HORAVA-LIFSHITZ GRAVITY FROM AN RG PERSPECTIVEgies/Conf/SIFT2015/DOdorico.pdf · •Dynamics of...

HORAVA-LIFSHITZ GRAVITY FROM AN RG PERSPECTIVE

GIULIO D’ODORICO

G. D., F. Saueressig, M. Schutten, Phys.Rev.Lett. 113 (2014) 171101, arXiv:1406.4366 G. D., J. W. Goossens, F. Saueressig, JHEP 1510 (2015) 126, arXiv:1508.00590

SIFT 2015, 5-7 November, Jena, Germany

WHYHORAVA-LIFSHITZ

GRAVITY?

Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015

Theory Space: Quantum Einstein GravitySymmetry: Diffeomorphisms

FP Structure of Quantum Gravity




GFP


• Critical properties can be determined using perturbative methods



GFP


• Critical properties can be determined using perturbative methods

• Unfortunately gravity is perturbatively nonrenormalizable



GFP

[GN ] = −2


Perturbative Quantum Gravity


• Dynamics of General Relativity governed by Einstein-Hilbert action

• Newton’s constant has negative mass-dimension

SEH =1

16πGN

�d4x

√g�−R+ 2Λ

�

GN





• Perturbative quantization of General Relativity:

SEH =1

16πGN

�d4x

√g�−R+ 2Λ

�

GN

�-loop-diagram diverges ∝ E2 (GN E2)�

�-loop counterterms have 2�+ 2-derivatives






• Gravity + matter :

• Pure gravity:

SEH =1

16πGN

�d4x

√g�−R+ 2Λ

�

GN



∆S1−loop ∝�

d4x√g�CαβµνC

αβµν�

∆S2−loop ∝�

d4x√g�Cµν

ρσCρσαβCαβ

µν�






• Gravity + matter :

• Pure gravity:

SEH =1

16πGN

�d4x

√g�−R+ 2Λ

�

GN



∆S1−loop ∝�

d4x√g�CαβµνC

αβµν�

∆S2−loop ∝�

d4x√g�Cµν

ρσCρσαβCαβ

µν�

General Relativity is perturbatively non-renormalizable



Asymptotic Safety


Asymptotic Safety

• UV-completion: 1. New physics


Asymptotic Safety

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


Asymptotic Safety


Asymptotically Safe Theory:

‣ Has a (non-gaussian) RG fixed point

‣ The UV Critical Surface is finite dimensional


Asymptotic Safety


• Generalized, nonperturbative renormalizability requirement

Asymptotically Safe Theory:

‣ Has a (non-gaussian) RG fixed point

‣ The UV Critical Surface is finite dimensional





• Critical properties “easy” to determine with perturbative methods




GFP

[GN ] = −2


• Critical properties “easy” to determine with perturbative methods




GFP

[GN ] = −2

NGFPβ


Horava-Lifshitz Gravity in a nutshell


• Anisotropic field theories: change dispersion relation





S =

� �φ2 − φ∆zφ+

N�

n=1

gnφn

�dt ddx



• This corresponds to an anisotropic scaling


S =

� �φ2 − φ∆zφ+

N�

n=1

gnφn

�dt ddx

t → b t,

x → b1/z x




• Higher order in spatial derivatives decreases the degree of divergence in loop integrals


S =

� �φ2 − φ∆zφ+

N�

n=1

gnφn

�dt ddx

t → b t,

x → b1/z x




• Higher order in spatial derivatives decreases the degree of divergence in loop integrals

• Two time derivatives: naive unitarity maintained


S =

� �φ2 − φ∆zφ+

N�

n=1

gnφn

�dt ddx

t → b t,

x → b1/z x




• Perturbatively renormalizable quantum theory of gravity




• Anisotropic scaling between space and time has a natural formulation in ADM variables:


ds2 = N2dt2 + σij

�dxi +N idt

� �dxj +N jdt

�




• Field content:


ds2 = N2dt2 + σij

�dxi +N idt

� �dxj +N jdt

�

{N(t, x) , N i(t, x) , σij(t, x)}




• Field content:

