Running Couplings in Topologically Massive Gravity

Transcription

1 Running Couplings in Topologically Massive Gravity Roberto Percacci 1 Ergin Sezgin 2 1 SISSA, Trieste 2 Texas A& M University, College Station, TX ERG Corfu arxiv: [hep-th] - C& QG 2010

2 Outline 1 TMG The theory 2 Calculations Wilsonian beta functions RG flow of TMG Beta functions of TMG 3 Results Ascending root cutoff Descending root cutoff Spectrally balanced cutoff 4 Conclusions Conclusions

3 The theory Topologically massive gravity Action S(γ) = Z d 3 x ( γ 2Λ + R + 1 2µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτγ τ νρ) ) Z = 1 16πG Dimensionless combinations of couplings ν = µg ; τ = ΛG 2 ; φ = µ/ Λ ν 2 = τφ 2

4 The theory Background field expansion γ µν = g µν + h µν Gauge fixing S GF = Z 2α d 3 x gχ µ g µν χ ν, S gh = Later will set β = 2α+1 3 χ ν = µ h µν β νh. d 3 x g C µ ( δ ν µ + 1 β 2 µ ν + R µ ν ) C ν

5 Wilsonian beta functions Modified generating functional { e Wk[J] = Dφ exp S[φ] S k [φ] } dx Jφ IR Cutoff S k [φ] = 1 2 dp φ( p) R k (p 2 )φ(p) = 1 φ 2 2 nr k (λ n ). n Effective average action Γ k [φ] = W k [J] dx Jφ S k [φ],

6 Wilsonian beta functions One loop evaluation For the EA Γ (1) = S Tr log ( S (2)) Similarly Γ (1) k = S Tr log ( S (2) + R k ) therefore k dγ(1) k dk = 1 2 Tr ( S (2) + R k ) 1 k dr k dk

7 Wilsonian beta functions Derivative expansion Γ k (φ, g i ) = k dγ k(φ, g i ) dk n=0 = i n=0 i g (n) i (k)o (n) (φ) β (n) i i (k)o (n) (φ) i

8 RG flow of TMG One loop flow eqn for TMG k dγ(1) k dk = 1 ( δ 2 2 Tr (S + S GF ) δh µν δh ρσ ) 1 + R µνρσ k dr ρσµν dk Tr ( δ 2 S gh δ C µ δc ν + R µ ν ) 1 k drµ ν dk Beta functions of G, µ, Λ can be read off calculating k dγ(1) k S 3 dk on

9 RG flow of TMG Second variations Take g metric on S 3 S (2) + S GF = 1 4 Z d 3 x g [h µν ( 2R3 + 2Λ ) h µν + 2(1 α) h µν µ ρ h ρν α ( + 2 β + 1 ) ) h µ ν (β + 1)2 h µν (1 h h+ 1 h(r 6Λ)h α 8α ( ( µ ελµν h λσ µ R ) ) ] h σν µ σ ρ h ρν 3

10 RG flow of TMG Decomposition h µν = h T µν + µ ξ ν + ν ξ µ + µ ν σ 1 3 g µν σ g µνh µ h T µν = 0 ; g µν h T µν = 0 ; µ ξ µ = 0 likewise for ghosts Field redefinitions C µ = V µ + µ S ; µ V µ = 0 + R 3 ξ µ = ˆξ µ ; ( + R ) σ = ˆσ ; 2 S = Ŝ

11 RG flow of TMG 2µν ρσ S (2) + S GF = Z [ hµν TT µνρσ 4 2 hρσ TT + c 1ˆξµ µν ˆξ ] 1 ν + c σˆσ σˆσ + c h h h h = ( 2R3 ) + 2Λ δ (ρ (µ δ ν) σ) + 1 ( µ ε (µ λ(ρ δ σ) ν) λ R ) 3 1µ ν σ = + 1 α 3 R + 2αΛ = + 2 α 4 α R + 6αΛ 4 α h = + R 4 α + 6Λ 4 α c 1 = 2 α, c σ = 2(4 α) 9α, c h = 4 α 18

