An Introduction to Supersymmetry

An Introduction to Supersymmetry Voja Radovanović Faculty of Physics University of Belgrade Belgrade, 2015. godine 1 Notation and conventions 1.1 Lorentz and SL(2, C) symmetries In these lectures we use mostly minus metric, i.e. g µν = diag(1, 1, 1, 1). Lorentz transformations leave metric unchanged, Λ T gλ = g. The proper orthochronous Lorentz transformations, L +, consisting of boosts and rotations, satisfy the following two conditions: det Λ = 1, Λ 0 0 1. (1) These transformations are a subgroup of Lorentz group. Lorentz algebra has six generators, M µν = M νµ. The commutation relations are [M µν, M ρσ ] = i(g µσ M νρ + g νρ M µσ g µρ M νσ g νσ M µρ ). The generators of rotations are J i = 1 2 ϵ ijkm jk, while generators corresponding to the boosts are N k = M 0k. Further, one can introduce the following complex combinations A i = 1 2 (J i + in i ) and B i = 1 2 (J i in i ). It is easy to prove that [A i, A j ] = iϵ ijl A l, [B i, B j ] = iϵ ijl B l, [A i, B j ] = 0. This is a well known result which gives a connection between the Lorentz algebra and two SU(2) algebras. Irreducible representations (i. e. fields) of the Lorentz group are classified by two quantum numbers (j 1, j 2 ) which come from above two SU(2) groups. L + is double connected (due to SO(3) subgroup) and non-compact group. To discuss its representations one has to consider its universally covering group, SL(2, C). SL(2, C) is the group of 2 2 complex matrices with unit determinant. Pauli matrices are given by σ µ = (1, σ), σ µ = (1, σ), where σ 1 = ( ) 0 1, σ 1 0 2 = ( ) ( ) 0 i 1 0 and σ i 0 3 = 0 1 1.

To find the connection between SL(2, C) and L + we define one-to-one map between Minkowski coordinates, x µ and 2 2 Hermitian matrices: ( ) X = x µ σ µ = X x0 + x = 3 x 1 ix 2. (2) x 0 + ix 2 x 0 x 3 Let us now transform X to another Hermitian matrix in the following way X X = MXM, where M SL(2, C). This transformation preserves the interval, i.e. det X = det X x 2 = x 2. From it follows that x µ = 1 2 Tr( σµ X ) = 1 2 Tr ( σ µ Mσ ν M )x ν = Λ µ νx ν, (3) Λ µ ν = 1 2 Tr ( σ µ Mσ ν M ). The mapping M Λ(M) is 2 1 homomorphism between SL(2, C) and L +. This mapping is not an isomorphism since matrices ±M correspond to Lorentz matrix Λ. SL(2, C) is simply connected, so it is the universal covering group of L +. 1.2 Representations A left-handed Weyl spinor, ψ α (x) is two component object, which transforms in the following way ψ α(λx) = Mα β ψ β (x) (4) under SL(2, C) transformations. The matrix M belongs SL(2, C) group. Indices α and β take values 1, 2. In Quantum Field Theory ψ α (x) is an operator. Its transformation law under Lorentz transformations is U 1 (Λ)ψ α (x)u(λ) = Mα β ψ β (Λ 1 x). (5) Let us define ψ α = (ψ α ), where dagger denotes Hermitian conjugation of the field operator. If ψ α is a classical Grassmann field, we use complex conjugation, i.e. ψ α = (ψ α ). The complex conjugation is defined in a such way that it has properties similar to the hermitian conjugation. For example, both of them reverse the order of fermions. The complex conjugate representation is ψ α = (M ) β ψ α β. (6) ψ β is a right handed Weyl spinor or dotted spinor. Left-handed spinor ψ α is also called undotted spinor. These two representations of SL(2, C) are inequivalent. Now, we introduce invariant tensors for SL(2, C). Antisymmetric tensors are defined by ( ) 0 1 ϵ αβ = ϵ α β = (7) 1 0 2

ϵ αβ = ϵ α β = (ϵ αβ ) 1 = (ϵ α β) 1 = ( 0 ) 1 1 0. (8) It is easy to prove ϵ αβ ϵ βγ = δ γ α and ϵ α βϵ β γ = δ γ α. From ϵ αβ Mγ α M β δ = ϵ γδ det M = ϵ γδ (9) it follows that ϵ αβ is an invariant tensor. Similarly, ϵ αβ, ϵ α β and ϵ α β are invariant tensors. The invariant tensors are used for lowering or raising spinor indices: Let us find the transformation law for ψ α : ψ α = ϵ αβ ψ β, ψ α = ϵ αβ ψ β, (10) ψ α = ϵ α β ψ β, ψ α = ϵ α β ψ β. (11) ψ α = ϵ αβ ψ β = ϵ αβ M γ β ϵ γδψ δ = (ϵ 1 Mϵ) α δψ δ = (M T 1 ) α δψ δ. (12) Spinors ψ α and ψ α transform according to equivalent representations. Similarly, we get ψ α = (M 1 ) α β ψ β, (13) i. e. spinors ψ α and ψ α transform under equivalent representations. The fields ψ α and ψ α are right-handed Weyl spinors. Undotted and dotted representation are very often called fundamental and antifundamental representations. The left-handed (or undotted ) spinors are ( 1, 0) 2 irreducible representation, while right-handed (or dotted) spinors are (0, 1). 2 Multiplying these spinors with each other we get higher spinors. They have dotted and undotted indices, ψ α1...α n α 1... α m and transform as follows ψ α 1...α n α 1... α m = M β 1 α 1... M β n α n (M ) β 1 α 1... (M ) β m α m ψ β1...β n β1... β m. These spinors are generally reducible. The ( irreducible ) spinors are ψ (α1,...,α n )( α 1,..., α m ), where (..) n denotes symmetrization. They belong to, m irreducible representation of SL(2, C). Let us 2 2 list some important irreducible representations: (0, 0) scalar or singlet representation (1/2, 0) left handed Weyl spinor (0, 1/2) right-handed Weyl spinor (1/2, 1/2) vector (1, 0) self-dual tensor (0, 1) antiself-dual tensor. From Mσ µ M = (Λ 1 ) µ ρσ ρ follows that matrices σ µ are invariant tensors. The matrix σ µ has the following spinorial indices σ µ α α. (14) 3

