One-loop renormalization in a toy model of Hořava-Lifshitz gravity

Transcription

1 1/0 Università di Roma TRE, Max-Planck-Institut für Gravitationsphysik One-loop renormalization in a toy model of Hořava-Lifshitz gravity Based on (hep-th: ) with Dario Benedetti Filippo Guarnieri Berlin,

2 /0 1 Motivations Hořava-Lifshitz gravity 3 Quantization procedure 4 β-functions 5 Conclusions and discussion

3 3/0 Motivations General relativity is perturbatively non-renormalizable. String theory? Asymptotic safety scenario? Power-counting renormalizability can be reconciled with unitarity at the expenses of a scale anisotropy between space and time. Hořava-Lifshitz gravity. Not much is known about the behaviour Hořava-Lifshitz gravity in the UV. Difficulties coming from the large number of invariants and working on an anisotropic curved background.

4 4/0 Hořava-Lifshitz gravity General relativity is power-counting non-renormalizabile in d >. [G] = d. Higher-derivative theories can cure the perturbative non-renormalizability, but they suffer from a lack of unitarity. S[g µν] hd = 1 d d x g {R γ R β R µν R µν Λ}, (1) 16 π G The propagator of spin- modes contains a poltergeist which spoils unitarity. Π(p) = 1 p β G p 4, p ghost = (β G) 1. ()

5 5/0 Hořava-Lifshitz gravity Power-counting renormalizability can be obtained by means of anisotropic scale dimensions for space and time coordinates, emulating what happens in condensed matter. [x] = 1, [t] = z, [ x] = 1, [ t ] = z. (3) As an example, for a Lifshitz scalar field theory we have S[φ] = dt d d x { φ(t, x) ( t G z x ) φ(t, x)}. (4) The propagator of the Lifshitz scalar field is 1 ω G(k ) z, (5) and a term c x in the action acts now (for z > 1) as a relevant perturbation.

6 6/0 Hořava-Lifshitz gravity Introducing a scale anisotropy we can gain perturbative-power counting and unitarity, but we lose Lorentz invariance. Invariance under diffeomorphisms is substituted by the invariance under a foliation-preserving diffeomorphisms, i.e. the reparameterization x i = x i (x j, t), t = t(t). (6) We have to build invariants under the new symmetry group. We employ the ADM decomposition, with g ij the spatial metric, N i the shift vector and N the lapse function. A natural spacetime topology in presence of a foliation F is M = N Σ. (7) where the leaf Σ is a generic d-dimensional manifold. We will consider N = R.

7 7/0 Hořava-Lifshitz gravity A kinetic action in d + 1 dimensions reads S K [N, N i, g ij ] = κ dt d d x ( g N K ij K ij λk ), (8) being K ij = 1 N ( t g ij D i N j D j N i ), (9) where D i is the spatial covariant derivative. At this stage the dynamical critical exponent z enters only in the dimension of the integration measure [dt d d x] = d z. (10) As a consequence, Newton s constant κ is a marginal parameter for z = d, since [κ] = z d. (11)

8 8/0 Hořava-Lifshitz gravity We can add a potential term by constructing spatial invariants S[N, N i, g ij ] = S K [N, N i, g ij ] + κ dt d d x g N V (g ij ). (1) In 3+1 dimensions with z = 3 the potential V (g ij ) will contain marginal operators like R 3, R ij R jk R k i, R R ij R ij D R D k R ij D k R ij (13) plus additional relevant terms as R, R ij R ij, D R D i D j R ij, (14) The number of invariants is reduced if the system features detailed balance, that is, if the d + 1-dimensional potential is related to a d-dimensional potential by means of a variational principle.

9 9/0 The + 1 dimensional case The + 1-dimensional case is easier to study, since for z = the potential contains fewer invariants. Furthermore, the Ricci is the only independent component of the Riemann tensor study and there are no gravitons. The theory with detailed balance has no potential. The most general action (non-projectable and without detailed balance) in + 1 dimensions with z = is S[N, N i, g ij ] = dt d xn { ( g λ K κ K ij K ij Λ + c R + γ R ) + c 1 D R +c a i a i + c 3 (a i a i ) + c 4 R a i a i + c 5 a i a i D j a j +c 6 (D j a j ) + c 7 (D i a j )(D i a j ) }, (15) where a i is the acceleration vector, a i = D i log N.

10 10/0 The + 1-dimensional case We will consider two simplifications: Projectable case, that is, a constant lapse function over the leaf, N N(t), so that S[N, N i, g ij ] = κ dt d xn { g λ K K ij K ij Λ + c R + γ R }. (16) Conformal reduction, that is, a scalar toy model in which we integrate only the conformal degree of freedom of the spatial metric, g ij = e φ(x) g ij, (17) where g ij is considered as a constant background. We will chose Σ = S. Questions?

11 11/0 Quantization procedure We will employ as usual the background field method: g ij g ij + ɛ h ij ; N N + ɛ n ; N i N i + ɛ n i, (18) and set N = 1 and N j = 0 for the background lapse and shift. The perturbative parameter ɛ will be set at a later stage. The metric fluctuation can be decomposed in a trace and traceless parts as h ij = ĥij + 1 g ij h, g ij ĥ ij = 0. (19) On a sphere S the traceless part ĥij contains just longitudinal components. In our toy model we neglect those contribution and integrate only the trace term.

12 1/0 Gauge-fixing and ghosts We employ as a gauge choices n = 0 and n i = 0. The gauge-fixing action reads S gf = 1 α dt d x g n + 1 β dt d x g n i n i. (0) Taking the limit α 0 and β 0 leads to a complete decoupling of n and n i in the second variation of the action. For the ghost sector, in order to avoid positivity problems of the Faddeev-Popov operator M = t we will employ the squared root of its determinant, det( M ), which corresponds to the ghost action S gh = dt N d x } g { c t c + c i t ci + b t b + b i t bi, (1) where c i and c Grassmannian complex fields and b i and b real bosonic fields.

