Scattering in Topologically Massive Gravity, Chiral Gravity and the corresponding Anyon-Anyon Potential Energy
aa r X i v : . [ h e p - t h ] N ov Scattering in Topologically Massive Gravity, Chiral Gravity and thecorresponding Anyon-Anyon Potential Energy
Suat Dengiz, ∗ Ercan Kilicarslan, † and Bayram Tekin ‡ Department of Physics,Middle East Technical University, 06800, Ankara, Turkey (Dated: November 22, 2013)We compute the tree-level scattering amplitude between two covariantly conserved sourcesin generic Cosmological Topologically Massive Gravity augmented with a Fierz-Pauli termthat has three massive degrees of freedom. We consider the Chiral Gravity limit in the anti-deSitter space as well as the limit of Flat-Space Chiral Gravity. We show that Chiral Gravitycannot be unitarily deformed with a Fierz-Pauli mass. We calculate the non-relativisticpotential energy between two point-like spinning sources. In addition to the expected mass-mass and spin-spin interactions, there are mass-spin interactions due to the presence of thegravitational Chern-Simons term which induces spin for any massive object and turns it toan anyon. We also show that the tree-level scattering is trivial for the Flat-Space ChiralGravity.
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I. INTRODUCTION
Gravity in 2 + 1 dimensions is a fertile theoretical ground in which plethora of interestingstructures and phenomena have been found [1–3]. In spite of its apparent simplicity, with avanishing Weyl tensor and no local degrees of freedom, Cosmological Einstein’s theory has blackhole solutions and the associated thermodynamics [4]. Ultimately, of course, the goal is to find aquantum gravity theory in this lower dimensional setting and hopefully learn about the structure ofquantum gravity to build one in the more relevant 3 + 1-dimensions. The natural question is, afterabout 30 years of research, whether or not we are any closer to quantum gravity in 2+1 dimensions.The answer depends on one’s level of optimism, while no quantum version of 2 + 1-dimensionalEinstein’s theory exists as of now, the situation is much better when Cosmological Einstein’s theoryis augmented with a tuned gravitational Chern-Simons term. In this case, "Chiral Gravity" [5, 6]which is potentially a viable quantum gravity theory is conjectured to have a dual unitary boundaryconformal field theory (CFT). Generically, Einstein’s Gravity plus the gravitational Chern-Simonsterm, that is the Topologically Massive Gravity (TMG) [7], has a single massive spin-2 excitationin both flat and (A)dS spacetimes. In the Chiral Gravity limit, massive graviton disappears andone is left with a BTZ black hole with positive energy and a boundary chiral CFT from which onecan relate entropy to microscopic states, a situation which seems to be lacking in CosmologicalEinstein’s theory in 2+1 dimensions.We started this work with the following question in mind: Suppose Chiral Gravity is a viabletheory, how then two covariantly conserved charges scatter at the lowest order with a single gravitonexchange in this theory. Of course, one immediately realizes that Chiral Gravity cannot be easilycoupled to a generic matter source but only to null matter. We then ask the more general question:How do two covariantly conserved particles scatter at the lowest order in Cosmological TMG ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] augmented