Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0080 Acta Polytechnica 62(1):80–84, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague ON THE WESS-ZUMINO MODEL: A SUPERSYMMETRIC FIELD THEORY Daya Shankar Kulshreshthaa, ∗, Usha Kulshreshthab a University of Delhi, Department of Physics and Astrophysics, Delhi – 110007, India b University of Delhi, Kirori Mal College, Department of Physics, Delhi – 110007, India ∗ corresponding author: dskulsh@gmail.com Abstract. We consider the free massless Wess-Zumino Model in 4D which describes a supersymmetric field theory that is invariant under the rigid or global supersymmetry transformations where the transformation parameter ϵ (or ϵ̄) is a constant Grassmann spinor. We quantize the theory using the Hamiltonian and path integral formulations. Keywords: Wess-Zumino Model, supersymmetric field theories, Hamiltonian and path integral quantization. 1. Introduction Supersymmetry (SUSY) is a symmetry that rotates bosons into fermions and fermions into bosons. It is one of the beautiful symmetries of nature. Also, a field theory (FT) which remains invariant under the rigid or global supersymmetry transformations (where the transformation paremeter is a constant Grassman spinor) and which also satisfies the super Poincare algebra (SPA) is usually referred to as a supersym- metric field theory (SFT). In this article, we consider the free massless Wess-Zumino Model (WZM) in 4D which describes a SFT. It may be important to men- tion here that the WZM is the first known example of an interacting 4D quantum field theory with linearly realised SUSY, studied by Wess and Zumino using the dynamics of a single chiral superfield (composed of a complex scalar and a spinor fermion). It may be important to mention that the WZM represents a typical SFT which is of central importance in the theory of SUSY, supergravity and superstring theory (SST) and for further details we refere to the work of Refs. [1–8]. The WZM describes an example of a non-manifest supersymmetry [5]. One could of course go to the formalism of superspace and superfields to construct a theory that has a manifest supersymmetry [5]. Tak- ing this theory as an example, it is possible to formu- late supersymmertic field theories in different dimen- sions including in higher dimensions. The WZM also provides a basic framework for the study of Ramond Nievue Schwarz (RNS) SST [8] which is an example of a SST with non-manifest SUSY. Further, start- ing with the WZM, it is also possible to construct a supergravity theory [1–6, 8]. SPA is a graded Lie algebra that includes anti- commutation relations (ACR’s) involving the super- charge Qa – the generator of the SUSY transforma- tions. WZM is one of the simplest examples of a SFT. In this article, we discuss the supersymmetry of WZM and present some remarks with respect to the rigid or global supersymmetry versus the local supersym- metry (which happens to be a Supergravity theory). Finally we consider the constraint quantization of this theory [7]. It is important to mention that the su- persymmetry has profound applications in conformal hadron physics from light-front holography where it even has some observational prospects [9–11]. As mentioned above, the supersymmetry is a sym- metry that relates bosonic and fermionic variables (or the bosons and fermions) so that: δB = ϵ̄F , δF = ϵ ∂B ; ∂ ≡ ∂µ (1) Here, δ is bosonic, B is bosonic and F is fermionic. The transformation parameter ϵ (or ϵ̄) is a constant Grassman spinor and is fermionic. Grassman vari- ables are anti-commuting. Supergravity theory on the other hand is a theory that has “local supersymmetry” and it is invariant under local Susy transformations