric Field Theories SUSY Lagrangians Breaking global SUSY 4 Conclusions Summary of this Lecture

4 The rudiments of Supersymmetry 2/ 53 Outline 1 Introduction Motivation The Gauge Hierarchy Problem (GHP) 2 The SUSY fundamentals Introducing SUSY The Algebra of Supersymmetry Supermultiplets 3 Supersymmetric Field Theories SUSY Lagrangians Breaking global SUSY 4 Conclusions Summary of this Lecture

5 The rudiments of Supersymmetry 2/ 53 Outline 1 Introduction Motivation The Gauge Hierarchy Problem (GHP) 2 The SUSY fundamentals Introducing SUSY The Algebra of Supersymmetry Supermultiplets 3 Supersymmetric Field Theories SUSY Lagrangians Breaking global SUSY 4 Conclusions Summary of this Lecture

6 The rudiments of Supersymmetry 3/ 53 Introduction Motivation Open Questions Although SM is in agreement with experimental observation fundamental questions still remain unanswered

7 The rudiments of Supersymmetry 3/ 53 Introduction Motivation 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?...

8 The rudiments of Supersymmetry 3/ 53 Introduction Motivation 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?... Need move Beyond the SM physics!

9 The rudiments of Supersymmetry 4/ 53 Introduction Motivation Why Supersymmetry? Good theoretical reasons to believe that SUSY will be the next big discovery! Some believe that SUSY is the low energy manifestation of a unified description valid at Planckian energies! Its mathematical beauty and its less divergent character, as a QFT, qualifies it as a powerful tool to build theoretical models. It is the only known symmetry that treats bosons and fermions on equal footing....

10 The rudiments of Supersymmetry 4/ 53 Introduction Motivation Why Supersymmetry? Good theoretical reasons to believe that SUSY will be the next big discovery! Some believe that SUSY is the low energy manifestation of a unified description valid at Planckian energies! Its mathematical beauty and its less divergent character, as a QFT, qualifies it as a powerful tool to build theoretical models. It is the only known symmetry that treats bosons and fermions on equal footing....

11 The rudiments of Supersymmetry 5/ 53 Introduction Motivation Why Supersymmetry? Supersymmetry resolves the GHP and it is an indispensable ingredient of String Theories! Historical note! 1970, Supersymmetry ( SUSY ) was invoked to explain the masslessness of the 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.

12 The rudiments of Supersymmetry 5/ 53 Introduction Motivation Why Supersymmetry? Supersymmetry resolves the GHP and it is an indispensable ingredient of String Theories! Historical note! 1970, Supersymmetry ( SUSY ) was invoked to explain the masslessness of the 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.

13 The rudiments of Supersymmetry 6/ 53 Introduction Motivation Why Supersymmetry? 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!

14 The rudiments of Supersymmetry 7/ 53 Introduction The Gauge Hierarchy Problem (GHP) How the Higgs stays light? The presence of a Higgs boson poses a severe problem if a theory at a highier scale, Λ, couples to the SM! V ( H ) = µ 2 H 2 + λ 2 H 4, H = 175 GeV The quartic coupling is bounded by theory λ < O(1), ( Unitarity,...), and Higgs mass is bounded : m H 2 = 2 λ H 2 < ( few hundred GeV ) 2 Higgs mass is subject to radiative corrections and its value changes! m H 2 = 2 λ v 2 + α Λ 2 SM corrections pose no problem, Λ M W and Higgs stays light. In the presence of a highier scale theory, Λ M W, Gravity is one example, m H Λ and Higgs mass is driven to high values!

15 The rudiments of Supersymmetry 7/ 53 Introduction The Gauge Hierarchy Problem (GHP) How the Higgs stays light? The presence of a Higgs boson poses a severe problem if a theory at a highier scale, Λ, couples to the SM! V ( H ) = µ 2 H 2 + λ 2 H 4, H = 175 GeV The quartic coupling is bounded by theory λ < O(1), ( Unitarity,...), and Higgs mass is bounded : m H 2 = 2 λ H 2 < ( few hundred GeV ) 2 Higgs mass is subject to radiative corrections and its value changes! m H 2 = 2 λ v 2 + α Λ 2 SM corrections pose no problem, Λ M W and Higgs stays light. In the presence of a highier scale theory, Λ M W, Gravity is one example, m H Λ and Higgs mass is driven to high values!

16 The rudiments of Supersymmetry 8/ 53 Introduction The Gauge Hierarchy Problem (GHP) To avoid Higgs mass from becoming large a fine - tuning of α is necessary Gravity = α < m 2 W /m 2 Planck GUTs = α < m 2 W /m 2 GUT Couplings should be tuned to many decimal places, Unnatural! Unless loop contributions conspire to cancelling each other! For a Dirac fermion ( 4 d.o.f ) coupled to Higgs corrections are δ m H 2 = λ2 [ f 2 Λ 2 16 π mf 2 ln( Λ/m f ) ]

17 The rudiments of Supersymmetry 8/ 53 Introduction The Gauge Hierarchy Problem (GHP) To avoid Higgs mass from becoming large a fine - tuning of α is necessary Gravity = α < m 2 W /m 2 Planck GUTs = α < m 2 W /m 2 GUT Couplings should be tuned to many decimal places, Unnatural! Unless loop contributions conspire to cancelling each other! For a Dirac fermion ( 4 d.o.f ) coupled to Higgs corrections are δ m H 2 = λ2 [ f 2 Λ 2 16 π mf 2 ln( Λ/m f ) ]

18 The rudiments of Supersymmetry 9/ 53 Introduction The Gauge Hierarchy Problem (GHP) A complex scalar boson coupled to Higgs yields quadratic corrections with opposite sign δ m H 2 = λ2 [ b Λ 2 16 π 2 2 mb 2 ln( Λ/m b ) ] If λ f = λ b and two complex scalars, to match the d.o.f of the Dirac fermion, the quadratic divergences of the two graphs cancel! This occurs automatically is supersymmetric theories even if they are broken softly!

