Calculation of decay constant using gradient flow, towards the Kaon bag parameter. University of Tsukuba, A. Suzuki and Y.

Transcription

1 Calculation of decay constant using gradient flow, towards the Kaon bag parameter University of Tsukuba, A. Suzuki and Y. Taniguchi

2 Contents Goal : Calculation of B K with Wilson fermion using gradient flow Why gradient flow? cf. measurement of topological susceptibility WHOT-QCD, Phys. Rev. D 95, (2017) B K is in preparation we report calculation of m PCAC, m π, f π using GF (as an exercise!) technical details of GF method its matching factor (GF scheme MS scheme)

3 Back ground B K gives QCD contribution to the Kaon mixing phenomenon B K = 8 3 f K 2 m K 2 1 K 0 O Δs=2 K 0, O Δs=2 = s γ L μd s γ L μd Contamination from the breaking of the chiral symmetry gives as an undesirable contribution O Δs=2(R) = Z O Δs=2 + Δ 2 O VV AA + Δ 3 O SS PP + Δ 4 O SS+PP + Δ 5 O TT Hard to subtract them Δ i O 10 2, O ΓΓ O Δs=2 D. Becirevic, V. Gim enez, V. Lubicz, G. Martinelli, M. Papinutto and J. Reyes, hep-lat/ Gradient flow supports to measure observables related with the chiral symmetry?

4 Gradient flow diffusion equation in 5dim (fictitious time t) t B μ t, x = D ν G νμ t, x + D μ ν B ν t, x t χ t, x = D 2 χ t, x μ B μ t, x t χ t, x = χ t, x D 2 + μ B μ t, x B μ t = 0 = A μ, χ t = 0 = ψ, χ t = 0 = ψ G. F. removes the UV divergence on the lattice? We can forget the detail of lattice fermion! Ex. Topological susceptibility χ t, x dy e x y 2 4t ψ y

5 Why gradient flow? Measurement of topological susceptibility Y. Taniguchi, K. Kanaya, H. Suzuki, T. Umeda, Phys Rev D Gluonic definition Q t = 1 64π 2 d4 x ε μνρσ G μν a (t, x)g ρσ a (t, x) Fermionic definition (from chiral WT id.) Q 2 = m2 n f 2 P 0 P 0 n f P a P a P a = d 4 x ψ x T a γ 5 ψ(x)

6 Why gradient flow? Measurement of topological susceptibility Y. Taniguchi, K. Kanaya, H. Suzuki, T. Umeda, Phys Rev D Although they use Wilson fermion, two definitions have good agreement! Gradient flow and B K have good chemistry? O Δs=2 + Δ 2 O VV AA + Δ 3 O SS PP + Δ 4 O SS+PP + Δ 5 O TT

7 Road to B K Our goal is B K = 8 3 f K 2 m K 2 1 K 0 O Δs=2 K 0 3pt. function Many success for 1pt. Function Yang-Mills flow and Fermion flow ex. EM tensor on the lattice The next step is 2pt. Function! Target : PCAC mass, pion mass and decay constant from : A μ t, 0, x π t, 0, y, π t, 0, x π t, 0, y towards denominator of B K (f K and m K ) Calculation of 3pt. Function is in preparation

8 Strategy of calculation Ⅰ. A μ t, 0, x π t, 0, y at flow time t 0 non-perturbatively renormalized in GF scheme Ⅱ. C A t C π (t) A μ t, 0, x π t, 0, y Ⅲ. transfer from GF scheme to MS scheme matching factor C Γ (t) have been calculated K. Hieda and H. Suzuki, arxiv: [hep-lat]. lim C A t C π (t) A μ t, 0, x π t, 0, y t 0 screen off t dim6 op. term

9 Some techniques Propagator at non-zero flow time is nontrivial χ t χ (t) D t + m 1 We have to flow the fermion field back to flow time 0 χ t, x = y K t, x; 0, y ψ(y) standard flow : 0 t χ t, x = ψ y y K t, x; 0. y adjoin flow : t 0 Lüscher, JHEP 1304, 123 (2013) Otherwise? Hadron mass depends on the flow time (N. Ukita, 2016)

10 Lattice setup NP O(a) improved Wilson fermion Iwasaki gauge action flavors Lattice size : β = a = fm 1/a GeV Heavy ud quark masses, m π /m ρ Almost physical s quark m ηss /m φ 0. 74

11 PCAC mass m ud = ± MeV renormalized in MS at 2GeV fit with second order poly.

12 PCAC mass m s = ± MeV renormalized in MS at 2GeV

13 Pion mass Flow eq. does not effect to the pion mass χ t χ (t) D t + m 1

14 Pion decay constant Lattice artifact O(a 2 /t) is severe! Window region Blue : A + Bt, f π = ± 44.1 MeV Green : A + Bt + C/t, f π = ± 43.2 MeV

15 Progress of B K Calculation of K 0 O Δs=2 K 0 is in preparation We calculated the matching factor p 1 p 2 O ± = s1 γ L μd 2 s3 γ L μd 4 ± s1 γ L μd 4 s3 γ L μd 2 t p 3 p 4 1st order perturbation set p i = 0 for simplicity

16 Matching factor of 4-fermi op. Z GF MS(±) means Heat kernel K t x = e ip x e tp2 p = 1 + g2 4π 2 3 ±N c 1 N c log 8μ 2 t γ log 4 + N c 2 6N c + 5 2N c + 2C F log(432)

17 Summary and outlook We calculate 2 pt. function as a step towards the calculation of B K Specifically, PCAC mass, pion mass and decay constant They are renormalized in MS at 2GeV We should work more about fitting We have to take the flow time to 0 limit! decay constant suffer from the Lattice artifacts We calculate the matching factor of O ± Calculation of B K is still on its way!

18 Road to B K Our goal is B K = 8 3 f K 2 m K 2 1 K 0 O Δs=2 K 0 3pt. function Many success for 1pt. Function Yang-Mills flow and Fermion flow ex. EM tensor on the lattice (WHOT-QCD, Lattice2016)

19 Some techniques (WHOT-QCD, Lattice2016) O a 2 /t Window region O t 2 Window region works well to extract the physics