where the transformation parameter depends on the spacetime xµ. So the transformation parameter for supergravity: ϵ(xµ) or ϵ̄(xµ) depends on xµ and hence supergravity is a “gauge theory” of gravity. In con- trast to this the WZM is a supersymmetric FT with rigid or global (not local) Supersymmetry. Let us us consider two consecutive infinitesimal rigid supersymmetry transformations of a bosonic field B: δ1 B = ϵ̄1F , δ2F = ϵ2∂B (2) This then implies that the two internal SUSY trans- formations lead us to a spacetime translation: {δ1,δ2}B = aµ∂µB ; aµ = (ϵ̄2γµϵ1) (3) Presence of a spacetime derivative of B on right hand side (RHS) of above equation suggests that the Susy is an extension of the Poincare spacetime symmetry: {Qa,Q̄b} = 2(γµ)abPµ (4) 80 https://doi.org/10.14311/AP.2022.62.0080 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 62 no. 1/2022 On the Wess-Zumino Model: A SFT Supercharge Qa (a = 1, 2, 3, 4 in 4D) is the gener- ator of SUSY transformations. It is related to the generator of spacetime translations Pµ and therefore is not an internal symmetry generator. The SUSY transformation is an extention of the Poincare space- time symmetry. Supercharge Qa is a spinor. It is fermionic and anti-commuting. Poincare algebra (PA) after including the supersymmetry becomes the SPA. 2. The Wess-Zumino Model The WZM is defined (on-shell) by the Lagrangian density [5]: L := [ 1 2 (∂µA)∂µA− 1 2 m2A2 + 1 2 (∂µB)∂µB − 1 2 m2B2 −mgA(A2 + B2) + ψ̄(iγν∂ν −m)ψ −g (ψ̄ψ A + iψ̄γ5ψB) − 1 2 g2(A2 + B2)2 ] (5) Here A is a scalar field, B is a pseudoscalar field, ψ is a spin-1/2 Majorana field (ψ = ψC = Cψ̄T ). C is the charge conjugation matrix and A = A† and B = B†. All the fields here have the same mass m and they couple with the same strength g. This is in contrast to the non-SUSY FT’s. This is due to the fact that states of a particular representation of the super Poincare algebra (SPA) are characterized by the eigenvalue m2 of P 2 (= PµP µ) and different values of spin s. Actually, all the fields belong to the same mass multiplet in SPA. Pauli-Ljubanski polarization vector is defined as: Wµ := 1 2 ϵµνρσP νMρσ (6) Here P 2 = PµP µ W 2 = WµW µ (7) are Casimir operators of PA that satisfy:[ P 2,Mµν ] = 0[ P 2,Pµ ] = 0[ W 2,Mµν ] = 0[ W 2,Pµ ] = 0 (8) We further have: P 2 = m2 > 0 W 2 = −m2s(s + 1). (9) Where, m2 and (−m2s(s+1)) are the eigenvalues of P 2 and W 2. Here s denotes the spin of the representation which assumes discrete values: s = 0, 1/2, 1, 3/2, . . . This representation is specified in terms of the mass m and spin s. Physically a state in a representation (m,s) corresponds to a particle of rest mass m and spin s. Also, since the spin projection S3 can take any value from −s to +s, (massive particles fall into (2s + 1)-dimensional multiplets). In WZM, all the fields namely, A, B, ψ, ψ̄ have the same mass m and they couple with the same strength g (in the unbroken SUSY) – in contrast to the non- supersymmetric field theories. States of a particular representation of SPA are characterized by the eigen- value m2 of Casimir operator P 2 and different values of spin s. W µ is proportional to P µ (generator of the Poincare group): W µ = λP µ (10) and W0 = λP0 = −→ P · −→ J (11) where P µ = (P0 , −→ P ). (12) The constant of proportionality λ in Wµ = λPµ is called Helicity and it is defined by: λ := −→ P · −→ J P0 (13) for massless particles with λ := ±s where s = 0, 1/2, 1, . . . is the spin of representation. N = 1 is called as the Minimal Supersymmetry and N > 1 is called the Extended Supersymmetry. For simplicity we set (g = 0) yielding the La- grangian density of the free WZM [5]: L := [ 1 2 ∂µA∂ µA− 1 2 m2A2 + 1 2 ∂µB∂ µB − 1 2 m2B2 + ψ̄(iγν∂ν −m)ψ ] (14) Theory is seen to be invariant (up to a total derivative) under the rigid SUSY transformation [5]: δA = ϵ̄ ψ δB = −i ϵ̄ γ5 ψ δψ = − (iγν∂ν + m) (A− iγ5 B) ϵ δψ̄ = ϵ̄ (A− iγ5 B) (iγν ←− ∂ ν −m) (15) Here ϵ is a constant Grasmann variable (which does not depend on spacetime x ≡ xµ) implying a global or rigid SUSY transformations. However, δψ and δψ̄ here, are seen to depend on spacetime derivatives of A and B. This implies that this is an extention of Poincare spacetime symmetry (different than an internal symmetry). Supercurrent jµ of the theory could be easily calcu- lated to be [5]: jµ = [ i 2 ϵ̄(A− iγ5 B) (iγν ←− ∂ ν −m)γµ ψ ] ≡ [ 1 β ϵ̄ kµ ] (16) 81 Daya Shankar Kulshreshtha, Usha Kulshreshtha Acta Polytechnica Here β is a real constant which could be suitably choosen. The spinor charges Qa are defined by [5]: Qa := ∫ d3x k0a k0a = i 2 β [ {(A− iγ5 B)(iγν ←− ∂ ν −m)}γ0ψ ] a (17) Here k0a are the spinor charge densities with a = 1, 2, 3, 4. Spinor charges and spinor charge densities being fermionic satisfy SPA and the spinor charges are seen to satisfy the anti-commutation relation (ACR) [5]: {Qa,Q̄b} = 2Pµ(γµ)ab (18) This explicitly shows that the WZM obeys the SPA and it is a supersymmetric FT having a rigid or global SUSY. Also, the supersymmetry of the theory is a non- manifest supersymmetry. We now set m = 0 for making the fields to be massless, so that the free massless WZM is defined by the Lagrangian density: L := [ 1 2 ∂µA∂ µA + 1 2 ∂µB∂ µB + ψ̄(iγν∂ν )ψ ] (19) This is the simplest example of a supersymmetric FT in 4D with a non-manifest supersymmetry. We obtain the free WZM by setting g = 0 and it is seen to be invariant, up to a total derivative, under the rigid SUSY transformations [5]: δA = ϵ̄ ψ δB = −i ϵ̄ γ5 ψ δψ = − (iγν∂ν ) (A− iγ5 B) ϵ δψ̄ = ϵ̄ (A− iγ5 B) (iγν ←− ∂ ν ) (20) Here ϵ is a constant Grasmann variable (which does not depend on spacetime x ≡ xµ). Here, δψ and δψ̄ are seen to depend on spacetime derivatives of A and B which implies that this is an extention or gener- alization of the Poincare spacetime symmetry. (ϵ, ϵ̄) being constant, implies that the symmetry is a rigid or global Susy. It is also possible to consider it as a theory of a single complex scalar field and a fermionic field by combining the fields A and B as follows: ϕ(x) := (A + iB)/2 ϕ⋆(x) = (A− iB)/2 (21) implying therefore: δϕ = ϵ̄ψ̄ and δϕ⋆ = ϵψ and δψA = 2i(σµϵ̄)A∂µϕ⋆(x) δψ̄Ȧ = −2i(σ̄µϵ)Ȧ∂µϕ(x) (22) Since A is a scalar field and B is a pseudoscalar field, the complex combination ϕ(x) transforms under the parity transformation like complex conjugation. Here, ψ and ψ̄ are not independent fields as they are the Ma- jorana spinor fields in the Weyl formulation. Hence the transformations of δψ and δψ̄ are not independent and one could be obtained from the other. Supercur- rent jµ of the theory is obtained as: j µ = [ i 2 ϵ̄(A− iγ5 B) (iγν ←− ∂ ν )γµ ψ ] ≡ [ 1 β ϵ̄ k µ ] (23) Here β is a real constant. Spinor charge Qa are: Qa := ∫ d3x k0a k0a = i 2 β [ {(A− iγ5 B) (iγν ←− ∂ ν )}γ0 ψ ] a (24) Here k0a are the spinor charge densities with a = 1, 2, 3, 4. WZM being a supersymmetric FT, spinor charges and the spinor charge densities are seen to satisfy SPA and the spinor charges satisfy the ACR: {Qa,Q̄b} = 2Pµ(γµ)ab (25) This implies that the WZM obeys SPA and it is a su- persymmetric FT with a rigid Susy. SPA reads [5]: [Pµ,Pν ] = 0 (26) [Mµν,Pρ] = −i (ηµρ Pν −ηνρ Pµ) (27) [Mµν,Mρσ ] = − i(ηµρMνσ + ηνσMµρ) + i (ηµσMνρ + ηνρMµσ ) (28) [Pµ,Qa] = 0 (29) [Mµν,Qa] = −(σ4µν )ab Qb (30) σ4µν := i 4 [γµ,γν ] {Qa,Q̄b} = 2(γµ)ab Pµ {Qa,Qb} = −2 (γµ C)ab Pµ {Q̄a,Q̄b} = 2 (C−1γµ)ab Pµ (31) SPA has 14 generators: 4 generators of Lorentz trans- lations Pµ , 6 