19 The rudiments of Supersymmetry 9/ 53 Introduction The Gauge Hierarchy Problem (GHP) A complex scalar boson coupled to Higgs yields quadratic corrections with opposite sign δ m H 2 = λ2 [ b Λ 2 16 π 2 2 mb 2 ln( Λ/m b ) ] If λ f = λ b and two complex scalars, to match the d.o.f of the Dirac fermion, the quadratic divergences of the two graphs cancel! This occurs automatically is supersymmetric theories even if they are broken softly!

20 The rudiments of Supersymmetry 10/ 53 Introduction The Gauge Hierarchy Problem (GHP) In Supersymmetry: 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 SUSY 2

21 The rudiments of Supersymmetry 11/ 53 Introduction The Gauge Hierarchy Problem (GHP) Leading quadratic corrections, Λ 2, cancel, next to leading ln Λ survive δm H 2 = ( m B 2 m F 2 ) ln Λ M SUSY 2 ( ln Λ ) corrections are not dangerous ( large ) due to their logarithmic nature! To keep corrections of the order of the EW scale = M SUSY O(1 TeV ) Important for collider searches, supersymmetric particles have masses in the TeV range, supersymmetry is likely to be discovered at the LHC! Supersymmetry controls the UV corrections more efficiently than an ordinary field theory! Due to this it also ameliorates the cosmological constant problem!

22 The rudiments of Supersymmetry 11/ 53 Introduction The Gauge Hierarchy Problem (GHP) Leading quadratic corrections, Λ 2, cancel, next to leading ln Λ survive δm H 2 = ( m B 2 m F 2 ) ln Λ M SUSY 2 ( ln Λ ) corrections are not dangerous ( large ) due to their logarithmic nature! To keep corrections of the order of the EW scale = M SUSY O(1 TeV ) Important for collider searches, supersymmetric particles have masses in the TeV range, supersymmetry is likely to be discovered at the LHC! Supersymmetry controls the UV corrections more efficiently than an ordinary field theory! Due to this it also ameliorates the cosmological constant problem!

23 The rudiments of Supersymmetry 11/ 53 Introduction The Gauge Hierarchy Problem (GHP) Leading quadratic corrections, Λ 2, cancel, next to leading ln Λ survive δm H 2 = ( m B 2 m F 2 ) ln Λ M SUSY 2 ( ln Λ ) corrections are not dangerous ( large ) due to their logarithmic nature! To keep corrections of the order of the EW scale = M SUSY O(1 TeV ) Important for collider searches, supersymmetric particles have masses in the TeV range, supersymmetry is likely to be discovered at the LHC! Supersymmetry controls the UV corrections more efficiently than an ordinary field theory! Due to this it also ameliorates the cosmological constant problem!

24 The rudiments of Supersymmetry 12/ 53 Introduction The Gauge Hierarchy Problem (GHP) Why vacuum energy is so tiny? Vacuum energy of zero point fluctuations in gravity Vacuum energy T µν = mplanck 4 g µν E vac m Planck 4 Observations point to a much lower value E vac ( 10 3 ev ) mplanck 4! In Supersymmetry leading contributions cancel between fermions and bosons even if SUSY is broken ( ) 2 Evac S MSUSY 2 m 2 MSUSY Planck = m Planck 4 = m Planck 4! m Planck Much improvement but still far from m 4 Planck

25 The rudiments of Supersymmetry 12/ 53 Introduction The Gauge Hierarchy Problem (GHP) Why vacuum energy is so tiny? Vacuum energy of zero point fluctuations in gravity Vacuum energy T µν = mplanck 4 g µν E vac m Planck 4 Observations point to a much lower value E vac ( 10 3 ev ) mplanck 4! In Supersymmetry leading contributions cancel between fermions and bosons even if SUSY is broken ( ) 2 Evac S MSUSY 2 m 2 MSUSY Planck = m Planck 4 = m Planck 4! m Planck Much improvement but still far from m 4 Planck

26 The rudiments of Supersymmetry 13/ 53 The SUSY fundamentals 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

27 The rudiments of Supersymmetry 13/ 53 The SUSY fundamentals 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

28 The rudiments of Supersymmetry 13/ 53 The SUSY fundamentals 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 The rudiments of Supersymmetry 14/ 53 The SUSY fundamentals Introducing SUSY The combined Hamiltonian H = H B + H F States n, s = n s have 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

30 The rudiments of Supersymmetry 14/ 53 The SUSY fundamentals Introducing SUSY The combined Hamiltonian H = H B + H F States n, s = n s have 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

31 The rudiments of Supersymmetry 15/ 53 The SUSY fundamentals 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 Supersymmetry!

32 The rudiments of Supersymmetry 15/ 53 The SUSY fundamentals 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 Supersymmetry!

33 The rudiments of Supersymmetry 16/ 53 The SUSY fundamentals 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

34 The rudiments of Supersymmetry 16/ 53 The SUSY fundamentals 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

35 The rudiments of Supersymmetry 16/ 53 The SUSY fundamentals 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 The rudiments of Supersymmetry 17/ 53 The SUSY fundamentals 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!

37 The rudiments of Supersymmetry 17/ 53 The SUSY fundamentals 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!

38 The rudiments of Supersymmetry 17/ 53 The SUSY fundamentals 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 The rudiments of Supersymmetry 17/ 53 The SUSY fundamentals 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 The rudiments of Supersymmetry 18/ 53 The SUSY fundamentals The Algebra of Supersymmetry In Particle Theory SUSY may play a fundamental role! It is a Femion, Boson symmetry enlarging the Poincare symmetry. 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 ) ) Coleman - Mandula Theorem : There is no fusion of internal with space-time symmetries, i.e. the maximal symmetry of the S - matrix is Poincare Internal symmetries unless Haag-Lopuszanski-Sohnius : The symmetry algebra is promoted to a Graded Lie Algebra ( GLA ) with Even E and Odd O elements closing the GLA {, } anticommutator [ E, E ] = E, [ E, O ] = O, { O, O } = E

41 The rudiments of Supersymmetry 18/ 53 The SUSY fundamentals The Algebra of Supersymmetry In Particle Theory SUSY may play a fundamental role! It is a Femion, Boson symmetry enlarging the Poincare symmetry. 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 ) ) Coleman - Mandula Theorem : There is no fusion of internal with space-time symmetries, i.e. the maximal symmetry of the S - matrix is Poincare Internal symmetries unless Haag-Lopuszanski-Sohnius : The symmetry algebra is promoted to a Graded Lie Algebra ( GLA ) with Even E and Odd O elements closing the GLA {, } anticommutator [ E, E ] = E, [ E, O ] = O, { O, O } = E

42 The rudiments of Supersymmetry 19/ 53 The SUSY fundamentals The Algebra of Supersymmetry The minimal extension is the N=1 SUSY including two Weyl-type spinorial generators Q, Q, in addition to P m, M mn { Q α, Q α } = 2 ( σ m ) α α [ ] Pm [ Q, P m ] = Q, P m = 0 [ Q, M mn ] = σ mn Q Highier Supersymmeries have more spinorial charges, N > 1! Q, Q generate SUSY transformations and transform bosonic states to 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!