12 m σ = m h = m S = n 2 + 2n + 1 RG flow of TMG Eigenvalues and multiplicities λ T ± n = R 6 (n2 + 2n + 2) 2Λ ± 1 µ λ ξ n = R 6 λ σ n = R 6 λ h n = R 6 ( ) n 2 + 2n 3 + 2α 2αΛ, n 2, ( n 2 + 2n 6(2 α) 4 α ( n 2 + 2n 6 4 α m T+ n ) ( ) R 3/2 n(n + 1)(n + 2), n 2, 6 ) 6αΛ 4 α, n 2, 6Λ 4 α, n 0 = m T n = n 2 + 2n 3, m ξ n = m V n = 2(n 2 + 2n),

13 RG flow of TMG Cutoff R k = Z O = Z R k ( 2 ) 2 c 1 1 c 1 R k ( 1 ) c σ σ c h h c σ R k ( σ ) c h R k ( h )

14 RG flow of TMG Collecting Tr(Zc i ( i + R k ( i ))) 1 d dt (Zc ir k ( i )) = Tr(W) W(x) = tr k (x) x + R k (x) k dγ k dk = 1 2 [Tr 2W( 2 ) + Tr 1 W( 1 ) + Tr 0 W( σ ) + Tr 0 W( h )] [Tr 1 W( V ) + Tr 0 W( S )]

15 RG flow of TMG Choice of R k R k (x) = (k 2 x)θ(k 2 x) ; W(x) = 2θ(k 2 x) k dγ k dk = ± + n n m T ± n θ(1 λ T n ± ) + n m ξ nθ(1 λ ξ n) mnθ(1 σ λ σ n) + mnθ(1 h λ h n) n 2 n m V n θ(1 λ V n ) 2 n m S n θ(1 λ S n) (here λ = λ/k 2 )

16 RG flow of TMG Euler-Maclaurin formula n=n 0 F(n) = Perform integrals n 0 dx F(x) F(n 0) B 2 2! F (n 0 ) B 4 4! F (n 0 ) + R n 0 dx m(x)θ(1 λ(x)) = where λ nmax = k 2 or λ nmax = 1 nmax n 0 dx m(x)

17 RG flow of TMG What is n max? 2 Λ 1 2 Λ n n λ T+ n Figure: (solid curve) and λ T n (dashed curve) as functions of n, for R = Λ = Right: large µ regime (here µ = 3). Left: small µ regime (here µ = 0.3).

18 RG flow of TMG Figure: The real (left) and imaginary (right) parts of the roots of the T+ equation λ n = 1, for R = Λ = 0.01, as functions of µ. The solutions of the equation λ T n = 1 are obtained by the reflection µ µ.

19 RG flow of TMG Figure: The real (left) and imaginary (right) parts of the roots of the equation λ T n = 1, for R = Λ = 0.01, as functions of µ. The T+ solutions of the equation λ n = 1 are obtained by the reflection µ µ.

20 Beta functions of TMG Euler-Maclaurin gives k dγ k dk = [ C 0 R 3/2 + C 2 R 1/2 + C 3/ F(n 0) B ] 2 2! F (n 0 ) compare with V(S 3 ) = 2π 2 ( 6 R ( 2Λ Γ k = V(S 3 ) = 3 6π 4 ) 3/2 ; tr(ωdω ω3 ) = 32π 2 16πG 1 ) 16πG R πGµ R3/ ( 2Λ k 3 16πG R 3/2 1 k 16πG R 1/ πGµ +... )

21 Beta functions of TMG R-independent terms n 0 C 3/2 F(n 0 ) F (n 0 ) hµν T ˆξ µ ˆσ h C µ Ĉ [ C 3/ F(n 0) B ] 2 2! F (n 0 ) = 0 β ν = 0

22 Beta functions of TMG Measure couplings in units of cutoff Beta functions G = Gk 1, Λ = Λk 2, µ = µk 1 8π G ( k d Λ dk Λ G k d G ) dk 1 d G k 16π G 2 dk ( π µ G G k d G ) d µ + k dk 1 µ dk = 3 Λ 8π G + A 16π = = π G + B 16π

23 Beta functions of TMG Beta functions of G and Λ β G = G + B( µ) G 2, β Λ = 2 Λ G ( ) A( µ, Λ) + 2B( µ) Λ Since ν = µg = µ G is constant can replace µ by ν/ G

