Lecture 2 The Wess-Zumino Model

Transcript

1 Lecture 2 The Wess-Zumino Model

2 Outline Review: two component spinors. The simplest SUSY Lagrangian: the free Wess-Zumino model. The SUSY algebra and the off-shell formalism. Noether theorem for SUSY: the supercurrent. Interactions in the WZ-model: the superpotential. Reading: Terning , A.1-2

3 Two component spinors The massless Dirac Lagrangian, in four-component and two-component forms: L = iψγ µ µ ψ = iψ L σµ µ ψ L + iψ R σµ µ ψ R with γ µ = ( 0 σ µ σ µ 0 ), ψ = ( ψl ψ R ), ψ = ψ γ 0 = ( ψ R ψ L ) Convention: focus on the L-component. In other words: the index L is always implied.

4 Aside: Lorentz-transformations Under Lorentz transformation (with rotation angles θ and boost parameters β), the L and R helicities do not mix: ψ L (1 i θ σ 2 β σ 2 )ψ L ψ R (1 i θ σ 2 + β σ 2 )ψ R A simple Lorentz invariant (the Majorana mass term): χ T L ( iσ 2)ψ L = χ α ɛ αβ ψ β = χ β ψ β = χψ Check: the transposed spinor transforms as ψ T L ψ T L (1 i θ σt 2 β σt 2 ) = ψt L σ 2(1 + i θ σ 2 + β σ 2 )σ 2 Notation: ψ α = ɛ αβ ψ β, ( ψ α = ɛ αβ ) ψ β 0 1 ɛ αβ = iσ 2 =, ɛ 1 0 αβ = iσ 2 = ( )

5 Signs: careful with ordering of indices, ordering of fermions, upper/lower indices... Sample manipulation: χψ = χ α ψ α = χ α ɛ αβ ψ β = ɛ αβ ψ β χ α = ψ β ɛ βα χ α = ψχ The complex conjugate of the L -spinor transforms like ψ R : σ 2 ψ L σ 2(1 + i θ σ 2 β σ 2 )ψ L = (1 i θ σ 2 + β σ 2 )σ 2ψ L Example: since χ R ψt L σ 2 (as far as Lorentz is concerned), the Majorana mass term (above) is equivalent to χ R ψ L = χ α ψ α Conjugate spinors have dotted indices with SW to NE contraction ψ L χ R = ψ α χ α

6 The Pauli matrices have transformation properties according to the index structure σ µ α α, αα σµ Example: a Lorentz-vector is formed from two-component spinors by the combination: χ σ µ ψ = χ α σµ αα ψ α

7 The free Wess Zumino model The simplest representation of the superalgebra: the massless chiral supermultiplet. Particle content: a massless complex scalar field and a massless two component fermion (a Weyl fermion). Goal: present a field theory with this particle content, and show that it realizes the superalgebra. Proposal: where Convention for spacetime metric: S = d 4 x (L s + L f ) L s = µ φ µ φ, L f = iψ σ µ µ ψ. g µν = η µν = diag(1, 1, 1, 1)

8 The SUSY transformation Generic infinitesimal symmetry transformation φ φ + δφ ψ ψ + δψ SUSY changes bosons into fermions so we consider the transformation Remarks: δφ = ɛ α ψ α = ɛψ ɛ α is an infinitesimal parameter with spinorial indices. Mass dimensions: dim(φ) = 1, dim(ψ) = 3 2 dim(ɛ) = 1 2. SUSY changes fermions into bosons. Since the SUSY transformation parameter ɛ α has dimension ( 1 2 ), the transformation must include a derivative (recall dim( ) = 1). Lorentz invariance determines the form as δψ α = i(σ ν ɛ ) α ν φ

9 SUSY of free WZ-model The variation of the scalar Lagrangian under δφ = ɛψ: δl s = µ δφ µ φ + µ φ µ δφ = ɛ µ ψ µ φ + ɛ µ ψ µ φ The variation of the fermion Lagrangian under δψ α = i(σ ν ɛ ) α ν φ, δψ α = i(ɛσν ) α ν φ gives δl f = iδψ σ µ µ ψ + iψ σ µ µ δψ = ɛσ ν ν φ σ µ µ ψ + ψ σ µ σ ν ɛ µ ν φ = ɛ µ ( ψ µ φ ɛ µ ψ µ φ + µ ɛσ µ σ ν ψ ν φ ɛψ µ φ + ɛ ψ µ φ ). The sum of the two variations is a total derivative so the action is invariant: δs = 0