43 The rudiments of Supersymmetry 19/ 53 The SUSY fundamentals The Algebra of Supersymmetry The minimal extension is the N=1 SUSY including two Weyl-type spinorial generators Q, Q, in addition to P m, M mn { Q α, Q α } = 2 ( σ m ) α α [ ] Pm [ Q, P m ] = Q, P m = 0 [ Q, M mn ] = σ mn Q Highier Supersymmeries have more spinorial charges, N > 1! Q, Q generate SUSY transformations and transform bosonic states to 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!

44 The rudiments of Supersymmetry 19/ 53 The SUSY fundamentals The Algebra of Supersymmetry The minimal extension is the N=1 SUSY including two Weyl-type spinorial generators Q, Q, in addition to P m, M mn { Q α, Q α } = 2 ( σ m ) α α [ ] Pm [ Q, P m ] = Q, P m = 0 [ Q, M mn ] = σ mn Q Highier Supersymmeries have more spinorial charges, N > 1! Q, Q generate SUSY transformations and transform bosonic states to 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 The rudiments of Supersymmetry 19/ 53 The SUSY fundamentals The Algebra of Supersymmetry The minimal extension is the N=1 SUSY including two Weyl-type spinorial generators Q, Q, in addition to P m, M mn { Q α, Q α } = 2 ( σ m ) α α [ ] Pm [ Q, P m ] = Q, P m = 0 [ Q, M mn ] = σ mn Q Highier Supersymmeries have more spinorial charges, N > 1! Q, Q generate SUSY transformations and transform bosonic states to 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 The rudiments of Supersymmetry 20/ 53 The SUSY fundamentals The Algebra of Supersymmetry Spinology and Conventions The fundamental spinor representations of the Poincare symmetry are the Weyl spinors A Left-handed Weyl spinor ψ has two components ψ α ( ) ψ1 ψ = ψ 2 The operation ψ i σ 2 ψ defines a Right-handed antispinor with components ψ α ( ) ψ2 ψ = ψ 1 Invariant mass terms ( indices raised lowered by ɛ αβ, ɛ αβ... ) ψχ ψ α χ α, ψ χ ψ α χ α Invariant kinetic terms ( with σ m (1, σ) and σ m (1, σ ) ) i ψ σ m mψ, i ψ σ m mψ

47 The rudiments of Supersymmetry 20/ 53 The SUSY fundamentals The Algebra of Supersymmetry Spinology and Conventions The fundamental spinor representations of the Poincare symmetry are the Weyl spinors A Left-handed Weyl spinor ψ has two components ψ α ( ) ψ1 ψ = ψ 2 The operation ψ i σ 2 ψ defines a Right-handed antispinor with components ψ α ( ) ψ2 ψ = ψ 1 Invariant mass terms ( indices raised lowered by ɛ αβ, ɛ αβ... ) ψχ ψ α χ α, ψ χ ψ α χ α Invariant kinetic terms ( with σ m (1, σ) and σ m (1, σ ) ) i ψ σ m mψ, i ψ σ m mψ

48 The rudiments of Supersymmetry 20/ 53 The SUSY fundamentals The Algebra of Supersymmetry Spinology and Conventions The fundamental spinor representations of the Poincare symmetry are the Weyl spinors A Left-handed Weyl spinor ψ has two components ψ α ( ) ψ1 ψ = ψ 2 The operation ψ i σ 2 ψ defines a Right-handed antispinor with components ψ α ( ) ψ2 ψ = ψ 1 Invariant mass terms ( indices raised lowered by ɛ αβ, ɛ αβ... ) ψχ ψ α χ α, ψ χ ψ α χ α Invariant kinetic terms ( with σ m (1, σ) and σ m (1, σ ) ) i ψ σ m mψ, i ψ σ m mψ

49 The rudiments of Supersymmetry 21/ 53 The SUSY fundamentals The Algebra of Supersymmetry In the Weyl basis of the gamma matrices ( ) ( ) γ σ =, γ = 1 0 σ 0 L,R projection operators are diagonal P L = 1 + γ ( = ), P R = 1 γ 5 2 = ( ) and the L and R -handed components of a Dirac fermion are Weyl spinors, ψ and φ ( ) ψ ψ D = φ The charge conjugation matrix C = i γ 2 γ 0 and upon charge conjugation ( ) ( ψ = ψ C = C ψ T ψ φ equivalent to ψ = = ψ φ C = ψ A Majorana fermion is self conjugate ψ = ψ C, Fermions = Antifermions! ( ) ψ ψ M = ψ )

50 The rudiments of Supersymmetry 21/ 53 The SUSY fundamentals The Algebra of Supersymmetry In the Weyl basis of the gamma matrices ( ) ( ) γ σ =, γ = 1 0 σ 0 L,R projection operators are diagonal P L = 1 + γ ( = ), P R = 1 γ 5 2 = ( ) and the L and R -handed components of a Dirac fermion are Weyl spinors, ψ and φ ( ) ψ ψ D = φ The charge conjugation matrix C = i γ 2 γ 0 and upon charge conjugation ( ) ( ψ = ψ C = C ψ T ψ φ equivalent to ψ = = ψ φ C = ψ A Majorana fermion is self conjugate ψ = ψ C, Fermions = Antifermions! ( ) ψ ψ M = ψ )

51 The rudiments of Supersymmetry 22/ 53 The SUSY fundamentals Supermultiplets The building blocks : The supersymmetry representations are called supermultiplets 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 includes a boson and a fermion with spin differing by 1/2. They are connected by SUSY transformations SUSY : φ ψ, A µ λ, g µν ψ µ Besides the physical d.o.f. they include auxiliary fields to match the number of bosonic and fermionic components off-shell. The most elegant description of supermultiplets is done through the notion of superspace!