with a Fierz-Pauli term. Fierz-Pauli term is somewhat a headache because of itsnon-covariance but at the linearized level it helps to find the propagator without worrying aboutgauge-fixing issues. And also, in 2+1 dimensions Fierz-Pauli massive gravity has a remarkablenon-linear extension dubbed as "New Massive Gravity [8] and even an infinite order extension ofthe Born-Infeld type [9].Our task in this work is to find first a formal expression for the tree-level scattering in Cosmologi-cal TMG with a Fierz-Pauli term and then consider the flat space limit. In the explicit computationof the non-relativistic potential energy, we will recover the gravitational anyons which were foundby Deser [10] as solutions of the linearized field equations in flat space TMG. It will be clear thatbecause of the gravitational Chern-Simons term a point-like structureless, spinless particle withmass m acts as if it has a spin κm/µ where κ is the Newton’s constant and µ is the gravitationalChern-Simons parameter. These gravitational anyons are analogs of their Abelian counterpartswhere any charged particle picks up a magnetic flux when coupled to an Abelian Chern-Simonsterm [11].The lay-out of the paper is as follows: We first find the particle content of the CosmologicalTMG with a Fierz-Pauli term in Section-II. That section is a generalization of the flat spaceversion of the same theory [12]. Moreover in that section we write the linearized field equationsin a form which can be formally solved by the Green’s function technique. Section III is devotedto a derivation of the tree-level scattering amplitude using the tensor decomposition of the spin-2field in terms of its irreducible parts. In that section, we also give the Chiral Gravity limit of theamplitude. In section IV we limit ourselves to the flat space, where Green’s functions are explicitlycomputable and the notion of potential energy makes sense as a non-relativistic approximation.There we consider two sources with mass and spin that interact via the full TMG+ Fierz-Paulitheory and find the anyon structure of the sources. II. PARTICLE SPECTRUM FOR TMG WITH A FIERZ-PAULI MASS IN (A)DSBACKGROUNDS
The Lagrangian density of TMG with a Fierz-Pauli mass term is L = √− g ( κ ( R − − m κ ( h µν − h ) + 12 µ η µνα Γ βµσ (cid:16) ∂ ν Γ σαβ + 23 Γ σνλ Γ λαβ (cid:17) + L matter ) , (1)where κ is the 2+1-dimensional Newton’s constant with mass dimension − µ is a dimensionlessparameter. η µνα tensor is defined in terms of the Levi-Civita symbol as ǫ µνα / √− g . We willwork with the mostly plus signature. Generically, this parity non-invariant spin-2 theory has three modes all with different masses about its flat and (A)dS backgrounds. There are constraints on theparameters coming from the tree-level unitarity. Observe that when the Fierz-Pauli term vanishes,the theory reduces to TMG with a single massive spin-2 mode with M graviton = µ /κ + Λ. On theother hand, µ → ∞ theory corresponds to the Fierz-Pauli massive gravity with two massive spin-2excitations, with mass m . In the full theory, with all the parameters non-vanishing, the masses ofthe three modes and the unitarity regions were studied in [12] for flat spacetimes. Here, in thissection, we generalize this result to the (A)dS backgrounds.To find the particle spectrum about the (A)dS vacuum and the propagator, let us consider thefield