Gravitational Instantons and Anomalous Chiral Symmetry Breaking

Gravitational instantons and anomalous chiral symmetry breaking

Yu Hamada (Kyoto Univ.) Based on [arXiv:2009.08728] w/ Jan M. Pawlowski (Heidelberg U.) Masatoshi Yamada (Heidelberg U.)

Exact Renormalization Group 2020 (3Nov. 2020)

1 Asymptotic safety of gravity [Weinberg ’76]

• Existence of non-trivial UV ﬁxed point for gravity Status of the asymptotic safety paradigm for quantum gravity and matter 3 g(k) ≡ Gk2 • eg.) Einstein-Hilbert truncation: [arXiv:1709.03696] [Reuter ’98] [Souma ’99] 1.5 g* 1 d4x g R Γk = ( − 2Λ) ] 1.0 16πG ∫ [ g(k) 0.5 • We can take continuum limit: k → ∞. 0.0 105 109 1013 1017 1021 1025 • UV-complete theory space k/GeV M = critical surface of the ﬁxed pt. pl

• Fixed pt. still exists even when other operators such as n μν R , RμνR , … are included.

→ non-perturbatively renormalizable quantum theory for gravity 2 Fig. 1 Beta functions and the corresponding running for a UV attractive free/interacting ﬁxed point (lower/upper panels). As the ﬁxed points are UV attractive, the value of the coupling at one reference scale is an initial condition that can be chosen freely. Upper panels: beta function and scale dependence of the Newton coupling according to Eq. (20).

the scaling of couplings is determined by the sum of canonical scaling and quantum scaling which is a consequence of loop e↵ects. In fact, a ﬁxed point can arise from a balance between canonical and quantum scaling, as

= d g + ⌘ (g ), (2) gi g¯i i i i

where ⌘i is an anomalous scaling dimension that arises as a consequence of quantum ﬂuctuations. An interacting ﬁxed point lies at

gi⇤ = ⌘i/dg¯i . (3)

4 For instance, in d =4 ✏,the4 theory has a fixed point at 4⇤ = 16⇡2✏/3, the Wilson-Fisher fixed point [7] playing an important role in sta- tistical physics. In that setting, interacting fixed points encode the scaling exponents near a continuous (second or higher order) phase transition, where scale invariance is due to a diverging correlation length at criticality. QFTs live in theory space, which is the infinite-dimensional space of all couplings that are compatible with the symmetries of the model. Asymptotic safety/freedom is the existence of a fixed point in this space, i.e., for a model to become asymptotically safe, all couplings have to reach a scale-invariant fixed point. Thus, determining whether a model can become asymptotically safe/free, requires us to explore the RG flow of all infinitely many couplings. On the other hand, in perturbatively renormalizable models with asymptotic freedom one typically does not think about higher-order couplings, but restrict the setting to the perturbatively renormalizable ones, thus seemingly working PhenomenologicalStatus of the asymptotic implication? safety paradigm for quantum gravity and matter 3 [arXiv:1709.03696] g • Gravity is strong coupling for k ≳ Mpl 1.5 *

] 1.0 [ • could cause chiral symmetry breaking g(k) of fermions: 0.5 ψψ 0.0 ⟨ ¯ ⟩ ≠ 0 105 109 1013 1017 1021 1025 k/GeV → All fermions acquire dynamical masses ∼ Mpl → inconsistent with light fermions in our world → phenomenologically exclude asymptotic safety scenario

[Eichhorn-Gies, ’11] [Eichhorn, ’12] [Meibohm-Pawlowski, ’16]

[Eichhorn-Lippoldt, ’16] [Eichhorn-Held, ’17] This talk: GravitationalFig. 1 Beta functions instantons and the can corresponding trigger chiral running sym. for a breaking! UV attractive free/interacting ﬁxed point (lower/upper panels). As the ﬁxed points are UV attractive, the value of the coupling at one reference scale is an initial condition that can be chosen freely. Upper 3 panels: beta function and scale dependence of the Newton coupling according to Eq. (20).

