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Home , Clifford algebra, Generator (mathematics), Graded Lie algebra, Superspace, Supersymmetry algebra

(...) an invariance of the theory under interchange of and [9] Introduction to

m {Qα, Qα˙ } = 2(σ )αα˙ Pm

Contents

1 in four dimensional space 2 1.1 Clifford Algebra ...... 2 1.2 Dirac Representation of SO(3, 1) ...... 2 1.3 Transformation of Weyl Spinors [4] ...... 3

2 (N =1) : From Coleman-Mandula to Haag-Lopuszanski-Sohnius 3

3 Chiral Component Representation 5

4 6 4.1 Representation of supersymmetry generators ...... 6 4.2 Superfields ...... 7

5 Chiral Superfields 8 5.1 K¨ahlerpotential ...... 8 5.2 Superpotential ...... 10

6 Guide to the Literature 11

D.Schmidt Talk given at the PhD/Diploma Seminar Electroweak Breaking and Supersymmetry Heidelberg, March 29 - 31, 2010

1 1 Spinors in four dimensional space time

• theory classifies all particles as members of irreducible represen- tations of the underlying symmetry group, i.e. particles are classified according to their transformation law under the symmetry group [1].

• symmetry group of four dimensional space time: SO(3, 1) (translations not involved; → Poincar´egroup)

• In what representation of the Lorentz group live fermions?

1.1 Clifford Algebra µ ν µν µν •{γ , γ } = 2 g 1; we use the signature g = gµν = diag (−, +, +, +).

• specialµ four dimensional¶ representation using (hermitian) Pauli σ matrices: 0 σµ γµ = σµ 0 µ ¶ −1 0 Attention: σ0 = in the given metric gµν . 0 −1

1.2 Representation of SO(3, 1) µ ¶ σµσν − σν σµ 0 • J µν ≡ i [γµ, γν ] = i satisfies 4 4 0 σµσν − σν σµ of SO(3,1): [J µν ,J ρσ] = i(ηνρJ µσ − ηµρJ νσ − ηνσJ µν + ηµσJ νρ). Remark~: In all representations of the Lorentz group, the corresponding generators are Bose type symmetry generators, i.e. they act on particles of equal [7].

µ • Transformation law of a Dirac Spinor (ΨD)a(x ) a = 1, 2, 3, 4 [2]:

µ Λ −1 ν ν (ΨD)a(x ) −→ S(Λ)ab(ΨD)b((Λ )µ x )

i ω M µν with a general Λ = e 2 µν and corresponding i ω J µν spinor transformation S(Λ) = e 2 µν . Avoid confusion: A generic element Λ of the SO(3, 1) can always i ω M µν µν be written as Λ = e 2 µν with M being the generators of the cor- responding Lie algebra. S(Λ) is a special representation of Λ, the spinor representation, for which the generators are J µν . • Dirac spinor representation completely reducible because generators J µν are diagonal. Get irreducible representation with the known left- and right-handed pro- jection opertors PL and PR: µ ¶ µ ¶ PL (ΨD)a Ψα (ΨD)a = ≡ α˙ . PR (ΨD)a χ

2 1 • Ψα ; α = 1, 2 left-handed Weyl spinor (( 2 , 0) representation of Lorentz group) α˙ 1 χ ;α ˙ = 1, 2 right-handed Weyl spinor ((0, 2 ) representation of Lorentz group)

1.3 Transformation of Weyl Spinors [4] 4 m • v ∈ R → ˆv = vmσ hermitian • M ∈ SL(2, C) → ˆv0 = MˆvM† is hermitian: expansion in σ matrices → 0 0 m ˆv = vmσ → v02 = det ˆv0 = det ˆv = v2 , because det M = 1. Result: Any M ∈ SL(2, C) defines via ˆv → ˆv0 = MˆvM† a Lorentz 0 n transformation vm → vm = Λm vn . • Weyl spinors transform under SL(2, C) (Weyl spinor has complex components.):

0 β α˙ α˙ 0 ∗−1 α˙ β˙ ~~ Ψα → Ψα = Mα Ψβ and χ → χ = (M )β˙ χ

• index zoo: αβ Lowering and raising indices with antisymmetric ² = (² ) = iσ2:

β α˙ α˙ β˙ Ψα = ²αβΨ and χ = ² χβ˙ .

