Supersymmetry and All That

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Supersymmetry and All That

  1. 1. Supersymmetry and All That : A simple example in 1–dimension ¨ Kayhan ULKER Feza G¨rsey Institute* u ˙ Istanbul, Turkey September 2, 2011 ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 1 / 32
  2. 2. Outline : Very brief history N=1 SUSY in 1 dimension Component formalism Superfield formalism N=2 SUSY in 1 dimension How to extend An alternative way of obtaining the action ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 2 / 32
  3. 3. SUPERSYMMETRY Brief history: Supersymmetry 1967 Coleman-Mandula no-go theorem: It is not possible to extend the Poincare group (Pµ , Jµν ) in a non-trivial way (i.e. the only way :[Pµ , Ω] = 0 = [Jµν , Ω]) 1971 Golfand-Likhtman - Birth of SUSY If a Lie group has a graded structure it is possible to extend the Poincare group. ⇒ Superalgebra (Z2 graded structure ) SUSY : Fermion → Boson , Boson → Fermion 1970’s Superstring theories 1974 Wess-Zumino Model First renormalizable theory in 4-dim.s ⇒ SUSY becomes popular ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 3 / 32
  4. 4. SUPERSYMMETRY Brief history: 1975 Haag-Lopusanski-Sohnius Poincar´ + SUSY is the only possible extension in 4-dim.s that can e appear in nontrivial QFTs ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 4 / 32
  5. 5. SUPERSYMMETRY Brief history: 1975 Haag-Lopusanski-Sohnius Poincar´ + SUSY is the only possible extension in 4-dim.s that can e appear in nontrivial QFTs ••• ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 4 / 32
  6. 6. SUPERSYMMETRY Brief history: 1975 Haag-Lopusanski-Sohnius Poincar´ + SUSY is the only possible extension in 4-dim.s that can e appear in nontrivial QFTs ••• 2006 find k supersymm @SPIRES ⇒41564 papers ! ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 4 / 32
  7. 7. SUPERSYMMETRY Brief history: 1975 Haag-Lopusanski-Sohnius Poincar´ + SUSY is the only possible extension in 4-dim.s that can e appear in nontrivial QFTs ••• 2006 find k supersymm @SPIRES ⇒41564 papers ! 2011 find k supersymm @SPIRES ⇒52384 papers ! ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 4 / 32
  8. 8. SUPERSYMMETRY Brief history: 1975 Haag-Lopusanski-Sohnius Poincar´ + SUSY is the only possible extension in 4-dim.s that can e appear in nontrivial QFTs ••• 2006 find k supersymm @SPIRES ⇒41564 papers ! 2011 find k supersymm @SPIRES ⇒52384 papers ! 201? LHC ⇒??? ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 4 / 32
  9. 9. SUPERSYMMETRY Basic properties Supersymmetric theories are highly restricted : Bosons and fermions can only be related to each other by fermionic symmetry operators Q of spin–1/2 (not spin–3/2 or higher). Q|fermion >= |boson > , Q|boson >= |fermion > Only in the presence of SUSY, multiplets can contain particles of different spin. Particles in the same supermultiplet have the same mass and coupling constant. no. of bosons = no. of fermions in a supersymmetric theory. One can write SUSY transformations and supersymmetric actions in two different ways : ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 5 / 32
  10. 10. SUPERSYMMETRY Basic properties I Component field formulation : Decide what fields you want to study. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 6 / 32
  11. 11. SUPERSYMMETRY Basic properties I Component field formulation : Decide what fields you want to study. Write most general transformation that maps bosons to fermions and fermions to bosons by studying dimension (and symmetries) of the fields. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 6 / 32