= d g + ⌘ (g ), (2) gi g¯i i i i

where ⌘i is an anomalous scaling dimension that arises as a consequence of quantum ﬂuctuations. An interacting ﬁxed point lies at

gi⇤ = ⌘i/dg¯i . (3)

• Gravity is a gauge theory with Lorentz sym: SO(4) ≃ SU(2) × SU(2)

μ i ab μ a μ ℒ ⊃ ψ¯ γ ∂μ − σabωμ ψ γ ≡ γ ea ( 2 ) ab i a b σ ≡ [γ , γ ] Spin connection 4 ~ SO(4) gauge ﬁeld

• instanton-like conﬁguration with winding # for the SU(2) sector →gravitational instanton

• solution to the vacuum Einstein eq.

analogue of Yang-Mills instanton

4 [Hawking ’76] Example of grav. instanton [Eguchi-Hanson ’78]

• Eguchi-Hanson metric → not considered in this talk • Taub-NUT metric

• K3-surface (Calabi-Yau manifold) → compact and closed manifold → In path integral, it appears as a ﬂuctuation around ﬂat space: [Hebecker-Henkenjohann 1906.07728]

worm-hole like throat

ﬂat spacetime 5 ’t Hooft vertex [’t Hooft ’86]

μ A • axial anomaly of fermions : ⟨∂ Jμ ⟩ ∼ RR˜ → Grav. instanton induces an effective vertex called ’t Hooft vertex.

(similarly to QCD instanton) K3

worm-hole like throat ρ ∼ 1tH ﬂat spacetime ﬂat spacetime

1 − γ5 1tH = λtH det ψ¯s ψt + (h . c.) U(1)A operator st 2

This cannot be induced by Taub-NUT, Eguchi-Hanson metrics. (APS-index is 0) 6 ’t Hooft vertex and asymptotically safe gravity

• dim. analysis based on the dilute gas approx.

−2 1 ⇒ λtH(k) ∼ k exp − ( gN(k) ) ρ ∼ k−1

Status of the asymptotic safety paradigm for quantum gravity and matter 3 • assume k is an energy scale [arXiv:1709.03696]

1.5 g*

UV (k ≫ Mpl ) : active (scale invariant) ] 1.0 [ g(k) 0.5 IR (k M ) : exponentially suppress ≪ pl 0.0 105 109 1013 1017 1021 1025 k/GeV

What does this affect？→ causes chiral sym. breaking: ⟨ψ¯ ψ⟩ ≠ 0

Fig. 1 Beta functions and the corresponding running for a UV attractive free/interacting ﬁxed point (lower/upper panels). As the ﬁxed points are UV attractive, the value of the coupling at one reference scale is an initial condition that can be chosen freely. Upper panels: beta function and scale dependence of the Newton coupling according to Eq. (20).

= d g + ⌘ (g ), (2) gi g¯i i i i

where ⌘i is an anomalous scaling dimension that arises as a consequence of quantum ﬂuctuations. An interacting ﬁxed point lies at

gi⇤ = ⌘i/dg¯i . (3)

• 2-ﬂavor NJL model + gluon and gravity

Γ = d4x g ψ¯ i∇ψ + λ ℒ + λ ℒ k ∫ [ q q top top]

1 1 + d4x g R + d4x g (Fa )2 2 μν 16πG ∫ 4gs ∫

2 a 2 σ − π channel : ℒq = (ψ¯ ψ) − (ψ¯ γ5σ ψ)

1 1 − γ5 ’t Hooft vertex : ℒtop = det ψ¯s ψt + (h . c.) 2 st 2

• SU(2)L × SU(2)R symmetry (broken down to SU(2)V )

• RG analysis for running coupling: λtop, λq, gN, gs

• Chiral sym. breaking = λq → ∞ 8 RG ﬂow eq. for four-fermion couplings

• Functional renormalization group (Wetterich equation):

−1 1 (2) ∂tΓk = Tr Γ + Rk ∂tRk 2 [( k ) ] Rk : cutoff function

• RGE in the absence of grav. instantons: [Eichhorn-Gies '11] [Braun-Leonhardt-Pospiech '18]

7 2 8 ∂tλ¯q = 2λ¯q − λ¯q − (λ¯q + λ¯ )λ¯ + ⋯ λ¯ , λ¯ :dim. less 2π2 π2 top top q top

13 2 1 ∂tλ¯ = 2λ¯ + λ¯ + 4λ¯ + λ¯q λ¯q + ⋯ top top 3π2 top 6π2 ( top )

We have omitted contributions from graviton and gluon exchanges. 9 RG ﬂow of four-fermion couplings

• It is difﬁcult to derive RG eq. in the presence of grav. instantons. → write down naive RG eq. leaving coefﬁcients as free parameters

• Recall that, based on the dilute approx., a single instanton contribution is −2 1 λtop(k) ∼ k exp − ( gN(k) )