equations coming from the variation of (1):1 κ ( R µν − g µν R + Λ g µν ) + 1 µ C µν + m κ ( h µν − g µν h ) = τ µν , (2)where C µν is the symmetric, traceless and divergence-free Cotton tensor, C µν = η µαβ ∇ α (cid:16) R νβ − δ ν β R (cid:17) , (3)which vanishes if and only if the spacetime is conformally flat. In what follows it pays to rewriteit in an explicitly symmetric form with the help of the Bianchi identity ( ∇ µ G µν = 0) C µν = 12 η µαβ ∇ α G ν β + 12 η ναβ ∇ α G µβ . (4)Let us consider the linearization of (2) about an (A)dS background g µν = ¯ g µν + h µν where ¯ R µναβ =Λ(¯ g µα ¯ g νβ − ¯ g µβ ¯ g να ), which is the vacuum of (2) with m = 0, τ µν = 0:1 κ G Lµν + 12 µ η µαβ ¯ ∇ α G Lν β + 12 µ η ναβ ¯ ∇ α G Lµ β + m κ ( h µν − ¯ g µν h ) = T µν . (5)Here T µν = τ µν + Θ( h , h , ... ) and satisfies the background covariant conservation ¯ ∇ µ T µν = 0.The 2+1-dimensional linearized curvature tensors that we need are [13] G Lµν = R Lµν −
12 ¯ g µν R L − h µν ,R Lµν = 12 ( ¯ ∇ σ ¯ ∇ µ h νσ + ¯ ∇ σ ¯ ∇ ν h µσ − ¯ (cid:3) h µν − ¯ ∇ µ ¯ ∇ ν h ) ,R L = (¯ g µν R µν ) L = − ¯ (cid:3) h + ¯ ∇ µ ¯ ∇ ν h µν − h, (6)where h = ¯ g µν h µν . To be able to identify the excitations, we have to write (5) as a source-coupled(higher derivative) wave-type equation of the form( ¯ (cid:3) − − m )( ¯ (cid:3) − − m )( ¯ (cid:3) − − m ) h µν = ˜ T µν , (7)where m i correspond to the masses of the excitations. Note that in (A)dS backgrounds ( ¯ (cid:3) − h µν = 0 is the wave equation for a massless spin-2 particle, hence the shifts in (7). Sincethe Levi-Civita tensor mixes various excitations in (5), we need to manipulate the equation to“diagonalize” it. But, before that let us note a special point in the parameter space. The divergenceof (5) gives m ( ¯ ∇ µ h µν − ¯ ∇ ν h ) = 0 , (8)yielding, for m = 0, R L = − h, h = κ Λ − m T, G L ≡ ¯ g µν G Lµν = Λ κ Λ − m T. (9)Observe that at the "partially massless point" m = Λ and so h is not fixed and a new higherderivative gauge invariance of the form δ ξ h µν = ¯ ∇ µ ¯ ∇ ν ξ + Λ ξ ¯ g µν appears reducing the number ofthe degrees of freedom by one from three to two [12, 14, 15]. Needless to say that this particulartheory has no flat space limit and it shall not be considered in the rest of the paper.Now let us try to bring (5) to the form (7) where the masses of the excitations become apparent.For this purpose, applying η µσρ ¯ ∇ σ to equation (5), gives1 κ η µσρ ¯ ∇ σ G Lµν − µ ¯ (cid:3) G Lν ρ + 3Λ µ G Lν ρ + m κ η µσρ ¯ ∇ σ ( h µν − ¯ g µν h ) = η µσρ ¯ ∇ σ T µν + Λ µ δ ρν G L + 12 µ ¯ ∇ ν ¯ ∇ ρ G L − µ δ ρν ¯ (cid:3) G L , (10)where we have made use of the identity η µσρ η ναβ = (cid:20) − δ µν (cid:16) δ σα δ ρβ − δ σβ δ ρα (cid:17) + δ µα (cid:16) δ σν δ ρβ − δ σβ δ ρν (cid:17) − δ µβ (cid:16) δ σν δ ρα − δ σα δ ρν (cid:17)(cid:21) . (11)Using the field equation (2) to eliminate the first term in (10) one obtains (cid:16) ¯ (cid:3) − − µ κ (cid:17) G Lρν − µ m κ ( h ρν − ¯ g ρν h ) − µm κ η µσρ ¯ ∇ σ ( h µν − ¯ g µν h )= µ η ρµσ ¯ ∇ µ T σν + µ η ν µσ ¯ ∇ µ T σρ − µ κ T ρν − Λ¯ g ρν G L −
