ASYMPTOTIC FREEDOM IN HORAVA-LIFSHITZ …users.uoa.gr/~papost/TALKS_FILES/Dodorico.pdfASYMPTOTIC FREEDOM IN HORAVA-LIFSHITZ GRAVITY GIULIO D’ODORICO ERG 2014 Conference, Lefkada,

ASYMPTOTIC FREEDOM IN HORAVA-LIFSHITZ GRAVITY

GIULIO D’ODORICO

ERG 2014 Conference, Lefkada, Greece

Based on: G.D., F. Saueressig, M. Schutten, arXiv:1406.4366 and G.D., Frank Saueressig, in preparation

http://arxiv.org/abs/1406.4366


OVERVIEW

Giulio D’Odorico ERG 2014 - 23 September 2014


Theory Space: Quantum Einstein GravitySymmetry: Diffeomorphisms

FP Structure of Quantum Gravity



NGFPβ


GFP

• Critical properties “easy” to determine with perturbative methods



NGFPβ


GFP

• Critical properties “easy” to determine with perturbative methods

• Unfortunately gravity is perturbatively nonrenormalizable



NGFPβ


GFP

[GN ] = −2

Horava-Lifshitz Gravity in a nutshell


• Anisotropic field theories: change dispersion relation

• Decreases degree of divergence in loop integrals



t → b t,

x → b1/z xS =

� �φ̇2 − φ∆zφ+

N�

n=1

gnφn

�dt ddx



• Gravity: Folation-Preserving Diffeomorphisms

• Natural formulation in ADM variables:



t → b t,

x → b1/z xS =

� �φ̇2 − φ∆zφ+

N�

n=1

gnφn

�dt ddx

ds2 = N2dt2 + σij

�dxi +N idt

� �dxj +N jdt

�

t → f(t)

x → ζ(t,x)



• Gravity: Folation-Preserving Diffeomorphisms

• Natural formulation in ADM variables:



t → b t,

x → b1/z xS =

� �φ̇2 − φ∆zφ+

N�

n=1

gnφn

�dt ddx

ds2 = N2dt2 + σij

�dxi +N idt

� �dxj +N jdt

�

t → f(t)

x → ζ(t,x)

N(t, x) = N(t) Projectable HL Gravity


Theory Space: Horava-LifshitzSymmetry: Foliation Preserving Diffs

NGFP

Subspace: Quantum Einstein GravitySymmetry: Diffs

GFP

aGFPβ


• Is the theory asymptotically free?

• Does it reproduce the correct phenomenology?

• Does ti resolve previous issues?

Questions


• Is the theory asymptotically free?

Questions


MATTER-INDUCED FLOWS IN

PROJECTABLE HL GRAVITY


Ansatz


• Projectable Horava-Lifshitz action plus anisotropic scalar

Ansatz


Γk[N,Ni,σ,φ] = ΓHL

k [N,Ni,σ] + SLS[N,Ni,σ,φ]


Ansatz




ΓHL

k = 1

16πGk

�dtddxN

√σ�KijK

ij − λkK2 + Vk

�


• Potential V is a function of the intrinsic curvatures:

Ansatz




ΓHL

k = 1

16πGk

�dtddxN

√σ�KijK

ij − λkK2 + Vk

�

V (d=2)k = g0 + g1 R+ g2 R

2

V (d=3)k = g0 + g1R+ g2R

2 + g3RijRij − g4R∆xR

− g5Rij∆xRij + g6R

3 + g7RRijRij + g8R

ijR

jkR

ki


• Minimally coupled anisotropic scalars with covariant derivatives

Ansatz


SLS ≡ 12

�dtddxN

√σφ [∆t + (∆x)

z]φ

∆t ≡ − 1N

√σ∂t N

−1√σ ∆x ≡ −σij∇i∇j




• Minimally coupled anisotropic scalars with covariant derivatives

• We will consider the beta functions for large-n (with n the number of scalars)

Ansatz


SLS ≡ 12

�dtddxN

√σφ [∆t + (∆x)

z]φ

∆t ≡ − 1N

√σ∂t N

−1√σ ∆x ≡ −σij∇i∇j



ANISOTROPIC HEAT-KERNELS


HK for Anisotropic Operators


• Need the Heat Kernel for anisotropic operator:

• It is possible to reduce the problem to an Off-Diagonal Heat Kernel computation.



D2 ≡ ∆t + (∆x)z∂tΓk = Tr O

�D

2�

[ Benedetti, Groh, Machado, Saueressig JHEP 06 (2011) 079 ][ Anselmi, Benini JHEP 10 (2007) 099 ]

• Need the Heat Kernel for anisotropic operator:

• It is possible to reduce the problem to an Off-Diagonal Heat Kernel computation.