  12. 12. SUPERSYMMETRY Basic properties I Component field formulation : Decide what fields you want to study. Write most general transformation that maps bosons to fermions and fermions to bosons by studying dimension (and symmetries) of the fields. Fix the coefficients in the transformation so that SUSY algebra is satisfied. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 6 / 32
  13. 13. SUPERSYMMETRY Basic properties I Component field formulation : Decide what fields you want to study. Write most general transformation that maps bosons to fermions and fermions to bosons by studying dimension (and symmetries) of the fields. Fix the coefficients in the transformation so that SUSY algebra is satisfied. Write the most general action that we know from field theory including kinetic, mass, interaction terms. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 6 / 32
  14. 14. SUPERSYMMETRY Basic properties I Component field formulation : Decide what fields you want to study. Write most general transformation that maps bosons to fermions and fermions to bosons by studying dimension (and symmetries) of the fields. Fix the coefficients in the transformation so that SUSY algebra is satisfied. Write the most general action that we know from field theory including kinetic, mass, interaction terms. Fix the coefficients in the action so that it is invariant under SUSY. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 6 / 32
  15. 15. SUPERSYMMETRY Basic properties I Component field formulation : Decide what fields you want to study. Write most general transformation that maps bosons to fermions and fermions to bosons by studying dimension (and symmetries) of the fields. Fix the coefficients in the transformation so that SUSY algebra is satisfied. Write the most general action that we know from field theory including kinetic, mass, interaction terms. Fix the coefficients in the action so that it is invariant under SUSY. This is a tedious but a straightforward way to construct. (See for instance book by West for details.) ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 6 / 32
  16. 16. SUPERSYMMETRY Basic properties II Superspace formulation : In 1920’s we realized that in nature we have bosonic (commuting) fields and fermionic (anticommuting) fields. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 7 / 32
  17. 17. SUPERSYMMETRY Basic properties II Superspace formulation : In 1920’s we realized that in nature we have bosonic (commuting) fields and fermionic (anticommuting) fields. So why not to have anticommuting coordinates in addition to our commuting coordinates (x, y , z, t) ! ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 7 / 32
  18. 18. SUPERSYMMETRY Basic properties II Superspace formulation : In 1920’s we realized that in nature we have bosonic (commuting) fields and fermionic (anticommuting) fields. So why not to have anticommuting coordinates in addition to our commuting coordinates (x, y , z, t) ! Indeed, from QFT we know that 4–dimensional space–time x µ is parametrized by Poincar´/Lorentz coset space. e ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 7 / 32
  19. 19. SUPERSYMMETRY Basic properties II Superspace formulation : In 1920’s we realized that in nature we have bosonic (commuting) fields and fermionic (anticommuting) fields. So why not to have anticommuting coordinates in addition to our commuting coordinates (x, y , z, t) ! Indeed, from QFT we know that 4–dimensional space–time x µ is parametrized by Poincar´/Lorentz coset space. e Let our super–space–time is parametrized by superPoincar´/Lorentz e coset space. ¯˙ (xµ , θα , θα ) ⇒ 4+4 dimensional SUPERSPACE ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 7 / 32