52 The rudiments of Supersymmetry 23/ 53 The SUSY fundamentals Supermultiplets Superspace & Superfields Minkowski space-time : Coordinates are x m G(a) = exp ( i a m P m ) generates translations by a m G(a) G(x) = G(x ) with x m = xm m + am m Fields transform as G(a) φ(x)g(a) = φ(x ) Superspace : Coordinates are z m = ( x m, θ, θ ) with θ, θ Weyl spinors G(a, ξ, ξ) = exp ( i a m P m + i ξ Q + i ξ Q ) generates translations by a m, ξ, ξ G(a, ξ, ξ) G(x, θ, θ) = G(x, θ, θ ) with x m = x m + a m + i (ξ σ m θ θ σ m ξ), θ = θ + ξ, θ = θ + ξ Superfields transform as G(a, ξ, ξ) Φ(x, θ, θ)g(a, ξ, ξ) = Φ(x, θ, θ )

53 The rudiments of Supersymmetry 23/ 53 The SUSY fundamentals Supermultiplets Superspace & Superfields Minkowski space-time : Coordinates are x m G(a) = exp ( i a m P m ) generates translations by a m G(a) G(x) = G(x ) with x m = xm m + am m Fields transform as G(a) φ(x)g(a) = φ(x ) Superspace : Coordinates are z m = ( x m, θ, θ ) with θ, θ Weyl spinors G(a, ξ, ξ) = exp ( i a m P m + i ξ Q + i ξ Q ) generates translations by a m, ξ, ξ G(a, ξ, ξ) G(x, θ, θ) = G(x, θ, θ ) with x m = x m + a m + i (ξ σ m θ θ σ m ξ), θ = θ + ξ, θ = θ + ξ Superfields transform as G(a, ξ, ξ) Φ(x, θ, θ)g(a, ξ, ξ) = Φ(x, θ, θ )

54 The rudiments of Supersymmetry 24/ 53 The SUSY fundamentals Supermultiplets Expanding Φ(x, θ, θ) in θ, θ no more than two powers of θ, θ can appear, the series terminates! Φ(x, θ, θ) = A(x) + θψ(x) + θ Σ(x) + θσ m θ V m (x) + θθ M(x) + θ θ N(x) + θθθ λ(x) + θ θθ ξ(x) + θθθ θ D(x) The fields A, Ψ, Σ, V m, M, N, ξ, λ, D are the components of Φ(x, θ, θ) in increasing mass dimension, D has the highiest! They transform to one another by SUSY transformations. Since D carries the highiest dimension and the parameters of SUSY transformations have mass dimension 1/2 δ SUSY D = m K m The integral of the highiest dimensionality component of any superfield is supersymmetric invariant! d 4 x D = SUSY invariant

55 The rudiments of Supersymmetry 24/ 53 The SUSY fundamentals Supermultiplets Expanding Φ(x, θ, θ) in θ, θ no more than two powers of θ, θ can appear, the series terminates! Φ(x, θ, θ) = A(x) + θψ(x) + θ Σ(x) + θσ m θ V m (x) + θθ M(x) + θ θ N(x) + θθθ λ(x) + θ θθ ξ(x) + θθθ θ D(x) The fields A, Ψ, Σ, V m, M, N, ξ, λ, D are the components of Φ(x, θ, θ) in increasing mass dimension, D has the highiest! They transform to one another by SUSY transformations. Since D carries the highiest dimension and the parameters of SUSY transformations have mass dimension 1/2 δ SUSY D = m K m The integral of the highiest dimensionality component of any superfield is supersymmetric invariant! d 4 x D = SUSY invariant

56 The rudiments of Supersymmetry 24/ 53 The SUSY fundamentals Supermultiplets Expanding Φ(x, θ, θ) in θ, θ no more than two powers of θ, θ can appear, the series terminates! Φ(x, θ, θ) = A(x) + θψ(x) + θ Σ(x) + θσ m θ V m (x) + θθ M(x) + θ θ N(x) + θθθ λ(x) + θ θθ ξ(x) + θθθ θ D(x) The fields A, Ψ, Σ, V m, M, N, ξ, λ, D are the components of Φ(x, θ, θ) in increasing mass dimension, D has the highiest! They transform to one another by SUSY transformations. Since D carries the highiest dimension and the parameters of SUSY transformations have mass dimension 1/2 δ SUSY D = m K m The integral of the highiest dimensionality component of any superfield is supersymmetric invariant! d 4 x D = SUSY invariant

57 The rudiments of Supersymmetry 25/ 53 The SUSY fundamentals Supermultiplets Chiral & Antichiral multiplets The general superfield involves many components. Multiplets with fewer components can be constructed! Covariant derivatives : D α = i θ α + ( σm θ ) α x m, D α = i θ α ( θ σm ) α x m Commute with SUSY transformations : [ δ SUSY, D ] = [ δ SUSY, D ] = 0 Chiral superfield : DΦ = 0, Antichiral superfield : DΦ = 0 These properties are preserved by SUSY transformations!

58 The rudiments of Supersymmetry 26/ 53 The SUSY fundamentals Supermultiplets A chiral superfield contains a complex scalar A a Left - handed Weyl fermion ψ and an auxiliary complex field F. Its particle content is 2 spin-0 and 2 - spin 1/2 states. In terms of the variable y m x m i θ σ m θ Φ = A(y) + 2 θ ψ(y) + θ θ F (y) The θθ component field F, or last component, carries the highiest dimensionality! Under infinitesimal SUSY transformations by ξ, ξ : δ A = 2 ξ ψ δ ψ = 2 ξ F i 2 σ m ξ m A δ F = i 2 ξ σ m m ψ The last component transform as a total derivative = d 4 x F = SUSY invariant