Raising the spinorial indices on σ µ we obtain It is easy to show The matrices σ µ and σ µ satisfy the following identites: Transformation laws for quantum Weyl fields are: where M SL(2, C). ( σ µ ) αβ = ϵ α γ ϵ βγ σ µ γ γ = (σµ ) β α. (15) ( σ µ ) αβ = (1, σ) αβ. (16) Tr(σ µ σ ν ) = 2g µν (17) σ µ α α ( σ µ) ββ = δ β αδ β α (18) σ µ α α ( σ µ) ββ = 2ϵ α βϵ αβ (19) (σ µ σ ν + σ ν σ µ ) β α = 2g µν δ β α. (20) U 1 (Λ)ψ α (x)u(λ) = M β α ψ β (Λ 1 x), ( 1 2, 0) U 1 (Λ)ψ α (x)u(λ) = (M 1 ) α β ψ β (Λ 1 x), ( 1 2, 0) U 1 (Λ) ψ α (x)u(λ) = (M ) β α ψ β(λ 1 x), (0, 1 2 ) U 1 (Λ) ψ α (x)u(λ) = (M 1 ) α β ψ β(λ 1 x), (0, 1 2 ), (21) 1.3 Dirac and Majorana spinors Weyl representation of γ µ matrices is given by ( ) 0 σ γ µ µ = σ µ 0. (22) γ 5 matrix is defined by Dirac spinor is γ 5 = iγ 0 γ 1 γ 2 γ 3 = Ψ D = ( ) φα χ α ( ) 1 0 0 1 and it belong to the reducible representation (0, 1 2 ) ( 1 2, 0). Adjoint Dirac spinor is ψ D = ψ β = ( χ α φ α ). (23), (24) (25) 4

where β = ( ) ( ) 0 1 = 0 δ β α 1 0 δα β 0. (26) The matrices γ 0 and β are numerically equal, but have different spinorial structure. Majorana spinor satisfies the condition Ψ M = C Ψ T M, were C = iγ0 γ 2 is the charge conjugation matrix. It is easy to show that ( ) ϵαβ 0 C = α β. 0 ϵ Majorana condition leads to ψ α = χ α, so Majorana spinor is ( ) ψα Ψ M = ψ α. (27) 1.4 Generators in (anti)fundamental representations Under Lorentz transformation Dirac spinors transform as ψ D(Λx) = e i 4 ω µνσ µν ψ D (x). (28) The matrices Σ µν are where 1 2 Σµν = i ( ) σ 4 [γµ, γ ν µν 0 ] = i 0 σ µν, (29) σ µν = 1 4 (σµ σ ν σ ν σ µ ) σ µν = 1 4 ( σµ σ ν σ ν σ µ ). Spinorial indices of these matrices are (σ µν ) α β by M β α = (M ) β α = (M 1T ) α β = (M 1 ) α β = and ( σ µν ) α β. Therefore, the M matrices are given (e 1 2 ω µνσ µν) β α (e β 1 ωµν σµν) 2 α (e 1 2 ωµνσµν) α β (e 1 2 ω µν σ µν) α β,,. (30) Matrices iσ µν and i σ µν are generators of Lorentz group, and they satisfy 1 1 We use the convention in which ϵ 0123 = ϵ 0123 = +1 ϵ µνρσ σ ρσ = 2iσ µν ϵ µνρσ σ ρσ = 2i σ µν. (31) 5

σ µν and σ µν are a self-dual tensor and an anti-self-dual tensor, respectively. We can define (σ µν ϵ) αβ and (ϵ σ µν ) α β by (σ µν ϵ) αβ = (σ µν ) α γ ϵ γβ, (32) They satisfy 1.5 Bilinear quantities Weyl spinors anticommute, i.e Let us define the product of Weyl spinors in the following way (ϵ σ µν ) α β = ϵ α γ ( σ µν ) γ β. (33) (σ µν ϵ) αβ = (σ µν ϵ) βα (ϵ σ µν ) α β = (ϵ σ µν ) β α. (34) ψ α χ β = χ β ψ α. (35) ψχ = ψ α χ α = ψ 1 χ 1 + ψ 2 χ 2 = ψ 2 χ 1 ψ 1 χ 2. (36) Similarly, we introduce ψ χ = ψ α χ α. (37) In our convection dotted indices are contracted from lower left to upper right, while undotted indices are contracted from upper left to lower right. ψχ and ψ χ are invariant under SL(2, C). Four-vectors and tensors are defined as ψσ µ χ = ψ α σ µ α α χ α, ψ σ µ χ = ψ α ( σ µ ) αα χ α, ψσ µν χ = ψ α (σ µν ) α β χ β, ψ σ µν χ = ψ α ( σ µν ) α β χ β. (38) The bilinear quantities defined above satisfy the following properties: φχ = χφ, φ χ = χ φ, φσ µ χ = χ σ µ φ, φσ µ σ ν χ = χσ ν σ µ φ, φ σ µ σ ν χ = χ σ ν σ µ φ, φσ µν χ = χσ µν φ, φ σ µν χ = χ σ µν φ, φσ µν φ = 0, φ σ µν φ = 0. (39) 6

Pauli matrices are hermitian. Their complex conjugation gives (σ µ α β ) = ϵ α βϵ βα ( σ µ ) βα = σ µ αβ = σµ β α, (40) ( σ µ αβ ) = ϵ αγ ϵ β δ(σ µ ) γ δ = σ µα β = σ µ βα. (41) Taking complex conjugation of Lorentz generators we obtain The following relations are useful: ((σ µν ) α β ) = ( σ µν ) β α, (( σ µν ) α β) = (σ µν ) β α. (42) ((σ µν ϵ) αβ ) = (ϵ σ µν ) β α, (43) ((ϵ σ µν ) α β) = (σ µν ϵ) βα. (44) Hermitian conjugation changes the order of spinors. For example: The following identities can be shown similarly: (ψχ) = (ψ α χ α ) = χ α ψ α = ψ α χ α = ψ χ. (45) ( ψ χ) = ψχ, (ψσ µ χ) = χσ µ ψ, ( ψ σ µ χ) = χ σ µ ψ, (ψσ µν χ) = ψ σ µν χ, ( ψ σ µν χ) = ψσ µν χ. (46) 1.6 Identites with σ matrices The list of useful identities with σ matrices is given below: Tr(σ µ σ ν ) = 2g µν (47) σ µ α α ( σ µ) ββ = δ β αδ β α (48) σ µ α α ( σ µ) ββ = 2ϵ α βϵ αβ (49) (σ µ σ ν + σ ν σ µ ) β α = 2g µν δ β α (50) ( σ µ σ ν + σ ν σ µ ) α β = 2g µν δ β α (51) (σ µ σ ν ) β α = g µν δ β α + 2(σ µν ) β α (52) ( σ µ σ ν ) α β = g µν δ β α + 2( σµν ) α β. (53) 7

From it follows γ µ γ ν γ ρ = (g µν g ρσ g νρ g νσ + g µσ g ρν )γ σ + iϵ σµνρ γ 5 γ σ (54) The trace identities are: σ µ σ ν σ ρ = g µν σ ρ g µρ σ ν + g νρ σ µ iϵ σµνρ σ σ σ µ σ ν σ ρ = g µν σ ρ g µρ σ ν + g νρ σ µ + iϵ σµνρ σ σ. (55) Tr(σ µν σ ρσ ) = 1 2 (gνρ g µσ g µρ g νσ + iϵ µνρσ ) (56) 1.7 Fierz identities Tr( σ µν σ ρσ ) = 1 2 (gνρ g µσ g µρ g νσ iϵ µνρσ ). (57) Fierz identities are very helpful in calculations with spinors. The following relations can be easily obtained θ α θ β = 1 2 ϵαβ θθ (58) θ α θ β = 1 2 ϵ αβθθ θ α θ β = 1 2 ϵ α β θ θ θ α θ β = 1 2 ϵ α β θ θ. (59) With a help of the previous identities the following Fierz rearrangement formulas can be proven (θϕ)(θψ) = 1 (θθ)(ϕψ), (60) 2 ( θ ϕ)( θ ψ) = 1 ( θ θ)( ϕ ψ), 2 (θσ µ θ)(θσ ν 1 θ) = 2 gµν (θθ)( θ θ), (θσ µ θ)θα = 1 2 σµ α β θ β(θθ), (θσ µ θ) θ α = 1 2 σµ β α θβ ( θ θ), ( θ λ)(χσ µ θ) 1 = ( θ θ)(χσ µ λ), 2 (θλ)( χ σ µ θ) = 1 2 (θθ)( χ σµ λ), (θσ ν ψ)(θσ µ λ) = 1 2 (θθ)( ψ σ ν σ µ λ), (61) 8

