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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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Theory Space: Quantum Einstein GravitySymmetry: Diffeomorphisms
FP Structure of Quantum Gravity
GFP
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Critical properties can be determined using perturbative methods
Theory Space: Quantum Einstein GravitySymmetry: Diffeomorphisms
FP Structure of Quantum Gravity
GFP
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Critical properties can be determined using perturbative methods
• Unfortunately gravity is perturbatively nonrenormalizable
Theory Space: Quantum Einstein GravitySymmetry: Diffeomorphisms
FP Structure of Quantum Gravity
GFP
[GN ] = −2
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Perturbative Quantum Gravity
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• 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 Quantum Gravity
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Dynamics of General Relativity governed by Einstein-Hilbert action
• Newton’s constant has negative mass-dimension
• 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
Perturbative Quantum Gravity
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Dynamics of General Relativity governed by Einstein-Hilbert action
• Newton’s constant has negative mass-dimension
• Perturbative quantization of General Relativity:
• Gravity + matter :
• Pure gravity:
SEH =1
16πGN
�d4x
√g�−R+ 2Λ
�
GN
�-loop-diagram diverges ∝ E2 (GN E2)�
�-loop counterterms have 2�+ 2-derivatives
∆S1−loop ∝�
d4x√g�CαβµνC
αβµν�
∆S2−loop ∝�
d4x√g�Cµν
ρσCρσαβCαβ
µν�
Perturbative Quantum Gravity
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Dynamics of General Relativity governed by Einstein-Hilbert action
• Newton’s constant has negative mass-dimension
• Perturbative quantization of General Relativity:
• Gravity + matter :
• Pure gravity:
SEH =1
16πGN
�d4x
√g�−R+ 2Λ
�
GN
�-loop-diagram diverges ∝ E2 (GN E2)�
�-loop counterterms have 2�+ 2-derivatives
∆S1−loop ∝�
d4x√g�CαβµνC
αβµν�
∆S2−loop ∝�
d4x√g�Cµν
ρσCρσαβCαβ
µν�
General Relativity is perturbatively non-renormalizable
Perturbative Quantum Gravity
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Asymptotic Safety
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Asymptotic Safety
• UV-completion: 1. New physics
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Asymptotic Safety
• UV-completion: 1. New physics 2. Nonperturbative “self-healing”
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Asymptotic Safety
• UV-completion: 1. New physics 2. Nonperturbative “self-healing”
Asymptotically Safe Theory:
‣ Has a (non-gaussian) RG fixed point
‣ The UV Critical Surface is finite dimensional
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Asymptotic Safety
• UV-completion: 1. New physics 2. Nonperturbative “self-healing”
• Generalized, nonperturbative renormalizability requirement
Asymptotically Safe Theory:
‣ Has a (non-gaussian) RG fixed point
‣ The UV Critical Surface is finite dimensional
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Theory Space: Quantum Einstein GravitySymmetry: Diffeomorphisms
FP Structure of Quantum Gravity
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Critical properties “easy” to determine with perturbative methods
• Unfortunately gravity is perturbatively nonrenormalizable
Theory Space: Quantum Einstein GravitySymmetry: Diffeomorphisms
FP Structure of Quantum Gravity
GFP
[GN ] = −2
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Critical properties “easy” to determine with perturbative methods
• Unfortunately gravity is perturbatively nonrenormalizable
Theory Space: Quantum Einstein GravitySymmetry: Diffeomorphisms
FP Structure of Quantum Gravity
GFP
[GN ] = −2
NGFPβ
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Horava-Lifshitz Gravity in a nutshell
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Anisotropic field theories: change dispersion relation
Horava-Lifshitz Gravity in a nutshell
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Anisotropic field theories: change dispersion relation
Horava-Lifshitz Gravity in a nutshell
S =
� �φ2 − φ∆zφ+
N�
n=1
gnφn
�dt ddx
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Anisotropic field theories: change dispersion relation
• This corresponds to an anisotropic scaling
Horava-Lifshitz Gravity in a nutshell
S =
� �φ2 − φ∆zφ+
N�
n=1
gnφn
�dt ddx
t → b t,
x → b1/z x
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Anisotropic field theories: change dispersion relation
• This corresponds to an anisotropic scaling
• Higher order in spatial derivatives decreases the degree of divergence in loop integrals
Horava-Lifshitz Gravity in a nutshell
S =
� �φ2 − φ∆zφ+
N�
n=1
gnφn
�dt ddx
t → b t,
x → b1/z x
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Anisotropic field theories: change dispersion relation
• This corresponds to an anisotropic scaling
• Higher order in spatial derivatives decreases the degree of divergence in loop integrals
• Two time derivatives: naive unitarity maintained
Horava-Lifshitz Gravity in a nutshell
S =
� �φ2 − φ∆zφ+
