Ultraviolet Dynamics of Fermions and Gravity Cold Quantum Cofgee on - - PowerPoint PPT Presentation

Ultraviolet Dynamics of Fermions and Gravity

Cold Quantum Cofgee on June 26, 2018 Marc Schifger, Heidelberg University with A. Eichhorn and S. Lippoldt arXiv 18xx.xxxxx with A. Eichhorn, S. Lippoldt, J. M. Pawlowski and M. Reichert arXiv 1807.xxxxx

Motivation for Quantum Gravity

Why quantum gravity?

General Relativity:

• several high precision tests
• latest confjrmation:

Existence of black holes by LIGO collaboration

• Simplest version of a black hole:

Schwarzschild black hole physical singularity at r . GR is not a fundamental theory

• For E

MPL

c GN :

QG efgects are expected

[LIGO Collaboration, 2016] 2

Why quantum gravity?

General Relativity:

• several high precision tests
• latest confjrmation:

Existence of black holes by LIGO collaboration

• Simplest version of a black hole:

Schwarzschild black hole physical singularity at r = 0. ⇒ GR is not a fundamental theory

• For E

MPL

c GN :

QG efgects are expected

Source: Northern Arizona University 2

Why quantum gravity?

General Relativity:

• several high precision tests
• latest confjrmation:

Existence of black holes by LIGO collaboration

• Simplest version of a black hole:

Schwarzschild black hole physical singularity at r = 0. ⇒ GR is not a fundamental theory

• For E > MPL =

√

c GN :

QG efgects are expected

Source: Northern Arizona University 2

Why quantum gravity with matter?

Standard Model:

• well tested low energy model
• formulated as QFT
• Triviality problem in scalar

and abelian gauge theories

[J. Fröhlich, 1982] [M. Gockeler et al., 1997] , [H. Gies and J. Jäckel, 2004]

• Singularities might carry over to

SM SM is only efgective theory breakdown beyond Planck scale QG might cure Landau poles

[D. Buttazzo, 2013] 3

Why quantum gravity with matter?

Standard Model:

• well tested low energy model
• formulated as QFT
• Triviality problem in scalar ϕ4 and

abelian gauge theories

[J. Fröhlich, 1982] [M. Gockeler et al., 1997] , [H. Gies and J. Jäckel, 2004]

• Singularities might carry over to

SM SM is only efgective theory breakdown beyond Planck scale QG might cure Landau poles

Image credits: Fleur Versteegen 3

Why quantum gravity with matter?

Standard Model:

• well tested low energy model
• formulated as QFT
• Triviality problem in scalar ϕ4 and

abelian gauge theories

[J. Fröhlich, 1982] [M. Gockeler et al., 1997] , [H. Gies and J. Jäckel, 2004]

• Singularities might carry over to

SM → SM is only efgective theory → breakdown beyond Planck scale → QG might cure Landau poles

Image credits: Fleur Versteegen 3

Why quantum gravity with matter?

Standard Model:

• well tested low energy model
• formulated as QFT
• Triviality problem in scalar ϕ4 and

abelian gauge theories

[J. Fröhlich, 1982] [M. Gockeler et al., 1997] , [H. Gies and J. Jäckel, 2004]

• Singularities might carry over to

SM → SM is only efgective theory → breakdown beyond Planck scale → QG might cure Landau poles

Image credits: Fleur Versteegen

compatibility with SM in IR provides test for quantum theory of gravity

3

Outline

Motivation for Quantum Gravity Asymptotically Safe Quantum Gravity Efgective Universality for Gravity and Matter Induced Couplings at UV Fixed Point

4

Asymptotically Safe Quantum Gravity

How to quantize Gravity?

• [GN] = 2 − d

⇒ GR is perturbatively non-renormalizable in d = 4

[G. ’t Hooft and M. J. G. Veltman, 1974] [M. H. Gorofg and A. Sagnotti, 1986]

• Efgective fjeld theory approach: Loss of predictivity at MPL

[J. F. Donoghue and B. R. Holstein, 2015] 6

How to quantize Gravity?