• Symmetry: Foliation-Preserving Diffeomorphisms


ds2 = N2dt2 + σij

�dxi +N idt

� �dxj +N jdt

�

{N(t, x) , N i(t, x) , σij(t, x)}

t → f(t)

x → ζ(t,x)Diff(M,Σ) ⊂ Diff(M)




• Field content:

• Symmetry: Foliation-Preserving Diffeomorphisms

• Projectable HL Gravity


ds2 = N2dt2 + σij

�dxi +N idt

� �dxj +N jdt

�

{N(t, x) , N i(t, x) , σij(t, x)}

N(t, x) = N(t)

t → f(t)

x → ζ(t,x)Diff(M,Σ) ⊂ Diff(M)


Horava-Lifshitz gravity in a nutshell



• Weakend symmetry requirements ⇒ more terms in action




• Gravitational part of the action

SHL =1

16πG

�dtddxN

√σ�KijK

ij − λK2 + V�





SHL =1

16πG

�dtddxN

√σ�KijK

ij − λK2 + V�


Kij ≡ (2N)−1 [∂tσij −∇iNj −∇jNi]




• Potential depends on the version of Horava-Lifshitz gravity considered

SHL =1

16πG

�dtddxN

√σ�KijK

ij − λK2 + V�

Vd=3 =g0 + g1R+ g2,1R2 + g2,2RijR

ij − g3,1R∆xR− g3,2Rij∆xRij

+ g3,3R3 + g3,4RRijR

ij + g3,5RijR

jkR

ki







• Interesting case at criticality z = d:

SHL =1

16πG

�dtddxN

√σ�KijK

ij − λK2 + V�

Vd=3 =g0 + g1R+ g2,1R2 + g2,2RijR



ij + g3,5RijR

jkR

ki








‣ Newton’s constant is dimensionless

SHL =1

16πG

�dtddxN

√σ�KijK

ij − λK2 + V�

Vd=3 =g0 + g1R+ g2,1R2 + g2,2RijR



ij + g3,5RijR

jkR

ki








‣ Newton’s constant is dimensionless‣ Two time-derivatives ⇒ naive unitarity

SHL =1

16πG

�dtddxN

√σ�KijK

ij − λK2 + V�

Vd=3 =g0 + g1R+ g2,1R2 + g2,2RijR



ij + g3,5RijR

jkR

ki








‣ Newton’s constant is dimensionless‣ Two time-derivatives ⇒ naive unitarity

Horava-Lifshitz gravity is power-counting renormalizable

SHL =1

16πG

�dtddxN

√σ�KijK

ij − λK2 + V�

Vd=3 =g0 + g1R+ g2,1R2 + g2,2RijR



ij + g3,5RijR

jkR

ki



Theory Space: Horava-LifshitzSymmetry: Foliation Preserving Diffs

NGFP

Subspace: Quantum Einstein GravitySymmetry: Diffs

GFP

aGFPβ



Theory Space: Horava-LifshitzSymmetry: Foliation Preserving Diffs

NGFP

Subspace: Quantum Einstein GravitySymmetry: Diffs

GFP

aGFPβ



QCD: Asymptotic Freedom


• Scale-dependence encoded in beta functions


k∂kα(k) = β(α)



• Consider SU(3) Yang-Mills theory:


k∂kα(k) = β(α)

β(αs) = −�11− 2nf

3

�α2s

2π





k∂kα(k) = β(α)

β(αs) = −�11− 2nf

3

�α2s

2π

2 4 6 8 10E0.0

0.2

0.4

0.6

0.8

1.0!




• α decreases with increasing energy• α = 0 is a UV-attractive fixed point of the RG


k∂kα(k) = β(α)

β(αs) = −�11− 2nf

3

�α2s

2π

2 4 6 8 10E0.0

0.2

0.4

0.6

0.8

1.0!





• QCD is asymptotically free


k∂kα(k) = β(α)

β(αs) = −�11− 2nf

3

�α2s

2π

2 4 6 8 10E0.0

0.2

0.4

0.6

0.8

1.0!






• Beta function changes sign if


k∂kα(k) = β(α)

β(αs) = −�11− 2nf

3

�α2s

2π

2 4 6 8 10E0.0

0.2

0.4

0.6

0.8

1.0!