N�
n=1
gnφn
�dt ddx
t → b t,
x → b1/z x
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Horava-Lifshitz Gravity in a nutshell
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Perturbatively renormalizable quantum theory of gravity
Horava-Lifshitz Gravity in a nutshell
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Perturbatively renormalizable quantum theory of gravity
• Anisotropic scaling between space and time has a natural formulation in ADM variables:
Horava-Lifshitz Gravity in a nutshell
ds2 = N2dt2 + σij
�dxi +N idt
� �dxj +N jdt
�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Perturbatively renormalizable quantum theory of gravity
• Anisotropic scaling between space and time has a natural formulation in ADM variables:
• Field content:
Horava-Lifshitz Gravity in a nutshell
ds2 = N2dt2 + σij
�dxi +N idt
� �dxj +N jdt
�
{N(t, x) , N i(t, x) , σij(t, x)}
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Perturbatively renormalizable quantum theory of gravity
• Anisotropic scaling between space and time has a natural formulation in ADM variables:
• Field content:
• Symmetry: Foliation-Preserving Diffeomorphisms
Horava-Lifshitz Gravity in a nutshell
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)
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Perturbatively renormalizable quantum theory of gravity
• Anisotropic scaling between space and time has a natural formulation in ADM variables:
• Field content:
• Symmetry: Foliation-Preserving Diffeomorphisms
• Projectable HL Gravity
Horava-Lifshitz Gravity in a nutshell
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)
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Horava-Lifshitz gravity in a nutshell
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Horava-Lifshitz gravity in a nutshell
• Weakend symmetry requirements ⇒ more terms in action
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
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�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
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�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Kij ≡ (2N)−1 [∂tσij −∇iNj −∇jNi]
Horava-Lifshitz gravity in a nutshell
• Weakend symmetry requirements ⇒ more terms in action
• Gravitational part of the action
• 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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Kij ≡ (2N)−1 [∂tσij −∇iNj −∇jNi]
Horava-Lifshitz gravity in a nutshell
• Weakend symmetry requirements ⇒ more terms in action
• Gravitational part of the action
• Potential depends on the version of Horava-Lifshitz gravity considered
• Interesting case at criticality z = d:
SHL =1
16πG
�dtddxN
√σ�KijK
ij − λK2 + V�
Vd=3 =g0 + g1R+ g2,1R2 + g2,2RijR
ij − g3,1R∆xR− g3,2Rij∆xRij
+ g3,3R3 + g3,4RRijR
ij + g3,5RijR
jkR
ki
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Kij ≡ (2N)−1 [∂tσij −∇iNj −∇jNi]
Horava-Lifshitz gravity in a nutshell
• Weakend symmetry requirements ⇒ more terms in action
• Gravitational part of the action
• Potential depends on the version of Horava-Lifshitz gravity considered
• Interesting case at criticality z = d:
‣ Newton’s constant is dimensionless
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Kij ≡ (2N)−1 [∂tσij −∇iNj −∇jNi]
Horava-Lifshitz gravity in a nutshell
• Weakend symmetry requirements ⇒ more terms in action
• Gravitational part of the action
• Potential depends on the version of Horava-Lifshitz gravity considered
• Interesting case at criticality z = d:
‣ 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 − g3,1R∆xR− g3,2Rij∆xRij
+ g3,3R3 + g3,4RRijR
ij + g3,5RijR
jkR
ki
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Kij ≡ (2N)−1 [∂tσij −∇iNj −∇jNi]
Horava-Lifshitz gravity in a nutshell
• Weakend symmetry requirements ⇒ more terms in action
• Gravitational part of the action
• Potential depends on the version of Horava-Lifshitz gravity considered
• Interesting case at criticality z = d:
‣ 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 − g3,1R∆xR− g3,2Rij∆xRij
+ g3,3R3 + g3,4RRijR
ij + g3,5RijR
jkR
ki
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Kij ≡ (2N)−1 [∂tσij −∇iNj −∇jNi]
Theory Space: Horava-LifshitzSymmetry: Foliation Preserving Diffs
NGFP
Subspace: Quantum Einstein GravitySymmetry: Diffs
GFP
aGFPβ
FP Structure of Quantum Gravity
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Theory Space: Horava-LifshitzSymmetry: Foliation Preserving Diffs
NGFP
Subspace: Quantum Einstein GravitySymmetry: Diffs
GFP
aGFPβ
FP Structure of Quantum Gravity
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
QCD: Asymptotic Freedom
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Scale-dependence encoded in beta functions
QCD: Asymptotic Freedom
k∂kα(k) = β(α)
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Scale-dependence encoded in beta functions
• Consider SU(3) Yang-Mills theory:
QCD: Asymptotic Freedom
k∂kα(k) = β(α)
β(αs) = −�11− 2nf
3
�α2s
2π
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Scale-dependence encoded in beta functions
• Consider SU(3) Yang-Mills theory:
QCD: Asymptotic Freedom
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!