10 Remark: in the manipulations we needed the Pauli identities: [ σ µ σ ν + σ ν σ µ] β = 2η µν δ β α α [ σ µ σ ν + σ ν σ µ] β = 2η µν δ β α α These are the two component versions of the Dirac algebra {γ µ, γ ν } = 2η µν

11 SUSY Commutators Consistency: the commutator of two SUSY transformations must itself be a symmetry transformation. We need to show that it is in fact a translation, as the SUSY algebra indicates. The commutator, acting on the complex scalar: (δ ɛ2 δ ɛ1 δ ɛ1 δ ɛ2 )φ = i(ɛ 1 σ µ ɛ 2 ɛ 2σ µ ɛ 1 ) µφ The commutator, acting on the fermion: (δ ɛ2 δ ɛ1 δ ɛ1 δ ɛ2 )ψ α = i(σ ν ɛ 1 ) α ɛ 2 ν ψ + i(σ ν ɛ 2 ) α ɛ 1 ν ψ = i(ɛ 1 σ µ ɛ 2 ɛ 2σ µ ɛ 1 ) µψ α +i(ɛ 1α ɛ 2 σµ µ ψ ɛ 2α ɛ 1 σµ µ ψ). (Reorganized using the Fierz identity: χ α (ξη) = ξ α (χη) (ξχ)η α ) The last term vanishes upon imposing the fermion equation of motion. With this caveat, the commutator is a translation, with details the same for the two fields. Thus the SUSY algebra closes on-shell.

12 Counting Degrees of Freedom The fermion e.o.m. projects out half of the degrees of freedom: ( ) ( ) 0 0 σ µ ψ1 p µ ψ =, p 0 2p ψ µ = (p, 0, 0, p) 2 Counting degrees of freedom shows that SUSY is not manifest off-shell: off-shell on-shell φ, φ 2 d.o.f. 2 d.o.f. ψ α, ψ α 4 d.o.f. 2 d.o.f. Restore SUSY off-shell: add an auxiliary boson field F with Lagrangian Recount degrees of freedom L aux = F F off-shell on-shell F, F 2 d.o.f. 0 d.o.f.

13 Maintain SUSY off-shell: transform the auxiliary field, and modify the transformation of the fermion δf = iɛ σ µ µ ψ, δf = i µ ψ σ µ ɛ δψ α = i(σ ν ɛ ) α ν φ + ɛ α F, δψ α = +i(ɛσν ) α ν φ + ɛ α F Transformation of the Lagrangian: δl aux = i µ ψ σ µ ɛ F iɛ σ µ µ ψ F δ new L f = δ old L f + iɛ σ µ µ ψf + iψ σ µ µ (ɛf) = δ old L f + iɛ σ µ µ ψf i( µ ψ )σ µ ɛf + µ (iψ σ µ ɛf) The last term is a total derivative so the action S new = d 4 x L free = d 4 x (L s + L f + L aux ) is invariant under SUSY transformations: δs new = 0

14 Off-shell SUSY commutator The previous computation, without using e.o.m. (δ ɛ2 δ ɛ1 δ ɛ1 δ ɛ2 )ψ α = i(ɛ 1 σ µ ɛ 2 ɛ 2σ µ ɛ 1 ) µψ α +i(ɛ 1α ɛ 2 σµ µ ψ ɛ 2α ɛ 1 σµ µ ψ) +δ ɛ2 ɛ 1α F δ ɛ1 ɛ 2α F = i(ɛ 1 σ µ ɛ 2 ɛ 2σ µ ɛ 1 ) µψ α The point: the additional term in the SUSY transformation of the fermion, the depending on the auxiliary field, is precisely such that the last two lines cancel. Conclusion: the SUSY algebra closes for off-shell fermions.

15 Off-shell SUSY commutator II Issue: the commutator of SUSY transformations must also close when acting on the auxiliary field. Computation: (δ ɛ2 δ ɛ1 δ ɛ1 δ ɛ2 )F = δ ɛ2 ( iɛ 1 σµ µ ψ) δ ɛ1 ( iɛ 2 σµ µ ψ) = iɛ 1 σµ µ ( iσ ν ɛ 2 νφ + ɛ 2 F) +iɛ 2 σµ µ ( iσ ν ɛ 1 νφ + ɛ 1 F) = i(ɛ 1 σ µ ɛ 2 ɛ 2σ µ ɛ 1 ) µf ɛ 1 σµ σ ν ɛ 2 µ ν φ + ɛ 2 σµ σ ν ɛ 1 µ ν φ = i(ɛ 1 σ µ ɛ 2 ɛ 2σ µ ɛ 1 ) µf Conclusion: the SUSY algebra closes (δ ɛ2 δ ɛ1 δ ɛ1 δ ɛ2 )X = i(ɛ 1 σ µ ɛ 2 ɛ 2σ µ ɛ 1 ) µx for all the fields in the off-shell supermultiplet X = φ, φ, ψ, ψ, F, F