  20. 20. SUPERSYMMETRY Basic properties II Superspace formulation : In 1920’s we realized that in nature we have bosonic (commuting) fields and fermionic (anticommuting) fields. So why not to have anticommuting coordinates in addition to our commuting coordinates (x, y , z, t) ! Indeed, from QFT we know that 4–dimensional space–time x µ is parametrized by Poincar´/Lorentz coset space. e Let our super–space–time is parametrized by superPoincar´/Lorentz e coset space. ¯˙ (xµ , θα , θα ) ⇒ 4+4 dimensional SUPERSPACE Supersymmetric actions can then be written directly in terms of ¯ SUPERFIELDS Φ(x, θ, θ) and their super derivatives. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 7 / 32
  21. 21. SUPERSYMMETRY Basic properties II Superspace formulation : In 1920’s we realized that in nature we have bosonic (commuting) fields and fermionic (anticommuting) fields. So why not to have anticommuting coordinates in addition to our commuting coordinates (x, y , z, t) ! Indeed, from QFT we know that 4–dimensional space–time x µ is parametrized by Poincar´/Lorentz coset space. e Let our super–space–time is parametrized by superPoincar´/Lorentz e coset space. ¯˙ (xµ , θα , θα ) ⇒ 4+4 dimensional SUPERSPACE Supersymmetric actions can then be written directly in terms of ¯ SUPERFIELDS Φ(x, θ, θ) and their super derivatives. This is an elegant way to construct but may not work for every case. (See any book on SUSY for details.) ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 7 / 32
  22. 22. SUPERSYMMETRY SUSY in 1-dimension A Simple Example in 1–Dimension : Let us consider, A real scalar field φ(t), A real fermionic field ψ(t). For each t, ψ(t), is an independent Grassmann variable : ψ(t1 )ψ(t2 ) = −ψ(t2 )ψ(t1 ) ⇒ (ψ(t))2 = 0 ˙ Assume that dψ(t)/dt ≡ ψ(t) exists then we also have ˙ ˙ ˙ ˙ ˙ ˙ ψ(t1 )ψ(t2 ) = −ψ(t2 )ψ(t1 ) , ψ(t)ψ(t) = −ψ(t)ψ(t) ˙ Therefore ψ and ψ anticommute with themselves and with each other at equal time t : ˙ ˙ ˙ {ψ(t), ψ(t)} = 0 , {ψ(t), ψ(t)} = 0 , {ψ(t), ψ(t)} = 0 ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 8 / 32
  23. 23. SUPERSYMMETRY SUSY in 1-dimension Note the difference between φ and ψ ˙ 1 d dtφ(t)φ(t) = dt (φ(t)φ(t)) 2 dt but ˙ 1 d dtψ(t)ψ(t) = dt (ψ(t)ψ(t)) 2 dt ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 9 / 32
  24. 24. SUPERSYMMETRY SUSY in 1-dimension As an action for these fields we take, 1 ˙2 i ˙ I = dt φ + ψψ . 2 2 Note that in this action, 1 ˙2 2φ term is a truncation of the Klein-Gordon action to an (x, y , z) independent field, i ˙ 2 ψ ψ term is a truncation of the Dirac action for a real spinor to one of its component that is also independent of (x, y , z). For = 1 the dimension of the action is zero ([I ] = 0), therefore taking [t] = −1 we get the dimensions of the fields as 1 [φ] = − , [ψ] = 0 2 ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 10 / 32
  25. 25. SUPERSYMMETRY SUSY Transformations SUSY Transformations We look for a symmetry such that δξ (boson) = ξ(somethingfermionic) δξ (fermion) = ξ(somethingbosonic) It is clear that ξ must be anticommuting ! Let δξ φ = iξψ. Then [ξ] = −1/2. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 11 / 32
  26. 26. SUPERSYMMETRY SUSY Transformations SUSY Transformations We look for a symmetry such that δξ (boson) = ξ(somethingfermionic) δξ (fermion) = ξ(somethingbosonic) It is clear that ξ must be anticommuting ! Let δξ φ = iξψ. Then [ξ] = −1/2. Let δξ ψ = iξφ. But this is not possible due to dimensional analysis ! ˙ Therefore, let us consider δξ ψ = ξf (φ, φ) such that [f ] = 1/2. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 11 / 32