12 ¯ ∇ ν ¯ ∇ ρ G L + 12 ¯ g ρν ¯ (cid:3) G L . (12)To transform (12) into a wave-type equation, one should write the Fierz-Pauli part in terms of thelinearized Einstein tensor G Lµν and its contractions. For this purpose, let us define a new tensor B µν ≡ η µαβ ¯ ∇ α G Lν β , (13)with ¯ g µν B µν = B = 0 and ¯ ∇ µ B µν = 0. Then, by plugging (13) into (12) and hitting with η ραβ ¯ ∇ α ,one arrives at − κ η ραβ ¯ ∇ α B ρν + 12 µ η ραβ η µσρ ¯ ∇ α ¯ ∇ σ B µν + 12 µ η ραβ η µσρ ¯ ∇ α ¯ ∇ σ B νµ + m κ η ραβ η µσρ ¯ ∇ α ¯ ∇ σ ( h µν − ¯ g µν h ) = η ραβ η µσρ ¯ ∇ α ¯ ∇ σ T µν . (14)After a somewhat lengthy but straightforward calculation, one can find the following expression m κ ( h βν − ¯ g βν h ) = − m κ ( ¯ (cid:3) − − G Lβν + Λ κ ( ¯ (cid:3) − − ¯ g βν G L − m κ ( ¯ (cid:3) − − (cid:16) ¯ g βν ¯ (cid:3) − ¯ ∇ β ¯ ∇ ν (cid:17) h − Λ( ¯ (cid:3) − − ¯ g βν T, (15)where the inverse of an operator is a short-hand notation for the corresponding Green’s function.By inserting (15) into (12) and using (9), one finds ( ¯ (cid:3) − − µ κ ) + 2 µ m κ ( ¯ (cid:3) − − − µ m κ ( ¯ (cid:3) − − ! G Lρν = µ η ρµσ ¯ ∇ µ T σν + µ η ν µσ ¯ ∇ µ T σρ − µ κ T ρν + µ m κ ( ¯ (cid:3) − − T ρν − µ m κ Λ(1 − m Λ ) ( ( ¯ (cid:3) − − (cid:16) − m ( ¯ (cid:3) − − (cid:17) − κ Λ µ m ) × (cid:16) ¯ g ρν ( ¯ (cid:3) − − ¯ ∇ ρ ¯ ∇ ν (cid:17) T, (16)where G Lρν = −
12 ( ¯ (cid:3) − h ρν + 12 ¯ ∇ ρ ¯ ∇ ν h. (17)Equation (16) is almost in the desired form, except, there is an h on the left-hand side. But thiscan be remedied with the help of h = κ Λ − m T . Then (16) reads as O h ρν = ˜ T ρν , (18)as needed. To study the linearized gravitational degrees of freedom, let us stay away from thesources and set T ρν = 0, which gives ˜ T ρν = 0. Then higher-order wave-type equation for Cosmo-logical TMG with a Fierz-Pauli mass term boils down to "(cid:16) ¯ (cid:3) − − µ κ (cid:17) ( ¯ (cid:3) − + 2 µ m κ ( ¯ (cid:3) − − µ m κ h ρν = 0 , (19)which, in flat space, reduces to the known form [12] " ( ∂ ) − µ κ ( ∂ ) + 2 µ m κ ∂ − µ m κ h ρν = 0 , (20)that has three distinct real roots corresponding to the masses of the excitations only if µ /m κ ≥ / M − (Λ + µ κ ) M + 2 µ m κ M − µ m κ = 0 , (21)where again the 3 roots M i are the masses of the excitations. In general there are complex rootsunless 1 + 9 κ Λ µ (1 − m
4Λ ) ≥ Λ m (1 + κ Λ µ ) , (22)which guarantees the non-negativity of the discriminant. The explicit forms of all the 3 roots aresomewhat cumbersome to write, therefore we shall not display them here. But let us note thefollowing cases:1. When µ κ = − Λ, which is the Chiral Gravity limit, there are two tachyonic excitations unless m = 0. This means that Chiral Gravity cannot be deformed unitarily with a Fierz-Paulimass.2. In the m = 0 limit caution must be exercised: Namely there are ostensibly two solutions M = 0 and m = Λ + µ κ . The latter is the well-known massive mode while the first solutiondoes not actually exist. This can be seen from the equations if one started with m = 0 inthe beginning [16, 17] .3. As a specific example, let us take m = , and µ κ = 3Λ then all the 3 roots are equal M = M = M = 4Λ3 , (23)which obey the Higuchi bound [18] in dS space ( M > Λ >
0) but do not obey theBreitenlohner-Freedman bound [19] in AdS ( M > Λ). At this specific point, the factthat helicity+2 and helicity-2 modes have the same mass in dS does not show that paritysymmetry is restored.4. In the µ → ∞ limit, which is the Einstein-Fierz-Pauli theory one gets two excitations withthe same mass m .Having found the spectrum of the full theory, we now move on to calculate the tree-levelscattering amplitude between two conserved currents ( ¯ ∇ µ T µν = 0) . III. SCATTERING AMPLITUDE IN GENERIC TMG PLUS FIERZ-PAULI THEORY