16 Noether s Theorem Noether s theorem: corresponding to every continuous symmetry, there is a conserved current. An infinitesimal transformation X X + δx of the field X that leaves the action invariant, transforms the Lagrangian to a total derivative: δl = L(X + δx) L(X) = µ V µ Identification of conserved current: ( µ V µ = δl = L X δx + L ( ) L = µ ( µ X) δx ɛ µ J µ = µ ( L Ingredient: the equation of motion ( ) L µ ( µ X) ( µ X) ) δ( µ X) ( µ X) δx V µ ) = 0 = L X

17 The conserved supercurrent, J µ α: The Supercurrent ɛj µ + ɛ J µ = L ( µ X) δx V µ = δφ µ φ + δφ µ φ + iψ σ µ δψ V µ = ɛψ µ φ + ɛ ψ µ φ + iψ σ µ ( iσ ν ɛ ν φ + ɛf) ɛσ µ σ ν ψ ν φ + ɛψ µ φ ɛ ψ µ φ iψ σ µ ɛf = 2ɛψ µ φ ɛσ µ σ ν ψ ν φ + ψ σ µ σ ν ɛ ν φ = ɛσ ν σ µ ψ ν φ + ψ σ µ σ ν ɛ ν φ Results for the supercurrents: J µ α = (σ ν σ µ ψ) α ν φ, J µ α = (ψ σ µ σ ν ) α ν φ.

18 The Supercharges The Noether charge generate the transformations of the corresponding symmetry (see e.g. PS 9.97 for signs and normalization). The conserved supercharges: Q α = 2 d 3 x J 0 α, Q α = 2 d 3 x J 0 α generate SUSY transformations when acting on any (string of) fields: [ ɛq + ɛ Q, X ] = i 2 δx Commutators of the supercharges acting on fields give: [ ɛ 2 Q + ɛ 2 Q, [ ɛ 1 Q + ɛ 1 Q, X ]] [ɛ 1 Q + ɛ 1 Q, [ ɛ 2 Q + ɛ 2 Q, X ]] = 2(δ ɛ2 δ ɛ1 δ ɛ1 δ ɛ2 )X = 2(ɛ 2 σ µ ɛ 1 ɛ 1σ µ ɛ 2 ) i µx

19 Reorganize (using the Jacobi identity) and identify i µ = P µ [ [ɛ2 Q + ɛ 2 Q, ɛ 1 Q + ɛ 1 Q ], X] = 2(ɛ 2 σ µ ɛ 1 ɛ 1σ µ ɛ 2 ) [P µ, X] This is an operator equation, since X is arbitrary [ ɛ2 Q + ɛ 2 Q, ɛ 1 Q + ɛ 1 Q ] = 2(ɛ 2 σ µ ɛ 1 ɛ 1σ µ ɛ 2 ) P µ Since ɛ 1 and ɛ 2 are arbitrary: [ ɛ2 Q, ɛ 1 Q ] = 2ɛ 2 σ µ ɛ 1 P µ [ ɛ2 Q, ɛ 1 Q ] = [ ɛ 2 Q, ɛ 1 Q ] = 0 Extracting the arbitrary ɛ 1 and ɛ 2 : {Q α, Q α } = 2σµ α α P µ {Q α, Q β } = {Q α, Q β} = 0 This is precisely the SUSY algebra, which is thus realized by the free WZ-model.

20 The interacting Wess Zumino model So far, we considered the free Wess-Zumino model for a single chiral field. Lagrangian for multicomponent generalization L free = µ φ j µ φ j + iψ j σ µ µ ψ j + F j F j Offshell SUSY transformations for multicomponent model δφ j = ɛψ j δφ j = ɛ ψ j δψ jα = i(σ µ ɛ ) α µ φ j + ɛ α F j δψ j α = i(ɛσµ ) α µ φ j j + ɛ α F δf j = iɛ σ µ µ ψ j δf j = i µ ψ j σ µ ɛ Next: introduce interactions, while preserving SUSY. Guiding principle: the off-shell SUSY transformations are independent of interactions. Interpretation: the SUSY transformations is the realization of the algebra (the same for all interactions), while the interactions specify the e.o.m. s (not needed at the level of off-shell algebra).