  27. 27. SUPERSYMMETRY SUSY Transformations SUSY Transformations We look for a symmetry such that δξ (boson) = ξ(somethingfermionic) δξ (fermion) = ξ(somethingbosonic) It is clear that ξ must be anticommuting ! Let δξ φ = iξψ. Then [ξ] = −1/2. Let δξ ψ = iξφ. But this is not possible due to dimensional analysis ! ˙ Therefore, let us consider δξ ψ = ξf (φ, φ) such that [f ] = 1/2. ˙ If the transformation is linear the only possible choice is f ∼ ξ φ. Indeed, in order to get δξ I = 0 we find ˙ δξ ψ = −ξ φ. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 11 / 32
  28. 28. SUPERSYMMETRY SUSY Transformations Summary : ˙ SUSY transformations : δξ φ = iξψ , δξ ψ = −ξ φ . 1 ˙2 i ˙ I = dt 2φ + 2 ψ ψ is invariant (i.e. superysmmetric). (φ , ψ) real scalar supermultiplet. # of bosons = # of fermions ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 12 / 32
  29. 29. SUPERSYMMETRY SUSY Algebra SUSY Algebra : To get the algebra let us study the commutator of two SUSY transformations: ˙ [δη , δξ ]φ = (δη δξ − δξ δη )φ = 2iηξ φ ˙ [δη , δξ ]ψ = 2iηξ ψ We get, d [δη , δξ ] = 2iηξ dt for constant parameters ξ, η. Note that right hand side is a translation over a distance t0 = 2iηξ ! ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 13 / 32
  30. 30. SUPERSYMMETRY SUSY Algebra Let us obtain the algebra in a tricky way. We have d i ≡ H , δξ ≡ iξQ , δη ≡ iηQ dt and H ve Q denotes the generators of translation and SUSY. The commutator of SUSY transformations can be written in terms of anticommutators as [δη , δξ ] = −(ξQηQ − ηQξQ) = ξη(QQ + QQ) = ξη{Q, Q} With the help of above definitions we get {Q, Q} = 2H Jacobi identity gives the other relation : [Q, H] = 0 ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 14 / 32
  31. 31. SUPERSYMMETRY SUSY Algebra Therefore, SUSY algebra in one dimension is {Q, Q} = 2H , [H, Q] = 0 , [H, H] = 0. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 15 / 32
  32. 32. SUPERSYMMETRY SUSY Algebra Therefore, SUSY algebra in one dimension is {Q, Q} = 2H , [H, Q] = 0 , [H, H] = 0. Note that, algebra contains both commutators and anticommutators, ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 15 / 32
  33. 33. SUPERSYMMETRY SUSY Algebra Therefore, SUSY algebra in one dimension is {Q, Q} = 2H , [H, Q] = 0 , [H, H] = 0. Note that, algebra contains both commutators and anticommutators, therefore, Q and H generators form a graded Lie algebra as promised before. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 15 / 32
  34. 34. SUPERSYMMETRY SUSY Algebra REMARK : √ In 1927 Dirac ∼ γ µ Dµ : Dirac equation ⇒ prediction : for every fermionic particle there should be a fermionic antiparticle . (In 1932 positron is discovered.) ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 16 / 32
  35. 35. SUPERSYMMETRY SUSY Algebra REMARK : √ In 1927 Dirac ∼ γ µ Dµ : Dirac equation ⇒ prediction : for every fermionic particle there should be a fermionic antiparticle . (In 1932 positron is discovered.) √ 1970’s H ∼ Q : Supersymmetry ⇒ prediction : for every particle there should be a superpartner . (201?, will LHC find one? ) ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 16 / 32
  36. 36. SUPERSYMMETRY Superspace Grassmann Algebra : Let θi , i = 1, 2 · · · n to be n Grassmann numbers that satisfies, θi θj + θj θi = 0 , ∀ i, j → θi θi = 0 Definition of derivative and integration is given as ∂ θj = δij , dθ1 dθ2 · · · dθn θn · · · θ2 θ1 = 1 ∂θi Note that integral operates as a derivative! The above relations simplify a lot for one θ : d dθθ = 1 , dθc = 0 → dθ ≡ dθ ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 17 / 32