Equation (16) is of the form O h µν = ˜ T µν , where O is a complicated operator. Since notevery component of h µν is dynamical, it pays to decompose the spin-2 field in terms of transversehelicity-2 ( h T Tµν ), helicity-1 ( V µ ) and helicity-0 components ( φ, ψ ) as h µν ≡ h T Tµν + ¯ ∇ ( µ V ν ) + ¯ ∇ µ ¯ ∇ ν φ + ¯ g µν ψ. (24)Taking the trace of (24) and the divergence of (8) lead to an elimination of φ and a relation between h and ψ as h = 1Λ ( ¯ (cid:3) + 3Λ) ψ. (25)Then the trace of the field equations (9) yield ψ = κ − m Λ ( ¯ (cid:3) + 3Λ) − T. (26)To relate h T Tµν to the source, one needs to use the Lichnerowicz operator, △ (2) L acting on thesymmetric spin-2 tensors as △ (2) L h µν = − ¯ (cid:3) h µν − R µρνσ h ρσ + 2 ¯ R ρ ( µ h ν ) ρ , (27)with the following properties [20] △ (2) L ∇ ( µ V ν ) = ∇ ( µ △ (1) L V ν ) , △ (1) L V µ = ( − (cid:3) + Λ) V µ , ∇ µ △ (2) L h µν = △ (1) L ∇ µ h µν , ∇ µ △ (1) L V µ = △ (0) L ∇ µ V µ , △ (2) L g µν φ = g µν △ (0) L φ, △ (0) L φ = − (cid:3) φ. (28)Then, with the help of (28), transverse-traceless part of the linearized Einstein tensor G T TL ρν canbe written in terms of Lichnerowicz operator as G T TL ρν = 12 (cid:16) △ (2) L − (cid:17) h T Tρν . (29)Substituting (29) into (16) gives a relation between the transverse-traceless field and the sources h T Tρν = µ O − ( ¯ (cid:3) − η ρµσ ¯ ∇ µ T T Tσν + µ O − ( ¯ (cid:3) − η νµσ ¯ ∇ µ T T Tσρ − µ κ O − ( ¯ (cid:3) − T T Tρν + 2 µ m κ O − ( ¯ (cid:3) − T T Tρν , (30)where the scalar Green’s function O − is O − ≡ (h ( ¯ (cid:3) − (cid:16) ¯ (cid:3) − − µ κ (cid:17) + 2 µ m κ ( ¯ (cid:3) − − µ m κ i × (cid:16) △ (2) L − (cid:17)) − . (31)Additionally, doing a similar tensor decomposition (24) for T ρν , one can write the transverse-traceless part T T Tρν [21] as T T Tρν ≡ T ρν −
12 ¯ g ρν T + 12 (cid:16) ¯ ∇ ρ ¯ ∇ ν + Λ¯ g ρν (cid:17) × ( ¯ (cid:3) + 3Λ) − T. (32)With all the above results, we are ready to write the tree-level scattering amplitude between twosources as A = 14 ˆ d x p − ¯ gT ′ ρν ( x ) h ρν ( x )= 14 ˆ d x p − ¯ g ( T ′ ρν h T T ρν + T ′ ψ ) . (33)Finally plugging (26), (29) and (32) into (33), one gets the amplitude as4 A = 2 µT ′ ρν O − ( ¯ (cid:3) − η ρµσ ¯ ∇ µ T σν − µ κ T ′ ρν O − ( ¯ (cid:3) − (cid:3) − − m ) T ρν − µ κ T ′ ρν O − ( ¯ (cid:3) − (cid:3) − − m )( ¯ ∇ ρ ¯ ∇ ν + Λ¯ g ρν ) × (cid:16) ¯ (cid:3) + 3Λ (cid:17) − T + µ κ T ′ O − ( ¯ (cid:3) − (cid:3) − − m ) T + κ − m Λ T ′ ( ¯ (cid:3) + 3Λ) − T, (34)where for notational simplicity we have suppressed the integral signs.The pole structure of the full theory is highly complicated: there are apparently 4 poles. Butin the most general case, it is hard to see from the scattering amplitude, whether the fourth pole,besides the 3 which are exactly the roots of the cubic equation (21), is unphysical or not. Inany case, the introduction of the Fierz-Pauli term served its purpose of making the propagatorinvertible and hence we can now set it to zero and consider the most promising limit of the generaltheory that is the Chiral Gravity with m = 0 , µ /κ = − Λ. Strictly speaking one must keep h = 0, to get the Chiral Gravity. Hence, we must also set T = 0. Then the amplitude in Chiralgravity reads 4 A = 2 µT ′ ρν ((cid:16) ¯ (cid:3) − (cid:17) × (cid:16) △ (2) L − (cid:17)) − η ρµσ ¯ ∇ µ T σν − µκ T ρν ! , (35)recall that the theory is valid in AdS with Λ <