21 The most general set of renormalizable interactions: L int = 1 2 W jk ψ j ψ k + W j F j + h.c., with W jk linear in φ i, φ i and W j quadratic in φ i, φ i. Recall: ψ j ψ k = ψ α j ɛ αβψ β k is symmetric under j k. So W jk is symmetric under j k (without loss of generality). Remark: a potential U(φ j, φ j ) would break SUSY, since a SUSY transformation gives δu = U φ j ɛψ j + U φ j ɛ ψ j which is linear in ψ j and ψ j with no derivatives or F dependence. Such terms cannot be canceled by any other term in δl int.

22 SUSY Conditions First focus on variations of L int with four spinors: δl int 4 spinor = 1 2 W jk φ n (ɛψ n )(ψ j ψ k ) 1 2 W jk φ n (ɛ ψ n )(ψ j ψ k ) + h.c. The vanishing of the second term requires that W jk is analytic (a.k.a. holomorphic): The Fierz identity W jk φ n = 0 (ɛψ j )(ψ k ψ n ) + (ɛψ k )(ψ n ψ j ) + (ɛψ n )(ψ j ψ k ) = 0 so the first term vanishes exactly when W jk / φ n is totally symmetric under the interchange of j, k, n. This condition amounts to the existence of a superpotential W such that: W jk = 2 φ j φ k W

23 Next, the terms in the SUSY variation with derivatives of the fields: δl int = iw jk µ φ k ψ j σ µ ɛ iw j µ ψ j σ µ ɛ + h.c. = i µ ( W φ j ) ψ j σ µ ɛ iw j µ ψ j σ µ ɛ + h.c. This term is a total derivative exactly if W j = W φ j All the remaining terms in the SUSY variation depend on the auxiliary field: δl int F,F = W jk F j ɛψ k + W j φ k ɛψ k F j These cancel automatically, if the previous conditions are satisfied. Summary: the interaction term L int = W φ i j ψ i ψ j + W φ i F i + h.c. transforms into a total derivative, for any holomorphic superpotential W.

24 Remark on Renormalizability The original ansatz for L int was motivated by renormalizability. Yet, the computation establishing SUSY did not rely on the functional form of W (other than it must be holomorphic). For renormalizable interactions W (φ) = E i φ i M ij φ i φ j yijk φ i φ j φ k where M ij, y ijk are are symmetric under interchange of indices. Often we additionally take E i = 0 so SUSY is unbroken in the minimum φ i = 0.

25 Integrate out auxillary fields The action is quadratic in the auxiliary field F and there are no derivatives: L F = F j F j + W j F j + W j F j The path integral can be performed exactly, by solving its algebraic equation of motion: Insert in L: F j = W j, F j = W j L = µ φ( j µ φ j + iψ j σ µ µ ψ j 1 2 W jk ψ j ψ k + W jk ψ j ψ k) W j Wj Remark: the off-shell SUSY transformation of the fermions ψ i depended on the auxiliary fields F i so, after these are integrated out, it depends on the choice of superpotential W.

26 WZ Lagrangian The interacting Wess Zumino model (with renormalizable superpotential): L WZ = µ φ j µ φ j + iψ j σ µ µ ψ j 1 2 M jk ψ j ψ k 1 2 M jk ψ j ψ k 1 2 yjkn φ j ψ k ψ n 1 2 y jkn φ j ψ k ψ n V (φ, φ ) The scalar potential: V (φ, φ ) = W j Wj = F jf j = Mjn M nk φ j φ k M jm yknm φ jφ k φ n M jm yknm φ j φ k φ n yjkm ynpmφ j φ k φ n φ p

27 Features: The scalar potential V (φ, φ ) = F j F j 0 as required by SUSY. The quartic coupling is y 2, as required to cancel the Λ 2 divergence in the φ-mass. The cubic coupling 2 quartic coupling M 2 as required to cancel the log Λ divergence.

28 Linearized equations of motion Equations of motion, keeping just the quadratic term in the superpotential: µ µ φ j = M jn M nk φ k +... ; iσ µ µ ψ j = M jk ψ k +... ; iσ µ µ ψ j = M jk ψ k +... Multiplying fermion equations by iσ ν ν and iσ ν ν, and using the Pauli identity, we obtain µ µ ψ j = Mjn M nk ψ k +... ; µ µ ψ k = ψ j Mjn M nk +... Conclusion: scalars and fermions have the same mass eigenvalues, as required by SUSY. Diagonalizing the mass matrix gives a collection of massive chiral supermultiplets.