24 Ascending root cutoff Ascending root cutoff For µ > Λ n 2

25 Ascending root cutoff Beta function coefficients A( Λ, µ) = 16 3π + 9(2 3 cos 2θ 3 cos 4θ + 8(cos θ) 3 sin θ) π(cos 3θ) 3 + 8(3 + 11α 2α2 ) π(4 α) Λ + 48(cos θ 3 sin θ) Λ, π sin 6θ 4(1 + α)(11 2α) B( µ) = π(4 α) 2( 3 sin θ cos θ) + 22( 3 sin 5θ + cos 5θ)) 3π sin 6θ, θ = 1 3 arctan 4 µ

26 Ascending root cutoff 1.0 G 0.10 G Figure: The flow in the Λ- G plane for α = 0, ν = 5. Right: enlargement of the region around the origin, showing the Gaussian FP. The beta functions become singular at G =

27 Ascending root cutoff 0.30 G Ν Figure: Position of the FP (left) and eigenvalues of the stability matrix (right) for the nontrivial FP with α = 0, 1 < ν < 40. Note that for this range of ν the singularity is always above the FP. In the left panel, ν grows from right to left. Note that Λ > 0 in this scheme. For large ν, G tends to and the eigenvalues tend to 1 and

28 Descending root cutoff Descending root cutoff For µ < Λ n 2

29 Descending root cutoff 1.0 G 0.10 G Figure: The flow in the Λ- G plane for α = 0, ν = 0.1. Right: enlargement of the region around the origin, showing that there is no Gaussian FP. The beta functions diverge on the Λ axis.

30 Descending root cutoff 0.8 G Ν Figure: Position of the FP (panel) and eigenvalues of the stability matrix (panel) for the nontrivial FP with α = 0, 10 6 < ν < 0.5, in the descending root cutoff scheme. In the left panel, ν decreases from right to left. Λ changes sign for ν = The rightmost point (ν = 0.5) has µ 3 > 27/4.

31 Spectrally balanced cutoff Spectrally balanced cutoff For any µ choose same n max for λ TT+ and λ TT. 2 Λ n 2

32 Spectrally balanced cutoff A( µ, Λ) = 16 3π π(cosh η) 3 + 8(3 + 11α 2α2 ) 16 3 Λ + π(4 α) π(cosh 3η + 2 cosh η) Λ B( µ) = 4(11 + 9α 2α2 ) 3π(4 α) η = 1 3 arctanh 8 3 9π 1 4 µ2 27 ( ) cosh 2η cosh 3η + 2 cosh η

33 Spectrally balanced cutoff 1.0 G Ν Figure: Left: The position of the FP in the Λ- G plane, varying ν from (upper left) to 1000 (lower right). The point with coordinates (0,0.2005) is the limit ν. Right: The eigenvalues of M as functions of ν. For ν = they are 1 and while for ν = 1000 they are and

34 Conclusions General features Simple form of R k implies spectrally unbalanced cutoff. No choice of roots is good for all µ: ascending root good for large µ (small G), descending root for small µ (large G) Spectrally balanced cutoff good for all µ Qualitative picture consistent Agreement with heat kernel calculation for µ

35 Conclusions Scheme independent features β ν = 0 expected for topological reasons GFP with crit exp 1 and 2 NGFP with crit exp 1 and 2 G is positive

36 Conclusions Scheme dependent features position of NGFP in particular, there is residual uncertainty on sign of cosmological constant at NGFP for large µ scaling exponents at NGFP (slightly)

37 Conclusions Gauge dependence EOM gives R = 6Λ, but we did not use this. Off shell EA gauge dependent, so off shell beta functions are gauge dependent. On shell Hilbert action is proportional to V(S 3 Λ ) 16πG 1 ΛG 1 τ so expect β τ to be gauge independent. Indeed β τ A + 6B Λ = 16 3π (1 + 3 Λ) + µ-dependent terms

38 Conclusions Asymptotic safety In d 4 large value of G consistency of truncation For large µ, TMG is an example of asymptotically safe theory which can be studied perturbatively Since Riemann can be expressed in terms of Ricci, higher derivative terms can be eliminated order by order in perturbation theory