7 2 8 ∂tλ¯q = 2λ¯q − λ¯q − (λ¯q + λ¯ )λ¯ 2π2 π2 top top

13 2 1 ∂tλ¯ = 2λ¯ + λ¯ + 4λ¯ + λ¯q λ¯q top top 3π2 top 6π2 ( top ) γ(i) : free parameter β gN 1 1 +γ(1) exp − + γ(2) λ¯ exp − 2 top gN ( gN ) ( gN )

10 Result of the ﬂow γ(1) = 30, γ(2) = 1

• Gravitational instantons affect RG ﬂow at k ∼ Mpl

10 0.4

0.3 5 10 15 20 25

0.2 -10

0.1 -20

0.0 -30 1011 1013 1015 1017 1019 1021

10 • λ¯ diverges at k ∼ 1(0.1) M q pl 0

→ χ-sym breaking ! -10

-20

-30 1 2 3 4 5 6 11 Result of the ﬂow γ(1) = 30, γ(2) = 1

20 20

15 15

10 10

5 5

0 0

-10 -5 0 5 -10 -5 0 5

20 20

15 15

10 10

5 5

0 0

-10 -5 0 5 -10 -5 0 5 12 Parameter space

i • How large free parameters γ( ) cause the chiral sym. breaking:

50 10

40 8

30 6

20 4

10 2

0 0 -40 -20 0 20 40 -20 -10 0 10 20

• Sym. breaking is universal for |γ(1) | ≳ 12 .

• This region leads to heavy fermions, and hence phenomenologically excluded. 13 Summary

• Gravity is strong coupling in the asymptotic safety scenario.

• Gravitational instantons induce the ’t Hooft vertex for fermions.

• FRG analysis in NJL-like model → Chiral sym. breaking ⟨ψ¯ ψ⟩ ≠ 0 occurs universally.

• Such parameter space is phenomenologically excluded.

Future works: i • Parameters γ( ) are calculable in principle →We can constrain UV theories. • Probe topological structure of spacetime using matter. 14 Back up

15 Divergence of λq → signal of ⟨ψ¯ ψ⟩ ≠ 0

• Bosonization : ϕ ∼ ψ¯ ψ ϕ 2 λq (ψ¯ ψ) 1 V(ϕ) = m2ϕ2 + ⋯ m2 = λq

χ -symmetric 2nd order phase transition χ -breaking

[Yamada-san's slide] 16 ’t Hooft vertex and asymptotically safe gravity

• dim. analysis based on the dilute gas approx. 2 2 ρ λtH ∼ ρ exp − ( G ) ρ Status of the asymptotic safety paradigm for quantum gravity and matter 3 • assume an energy scale k [arXiv:1709.03696] −1 2 g ρ → k G → G(k) = gN(k)/k 1.5 *

] 1.0 [ g(k) −2 1 ⇒ λtH(k) ∼ k exp − 0.5 ( gN(k) ) 0.0 105 109 1013 1017 1021 1025 k/GeV UV (k ≫ Mpl ) : instantons are active (scale invariant)

IR (k ≪ Mpl ) : exponentially suppress

What does this affect？→ causes chiral sym. breaking: ⟨ψ¯ ψ⟩ ≠ 0 17

= d g + ⌘ (g ), (2) gi g¯i i i i

where ⌘i is an anomalous scaling dimension that arises as a consequence of quantum ﬂuctuations. An interacting ﬁxed point lies at

gi⇤ = ⌘i/dg¯i . (3)

1 a −S[ψ,ψ¯] S ψ ψ iψ ψ λ ψ ψ 2 ψγ σ ψ 2 Z = 7ψ7ψ¯e [ , ¯] = ¯∂ + σ [( ¯ ) − ( ¯ 5 ) ] ∫ 2

4 1 2 2⃗ −∫ d x mϕ ϕ multiply: 1 = N 7ϕ e [ 2 ] ∫ i Shift: ϕ → ϕ + (ψ¯ ψ) with m2 = 1/λ 2 σ σ 2mσ τ⃗ = (i, γ5) i 4 1 2 2⃗ ⃗ → S[ψ, ψ¯, ϕ] = d x iψ¯∂ψ + mσ ϕ + ψ¯ (τ⃗ ⋅ ϕ )ψ ∫ [ 2 2 ]

1 EOM: ϕ ⃗ = τ⃗ (ψ¯ ψ) でもとに戻る 2 2mσ

∴ ⟨|ϕ|⟩ ≠ 0 ⇒ ⟨ψ¯ ψ⟩ ≠ 0 ϕ ⃗を回す方向がpion: NG mode 19