59 The rudiments of Supersymmetry 27/ 53 The SUSY fundamentals Supermultiplets An antichiral superfield includes a complex scalar, a Right - handed Weyl fermion and an auxiliary complex field. If Φ is a chiral superfield with components ( A, ψ, F ) its Hermitian superfield Φ is antichiral with components ( A, ψ, F ) Products of chiral superfields : If Φ ( A, ψ, F ) and Φ ( A, ψ, F ) are chiral superfields their product Φ = Φ Φ is also a chiral field with components A = A A ψ = A ψ + A ψ F = A F + A F ψ ψ The product Φ Φ is a real superfield which includes kinetic terms in its last θ 2 θ 2 component, producing SUSY invariant kinetic terms upon d 4 x! Φ Φ 2 2 θ θ = 1 4 A A 1 4 A A m A 2 + i 2 ( ψσm mψ h.c. ) + FF

60 The rudiments of Supersymmetry 28/ 53 The SUSY fundamentals Supermultiplets Vector multiplets A Hermitian superfield V defines a vector multiplet Vector superfield : V = V including, among other components, a vector field A µ a Weyl fermion λ and an auxiliary field D as its last θ 2 θ 2 component. For any chiral field Φ ( A, ψ, F ) the transformation V = V + Φ + Φ defines a gauge transformation with gauge parameter Λ = 2 Im A Under the gauge transformation A µ = A µ + µ Λ, λ, D = themselves The remaining components are not gauged invariant and can be gauged way! This defines the Wess - Zumino gauge

61 The rudiments of Supersymmetry 29/ 53 The SUSY fundamentals Supermultiplets In the Wess - Zumino gauge V = ( θ σ µ σµ θ ) A µ + i θθθ λ i θ θ θ λ θθ θ θ D and V n = 0 for n 3 A µ The vector field A µ and its partner, gaugino, λ are the physical d.o.f. describing 2 spin-1 and 2 spin-1/2 states. Under infinitesimal SUSY transformations λ, D and the field strength F µν = µ A ν ν A µ transform to each other : δ F µν = i ( ξσ ν µ λ + ξ σ ν µ λ ) (ν µ) δ λ = i ξ D + i σ µν F µν δ D = ξσ µ µ λ ξ σ µ µ λ

62 The rudiments of Supersymmetry 29/ 53 The SUSY fundamentals Supermultiplets In the Wess - Zumino gauge V = ( θ σ µ σµ θ ) A µ + i θθθ λ i θ θ θ λ θθ θ θ D and V n = 0 for n 3 A µ The vector field A µ and its partner, gaugino, λ are the physical d.o.f. describing 2 spin-1 and 2 spin-1/2 states. Under infinitesimal SUSY transformations λ, D and the field strength F µν = µ A ν ν A µ transform to each other : δ F µν = i ( ξσ ν µ λ + ξ σ ν µ λ ) (ν µ) δ λ = i ξ D + i σ µν F µν δ D = ξσ µ µ λ ξ σ µ µ λ

63 The rudiments of Supersymmetry 30/ 53 Supersymmetric Field Theories SUSY Lagrangians Ungauged Lagrangians The recipe to construct ungauged SUSY Lagrangians involving N chiral multiplets is easy Kinetic terms are included in the θ 2 θ 2, or D - terms, of Φ i Φ i ( Φ i Φ i ) θ 2 θ 2 d 2 θ d 2 θ Φ i Φ i Non - gauge interaction terms are included in the θ 2, F -terms, of a chiral field called superpotential, W ( Φ i ), function of Φ i s W (Φ i ) = λ ijk 3! Φ i Φ j Φ k + m ij 2! Φ i Φ j for renormalizable theories up to cubic terms are kept, W ( Φ i ) θ 2 + h.c. d 2 θ W ( Φ i ) + h.c. θ 2

64 The rudiments of Supersymmetry 31/ 53 Supersymmetric Field Theories SUSY Lagrangians Auxiliary F - fields arise from the kinetic and superpotential terms ( ) F i F i + F W (A i ) i + h.c. A i and are eliminated by their eqs. of motion F i = W (A i) A i The SUSY Lagrangian is : i L = µ A i 2 + i ψi σ µ µ ψ i W A i ( 2 ) W ψ i ψ j + h.c. A i A j A positive definite scalar potential arises : V = F i F i = W A i 2 0

65 The rudiments of Supersymmetry 31/ 53 Supersymmetric Field Theories SUSY Lagrangians Auxiliary F - fields arise from the kinetic and superpotential terms ( ) F i F i + F W (A i ) i + h.c. A i and are eliminated by their eqs. of motion F i = W (A i) A i The SUSY Lagrangian is : i L = µ A i 2 + i ψi σ µ µ ψ i W A i ( 2 ) W ψ i ψ j + h.c. A i A j A positive definite scalar potential arises : V = F i F i = W A i 2 0

66 The rudiments of Supersymmetry 31/ 53 Supersymmetric Field Theories SUSY Lagrangians Auxiliary F - fields arise from the kinetic and superpotential terms ( ) F i F i + F W (A i ) i + h.c. A i and are eliminated by their eqs. of motion F i = W (A i) A i The SUSY Lagrangian is : i L = µ A i 2 + i ψi σ µ µ ψ i W A i ( 2 ) W ψ i ψ j + h.c. A i A j A positive definite scalar potential arises : V = F i F i = W A i 2 0

67 The rudiments of Supersymmetry 32/ 53 Supersymmetric Field Theories SUSY Lagrangians Bosons and Fermions come in pairs with same mass as a manifestation of supersymmetry m B = m F as long as the vacuum energy vanishes, V 0 = 0. If V 0 0, due to some F i 0, the mass degeneracy is lifted! Exact SUSY : V 0 = 0 and SB SUSY : V 0 0 and m B = m F m B m F Only broken Supersymmetry, spontaneously or other, can be realized in nature to lift mass degeneracies!

68 The rudiments of Supersymmetry 32/ 53 Supersymmetric Field Theories SUSY Lagrangians Bosons and Fermions come in pairs with same mass as a manifestation of supersymmetry m B = m F as long as the vacuum energy vanishes, V 0 = 0. If V 0 0, due to some F i 0, the mass degeneracy is lifted! Exact SUSY : V 0 = 0 and SB SUSY : V 0 0 and m B = m F m B m F Only broken Supersymmetry, spontaneously or other, can be realized in nature to lift mass degeneracies!