0 † m † ~~ Observe from ˆv = MˆvM = vmMσ M and that the index structure m m αα˙ α˙ β˙ αβ m of the Pauli σ matrices is given as: (σ )αα˙ and (σ ) = ² ² (σ )ββ˙ with latin letters as space time indices and greek letters as spinor indices. † α˙ † Attention to dotted and undotted spinor indices: Ψα = Ψα˙ and χ = χα . Kinetic term for spinor

m αβ˙ α m β˙ iΨd/Ψ = iΨα˙ (σ ) ∂mΨβ + iχ (σ )αβ˙ ∂mχ results in equation of motion for a massless left-handed Weyl spinor: m αβ˙ i(σ ) ∂mΨβ = 0.

2 (N =1) Supersymmetry Algebra: From Coleman-Mandula to Haag-Lopuszanski- Sohnius

• Coleman-Mandula no-go theorem for (...) an invariance of the theory un- der interchange of fermions and bosons : Any symmetry group (for which the is the bilinear operation in the Lie algebra) of a quantum field theory (QFT) is locally isomorphic to the direct product of an inter- nal symmetry group, i.e. a group for which the generators have no elements between particles of different spin, and the Poincar´egroup [6]. −→~ Symmetry between fermions and bosons impossible?

3 • Bypass Coleman-Mandula theorem: no-go theorem holds for Lie alge- bras with commutator as bilinear operation. Consider Lie algebras whose generators obey commutator and anticommutator relations.

• new generator (N =1): Qα and its hermitian conjugate Qα˙ • Consider the following graded Lie algebra1:

– vanishing anticommutators (due to spinor character):

{Qα,Qβ} = {Qα˙ , Qβ˙ } = 0 – transformation under Lorentz group: mn i m n n m β [M ,Qα] = (σ σ − σ σ )α Qβ |4 {z } transformation of left-handed Weyl spinor 1 1 Qα in ( 2 , 0) representation, Qα˙ in (0, 2 ) representation of Lorentz group – All operators should commute with space time translations for energy m momentum conversation: [P ,Qα] = 0. m – The new anticommutator: {Qα, Qα˙ } = 2(σ )αα˙ Pm 1 1 Motivation: QαQα˙ transforms in ( 2 , 2 ) representation of the Lorentz group what is the spinor description of a four vector that has to be proportional to P m for energy momentum conversation. The only objects which have space time and spinor indices in their structures are the Pauli σ matrices. m αα˙ m With (σ ) · (σ )αα˙ = −4 the new anticommutator takes on the form P m = − 1 (σm)αα˙ {Q , Q } and we find 8 ³ α α˙ ´ 0 1 † † † † H = P = + 8 Q1Q1 + Q1Q1 + Q2Q2 + Q2Q2 , what shows ↑ 0 σ =−12×2 that the expectation value of the Hamiltonian H is not negative in any state |ηi constructed by applying creation operators, which are proportional to Qα˙ , to the vaccum state |Ωi : hη|H|ηi = 0 ; |Ωi is defined as usual: Qα|Ωi = 0 with annihilation operators being proportional to Qα [4]. • Haag, Lopuszanski and Sohnius: The given graded Lie algebra is the unique graded Lie algebra of symmetries consistent with QFT. → (N =1) Supersymmetry Algebra [7] • (global) infinitesimal supersymmetry transformation:

α α˙ α α˙ δξ = ξ Qα − ξ Qα˙ = ξ Qα + ξα˙ Q ↑ indices with spinor parameter ξ. The commutator of two supersymmetry transformations leads to space time translations (Pm = i∂m):

1 m Action S invariant under supersymmetry transformationR → get supercurrents Jα after 3 0 applying Noether’s theorem → Qα = d x Jα fulfill the given graded Lie algebra [4].

