Thermodynamics of 2+1 flavor QCD with the SF t X method based on the - PowerPoint PPT Presentation

2020/5/20 @ R-CCS Thermodynamics of 2+1 flavor QCD with the SF t X method based on the gradient flow SF t X method : small flow-time expansion method WHOT-QCD Collaboration: K. Kanaya, Y. Taniguchi, A. Baba, A. Suzuki (Univ. Tsukuba) S. Ejiri, S. Itagaki, R. Iwami, M. Shirogane, N. Wakabayashi (Niigata Univ.) M. Kitazawa, A. Kiyohara (Osaka Univ.) T. Umeda (Hiroshima Univ.) H. Suzuki (Kyushu Univ.) Tomonaga Center for the History of the Universe

<latexit sha1_base64="v8N41LklpMGlmb/zClCoKW1og=">AC83ichZHNbtNAEMf/dvko4aNpuSC4GEIRB6jWubRCILWAEJdKbUPaojgxtrtJV/WX1uIYvkFOHDl0BMVqKqPgUXoBDHwFxbCUuHJg4kaK0Asby7n9m5zc7u+vGvkgUY0eaPnbu/IWL45dKl69cvTZRnpxaTaJUerzuRX4k10n4b4IeV0J5fP1WHIncH2+5m4962vdblMRBS+UtsxbwZOJxRt4TmKQnb5Q83OMksGxuvFPG8s2FaQNo3SE+Oh1ZaOl5l5Vu3YrFXNLREq6/ZG6/lb64FRErmjRd2RoQVpvlQFfxp3Bjh8ktZ8SxyxU2wozgpzICrzC48P3wBYisp7sLCBCB5SBOAIoUj7cJDQ14AJhphiTWQUk6REsc6Ro0RsSlmcMhyKbtHYIa8xiIbk92omBe3RLj79kgD0+w72fH7Bs7YD/Y7/WyoavV62aXb7LI/tifc3ar/+SwU0K2wOqX/2rNDGXNGroN7jItI7hdfnu+8+HtcerUxn9gu+0n9f2JH7CudIOyeF+W+coOSvQA5unrPitWqzMm6WV6iafo2zhu4Q7u03PYh4vsYQ67Xui3dQq2l091Xf0Xf1zP1XBsx1jJh+Ad4kb+</latexit> <latexit sha1_base64="q+izDtCNG7wEQYgYDtfW3wP6hyA=">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</latexit> <latexit sha1_base64="q+izDtCNG7wEQYgYDtfW3wP6hyA=">AC83ichZHNThRBEMdrBhVYRVa8GD0wuGA8AKnZi4RoAmgIFxNgXT6yszvODL1Lh/lKT8GnMwLcODKgZNEY4zhKbz4Ahx4BOMREg96sHZ2k81CkJpM97+q61d3W2HLo8k4pmi9t26fad/YDB397Q/eH8g5G1KIiFw8pO4AZiw7Yi5nKflSWXLtsIBbM82Xr9s7r1vp6k4mIB/47uReyqmc1fF7njiUpZOYPSmaSGMLTNt+maWXeNLy4quVeaVNGXVhOoqdJsWFirZga3JfG2Fbtza4xqWEFGl0UyIMPw47aqMv4xrPXw3uWb1OGa+gNOYmXZV6B1RmJt/efJ+9M/ScpD/AgZsQAOxOABAx8kaRcsiOirgA4IcWqkFBMkOLZOoMUcsTGlMUow6LoDo0N8iqdqE9+q2aU0Q7t4tIviNRgAk/xK57jD/yGP/HvtbWSrEarlz2a7TbLQnN4/1Hp942UR7OE7S71354l1GEm65VT72EWaZ3CafPND4fnpdnVieQZHuMv6v8jnuF3OoHfvHA+r7DVI8jRA+iXr/uqWCtO6RX6CUWoG0D8ASewnO67xcwB0uwDGXa90J5rBSUcTVWj9Rj9VM7VU6zEPoMfXkH/zqwSA=</latexit> <latexit sha1_base64="Xb+SJBMLuKD6YQK7tJfnfCwTSI4=">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</latexit> Gradient Flow Narayanan-Neuberger (2006), Lüscher (2010-) Gradient Flow t (example) Yang-Mills theory in the continuum B µ ( t,x ) Original theory: gauge field A µ ( x ) in D =4 dim. space-time, S YM [ A µ ] = − 1 1 Z Z d D x tr[ F µ ν F µ ν ] = d D x F a µ ν F a µ ν 2 g 2 2 g 2 0 0 Introduce a fictitious "time" t , and evolve ("flow") the field A µ by with . This is a kind of diffusion equation. A µ ( x ) t = 0 Its perturbative solution reads B µ ~ smeared A µ over a physical range of √ (8 t ) . ("8" = 2 x D with D =4) Quantum expectation values = path-integration over the original fields A µ def. def. Flowed operators are free from UV divergences and short-distance singularities. Lüscher-Weisz (2011)