13 13/0 One-loop effective action We will focus on evaluating only the one-loop correction to the effective action, Γ = S tot + ħ S 1 loop + O(ħ ), () being and where S tot = S + S gf + S gh, (3) S 1 loop = 1 STr ln S() tot. (4) The Hessian can be evaluated by the reduced second variation of the action, i.e. δ S = 1 κ dt d x {( g λ 1 ) } ( t h) + γ h (D 4 + R D + R ) h. (5) The perturbative parameter is chosen so to normalize the kinetic operator, so that in our case κ ɛ =. (6) (λ 1 ) 1

14 14/0 Heat kernel techniques We will employ heat-kernel techniques to evaluate the one-loop term, i.e. S 1 loop = 1 Tr ln(s() ) = µ 4 1 Λ 4 ds s 1 µ 4 1 Λ 4 dt d x ds s Tr H(x, s; S() ) = { } ĝ a 0 + s 1 a 1 + s a + O(s 3 ), (7) where a = b 1 K ij K ij + b K + b 3 R +, and µ is a renormalization scale. For the coefficients b 1 and b we can use the result of Baggio, de Boer and Holsheimer ( ) for an anisotropic differential operator D action on a scalar field, i.e. D bbh = 1 N g 1 t g t + D 4, (8) N whereas our operator reads instead D = (λ 1 ) 1 g t g t + γ (D + R). (9)

15 15/0 Heat-kernel techniques As already mentioned, using the expression of ɛ in the background field decomposition the kinetic part normalizes, i.e. D = 1 g t g t + γ λ 1 (D + R). (30) The normalization of the spatial part can be obtained by working with an auxiliary metric, ( ) 1 λ 1 ĝ ij = g ij, (31) γ so that we have ˆD = 1 ĝ t ĝ t + ( ˆD + ˆR). (3) We can use the results of Baggio et.al. and obtain for b 1 and b a = 1 56 π ( ˆK ij ˆK ij 1 ) ˆK +. (33)

16 16/0 Heat-kernel techniques For the coefficient b 3 we can use the well known results for higher-derivative operators in the isotropic case (see Gusynin, Nucl.Phys. B333). In particular, the heat kernel coefficient of R for an operator (D + X) vanishes in two dimensions, so that we have no R term in our one-loop result. The one-loop correction is then equal to 1 ˆTr ln( ˆD) = 1 dt d x ĝ + ln { (Λ 4 µ 4 ) 1 16 π + (Λ µ ) 14 ˆR 48 π3/ ( ) { Λ 1 1 ˆK ij ˆK ij + 1 } ( ) } 1 ˆK + O µ 16 π 4 8 Λ. (34) We are interested only in the logarithmic divergence, which rewritten in terms of the physical metric reads ( ) 1 S 1 loop = 1 λ 1 ( ) Λ ln dt d x g {K log ij K ij 1 } 18 π γ µ K. (35)

17 β-functions We can obtained the β-functions by expressing the i-th bare coupling g b,i as g b,i = g R,i + δg i, being g R,i a renormalized coupling δg i a counterterm, so that ( ) 1 = κ 1 λ 1 ( ) Λ ln, 18 π γ µ κ R λ R κ R γ R κ R = λ κ 1 56 π = γ κ. ( λ 1 γ ) 1 ln ( ) Λ, µ (36) 17/0 The renormalized couplings then read ( ( κ R = κ 1 + κ λ 1 56 π γ ) 1 ln ( ) ) Λ + O(ħ ), µ λ R = λ + 1 κ (λ 56 π γ 1/ 1 ) 3 ( ) Λ ln + O(ħ ), µ ( ( ) 1 γ R = γ 1 + κ λ 1 ( ) ) Λ ln + O(ħ ). 56 π γ µ (37)

18 18/0 β-functions As usual the β-functions are obtained stating µ g b = 0, so that ( ) 1 β κ = µ µ κ R = κ4 λ 1, 56 π γ ( ) λ 1 β λ = µ µ λ R = κ β κ, β γ = µ µ γ R = γ κ β κ. (38) Solving the system we find k R (µ) = 56 π b 1/ (ln µ µ 0 + C), λ R (µ) = 1 + C 1 ln µ µ 0 + C, (39) γ R (µ) = C ln µ µ 0 + C. where C and b = C 1 /C are integration constants.

19 19/0 β-functions Although the Newton s constant tends to zero in the UV, the interaction of the theory is defined by ɛ, whose running reads ɛ R = κ R λ R 1 = 56 π C 1/ C 3/ 1 Consequently, at one loop the theory is not asymptotically free.. (40) However, it is interesting to note that λ tends to 1/. For this value the kinetic action is invariant under anisotropic Weyl transformations g ij e φ(t,x) g ij, N e z φ(t,x) N, N i e φ(t,x) N i. (41) We expect the running of λ to the conformal point to be a feature of the full model.

20 0/0 Conclusions and discussion Hořava-Lifshitz gravity in d + 1 dimensions with z = d features simultaneously power-counting renormalizability and unitarity. Because of the large number of couplings and complications coming from the anisotropic character of the background the UV has not been explored. We studied a simpler case, i.e. Hořava-Lifshitz gravity in +1 dimensions with z =. We focused on a conformally reduced version of the projectable case. At one-loop the Newton s constant runs to zero in UV and λ to its conformal value λ = 1. The interaction strength, however, is constant along the flow. What happens at two loops? What happens with gravitons? We expect λ to run to its conformal value also in the full model. Thanks for the attention.