4 ³ ´ α m β˙ β m α˙ [δξ1 , δξ2 ] = 2i ξ1 (σ )αβ˙ ξ 2 − ξ2 (σ )βα˙ ξ 1 ∂m .

Remark: ξ does not depend on space time, i.e. we consider only global supersymmetry. The theory of local supersymmetry transformations with ξ = ξ(xm), called , is beyond the scope of this introduction.

• Find a representation of this algebra on four dimensional space time to describe a QFT !

3 Chiral Field Component Representation

• theoryR of a complex scalar z and a left-handed Weyl spinor Ψα: 4 m m S = d x (−∂mz ∂ z − iΨ σ ∂mΨ) • supersymmetry transformations: For a supersymmetric theory the action ! S has to be invariant under a supersymmetry transformation δξ, δξS = 0. Therefore we have to define the following transformations: √ √ α α˙ δξz√≡ 2 ξ Ψα δξz ≡√ 2 ξ Ψα˙ m α˙ α m δξΨα ≡ 2 i(σ )αα˙ ξ ∂mz δξΨα˙ ≡ − 2 iξ (σ )αα˙ ∂mz Observe: In the first row bosons are transformed into fermions, in the second row fermions are transformed into bosons. • Consistency: The given definitions above must close the supersymmetry algebra. Problem for left-handed Weyl spinor:

√ √ m α˙ m α˙ [δξ1 , δξ2 ]Ψα = δξ1 2 i(σ )αα˙ ξ 2 ∂mz − δξ2 2 i(σ )αα˙ ξ 1 ∂mz √ √ √ √ m α˙ β m α˙ β = 2 i(σ )αα˙ ξ ∂m 2 ξ Ψβ − 2 i(σ )αα˙ ξ ∂m 2 ξ Ψβ ³ 2 1 ´ 1 2 m α˙ β m α˙ β = 2i (σ )αα˙ ξ 2 ξ1 ∂m − (σ )αα˙ ξ 1 ξ2 ∂m Ψβ ³ ´ (?) α m β˙ β m β˙ = 2i −ξ1 (σ )ββ˙ ξ 2 ∂mΨβ + ξ1 (σ )ββ˙ ξ 2 ∂mΨα − (1 ↔ 2) ³ ´ (??) β˙ m ββ˙ α β m β˙ = 2i ξ 2 (σ ) ξ1 ∂mΨβ + ξ1 (σ )ββ˙ ξ 2 ∂mΨα − (1 ↔ 2) ³ ´ α β˙ m ββ˙ β m β˙ = 2i −ξ1 ξ 2 (σ ) ∂mΨβ + ξ1 (σ )ββ˙ ξ 2 ∂mΨα − (1 ↔ 2) ³ ´ β m β˙ β m β˙ = 2i ξ1 (σ )ββ˙ ξ 2 − ξ2 (σ )ββ˙ ξ 1 ∂mΨα ³ ´ α β˙ m ββ˙ α β˙ m ββ˙ − 2i ξ1 ξ 2 (σ ) ∂mΨβ − ξ2 ξ 1 (σ ) ∂mΨβ

The algebra closes only for Ψβ fulfilling the equation of motion (eom): m αβ˙ (σ ) ∂mΨβ = 0. (?) Use Fierz rearrangement idendity: χα(ξς) = −ξα(ςχ) − ςα(χξ); α fixed, not summed m α˙ β with χ = (σ )αα˙ ξ 2 , ξ = ξ1 and ς = ∂mΨβ . (??) Use χσmξ† = −ξ†σ m.

5 • Counting the degrees of freedom (dof): Without use of the eom, there are two independent complex dof for the Weyl spinor, i.e. four independent real dof. For the complex scalar there are two independent real dof. Same amount of fermion and boson dof in one representation → addition of two real boson dof such that no dof are added with the use of the eom → auxiliary (no kinetic terms) field F which transforms into equation of motion of spinor:

√ m αα˙ δξF ≡ 2iξα˙ (σ ) ∂mΨα √ √ m α˙ δξΨα ≡ 2i(σ )αα˙ ξ ∂mz + 2ξαF

R 4 m m S = d x (−∂mz ∂ z − iΨ σ ∂mΨ + F F ) is supersymmetric invariant without use of eom.

