Jackiw-Pi Model: A Superfield Approach

Jackiw-Pi Model: A Superfield Approach Saurabh Gupta The Institute of Mathematical Sciences CIT Campus, Chennai, India July 29, 2013 Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 1 / 31

This talk is based on S. Gupta, R. Kumar, R. P. Malik, arxiv:1108.1547 [hep-th]. S. Gupta, R. Kumar, Mod. Phys. Lett. A 28, 1350011 (2013). Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 2 / 31

Introduction Jackiw-Pi Model: Main Features It is a non-abelian model in (2+1)-dimensions (3D) of spacetime that generate mass for the gauge field. Gauge invariance and parity are respected together. Mass and gauge invariance co-exists together. Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 3 / 31

3D Massive Gauge Theory Jackiw-Pi Model: Lagrangian Density Lagrangian density for Jackiw-Pi (JP) model is 1 L 0 = 1 4 Fµν F µν 1 4 ( G µν +g F µν ρ ) (G µν +g F µν ρ ) + m 2 εµνη F µν φ η, where F µν = µ A ν ν A µ g (A µ A ν ) is derived from Maurer-Cartan equation F (2) = da (1) +ig (A (1) A (1) ) 1 2! (dxµ dx ν )F µν T G µν = D µ φ ν D ν φ µ is obtained from G (2) = dφ (1) +ig [φ (1) A (1) +A (1) φ (1) ] 1 2! (dxµ dx ν )G µν T 1 R. Jackiw, S. -Y. Pi, Phys. Lett. B 403, 297 (1997) Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 4 / 31

3D Massive Gauge Theory Jackiw-Pi Model: Notations and Conventions A µ,φ µ : one-form fields ρ: auxiliary field g: coupling constant m: mass parameter Notations and conventions: P Q = P a Q a, P Q = f abc P a Q b T c in the SU(N) Lie algebraic space. The generators T a satisfy [T a,t b ] = if abc T c. The structure constants f abc is totally antisymmetric in a,b,c. D µ φ ν = µ φ ν g (A µ φ ν ). Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 5 / 31

3D Massive Gauge Theory Jackiw-Pi Model: Symmetries Endowed with two types of symmetries Usual Yang-Mills (YM) symmetry (δ 1 ): [with gauge parameter Λ(x)] δ 1 A µ = D µ Λ, δ 1 φ µ = g (φ µ Λ), δ 1 ρ = g (ρ Λ), δ 1 F µν = g (F µν Λ), δ 1 G µν = g (G µν Λ). The Lagrangian density transforms as: δ 1 L 0 = 0. Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 6 / 31

3D Massive Gauge Theory Jackiw-Pi Model: Symmetries Endowed with two types of symmetries Usual Yang-Mills (YM) symmetry (δ 1 ): [with gauge parameter Λ(x)] δ 1 A µ = D µ Λ, δ 1 φ µ = g (φ µ Λ), δ 1 ρ = g (ρ Λ), δ 1 F µν = g (F µν Λ), δ 1 G µν = g (G µν Λ). The Lagrangian density transforms as: δ 1 L 0 = 0. Non-Yang-Mills (NYM) symmetry (δ 2 ): [with gauge parameter Ω(x)] δ 2 A µ = 0, δ 2 φ µ = D µ Ω, δ 2 ρ = + Ω, δ 2 F µν = 0. Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 6 / 31

3D Massive Gauge Theory Jackiw-Pi Model: Symmetries Endowed with two types of symmetries Usual Yang-Mills (YM) symmetry (δ 1 ): [with gauge parameter Λ(x)] δ 1 A µ = D µ Λ, δ 1 φ µ = g (φ µ Λ), δ 1 ρ = g (ρ Λ), δ 1 F µν = g (F µν Λ), δ 1 G µν = g (G µν Λ). The Lagrangian density transforms as: δ 1 L 0 = 0. Non-Yang-Mills (NYM) symmetry (δ 2 ): [with gauge parameter Ω(x)] δ 2 A µ = 0, δ 2 φ µ = D µ Ω, δ 2 ρ = + Ω, δ 2 F µν = 0. ] [m The Lagrangian density transforms as: δ 2 L 0 = µ 2 εµνη F νη Ω. Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 6 / 31