χ α (ξη) + ξ α (ηχ) + η α (χξ) = 0, (62) χ α ( ξ η) + ξ α ( η χ) + η α ( χ ξ) = 0. (63) As a example, let us check (60). Applying (58), the left hand side of (60) becomes (θϕ)(θψ) = 1 2 ϵαβ (θθ)ϕ α ψ β = 1 (θθ)(ϕψ). (64) 2 Remaining identities can be proven in a similar way. Let us generalize identities (58-59) to the case of two different spinors. We obtain θ α ψ α = 1 2 (θσµ ψ)(σµ ) α α (65) ψ α χ β = 1 2 ϵ αβψχ + 1 2 (ψσµν χ)(σ µν ϵ) αβ ψ α χ β = 1 2 ϵ α β ψ χ + 1 2 ( χ σµν ψ)(ϵ σµν ) β α. (66) Let us prove (65). The left hand side of (65) is a product of dotted and undotted spinors. Since we can assume ( 1 2, 0) (0, 1 2 ) = (1 2, 1 2 ) where A µ is a four-vector. Multiplying (67) by σ ν αα we obtain 1.8 Partial derivatives θ α ψ α = A µ σ µα α, (67) A µ = 1 2 θσµ ψ. Partial derivatives with respect to the anticommuting variables are denoted by α = θ α, α = θ α, α = θ α, α = θ α. (68) Indices on partial derivatives are lowered or raised by ϵ symbol. Note that, there is one extra minus sign: The following derivatives are frequently used: α = ϵ αβ β, α = ϵ αβ β α = ϵ α β β, α = ϵ α β β. (69) θ β θ α = αθ β = δ β α, θ β θ α = αθ β = ϵ βα (70) α θ β = δ β α, α θ β = ϵ β α (71) 9

Complex conjugation of derivative is defined by This is in accordance with definition of conjugation 2 Supersymmetric algebra 2.1 N = 1 Super Poincare algebra α (θθ) = 2θ α (72) α ( θ θ) = 2 θ α (73) α (θθ) = 2θ α (74) α ( θ θ) = 2 θ α (75) ( )(θθ) = α α (θθ) = 4 (76) ( )( θ θ) = α α ( θ θ) = 4. (77) ( α ) = α. (78) δ β α = ( αθ β ) = θ β( α ) = ( α ) θ β = α θ β. (79) To start with, let us state the Coleman-Mandula theorem: Only possible Lie symmetry of a nontrivial S matrix is a direct product of Poincare and internal symmetry. The assumptions in Coleman-Mandula theorem are the following: the particles are massive, theory is local, unitary and relativistic covariant. Thus, generators of the Lie symmetry are: momenta, P µ, angular momenta, M µν and bosonic generators B r. These bosonic generators are Lorentz scalars. This theorem can be evaded using the graded Lie symmetry instead of the usual Lie symmetry. The graded Lie algebra is a generalization of Lie algebra, containing bosonic and fermionic generators. The graded vector space (precisely Z 2 graded) is a direct sum V = L 0 L 1. The elements of L 0 are bosons or even elements, while the elements of L 1 are fermions or odd elements. The grading is defined by { 0, X L 0 X =. (1) 1, X L 1 Between elements of the graded vector space Lie product [, } is defined, with the following properties: 1. [X, Y } = ( ) X Y [Y, X}, 2. ( ) X Z [X, [Y, Z}} + ( ) Y X [Y, [Z, X}} + ( ) Z Y [Z, [X, Y }} = 0. (2) 10

The Lie product of two bosons or one fermion and one boson is a commutator. In the case two fermions Lie product is an anticommutator, i.e. [B 1, B 2 } = [B 1, B 2 ], [B 1, F 1 } = [B 1, F 1 ], [F 1, F 2 } = {F 1, F 2 }. (3) The second property is super-jacobi identity. Super Poincare algebra is an extension of Poincare algebra which includes the fermionic generators in addition to the bosonic ones. These fermionic generators transform bosonic state into fermionic and vice verse. The fermionic generators are left-handed and right-handed spinors, Q α and Q α. This statement is known as the Haag Loppusanski Sonius theorem. The Lie product of generators is a commutator or anticommutator, and it must be linear in terms of generators. First, we have to find [P µ, Q α ]. We assume that From this expression it follows that Applying super Jacobi identity we obtain [P µ, Q α ] = cσ µ Q β α β. (4) [P µ, Q β] = c σ µ βγ Q γ. (5) [P µ, [P ν, Q α ]] + [P ν, [Q α, P µ ]] + [Q α, [P µ, P ν ]] = 0 c 2 (σ νµ ) β α Q β = 0, and finally we conclude c = 0, i. e. [P µ, Q α ] = 0. Since Q α is a Weyl spinor, it transforms as follows ) β e i 2 ωµν M µν Q α e i 2 ωµν M µν = (e 1 2 ωµν σ µν Q β. Expanding the previous expression in the first order in ω µν we obtain In the same way, one can show that α [M µν, Q α ] = i(σ µν ) β α Q β. (6) [M µν, Q α ] = i( σ µν ) α β Q β. (7) Let us find the anticommutator {Q α, Q β }. Due to index structure this anticommutator has the form {Q α, Q β } = d(σ µν ) α β M µν, (8) where d is an arbitrary constant. Applying the super Jacobi identity [P µ, {Q a, Q β }] + {Q α, [Q β, P µ ]} {Q β, [P µ, Q α ]} = 0 11

we find d = 0. Thus, {Q α, Q β } = 0. Finally, we have to find {Q α, Q β}. Taking into account the index structure of this expression we find The factor 2 is fixed by a convection. Simple (or N = 1) Super Poincare algebra is {Q α, Q β} = 2σ µ α β P µ. (9) [P µ, P ν ] = 0, (10) [M µν, P ρ ] = i(g νρ P µ g µρ P ν ), (11) [M µν, M ρσ ] = i(g µρ M νσ + g νσ M µρ g νρ M µσ g µσ M νρ ), (12) {Q α, Q β } = { Q α, Q β} = 0, {Q α, Q β} = 2σ µ α β P µ, (13) [Q α, P µ ] = [ Q α, P µ ] = 0, [Q α, M µν ] = i(σ µν ) β α Q β, [ Q α, M µν ] = i( σ µν ) α β Q β. 2.2 Extended Super Poincare algebra Fermionic generators can have an additional index coming from some internal symmetry group: Q A α (A = 1,..., N ). In this case N = 1 supersymmetry is extended to the N extended supersymmetry: [P µ, P ν ] = 0, (14) [M µν, P ρ ] = i(g νρ P µ g µρ P ν ), [M µν, M ρσ ] = i(g µρ M νσ + g νσ M µρ g νρ M µσ g µσ M νρ ), {Q A α, Q B β } = ϵ αβ Z AB, { Q Ȧ α, Q Ḃ β } = ϵ α β Z AB, {Q A α, Q Ḃ β } = 2δAB σ µ α β P µ, [Q A α, P µ ] = [ Q Ȧ α, P µ ] = 0, [Q A α, M µν ] = i(σ µν ) α β Q A β, [ Q αa, M µν ] = i( σ µν ) α β Q βa [B r, B s ] = if rst B t [B r, Q A α] = (b r ) A CQ C α [B r, Q Ȧ α] = Q αc (b r ) C A (15) Central charge, Z AB = Z BA commutes with all generators in the algebra. It is a linear combination of the internal symmetry generators, Z AB = λ ABr B r. The largest possible internal 12