0. As expected the massive mode dropped out.
IV. FLAT SPACE CONSIDERATIONS
In this section, we will study the flat space limit of the scattering amplitude (34) in varioustheories and the corresponding Newtonian potential energy ( U ) between two localized conservedspinning point-like sources defined by T = m a δ (2) ( x − x a ) , T i = − J a ǫ ij ∂ j δ (2) ( x − x a ) , (36)where a = 1 , m a and J a refer to the mass and spin respectively. (Note that spin in 2 + 1dimensions is a pseudoscalar quantity which can be negative or positive.) A. Scattering of Anyons in TMG with a Fierz-Pauli term
In the Λ → A = − µT ′ ρν ∂ ∂ ( ∂ − µ κ ) + µ m κ ∂ − µ m κ η ρµσ ∂ µ T σν + 2 µ κ T ′ ρν ∂ − m ∂ ( ∂ − µ κ ) + µ m κ ∂ − µ m κ T ρν − µ κ T ′ ∂ − m ∂ ( ∂ − µ κ ) + µ m κ ∂ − µ m κ T. (37)As long as one is looking at the generic theory where the masses are distinct, one can fractionallydecompose the propagator as ∂ − m ∂ ( ∂ − µ κ ) + µ m κ ∂ − µ m κ ≡ X k =1 3 Y r =1 r = k ( M k − m )( M k − M r ) G k ( x , x ′ , t, t ′ ) , (38)where the scalar Green’s function is G k ( x , x ′ , t, t ′ ) = ( ∂ − M k ) − and M k = M k ( κ , µ , m ), k = 1 , ,
3, are the generic roots of the equation (21). Substituting (38) into (37) and using (36)and carrying out the time integrals yield4 U = X k =1 3 Y r =1 k = r ( M k − M r ) − ( µ M k κ (cid:16) κm µ J + κm µ J + J J (1 − m M k ) (cid:17) × ˆ d x ˆ d x ′ δ (2) ( x ′ − x ) ∂ i ∂ i ˆ G k ( x , x ′ ) δ (2) ( x − x )+ µ m m κ ( M k − m ) ˆ d x ˆ d x ′ δ (2) ( x ′ − x ) ˆ G k ( x , x ′ ) δ (2) ( x − x ) ) , (39)where the potential energy is defined as U = A /t (See [22]) and the time-integrated Green’s functionreads ˆ G k ( x , x ′ ) = ˆ dt ′ G k ( x , x ′ , t, t ′ ) = 12 π K ( M k | x − x ′ | ) . (40)Finally using the recurrence relation between the modified Bessel functions ~ ∇ K ( M k r ) = M k (cid:16) K ( M k r ) + K ( M k r ) (cid:17) , (41)where r = | x − x | , one obtains U = X k =1 3 Y r =1 k = r ( M k − M r ) − ( µ M k πκ (cid:16) J tot J tot − κ m m µ − m J J M k (cid:17) K ( M k r )+ µ M k πκ h m m M k (1 − m M k ) + (cid:16) J tot J tot − κ m m µ − m J J M k (cid:17)i × K ( M k r ) ) . (42)We defined the total spin as the original spin of the source plus the induced spin due to thegravitational Chern-Simons term, turning the source to an anyon [10]: J tota ≡ J a + κm a µ , a = 1 , . (43)Our result not only reveals the anyon structure of the sources, but it also describes how anyonsscatter at small energies. For other works on gravitational anyons see [23–26]. Observe thatdepending on the choice of ( J a , m a , m ), U can be either negative or positive or even it couldvanish.Let us now consider the short and large distance limits of the anyon-anyon potential energy.First, in short distances, since the Bessel function behave as K ( M k r ) ∼ − ln( M k r ) − γ E , K ( M k r ) ∼ M k r , (44)the potential energy reads U ∼ X k =1 3 Y r =1 k = r ( M k − M r ) − ( µ M k πκ (cid:16) J tot J tot − κ m m µ − m J J M k (cid:17) × r − µ M k πκ h m m M k (1 − m M k ) + (cid:16) J tot J tot − κ m m µ − m J J M k (cid:17)i × (cid:16) ln( M k r ) + γ E (cid:17)) , (45)where γ E is the Euler-Mascheroni constant. On the other side, for large distances, since the Besselfunctions decay as K n ( M k r ) ∼ r π M k r e − M k r , (46)the potential energy becomes U ∼ X k =1 3 Y r =1 k = r ( M k − M r ) − µ M k πκ h m m M k (1 − m M k ) + (cid:16) J tot J tot − κ m m µ − m J J M k (cid:17)i × r π M k r e − M k r . (47) B. Scattering of Anyons in TMG