• z, Ψα and F are the components of the chiral multiplet.

4 Superspace

• Supersymmetric transformations on the chiral multiplet in accordance with δξS = 0. Reproduce these transformations from the general supersymmetry trans- α α˙ formation δξ = ξ Qα − ξ Qα˙ . • Look at the anticommutator:

m {Qα, Qα˙ } = 2(σ )αα˙ Pm On the left hand side, we have the product of two supersymmetry trans- formations generated by Qα and Qα˙ which result on the right hand side in translations in four dimensional space time. Idea: Construct a space on which Qα and Qα˙ generate translations. Due to the spinor character of Qα and Qα˙ this space must have in additon to the space time coordinates two anticommuting coordinates Θα and Θα˙ . m This space with coordinates (x , Θα, Θα˙ ) is called superspace.

4.1 Representation of supersymmetry generators ∂ • On superspace we define ∂α ≡ ∂Θα and represent Qα and Qα˙ as translation operators [4]:

m α˙ Qα ≡ ∂α − i(σ )αα˙ Θ ∂m α m Qα˙ ≡ −∂α˙ + iΘ (σ )αα˙ ∂m .

→ Supersymmetry algebra motivated in section 2 justified: m {Qα,Qβ} = 0 and {Qα, Qα˙ } = 2(σ )αα˙ Pm

6 4.2 Superfields m • General complex superfield G(x , Θα, Θα˙ ) with supersymmetry transfor- mation¡ ¢ α α˙ δξG = ξ Qα + ξα˙ Q G. • actions for superfields → How to integrate over superspace? The integration of a function g = g(Θ) depending on an anticommut- number Θ is the same operation as differentiation with respect to Θ: 1 dΘ =b ∂1. 2 2 fact: ∂ Θ = −R4 [3] R normalization: d2Θ = − 1 ∂2 → d2ΘΘ2 = 1. R 4 R 4 2 2 1 4 2 2 action: S = d x d Θ d Θ G(x, Θ, Θ) = 16 d x∂ ∂ G(x, Θ, Θ) • achievment: action S for general superfield G is automatically supersym- metric invariant:

Z 4 2 2 ¡ α α˙ ¢ δξS = d x d Θ d Θ ξ Qα − ξ Qα˙ G   Z   = d4x d2Θ d2Θ  −∂ (ξαG) − ∂ (ξ α˙ G) − i(ξσmΘ − Θσmξ)∂ G | α {z α˙ } | {z m } total derivative in Θ and Θ total derivative in x = 0 .

• components of a general superfield G → How to differentiate on superspace [4]?

m α˙ Dα ≡ ∂α + i(σ )αα˙ Θ ∂m α m Dα˙ ≡ −∂α˙ − iΘ (σ )αα˙ ∂m

m with {Dα, Dα˙ } = −2i(σ )αα˙ ∂m ; {Dα,Qβ} = {Dα˙ ,Qβ} = = {Dα˙ , Qβ˙ } = 0 and (QαG)|Θ=Θ=0 = (DαG)|Θ=Θ=0 . ∗ Attention: ∂α = −∂α˙ [5]. • Define components of G via derivatives [3]:

components of vector multiplet z }| { 1 C = G| (σm) A = − [D , D ]G| Θ=Θ=0 αα˙ m 2 α α˙ Θ=Θ=0 2 Ψ = √1 D G| λ = − i D D G| α 2 α Θ=Θ=0 α 4 α Θ=Θ=0 Ψ = √1 D G| λ = − i D2D G| α˙ 2 α˙ Θ=Θ=0 α˙ 4 α˙ Θ=Θ=0 1 2 ˜ 1 α 2 F = − 4 D G|Θ=Θ=0 D = 8 D D DαG|Θ=Θ=0 1 2 F = − 4 D G|Θ=Θ=0 Higher derivatives vanish due to anticommuting variables and other or- derings of derivatives not independent of the chosen ones.