symmetry group is U(N). If Z = 0, SUSY algebra is invariant under Q A α U A B Q B α. If Z 0, {Q, Q} relation spoils this invariance. In this case the algebra is invariant under some subgroup of U(N). In N = 1 SUSY, central charge is zero, Z = 0: there is only one index, so Z can t be antisymmetric. In this case, U(1) symmetry is called R-symmetry: 3 Superspace and Superfields [Q α, R] = Q α, [ Q α, R] = Q α. (16) In this section we introduce the concept of superspace. Fields on superspace are called superfields. Supersymmetry is realized as a translation in superspace. A superspace is an extension of Minkowski space-time where we add four Grassmann coordinates θ α, θ α to the four bosonic Minkowski coordinates. These coordinates satisfy [x µ, x ν ] = [x µ, θ α ] = [x µ, θ α ] = 0 {θ α, θ α } = {θ α, θ β } = { θ α, θ β} = 0. (1) Thus, superspace is parametrized by the coordinates: x µ, θ α, θ α. For later convenience we calculate [ξq, θ Q] = ξ α Q α θ α Q α θ α Q α ξ α Q α = ξ α θ α Q α Q α + θ α ξ α Q α Q α = ξ α θ α {Q α, Q α } = ξ α θ α {Q α, Q α } = 2ξ α σ µ α α θ α P µ. (2) In this part of Lectures, SUSY generators are denoted by Q, Q. Any element of super Poincare group can be written in the form We use the notation Actually, G(x, θ, θ) is an element of the coset space e i(xµ P µ+θ α Q α+ θ α Q α) e 1 2 ωµν M µν. (3) G(x, θ, θ) = e i(xµ P µ +θ α Q α + θ α Q α). (4) Super Poincare group/lorentz group. A superspace can be realized as a coset space. For details, see [1, 5]. The product of two elements, G(a, ξ, ξ)g(x, θ, θ) is found applying Baker Campbell Hausdorff formula and (2). The result is e A e B = e A+B+ 1 2 [A,B]+... G(a, ξ, ξ)g(x, θ, θ) = G(x µ + a µ + iξσ µ θ iθσ µ ξ, θ + ξ, θ + ξ). (5) 13

The left multiplication in the group induces the following transformation of supercoordinates: x µ x µ + δx µ = x µ + a µ + iξσ µ θ iθσ µ ξ θ θ + ξ θ θ + ξ, where ξ is a constant parameter of SUSY transformation. Under infinitesimal translation x = x + a a classical scalar field φ(x) transforms as φ(x) φ (x) = φ(x a). The form-variation of the scalar field is given by δφ(x) = φ (x) φ(x) = a µ µ φ(x). (6) From δφ(x) = ia µ P µ φ we conclude that P µ = i µ is a representation of momenta in the space of functions. The momentum is represented as a differential operator in accordance to quantum mechanics. In QFT a scalar field is an operator in Hilbert space. Transformation law under translations is given by φ(x) φ (x) = e iaµ P µ φ(x)e iaµ P µ = φ(x a), (7) where P µ are four-momenta. Then we obtain δφ = φ (x) φ(x) = i[a µ P µ, φ] = a µ µ φ. (8) Generally, infinitesimal variation of field under transformations of symmetry is given by δφ = i[ d 3 xj 0, φ], (9) where j 0 is zero component of the Noether current density. Variations of a quantum and classical field are related by the correspondence principle. If the canonical Poisson bracket of two observable gives a third observable {A, B} = C, then after quantization the commutator of corresponding operators have to satisfy i[â, ˆB] = Ĉ. For details see the book [6]. A scalar superfield is a function of superspace coordinates, F = F (x, θ, θ). A superfield can be expanded in power series of θ and θ as F (x, θ, θ) = f(x) + θϕ(x) + θ χ(x) + θθm(x) + θ θn(x) + θσ µ θvµ (x) +θθ θ λ(x) + θ θθφ(x) + θθ θ θd(x). (10) This expansion consists of finite number of terms, due to the anticommuting properties of θ, θ coordinates. The coefficients f(x), ϕ(x), χ(x), m(x), n(x), v µ (x), λ(x), φ(x) and d(x) in the expansion (10) are usual fields in Minkowski space time. Superfield (10) contains 16 fermionic 14

and 16 bosonic degrees of freedom. There is an equal number of bosonic and fermionic degrees of freedom in any supersymmetric multiplet. A superfield has to satisfy some transformation properties. Under super Poincare transformation 2 x µ x = x µ δx µ = x µ a µ iξσ µ θ + iθσ µ ξ a classical scalar superfield transforms as θ θ = θ ξ θ θ = θ ξ, (11) The infinitesimal variation of the superfield is given by F (x, θ, θ) = F (x + δx, θ + ξ, θ + ξ). (12) δf = F (x, θ, θ) F (x, θ, θ) where we introduce the differential operators: = F (x + a µ + iξσ µ θ iθσ µ ξ, θ + ξ, θ + ξ) F (x, θ, θ) = (a µ + iξσ µ θ iθσ µ ξ) µ F + ξ α α F + ξ α α F = (a µ µ + ξ α Q α + ξ α Q α )F, (13) From 3 δf = i(a µ P µ + ξ α Q α + ξ α Q α )F it follows Q α = α + i(σ µ θ)α µ Q α = α + i σ µ αα θ α µ. (14) P µ = i µ, (15) Q α = iq α = i α σ µ α α θ α µ, (16) Q α = i Q α = i α ( σ µ θ) α µ. (17) One can check that Q α = (Q α ) = ϵ α β Q β = i Q α, where Q α = α (θσ µ ) α µ. The operators P µ, Q α and Q α satisfy super Poincare algebra, as we expect. For example, {Q α, Q α } = {Q α, Q α } = 2σ µ α α i µ = 2σ µ α α P µ. (18) Under supertanslation (11) the quantum superfield F (x, θ, θ) transforms as 2 Note minus signs in coordinate transformations. 3 The parameters are a, ξ, ξ. e i(aµ P µ +ξ α Q α + ξ α Q α) F (x, θ, θ)e i(aµ P µ +ξ α Q α + ξ α Q α ) = F (x + a µ + iξσ µ θ iθσ µ ξ, θ + ξ, θ + ξ) = F + δf. (19) 15