BRST Formalism: Introduction Becchi-Rouet-Stora-Tyutin (BRST) formalism is one of the most intuitive approaches to covariantly quantize the Gauge Theories. Local gauge symmetry BRST symmetry (s b ) Anti-BRST symmetry (s ab ) Properties of (anti-)brst symmetries Nilpotency [i.e. s 2 (a)b = 0] Fermionic nature Absolute anticommutativity [i.e. s b s ab +s ab s b = 0] Linear independence of s (a)b Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 7 / 31

Outline Introduction Jackiw-Pi (JP) model Symmetries (YM & NYM) BRST formalism Motivation Our work Brief overview of superfield formalism Superfield formalism applied to JP model Coupled Lagrangian densities Ghost symmetry and BRST algebra Summary and Future directions Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 8 / 31

Jackiw-Pi Model: Motivations Only BRST symmetries were known. No derivation of (anti-)brst transformation for auxiliary field. Derivation of Curci-Ferrari condition absolute anticommutativity 2 S. Gupta, R. Kumar, R. P. Malik, arxiv:1108.1547 [hep-th] Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 9 / 31

Jackiw-Pi Model: Motivations Only BRST symmetries were known. No derivation of (anti-)brst transformation for auxiliary field. Derivation of Curci-Ferrari condition absolute anticommutativity We applied superfield formalism 2 2 S. Gupta, R. Kumar, R. P. Malik, arxiv:1108.1547 [hep-th] Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 9 / 31

Bonora-Tonin Superfield Formalism: A Brief Synopsis In superfield formalism 3 to BRST Exterior derivative and the connections (basic fields) are generalized to the supermanifold parametrized by x µ,θ, θ. where θ, θ are Grassmannian variables (i.e. θ 2 = θ 2 = 0, θ θ + θθ = 0). 3 L. Bonora, M. Tonin, Phys. Lett. B, 98, 48 (1981) L. Bonora, P. Pasti, M. Tonin, Nuovo Cimento A, 63, 353 (1981) Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 10 / 31

Bonora-Tonin Superfield Formalism: A Brief Synopsis In superfield formalism 3 to BRST Exterior derivative and the connections (basic fields) are generalized to the supermanifold parametrized by x µ,θ, θ. where θ, θ are Grassmannian variables (i.e. θ 2 = θ 2 = 0, θ θ + θθ = 0). Now, these superfields (corresponding to each basic fields) are expanded along the Grassmannian directions, which contain the secondary fields. 3 L. Bonora, M. Tonin, Phys. Lett. B, 98, 48 (1981) L. Bonora, P. Pasti, M. Tonin, Nuovo Cimento A, 63, 353 (1981) Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 10 / 31

Bonora-Tonin Superfield Formalism: A Brief Synopsis In superfield formalism 3 to BRST Exterior derivative and the connections (basic fields) are generalized to the supermanifold parametrized by x µ,θ, θ. where θ, θ are Grassmannian variables (i.e. θ 2 = θ 2 = 0, θ θ + θθ = 0). Now, these superfields (corresponding to each basic fields) are expanded along the Grassmannian directions, which contain the secondary fields. Finally, using horizontality condition, the relationships between the basic and secondary fields can be determined. 3 L. Bonora, M. Tonin, Phys. Lett. B, 98, 48 (1981) L. Bonora, P. Pasti, M. Tonin, Nuovo Cimento A, 63, 353 (1981) Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 10 / 31

