Geometric Lagrangians for massive higher-spin fields Dario Francia Chalmers University of Technology - Göteborg Valencia - 3 rd RTN Workshop October, 4-2007 D. F. - in preparation
Introduction I: geometry for higher-spin fields? Central object in Maxwell, Yang-Mills (spin 1) and Einstein (spin 2) theories is the curvature : A µ F µ ν, h µν R µ ν, ρ σ. It gives us both dynamical informations and geometrical meaning. Is it possible to find any similar description for higher-spins?
Introduction II: massive fields - the Fierz system Free propagation of massive irreps of the Poincaré group can be described by spin-s, massive boson symmetric rank-s tensor ϕ µ1... µ s spin-(s + 1/2), massive fermion symmetric rank-s spinor-tensor ψ µ a 1... µ s satisfying the conditions [Fierz, 1939]: ( m 2 ) ϕ µ1... µ s = 0, (i γ α α m) ψ µ a 1... µ s = 0, α ϕ α µ2... µ s = 0, α ψ α a µ 2... µ s = 0, ϕ α α µ 3... µ s = 0, γ α ψ α a µ 2... µ s = 0. At the Lagrangian level the problem is how to implement the conditions Typically, known solutions { α ϕ α µ2...µ s = 0 ϕ α α µ 3...µ s = 0 { α ψα a µ 2...µ s = 0 γ α ψ α a µ 2... µ s = 0? involve auxiliary fields, are not simple deformations of the massless theory.
Introduction III: the examples of spin 1 and spin 2 Spin 1 and spin 2: massive theory as quadratic deformation of the geometric theory: Spin 1 [Proca] L(m = 0) = 1 4 F µνf µν L(m) = 1 4 F µνf µν 1 2 m 2 A µ A µ, ν F µν m 2 A µ = 0, µ ν F µν 0 µ A µ = 0 Spin 2 [Fierz-Pauli] L(m = 0) = 1 2 h µν (R µν 1 2 ηµν R) L(m) = 1 2 h µν {(R µν 1 2 ηµν R) m 2 (h µν η µν h }{{ α α ) } } F ierz P auli mass term µ (R µν 1 2 ηµν R) 0 µ (h µν ν h α α ) = 0 µ h µν ν h α α = 0 Fierz-Pauli constraint
Introduction IV: difficulties of the Fronsdal s theory Canonical description of higher-spin gauge fields encoded in the Fronsdal s equation (1978): F ϕ ϕ + 2 ϕ = 0 gauge invariant under δ ϕ = Λ iff Λ ( Λ α α ) 0; Lagrangian description iff ϕ ( ϕ α β α β ) 0. Gauge invariance with a traceless parameter the Einstein tensor does not need to be (and is not) divergenceless {F 1 2 η F } = 1 2 η F, not possible to extend the results of spin 1 and spin 2 [Aragone-Deser-Yang 1987]
Introduction V: what do we need to generalise the spin 1 and spin 2 cases? To try and reproduce the geometric construction we need the following: Candidate tensors to play the role of higher-spin curvatures. Candidate Ricci tensors and Dirac tensors, to define the free equations of motion. These have to be consistent with the known result for the non-geometric, constrained theory of Fronsdal. Bianchi identities to be satisfied by the generalised Ricci tensors. Suitable mass deformations, such that the on-shell consequence of the Bianchi identity imply that the system reduces to the Fierz form.
Plan I. Higher-spin curvatures II. Geometric massive theory I: bosons Ricci tensors, Bianchi identities, mass def ormations. III. Geometric massive theory II: fermions { the example of spin 3/2, mass def ormations. IV. Propagators and uniqueness
I. Higher-spin curvatures
Higher-spin curvatures Following de Wit and Freedman (1980), introduce higher-spin curvatures: Spin 1 [Maxwell]: F µν = µ A ν ν A µ s. t. δf µν = 0 under δ A µ = µ Λ ; (but also s = 3/2) F µ = 0 Spin 2 [Einstein]: R α βµν = νγ α βµ µγ α βν s. t. δr α βµν = 0 under δ h µν = µ Λ ν + ν Λ µ ; (but also s = 5/2) R α βαν R µν = 0 Spin 3 [de Wit - Freedman]: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα (but also s = 7/2) Γ (1) ρ,αβγ = ρϕ αβγ ( α ϕ ρβγ + β ϕ ραγ + γ ϕ αβρ ), Γ (2) ρσ,αβγ = ργ (1) σ,αβγ 1 2 ( αγ (1) σ,ρβγ +... ), Γ (3) ρστ,αβγ = ργ (2) στ,αβγ 1 3 ( αγ (2) στ,ρβγ +... ), s. t. δγ(3) ρστ,αβγ = 0. Curvature gauge-invariant top of this hierarchy of connexions: Γ (3) ρστ,αβγ R(3) ρστ,αβγ EoM?
