Non-Abelian tensor multiplet in four dimensions

Transcription

1 PASCOS th nternational Symposium on Particles Strings and Cosmology OP Publishing Non-Abelian tensor multiplet in four dimensions Hitoshi Nishino and Subhash Rajpoot, Department of Physics and Astronomy, California State University, 1250 Bellflower Blvd., Long Beach, CA 90840, USA. Conference Speaker Abstract. The long-standing problem with a non-abelian tensor with non-trivial consistent couplings in four dimensions has been solved. The key technique is double-fold: (1) Adding extra Chern-Simons terms for the field strength of non-abelian tensor, and (2) employing a compensator mechanism. We generalize this mechanism to supersymmetric system. Our system has three multiplets: (i) The usual non-abelian vector multiplet (VM) (A µ, λ ), (ii) A non- Abelian tensor multiplet (TM) (B µν, χ, ϕ ), and (iii) A compensator vector multiplet (CVM) (C µ, ρ ). The indices, J, are for the adjoint representation of a non-abelian group G. All of our fields are propagating with kinetic terms. The C µ -field plays the role of a compensator absorbed into the longitudinal component of B µν. We give both the component lagrangian and a corresponding superspace reformulation, reconfirming the total consistency of the system. 1. The Conventional Problem with Non-Abelian Tensors First, the conventional problem with non-abelian tensors will be discussed. A solution will be presented. Also presented will be a supersymmetrized physical non-abelian field with consistent interactions [1], both in component language and superfield language. The long-standing problem with a non-abelian tensor is described as follows. Let be the adjoint index of a non-abelian group G gauged by the a non-abelian vector field A µ, minimally coupled to the antisymmetric tensor B µν with the coupling constant g. Consider the conventional field strength 3 G (0) µνρ +3D µ B νρ +3( µ B νρ + gf JK A µ J B νρ K ), (1.1) where D µ is the usual gauge-covariant derivative with the structure constant f JK of the group G. Consider an action 0 d 4 x L 0 with the lagrangian 4 L (G(0) µνρ ) (F µν ) 2, (1.2) with F µν 2 µ A ν + gf JK A µ J A ν K. Obviously, the B -field equation is 5 δl 0 δb µν = +1 2 D ρg (0)µνρ. = 0. (1.3) 3 The formulation in this section is the so-called tensor hierarchy [2, 3, 4, 5], but we re-formulate their general expressions in terms of our objectives here. 4 We use the signature (, +, +, +) for four dimensions (4D) as in [1].. 5 The symbol = stands for a field equation, to be distinguished from an algebraic identity. We also use the symbol? = for an equality under question. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DO. Published under licence by OP Publishing Ltd 1

2 PASCOS th nternational Symposium on Particles Strings and Cosmology OP Publishing The problem is that the divergence of this B -field equation does not vanish: 0 =? δl0 D ν = + 1 δb µν 4 gf JK F J νρ G (0)µνρK 0, (1.4) unless F µν or G (0) µνρ vanishes trivially. This inconsistency arises already at the classical level. This is also one of the reasons, why topological formulations with vanishing field strengths F µν =. (0) 0, G. µνρ = 0 such as in [6] are easier to formulate for non-abelian tensors. An additional problem is related to the so-called local tensorial gauge transformation of the B -field: δ β B µν = +D µ β ν D ν β µ, (1.5) because the field strength G µν is not invariant under δ β : δ β G (0) µνρ = +3gf JK F µν J β ρ K 0. (1.6) This further implies the non-invariance of the action: δ β 0 0. These two problems are mutually related, because the non-vanishing of (1.4) is also re-casted into δ β Solution to the Problem The solution to the problem above is to introduce a non-trivial Chern-Simons (CS) term into the G -field strength: G µνρ +3( µ B νρ + gf JK A µ J B νρ K ) 3f JK C µ J F νρ K = +3D µ B νρ 3f JK C µ J F νρ K +G (0) µνρ 3f JK C µ J F νρ K, (2.1) where C µ is a compensator vector field, also carrying the adjoint index. The field strength for C is defined by H µν +D µ C ν D ν C µ + gb µν. (2.2) Now these field strengths G and H are invariant under the δ β -transformation δ β B µν = + D µ β ν D ν β µ δ β C µ = gβ µ (2.3a), (2.3b) which is the proper gauge transformation for B µν, and δ γ -transformation δ γ B µν = f JK F µν J γ K, (2.4a) δ γ C µ = D µ γ. (2.4b) which is the proper gauge transformation for C µ. As (2.3b) shows, C µ is a compensator field for the δ β -transformation. The role played by the C F -term in (2.1) is to cancel the unwanted term in (1.6). The C -field itself should have its own gauge transformation as the covariant gradient (2.4b). The contribution of δ γ (2D µ C ν ) in (2.2) is cancelled by the contribution of δ γ (gb µν ). n other words, we have the total invariances δ β (G µνρ, H µν ) = (0, 0), δ γ (G µνρ, H µν ) = (0, 0). (2.5) 2