• For a scalar we find:



D2 ≡ ∆t + (∆x)z

Tr e−sD2

� (4π)−(d+1)/2 s−(1+d/z)/2

�dtddxN

√σ×

�s

6

�e1 K

2 + e2 KijKij�+

�

n≥0

sn/z bn a2n

�

∂tΓk = Tr O�D

2�

N = 1, N i = 0

[ Benedetti, Groh, Machado, Saueressig JHEP 06 (2011) 079 ][ Anselmi, Benini JHEP 10 (2007) 099 ]

• The e coefficients are:

•The b coefficients come in two classes:

Anisotropic HK: Results


e1 =d− z + 3

d+ 2

Γ( d2z )

zΓ(d2 ), e2 = −d+ 2z

d+ 2

Γ( d2z )

zΓ(d2 )

0 ≤ n ≤ �d/2�bn =

Γ�d−2n2z + 1

�

Γ�d−2n

2 + 1�

n > �d/2�bn(d, z) ≡

(−1)k

Γ(d/2− n+ k)

� ∞

0dxxd/2−n+k−1 (∂x)

k e−xz

k = n+ 1− �d/2�


d = 2 d = 3

z = 1 z = 2 z = 3 z = 1 z = 2 z = 3 z = 4

b0 1√π2 Γ( 43 ) 1 4

3√πΓ( 74 )

23

43√πΓ( 118 )

b1 1 1 1 1 2√πΓ( 54 )

2√πΓ( 76 )

2√πΓ( 98 )

b2 1 0 0 1 1√πΓ( 34 )

1√πΓ( 56 )

1√πΓ( 78 )

b3 1 −2 0 1 − 2√πΓ( 54 ) − 1

2 − 12√πΓ( 58 )

b4 1 0 6 1 − 4√πΓ( 74 )

92√πΓ( 76 )

2√πΓ( 118 )

• z=1 reproduces standard covariant results


d = 2 d = 3

z = 1 z = 2 z = 3 z = 1 z = 2 z = 3 z = 4

b0 1√π2 Γ( 43 ) 1 4

3√πΓ( 74 )

23

43√πΓ( 118 )

b1 1 1 1 1 2√πΓ( 54 )

2√πΓ( 76 )

2√πΓ( 98 )

b2 1 0 0 1 1√πΓ( 34 )

1√πΓ( 56 )

1√πΓ( 78 )

b3 1 −2 0 1 − 2√πΓ( 54 ) − 1

2 − 12√πΓ( 58 )

b4 1 0 6 1 − 4√πΓ( 74 )

92√πΓ( 76 )

2√πΓ( 118 )

• z=1 reproduces standard covariant results

• z=d=2 reproduces Baggio, de Boer, Holsheimer, arXiv:1112.6416


d = 2 d = 3

z = 1 z = 2 z = 3 z = 1 z = 2 z = 3 z = 4

b0 1√π2 Γ( 43 ) 1 4

3√πΓ( 74 )

23

43√πΓ( 118 )

b1 1 1 1 1 2√πΓ( 54 )

2√πΓ( 76 )

2√πΓ( 98 )

b2 1 0 0 1 1√πΓ( 34 )

1√πΓ( 56 )

1√πΓ( 78 )

b3 1 −2 0 1 − 2√πΓ( 54 ) − 1

2 − 12√πΓ( 58 )

b4 1 0 6 1 − 4√πΓ( 74 )

92√πΓ( 76 )

2√πΓ( 118 )

http://arxiv.org/find/hep-th/1/au:+Holsheimer_K/0/1/0/all/0/1

http://arxiv.org/find/hep-th/1/au:+Holsheimer_K/0/1/0/all/0/1



RESULTS


• Defining

• The beta functions for the dimensionless couplings read

Beta Functions


gk ≡ Gk k2η , η ≡ d

2z − 12 φn ≡ 1

Γ(n)

� 1

0dx xn−1

βg =2 η g − 23 (4π)

−(d−1)/2 φη e2 g2 ,

βλ = − 23 (4π)