Expanding the first line of (19) we obtain From the second line in (19) we again obtain e i(aµ P µ +ξ α Q α + ξ α Q α) F (x, θ, θ)e i(aµ P µ +ξ α Q α + ξ α Q α) = F (x, θ, θ) + i[a µ P µ + ξ α Q α + ξ α Q α, F ]. (20) δf = (a µ µ + ξ α Q α + ξ α Q α )F. Under the SUSY infinitesimal transformation the variation of of quantum superfields is given by δ ξ F = i[ξ α Q α + ξ α Q α, F ] = (ξ α Q α + ξ α Q α )F. (21) Applying (10) we find SUSY variations for components in supermultiplet: δ ξ f = ξϕ + ξ χ, (22) δ ξ ϕ α = 2ξ α m + σ µ ξ ) α α α (v µ i µ f, (23) δ ξ χ α = 2 ξ α n + ξ α σ µ α α( vµ + i µ f ), (24) δ ξ m = ξ λ i 2 ξ σ µ µ ϕ, (25) δ ξ n = ξφ i 2 ξσµ µ χ, (26) δ ξ v µ = ξσ µ λ + φσµ ξ i 2 ξσ ν σ µ ν ϕ + i 2 ξ σ ν σ µ ν χ, (27) δ ξ λ α = 2 ξ α d i( σ µ ξ) α ( µ m) i 2 ( σν σ µ ξ) α µ v ν, (28) δ ξ φ α = 2ξ α d iσ µ α α ξ α µ n + i 2 σν α α σ µ αβ ξ β µ v ν, (29) δ ξ d = i 2 ξσµ µ λ + i 2 µφσ µ ξ. (30) Note that a SUSY variation of the highest component, d(x) is a total derivative. Let us apply two successive supersymmetric transformations on a superfield, δ η δ ξ F. Transformation with the parameter ξ, is followed by the second transformation with parameter η. The second transformation acts only on the field F, i.e. δ η δ ξ F = δ η (ξq + ξ Q)F = (ξq + ξ Q)(ηQ + η Q)F. A commutator of two SUSY transformations can be found easily. Result is given by [δ η, δ ξ ]F = 2i(ξσ µ η ησ µ ξ) µ F. (31) We see that commutator of two SUSY transformations is a translation, i.e. the supersymmetric algebra is closed. 16

3.1 Covariant Fermionic Derivatives Supercovariant derivatives are defined by D α = α iσ µ α α θ α µ, (32) D α = α + iθ α σ µ α α µ. (33) With a help of ( α ) = α and (σ µ α β ) θ β = σ µ β α θβ we get (D α ) = D α. Derivatives with upper indices are The following anticommutation relations are very useful: D α = ϵ αβ D β = α + i θ β σ µ βα µ D α = α iσ µ αα θ α µ. (34) {D α, D β } = { D α, D β} = {D α, Q β } = {D α, Q α } = { D α, Q α } = { D α, Q β} = 0, (35) One can check that: {Q α, Q α } = {D α, D α } = 2iσ µ α α µ. (36) D 2 = D α D α = α α 2iσ µ α α θ α µ α + θ θ, (37) D 2 = D α D α = α α 2i σ µ αα θ α µ α + θθ. (38) 3.2 Chiral and antichiral superfields An arbitrary superfield (10) is a reducible supersymmetric multiplet. One can impose some constraints on the superfield to get irreducible representations of supersymmetry. A chiral superfield Φ(x, θ, θ) is defined by the following constraint D α Φ(x, θ, θ) = 0, and it is a irreducible superfield. This constraint is compatible with supersymmetry, δ ξ ( D α Φ) = D α (δ ξ Φ), (39) since supercovariant derivative anticommutes with supercharges. We now want to solve this constraint. It can be proven that D α F = χ α + θ α σ µ α α (vµ + i µ f) + 2n θ α + (θθ) ( λ α i2 ) σµα α µϕ α + (θσ µ θ) ( σ µβ α φ β i ) 2 ( ν χ σ µ σ ν ) α The condition D α F = 0 implies + 2(θθ) θ α d i 2 ϵαβ (θθ)σ µ α α σν β β θ β µ v ν + i( θ θ)θ α σ µ α α µn i 2 (θθ)( θ θ)σ µ α α µφ α. (40) χ = 0, n = 0, v µ = i µ f, φ = 0, λ α = i 2 σµ α α µϕ α, d = 1 f. (41) 4 17

Introducing new notation: a general expression for a chiral superfield is f A, ϕ 2ψ, m F (42) Φ(x, θ, θ) = A(x) + 2θψ(x) + θθf (x) iθσ µ θ µ A(x) + i 2 (θθ)( µ ψ(x)σ µ θ) 1 4 (θθ)( θ θ) A(x). (43) The chiral supermultiplet consists of a complex scalar field A (2 d.o.f.), a left-handed Weyl-spinor ψ α (4 d.o.f.) and a complex scalar field F (2 d.o.f). Notice that the number of fermionic and bosonic d.o.f. is equal. By introducing chiral supercoordinate: we can rewrite the chiral superfield in the form: y µ = x µ i(θσ µ θ), (44) Φ(y, θ) = A(y) + 2θψ(y) + θθf (y). (45) From general transformation rules (22-30), one can easily find transformation laws for components of a chiral multiplet: δ ξ A = 2ξψ, δ ξ ψ α = 2F ξ α i 2 µ Aσ µ α α ξ α, δ ξ F = i 2 µ ψσ µ ξ. (46) The components A, ψ and F of a chiral multiplet transform into each other, so the multiplet is irreducible. Also, we see that θθ-component of a chiral superfield transforms into itself plus a divergence term. θθ component of superfield is called F term. An antichiral superfield Ψ(x, θ, θ) is determined by the condition D α Ψ(x, θ, θ) = 0. (47) If Φ(x, θ, θ) is a chiral superfield, than Φ (x, θ, θ) is antichiral. An antichiral superfield can be expressed in y = x µ + i(θσ µ θ) coordinate as follows Φ(x, θ, θ) = A (y ) + 2 θ ψ(y ) + θ θf (y ). (48) Rewriting the antichiral superfield in terms of x, θ, θ coordinates we find Φ(x, θ, θ) = A + 2 θ ψ + θ θf + i µ A θσ µ θ i ( θ θ)(θσ µ 1 µ ψ) 2 4 A (θθ)( θ θ). (49) 18