Higher-spin curvatures Following de Wit and Freedman (1980), introduce higher-spin curvatures: Spin 1 [Maxwell]: F µν = µ A ν ν A µ s. t. δf µν = 0 under δ A µ = µ Λ ; (but also s = 3/2) F µ = 0 Spin 2 [Einstein]: R α βµν = νγ α βµ µγ α βν s. t. δr α βµν = 0 under δ h µν = µ Λ ν + ν Λ µ ; (but also s = 5/2) R α βαν R µν = 0 Spin 3 [de Wit - Freedman]: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα (but also s = 7/2) Γ (1) ρ,αβγ = ρϕ αβγ ( α ϕ ρβγ + β ϕ ραγ + γ ϕ αβρ ), Γ (2) ρσ,αβγ = ργ (1) σ,αβγ 1 2 ( αγ (1) σ,ρβγ +... ), Γ (3) ρστ,αβγ = ργ (2) στ,αβγ 1 3 ( αγ (2) στ,ρβγ +... ), s. t. δγ(3) ρστ,αβγ = 0. Curvature gauge-invariant top of this hierarchy of connexions: Γ (3) ρστ,αβγ R(3) ρστ,αβγ EoM?
Higher-spin curvatures Following de Wit and Freedman (1980), introduce higher-spin curvatures: Spin 1 [Maxwell]: F µν = µ A ν ν A µ s. t. δf µν = 0 under δ A µ = µ Λ ; (but also s = 3/2) F µ = 0 Spin 2 [Einstein]: R α βµν = νγ α βµ µγ α βν s. t. δr α βµν = 0 under δ h µν = µ Λ ν + ν Λ µ ; (but also s = 5/2) R α βαν R µν = 0 Spin 3 [de Wit - Freedman]: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα Λ α α 0! Γ (1) ρ,αβγ = ρϕ αβγ ( α ϕ ρβγ + β ϕ ραγ + γ ϕ αβρ ), Γ (2) ρσ,αβγ = ργ (1) σ,αβγ 1 2 ( αγ (1) σ,ρβγ +... ), Γ (3) ρστ,αβγ = ργ (2) στ,αβγ 1 3 ( αγ (2) στ,ρβγ +... ), s. t. δγ(3) ρστ,αβγ 0. (but also s = 7/2) Curvature gauge-invariant top of this hierarchy of connexions: Γ (3) ρστ,αβγ R(3) ρστ,αβγ EoM?
Higher-spin curvatures Following de Wit and Freedman (1980), introduce higher-spin curvatures: Spin 1 [Maxwell]: F µν = µ A ν ν A µ s. t. δf µν = 0 under δ A µ = µ Λ ; (but also s = 3/2) F µ = 0 Spin 2 [Einstein]: R α βµν = νγ α βµ µγ α βν s. t. δr α βµν = 0 under δ h µν = µ Λ ν + ν Λ µ ; (but also s = 5/2) R α βαν R µν = 0 Spin 3 [de Wit - Freedman]: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα Λ α α 0! Γ (1) ρ,αβγ = ρϕ αβγ ( α ϕ ρβγ + β ϕ ραγ + γ ϕ αβρ ), Γ (2) ρσ,αβγ = ργ (1) σ,αβγ 1 2 ( αγ (1) σ,ρβγ +... ), Γ (3) ρστ,αβγ = ργ (2) στ,αβγ 1 3 ( αγ (2) στ,ρβγ +... ), s. t. δγ(3) ρστ,αβγ 0. (but also s = 7/2) (curvatures no constraints Fronsdal s theory is not geometric) Curvature gauge-invariant top of this hierarchy of connexions: Γ (3) ρστ,αβγ R(3) ρστ,αβγ EoM?