• [GN] = 2 − d

⇒ GR is perturbatively non-renormalizable in d = 4

[G. ’t Hooft and M. J. G. Veltman, 1974] [M. H. Gorofg and A. Sagnotti, 1986]

• Efgective fjeld theory approach: Loss of predictivity at MPL

[J. F. Donoghue and B. R. Holstein, 2015] 6

Asymptotic safety

• Asymptotic freedom

◮ all couplings vanish in the UV ◮ perturbative renormalizability

• Asymptotic safety

[S. Weinberg, 1979]

all dimensionless couplings enter a scale invariant regime interacting theory in the UV non-perturbative renormalizability

CMS Collaboration, 2017 7

Asymptotic safety

• Asymptotic freedom
• all couplings vanish in the UV
• perturbative renormalizability
• Asymptotic safety

[S. Weinberg, 1979]

◮ all dimensionless couplings enter a scale invariant regime ◮ interacting theory in the UV ◮ non-perturbative renormalizability

• A. Eichhorn, 2017

7

Search for Asymptotic safety

Study RG-fmow of dimensionless couplings gi = ¯ gi k−d ¯

gi

βgi(⃗ g) = k ∂kgi = −d ¯

gi gi

+ fi(⃗ g) dimensional quantum ⇒ balancing of dimensional term with quantum correction can lead to AS

8

Critical Exponents

• Linearized β-functions

βgi = βgi

• g=g∗ +

∑

j

(∂βgi ∂gj )

• g=g∗(gj − g∗

j ) + O

( (gj − g∗

j )2)

• Solution to linearized fmow equations

gi k gi

j

cjVi

j

k k

j

with eig M

i

9

Critical Exponents

• Linearized β-functions

βgi = βgi

• g=g∗ +

∑

j

(∂βgi ∂gj )

• g=g∗(gj − g∗

j ) + O

( (gj − g∗

j )2)

• Solution to linearized fmow equations

gi(k) = g∗

i +

∑

j

cjVi

j

( k k0 )−Θj with − eig (M) = Θi .

9

Critical Exponents

• Solution to linearized fmow equations

gi(k) = g∗

i +

∑

j

cjVi

j

( k k0 )−Θj with − eig (M) = Θi . Re(Θi) < 0

• irrelevant direction
• gi(k) k→∞

− − − → g∗

i

• cj drop out
• g∗

i is a prediction

dimensional

9

Critical Exponents

• Solution to linearized fmow equations

gi(k) = g∗

i +

∑

j

cjVi

j

( k k0 )−Θj with − eig (M) = Θi . Re(Θi) < 0

• irrelevant direction
• gi(k) k→∞

− − − → g∗

i

• cj drop out
• g∗

i is a prediction

dimensional Re(Θi) > 0

• relevant direction
• cj remain
• adjust cj to reach g∗

i

• one free parameter for each

relevant direction

9

Critical Exponents

• Solution to linearized fmow equations

gi(k) = g∗

i +

∑

j

cjVi

j

( k k0 )−Θj with − eig (M) = Θi . Re(Θi) < 0

• irrelevant direction
• gi(k) k→∞

− − − → g∗

i

• cj drop out
• g∗

i is a prediction

dimensional Re(Θi) > 0

• relevant direction
• cj remain
• adjust cj to reach g∗

i

• one free parameter for each

relevant direction test Number of relevant directions determines predictivity

9

Critical Exponents

[A. Eichhorn, 2017]

Re(Θi) < 0

• irrelevant direction
• g∗

i is a prediction

Re(Θi) > 0

• relevant direction
• one free parameter for each

relevant direction

9

Tool: Functional Renormalization Group

Non-Perturbative Renormalisation Group Equation [Wetterich, 1993], [Reuter, 1996] k ∂kΓk = 1 2 STr (( Γ(2)

k

+ Rk )−1 k ∂kRk ) = 1 2 Γk = scale dependent efgective action Rk = IR regulator

• exact 1-loop equation
• extract β-functions via projection
• truncation needed → not closed

[H. Gies, 2006] 10

Efgective Universality for Gravity and Matter

Avatars of the Newton coupling

• Einstein-Hilbert gravity minimally coupled to fermions, i.e.