7 • Use {Dα˙ ,Qβ} = 0 to show Dα˙ δξG = δξDα˙ G. If Dα˙ G = 0, then Dα˙ (δξG) = 0 . → All superfields G with Dα˙ G = 0 form a subrepresentation of the super- symmetry algebra because after a supersymmetry transformation δξ of G, we still have Dα˙ (δξG) = 0 [5].

5 Chiral Superfields

• superfield Φ chiral ⇔ Dα˙ Φ = 0

• Remaining independent components of Φ (see section 4.2): C,Ψα and F , i.e. the members of the chiral multiplet from the component approach in section 3.

• achievement: Supersymmetry transformations of the components of a chi- ral superfield are now calculated rather than assuming them to obtain a supersymmetric action (cf. section 3). • Special importance has supersymmetry transformation of F term of a chiral superfield Φ:

1 ¡ ¢ δ F = − ξαD − ξ α˙ D D2G| ξ 4 α α˙ Θ=Θ=0 1 = ξ α˙ D D2G| , because D D2 = 0 4 α˙ Θ=Θ=0 α 1 = − ξ α˙ [D2, D ]G| , because G is chiral 4 α˙ Θ=Θ=0 (?) m α˙ α = i(σ )αα˙ ξ ∂mD G| √ Θ=Θ=0 m α˙ α = 2 i(σ )αα˙ ξ ∂mΨ √ α˙ β˙ αβ m = 2 i² ² (σ )αα˙ ξβ˙ ∂mΨβ √ β˙α˙ βα m = 2 i² ² (σ )αα˙ ξβ˙ ∂mΨβ √ m ββ˙ ~~~ = 2 i ξβ˙ (σ ) ∂mΨβ .

Observation: F term changed by total derivative after supersymmetry transformation. → Action for F term of any chiral superfield Φ is super- symmetric invariant [8]. 2 m α (?) Use [D , Dα˙ ] = −4i (σ )αα˙ D ∂m.

5.1 K¨ahlerpotential

• Consider two superfields Φ and Φ with Dα˙ Φ = 0 (chiral) and DαΦ = 0 (antichiral) and calculate the action for ΦΦ.

8 Z 4 2 2 £ ¤ S = d x d Θ d Θ ΦΦ |Θ=Θ=0 Z 1 2 £ ¤ = d4x D D2 ΦΦ | 16 Θ=Θ=0 Z 1 α˙ £ ¤ = d4x D D ΦD2Φ | ; D DαΦ = 0 because Φ is antichiral 16 α˙ Θ=Θ=0 α   Z 1 4  α˙ 2 α˙ 2  = d x (Dα˙ (D Φ)(D Φ)) + Dα˙ (Φ D D Φ) |Θ=Θ=0 16 ↑  Θ=Θ=0  Z 1 4  2 2 α˙ 2 α˙ 2 2 2  = d x (D Φ)(D Φ) − (D Φ)Dα˙ (D Φ) + (Dα˙ Φ)D (D Φ) + Φ D D Φ |Θ=Θ=0 16 ↑ {D ,Dα˙ }=0  α˙  Z 1 4  2 2 α˙ 2 α˙ 2 2 2  = d x (D Φ)(D Φ) − (D Φ)Dα˙ (D Φ) − (D Φ)Dα˙ (D Φ) + ΦD D Φ |Θ=Θ=0 16 ↑ indices Z h i 1 2 α˙ 2 = d4x (D Φ)(D2Φ) − 2(D Φ)D (D2Φ) + Φ D D2Φ | 16 α˙ Θ=Θ=0 µ ¶ Z √ √ (?) 1 4 m 1 α˙ m 1 α = d x (−4)(−4) FF + 16 Φ∂ ∂mΦ − 2 2√ D Φ 4i (σ )αα˙ ∂m 2√ D Φ 16 2 2   Z 1 4  m α˙ m α = d x 16 FF − 16 ∂ Φ∂mΦ − 16i Ψ (σ )αα˙ ∂mΨ  16 ↑ part. int. Z 4 ¡ m m ¢ = d x FF − ∂ z∂mz − iΨσ ∂mΨ

→ The free action of a complex scalar z, a left-handed Weyl spinor Ψ and an auxiliary field F results from integrating the product of the corre- sponding chiral and antichiral superfield over superspace. (?) Use chirality and the following relations:

2 2 2 α˙ m {D ,D } = 2 Dα˙ D D + 16 ∂ ∂m 2 m α [D , Dα˙ ] = −4i (σ )αα˙ D ∂m .

