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Gauge and Parametrization dependence in ... , Kyoto, Aug. 7, 2017, N. Ohta 1

Gauge and Parametrization dependence in Renormalization group approach to Quantum Gravity Nobuyoshi Ohta (Kindai Univ.)

Based on

• 1. NO, R. Percacci and A. D. Pereira,“Gauges and functional measures in quantum gravity I: Einstein

theory,” JHEP 1606 (2016) 115 [arXiv:1605.00454 [hep-th]],

• 2. “Gauges and functional measures in quantum gravity II: Higher derivative gravity,” arXiv:1610.07991

[hep-th].

Gauge and Parametrization dependence in ... , Kyoto, Aug. 7, 2017, N. Ohta 2

1 Introduction

Gravitiational wave is discovered! The next step is Quantum gravity. However

• If the spacetime metric itself is whole dynamical quantity, there is

no notion of spacetime, distance etc. and it is quite difficult how to ”quantize” the theory. ⇒ background field formalism Still Einstein theory is non-renormalizable but it is only a low-energy effective theory!

• Higher-order terms always appear in quantum theory e.g. quantized

Einstein and string theories! ⇒ Possible UV completion?

• In 4D, quadratic (higher derivative) theory is renormalizable but non-

unitary! (Stelle) ⇒ No way out?

• In this situation, the only possible way to make sense of the quantum

effects in gravity seems to be the asymptotic safety.

Gauge and Parametrization dependence in ... , Kyoto, Aug. 7, 2017, N. Ohta 3

2 Asymptotic safety and Wilsonian action

Effective action describing physical phenomenon at a momentum scale k = integrate out all fluctuations of the fields with momenta larger than k. ⇒ effective average action Γk (Note: Γ0 is the effective action.) k: the lower limit of the functional integration (the infrared cutoff).

• We consider effective theory at an energy scale k. This is divergent

itself. Most important fact

✓ ✏

The dependence of the effective action on k gives the Wilsonian RG flow, which is free from any divergence, giving finite quantum theory.

✒ ✑

k∂kΓk(Φ) = 1 2tr [( δ2Γk ∂ΦAΦB + Rk )−1 k∂kRk ] . Exact renormalization group equation! Rk: the cutoff function.

Gauge and Parametrization dependence in ... , Kyoto, Aug. 7, 2017, N. Ohta 4

• FRGE gives flow of the effective action in the

theory space defined by suitable bases Oi. Γk = ∑

i

gi(k)Oi ⇒ dΓk dt = ∑

i

βiOi βi = dgi dt

• We can set initial conditions at some point and

then flow to k → ∞.

Figure 1: RG flow

• Behaviors for k → ∞

– Couplings go to infinity Failure as quantum theory (Landau pole in QED) – Couplings do not go to infinity ∗ go to fixed points (FPs) If all couplings go to finite FPs, physical quantities are well de- fined, giving the UV complete theory ⇒ Asymptotic safety ∗ becomes limit cycle This case is rare and is not considered.

• Starting from suitable initial condition, we integrate it down to k → 0,

which gives full quantum effective action, from which we can obtain correlation functions!

Gauge and Parametrization dependence in ... , Kyoto, Aug. 7, 2017, N. Ohta 5

The theories on the same trajectory belong to the same universality

• class. ⇒ The trajectories with the same FP make a surface, called critical

surface of dimension given by the number of relevant operators. In the ideal case, we also require that the number of relevant operators (only which are retained) are finite. ⇒ Predictability In pertubation with Gaussian fixed point, the relevant operators are precisely renormalizable interactions. Asymptotic safety

✓ ✏

We can define quantum theory if we can define nonperturbative RG flow and the couplings approach a fixed point in the ultraviolet energy. (Weinberg) · · · nonperturbative renomalizability

✒ ✑

There is accumulating evidence (up to 34th order in R) that there are always nontrivial fixed points. ⇒ Asymptotic safety program may be the right direction. The problems studied here

✓ ✏

We want to study gauge and parametrization dependence

✒ ✑

Gauge and Parametrization dependence in ... , Kyoto, Aug. 7, 2017, N. Ohta 6

The quantum theory of any gauge theory including gravity is defined by fixing the gauge. ⇒ How does it affect the result?

3 Guage and parametrization dependence

Gauge and parametrization dependence.

✓ ✏

There are three kinds that we can consider:

• 1. Linear split defined by

gµν = (¯ gµν + hµν)Det(δα

β + hα β)m

• 2. Inverse split defined by

gµν = (¯ gµν − hµν)Det(δα

β + hα β)m.

• 3. Exponential split defined by

gµν = ¯ gµρ(eh)ρ

νemh, h = hµ µ.

m is called weight.

✒ ✑

Gauge and Parametrization dependence in ... , Kyoto, Aug. 7, 2017, N. Ohta 7

More generally we parametrize: gµν = ¯ gµν + δg(1)

µν + δg(2) µν

δg(1)

µν = hµν + m¯

gµνh , δg(2)

µν = ωhµρhρ ν + mhhµν + m

( ω − 1 2 ) ¯ gµνhαβhαβ + 1 2m2¯ gµνh2 . ω = 0 ⇒ linear expansion of metric ω = 1/2 ⇒ exponential expansion ω = 1 ⇒ linear expansion of the inverse metric. One-loop divergences in the Einstein gravity with gauge-fixing S = ZN ∫ ddx√g(2Λ − gµνRµν(g)) + ZN 2a ∫ ddx √¯ g( ¯ ∇αhα

µ −

¯ b + 1 d ¯ ∇µh)2 , Results Γk = ∫ ddx √¯ g [ A1 16πdkd + B1 16π(d − 2)kd−2 ¯ R + C1 d − 4kd−4 ¯ R2 + . . . ] , Keep a arbitrary. Choice of ω = 1

2 and b → ∞ (unimodular gauge) minimize the dependence

Gauge and Parametrization dependence in ... , Kyoto, Aug. 7, 2017, N. Ohta 8

• n the gauge and parametrization.

A1 = 16π(d − 3) (4π)d/2Γ(d/2), B1 = d5 − 4d4 − 9d3 − 48d2 + 60d + 24 (4π)d/2−13(d − 1)d2Γ (d

2

) , C1 = 5d8 − 37d7 − 17d6 − 743d5 + 1668d4 + 684d3 + 16440d2 − 13680d − 8640 (4π)d/21440(d − 1)2d3Γ (d

2

) . independently of m, a, Λ. In particular it is independent of gauge param- eters! Most interesting property: Duality In any dimension, for any value of ˜ Λ and in any gauge, the functions A1, B1 and C1 have the following property: A1(ω, m) = A1 ( 1 − ω, −m − 2 d ) , and so on This persists if we include higher derivative terms R2, R2

µν, and probably

any function of these. For how large class of theory is this true, and what is its physical mean- ing is yet to be studied.

Gauge and Parametrization dependence in ... , Kyoto, Aug. 7, 2017, N. Ohta 9

4 Conclusions

We have studied a functional renormalization group equation for arbi- trary gauge and parametrization up to R2 of the scalar curvature and R2

µν.

There are ultraviolet fixed points essential for Asymptotic Safety in all theories studied so far. There is indication that the critical UV surface is 3 dimensional. We have studied the dependence of gauge and parametrization of the metric, and found that

• 1. the dependence is minimal for the exponential parametrization,
• 2. there exists a new DUALITY.

We believe that this is a good step toward the realization of asymptotic safety. Possible future directions:

• Extending the analysis to more general theory (extend the theory

space)

• What is the physical meaning of the DUALITY.