The Jackiw-Pi model and all that...

Transcription

1 In honor of the 70th birthday of Prof. Olivier Piguet The Jackiw-Pi model and all that... O.M. Del Cima Departamento de Física - UFV O.M. Del Cima, J.Phys.A44 (2011) (fast track communication); Phys.Lett.B720 (2013) 254 (published in honor of the 70th birthday of Prof. Olivier Piguet) O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

2 ... where it started in space-time... O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

3 O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

4 O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

5 years later... O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

6 A D = 3 space-time brainstorm The study of gauge field theories in D = 3 space-time has raised a great deal of interest since the early works of Deser, Jackiw and Templeton. Over the last decades, this issue has also been motivated and well-supported in view of the possibilities in the description of condensed matter phenomena (quantum Hall effect, high-t c superconductivity, graphene...). Meantime, one of the central problems in the framework of gauge field theories is the issue of gauge field mass. Gauge symmetry is not, in principle, conflicting with the presence of massive gauge bosons. O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

7 A D = 3 space-time brainstorm In D = 2 space-time, the well-known Schwinger model puts in evidence the presence of a massive photon without the breaking of gauge symmetry (J. Schwinger, Phys.Rev.125 (1962) 397; Phys.Rev.128 (1962) 2425). Compatibility between gauge symmetry and massive gauge fields has also arisen in D = 3 space-time. A topological (Chern-Simons) mass term added to the Yang-Mills term, shifts the photon mass to a non-vanishing value without breaking gauge invariance, however parity symmetry is lost (S. Deser, R. Jackiw and S. Templeton, Ann.Phys.(NY)140 (1982) 372). Could gauge symmetry, massive gauge bosons and parity symmetry live together in D = 3 space-time? O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

8 A D = 3 space-time brainstorm The Jackiw-Pi model By breaking the Yang-Mills paradigm non-abelian generalizations of Abelian models Jackiw and Pi overcame the challenge to implement both gauge and parity invariance, and massive gauge bosons, in D = 3 space-time. They proposed a non-yang-mills gauge model in D = 3 space-time for a pair of vector fields with opposite parity transformations, which generates a mass-gap through a mixed Chern-Simons-like term preserving parity (R. Jackiw and S.-Y. Pi, Phys.Lett.B403 (1997) 297). The physical states consistency has been demonstrated in the Hamiltonian framework (Ö.F. Dayi, Mod.Phys.Lett.A13 (1998) 1969). O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

9 The challenge was launched The Jackiw-Pi model... it seems impossible to resolve this problem by a gauge-fixing term.... the existence of a Goldstone field, which shifts by a constant under a symmetry transformations, hints at some kind of symmetry breaking. R. Jackiw, Non-Yang-Mills gauge theories, hep-th/ straightforward perturbation theory cannot be carried out. Targets to focus R. Jackiw and S.-Y. Pi, Phys.Lett.B403 (1997) 297. Find the gauge-fixing through BRST approach and all that... (target achieved). Spontaneous symmetry breaking? (target focused). Perturbative quantization through the algebraic method of renormalization. Quantum scale invariance? (target focused). O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

10 The Jackiw-Pi model The model The classical action of the Jackiw-Pi model is given by: { 1 Σ inv = Tr d 3 x 2 F µν F µν + 1 ( G µν + g[f µν, ρ] )( G µν + g[f µν, ρ] ) 2 } mɛ µνρ F µν φ ρ, (1) such that, F µν = µ A ν ν A µ + g[a µ, A ν ], G µν = D µ φ ν D ν φ µ and D µ = µ +g[a µ, ], where A µ and φ µ are vector fields with opposite parity transformations, ρ is a scalar, g is a coupling constant and m a mass parameter, also, means any field. O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

11 The Jackiw-Pi model The model The Lie group is a simple compact, so that every field, X = X a τ a, is Lie algebra valued, with the matrices τ being the generators of the group in the adjoint representation and obey [τ a, τ b ] = f abc τ c and Tr(τ a τ b ) = 1 2 δ ab (a, b, c = 1, 2,..., N 2 1). Gauge symmetries The action Σ inv (1) is invariant under two sets of gauge transformations, δ θ and δ χ : δ θ A µ = D µ θ, δ θ φ µ = g[φ µ, θ] and δ θ ρ = g[ρ, θ] ; (2) δ χ A µ = 0, δ χ φ µ = D µ χ and δ χ ρ = χ, (3) where θ and χ are Lie algebra valued infinitesimal local parameters. O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

