Gauge and Parametrization dependence in Renormalization group - PowerPoint PPT Presentation

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 Γ k (Note: Γ 0 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. ✒ ✑ [( δ 2 Γ k ) − 1 ] k∂ k Γ k (Φ) = 1 2 tr ∂ Φ A Φ B + R k k∂ k R k . Exact renormalization group equation! R k : 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 O i . ⇒ d Γ k β i = dg i ∑ ∑ Γ k = g i ( k ) O i dt = β i O i dt i i • 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 β + h α β ) m g µν = (¯ g µν + h µν )Det( δ α 2. Inverse split defined by g µν = (¯ β + h α g µν − h µν )Det( δ α β ) m . 3. Exponential split defined by g µρ ( e h ) ρ ν e mh , h = h µ g µν = ¯ µ . m is called weight. ✒ ✑

Gauge and Parametrization dependence in ... , Kyoto, Aug. 7, 2017, N. Ohta 7 More generally we parametrize: g µν + δg (1) µν + δg (2) g µν = ¯ µν δg (1) µν = h µν + m ¯ g µν h , ( ) ω − 1 g µν h αβ h αβ + 1 g µν h 2 . δg (2) µν = ωh µρ h ρ 2 m 2 ¯ ν + mhh µν + m ¯ 2 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 ¯ d d x √ g (2Λ − g µν R µν ( g )) + Z N d d x √ ¯ b + 1 ∫ ∫ ∇ µ h ) 2 , g ( ¯ ¯ ∇ α h α S = Z N µ − 2 a d Results [ A 1 d d x √ ¯ B 1 C 1 ] ∫ 16 πdk d + 16 π ( d − 2) k d − 2 ¯ d − 4 k d − 4 ¯ R 2 + . . . Γ k = g R + , 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 on the gauge and parametrization. 16 π ( d − 3) A 1 = (4 π ) d/ 2 Γ( d/ 2) , B 1 = d 5 − 4 d 4 − 9 d 3 − 48 d 2 + 60 d + 24 , ( d ) (4 π ) d/ 2 − 1 3( d − 1) d 2 Γ 2 C 1 = 5 d 8 − 37 d 7 − 17 d 6 − 743 d 5 + 1668 d 4 + 684 d 3 + 16440 d 2 − 13680 d − 8640 . ( d ) (4 π ) d/ 2 1440( d − 1) 2 d 3 Γ 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 A 1 , B 1 and C 1 have the following property: ( ) 1 − ω, − m − 2 A 1 ( ω, m ) = A 1 , and so on d This persists if we include higher derivative terms R 2 , R 2 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 R 2 of the scalar curvature and R 2 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.