Higher-spin curvatures Following de Wit and Freedman (1980), introduce higher-spin curvatures: Spin 1 [Maxwell]: F µν = µ A ν ν A µ s. t. δf µν = 0 under δ A µ = µ Λ ; (but also s = 3/2) F µ = 0 Spin 2 [Einstein]: R α βµν = νγ α βµ µγ α βν s. t. δr α βµν = 0 under δ h µν = µ Λ ν + ν Λ µ ; (but also s = 5/2) R α βαν R µν = 0 Spin 3 [de Wit - Freedman]: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα Λ α α 0! Γ (1) ρ,αβγ = ρϕ αβγ ( α ϕ ρβγ + β ϕ ραγ + γ ϕ αβρ ), Γ (2) ρσ,αβγ = ργ (1) σ,αβγ 1 2 ( αγ (1) σ,ρβγ +... ), Γ (3) ρστ,αβγ = ργ (2) στ,αβγ 1 3 ( αγ (2) στ,ρβγ +... ), s. t. δγ(3) ρστ,αβγ 0. (but also s = 7/2) (curvatures no constraints Fronsdal s theory is not geometric) Curvature gauge-invariant top of this hierarchy of connexions: Γ (3) ρστ,αβγ R(3) ρστ,αβγ EoM?
II. Geometric massive theory I: bosons
Candidate Ricci tensors (s = 3) Spin-3 curvature: saturate three indices to restore the symmetries of ϕ µ1 µ 2 µ 3 : R ρ1 ρ 2 ρ 3, µ 1 µ 2 µ 3 R, restore dimensions of a relativistic wave operator, introducing non-localities: R 1 R, so defining the simplest candidate Ricci tensor. [D.F. - A. Sagnotti, 2002] This possibility is highly non-unique infinitely many -more singular- ones: 1 R A ϕ (a) 1 R + a 2 R, 2 A ϕ (a) = 0 Meaning? Lagrangian origin (i.e. Bianchi identity)?
Geometric Lagrangians for massless bosons Spin s: the most general candidate Ricci tensor A ϕ (a 1,... a k,... ) is such that, for almost all choices of a 1,... a k,... : (CONSISTENCY ) the equation A ϕ = 0 implies the compensator equation F 3 3 α ϕ = 0, with δ α ϕ = Λ Fronsdal form, after partial gauge-fixing. (LAGRANGIAN) it is possible to define an identically divergenceless Einstein tensor E ϕ s.t. L = 1 2 ϕ E ϕ E ϕ = 0 A ϕ = 0,
Mass deformation for bosons I Spin 2: R µν 1 2 η µν R m 2 (h µν η µν h ) = 0. M h = h µν η µν h }{{} F ierz P auli mass term, h µ µ h = 0, }{{} F ierz P auli constraint Spin s: General idea: higher traces should appear in the mass term : E ϕ (a 1,... a k,... ) m 2 M ϕ = 0 M ϕ = λ k η k ϕ [k], }{{} generalised F P mass term ϕ ϕ = 0, }{{} F P constraint How?
Mass deformation for bosons II: spin 4 µ ϕ ϕ ϕ We look for λ 1, λ 2 s.t. in this way M ϕ = {ϕ + λ 1 η ϕ + λ 2 η 2 ϕ } = µ ϕ + k η µ ϕ ; {E ϕ ({a 1, a 2 }) m 2 M ϕ (λ 1, λ 2 )} = 0 µ ϕ + k η µ ϕ = 0 µ ϕ = 0 µ ϕ = 0 The condition µ ϕ = 0 implies in turn A ϕ ({a 1, a 2 }) = 0 {a 1, a 2 } From the resulting equation (λ 1 = 1, λ 2 = 1) A ϕ ({a 1, a 2 }) m 2 (ϕ η ϕ ϕ ) = 0 ϕ = 0 ϕ = 0 A ϕ ({a 1, a 2 }) = ϕ {a 1, a 2 } the last condition needed to put the system in the Fierz form.
Mass deformation for bosons III The general solution is L (m) = 1 2 ϕ {E ϕ ({a 1,... a k... }) m 2 M ϕ }, where, for s = 2n or s = 2n + 1 M ϕ = ϕ η ϕ η 2 ϕ 1 3 η 3 ϕ 1 (2 n 3)!! η n ϕ [n]. The structure of the mass term is to be understood such that M ϕ = µ ϕ + k 1 η µ ϕ +... + k m η [m] µ [m] ϕ +...., so that M ϕ = 0 implies the basic, Fierz-Pauli constraint µ ϕ = ϕ ϕ = 0, together with all its consistency conditions: µ [m] ϕ = 0 m. The massive theory is not unique: The Fierz-Pauli constraint implies A ϕ ({a k}) = 0, ({a k }) Under this condition all traces of ϕ can be shown to vanish on-shell this implies in turn A ϕ ({a k }) = ϕ, ({a k }), and then any Lagrangian equation can be reduced to the Fierz system.