Jackiw-Pi Model: Superfield Formalism In superfield formalism 3D manifold = (3,2) D supermanifold which is parametrized by superspace variables Z M = (x µ,θ, θ) with x µ (= 0,1,2) and (θ, θ) are pair of Grassmannian variables (with θ 2 = θ 2 = 0, θ θ+ θθ = 0) Exterior derivative and connection are generalized as d d = dx µ µ +dθ θ +d θ θ, A (1) Ã(1) = dx µ Bµ (x,θ, θ)+dθ F(x,θ, θ)+d θ F(x,θ, θ) θ = ( / θ), θ = ( / θ) are the Grassmannian derivatives B µ (x,θ, θ), F(x,θ, θ), F(x,θ, θ) are the superfields corresponding to A µ (x),c(x), C(x), respectively. Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 11 / 31

Superfield expansion B µ (x,θ, θ) = A µ (x)+θ R µ (x)+ θ R µ (x)+i θ θ S µ (x), F(x,θ, θ) = C(x)+i θ B 1 (x)+i θ B 1 (x)+i θ θ s(x), F(x,θ, θ) = C(x)+i θ B 2 (x)+i θ B 2 (x)+i θ θ s(x). Here the secondary fields R µ (x), Rµ (x), s(x), s(x) are fermionic and S µ (x), B 1 (x), B1 (x), B 2 (x), B2 (x) are bosonic in nature. Horizontality condition (HC) gives the relationship between basic and secondary field of the theory d Ã(1) = d A (1) F µν (x,θ, θ) = F µν (x) Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 12 / 31

Jackiw-Pi Model: Horizontality Condition The kinetic term ( 1 4 Fµν F µν ) remains invariant under the gauge transformations, therefore must be independent of Grassmannian variables when generalized to (3, 2)-D supermanifold. Mathematically, 1 4 Fµν F µν = 1 4 F MN F MN Using HC, we find the relationships amongst the basic, auxiliary and secondary fields of the theory, as follows R µ = D µ C, Rµ = D µ C, B1 = i 2 g(c C), B2 = i 2 g( C C), S µ = D µ B + ig (D µ C C) D µ B ig (Dµ C C), s = g ( B C), s = +g (B C), B + B = ig (C C). Here we have made the choices B 1 = B and B 2 = B [Identified with Nakanishi-Lautrup type auxiliary fields] Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 13 / 31

Plugging these values of secondary fields in the superfield expansion of basic fields, we get the (anti-)brst symmetry transformations only for the gauge and (anti-)ghost fields. B (h) µ (x,θ, θ) = A µ (x)+θ D µ C(x)+ θ Dµ C(x) [ + θ θ id µ B g(d µ C C) ] (x) A µ (x)+θ[s ab A µ (x)]+ θ[s b A µ (x)]+θ θ[s b s ab A µ (x)], F (h) (x,θ, θ) = C(x)+θ [i B(x)]+ θ [ g ] 2 (C C)(x) θ θ [ig( B C)(x)] C(x)+θ[s ab C(x)]+ θ[s b C(x)]+θ θ[s b s ab C(x)], F (h) (x,θ, θ) = C(x)+θ [ g C)(x)] 2 ( C + θ [ib(x)] + θ θ [ig(b C)(x)] C(x)+θ[s ab C(x)]+ θ[sb C(x)]+θ θ[sb s ab C(x)], Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 14 / 31

where (h), as the superscript, denotes the expansions of the superfields after the application of HC Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 15 / 31

where (h), as the superscript, denotes the expansions of the superfields after the application of HC What about vector field (φ µ ) and auxiliary field (ρ)?? Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 15 / 31

Jackiw-Pi Model: GIRs The (anti-)brst symmetry transformations for φ µ and ρ are derived using Gauge Invariant Restrictions (GIRs) Following quantities remain invariant under YM gauge transformations (δ 1 ), i.e. δ 1 (F µν φ η ) = 0, δ 1 (F µν ρ) = 0. Thus, we have following GIRs: F (h) µν (x,θ, θ) φ η (x,θ, θ) = F µν (x) φ η (x), F (h) µν (x,θ, θ) ρ(x,θ, θ) = F µν (x) ρ(x). where φ η and ρ are superfields corresponding φ η and ρ, respectively. Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 16 / 31