• Beta function changes sign if

• Too many flavors destroy asymptotic freedom


nf > 33/2

k∂kα(k) = β(α)

β(αs) = −�11− 2nf

3

�α2s

2π

2 4 6 8 10E0.0

0.2

0.4

0.6

0.8

1.0!


QED: Landau Pole



QED: Landau Pole


k∂kα(k) = β(α)


• QED:

QED: Landau Pole

β(α) =2α2

3π

2 4 6 8 10E0

20

40

60

80

100!


k∂kα(k) = β(α)


• QED:

• α decreases with decreasing energy

QED: Landau Pole

β(α) =2α2

3π

2 4 6 8 10E0

20

40

60

80

100!


k∂kα(k) = β(α)


• QED:


• α = 0 is an IR attractive fixed point of the renormalization group flow

QED: Landau Pole

β(α) =2α2

3π

2 4 6 8 10E0

20

40

60

80

100!


k∂kα(k) = β(α)


• QED:


• α = 0 is an IR attractive fixed point of the renormalization group flow

• At high energies α diverges at a Landau pole

QED: Landau Pole

β(α) =2α2

3π

2 4 6 8 10E0

20

40

60

80

100!


k∂kα(k) = β(α)

• Is the theory asymptotically free?

• Does it reproduce the correct phenomenology?

• Does it resolve previous issues?

Questions


• Is the theory asymptotically free?

Questions


COVARIANT EFFECTIVE ACTIONS

IN HL GRAVITY


Anisotropic scalar field


• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar


Γk[N,Ni,σ,φ] = ΓHL

k [N,Ni,σ] + SLS[N,Ni,σ,φ]



• Gravitational sector




ΓHL

k = 1

16πGk

�dtddxN

√σ�KijK

ij − λkK2 + Vk

�







ΓHL

k = 1

16πGk

�dtddxN

√σ�KijK

ij − λkK2 + Vk

�

Potential V is a function of the intrinsic curvatures




‣ Projectable version




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

Potential V is a function of the intrinsic curvatures




‣ Non-projectable version




ΓHL

k = 1

16πGk

�dtddxN

√σ�KijK

ij − λkK2 + Vk

�








ΓHL

k = 1

16πGk

�dtddxN

√σ�KijK

ij − λkK2 + Vk

�

∆V = u1 ai ai + u2,1 R∇i a

i + u2,2 ai ∆x ai +

u3,1(∆xR)∇i ai + u3,2 ai (∆x)

2 ai + . . .








ΓHL

k = 1

16πGk

�dtddxN

√σ�KijK

ij − λkK2 + Vk

�

∆V = u1 ai ai + u2,1 R∇i a

i + u2,2 ai ∆x ai +

u3,1(∆xR)∇i ai + u3,2 ai (∆x)

2 ai + . . .

ai ≡ ∇i lnN










• Matter sector




SLS ≡ 12

�dtddxN

√σφ [∆t + (∆x)

z]φ



• Matter sector

‣ Two covariant derivatives (orthogonal and tangent to the slice)




SLS ≡ 12

�dtddxN

√σφ [∆t + (∆x)

z]φ

∆t ≡ − 1

N√σ∂t N

−1√σ ∂t , ∆x ≡ − 1

N√σ∂i σ

ijN√σ ∂j



• Matter sector


‣ Anisotropic Laplacian




SLS ≡ 12

�dtddxN

√σφ [∆t + (∆x)

z]φ

∆t ≡ − 1

N√σ∂t N

−1√σ ∂t , ∆x ≡ − 1

N√σ∂i σ

ijN√σ ∂j

D2 ≡ ∆t + (∆x)z



• Matter sector


‣ Anisotropic Laplacian

‣ Reduces to the standard one for




SLS ≡ 12

�dtddxN

√σφ [∆t + (∆x)

