Calculation of decay constant using gradient flow, towards the Kaon bag parameter University of Tsukuba, A. Suzuki and Y. Taniguchi
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, 054502 (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)
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/0401033. Gradient flow supports to measure observables related with the chiral symmetry?
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
Why gradient flow? Measurement of topological susceptibility Y. Taniguchi, K. Kanaya, H. Suzuki, T. Umeda, Phys Rev D.95.054502 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)
Why gradient flow? Measurement of topological susceptibility Y. Taniguchi, K. Kanaya, H. Suzuki, T. Umeda, Phys Rev D.95.054502 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
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
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:1606.04193 [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
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)
Lattice setup NP O(a) improved Wilson fermion Iwasaki gauge action 2 + 1 flavors Lattice size : 28 3 56 β = 2. 05 a = 0. 0701 29 fm 1/a 2. 79 GeV Heavy ud quark masses, m π /m ρ 0. 63 Almost physical s quark m ηss /m φ 0. 74
PCAC mass m ud = 71.210 ± 0.391 MeV renormalized in MS at 2GeV fit with second order poly.
PCAC mass m s = 119.96 ± 0.412 MeV renormalized in MS at 2GeV
Pion mass Flow eq. does not effect to the pion mass χ t χ (t) D t + m 1
Pion decay constant Lattice artifact O(a 2 /t) is severe! Window region Blue : A + Bt, f π = 114.5 ± 44.1 MeV Green : A + Bt + C/t, f π = 104.2 ± 43.2 MeV
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
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 + 1 + γ log 4 + N c 2 6N c + 5 2N c + 2C F log(432)
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!
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)
Some techniques (WHOT-QCD, Lattice2016) O a 2 /t Window region O t 2 Window region works well to extract the physics