S = − 1 16 πGN ∫ d4x √g (R − 2Λ) +

Nf

∑

i=1

∫ d4x √g ¯ ψi / ∇ ψi G h Gh

• two difgerent ”avatars” of the Newton coupling

12

Avatars of the Newton coupling

• Einstein-Hilbert gravity minimally coupled to fermions, i.e.

S = − 1 16 πGN ∫ d4x √g (R − 2Λ) +

Nf

∑

i=1

∫ d4x √g ¯ ψi / ∇ ψi ∼ √ G3h ∼ √ Gh ¯

ψψ

• two difgerent ”avatars” of the Newton coupling

12

Efgective universality

• Classically: Difgeomorphism invariance

⇒ there should only be one Newton coupling

• On the quantum level

GN 2-loop universality is lost

[Weinberg, 1995]

Gauge fjxing, Regulator

set of identities (mSTI’s) relates avatars

• Efgective universality:

Quantitative agreement of difgerent avatars of the Newton coupling

[A. Eichhorn, P. Labus, J. M. Pawlowski and M. Reichert, 2018]

• Compare both avatars on the level of their
• functions at

G h Gh

G h

G G G G

Gh

G G G G

13

Efgective universality

• Classically: Difgeomorphism invariance

⇒ there should only be one Newton coupling

• On the quantum level

◮ [GN] = −2 ⇒ 2-loop universality is lost

[Weinberg, 1995]

◮ Gauge fjxing, Regulator

set of identities (mSTI’s) relates avatars

• Efgective universality:

Quantitative agreement of difgerent avatars of the Newton coupling

[A. Eichhorn, P. Labus, J. M. Pawlowski and M. Reichert, 2018]

• Compare both avatars on the level of their
• functions at

G h Gh

G h

G G G G

Gh

G G G G

13

Efgective universality

• Classically: Difgeomorphism invariance

⇒ there should only be one Newton coupling

• On the quantum level

◮ [GN] = −2 ⇒ 2-loop universality is lost

[Weinberg, 1995]

◮ Gauge fjxing, Regulator

⇒ set of identities (mSTI’s) relates avatars

• Efgective universality:

Quantitative agreement of difgerent avatars of the Newton coupling

[A. Eichhorn, P. Labus, J. M. Pawlowski and M. Reichert, 2018]

• Compare both avatars on the level of their β-functions at (µ∗, λ∗

3, G3h = Gh ¯ ψψ)

βG3h =2G − 3.4G2 + 0.37G3 + O(G4) βGh ¯

ψψ =2G − 2.8G2 + 0.42G3 + O(G4) 13

Efgective universality for fermions and gravity

Measure of efgective universal- ity ε(G, µ, λ3) =

• ∆βGi − ∆βGj

∆βGi + ∆βGj

• Gi=Gj

with ∆βGi = βGi − 2 Gi

[A. Eichhorn, P. Labus, J. M. Pawlowski and M. Re- ichert, 2018] 14

Efgective universality for fermions and gravity

Measure of efgective universal- ity ε(G, µ, λ3) =

• ∆βGi − ∆βGj

∆βGi + ∆βGj

• Gi=Gj

with ∆βGi = βGi − 2 Gi

[A. Eichhorn, P. Labus, J. M. Pawlowski and M. Re- ichert, 2018] [A. Eichhorn, S. Lippoldt, MS, in preparation] [A. Eichhorn, S. Lippoldt, J. M. Pawlowski, M. Reichert, MS, in preparation] 14

Efgective universality for all avatars of the Newton coupling

• Compare fermions,

scalars, vector fjelds and gravity

15

Efgective universality for all avatars of the Newton coupling

Zoomed in plot region!