12 The Jackiw-Pi model BRST symmetries The corresponding BRST transformations of the fields A µ, φ µ and ρ, stemming from the symmetries (2) and (3), are given by: sa µ = D µ c, sφ µ = D µ ξ + g[φ µ, c], sρ = ξ + g[ρ, c], sc = gc 2 and sξ = g[ξ, c], (4) where c and ξ are the Faddeev-Popov ghosts, with Faddeev-Popov charge (ghost number) one. In order to implement the gauge-fixing following the BRST procedure, we introduce two sorts of antighosts ( c and ξ) and Lautrup-Nakanishi fields (b and π), such that s c = b, sb = 0 ; (5) s ξ = π, sπ = 0 ; (6) where the multiplier fields and the Faddeev-Popov antighosts (with ghost number minus one) belong to BRST-doublets. O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

13 The Jackiw-Pi model The gauge-fixing By adopting the covariant linear gauge conditions: δσ gf δb = µ A µ + αb, (7) δσ gf δπ = µ φ µ + βπ, (8) it follows that the BRST-trivial gauge-fixing action compatible with then reads { Σ gf = s Tr d 3 x c µ A µ + ξ µ φ µ + α 2 cb + β } 2 ξπ { = Tr d 3 x b µ A µ c µ D µ c + π µ φ µ ξ µ( D µ ξ + g[φ µ, c] ) + α 2 b2 + β } 2 π2. (9) O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

14 The Jackiw-Pi model The antifield action Let us introduce the action in which the nonlinear BRST transformations are coupled to the antifields, so as to control, at the quantum level, the renormalization of those transformations: Σ ext = Tr d 3 x { A µsa µ + φ µsφ µ + ρ sρ + c sc + ξ sξ }, (10) where the antifields are BRST invariant, namely, The tree-level action sa µ = sφ µ = sρ = sc = sξ = 0. (11) The total the tree level action for the Jackiw-Pi model, Γ (0), is given by: Γ (0) = Σ inv + Σ gf + Σ ext, (12) which is invariant under the BRST transformations given by the equations (4), (5), (6) and (11). O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

15 The Jackiw-Pi model The parity-even mass term: BRST local non invariant An interesting feature of the Jackiw-Pi action Γ (0) is that it is not BRST local invariant thanks to the parity-even mass term: Σ m = Tr d 3 x { mɛ µνρ F µν φ ρ }, (13) then sσ m = m Tr d 3 x {ɛ ρµν ρ (F µν ξ)}, (14) which is invariant only up to a total derivative, possibly indicating that at the quantum level the β-function associated to the mass parameter m vanishes 1. 1 O.M. Del Cima, D.H.T. Franco, J.A. Helayël-Neto and O. Piguet; JHEP9802 (1998) 002; JHEP9804 (1998) 010; Lett.Math.Phys.47 (1999) 265 O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

16 Spectral analysis Unitarity and causality In quantum field theory, unitarity and causality are essential physical requirements a. Unitarity (of the S-matrix) reflects the fundamental principle of probability conservation meaning the absence of negative-norm 1-particle states in the spectrum. Even though we have to introduce in certain instances the artificial device of an indefinite metric in Hilbert space (Faddeev-Popov ghosts) Causality principle establishes a time correlation among the cause and its subsequent effect, requiring that the change in the interaction law in any space-time region can influence the evolution of the system only at subsequent times. a N.N. Bogoliubov and D.V. Shirkov, Quantum Fields, (1983) Benjamin/Cummings. O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

17 Spectral analysis The propagators By switching off the coupling constant g we get the free part of the action, Σ inv + Σ gf, as follows: { 1 Σ free = Tr d 3 x 2 F µν F g=0 µν G µν G g=0 µν mɛ µνρ F µν φ g=0 ρ + b µ A µ + α 2 b2 + π µ φ µ + β } 2 π2 c µ µ c ξ µ µ ξ, (15) { 1 = d 3 x 2 Aa µ Θ µν A a ν φa µ Θ µν φ a ν + ma a µσ µν φ a ν 1 2 ba µ A a µ α 4 ba b a 1 2 πa µ φ a µ β 4 πa π a ca c a + 1 } 2 ξ a ξ a, where the projectors Θ µν, Ω µν and Σ µν read: Θ µν = η µν µ ν, Ωµν = µ ν and Σ µν = ɛ µρν ρ. (16) O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