69 The rudiments of Supersymmetry 33/ 53 Supersymmetric Field Theories SUSY Lagrangians Gauged SUSY Lagrangians Generalize gauge transformations for chiral multiplets Φ e i g Λ Φ Λ Λ a T a, Λ a = chiral superfields, T a = group generators. Φ Φ is not gauge invariant, Introduce gauge multiplet V V a T a with V a = ( A a µ, λ a, D a ) to make Φ e 2 g V Φ be gauge invariant. The gauge multiplet should transform as In the Abelian case this reads e 2 g V e i g Λ e 2 g V e i g Λ V V + i 2 ( Λ Λ )

70 The rudiments of Supersymmetry 34/ 53 Supersymmetric Field Theories SUSY Lagrangians The gauge interactions of the chiral fields can be easily read in the Wess-zumino gauge since V n = 0, n 3. Φ e 2 g V Φ = Φ Φ + 2 g Φ V Φ + g 2 Φ V 2 Φ The SUSY Yang-Mills Lagrangian L gauge = 1 4 G (a) 2 i µν + 2 λ(a) σ µ D µ λ (a) D(a)2 with D µ λ (a) = µ λ (a) + g f abc A b µ λ (c) covariant derivative and G µν (a) the non-abelian gauge field strength is both gauge and SUSY invariant! Φ i G (a) A supersymmetric gauge Lagrangian with chiral multiplets Φ i put in some representation Φ, reducible in general, is L = L gauge + d 2 θ d 2 θ Φ e 2 g V Φ + ( d 2 θ W ( Φ ) + h.c. )

71 The rudiments of Supersymmetry 34/ 53 Supersymmetric Field Theories SUSY Lagrangians The gauge interactions of the chiral fields can be easily read in the Wess-zumino gauge since V n = 0, n 3. Φ e 2 g V Φ = Φ Φ + 2 g Φ V Φ + g 2 Φ V 2 Φ The SUSY Yang-Mills Lagrangian L gauge = 1 4 G (a) 2 i µν + 2 λ(a) σ µ D µ λ (a) D(a)2 with D µ λ (a) = µ λ (a) + g f abc A b µ λ (c) covariant derivative and G µν (a) the non-abelian gauge field strength is both gauge and SUSY invariant! Φ i G (a) A supersymmetric gauge Lagrangian with chiral multiplets Φ i put in some representation Φ, reducible in general, is L = L gauge + d 2 θ d 2 θ Φ e 2 g V Φ + ( d 2 θ W ( Φ ) + h.c. )

72 The rudiments of Supersymmetry 34/ 53 Supersymmetric Field Theories SUSY Lagrangians The gauge interactions of the chiral fields can be easily read in the Wess-zumino gauge since V n = 0, n 3. Φ e 2 g V Φ = Φ Φ + 2 g Φ V Φ + g 2 Φ V 2 Φ The SUSY Yang-Mills Lagrangian L gauge = 1 4 G (a) 2 i µν + 2 λ(a) σ µ D µ λ (a) D(a)2 with D µ λ (a) = µ λ (a) + g f abc A b µ λ (c) covariant derivative and G µν (a) the non-abelian gauge field strength is both gauge and SUSY invariant! Φ i G (a) A supersymmetric gauge Lagrangian with chiral multiplets Φ i put in some representation Φ, reducible in general, is L = L gauge + d 2 θ d 2 θ Φ e 2 g V Φ + ( d 2 θ W ( Φ ) + h.c. )

73 The rudiments of Supersymmetry 35/ 53 Supersymmetric Field Theories SUSY Lagrangians Auxiliary D - fields arise from the YM kinetic and gauge interaction terms and are eliminated by their eqs. of motion. 1 2 D(a)2 + D (a) i g A T (a) A D and F type auxiliary fields are D (a) = i g A T (a) A, F i W (A) = A i The complete SUSY Lagrangian L = 1 4 G (a) 2 + Dµ A 2 + ( i 2 λ(a) σ µ D µ λ (a) + i 2 Ψ σ µ D µ Ψ + h.c. ) µν ( 1 2 W ij ψ i ψ j + i 2 g A T (a) Ψ λ (a) + h.c. ) F i D (a) 2

74 The rudiments of Supersymmetry 35/ 53 Supersymmetric Field Theories SUSY Lagrangians Auxiliary D - fields arise from the YM kinetic and gauge interaction terms and are eliminated by their eqs. of motion. 1 2 D(a)2 + D (a) i g A T (a) A D and F type auxiliary fields are D (a) = i g A T (a) A, F i W (A) = A i The complete SUSY Lagrangian L = 1 4 G (a) 2 + Dµ A 2 + ( i 2 λ(a) σ µ D µ λ (a) + i 2 Ψ σ µ D µ Ψ + h.c. ) µν ( 1 2 W ij ψ i ψ j + i 2 g A T (a) Ψ λ (a) + h.c. ) F i D (a) 2

75 The rudiments of Supersymmetry 36/ 53 Supersymmetric Field Theories SUSY Lagrangians 2 G µν gauge kinetic + λ (a) σ µ D µ λ (a) gaugino kinetic + D µ A 2 scalar kinetic + Ψ σ µ D µ Ψ fermion kinetic +

76 The rudiments of Supersymmetry 37/ 53 Supersymmetric Field Theories SUSY Lagrangians Unconventional gauge interactions : g A T (a) Ψ λ (a) + h.c. D (a) 2

77 The rudiments of Supersymmetry 38/ 53 Supersymmetric Field Theories SUSY Lagrangians Non - gauge ( superpotential ) interactions : With a superpotential 2 F i 2 = W A i Scalar mass terms : W (Φ) = M ik 2 Φ i Φ j + y ijk 3! Φ i Φ j Φ k A i M ki M kj A j Scalar interactions : W ij ψ i ψ j + h.c. Fermion mass terms : M ij ψ i ψ j + h.c. Yukawa interactions :

78 The rudiments of Supersymmetry 39/ 53 Supersymmetric Field Theories SUSY Lagrangians Vertices are related by SUSY transformations!