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Scale-dependence encoded in beta functions
• Consider SU(3) Yang-Mills theory:
• α decreases with increasing energy• α = 0 is a UV-attractive fixed point of the RG
QCD: Asymptotic Freedom
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!
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Scale-dependence encoded in beta functions
• Consider SU(3) Yang-Mills theory:
• α decreases with increasing energy• α = 0 is a UV-attractive fixed point of the RG
• QCD is asymptotically free
QCD: Asymptotic Freedom
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!
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Scale-dependence encoded in beta functions
• Consider SU(3) Yang-Mills theory:
• α decreases with increasing energy• α = 0 is a UV-attractive fixed point of the RG
• QCD is asymptotically free
• Beta function changes sign if
QCD: Asymptotic Freedom
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!
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Scale-dependence encoded in beta functions
• Consider SU(3) Yang-Mills theory:
• α decreases with increasing energy• α = 0 is a UV-attractive fixed point of the RG
• QCD is asymptotically free
• Beta function changes sign if
• Too many flavors destroy asymptotic freedom
QCD: 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!
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
QED: Landau Pole
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Scale-dependence encoded in beta functions
QED: Landau Pole
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
k∂kα(k) = β(α)
• Scale-dependence encoded in beta functions
• QED:
QED: Landau Pole
β(α) =2α2
3π
2 4 6 8 10E0
20
40
60
80
100!
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
k∂kα(k) = β(α)
• Scale-dependence encoded in beta functions
• QED:
• α decreases with decreasing energy
QED: Landau Pole
β(α) =2α2
3π
2 4 6 8 10E0
20
40
60
80
100!
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
k∂kα(k) = β(α)
• Scale-dependence encoded in beta functions
• QED:
• α decreases with decreasing energy
• 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!
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
k∂kα(k) = β(α)
• Scale-dependence encoded in beta functions
• QED:
• α decreases with decreasing energy
• 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!
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
k∂kα(k) = β(α)
• Is the theory asymptotically free?
• Does it reproduce the correct phenomenology?
• Does it resolve previous issues?
Questions
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Is the theory asymptotically free?
Questions
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
COVARIANT EFFECTIVE ACTIONS
IN HL GRAVITY
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Anisotropic scalar field
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar
Anisotropic scalar field
Γk[N,Ni,σ,φ] = ΓHL
k [N,Ni,σ] + SLS[N,Ni,σ,φ]
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar
• Gravitational sector
Anisotropic scalar field
Γk[N,Ni,σ,φ] = ΓHL
k [N,Ni,σ] + SLS[N,Ni,σ,φ]
ΓHL
k = 1
16πGk
�dtddxN
√σ�KijK
ij − λkK2 + Vk
�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar
• Gravitational sector
Anisotropic scalar field
Γk[N,Ni,σ,φ] = ΓHL
k [N,Ni,σ] + SLS[N,Ni,σ,φ]
ΓHL
k = 1
16πGk
�dtddxN
√σ�KijK
ij − λkK2 + Vk
�
Potential V is a function of the intrinsic curvatures
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar
• Gravitational sector
‣ Projectable version
Anisotropic scalar field
Γk[N,Ni,σ,φ] = ΓHL
k [N,Ni,σ] + SLS[N,Ni,σ,φ]
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar
• Gravitational sector
‣ Non-projectable version
Anisotropic scalar field
Γk[N,Ni,σ,φ] = ΓHL
k [N,Ni,σ] + SLS[N,Ni,σ,φ]
ΓHL
k = 1
16πGk
�dtddxN
√σ�KijK
ij − λkK2 + Vk
�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar
• Gravitational sector
‣ Non-projectable version
Anisotropic scalar field
Γk[N,Ni,σ,φ] = ΓHL
k [N,Ni,σ] + SLS[N,Ni,σ,φ]
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 + . . .