Jackiw-Pi Model: GIRs These bosonic superfields can be expanded on the (3, 2) dimensional supermanifold as follows φ µ (x,θ, θ) = φ µ (x)+θ P µ (x)+ θ P µ (x)+i θ θ Q µ (x), ρ(x,θ, θ) = ρ(x)+θ P(x)+ θ P(x)+i θ θ Q(x), where the secondary fields (P µ, P µ,p, P) are fermionic and (Q µ,q) are bosonic in nature. These secondary fields are determined with the help of GIRs. P µ = g(φ µ C), Q µ = i [ g 2 (φ µ C) C ig(φ µ B) ], P µ = g(φ µ C), Q = i [ g 2 (ρ C) C ig(ρ B) ], P = g(ρ C), P = g(ρ C). Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 17 / 31

Substituting the above values into the super-expansion, we have φ (h,g) µ (x,θ, θ) = φ µ (x) θ [g (φ µ C)](x) θ [g (φ µ C)](x) + θ θ [g 2 (φ µ C) C ig (φ µ B)](x) φ µ (x)+θ(s ab φ µ (x))+ θ(s b φ µ (x))+θ θ(s b s ab φ µ (x)) ρ (h,g) (x,θ, θ) = ρ(x) θ [g (ρ C)](x) θ [g (ρ C)](x) + θ θ [g 2 (ρ C) C ig (ρ B)](x) ρ(x)+θ (s ab ρ(x))+ θ (s b ρ(x))+θ θ (s b s ab ρ(x)) where the superscripts (h, g) on the above superfields refers to the super-expansions of the superfields obtained after the application of HC condition and gauge invariant restrictions. Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 18 / 31

(Anti-)BRST symmetries: Geometrical Interpretation (Anti-)BRST symmetry transformations can be given as s b Ψ(x) = θ Ψ (h,g) θ=0, s ab Ψ(x) = θ Ψ (h,g) θ=0, s b s ab Ψ(x) = θ θ Ψ (h,g) where Ψ(x) is the generic 3D field of the theory and Ψ (h,g) (x,θ, θ) is the corresponding superfields after the application of HC and/or GIRs. Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 19 / 31

Nilpotency and absolute anticommutativity check Nilpotency of (anti-)brst transformations ( 2 θ = 0, 2 θ = 0) s 2 (a)b = 0 Absolute anticommutativity is ensured by ( θ θ + θ ) Ψ(h,g) (x,θ, θ) = 0 (s b s ab +s ab s b ) = 0 θ Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 20 / 31

Thus, we have obtained following off-shell nilpotent and absolutely anticommuting (anti-)brst symmetry transformations g s ab A µ = D µ C, sab C = 2 ( C C), s ab B = g(b C), s ab ρ = g(ρ C), s ab φ µ = g(φ µ C), s ab C = i B, s ab B = 0, s b A µ = D µ C, s b C = g 2 (C C), s b B = g( B C), s b ρ = g(ρ C), s b φ µ = g(φ µ C), s b C = ib, sb B = 0. Absolutely anticommutativity of the (anti-)brst symmetry transformations for A µ,φ µ and ρ is true only when CF-condition [B + B +ig(c C) = 0] is satisfied. Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 21 / 31

Coupled Lagrangian densities The expression for the coupled Lagrangian densities can be written as [ i L B = L 0 +s b s ab 2 A µ A µ +C C + 1 ] 2 φ µ φ µ, [ i L B = L 0 s ab s b 2 A µ A µ +C C + 1 ] 2 φ µ φ µ. The above coupled Lagrangian densities can be, explicitly, given as L B = 1 4 Fµν F µν 1 ( G µν +gf µν ρ ) (G µν +gf µν ρ ) 4 + m 2 εµνη F µν φ η +B ( µ A µ )+ 1 2 (B B + B B) i µ C Dµ C Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 22 / 31