79 The rudiments of Supersymmetry 39/ 53 Supersymmetric Field Theories SUSY Lagrangians Vertices are related by SUSY transformations! Essential for cancellations among graphs! δm H 2 = 0 if SUSY is exact! δm 2 H δm H 2 m2 soft ln ( Λ/m soft ) if SUSY is broken spontaneously or softly

80 The rudiments of Supersymmetry 40/ 53 Supersymmetric Field Theories SUSY Lagrangians The Wess - Zumino model Describes one chiral multiplet Φ = ( A, ψ, F ) with superpotential, W ( Φ ) = m 2 Φ + λ 3! Φ3 Defining the Majorana fermion Ψ, two d.o.f.! ( ) ψ Ψ = ψ the SUSY Lagrangian is : L SUSY = D µ A 2 + i 2 Ψ γ µ µ Ψ Kinetic terms 1 2 [ ( m + λ A )Ψ LΨ R + h.c.] m A + λ 2 2 A 2 Fermion mass and Yukawa terms Potential terms

81 The rudiments of Supersymmetry 41/ 53 Supersymmetric Field Theories SUSY Lagrangians The potential is positive definite V = m A + λ 2 2 A 2 0 There are two vacua, at A = 0 or A = 2m/λ, symmetric about the point A = m/λ. The vacuum energy vanishes, V min = 0 and SUSY is unbroken! The model describes : A complex scalar boson A of mass m, two d.o.f. A Majorana fermion Ψ of mass m, two d.o.f!

82 The rudiments of Supersymmetry 42/ 53 Supersymmetric Field Theories SUSY Lagrangians Renormalization effects : Neither mass nor Yukawa coupling get renormalized. Only wave function renormalizations with Z B = Z F No - Renormalization Theorem : Superpotential parameters are not renormalized. The only infinities are those associated with the wave function renormalization of the chiral and vector multiplets and renormalization of the gauge couplings!

83 The rudiments of Supersymmetry 42/ 53 Supersymmetric Field Theories SUSY Lagrangians Renormalization effects : Neither mass nor Yukawa coupling get renormalized. Only wave function renormalizations with Z B = Z F No - Renormalization Theorem : Superpotential parameters are not renormalized. The only infinities are those associated with the wave function renormalization of the chiral and vector multiplets and renormalization of the gauge couplings!

84 The rudiments of Supersymmetry 43/ 53 Supersymmetric Field Theories Breaking global SUSY Spontaneous Breaking of Global SUSY SSB of SUSY occurs if Q α vacuum > 0 or Q α vacuum > 0 = E vacuum > 0 Lift of the vacuum energy signals SSB of global Supersymmetry! Tree level vacuum energy : E vacuum V min = F i 2 + Da 2 and spontaneous breaking occurs for F i = 0, F - type breaking, ( or O Raifertaight breaking ) Need special form of superpotential. D a = 0, D - type breaking, ( or Fayet - Iliopoulos breaking ) Need a U(1) gauge symmetry. 2

85 The rudiments of Supersymmetry 43/ 53 Supersymmetric Field Theories Breaking global SUSY Spontaneous Breaking of Global SUSY SSB of SUSY occurs if Q α vacuum > 0 or Q α vacuum > 0 = E vacuum > 0 Lift of the vacuum energy signals SSB of global Supersymmetry! Tree level vacuum energy : E vacuum V min = F i 2 + Da 2 and spontaneous breaking occurs for F i = 0, F - type breaking, ( or O Raifertaight breaking ) Need special form of superpotential. D a = 0, D - type breaking, ( or Fayet - Iliopoulos breaking ) Need a U(1) gauge symmetry. 2

86 The rudiments of Supersymmetry 43/ 53 Supersymmetric Field Theories Breaking global SUSY Spontaneous Breaking of Global SUSY SSB of SUSY occurs if Q α vacuum > 0 or Q α vacuum > 0 = E vacuum > 0 Lift of the vacuum energy signals SSB of global Supersymmetry! Tree level vacuum energy : E vacuum V min = F i 2 + Da 2 and spontaneous breaking occurs for F i = 0, F - type breaking, ( or O Raifertaight breaking ) Need special form of superpotential. D a = 0, D - type breaking, ( or Fayet - Iliopoulos breaking ) Need a U(1) gauge symmetry. 2

87 The rudiments of Supersymmetry 44/ 53 Supersymmetric Field Theories Breaking global SUSY D - type breaking Need a U(1) gauge symmetry and a U(1) vector multiplet V Add to SUSY Lagrangian a term This is U(1) and SUSY invariant! δ L = 2 ξ V = ξ D The presence of the ξ - term alters the eqs. of motion for the auxiliary field D resulting to D = 0. ξ sets the order parameter of global SUSY breaking and lifts the vacuum energy by E vacuum ξ 2. Example: Two chiral multiplets Φ 1, Φ 2 with U(1) charges +1, -1 and a superpotential W (Φ) = m Φ 1 Φ 2

88 The rudiments of Supersymmetry 44/ 53 Supersymmetric Field Theories Breaking global SUSY D - type breaking Need a U(1) gauge symmetry and a U(1) vector multiplet V Add to SUSY Lagrangian a term This is U(1) and SUSY invariant! δ L = 2 ξ V = ξ D The presence of the ξ - term alters the eqs. of motion for the auxiliary field D resulting to D = 0. ξ sets the order parameter of global SUSY breaking and lifts the vacuum energy by E vacuum ξ 2. Example: Two chiral multiplets Φ 1, Φ 2 with U(1) charges +1, -1 and a superpotential W (Φ) = m Φ 1 Φ 2

89 The rudiments of Supersymmetry 44/ 53 Supersymmetric Field Theories Breaking global SUSY D - type breaking Need a U(1) gauge symmetry and a U(1) vector multiplet V Add to SUSY Lagrangian a term This is U(1) and SUSY invariant! δ L = 2 ξ V = ξ D The presence of the ξ - term alters the eqs. of motion for the auxiliary field D resulting to D = 0. ξ sets the order parameter of global SUSY breaking and lifts the vacuum energy by E vacuum ξ 2. Example: Two chiral multiplets Φ 1, Φ 2 with U(1) charges +1, -1 and a superpotential W (Φ) = m Φ 1 Φ 2