• Compare fermions,

scalars, vector fjelds and gravity

• Efgective universality for

all avatars of the Newton coupling

15

Efgective universality for all avatars of the Newton coupling

• Compare fermions,

scalars, vector fjelds and gravity

• Efgective universality for

all avatars of the Newton coupling Implications

• Highly non-trivial cancellations
• Not expected for truncation artifact

→ strong hint for physical nature of asymptotically safe fjxed point

15

Induced Couplings at UV Fixed Point

Induced Couplings at UV Fixed Point

• Naive expectation:

Flow equation generates all interactions that are compatible with symmetry

• For gravity-matter systems:

∃ chiral (shift) symmetric matter interactions

• Corresponding coupling does not

have GFP if GN

• Explicit computations confjrm

expectation

[A. Eichhorn and H. Gies, 2011] [A. Eichhorn and A. Held, 2017] 17

Induced Couplings at UV Fixed Point

• Naive expectation:

Flow equation generates all interactions that are compatible with symmetry

• For gravity-matter systems:

∃ chiral (shift) symmetric matter interactions

• Corresponding coupling does not

have GFP if GN ̸= 0

• Explicit computations confjrm

expectation

[A. Eichhorn and H. Gies, 2011] [A. Eichhorn and A. Held, 2017] [A. Eichhorn, A. Held and J. M. Pawlowski, 2016] 17

Induced Couplings at UV Fixed Point

• Naive expectation:

Flow equation generates all interactions that are compatible with symmetry

• For gravity-matter systems:

∃ chiral (shift) symmetric matter interactions

• Corresponding coupling does not

have GFP if GN ̸= 0

• Explicit computations confjrm

expectation

[A. Eichhorn and H. Gies, 2011] [A. Eichhorn and A. Held, 2017] [A. Eichhorn, A. Held and J. M. Pawlowski, 2016] 17

Induced non-minimal coupling

S =SEH + SDirac +σ ∫ d4x √g Rµν ( ¯ ψiγµ ← → ∇ νψi)

• features shifted Gaussian fjxed

point (sGFP) A G A G

• for GN

18

Induced non-minimal coupling

S =SEH + SDirac +σ ∫ d4x √g Rµν ( ¯ ψiγµ ← → ∇ νψi)

• σ features shifted Gaussian fjxed

point (sGFP) βσ =A0(G..) + A1(G..) σ + O(σ2)

• for GN

−0.5 0.0 0.5 1.0 σh ¯

ψψ

1 βσh ¯

ψψ

Gh ¯

ψψ = 0

Gh ¯

ψψ = 2

GFP for Gh ¯

ψψ = 0

sGFP for Gh ¯

ψψ = 0

18

Induced non-minimal coupling

S =SEH + SDirac +σ ∫ d4x √g Rµν ( ¯ ψiγµ ← → ∇ νψi)

• σ features shifted Gaussian fjxed

point (sGFP) βσ =A0(G..) + A1(G..) σ + O(σ2)

• σ∗ ̸= 0 for G∗

N ̸= 0 −0.5 0.0 0.5 1.0 σh ¯

ψψ

1 βσh ¯

ψψ

Gh ¯

ψψ = 0

Gh ¯

ψψ = 2

GFP for Gh ¯

ψψ = 0

sGFP for Gh ¯

ψψ = 0

1 2 3 4 5 Gh ¯

ψψ

−0.3 −0.2 −0.1 0.0 σ∗

h ¯ ψψ

G3h = Gh ¯

ψψ

G3h = 0

18

Induced non-minimal coupling

S =SEH + SDirac +σ ∫ d4x √g Rµν ( ¯ ψiγµ ← → ∇ νψi)