generators of Poincare transformations Mµν and 4 spinor charges Qa (the Majorana spinors). Here the indices a and b run from 1 to 4 in 4D. 3. Free massless WZM We now set m = 0 for making the fields to be mass- less, so that the free massless WZM is defined by the Lagrangian density: L := [ 1 2 ∂µA∂ µA + 1 2 ∂µB∂ µB + ψ̄(iγν∂ν )ψ ] (32) 82 vol. 62 no. 1/2022 On the Wess-Zumino Model: A SFT We break up the Lagrangian density of the free mass- less WZM into bosonic and fermionic parts: L = LB + LF LB = 1 2 ∂µA∂ µA + 1 2 ∂µB∂ µB LF = ψ̄(iγν∂ν )ψ (33) Further, LF (= LF ) could be written in two differ- ent looking but conceptually equivalent forms (which differ by a total derivative (t.d.)) as follows: LF1 = iψ̄γ µ∂µψ LF2 = i 2 [ ψ̄γµ(∂µψ) − (∂µψ̄)γµψ ] (34) LF1 −L F 2 = i 2 ∂µ(ψ̄γµψ) = i 2 ∂µj µ jµ = (ψ̄γµψ) (35) Theory described by LF1 is seen to possess a set of two second class constraints: ρ1 = (π + iψ̄γ0) ≈ 0 ρ2 = π̄ ≈ 0 (36) Here, Fermi fields ψ and ψ̄ are to be treated as inde- pendent fields. Theory described by LF2 is also seen to possess a set of two second class constraints: χ1 = (π + i 2 ψ̄γ0) ≈ 0 χ2 = (π̄ + i 2 γ0ψ) ≈ 0. (37) The Fermi fields ψ and ψ̄ in this later case are not independent fields. This is consistent with the defini- tion of Majorana spinor fields (we remind ourselves here that in WZM, the fermionic fields are Majorana spinor fields). We now study the Hamiltonian formulation of the theory [7]. The canonical momenta following from the Lagerangian density of WZM defined by L := (LB +LF ) with LF = LF2 (working with the signature ηµν := diag(+1,−1,−1,−1)) are: ΠA := ∂L ∂(∂0A) = ∂0A ΠB := ∂L ∂(∂0B) = ∂0B (38) π := ∂L ∂(∂0ψ) = − i 2 ψ̄γ0 π̄ := ∂L ∂(∂0ψ̄) = − i 2 γ0ψ (39) Theory thus has 2 primary constraints (PC’s): χ1 = (π + i 2 ψ̄γ0) ≈ 0 χ2 = (π̄ + i 2 γ0ψ) ≈ 0 (40) In principle, χ1,χ2 represent an infinite number of PC’s which could be labeled say by α,β (which run from one to infinity). We however, ignore these fur- ther labelings in our considerations. The canonical Hamiltonian density of the theory is obtained as: Hc =(∂0A) ΠA + (∂0B) ΠB + (∂0ψα) πα + (∂0ψ̄α) π̄α −LB −LF (41) Hc = 1 2 [ Π2A + Π 2 B − iψ̄γk∂ kψ + i(∂kψ̄)γkψ ] (42) The total Hamiltonian density is: HT := Hc + χ1 u + χ2 v (43) Demanding that the constraints χ1 and χ2 are pre- served in the course of time one does not get any secondary constraints and therefore these are the only 2 constraints that the theory possesses. Non-vanishing matrix elements of the 2 × 2 matrix of the PB’s of these above constraints among themselves are: R12 = −R21 = iγ0δ(x1 −y1)δ(x2 −y2)δ(x3 −y3). (44) The non-vanishing equal-time (ET) commutation relations (CR’s) (denoted by a square bracket) and ET anti-commutation relations (ACR’s) (denoted by a curly bracket) of the bosonic and ferminic variables of the theory are found to be: [A(x,t), ΠA(y,t)] = i δ(x−y) (45) [B(x,t), ΠB (y,t)] = i δ(x−y) (46) {ψα(x,t), ψ̄β (y,t)} = γ0δαβ δ(x−y) (47) δ(x−y) := δ(x1 −y1)δ(x2 −y2)δ(x3 −y3) (48) These relations appear to be similar to the usual ones. However, the fermionic spinor field ψ here is not a Dirac spinor but it is a Majorana spinor having real components: (ψ = ψC ). We need to remember here that the Dirac spinor is a 4-component spinor which has complex elements and it could be expressed in terms of two, 2-component Weyl spinors having complex elements. However, if the elements of these Weyl spinors are taken as real (ψ = ψC ) then it becomes a Majorana spinor (having real elements). In path integral quantization (PIQ) [7], transition to quantum theory is made by writing the vacuum to vacuum transition amplitude for the theory, called the generating functional Z[Jk] of the theory which in the