• Generalizing the approach of getting a supersymmetric invariant action from a real valued function K(Φ, Φ) of an antichiral superfield Φ and a chiral superfield Φ leads to K¨ahlergeometries and the K¨ahlerpotential [3]. K(Φ, Φ) = Φ Φ is just a special form of the K¨ahlerpotential for one antichiral superfield Φ and one chiral superfield Φ. • Remark: It can be shown that for a general superfield G the only compo- nent whose supersymmetric transformation is a total derivative and hence

9 whose action is supersymmetric invariant is the D˜ component (cf. section 4.2). Since the action of the K¨ahlerpotential is supersymmetric invariant, the K¨ahlerpotential must be the D˜ term of a general superfield G [8].

5.2 Superpotential • G general superfield → action for D˜ component supersymmetric invariant → K¨ahlerpotential ~~~ Φ chiral superfield −−−→ action for F component supersymmetric invariant → ? potential ?

• Observe from definition of chiral superfield:P Q Φi ; i = 1, ... , n chiral superfields ⇒ i Φi and i Φi chiral superfields. → Any holomorphic function W (Φ) of chiral superfields Φ = Φi ; i = 1, ... , n is a chiral superfield, i.e. Dα˙ W (Φ) = 0. • W (Φ) no Θ dependence → only d2Θ integration for action S:

Z S = d4x d2Θ W (Φ) Z 1 δ S = − d4x D2ξαD W (Φ) ξ 4 α Z µ ¶ 1 = d4x ξαD − D2W (Φ) α 4 2 = 0 , because DαD = 0

1 2 Recall from section 4.2: [W (Φ)]F = − 4 D W (Φ)|Θ=Θ=0 → The action for the F term of the chiral superfield W (Φ) is supersymmet- ric invariant. The holomorphic function W (Φ) is called the superpotential.

10 6 Guide to the Literature

1. A Modern Introduction to ; M. Maggiore • Chapter 2 implements the basics of group theory and the represen- tation of the Lorentz group.

2. Quantum Field Theory I ; Lecture Notes by A. Hebecker • Chapter 8 reveals SL(2, C) as the true symmetry group of four di- mensional space time (universal covering groups).

3. Supersymmetry; Lecture Notes by G. Nibbelink • Chapter 1 to chapter 6 are the main input for this talk. • Knowledge of QFT is presumed.

4. Supersymmetry and Supergravity; J.Wess, J. Bagger • THE introduction to supersymmetry and beyond. • few text, mainly calulations, therefore very formal and technical • useful review about spinors and their index structure in APPENDIX A and APPENDIX B

5. Beyond The Standard Model; Lecture Notes by A. Hebecker • short introduction into supersymmetry in chapter 4

6. All Possible Symmetries of the S Matrix; S. Coleman, J. Mandula; Phys- ical Review 159, 5, (1967) • mathematical formulation and proof of the no-go theorem

7. All Possible Generators Of of the S-Matrix; R.Haag, J.T. Lopuszanski, M. Sohnius; Nuclear Physics B88 (1975) 257-274 • motivation for the supersymmetric algebra and its extension to N =L • general classification of Bose and Fermi type symmetry generators

8. The Quantum Theory of Fields, Volume III Supersymmetry; S. Weinberg • Chapter 26.3 and chapter 26.4 provide explanation of D and F terms. • One must get used to the the author’s unique notation.

9. The Supersymmetric World. The Beginnings of the Theory; G. Kane, M. Shifman • story of the historical development of supersymmetry seen through the eyes of the contributors ,→

11 Outlook

12