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar
• Gravitational sector
‣ Non-projectable version
Anisotropic scalar field
Γk[N,Ni,σ,φ] = ΓHL
k [N,Ni,σ] + SLS[N,Ni,σ,φ]
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Anisotropic scalar field
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar
Anisotropic scalar field
Γk[N,Ni,σ,φ] = ΓHL
k [N,Ni,σ] + SLS[N,Ni,σ,φ]
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar
• Matter sector
Anisotropic scalar field
Γk[N,Ni,σ,φ] = ΓHL
k [N,Ni,σ] + SLS[N,Ni,σ,φ]
SLS ≡ 12
�dtddxN
√σφ [∆t + (∆x)
z]φ
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar
• Matter sector
‣ Two covariant derivatives (orthogonal and tangent to the slice)
Anisotropic scalar field
Γk[N,Ni,σ,φ] = ΓHL
k [N,Ni,σ] + SLS[N,Ni,σ,φ]
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar
• Matter sector
‣ Two covariant derivatives (orthogonal and tangent to the slice)
‣ Anisotropic Laplacian
Anisotropic scalar field
Γk[N,Ni,σ,φ] = ΓHL
k [N,Ni,σ] + SLS[N,Ni,σ,φ]
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Horava-Lifshitz gravity minimally coupled to an anisotropic scalar
• Matter sector
‣ Two covariant derivatives (orthogonal and tangent to the slice)
‣ Anisotropic Laplacian
‣ Reduces to the standard one for
Anisotropic scalar field
Γk[N,Ni,σ,φ] = ΓHL
k [N,Ni,σ] + SLS[N,Ni,σ,φ]
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
One-loop Effective Action
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• One-loop effective action found by integrating out the scalar
One-loop Effective Action
Γeff [N,Ni,σij ] = Sbare + 12Tr log
δ2SLS
δφ δφ
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• One-loop effective action found by integrating out the scalar
• The operator trace
One-loop Effective Action
Γeff [N,Ni,σij ] = Sbare + 12Tr log
δ2SLS
δφ δφ
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
1
2Tr logO = −1
2
� ∞
0
ds
sTre−sO
• One-loop effective action found by integrating out the scalar
• The operator trace
• Relates the one-loop determinant to the heat-kernel
One-loop Effective Action
Γeff [N,Ni,σij ] = Sbare + 12Tr log
δ2SLS
δφ δφ
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
1
2Tr logO = −1
2
� ∞
0
ds
sTre−sO
KO(s) ≡ Tr e−sO , ∂sKO(s) +OKO(s) = 0
• One-loop effective action found by integrating out the scalar
• The operator trace
• Relates the one-loop determinant to the heat-kernel
• Divergences are encoded in the Seeley-deWitt expansion
One-loop Effective Action
Γeff [N,Ni,σij ] = Sbare + 12Tr log
δ2SLS
δφ δφ
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
1
2Tr logO = −1
2
� ∞
0
ds
sTre−sO
KO(s) ≡ Tr e−sO , ∂sKO(s) +OKO(s) = 0
ANISOTROPIC HEAT-KERNELS
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
AHK: dimensional analysis
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Need the Heat Kernel for anisotropic operator:
AHK: dimensional analysis
D2 ≡ ∆t + (∆x)z
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Need the Heat Kernel for anisotropic operator:
• Scaling dimensions
AHK: dimensional analysis
D2 ≡ ∆t + (∆x)z
[t] = z , [x] = 1 , [s] = 2z , [N ] = 0 , [σij ] = 0 ,�N i
�= z − 1
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Need the Heat Kernel for anisotropic operator:
• Scaling dimensions
• The Seeley-deWitt expansion takes the following form
AHK: dimensional analysis
D2 ≡ ∆t + (∆x)z
[t] = z , [x] = 1 , [s] = 2z , [N ] = 0 , [σij ] = 0 ,�N i
�= z − 1
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
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
• Need the Heat Kernel for anisotropic operator:
• Scaling dimensions
• The Seeley-deWitt expansion takes the following form
‣ l = number of shift vectors‣ m = number of time derivatives‣ n = number of spatial derivatives