18 Spectral analysis The propagators The generating functional for the connected Green functions (Z c [J]) is defined by means of the vertex functional (Γ (0) ) through the Legendre transformation: Z c [J i ] = Γ (0) [Φ i ] + Tr d 3 x ( A µ J µ A + φ µj µ φ + bj b + πj π + J c c + J c c + J ξ ξ + J ξ ξ ), (17) where Φ i = (A µ, φ µ, b, π, c, c, ξ, ξ) and J i = (J µ A, Jµ φ, J b, J π, J c, J c, J ξ, J ξ). The tree-level propagators for all the fields are defined by: δ 2 Z c T Φ i (x)φ j (y) = i δj i (x)δj j (y). (18) O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

19 Spectral analysis The propagators The tree-level propagators (18) in momenta space read: { ( A a µ(k)a b ν(k) = iδ ab 1 k 2 m 2 η µν k ) µk ν k 2 2α ( ) } kµ k ν k 2 k 2, { ( φ a µ(k)φ b ν(k) = iδ ab 1 k 2 m 2 η µν k ) µk ν k 2 2β ( ) } kµ k ν k 2 k 2, A a µ(k)φ b ν(k) = δ ab m k 2 (k 2 m 2 ) ɛ µρνk ρ, A a µ(k)b b (k) = δ ab 2 k 2 k µ, φ a µ(k)π b (k) = δ ab 2 k 2 k µ, b a (k)b b (k) = 0, π a (k)π b (k) = 0, c a (k) c b (k) = iδ ab 2 k 2, ξa (k) ξ b (k) = iδ ab 2 k 2. (19) O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

20 Spectral analysis Unitarity and causality We will now discuss the spectrum and tree-level unitarity of the model. Coupling the propagators to external currents, J a Φ i, compatible with the symmetries of the model, and taking the imaginary part of the residues of the transition amplitudes, A Φi Φ j, at the poles, we can probe the necessary conditions for unitarity IRes A Φi Φ j k 2 > 0 and count the degrees of freedom described by the fields, Φ a i = (A a µ, φ a µ, b a, π a, c a, c a, ξ a, ξ a ). The transition amplitudes in momentum space are written as: A Φi Φ j = J a Φ i (k) Φ a i (k)φ b j (k) J b Φ j (k). (20) O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

21 Spectral analysis Unitarity and causality Let us analyze first the case of the propagators of the vector fields A a µ and φ a µ. The vector currents, J aµ aµ A and Jφ, can be expanded in terms of a three-dimensional complete basis in the momentum space as follows: J aµ A = X a A kµ + Y a A k µ + Z a A εµ and J aµ φ = X a φ kµ + Y a φ k µ + Z a φ εµ, (21) fulfilling the current conservation conditions: k µ J aµ A = 0 and k µj aµ φ = 0, (22) where k µ = (k 0, k), k µ = (k 0, k) and ε µ = (0, ε) are linearly independent vectors satisfying the constraints: k µ ε µ = k µ ε µ = 0 and ε µ ε µ = 1, (23) such that for a massive pole, k µ k µ = k µ kµ = m 2, and for a massless one, k µ k µ = k µ k µ = 0. O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

22 Spectral analysis Unitarity and causality The transition amplitudes for the vector fields A a µ and φ a µ are given by: 1 A AA = i k 2 m 2 J aµ A J Aµ a, (24) 1 A φφ = i k 2 m 2 J aµ φ Jφµ a, (25) m A Aφ = k 2 (k 2 m 2 ) ɛ µρνj aµ A kρ Jφ aν, (26) where use has been made of the current conservation conditions (22). Causality: poles Analyzing the amplitudes above, it can be verified that the amplitudes A AA (24) and A φφ (25) have single massive poles at k 2 = m 2, whereas the amplitude A Aφ (26) has two poles, a massive and a massless, at k 2 = m 2 and k 2 = 0, respectively. O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