presence of the external sources Jk for the present theory is [7]: Z[Jk] = ∫ [dµ] exp [ i ∫ dxdy[JkΦk + ΠA∂0A + ΠB∂0B + π∂0ψ + π̄∂0ψ̄ + Πu∂0u + Πv∂0v −HT ] ] (49) 83 Daya Shankar Kulshreshtha, Usha Kulshreshtha Acta Polytechnica Here Φk ≡ (A,B,ψ,ψ̄,u,v) are the phase space variables of the theory with the corresponding respective canonical conjugate momenta: Πk ≡ (ΠA, ΠB,π, π̄, Πu, Πv ). The functional measure [dµ] of the theory (with the above generating functional Z[Jk]) is: [dµ] = [ [δ(x1 −y1)δ(x2 −y2)δ(x3 −y3)][dA] [dB][dψ][dψ̄][du][dv][dΠA] [dΠB ][dπ][dπ̄][dΠu][dΠv ] δ[(π + i 2 ψ̄γ0) ≈ 0] δ[(π̄ + i 2 γ0ψ) ≈ 0] ] (50) 4. Conclusions and summary Some important remarks may be helpful. In relativis- tic quantum mechanics, the Dirac equation (DE) is a single particle relativistic wave equation where ψ represents a wave function. In FT, DE is an Euler- Lagrange field equation which is obtained from the Dirac action or the Dirac Lagrangian by using the vari- ational principle. WZM is the simplest example of a supersymmetric field theory in 4D. This is also an example of a FT with non-manifest supersymmetry. Taking the ex- ample of free massless WZM, one could study many important theories in different dimensions including in higher dimsimensions. The theory also provides a basic framework for the study of Ramond Nievue Schwarz (RNS) superstring theory (SST) which is an example of a SST with non-manifest SUSY. Starting with the WZM, it is possible to construct a super- gravity theory by gauging its global (rigid) SUSY into a local SUSY through the Noether’s procedure. Just to summarize in brief, we have studied in this work, the WZM [5], which is a supersymmetric FT that has rigid or global supersymmetry. The theory has a supercharge Qa (a = 1, 2, 3, 4 in 4D) which is a Grassmann spinor having anti-commuting proper- ties. Theory is invariant under rigid supersymmetry transformations where the transformation parame- ter is a constant Grassmann spinor [5]. Finally, we have also studied the Hamiltonian and path integral quantization of the theory [7]. Acknowledgements It is matter of great pleasure to thank the organizers of the International Conference on Analytic and Algebraic Meth- ods in Physics XVIII (AAMP XVIII) – 2021 at Prague, Czechia, Prof. Vít Jakubský, Prof. Vladimir Lotoreichik, Prof. Matej Tusek, Prof. Miloslav Znojil and the entire team for a wonderful orgnization of the conference. References [1] J. Wess, B. Zumino. Supergauge transformations in four dimensions. 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String Theory and M-Theory: A Modern Introduction. Cambridge University Press, 2007. ISBN 9780521860697. [9] S. J. Brodsky. Supersymmetric and conformal features of hadron physics. Universe 4(11):120, 2018. https://doi.org/10.3390/universe4110120. [10] S. J. Brodsky, G. F. de Teramond, H. G. Dosch. Light-front holography and supersymmetric conformal algebra: A novel approach to hadron spectroscopy, structure, and dynamics, 2020. arXiv:2004.07756. [11] S. J. Brodsky. Color confinement and supersymmetric properties of hadron physics from light-front holography. Journal of Physics: Conference Series 1137:012027, 2019. https://doi.org/10.1088/1742-6596/1137/1/012027. 84 https://doi.org/10.1016/0550-3213(74)90355-1 https://doi.org/10.1016/0370-2693(76)90089-7 https://doi.org/10.1103/PhysRevD.13.3214 https://doi.org/10.1016/0370-1573(81)90157-5 https://doi.org/10.1142/7594 https://doi.org/10.3390/universe4110120 https://arxiv.org/abs/2004.07756 https://doi.org/10.1088/1742-6596/1137/1/012027 Acta Polytechnica 62(1):80–84, 2022 1 Introduction 2 The Wess-Zumino Model 3 Free massless WZM 4 Conclusions and summary Acknowledgements References