z]φ

∆t ≡ − 1

N√σ∂t N

−1√σ ∂t , ∆x ≡ − 1

N√σ∂i σ

ijN√σ ∂j

z → 1

D2 ≡ ∆t + (∆x)z


One-loop Effective Action


• One-loop effective action found by integrating out the scalar


Γeff [N,Ni,σij ] = Sbare + 12Tr log

δ2SLS

δφ δφ



• The operator trace



δ2SLS

δφ δφ


1

2Tr logO = −1

2

� ∞

0

ds

sTre−sO



• Relates the one-loop determinant to the heat-kernel



δ2SLS

δφ δφ


1

2Tr logO = −1

2

� ∞

0

ds

sTre−sO

KO(s) ≡ Tr e−sO , ∂sKO(s) +OKO(s) = 0



• Relates the one-loop determinant to the heat-kernel

• Divergences are encoded in the Seeley-deWitt expansion



δ2SLS

δφ δφ


1

2Tr logO = −1

2

� ∞

0

ds

sTre−sO

KO(s) ≡ Tr e−sO , ∂sKO(s) +OKO(s) = 0

ANISOTROPIC HEAT-KERNELS


AHK: dimensional analysis


• Need the Heat Kernel for anisotropic operator:


D2 ≡ ∆t + (∆x)z



• Scaling dimensions


D2 ≡ ∆t + (∆x)z

[t] = z , [x] = 1 , [s] = 2z , [N ] = 0 , [σij ] = 0 ,�N i

�= z − 1




• The Seeley-deWitt expansion takes the following form


D2 ≡ ∆t + (∆x)z

[t] = z , [x] = 1 , [s] = 2z , [N ] = 0 , [σij ] = 0 ,�N i

�= z − 1


Tr e−sD2

= (4π)−d+12 s−

1+d/z2

�dtddxN

√σ

�

l,m,n≥0

sl(1−z)

2z +m2 + n

2z trbl,m,n




‣ l = number of shift vectors‣ m = number of time derivatives‣ n = number of spatial derivatives


D2 ≡ ∆t + (∆x)z

[t] = z , [x] = 1 , [s] = 2z , [N ] = 0 , [σij ] = 0 ,�N i

�= z − 1


Tr e−sD2

= (4π)−d+12 s−

1+d/z2

�dtddxN

√σ

�

l,m,n≥0

sl(1−z)

2z +m2 + n

2z trbl,m,n






D2 ≡ ∆t + (∆x)z

[t] = z , [x] = 1 , [s] = 2z , [N ] = 0 , [σij ] = 0 ,�N i

�= z − 1

Invariants respecting foliation-preserving diffeomorphisms


Tr e−sD2

= (4π)−d+12 s−

1+d/z2

�dtddxN

√σ

�

l,m,n≥0

sl(1−z)

2z +m2 + n

2z trbl,m,n






D2 ≡ ∆t + (∆x)z

[t] = z , [x] = 1 , [s] = 2z , [N ] = 0 , [σij ] = 0 ,�N i

�= z − 1

Invariants respecting foliation-preserving diffeomorphisms

e2 KijKij

� �� b0,2,0

a2 R(3)

� �� b0,0,2

c1 aiai

� �� b0,0,2


Tr e−sD2

= (4π)−d+12 s−

1+d/z2

�dtddxN

√σ

�

l,m,n≥0

sl(1−z)

2z +m2 + n

2z trbl,m,n

Evaluating the Trace


• We want the asymptotic expansion of


KD2(s) = Tr e−sD2



• Notice that on general backgrounds:


[∆t,∆x] �= 0

KD2(s) = Tr e−sD2




• We can use the following algorithm:


[∆t,∆x] �= 0

KD2(s) = Tr e−sD2





‣ Step 1: Split the exponential using the inverse Campbell-Baker-Hausdorff (Zassenhaus) formula


[∆t,∆x] �= 0

KD2(s) = Tr e−sD2


e−s(A+B) = e−sAe−sBe−s2

2 [A,B]e−s3

6 ([A,[A,B]]+2[B,[A,B]]) · · ·




‣ Step 1: Split the exponential using the inverse Campbell-Baker-Hausdorff (Zassenhaus) formula

‣ This gives


[∆t,∆x] �= 0

KD2(s) = Tr e−sD2

Tr e−sD2

= Tr�e−s∆te−s(∆x)

z

C(∆t,∆x)�


e−s(A+B) = e−sAe−sBe−s2

2 [A,B]e−s3

6 ([A,[A,B]]+2[B,[A,B]]) · · ·




‣ Step 2: Do a Laplace transform


[∆t,∆x] �= 0

KD2(s) = Tr e−sD2

Tr e−sD2

=�

i

� ∞

0dv�Wi(v) Tr

�e−s∆t e−v∆x Ci(∆t)