23 Spectral analysis Unitarity and causality By considering the imaginary part of the residues at the poles k 2 = m 2 and k 2 = 0, we get: IRes A AA k 2 =m 2 = Z a A 2 > 0, (27) IRes A φφ k 2 =m 2 = Z a φ 2 > 0, (28) IRes A Aφ k 2 =m 2 = 0 and IRes A Aφ k 2 =0 = 0. (29) Unitarity: A a µ and φ a µ It can be concluded from (27) and (28) that the both vector fields, A a µ and φ a µ, carry 2(N 2 1) massive degrees of freedom with mass m, however, from (29) it follows that there are no massless degrees of freedom propagating associated to the vector fields. O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

24 Spectral analysis Unitarity and causality Let us now analize the propagators related to the fields b a, π a, c a, c a, ξ a and ξ a, given by Eqs.(19) (19). The transition amplitudes read: A Ab = 2 k 2 k µj aµ A J b a = 0, (30) A φπ = 2 k 2 k µj aµ φ Jπ a = 0, (31) A bb = 0, A ππ = 0, (32) A c c i 2 k 2 J c a Jc ā, A ξ ξ = i 2 k 2 J ξ a J ā ξ, (33) where the current conservation conditions (22) were applied in (30) and (31). O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

25 Spectral analysis Unitarity and causality Through the amplitudes displayed above, by considering their imaginary parts of the residues at the massless pole k 2 = 0: IRes A Ab k 2 =0 = IRes A φπ k 2 =0 = 0, IRes A bb k 2 =0 = IRes A ππ k 2 =0 = 0, (34) IRes A c c k 2 =0 = 2 Jc a J ā IRes A ξ ξ k 2 =0 = 2 J a ξ c < 0, J ā ξ < 0. (35) Unitarity: Faddeev-Popov ghosts There are no massless modes propagating in the Lautrup-Nakanishi fields sector (34), nevertheless, from (35) we see that the massless propagating (negative norm state) Faddeev-Popov ghosts carry, each of them, N 2 1 degrees of freedom taking care of the N 2 1 spurious degrees of freedom stemming from the longitudinal sector of each vector field, A a µ and φ a µ. O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

26 Spectral analysis Unitarity and causality From the previous results, it can be concluded that the Jackiw-Pi model is free from tachyons and ghosts at the classical level. Nevertheless, to have full control of the unitarity at tree-level, it is still necessary to study the behaviour of the scattering cross sections (σ) in the limit of high center of mass energies, by analizing the Froissart-Martin bound 2 : lim σ C ln s. s 2 M. Chaichian, J. Fischer and Yu.S. Vernov, Nucl.Phys.B383 (1992) 152; O.M. Del Cima, Mod.Phys.Lett.A9 (1994) O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

27 Symmetries Slavnov-Taylor identity, ghost and antighost equations and Ward identities The BRST invariance of the action Γ (0) (12) is expressed through the Slavnov-Taylor identity: { δγ (0) S(Γ (0) ) = Tr d 3 x δa µ δγ (0) δa µ + δγ(0) δφ µ δγ (0) δφ µ + δγ(0) δγ (0) δρ δρ } + δγ(0) δγ (0) δc + δγ(0) δγ (0) δc δξ + b δγ(0) + π δγ(0) δξ δ c δ ξ = 0, (36) which translates, in a functional way, the invariance of the classical model under the BRST symmetry. O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

28 Symmetries Slavnov-Taylor identity, ghost and antighost equations and Ward identities It is suitable to define, for later use, the linearized Slavnov-Taylor operator as below: { δγ (0) δ δ S Γ (0) = Tr + δγ(0) δρ d 3 x δa µ δ δρ + δγ(0) δρ } + b δ δ c + π δ δ ξ. δ δa µ + δγ(0) δa µ δ δρ + δγ(0) δc δa µ + δγ(0) δφ µ δ δc + δγ(0) δc δ δφ µ + δγ(0) δφ µ δ δc + δγ(0) δξ δφ µ δ δξ + δγ(0) δξ δ δξ (37) O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