  37. 37. SUPERSYMMETRY Superspace Superspace : Since in one dimension we have one Q and H we parametrize the space with θ and t where it is obvious that both are real and θ is Grassman variable. DEFINITION : The space with coordinates t and θ is called SUPERSPACE. Any function of t and θ (i.e. Φ(t, θ)) is called a SUPERFIELD. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 18 / 32
  38. 38. SUPERSYMMETRY Superspace Then since, θ2 = 0 in our model the simplest superfield is, Φ(t, θ) = φ(t) + iθψ(t) ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 19 / 32
  39. 39. SUPERSYMMETRY Superspace Then since, θ2 = 0 in our model the simplest superfield is, Φ(t, θ) = φ(t) + iθψ(t) φ(t) is real scalar. In order to write a uniform superfield ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 19 / 32
  40. 40. SUPERSYMMETRY Superspace Then since, θ2 = 0 in our model the simplest superfield is, Φ(t, θ) = φ(t) + iθψ(t) φ(t) is real scalar. In order to write a uniform superfield All components of Φ must be scalar. Therefore, since θ is a Grassmann variable, ψ must be a anticommuting field. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 19 / 32
  41. 41. SUPERSYMMETRY Superspace Then since, θ2 = 0 in our model the simplest superfield is, Φ(t, θ) = φ(t) + iθψ(t) φ(t) is real scalar. In order to write a uniform superfield All components of Φ must be scalar. Therefore, since θ is a Grassmann variable, ψ must be a anticommuting field. All components of Φ must be real. Thats why we have an i in front θψ . ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 19 / 32
  42. 42. SUPERSYMMETRY Superspace Then since, θ2 = 0 in our model the simplest superfield is, Φ(t, θ) = φ(t) + iθψ(t) φ(t) is real scalar. In order to write a uniform superfield All components of Φ must be scalar. Therefore, since θ is a Grassmann variable, ψ must be a anticommuting field. All components of Φ must be real. Thats why we have an i in front θψ . Since [Φ] = [φ] = −1/2 we must also have [ψ] = 0 and [θ] = −1/2 . ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 19 / 32
  43. 43. SUPERSYMMETRY Superspace SUSY transformations can be obtained with the help of a Hermitian operator, ∂ ∂ Q= + iθ ∂θ ∂t because ∂ ∂ ˙ ξQΦ = ξ + iθ Φ = iξψ + iθ(−ξ φ) = δφ + iθδψ ∂θ ∂t Note that this operator Q satisfies SUSY algebra : ∂ {Q, Q} = 2i ≡ H , [Q, H] = 0 , [H, H] = 0 ∂t ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 20 / 32
  44. 44. SUPERSYMMETRY Superspace The invariance of the action can be written as δI = dtdθξQ[· · · ] = 0. It is important to know other operators that commute with ξQ. One of them is d/dt’dir. The other one is defined as ∂ ∂ D= − iθ ∂θ ∂t and it is called super covariant derivative. From the definition we see that, [ξQ, D] = ξ{Q, D} = 0. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 21 / 32
  45. 45. SUPERSYMMETRY Superspace Let us construct the action by using ”what else can it be” method. In general we can write, ∂ I = dtdθF ( , D, Φ). ∂t ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 22 / 32
  46. 46. SUPERSYMMETRY Superspace Let us construct the action by using ”what else can it be” method. In general we can write, ∂ I = dtdθF ( , D, Φ). ∂t By dimensional analysis we see that [dtdθ] = −1/2 and therefore [F ] = 1/2. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 22 / 32