3 PASCOS th nternational Symposium on Particles Strings and Cosmology OP Publishing Accordingly, we also have the consistency problem (1.4) solved, by considering the lagrangian L (G µνρ ) (H µν ) (F µν ) 2. (2.6) The total action is also invariant δ β 1 = δ γ 1 = 0. The field equations for B and C -fields are δl 1 δb µν = D ρg µνρ 1 2 ghµν. = 0, (2.7a) δl 1 δc µ = D ν H µν f JK F ρσ J G µρσ K. = 0, (2.7b) The divergence of the B -field equation vanishes now: 0 =? δl1 D ν = + 1 δb µν 2 g δl1 δc µ.= 0, (2.8) where the last equality holds because of the C -field equation. n other words, the unwanted F G -term in (1.4) is now cancelled by the contribution of the C -field equation. Relevantly, the divergence of (2.7b) also vanishes, as it should without any inconsistency: 0 =? δl1 D µ = +f JK J δl1 F δc µν µ δb K µν.= 0. (2.9) We emphasize repeatedly that these invariances have never been accomplished without the peculiar CS terms both in (2.1) and (2.2) [2, 3, 4, 5]. 3. Component Formulation of N=1 TM The supersymmetrization of the purely bosonic system (2.6) has been accomplished in our recent paper [1]. We need three multiplets: (i) A tensor multiplet (TM) (B µν, χ, ϕ ), (ii) A Yang- Mills vector multiplet (YMVM) (A µ, λ ), and (iii) A compensating vector multiplet (CVM) (C µ, ρ ). Our total action d 4 x g 2 L has the lagrangian [1] L = 1 12 (G µνρ ) (χ D/ χ ) 1 2 (D µϕ ) g2 (ϕ ) 2 g(χ ρ ) 1 4 (H µν ) (ρ D/ ρ ) 1 4 (F µν ) (λ D/ λ ) 1 2 gf JK (λ χ J )ϕ K f JK (λ γ µ ρ J )D µ ϕ K f JK (λ γ µνρ ρ J )G µνρ K f JK (ρ γ µν χ J )F µν K 1 4 f JK (λ γ µν χ J )H µν K 1 2 f JK F µν H µν J ϕ K, (3.1) up to quartic-order terms O(φ 4 ), and the coupling constant g has the dimension of mass. The scalar ϕ has its mass g, while there is a mixture between χ and ρ with the same mass g. As has been mentioned after (2.4), C µ is a compensator field [7], absorbed into the longitudinal component of B µν. The kinetic term of the C -field becomes the mass term of B µν. Accordingly, the degrees of freedom (DOF) for the massive TM fields are B µν (3), χ with ρ (4) and ϕ (1). 3