3.3 Products of Chiral and Antichiral SF If Φ i and Φ j are two chiral superfields, then their product Φ i Φ j is again a chiral superfield since D α (Φ i Φ j ) = ( D α Φ i )Φ j + Φ i D α Φ j = 0. In components this product is ( Φ i Φ j = A i (y) + )( 2θψ i (y) + θθf i (y) A j (y) + ) 2θψ j (y) + θθf j (y) = A i A j (y) + 2θ(A i ψ j + A j ψ i )(y) + θθ(f i A j + F j A i ψ i ψ j )(y), (50) where we apply (60). So, from two chiral multiplets we get another chiral multiplet with components (A i, ψ i, F i ) (A j, ψ j, F j ) = (A i A j, A i ψ j + A j ψ i, A i F j + F i A j ψ i ψ j ). (51) And triple product of chiral multiplets is again a chiral multiplet: (A i, ψ i, F i ) (A j, ψ j, F j ) (A k, ψ k, F k ) = (A i A j, A i ψ j + A j ψ i, A i F j + F i A j ψ i ψ j ) (A k, ψ k, F k ) = (A i A j A k, A i ψ j A k + A j ψ i A k + A i A j ψ k, A i A j F k A i ψ j ψ k + ciclic ijk). (52) The product of a chiral and an anti-chiral superfied is ΦΦ = A A + 2A θψ + 2A θ ψ + θθa F + θ θf A + θσ µ θ( ia µ A + ia µ A + ψσ µ ψ) + i A (θθ)( µ ψσ µ θ) i + 2(θθ)( θ ψ)f (θθ)(ψσ µ θ) µ A 2 2 + i ( θ θ)(θσ µ ψ) µ A + 2( θ θ)(θψ)f i ( θ θ)(θσ µ µ ψ)a 2 2 ( + (θθ)( θ θ) 1 4 A A 1 4 A A i 2 µψσ µ ψ + F F + 1 2 µa µ A + i ) 2 ψσµ µ ψ. (53) The product ΦΦ is neither a chiral nor the anti-chiral field, since it do not satisfy any of constrains introduce above. This superfield is a real field. Its highest component θθ θ θ is known as D term and it transforms as space time derivative under SUSY. 3.4 Wess Zumino model We want to construct a Lagrangian that is SUSY invariant and renormalizable for chiral and anti-chiral fields. First, we consider only one superfield Φ. The ΦΦ θθ θ θ term is a kinetic term ΦΦ θθ θ θ = 1 4 A A 1 4 A A i 2 µψσ µ ψ + F F + 1 2 µa µ A + i 2 ψσµ µ ψ = A A + i ψ σ µ µ ψ + F F, (54) where we discard surface terms. The term ΦΦ θθ θ θ is SUSY invariant. However, it is itself a derivative term, so it does not contribute to the Lagrangian. Consider therefore the SUSY invariant term ΦΦ θθ : ΦΦ θθ = 2AF ψψ. (55) 19

There is one more renormalizable term, namely Φ 3 θθ : Φ 3 θθ = 3A 2 F 3Aψψ. (56) The higher order terms in chiral fields are not renormalizable. Thus, SUSY invariant Lagrangian is given by L = ΦΦ ( 1 θθ θ θ + 2 mφ2 θθ + 1 ) 3 λφ3 θθ + h.c., (57) where m and λ are coupling constants is SUSY invariant. In terms of component fields, the Lagrangian reads: L = A A + i ψ σ µ µ ψ + F F + m 2 (2AF ψψ) + m 2 (2A F ψ ψ) + λ(a 2 F Aψψ + A 2 F A ψ ψ). (58) This is the Lagrangian that is SUSY invariant and renormalizable. It is called Wess-Zumino model. This model was constructed in 1974 by J. Wess and B. Zumino. It was the first four dimensional supersymmetric model. The fields F and F are auxiliary fields, since Lagrangian does not contain their derivatives. These fields can be eliminated from Lagrangian using the equation of motion F = ma λa 2. In this way we obtain so called on-shell Lagrangian: L = µ A µ A + i ψ σ µ µ ψ ma + λa 2 2 1 2 mψψ 1 2 m ψ ψ λaψψ λa i ψ ψ. (59) More general, with more than one chiral superfield, Lagrangian is L = Φ i Φ i θθ θ θ + (W [Φ i ] θθ + h.c.), (60) where the superpotential W is an analytic function of the chiral superfields: W [Φ i ] = 1 2 m ijφ i Φ j + 1 3 λ ijkφ i Φ j Φ k. (61) The quantities m ij and λ ijk are some totally symmetric constant coefficients. Lagrangian in terms of component fields reads L = µ A i µ A i +i ψ ( i σ µ µ ψ i +Fi F i + m ij A i F j 1 ) 2 m ijψ i ψ j +λ ijk A i A j F k λ ijk A i ψ j ψ k +c.c. (62) The previous Lagrangian gives the following equations of motion A i = m ij F j + 2λ ijk A jf k λ ijk ψj ψk i σ µ µ ψ i = m ij ψj + 2λ ijk A j ψ k F i = m ij A j λ ijk A ja k. (63) 20

The auxiliary fields F i can be integrated out using equations of motion (63). In that way we obtain L = µ A i µ A i + i ψ i σ µ µ ψ i m ij A j + λ ijk A j A k 2 1 2 m ijψ i ψ j 1 2 m ij ψ i ψj λ ijk A i ψ j ψ k λ ijk A i ψ j ψk. (64) The last two terms are Yukawa interaction and we can define effective potential as: 3.5 Vector superfield V m ij A j + λ ijk A j A k 2 = F i 2. (65) An irreducble multiplet can be obtained imposing the reality condition V (x, θ, θ) = V (x, θ, θ). (66) This superfield is co called the vector superfield. It has the following form V (x, θ, θ) = C(x) + 2θχ(x) + 2 θ χ(x) + θθm(x) + θ θm (x) + θσ µ θvµ (x) i ) ( + (θθ) θ ( λ(x) σ µ µ χ(x) + ( θ θ)θ λ(x) i ) σ µ µ χ(x) 2 2 + 1 (D(x) 2 (θθ)( θ θ) 1 ) 2 C(x). (67) The reality condition implies C = C, v µ = v µ and D = D. A vector multiplet consists of 8 bosonic and 8 fermionic degrees of freedom. The canonical dimension of a vector field is one, [v µ ] = 1. Knowing [θ] = [ θ] = 1 2 implies and subsequently [V ] = 0, (68) [C] = 0, [χ] = [ χ] = 1 2, [M] = [M ] = 1, [λ] = [ λ] = 3, [D] = 2. (69) 2 Let iλ be a chiral SF: iλ = f + 2θφ + θθf i(θσ µ θ) µ f The following combination is a vector SF: i 2 θθ( θ σ µ µ φ) 1 4 (θθ)( θ θ) f. (70) iλ iλ = 2Ref + 2θφ + 2 θ φ + θθf + θ θf + 2(θσ µ θ) µ Imf i (θθ)( θ σ µ µ φ) i ( θ θ)(θσ µ µ φ) 1 2 2 2 (θθ)( θ θ) Ref. (71) We define a super-gauge transformation for the vector superfield in the following way: V (x, θ, θ) = V (x, θ, θ) + i(λ(x, θ, θ) Λ (x, θ, θ)), (72) 21

where Λ(x, θ, θ) is a chiral superfield. Under the super gauge transformations the components of a vector superfield transform as C C + 2Ref, χ χ + φ, M M + F, v µ v µ + µ (2Imf), λ λ, D D. (73) The fields λ and D are gauge invariant, while v µ transforms as the usual gauge field with U(1) gauge parameter 2Imf. This means that the super-gauge symmetry is larger then usual gauge symmetry. The super-gauge symmetry can be fixed by the following choice of parameters 2Ref = C, φ = χ, F = M. (74) This gauge choice is known as Wess-Zumino gauge. In the WZ gauge a vector superfield is given by V WZ = θσ µ θvµ + (θθ)( θ λ) + ( θ θ)(θλ) + 1 2 (θθ)( θ θ)d. (75) Supersymmetry does not preserve WZ gauge, i.e. a supersymmetric transformation of a vector superfield in Wess-Zumino gauge gives a superfield which is not in this gauge. However, a suitable combination of supersymmetry transformation followed by a super gauge transformation preserves the Wess-Zumino gauge. The fields in the vector multiplet are: a vector gauge field v µ, a gaugino, λ α and an auxiliary field, D. We can perform further transformations that leave us within the WZ gauge with iλ = iimf + θσ µ θ µ Imf i 4 (θθ)( θ θ) Imf. (76) This transformation will not change the conditions C = χ = M = 0, and gives v µ v µ + µ (2Imf), λ λ, D D, (77) which is just the ordinary U(1) gauge transformation. It easy to prove that V 2 W Z(x, θ, θ) = 1 2 (θθ)( θ θ)v µ (x)v µ (x) (78) V n W Z(x, θ, θ) = 0, n 3. (79) So, we find e V WZ = 1 + V WZ + 1 2 V WZ 2 = 1 + θσ µ θvµ + (θθ)( θ λ) + ( θ θ)(θλ) + 1 (D 2 (θθ)( θ θ) + 12 ) vµ v µ. (80) 22