L B = 1 4 Fµν F µν 1 4 ( G µν +gf µν ρ ) (G µν +gf µν ρ ) + m 2 εµνη F µν φ η B ( µ A µ )+ 1 2 (B B + B B) i D µ C µ C. Under BRST symmetry transformations s b L B = µ [B (D µ C)], s b L B = µ ( B µ C)+D µ [B + B +ig(c C)] µ C. Under anti-brst symmetry transformations s ab L B = µ [ B (D µ C)], s ab L B = µ (B µ C) Dµ [B + B +ig(c C)] µ C. Thus, the corresponding actions remain invariant on the constrained surface parametrized by CF condition. Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 23 / 31

(Anti-)BRST symmetries: Geometrical Interpretation Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 24 / 31

Conserved charges: Novel features Exploiting the Norther s theorem and the corresponding (anti-)brst charges are as follows [ ] Q ab = d 2 x B D 0 i C B C 2 g Ċ ( C C), Q b = d 2 x [B D 0 C Ḃ C i2 ] g C (C C). These charges are: Nilpotent [i.e. Q 2 (a)b = 0] Conserved [i.e. Q (a)b = 0] Generators of (anti-)brst symmetry transformations [i.e s r Φ = ±i [Φ, Q r ] ± (r = b,ab)] Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 25 / 31

Conserved charges: Novel features The nilpotent generators Q (a)b, even though produce the nilpotent (anti-)brst transformations for the basic fields, they are unable to generate the (anti-)brst transformations for the auxiliary field ρ of the theory. Even the nilpotency and absolutely anticommuting properties of the proper (anti-)brst symmetry transformations do not generate the (anti-)brst transformations for the auxiliary field ρ. Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 26 / 31

Ghost charge & BRST Algebra The coupled Lagrangian densities remain invariant under the following ghost transformations (s g ) s g C = +Σ C, s g C = Σ C, sg (A µ, φ µ, ρ, B, B) = 0, where Σ is a global infinitesimal scale parameter. The above ghost symmetry transformations s g lead to the following conserved Noether s charge as: Q g = i d 2 x [ C D 0 C C ] C. The (anti-)brst charge (Q (a)b ) and the ghost charge Q g satisfy the following standard BRST algebra Q 2 b = 0, Q2 ab = 0, i [Q g, Q b ] = Q b, i [Q g, Q ab ] = Q ab, {Q b, Q ab } = Q b Q ab +Q ab Q b = 0, Q 2 g 0, Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 27 / 31

Jackiw-Pi Model: Summary We derive the off-shell nilpotent and absolutely anticommuting (anti-)brst symmetry transformations for Jackiw-Pi model. With the help of Bonora-Tonin s superfield formalism we derive the (anti-)brst transformations for the gauge and (anti-)ghost fields. GIRs give the (anti-)brst symmetry transformations for ρ and φ η. Absolute anticommutativity of the (anti-)brst symmetry transformations is ensured by Curci-Ferrari type restriction which emerges naturally in Bonora-Tonin s superfield formalism. Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 28 / 31

Further work & Future directions We have studied NYM gauge transformations within the framework of augmented superfield formalism 4 Combination of YM and NYM symmetries together 4 Saurabh Gupta, R. Kumar, Mod. Phys. Lett. A 28 (2013) 1350011 Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 29 / 31

Acknowledgments The Institute of Mathematical Sciences, Chennai, India for financial support. Collaborators Prof. R. P. Malik (BHU, India) Mr. R. Kumar (BHU, India) Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 30 / 31

Thank You!! Saurabh Gupta (IMSc) 3D Jackiw-Pi Model July 29, 2013 31 / 31