4 PASCOS th nternational Symposium on Particles Strings and Cosmology OP Publishing Our action is invariant under global N = 1 supersymmetry [1] δ Q B µν = + (ɛγ µν χ ) 2f JK C µ J (δ Q A ν K ), δ Q χ = (γµνρ ɛ)g µνρ (γ µ ɛ)d µ ϕ f JK[ ] + ɛ(λ J ρ K ) (γ 5 γ µ ɛ)(λ J γ 5 γ µ ρ K ) (γ 5 ɛ)(λ J γ 5 ρ K ) (3.2a), (3.2b) δ Q ϕ = + (ɛχ ), (3.2c) δ Q C µ = + (ɛγ µ ρ ) + f JK (ɛγ µ λ J )ϕ K, (3.2d) δ Q ρ = (γµν ɛ)h µν gɛϕ 1 2 f JK (γ µν ɛ)f µν J ϕ K f JK[ + ɛ(λ J χ K ) (γ µ ɛ)(λ J γ µ χ K ) (γµν ɛ)(λ J γ µν χ K ) ] (γ 5 γ µ ɛ)(λ J γ 5 γ µ χ K ) (γ 5 ɛ)(λ J γ 5 χ K ), (3.2e) δ Q A µ = + (ɛγ µ λ ), δ Q λ = (γµν ɛ)f µν f JK (γ 5 ɛ)(ρ J γ 5 χ K ), (3.2f) (3.2g) up to O(φ 3 ) -terms. Our tensorial gauge transformation δ β, and δ γ -transformation are exactly the same as (2.3) and (2.4), while all the fermionic fields transform only under the usual non-abelian gauge transformation δ α, as the B and C -fields do, so that there is no problem with the δ β = 0 and δ γ = 0 of the field strengths as in (2.1) and (2.2), via (2.5). The δ Q -transformations of the field strengths reflect their CS terms: δ Q G µνρ = + 3(ɛγ µν D ρ χ ) + 3f JK (δ Q A µ J )H νρ K 3f JK (δ Q C µ J )F νρ K, (3.3a) δ Q H µν = 2(ɛγ µ D ν ρ ) + g(ɛγ µν χ ) + 2f JK D µ [(δ Q A ν J )ϕ K], (3.3b) δ Q F µν = 2(ɛγ µ D ν λ ). (3.3c) Since we have not added the D -auxiliary field, our YMVM and CVM have on-shell DOF 2+2, while off-shell DOF 3+4. However, our TM is in the off-shell formulation with the total off-shell DOF 4 + 4, because the off-shell DOF of each field are [(4 1) (4 2)]/2 = 3 for B µν, 4 for χ and 1 for ϕ. The field equations for all of our fields are 6 [1] + D/ λ 1 2 gf JK χ J ϕ K f JK (γ µ ρ J )D µ ϕ K 1 4 f JK (γ µν χ J )H µν K f JK (γ µνρ ρ J )G µνρ K. = 0, (3.4a) + D/ χ gρ gf JK λ H ϕ K 1 4 f JK (γ µν λ J )H µν K f JK (γ µν ρ J )F µν K. = 0, (3.4b) + D/ ρ gχ f JK (γ µ λ J )D µ ϕ K 6 These equations are fixed up to O(φ 3 ) -terms, due to the quartic fermion terms in the lagrangian. 4

5 PASCOS th nternational Symposium on Particles Strings and Cosmology OP Publishing 1 12 f JK (γ µνρ λ J )G µνρ K f JK (γ µν χ J )F µν K. = 0, (3.4c) + D ν F µ ν + gf JK ϕ J D µ ϕ K gf JK (λ J γ µ λ K ) + f JK H µν J D ν ϕ K 1 2 f JK G µρσ J H ρσ K f JK (χ J D µ ρ K ) f JK (ρ J D µ χ K ). = 0, (3.4d) + D ρ G µνρ gh µν 1 2 f JK D ρ (λ J γ µνρ ρ K ) + gf JK F µν J ϕ K 1 2 gf JK (λ J γ µν χ K ). = 0, (3.4e) + D 2 µϕ gf JK (λ J χ K ) g 2 ϕ 1 2 f JK F µν J H µν K. = 0, (3.4f) + D ν H µν 1 2 f JK F ρσ J G µρσ K 1 2 f JK (χ J D µ λ K ) 1 2 f JK (λ J D µ χ K ) gf JK (λ J γ µ ρ K ) f JK F µν J D ν ϕ K. = 0. (3.4g) n deriving each of these equations, we have also used other field equations. 4. Superspace Reformulation of N=1 TM We can re-formulate our theory in superspace, as an independent consistency-reconfirmation. Our superspace Bds for the superfield strengths F AB, G ABC and H AB are AG BCD) 1 4 T AB E G E CD) 1 4 f JK F AB J H CD) K 0, (4.1a) AH BC) 1 2 T AB D H D C) g G ABC 0, (4.1b) AF BC) 1 2 T AB D F D C) Our relevant superspace constraints at the mass dimensions 0 d 1 are 0. (4.1b) T αβ c = + 2(γ c ) αβ, G αβc = +2(γ c ) αβ ϕ, (4.2a) G αbc = (γ bc χ ) α, H αb = (γ b ρ ) α f JK (γ b λ J ) α ϕ K, (4.2b) F αb = (γ b λ ) α, α ϕ = χ α α χ β = 1 6 (γcde ) αβ G cde (γ c ) αβ c ϕ, (4.2c) 1 2 f JK[ ] + C αβ (λ J ρ K ) (γ 5 γ c ) αβ (λ J γ 5 γ c ρ K ) (γ 5 ) αβ (λ J γ 5 ρ K ) α ρ β = (γcd ) αβ H cd + g C αβ ϕ 1 2 f JK (γ cd ) αβ F cd J ϕ K 1 4 f JK[ + C αβ (λ J χ K ) + (γ c ) αβ (λ J γ c χ K ) 1 2 (γcd ) αβ (λ J γ cd χ K ), (4.2d) (γ 5 γ c ) αβ (λ J γ 5 γ c χ K ) (γ 5 ) αβ (λ J γ 5 χ K ), α λ β = (γcd ) αβ F cd 1 2 (γ 5) αβ f JK (ρ J γ 5 χ K ). (4.2e) (4.2f) 7 n this superspace section, we use the indices A = (a,α), B = (b,β), for superspace coordinates, where a, b, = 0, 1, 2, 3 (or α, β, = 1, 2, 3, 4) are for bosonic (or fermionic) coordinates. n superspace, we use the (anti)symmetrization convention, e.g., X AB) X AB ( 1) AB X BA, different from our component formulation. 5