We now consider the scattering of anyons and find the related potential energy for TMG withoutthe Fierz-Pauli mass. Taking m → → A = − µT ′ ρν ∂ − µ κ ) ∂ η ρµσ ∂ µ T σν + 2 µ κ T ′ ρν ∂ − µ κ ) ∂ T ρν − µ κ T ′ ∂ − µ κ ) ∂ T + κT ′ ∂ T, (48)which generically has a massive and a massless modes. The explicit computation of the potentialenergy follows along the same line of the previous section. One finally arrives at the anyon-anyonscattering potential energy in TMG: U = κm g π (cid:26)(cid:16) J tot J tot − m m m g (cid:17) K ( m g r ) + (cid:16) J tot J tot + m m m g (cid:17) K ( m g r ) (cid:27) , (49)where m g = µ /κ .0Let us now check the small and large distance behaviors of the potential energy: First of all,for small separations, one obtains U ∼ κ π (cid:16) J tot J tot − m m m g (cid:17) r − κm g π (cid:16) J tot J tot + m m m g (cid:17)(cid:16) ln( m g r ) + γ E (cid:17) . (50)At large distances (49) behaves as U ∼ κm g J tot J tot π s π m g r e − m g r , (51)which of course could be repulsive or attractive. For the specific case of the tuned spin J = − κm/µ ,there is no interaction at large separations. C. Scattering of Anyons in Flat-Space Chiral Gravity
In [28] as an example of the holographic correspondence between a gravitational theory in flatspace and a conformal field theory (CFT) in a lower dimensional space (which is akin to theAdS/CFT correspondence) a chiral gravity is constructed as a limit of TMG, which the authorsdubbed “Flat-Space Chiral Gravity” and showed that a pure gravitational Chern-Simons term withlevel k , i.e., S = k π ˆ d x √− g η µνα Γ βµσ (cid:16) ∂ ν Γ σαβ + 23 Γ σνλ Γ λαβ (cid:17) , (52)is dual to a CFT with a chiral charge c = 24.Here we consider the scattering amplitude and the Newtonian potential energy in Flat-SpaceChiral Gravity. To do so, let us note how Flat-Space Chiral Gravity arises from TMG: κ → ∞ , µ → πk . (53)From (34) and with T = 0 and m = 0, one obtains4 A = − πk T ′ ρν ∂ η ρµσ ∂ µ T σν . (54)We need to construct a covariantly conserved traceless source . To do this we can write theMinkowski space in null coordinates as ds = − dudv + dy , with u = t + x and v = t − x . Then thevector l µ ≡ ∂ µ u satisfies l µ = − δ µv and l µ l µ = 0. Therefore the null source should read T µν ∼ l µ l ν .Together with the condition ∇ µ T µν = 0, we have T µν = Eδ ( u ) δ ( y ) δ µv δ νv . Substitution of this in(54) yields a trivial scattering amplitude. V. CONCLUSION
We have studied the 2+1-dimensional Cosmological Topologically Massive gravity (CTMG)augmented by a Fierz-Pauli mass term in (A)dS and flat backgrounds in detail. We first found the After time integration of ( ∂ ) − , one gets the biharmonic Green’s function: ´ dt ( ∂ ) − ( x , x ′ , t, t ′ ) = − | x − x ′ | π (cid:16) ln | x − x ′ | ζ − (cid:17) . VI. ACKNOWLEDGMENTS
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