−(d−1)/2 φη (e1 + λ e2) g

βg̃0 = − 2 g̃0 +4g

(4π)(d−1)/2

�b0 φη+1 − 1

6 e2 φη g̃0�,

βg̃1 =�2z − 2

�g̃1 +

2g3(4π)(d−1)/2

�b1 φη+1−1/z − e2 φη g̃1

�

βg̃2 = − 23 g̃2 +

g5π

�1

8√π

Γ(5/6)Γ(1/3) + g̃2

�,

βg̃3 = − 23 g̃3 +

g5π

�1

4√π

Γ(5/6)Γ(1/3) + g̃3

�,

βg̃i =gπ

�15 g̃i −

12 ci

�, i = {4, 5, 6, 7, 8}

wave-functionrenormalization {

Newton and cosmological constants

Higher derivativecouplings

{

{

Anisotropic Gaussian FP


g∗ = 0 , λ∗ = 1d

20000 40000 60000 80000 100000t

gi!t"

ci

20000 40000 60000 80000 100000t

!0.5

!0.4

!0.3

!0.2

!0.1

G!t"

1 2 3 4 5 t

!2."10!81

!1."10!81

1."10!81

2."10!81

#!t"!1#3

g̃∗1 = g̃∗2 = g̃∗3 = 0 , g̃∗i =5

2ci

c4 = 1336 , c5 = 1

840 , c6 = − 1560 ,

c7 = 1105 , c8 = − 1

180

Detailed Balance Revisited


• The Detailed Balance condition states that



V ∝ δW [σ]

δσijHijkl

δW [σ]

δσkl


• In d=3 this implies that the potential is the square of the Cotton tensor



V ∝ δW [σ]

δσijHijkl

δW [σ]

δσkl

Vdb ∝ CijCij

Cij = �iklDk

�Rj

l −14Rδjl

�


• In d=3 this implies that the potential is the square of the Cotton tensor

• Constructing the fixed point potential from the fixed point values of the couplings, one can see that it does not respect Detailed Balance



V ∝ δW [σ]

δσijHijkl

δW [σ]

δσkl

Vdb ∝ CijCij

Cij = �iklDk

�Rj

l −14Rδjl

�

Vdb ∝Rij∆xRij − 3

8R∆xR

+ 3RijR

jkR

ki − 5

2RRijRij +12R

3 VSV∗ = g∗4R∆xR− g∗5Rij∆xR

ij+

g∗6R3 + g∗7RRijR

ij + g∗8RijR

jkR

ki

CONCLUSIONS


• Evaluation of anisotropic Heat Kernels

• Large-N study of HL fixed point

Achievements


• Evaluation of anisotropic Heat Kernels

• Large-N study of HL fixed point

Achievements

Open Questions

• Flow portrait including higher spins (working on)

• Applications to other systems?


THANK YOU

APPENDIX


• Basic fields: Lapse, Shift, Induced metric

• Induced metric σ describes intrinsic curvature on the slice

• Lapse and shift describe extrinsic curvature

• EH action in ADM form (Gauss-Codazzi relation)

More on ADM


gµν �→ {N , Ni , σij }

Kij =1

2N{−σ̇ij +∇iNj +∇jNi}

SEH =

� �(KijKij −K2) +R

�√σ N d3x dt

• Notice that on general backgrounds:• Laplace transform:

• Split the exponential using the inverse Baker-Hausdorf (Zassenhaus) formula:

• Do another Laplace transform involving only the spatial Laplacian, and use Baker-Hausdorf again

• End up with:

• After a scale transformation, resum the two Laplacians into a “fake” covariant one. Then use Off-Diagonal techniques

Evaluating the Trace


[∆t,∆x] �= 0

et(X+Y ) � etX etY e−t2

2 [X,Y ] et3

6 (2[Y,[X,Y ]+[X,[X,Y ]]) · · ·

[∆t, (∆x)z]

exp (−s1∆t − s2∆x)

Trf�D2

�→ Tr e−sD2

Theory Space

limk→0

Γk[ϕ] = Γ[ϕ]limk→∞

Γk[ϕ] = S[ϕ]

k → 0

BARE

EA

The EAA interpolates smoothly between the bare action and the full quantum EA


Asymptotic Safety

•Gravity is perturbatively nonrenormalizable

•UV-completion: 1. New physics 2. Nonperturbative “self-healing”

• Generalized, nonperturbative renormalizability requirement

[GN ] = −2

Asymptotically Safe Theory:

‣ Has a (non-gaussian) RG fixed point

‣ The UV Critical Surface is finite dimensional


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ASYMPTOTIC FREEDOM IN HORAVA-LIFSHITZ …users.uoa.gr/~papost/TALKS_FILES/Dodorico.pdfASYMPTOTIC FREEDOM IN HORAVA-LIFSHITZ GRAVITY GIULIO D’ODORICO ERG 2014 Conference, Lefkada,