�





‣ Step 2: Do a Laplace transform

‣ Step 3: Rescale the metric


[∆t,∆x] �= 0

KD2(s) = Tr e−sD2

Tr e−sD2

=�

i

� ∞

0dv�Wi(v) Tr

�e−s∆t e−v∆x Ci(∆t)

�

σij =s

vσij ∆x =

v

s∆x





‣ Step 4: Apply the Campbell-Baker-Hausdorff formula again


[∆t,∆x] �= 0

KD2(s) = Tr e−sD2

e−s∆te−s∆x = e−s(∆t+∆x)B(∆t, ∆x)





‣ Step 4: Apply the Campbell-Baker-Hausdorff formula again

‣ The result is of the off-diagonal heat-kernel type

‣ in terms of the “fake” Laplacian


[∆t,∆x] �= 0

KD2(s) = Tr e−sD2

e−s∆te−s∆x = e−s(∆t+∆x)B(∆t, ∆x)

Tr e−sD2

= Tr�O e−s∆(D)

�

∆(D) = ∆t + ∆x





‣ Step 5: Use the off-diagonal Heat-Kernel !


[∆t,∆x] �= 0

KD2(s) = Tr e−sD2


[ Benedetti, Groh, Machado, Saueressig JHEP 06 (2011) 079 ]

[ Anselmi, Benini JHEP 10 (2007) 099 ]

[ Groh, Saueressig, Zanusso arXiv:1112.4856 ]

http://arxiv.org/abs/arXiv:1112.4856

http://arxiv.org/abs/arXiv:1112.4856

Numerical Coefficients


• Extrinsic curvature (kinetic) terms


Tr e−sD2

� (4π)−d+12 s−

1+d/z2

�dtddxN

√σs

6

�e1 K

2 + e2 KijKij�

e1(d, z) =d− z + 3

d+ 2

Γ( d2z )

zΓ(d2 ), e2(d, z) = −d+ 2z

d+ 2

Γ( d2z )

zΓ(d2 )



• Checks:

‣ z = 1 :


Tr e−sD2

� (4π)−d+12 s−

1+d/z2

�dtddxN

√σs

6

�e1 K

2 + e2 KijKij�

e1(d, z) =d− z + 3

d+ 2

Γ( d2z )

zΓ(d2 ), e2(d, z) = −d+ 2z

d+ 2

Γ( d2z )

zΓ(d2 )

e1 = 1, e2 = −1



• Checks:

‣ z = 1 :

‣ z = d :


Tr e−sD2

� (4π)−d+12 s−

1+d/z2

�dtddxN

√σs

6

�e1 K

2 + e2 KijKij�

e1(d, z) =d− z + 3

d+ 2

Γ( d2z )

zΓ(d2 ), e2(d, z) = −d+ 2z

d+ 2

Γ( d2z )

zΓ(d2 )

e1 = 1, e2 = −1

e1 = −1

de2




• Projectable potential terms





Tr e−sD2

� (4π)−d+12 s−

1+d/z2

�dtddxN

√σ�

n≥0

snz bn(d, z)

�

i

a2n,i R(i)2n




Tr e−sD2

� (4π)−d+12 s−

1+d/z2

�dtddxN

√σ�

n≥0

snz bn(d, z)

�

i

a2n,i R(i)2n

basis of curvature monomials with 2n spatial derivatives

isotropic heat kernel coefficients



‣ Power-counting relevant and marginal terms


Tr e−sD2

� (4π)−d+12 s−

1+d/z2

�dtddxN

√σ�

n≥0

snz bn(d, z)

�

i

a2n,i R(i)2n



bn(d, z) =Γ(d−2n

2z + 1)

Γ(d−2n2 + 1)

, 0 ≤ n ≤ �d/2�



‣ Power-counting relevant and marginal terms

‣ Power-counting irrelevant terms


Tr e−sD2

� (4π)−d+12 s−

1+d/z2

�dtddxN

√σ�

n≥0

snz bn(d, z)

�

i

a2n,i R(i)2n



bn(d, z) =Γ(d−2n

2z + 1)