3.6 Lagrangian for Abelian gauge theory In order to construct a kinetic term for a vector superfield, we have to find a supersymmetric analog of the field strength. Abelian field strength superfields are W α = 1 4 D 2 D α V, W α = 1 4 D2 D α V. (81) One can check that both W α and W α are gauge invariant quantities: W α = 1 4 D 2 D α V 1 4 D 2 D α (V + iλ iλ ) = W α i 4 D 2 D α Λ = W α i 4 D β D βd α Λ = W α i 4 D ( β { D β, ) β D α } D α D Λ = W α + i 4 D β{ D β, D α }Λ = W α + i ( ) β D 2i(σ µ ) 4 α β µ Λ = W α 1 2 (σµ ) α β µ D βλ = W α (82) and satisfy D α W α = D α W α = 0. W α and W α are a chiral and an antichiral spinorial superfields. It can be proven that W α = λ α (y) + θ α D(y) + i(θθ)σ µ α α µ λ α (y) i(σ µν θ) α F µν (y), (83) W α = λ α (y ) + θ α D(y ) iϵ α β( σ µν θ) βf µν (y ) i( θ θ)( µ λ(y )σ µ ) α, (84) where F µν = µ v ν ν v µ. It can be shown that W α W α θθ = iλσ µ µ λ i µ λ σ µ λ + D 2 1 2 F µν F µν + i 1 2 F µν Fµν, (85) where F µν = 1ϵ 2 µνρσf ρσ is the dual tensor of F µν. The last term in (85) is a total derivative. Lagrangian for the pure super Abelian gauge theory is given by L = 1 ( W α θθ W α + 4 W ) α α W = 1 θ θ 4 F µν F µν + iλσ µ 1 µ λ + 2 D2, (86) up to a total derivative term. The first term in (86) is Maxwell action. Next, let us consider a coupling of the Abelian gauge superfield, V to chiral superfields, Φ i Under the Abelian super-gauge transformation the chiral and antichiral superfields transform as Φ i = e 2igt iλ Φ i, (87) 23

Φ = e 2igt iλ Φi, (88) where t i is (real) charge of superfield Φ i. The term Φ i Φ i is not super gauge invariant, but the term Φ i e 2gt iv Φ i is: Φ i e 2gt iv Φ i Φ i e 2igt iλ (z) e 2gt i(v +iλ iλ ) e 2igt iλ(z) Φ i (89) = Φ i e 2gt iv Φ i. (90) In the WZ gauge only three terms in expression Φ i e 2gt iv Φ i do not vanish: Φ i e 2gt iv Φ i W Z = Φ i Φ i + 2gt i Φi V W Z Φ i + 2g 2 t 2 i Φ i V 2 W ZΦ i. (91) It can be shown that the highest component of Φ i e 2gt iv Φ i is given (up to total derivative terms) by Φ i e 2gt iv W Z Φ i θθ θ θ = Fi F i + (D µi A i )(D µ A i ) + iψ i σ µ D ψ µi i where we introduce covariant derivatives as The action is S = d 4 x i + gt i D A i 2 g 2[( ψ i λ)ai + (ψ i λ)a i ], (92) D µi A i = ( µ + igt i v µ )A i (93) D µi ψ i = ( µ + igt i v µ )ψ i. (94) ( Φ i e 2gtiV Φ i θθ θ θ + 1 4 W α W α θθ + 1 ) 4 W α W α θ θ + W [Φ i ] θθ + W [ Φ i ] θ θ. (95) The superpotential W (Φ) has to respect super-gauge symmetry. Now, we can write down the full Lagrangian in components forms L = Fi F i + (D µ i A i) (D µi A i ) + iψ i σ µ D ψ µi i + gt i DA i A i 2gt i [( ψ i λ)ai + (ψ i λ)a i ] + 1 2 D2 1F 4 µνf µν + iλσ µ µ λ W +F i A i 1 2 W 2 A i A j ψ i ψ j + Fi W 1 2 W ψ 2 i ψj. (96) A i A i A j Here, we can use the equations of motion to integrate out auxiliary fields: F i = W A i F i = W A i D = gt i A i A i. (97) Substituting these results in (96) we obtain the on-shell Lagrangian: L = W 2 φ i + (D µ i A i) (D µi A i ) + iψ i σ µ D ψ µi i 1(gt 2 ia i A i ) 2 2gt i [( ψ i λ)ai + (ψ i λ)a i ] 1F 4 µνf µν + iλσ µ µ λ 1 2 W 2 A i A j ψ i ψ j 1 2 2 W A i A j In this theory A is a scalar, λ is a gaugino and ψ is a chiral fermion. 24 ψ i ψj. (98)

3.7 Super Quantum Electrodynamics Super QED is a supersymmetric extension of the standard QED. Let Φ + = (A +, ψ +, F + ) and Φ = (A, ψ, Φ ) be two chiral superfields with charges q and q. Under U(1) super gauge transformations they transform as follows Φ + Φ + = e 2iqΛ(z) Φ +, (99) Φ Φ = e 2iqΛ(z) Φ. (100) The action is given by S = d 8 z( Φ+ e 2qV Φ + + Φ e 2qV Φ + δ (2) ( θ)( 1 4 W α W α + mφ + Φ ) ( ) 1 ) + δ (2) (θ) 4 W α W α + m Φ + Φ. (101) Note that the mass terms mφ ± Φ ± are absent since they are not the super gauge invariant. The Lagrangian in component form is given by L = F + 2 + D µ A + 2 + iψ + σ µ D µ ψ + 2q[(ψ + λ)a + + h.c.] + qa +A + D + F 2 + D µ A 2 + iψ σ µ D µ ψ + 2q[(ψ λ)a + h.c.] qa A D + 1 2 D2 1 4 F µνf µν + iλσ µ µ λ + m(a+ F + A F + ψ + ψ ) + m(a +F + A F + ψ + ψ ). Substituting the equations of motion into Lagrangian in (102) we obtain the on-shell Lagrangian: where (102) F ± = ma (103) D = qa +A + + qa A (104) L = 1 4 F µνf µν + iλσ µ µ λ + iψ+ σ µ D µ ψ + + iψ σ µ D µ ψ + D µ A + 2 + D µ A 2 q 2 (A +A + A A ) m 2 (A +A + φ A ) m(ψ + ψ + ψ + ψ ) 2q[(ψ + λ)a + + ( ψ + λ)a+ (ψ λ)a ( ψ λ)a ], (105) We introduce Dirac spinor for matter fields: D = µ + iqa µ. (106) Ψ = ( (ψ+ ) α ( ψ ) α ) (107) 25