90 The rudiments of Supersymmetry 45/ 53 Supersymmetric Field Theories Breaking global SUSY Auxiliary fields : D = g ( A 1 2 A 2 2 ) ξ F 1 = m A 2, F 2 = m A 2 Potential : Case g ξ < m 2 : V = F F D 2 2 = ( m 2 + g ξ ) A ( m 2 g ξ ) A g 2 2 ( A 1 2 A 2 2 ) 2 + ξ2 2 0 Minimum of potential at A 1,2 = 0 = U(1) unbroken D = ξ, F 1,2 = 0 = SUSY is broken by the D- term and V min = ξ 2 /2 > 0

91 The rudiments of Supersymmetry 45/ 53 Supersymmetric Field Theories Breaking global SUSY Auxiliary fields : D = g ( A 1 2 A 2 2 ) ξ F 1 = m A 2, F 2 = m A 2 Potential : Case g ξ < m 2 : V = F F D 2 2 = ( m 2 + g ξ ) A ( m 2 g ξ ) A g 2 2 ( A 1 2 A 2 2 ) 2 + ξ2 2 0 Minimum of potential at A 1,2 = 0 = U(1) unbroken D = ξ, F 1,2 = 0 = SUSY is broken by the D- term and V min = ξ 2 /2 > 0

92 The rudiments of Supersymmetry 46/ 53 Supersymmetric Field Theories Breaking global SUSY Spin - 0 Masses 2 # of states States m 2 + g ξ m 2 g ξ Spin - 1/2 m F 2 B A 1 2 B A 2 4 F Dirac : Ψ = ( ψ1 ψ 2 2 F Goldstino : λ ) Spin B Gauge boson : A µ

93 The rudiments of Supersymmetry 46/ 53 Supersymmetric Field Theories Breaking global SUSY Spin - 0 Masses 2 # of states States m 2 + g ξ m 2 g ξ Spin - 1/2 m F 2 B A 1 2 B A 2 4 F Dirac : Ψ = ( ψ1 ψ 2 2 F Goldstino : λ ) Spin B Gauge boson : A µ

94 The rudiments of Supersymmetry 47/ 53 Supersymmetric Field Theories Breaking global SUSY Case g ξ > m 2 : Take ξ > 0, if negative the role of A 1, A 2 is interchanged Minimum of potential at A 1 = 0, A 2 = ( gξ m2 ) 1/2 = the gauge symmetry U(1) is broken! D = 0, F 1 = 0, F 2 = 0 = SUSY is broken, by the D- term, and also F 1 The vacuum energy is V min V min = m2 g 2 g ( 2 gξ m2 ) > 0 Since both D, F 1 non-zero the Goldstino χ is mixture of and the gaugino λ ψ 1

95 The rudiments of Supersymmetry 48/ 53 Supersymmetric Field Theories Breaking global SUSY Masses 2 # of states States Spin m 2 2 B A 1 2 g 2 u 2 1 B Re ( A 2 u ) 0 1 B Goldstone : Im ( A 2 ) Spin - 1/2 m g 2 u F ( ) ψ2 χ T 2 F Goldstino : χ 4 F Dirac : Ψ = Spin g 2 u 2 3 B Gauge boson : A µ

96 The rudiments of Supersymmetry 48/ 53 Supersymmetric Field Theories Breaking global SUSY Masses 2 # of states States Spin m 2 2 B A 1 2 g 2 u 2 1 B Re ( A 2 u ) 0 1 B Goldstone : Im ( A 2 ) Spin - 1/2 m g 2 u F ( ) ψ2 χ T 2 F Goldstino : χ 4 F Dirac : Ψ = Spin g 2 u 2 3 B Gauge boson : A µ

97 The rudiments of Supersymmetry 49/ 53 Supersymmetric Field Theories Breaking global SUSY F -type breaking The form of the superpotential forces at least one F- term F i for. Example : Three chiral multiplets Φ A, Φ B, Φ X coupled with a superpotential W = g Φ X ( Φ 2 A µ 2 ) + m Φ A Φ B The equations for the auxiliary fields : F X = g ( A2 µ 2 ), F A = m A, F B = ( 2 g A X + m B )

98 The rudiments of Supersymmetry 49/ 53 Supersymmetric Field Theories Breaking global SUSY F -type breaking The form of the superpotential forces at least one F- term F i for. Example : Three chiral multiplets Φ A, Φ B, Φ X coupled with a superpotential W = g Φ X ( Φ 2 A µ 2 ) + m Φ A Φ B The equations for the auxiliary fields : F X = g ( A2 µ 2 ), F A = m A, F B = ( 2 g A X + m B ) The potential has a flat direction along X : V = g 2 A 2 µ g A X + m B 2 + m 2 A 2

99 The rudiments of Supersymmetry 50/ 53 Supersymmetric Field Theories Breaking global SUSY In the range m 2 ( 4 µ 2 g 2 m 2 ) > 4 µ 4 g 4 potential is at the minimum of the and the F - terms A = B = 0, X = undetermined F X = g µ 2, F A,B = 0 Supersymmetry is broken by F X = 0. The vacuum energy is V min = g 2 µ 4 and the order parameter of SUSY breaking is f g µ 2. The Goldstino is the partner of the scalar X.

100 The rudiments of Supersymmetry 51/ 53 Supersymmetric Field Theories Breaking global SUSY Comments : In only F - type breaking of global SUSY some sfermions become lighter than their corresponding fermions, due to the relation Str M 2 J ( 1) 2 J (2J + 1) m J 2 = 0 which is preserved after SUSY breaking. In D - type breaking this problem is evaded since the r.h.s. receives D contributions! For D - type breaking an extra U(1) is required, a new neutral vector boson appears and new neutral current interactions are present! The presence of a U(1) vector multiplet may have disastrous consequences since quadratic loop corrections may emerge Λ 2 d 2 θ d 2 θ V = Λ 2 D This may be circumvented if U(1) is subgroup of a simple group. then the corrections are proportional to Trace Y = 0!.

101 The rudiments of Supersymmetry 52/ 53 Supersymmetric Field Theories Breaking global SUSY Further The Goldstino mode is not observed in nature! In local versions of Supersymmetry this is absorbed by a massless gravitino to make it massive. The Goldstino disappears and the gravitino provides for a Dark Matter candidate ( in addition...) Supergravity and String Theories provide additional mechanisms for SUSY breaking!