Γ(d−2n2 + 1)

, 0 ≤ n ≤ �d/2�

k = n+ 1− �d/2�

bn(d, z) =(−1)k

Γ(d/2− n+ k)

� ∞

0dx xd/2−n+k−1 (∂x)

k e−xz

, n > �d/2�




• Non-projectable potential terms





Tr e−sD2

� (4π)−d+12 s−

1+d/z2

�dtddxN

√σ s

1z c1(d, z) aia

i

c1(d, z) = −13

6

z − 1

d

Γ�d−22z + 1

�

Γ�d2

�



• Check for z = 1:


Tr e−sD2

� (4π)−d+12 s−

1+d/z2

�dtddxN

√σ s

1z c1(d, z) aia

i

c1(d, z) = −13

6

z − 1

d

Γ�d−22z + 1

�

Γ�d2

�

c1(d, 1) = 0



• Check for z = 1:

• Note: determining the c’s is computationally very intensive !


Tr e−sD2

� (4π)−d+12 s−

1+d/z2

�dtddxN

√σ s

1z c1(d, z) aia

i

c1(d, z) = −13

6

z − 1

d

Γ�d−22z + 1

�

Γ�d2

�

c1(d, 1) = 0



d = 2 d = 3

dim a2n z = 2 z = 3 a2n z = 2 z = 3 z = 4

K2 d−z2z

16

√π

16136Γ

�13

�16

2Γ( 34 )

15√π

115

Γ( 38 )

30√π

KijKij d−z2z − 1

6 −√π8 − 1

9Γ�13

�− 1

6 − 7Γ( 34 )

30√π

− 15 − 11Γ( 3

8 )60

√π

1 d+z2z 1

√π2 Γ

�43

�1

4Γ( 74 )

3√π

23

4Γ( 118 )

3√π

R d−2+z2z

16

16

16

16

Γ( 54 )

3√π

Γ( 76 )

3√π

Γ( 98 )

3√π

R2 d−4+z2z

160 0 0 1

120

Γ( 54 )

120√π

Γ( 76 )

120√π

Γ( 98 )

120√π

RijRij d−4+z2z − − − 1

60

Γ( 54 )

60√π

Γ( 76 )

60√π

Γ( 98 )

60√π

−R∆xRd−6+z

2z − − − 1336 − Γ( 5

4 )168

√π

− 1672 − Γ( 5

8 )672

√π

−Rij∆xRij d−6+z2z − − − 1

840 − Γ( 54 )

420√π

− 11680 − Γ( 5

8 )1680

√π

R3 d−6+z2z

1756 −2 0 − 1

560

Γ( 54 )

280√π

11120

Γ( 58 )

1120√π

RRijRij d−6+z2z − − − 1

105 − 2Γ( 54 )

105√π

− 1210 − Γ( 5

8 )210

√π

RijR

jkR

ki

d−6+z2z − − − − 1

180

Γ( 54 )

90√π

1360

Γ( 58 )

360√π

a2 d−2+z2z 0 − 13

12 − 136 0 − 13Γ( 5

4 )9√π

− 26Γ( 76 )

9√π

− 13Γ( 98 )

3√π


SCALAR-DRIVENRG FLOWS INHL GRAVITY




• The one-loop part of the affective action is


Γ1 = −1

2

� ∞

0

ds

se−sm2

Tr�e−sD2

�



• Substituting the Seeley-deWitt expansion


Γ1 = −1

2

� ∞

0

ds

se−sm2

Tr�e−sD2

�

Γ1 = − (4π)−d+12

2

�dtddxN

√σ

� ∞

0

ds

se−sm2

s−1+d/z

2

×�s6

�e1 K

2 + e2 KijKij�+

�

n≥0

sn/z bn�

i

a2n,iR(i)2n + s

1z c1 aia

i + . . .

�




• UV-divergences appear at lower boundary of s-integral


Γ1 = −1

2

� ∞

0

ds

se−sm2

Tr�e−sD2

�

Γ1 = − (4π)−d+12

2

�dtddxN

√σ

� ∞

0

ds

se−sm2

s−1+d/z

2

×�s6

�e1 K

2 + e2 KijKij�+

�

n≥0

sn/z bn�

i

a2n,iR(i)2n + s

1z c1 aia

i + . . .