29 Symmetries Slavnov-Taylor identity, ghost and antighost equations and Ward identities Another identities, the two ghost equations, G I Γ (0) δγ(0) δ c G II Γ (0) δγ(0) δ ξ follow from the gauge-fixing conditions, + µ δγ(0) = 0, (38) δa µ + µ δγ(0) = 0, (39) δφ µ δγ (0) δb = µ A µ + αb and δγ(0) δπ = µ φ µ + βπ, (40) and the Slavnov-Taylor identity (36), meaning that Γ (0) depends on the antighosts, c and ξ, and the antifields, A µ and φ µ, through the combinations à µ = A µ + µ c and φ µ = φ µ + µ ξ. (41) O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

30 Symmetries Slavnov-Taylor identity, ghost and antighost equations and Ward identities The Jackiw-Pi model presents two antighost equations, they are listed as below: { ] ]} δγ G I Γ (0) d 3 (0) x g [ c, δγ(0) g [ ξ, δγ(0) = I, (42) δc δb δπ where I g d 3 x { [A µ, A µ ] + [φ µ, φ µ ] + [ρ, ρ] [c, c] [ξ, ξ] + α[ c, b] + β[ ξ, π] } ; (43) { ]} δγ G II Γ (0) d 3 (0) x g [ ξ, δγ(0) = II, (44) δξ δb where II g d 3 x { [φ µ, A µ ] [ξ, c] ρ g + α[ ξ, b] }. (45) O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

31 Symmetries Slavnov-Taylor identity, ghost and antighost equations and Ward identities The Landau gauge Yang-Mills theories in the Landau gauge have very special features as compared to a generic linear gauge the existence, besides the Slavnov-Taylor identity, of the antighost equation a, which controls the dependence of the theory on the ghost c. In further quantization, the breakings, I and II due to the presence of the terms, d 3 x α[ c, b] and d 3 x β[ ξ, π], and d 3 x α[ ξ, b], respectively being nonlinear in the quantum fields will be subjected to renormalization. Therefore, the breakings, I and II have to be renormalized, which could spoil the usefulness of the antighost equations, by this reason, bearing in mind later renormalization of the model, we adopt from now on the Landau gauge α = β = 0. a A. Blasi, O. Piguet and S.P. Sorella, Nucl.Phys.B356 (1991) 154. O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

32 Symmetries Slavnov-Taylor identity, ghost and antighost equations and Ward identities As another feature of the Landau gauge (α = β = 0), thanks to the antighost equations and the Slavnov-Taylor identity, two rigid symmetries are identified by means of the Ward identities: W rig I Γ (0) = g d 3 x where Φ i = (A µ, φ µ, ρ, b, π, c, c, ξ, ξ, A µ, φ µ, ρ, c, ξ ). II Γ(0) = g + [φ W rig d 3 x µ, δγ(0) δa µ {[A µ, δγ(0) ] + δφ µ [ξ, δγ(0) δc { ]} [Φ i, δγ(0) = 0, (46) δφ i ] + i ] + 1 g [π, δγ(0) ] ] ] + [c, δγ(0) + [ ξ, δγ(0) δb δξ δ c } = 0. (47) δγ (0) δρ O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

33 The Jackiw-Pi model Conclusions The BRST symmetry of the model was established and all the difficulties found out in the literature concerning the gauge-fixing were by-passed. At tree-level the spectrum consistency (causality and unitarity) has been verified and we conclude that the Jackiw-Pi model are free from tachyons and ghosts. BRST invariance and Slavnov-Taylor identity together with the antighost equations, in the Landau gauge, allowed to find out two rigid (new) symmetries. An important issue is that the Jackiw-Pi even-parity mass term is not local BRST invariant. It could be conjectured that, at the quantum level, the β-function associated to the mass parameter m should be zero, β m = 0. O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

34 The Jackiw-Pi model... going beyond... Investigate the possible perturbatively ultraviolet finiteness (quantum scale invariance) in the framework of the algebraic renormalization method. Verify the behaviour of the scattering cross sections in the ultraviolet limit by analyzing the Froissart-Martin bound. Study the clash between the two gauge invariances (bifurcation effect), indicating a spontaneous symmetry breaking a. a Suggested by S. Deser in a private communication (see also S. Deser, S. Ertl and D. Grumiller, arxiv: [hep-th]). O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35

35 Olivier, thank you for everything you have done for all of us... all the best my friend, my colleague, my master... O.M. Del Cima (DPF-UFV) The Jackiw-Pi model and all that March / 35