6 PASCOS th nternational Symposium on Particles Strings and Cosmology OP Publishing All other components, such as G αβγ or T γ αβ etc. at d 1 are zero. Although most of technical details associated with superspace formulation are skipped here, we presented a rather independent confirmation for the total consistency of our N = 1 non- Abelian tensor multiplet in superspace. 5. Concluding Remarks n this talk, we have explained how to formulate the N = 1 supersymmetrization in 4D of a physical non-abelian tensor with consistent couplings [1]. This is the supersymmetrization of the special case [4] of the minimal tensor hierarchy [5], which, in turn, is a special case of more general hierarchy in [2][3]. Both the component and superspace formulations of our system are given, as the cross-verification of our system. Our CVM (C µ, ρ ) plays the role of a compensator multiplet, absorbed into the TM (B µν, χ, ϕ ), making the latter massive. There exists certain problem for the quantization of Stueckelberg theory [7] for non-abelian gauge groups [9]. This is because the longitudinal components of the gauge field do not decouple from the physical Hilbert space, so that the renormalizability and unitarity of the system are spoiled [9]. We take rather an optimistic standpoint about this potential problem for the following reasons. First, we mention that our theory is not renormalizable due to Pauli couplings. This feature is not necessarily a fatal drawback for our theory, because certain theories exist in 4D, such as non-linear sigma models that are not renormalizable, but are not rejected from the outset. Second, N = 1 supersymmetry may well improve quantum behavior of our theory, compared with non-supersymmetric systems. There is good chance that supersymmetries solve the quantum problem of non-abelian Stueckelberg theories. The importance of our result [1] is double-fold: (i) A new supersymmetric physical system with Stueckelberg mechanism that solves the problem with non-abelian tensor is presented. (ii) The problem with extra vector fields in the non-singlet representation of a non-abelian gauge group is now solved. We should also consider the possibility that N = 1 supersymmetry may well provide better quantum behavior compared with non-supersymmetric cases. Acknowledgments This work is supported in part by DOE grant grant # DE-FG02-10ER References [1] Nishino H and Rajpoot S 2012 Phys. Rev. D (Preprint ) [2] de Wit B and Samtleben H 2005 Fortsch. Phys (Preprint hep-th/ ) [3] de Wit B, Nicolai H and Samtleben H 2008 JHEP (Preprint ) [4] Chu C-S 2011 A Theory of Non-Abelian Tensor Gauge Field with Non-Abelian Gauge Symmetry G G (Preprint ) [5] Samtleben H, Sezgin E and Wimmer R 2011 Jour. High. Energy. Phys [6] Freedman D Z, Townsend P K 1981 Nucl. Phys. B ; Ogievetsky V and Polubarinov V 1967 Sov. J. Nucl. Phys [7] Stueckelberg E C G 1938 Helv. Phys. Acta ; For reviews, see, e.g., Ruegg H and Ruiz-Altaba M 2004 The Stuckelberg field. nt. Jour. Mod. Phys [8] Ferrara S, Freedman D Z and van Nieuwenhuizen P 1976 Phys. Rev. 13D 3214; Deser S and Zumino B 1976 Phys. Lett. 62B 335; van Nieuwenhuizen P1981 Supergravity Phys. Rept. 68C 189; Wess J and Bagger J 1992 Superspace and Supergravity, Princeton University Press [9] Kunimasa J M and Goto T 1967 Prog. Theor. Phys ; Slavnov A A 1972 Theor. Math. Phys ; Veltman M J G1968 Nucl. Phys. B7 637; Slavnov A A and Faddeev L D 1970 Theor. Math. Phys ; Vainshtein A and Khriplovich B 1971 Yad. Fiz ; Shizuya K 1977 Nucl. Phys. B ; Kafiev Y N 1982 Nucl. Pnys