�





• Renormalize in standard way


Γ1 = −1

2

� ∞

0

ds

se−sm2

Tr�e−sD2

�

Γ1 = − (4π)−d+12

2

�dtddxN

√σ

� ∞

0

ds

se−sm2

s−1+d/z

2

×�s6

�e1 K

2 + e2 KijKij�+

�

n≥0

sn/z bn�

i

a2n,iR(i)2n + s

1z c1 aia

i + . . .

�





• Renormalize in standard way

• From this we can read the matter-induced beta functions


Γ1 = −1

2

� ∞

0

ds

se−sm2

Tr�e−sD2

�

Γ1 = − (4π)−d+12

2

�dtddxN

√σ

� ∞

0

ds

se−sm2

s−1+d/z

2

×�s6

�e1 K

2 + e2 KijKij�+

�

n≥0

sn/z bn�

i

a2n,iR(i)2n + s

1z c1 aia

i + . . .

�


Beta functions for d=z=3


• Kinetic terms


βg =ns

5πg2 , βλ =

ns

15π(3λ− 1) g


• Kinetic terms



βg =ns

5πg2 , βλ =

ns

15π(3λ− 1) g

βg0 = − 2 g0 + nsgπ

�b0 +

15 g0

�

βg1 = − 43 g1 + ns

gπ

�b1 a2,i +

15 g1

�

βg2,i = − 23 g2,i + ns

gπ

�b2 a4,i +

15 g2,i

�,

βg3,i =nsgπ

�b3 a6,i +

15 g3,i

�,


• Kinetic terms




βg =ns

5πg2 , βλ =

ns

15π(3λ− 1) g

βg0 = − 2 g0 + nsgπ

�b0 +

15 g0

�

βg1 = − 43 g1 + ns

gπ

�b1 a2,i +

15 g1

�

βg2,i = − 23 g2,i + ns

gπ

�b2 a4,i +

15 g2,i

�,

βg3,i =nsgπ

�b3 a6,i +

15 g3,i

�,

βu1 = − 43 u1 + ns

gπ

�c1 +

15 ut1

�,

βu2,i = − 23 u2,i + ns

gπ

�c2,i +

15 u2,i

�,

βu3,i =nsgπ

�c3,i +

15 u3,i

�.


A parametric view of theory space

!40 !30 !20 !10 10 20G"

1d

1

#"

z$1

z$d



!40 !30 !20 !10 10 20G"

1d

1

#"

z$1

z$d



!40 !30 !20 !10 10 20G"

1d

1

#"

z$1

z$d

‣ z = 1: isotropic non-Gaussian fixed point (Asymptotic Safety)

g∗ = − 3(d−1)2 (4π)(d−1)/2 Γ

�d+12

�, λ∗ = 1



!40 !30 !20 !10 10 20G"

1d

1

#"

z$1

z$d

‣ z = 1: isotropic non-Gaussian fixed point (Asymptotic Safety)

‣ z = d: anisotropic Gaussian fixed point (Horava-Lifshitz)

g∗ = − 3(d−1)2 (4π)(d−1)/2 Γ

�d+12

�, λ∗ = 1

g∗ = 0 , λ∗ =1

d


Massless flows at criticality (d=z=3)

‣ Arrows point towards the infrared‣ The matter-induced anisotropic Gaussian fixed point is an infrared attractor!‣ Isotropic plane: no special properties

0.2 0.4 0.6 0.8 1.0 1.2!

"0.4

"0.2

0.2

0.4

g


CONCLUSIONS



Conclusions


• Two different universality classes for quantum gravity

Conclusions



• Scalar induced beta function at criticality similar to QCD

Conclusions

β(G) = − (c− ns)G2

5π



• Scalar induced beta function at criticality similar to QCD

• Asymptotic freedom requires

Conclusions

β(G) = − (c− ns)G2

5π

c > 0



Future directions


Future directions

• Anisotropic heat-kernel coefficients can be obtained for any spin (vectors, fermions and gravitons)


Future directions


• Which version of Horava-Lifshitz gravity in the gravitational sector?


Future directions



• Applications to other systems?


Future directions



• Applications to other systems?


THANK YOU

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