and the Majorana spinor for gaugino field λ M = ( λα λ α ) In terms of Dirac and Majorana spinors Lagrangian (105) reads L = 1 4 F µνf µν + i 2 λ Mγ µ µ λm + i Ψγ µ D µ Ψ m ΨΨ. (108) + D µ A + 2 + D µ A 2 m 2 ( A + 2 + A 2 ) q 2 ( A + 2 + A 2 ) 2 2q(A + λ M Ψ L A λm Ψ R + c.c). (109) 3.8 Non-Abelian Super Yang-Mills theories In the previous section we discussed the supersymmetric U(1) gauge theories. Now, we generalize the Abelian symmetry to the non Abelian one. Let us consider a simple connected group G. The hermitian generators of G are T a and they satisfy [T a, T b ] = if abc T c, (110) where f abc are structure constants. Chiral superfields transform under rigid transformations as follows Φ I = exp( igλ a T a ) IJ Φ J, where Λ a are constant real parameters and g is a coupling constant. The kinetic term Φ I Φ I is invariant under the rigid transformations. To localize this symmetry in the superspace, the parameters Λ a have to become chiral functions. The superfields, Φ I and Φ I transform under gauge transformation in the following way Φ I = exp( igλ) IJ Φ J, Φ I = Φ J exp( igλ ) JI, (111) where Λ = 2gΛ a (x, θ, θ)t a, Λ = 2gΛ a (x, θ, θ)t a are the chiral and antichiral superfields. The term Φ I Φ I is not gauge invariant. To recover invariance we introduce a vector (or gauge or Yang-Mills) superfield V = 2gV a T a. It is a Hermitian matrix transforming under the adjoint representation of G. In order for the term Φ I (e V ) IJ Φ J to be invariant under super-gauge transformation, the super gauge transformation of the vector superfield has to be given by e +V = e iλ e +V e +iλ. (112) Writing V = V + δv, where δv is an infinitesimal change of vector superfield in the first order in Λ, we obtain e V +δv e V = δv + 1 2 (V δv + δv V ) + 1 6 (δv V 2 + V δv V + V 2 δv ) +.... (113) On the other hand, from (112) it follows e V +δv e V = e V iλ iλ e V = iλ iλ + V iλ iλ V + i 2 V 2 Λ i 2 Λ V 2 +.... (114) 26

To solve these equations we assume that δv is a expansion in powers of V δv = δv (0) + δv (1) + δv (2) +.... (115) The term δv (n) is n th order in power of the vector superfield. Comparing (113) and (114) we find Collecting these terms together we obtain or in component notation δv (0) = i(λ Λ ), δv (1) = i 2 [V, Λ + Λ ], δv (2) = i 12 [V, [V, Λ Λ ]]. (116) δv = i(λ Λ ) + i 2 [V, Λ + Λ ] + i 12 [V, [V, Λ Λ ]] +..., (117) δv a = i(λ Λ ) a gf abc V b (Λ + Λ ) c i 3 g2 f dbc f aed V e V b (Λ Λ ) c + O(V 3 ). (118) It can be shown that we can impose the Wess Zumino gauge V WZ = (θσ µ θ)vµ + (θθ)(θλ) + ( θ θ)(θλ) + 1 2 (θθ)( θ θ)d. (119) We use the notation v µ = 2gv a µt a, D = 2gD a T a... where v a µ are the gauge fields. The Wess- Zumino gauge does not fix super gauge symmetry completely. The super gauge transformations determined by iλ = iimf + (θσ µ θ) µ Imf i 4 (θθ)( θ θ)imf (120) preserve WZ gauge. This residual symmetry corresponds to usual gauge symmetry. The vector superfield transformation is given by: δv WZ = i(λ Λ ) + i 2 [V WZ, Λ + Λ ]. (121) To find the Lagrangian for super Yang Mills fields we introduce two new superfields W α = 1 4 D 2 ( e V D α e +V ), W α = + 1 4 D2 ( e +V D α e V ). (122) It is clear that W α and W α are chiral and antichiral superfields, respectively. The superfield strength W α can be expanded in V : W α = 1 4 D 2 D α V 1 8 D 2 [D α V, V ] 1 24 D 2 [[D α V, V ], V ] +.... (123) 27

In the Abelian case this expression reduces to W α = 1 4 D 2 D α V, as we expected. It can be shown that W α transforms covariantly under the super gauge transformations: In the Wess-Zumino gauge we have Finally, we arrive at W α = 2gW a αt a, where W α e iλ W α e +iλ. (124) W α = 1 4 D 2 D α V WZ 1 8 D 2 [D α V WZ, V WZ ]. (125) W a α = λ a α(y) + θ α D a (y) + iθθ(σ µ D µ λa (y)) α i(σ µν θ) α F a µν(y). (126) The term Tr(W α W α ) is super gauge and Lorentz invariant. Its θθ component is invariant under supersymmetry. Therefore, the Lagrangian for the super-non-abelian gauge field is where L = 1 ( 16g 2 k Tr W α θθ W α + W α α W θ θ ) ) (127) = 1 4 F a µνf µν a + iλ a σ µ (D µ λ) a + 1 2 Da D a, (128) (D µ λ) a = µ λa gf abc v b µ λ c, F a µν = µ v a ν ν v a µ gf abc v b mv c ν, and the constant k is determined by Tr(T a T b ) = 1 2 kδab. Lagrangian (128) describes super-yang-mills theory without matter fields. The Lagrangian for matter sector is given by L matter = Φe V Φ + W (Φ) + W ( Φ), (129) (θθ)( θ θ) θθ where the superpotential W (Φ) is a gauge invariant quantity i.e. it belongs to a singlet representation of G. In the Wess-Zumino gauge the kinetic term for matter fields, Φe V Φ = Φ I (e V ) IJ Φ J is given by Φe V Φ = ΦΦ + ΦV Φ + 1 2 ΦV 2 Φ. (130) Its D-component is Φe V Φ (θθ)( θ θ) = F F + (D µ A) (D µ A) i ψ σ µ D µ ψ + ga DA 2gA (ψλ) 2g( ψ λ)a, (131) θ θ where: D µ A = ( µ + igv a µt a )A, D µ ψ = ( µ + igv a µt a )ψ. (132) 28

Let us stress that the fields A, ψ and F transform under fundamental, while D, λ and v µ under adjoint representation of G. Adding all terms together we obtain the off-shell Lagrangian: References L = 1F a 4 µνf a µν i λ a σ µ D µ λ a + (D µ A) (D µ A) i ψ σ µ D µ ψ +ga D a T a A 2gψ I λ a TIJ a A J 2g λ a ψi TIJ a A J ( ) W + F I A I 1 2 W 2 A I A J ψ I ψ J + h.c. + 1 2 Da D a + F I F I. (133) [1] J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton UP, (1991) [2] I. Buchbinder and S. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity, Taylor and Francis (1998) [3] P. Srivastrava, Supersymmetry, Superfields and Supergravity, Adam Hilger (1986) [4] D. Bailin and A. Love, Supersymmetric Gauge Field Theory and String Theory, IOP (1994) [5] B. Allanach and F. Quevedo, Supersymmetry and Extra Dimensions, Cambridge Lecture Notes (2015) [6] D. Freedman and A. van Proeyen, Supergravity, CUP (2012) [7] M. Dress, R. M. Godbole and P. Roy, Theory and Phenomenology of Sparticles, World Scientific (2008) 29