AHK: dimensional analysis
D2 ≡ ∆t + (∆x)z
[t] = z , [x] = 1 , [s] = 2z , [N ] = 0 , [σij ] = 0 ,�N i
�= z − 1
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
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
• Need the Heat Kernel for anisotropic operator:
• Scaling dimensions
• The Seeley-deWitt expansion takes the following form
‣ l = number of shift vectors‣ m = number of time derivatives‣ n = number of spatial derivatives
AHK: dimensional analysis
D2 ≡ ∆t + (∆x)z
[t] = z , [x] = 1 , [s] = 2z , [N ] = 0 , [σij ] = 0 ,�N i
�= z − 1
Invariants respecting foliation-preserving diffeomorphisms
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
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
• Need the Heat Kernel for anisotropic operator:
• Scaling dimensions
• The Seeley-deWitt expansion takes the following form
‣ l = number of shift vectors‣ m = number of time derivatives‣ n = number of spatial derivatives
AHK: dimensional analysis
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• We want the asymptotic expansion of
Evaluating the Trace
KD2(s) = Tr e−sD2
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• We want the asymptotic expansion of
• Notice that on general backgrounds:
Evaluating the Trace
[∆t,∆x] �= 0
KD2(s) = Tr e−sD2
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• We want the asymptotic expansion of
• Notice that on general backgrounds:
• We can use the following algorithm:
Evaluating the Trace
[∆t,∆x] �= 0
KD2(s) = Tr e−sD2
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• We want the asymptotic expansion of
• Notice that on general backgrounds:
• We can use the following algorithm:
‣ Step 1: Split the exponential using the inverse Campbell-Baker-Hausdorff (Zassenhaus) formula
Evaluating the Trace
[∆t,∆x] �= 0
KD2(s) = Tr e−sD2
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
e−s(A+B) = e−sAe−sBe−s2
2 [A,B]e−s3
6 ([A,[A,B]]+2[B,[A,B]]) · · ·
• We want the asymptotic expansion of
• Notice that on general backgrounds:
• We can use the following algorithm:
‣ Step 1: Split the exponential using the inverse Campbell-Baker-Hausdorff (Zassenhaus) formula
‣ This gives
Evaluating the Trace
[∆t,∆x] �= 0
KD2(s) = Tr e−sD2
Tr e−sD2
= Tr�e−s∆te−s(∆x)
z
C(∆t,∆x)�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
e−s(A+B) = e−sAe−sBe−s2
2 [A,B]e−s3
6 ([A,[A,B]]+2[B,[A,B]]) · · ·
• We want the asymptotic expansion of
• Notice that on general backgrounds:
• We can use the following algorithm:
‣ Step 2: Do a Laplace transform
Evaluating the Trace
[∆t,∆x] �= 0
KD2(s) = Tr e−sD2
Tr e−sD2
=�
i
� ∞
0dv�Wi(v) Tr
�e−s∆t e−v∆x Ci(∆t)
�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• We want the asymptotic expansion of
• Notice that on general backgrounds:
• We can use the following algorithm:
‣ Step 2: Do a Laplace transform
‣ Step 3: Rescale the metric
Evaluating the Trace
[∆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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• We want the asymptotic expansion of
• Notice that on general backgrounds:
• We can use the following algorithm:
‣ Step 4: Apply the Campbell-Baker-Hausdorff formula again
Evaluating the Trace
[∆t,∆x] �= 0
KD2(s) = Tr e−sD2
e−s∆te−s∆x = e−s(∆t+∆x)B(∆t, ∆x)
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• We want the asymptotic expansion of
• Notice that on general backgrounds:
• We can use the following algorithm:
‣ 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
Evaluating the Trace
[∆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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• We want the asymptotic expansion of
• Notice that on general backgrounds:
• We can use the following algorithm:
‣ Step 5: Use the off-diagonal Heat-Kernel !