  47. 47. SUPERSYMMETRY Superspace Let us construct the action by using ”what else can it be” method. In general we can write, ∂ I = dtdθF ( , D, Φ). ∂t By dimensional analysis we see that [dtdθ] = −1/2 and therefore [F ] = 1/2. A physically interesting action must at least be quadratic in fields.So we get ∂ F = K ( , D).Φ2 . ∂t ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 22 / 32
  48. 48. SUPERSYMMETRY Superspace Let us construct the action by using ”what else can it be” method. In general we can write, ∂ I = dtdθF ( , D, Φ). ∂t By dimensional analysis we see that [dtdθ] = −1/2 and therefore [F ] = 1/2. A physically interesting action must at least be quadratic in fields.So we get ∂ F = K ( , D).Φ2 . ∂t Since [Φ] = −1/2 we can only have [K ] = 3/2. So only by doing dimensional analysis we get only one solution for K : ∂ K∼ .D. ∂t ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 22 / 32
  49. 49. SUPERSYMMETRY Superspace In this simplest model we cannot write mass and interaction terms ( unlike in 4-dim.s) ! ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 23 / 32
  50. 50. SUPERSYMMETRY Superspace In this simplest model we cannot write mass and interaction terms ( unlike in 4-dim.s) ! If we want at most 2nd derivatives of the fields we have an unique solution : ∂Φ I = α dtdθ . (DΦ) ∂t ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 23 / 32
  51. 51. SUPERSYMMETRY Superspace In this simplest model we cannot write mass and interaction terms ( unlike in 4-dim.s) ! If we want at most 2nd derivatives of the fields we have an unique solution : ∂Φ I = α dtdθ . (DΦ) ∂t which gives i ˙ ˙ ˙ i ˙˙ ˙ I = dtdθ φ + iθψ iψ − iθφ = 0 + dtdθ −iθφφ + θψ ψ + 0 2 2 1 ˙˙ i ˙ = dtdθ θφφ + θψ ψ 2 2 ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 23 / 32
  52. 52. SUPERSYMMETRY Superspace In this simplest model we cannot write mass and interaction terms ( unlike in 4-dim.s) ! If we want at most 2nd derivatives of the fields we have an unique solution : ∂Φ I = α dtdθ . (DΦ) ∂t which gives i ˙ ˙ ˙ i ˙˙ ˙ I = dtdθ φ + iθψ iψ − iθφ = 0 + dtdθ −iθφφ + θψ ψ + 0 2 2 1 ˙˙ i ˙ = dtdθ θφφ + θψ ψ 2 2 This is the same action that we obtained before without using superspace techniques ! ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 23 / 32
  53. 53. SUPERSYMMETRY Extended SUSY : Extended SUSY : Now let us consider more then one SUSY in one dimension, i.e. we have N SUSY generators Q i , i = 1, 2, · · · , N so that N δ ξ φ = ξi Q i φ = i ξi ψi , ˙ δξ ψ = ξi Qi ψ = −ξi φ. i=1 The action, N 1˙˙ i ˙ I = dt φφ − ψi ψi 2 2 i=1 is still invariant under this extended SUSY transformations. But there is something unusual There are N fermions but still 1 boson ! ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 24 / 32
  54. 54. SUPERSYMMETRY Extended SUSY : Let us consider N=2 and check SUSY algebra, ˙ [δξ , δη ]φ = 2i(ξ1 η1 + ξ2 η2 )φ as expected but for instance for ψ1 we get ˙ ˙ [δξ , δη ]ψ1 = 2iξ1 η1 ψ1 + i(ξ1 η2 + ξ2 η1 )ψ2 and it doesn’t close on translation of ψ1 unless we use equation of motion ˙ of ψ2 : ψ2 = 0. Such a SUSY called onshell SUSY ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 25 / 32
  55. 55. SUPERSYMMETRY Extended SUSY : Since we have 2 fermions ψ1 , ψ2 and one boson φ, and algebra doesn’t close automatically to cure the problem let us introduce another boson field F such that 1˙˙ i ˙ 1 I = dt φφ − ψi ψi + F 2 2 2 2 Note that F doesn’t have a kinetic term and it is called auxiliary field. One can view (φ, ψ1 ) as one multiplet and (ψ2 , F ) as another such that ˙ δφ = iξ1 ψ1 , δψ1 = −ξ1 φ , ˙ δψ2 = ξ2 F , δF = iξ2 ψ2 with 1˙˙ i ˙ i ˙ 1 I1 = dt φφ − ψ1 ψ1 , I2 = dt − ψ2 ψ2 + F 2 2 2 2 2 i.e. N = 2 = 1 + 1 ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 26 / 32