• σ features shifted Gaussian fjxed

point (sGFP) βσ =A0(G..) + A1(G..) σ + O(σ2)

• σ∗ ̸= 0 for G∗

N ̸= 0 −0.5 0.0 0.5 1.0 σh ¯

ψψ

1 βσh ¯

ψψ

Gh ¯

ψψ = 0

Gh ¯

ψψ = 2

GFP for Gh ¯

ψψ = 0

sGFP for Gh ¯

ψψ = 0

1 2 3 4 5 Gh ¯

ψψ

−0.3 −0.2 −0.1 0.0 σ∗

h ¯ ψψ

G3h = Gh ¯

ψψ

G3h = 0

Asymptotic safety passes non-minimal test for a UV complete theory

• f gravity and matter

18

Induced non-minimal coupling

S =SEH + SDirac +σ ∫ d4x √g Rµν ( ¯ ψiγµ ← → ∇ νψi)

• σ features shifted Gaussian fjxed

point (sGFP) βσ =A0(G..) + A1(G..) σ + O(σ2)

• σ∗ ̸= 0 for G∗

N ̸= 0 −0.5 0.0 0.5 1.0 σh ¯

ψψ

1 βσh ¯

ψψ

Gh ¯

ψψ = 0

Gh ¯

ψψ = 2

GFP for Gh ¯

ψψ = 0

sGFP for Gh ¯

ψψ = 0

1 2 3 4 5 Gh ¯

ψψ

−0.3 −0.2 −0.1 0.0 σ∗

h ¯ ψψ

G3h = Gh ¯

ψψ

G3h = 0

Interacting nature of asymptotically safe fjxed point percolates into chiral symmetry-protected matter sector

18

Summary and Outlook

• Efgective universality

◮ hint for physical nature of asymptotically safe fjxed point ◮ guideline/ justifjcation for future/past truncations

• Induced couplings

◮ Non-minimal couplings are present at the fjxed point ◮ Symmetry protected matter sector is interacting in the UV

19

Summary and Outlook

• Efgective universality

◮ hint for physical nature of asymptotically safe fjxed point ◮ guideline/ justifjcation for future/past truncations

• Induced couplings

◮ Non-minimal couplings are present at the fjxed point ◮ Symmetry protected matter sector is interacting in the UV

• Extend analysis to

phenomenologically relevant numbers of scalars, fermions and vector fjelds

• Explore restrictions of the

gravitational parameter space under the inclusion of non-minimal couplings (weak gravity bound

[A. Eichhorn and A. Held, 2017] ) 19

Summary and Outlook

• Efgective universality

◮ hint for physical nature of asymptotically safe fjxed point ◮ guideline/ justifjcation for future/past truncations

• Induced couplings

◮ Non-minimal couplings are present at the fjxed point ◮ Symmetry protected matter sector is interacting in the UV

• Extend analysis to

phenomenologically relevant numbers of scalars, fermions and vector fjelds

• Explore restrictions of the

gravitational parameter space under the inclusion of non-minimal couplings (weak gravity bound

[A. Eichhorn and A. Held, 2017] )

Thank you for your attention!

19

Example for Asymptotic Safety

Example: Yang-Mills Theory in d = 4 + ϵ at 1-loop

[M. E. Peskin, 1980], [M. Creutz, 1979], [H. Gies, 2003]

[¯ g] = − ϵ 2 βg = ϵ 2 g − b0 g3 = g ( ϵ 2−b0 g2 ) .

0.0 0.5 1.0 1.5 gi −1.0 −0.5 0.0 0.5 βgi

20

Nontrivial cancellations

Scalar-Fish Fermion-Fish Scalar-Star Fermion-Star Scalar-Fish+ Scalar-Star Fermion-Fish+ Fermion-Star

• 0.8
• 0.6
• 0.4
• 0.2

0.0 0.2 0.4

• 40
• 30
• 20
• 10

μ

21