Evaluating the Trace
[∆t,∆x] �= 0
KD2(s) = Tr e−sD2
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
[ Benedetti, Groh, Machado, Saueressig JHEP 06 (2011) 079 ]
[ Anselmi, Benini JHEP 10 (2007) 099 ]
[ Groh, Saueressig, Zanusso arXiv:1112.4856 ]
Numerical Coefficients
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Extrinsic curvature (kinetic) terms
Numerical Coefficients
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 )
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Extrinsic curvature (kinetic) terms
• Checks:
‣ z = 1 :
Numerical Coefficients
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Extrinsic curvature (kinetic) terms
• Checks:
‣ z = 1 :
‣ z = d :
Numerical Coefficients
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Numerical Coefficients
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Projectable potential terms
Numerical Coefficients
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Projectable potential terms
Numerical Coefficients
Tr e−sD2
� (4π)−d+12 s−
1+d/z2
�dtddxN
√σ�
n≥0
snz bn(d, z)
�
i
a2n,i R(i)2n
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Projectable potential terms
Numerical Coefficients
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Projectable potential terms
‣ Power-counting relevant and marginal terms
Numerical Coefficients
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
bn(d, z) =Γ(d−2n
2z + 1)
Γ(d−2n2 + 1)
, 0 ≤ n ≤ �d/2�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Projectable potential terms
‣ Power-counting relevant and marginal terms
‣ Power-counting irrelevant terms
Numerical Coefficients
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
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�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Numerical Coefficients
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Non-projectable potential terms
Numerical Coefficients
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Non-projectable potential terms
Numerical Coefficients
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
�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Non-projectable potential terms
• Check for z = 1:
Numerical Coefficients
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Non-projectable potential terms
• Check for z = 1:
• Note: determining the c’s is computationally very intensive !
Numerical Coefficients
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Numerical Coefficients
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
SCALAR-DRIVENRG FLOWS INHL GRAVITY
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
One-loop Effective Action
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• The one-loop part of the affective action is
One-loop Effective Action
Γ1 = −1
2
� ∞
0
ds
se−sm2
Tr�e−sD2
�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• The one-loop part of the affective action is
• Substituting the Seeley-deWitt expansion
One-loop Effective Action
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 + . . .
�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• The one-loop part of the affective action is
• Substituting the Seeley-deWitt expansion
• UV-divergences appear at lower boundary of s-integral
One-loop Effective Action
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 + . . .
�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• The one-loop part of the affective action is
• Substituting the Seeley-deWitt expansion
• UV-divergences appear at lower boundary of s-integral
• Renormalize in standard way
One-loop Effective Action
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 + . . .
�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• The one-loop part of the affective action is
• Substituting the Seeley-deWitt expansion
• UV-divergences appear at lower boundary of s-integral
• Renormalize in standard way
• From this we can read the matter-induced beta functions
One-loop Effective Action
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 + . . .
�
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Beta functions for d=z=3
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Kinetic terms
Beta functions for d=z=3
βg =ns
5πg2 , βλ =
ns
15π(3λ− 1) g
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Kinetic terms
• Projectable potential terms
Beta functions for d=z=3
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
�,
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Kinetic terms
• Projectable potential terms
• Non-projectable potential terms
Beta functions for d=z=3
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
�.
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
A parametric view of theory space
!40 !30 !20 !10 10 20G"
1d
1
#"
z$1
z$d
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
A parametric view of theory space
!40 !30 !20 !10 10 20G"
1d
1
#"
z$1
z$d
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
A parametric view of theory space
!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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
A parametric view of theory space
!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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
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
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
CONCLUSIONS
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Conclusions
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Two different universality classes for quantum gravity
Conclusions
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Two different universality classes for quantum gravity
• Scalar induced beta function at criticality similar to QCD
Conclusions
β(G) = − (c− ns)G2
5π
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
• Two different universality classes for quantum gravity
• Scalar induced beta function at criticality similar to QCD
• Asymptotic freedom requires
Conclusions
β(G) = − (c− ns)G2
5π
c > 0
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Future directions
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Future directions
• Anisotropic heat-kernel coefficients can be obtained for any spin (vectors, fermions and gravitons)
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Future directions
• Anisotropic heat-kernel coefficients can be obtained for any spin (vectors, fermions and gravitons)
• Which version of Horava-Lifshitz gravity in the gravitational sector?
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Future directions
• Anisotropic heat-kernel coefficients can be obtained for any spin (vectors, fermions and gravitons)
• Which version of Horava-Lifshitz gravity in the gravitational sector?
• Applications to other systems?
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
Future directions
• Anisotropic heat-kernel coefficients can be obtained for any spin (vectors, fermions and gravitons)
• Which version of Horava-Lifshitz gravity in the gravitational sector?
• Applications to other systems?
Giulio D’Odorico SIFT 2015 - Jena, 5 November 2015
THANK YOU