  56. 56. SUPERSYMMETRY Extended SUSY : However, one can construct N=2 SUSY as N=2 ! (Like for the 4-dimensional case.) Let us write more general SUSY transformations by analyzing dimensions of the fields. δφ = iξi ψi , ˙ δψi = ξi φ + αij ξj F , ˙ δF = iξi βij ψj where α and β are real matrices. From the commutator algebra ˙ [δξ , δη ]φ = 2iηi ξi φ + (iηi (αij + αji )ξj F ) ˙ ˙ [δξ , δη ]ψi = i(ηi ξj − ξi ηj )ψj + iαij βkl (ηj ξk − ξj ηk )ψl we get αij + αji = 0 , αij βjk = δik so that [δξ , δη ] closes to translations. ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 27 / 32
  57. 57. SUPERSYMMETRY Extended SUSY : Finally, off–shell N=2 SUSY algebra in 1–dimension can be written as, δφ = iξi ψi , ˙ δψi = ξi φ + ij ξj F , δF = ξi ˙ ij ψj where 12 =− 21 = 1. Moreover, since we have two fermions now we can write mass term Im = −m dt (F φ + iψ1 ψ2 ) and an interaction term 1 2 Ig = g dt F φ + iψ1 ψ2 φ 2 ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 28 / 32
  58. 58. SUPERSYMMETRY Alternative way to obtain the action : An alternative way to obtain the action : Note that Fields they belong to a supersymmetric multiplet. One can move from the lowest member of the multiplet to highest one by SUSY transformation. Since Action is supersymmetric it also belongs to a SUSY multiplet. Therefore, one should be able to obtain the action by applying multiple SUSY variations to a lower dimensional integrated field polynomial. Of course, one can say that this observation is related with superspace. But, we need off–shell formulation to write superfields, and off–shell formulation does not always exist ! ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 29 / 32
  59. 59. SUPERSYMMETRY Alternative way to obtain the action : Let us show, how this method works for our simple model : We can write the SUSY transformation as, δξ = ξ1 Q1 + ξ2 Q2 For simplicity let us also define Q1 + iQ2 ¯ Q1 − iQ2 ψ1 + iψ2 − ¯ ψ1 √ iψ2 Q= √ , Q= √ , ψ= √ , ψ= 2 2 2 2 ¯ We can write Q and Q variations as Qφ = iψ, , Qψ = 0 , ¯ ˙ Q ψ = φ − iF , ˙ QF = ψ ¯ ¯ Qφ = i ψ, , ¯¯ Qψ = 0 , ¯ ˙ Qψ = φ + iF , ¯ ˙ ¯ QF = −ψ ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 30 / 32
  60. 60. SUPERSYMMETRY Alternative way to obtain the action : Then by analysis the dimensions of the fields and parameters, 3 1 1 1 [t] = −1, [m] = 1, [g ] = , [Q] = , [φ] = − , [ψ] = 0, [F ] = 2 2 2 2 it is easy to get, ¯ ¯ m 2 g 3 I = dt −(QQ)2 φ2 + QQ φ + φ 2 3! In other words, we can write the action by applying multiple super variations of the monomials of φ. This also true for the on–shell transformations except that one gets the action modulo equation of motion of fermion fields! A similar construction also works in 4 dimensions for WZ, N=1 and N=2 SYM (K.U, MPLA’XX) and even in much more complicated cases (H.Sonoda, K.U,2009). ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 31 / 32
  61. 61. SUPERSYMMETRY Alternative way to obtain the action : Reference Nearly all of this talk is from the excellent lectures : P. van Nieuwenhuizen ”Supersymmetry, Supergravity, Superspace and BRST Symmetry in a Simple Model ” arXiv: hep-th/0408179 The very minor part about cohomology is from some unpublished notes of mine, ”An Introduction to SUSY”, FGE 2005. One of the standard reference in 4–dimension is, J. Wess and J. Bagger, ”Supersymmetry and Supergravity”, (1992). ¨Kayhan ULKER (Feza G¨rsey Institute*) u Introduction to Supersymmetry September 2, 2011 32 / 32

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