III. Geometric massive theory II: fermions
The example of spin 3/2 Spin-3/2 massive theory as quadratic deformation of the geometric, Rarita-Schwinger theory: Spin 3/2 L(m = 0) = 1 2 ψ µ ( R µ 1 2 γµ R) + h.c. L(m) = 1 2 ψ µ {( R µ 1 2 γµ R) m (ψ µ γ µ ψ)} + h.c. µ ( R µ 1 2 γµ R) 0 µ (ψ µ γ µ ψ) = 0 µ ψ µ ψ = 0 (fermionic) Fierz-Pauli constraint
Mass deformation for fermions In the general case the fermionic analogue of the Fierz-Pauli constraint is µ ψ ψ ψ ψ = 0 Spin 5/2: We look for a mass term M ψ s. t. E ψ m (ψ λ 1 γ ψ λ 2 η ψ ) }{{} = 0, M ψ = µ ψ + ρ 1 γ µ ψ = 0. M ψ The unique solution is L = 1 2 ψ {E ψ m (ψ γ ψ η ψ )} + h.c.. Spin s + 1/2: The same procedure leads to the general solution L = 1 2 ψ {E ψ m (ψ s j=0 1 (2j 1)!! γ η j ψ [j] s i=1 1 (2i 3)!! η i ψ [i] )} + h.c. The generalised Fierz-pauli constraint implies the Fierz system for all geometric Einstein tensors (including those with more singular terms) issue of uniqueness (already at the simplest level).
IV. Propagators & uniqueness
Massless propagators Are all those solutions really equivalent? Propagator from Lagrangian equation with an external current: E ϕ (a 1,... a k... ) = J ϕ = G J Current exchange J ϕ = J G J consistency conditions on the polarisations flowing: almost all geometric theories give the wrong result, but one. The correct theory has a simple structure: The kinetic tensor has the compensator form A ϕ = F 3 3 γ(ϕ); It satisfies the identities : { A ϕ 1 2 A ϕ 0 A ϕ 0, and the Lagrangian is L = 1 2 ϕ (A ϕ 1 2 η A ϕ + η 2 B ϕ ) ϕ J [ D.F. - J. Mourad - A. Sagnotti, 2007]
Massive propagators - spin 4 Consider the Lagrangian L (m) = 1 2 ϕ {E ϕ (a 1, a 2 ) m 2 M ϕ } ϕ J, where J is a conserved current. The on-shell condition A ϕ (a 1, a 2 ) = 0 reduces the equation of motion to A ϕ (a 1, a 2 ) m 2 (ϕ η ϕ ϕ ) = J. where A ϕ (a 1, a 2 ) o.s. = ϕ 2 ϕ 3 4 2 ϕ = 0. The whole structure of the propagator is encoded in the coefficients of M ϕ Inverting the equation of motion J ϕ = 1 p 2 m {J J 6 2 D + 3 J J + 3 (D + 1)(D + 3) J J } while the corresponding computation for the massless case gives J ϕ = 1 p 2 {J J 6 D + 2 J J + 3 D(D + 2) J J } thus showing the (generalised) vdvz discontinuity for higher-spin fields.
KK reduction and uniqueness How to understand the origin of the Fierz-Pauli mass-term, for s = 2? KK reduction ( m 2 ): R µν 1 2 η µν R (h η h ) +..., How to perform a KK reduction of a non local theory? Maybe not clear ( 1 1 m 2 ), but still it is unambiguously defined the pure massive contribution of the resulting reduction: E ϕ = (ϕ + k 1 η ϕ + k 2 η 2 ϕ +... ) +..., Is it possible to find a geometric theory whose box term encodes the coefficients of the generalised FP mass term? up to spin 4 (at least) it is the one selected by the analysis of the current exchange. Why the mass term works well with all geometric tensors? Not too strange, also true for spin 2: the non-local theory defined by the eom R µν 1 2 η µν R + λ (η 2 ) R m 2 (h η h ), reduces to the Fierz system, and gives the correct current exchange!
Comments & Conclusions Foregoing locality, linear dynamics of higher-spin gauge fields in geometric fashion; infinitely many formulations are indeed available, at the free level. relationship with a parallel, local, unconstrained formulation, or analysis of the propagator, shows that there is one preferred geometric theory. [ D.F. - A. Sagnotti, 05, 06] Generalisation of the Fierz-Pauli mass term, for bosons and fermions, involving all (γ )traces of the field. Description of massive theory, for spin greater than two, that does not involve auxiliary fields (and no dimension-dependent coefficients). Mass term unique Issue of uniqueness: Massive theory degenerate memory of the correct theory, by KK analysis? The van Dam - Veltman - Zakharov discontinuity is shown to be present for any spin.