SF t X method based on GF H. Suzuki, PTEP 2013, 083B03 (2013) [E: 2015, 079201] Making use of the finiteness of the GF, H. Suzuki developed a general method to correctly calculate any renormalized observables non-perturbatively on the lattice . Small Flow- t ime eXpansion (SF t X) method physical obs.'s we want GF finite, continuum continuum physically well-defined t =0 t >0 t → 0 a → 0 We can safely evaluate lattice lattice their expectation values t =0, a >0 t >0, a >0 non-perturbatively GF by constructing corresponding operators on the lattice. Because we can construct a lattice operator directly from the continuum operator, this method is applicable also to observables whose base symmetry is broken on the lattice ( Poincaré inv. etc.) ➯ energy-momentum tensor

shear stress momentum pressure energy energy-momentum tensor In continuum, EMT is defined as the generator of Poincaré transformation. for YM theory source of the gravity   T 00 T 01 T 02 T 03 conserved Noether current associated with the Poincaré inv. T 10 T 11 T 12 T 13   a fundamental observable of the theory to extract   T 20 T 21 T 22 T 23 EoS (energy, pressute), momentum, shear stress, ...   T 30 T 31 T 32 T 33 fluctuation/correlation functions => specific heat, viscosity, ... On the lattice , the Poincaré invariance is explicitly broken. We have to fine-tune the renormalization and mixing coefficients of many operators to make the current conserved and to get the correct values of en. density etc. in the continuum limit. → 7 � Caracciolo et al., NP B309, 612 (1988); Ann.Phys. 197, 119 (1990) { T µ ν } R ( x ) = Z i O i µ ν ( x ) | lattice − VEV , i = 1 where � � F a µ ρ ( x ) F a F a ρσ ( x ) F a O 1 µ ν ( x ) ≡ νρ ( x ) , O 2 µ ν ( x ) ≡ δ µ ν ρσ ( x ) , ρ ρ , σ ← → ← → ψ ( x ) ← → � � O 3 µ ν ( x ) ≡ ¯ O 4 µ ν ( x ) ≡ δ µ ν ¯ ψ ( x ) γ µ D ν + γ ν D µ ψ ( x ) , D ψ ( x ) , / O 5 µ ν ( x ) ≡ δ µ ν m 0 ¯ ψ ( x ) ψ ( x ) , ← → � O 7 µ ν ( x ) ≡ δ µ ν ¯ F a µ ρ ( x ) F a O 6 µ ν ( x ) ≡ δ µ ν µ ρ ( x ) , ψ ( x ) γ µ D µ ψ ( x ) and, Lorentz non-covariant ones: allowed by the lattice rotation symmetry => ρ ���� ���� �������������� ��� ���� ���� �������������� ���

YM EMT with the SF t X method Small- t expansion Lüscher-Weisz, JHEP1102.051(2011) Suzuki, PTEP 2013, 083B03 [E: 2015, 079201] At small t , flowed operators can be expanded in terms of un-flowed operators. In QCD, the coefficients at small t can be calculated by perturbation theory thanks to AF. For YM EMT, with Inverting these, the correctly normalized EMT is given by matching coefficients to make t → 0 smoother by removing known small- t mixings & t -dep. in the continuum to match the renormalization schemes when the observable is scheme-dependent They are finite => safe to evaluate on the lattice.

A test in quenched QCD (FlowQCD Collab.) Kitazawa-Iritani-Asakawa-Hatsuda-Suzuki, Phys.Rev. D 94, 114512 (2016) Wilson plaquette gauge action both for S YM [ A µ ] and S YM [ B µ ]. Clover definition for G µv . Improved operator for E : O ( a 2 ) removed in the tree-level. a → 0 & t → 0 Lattice error expected at √ (8t) ≤ a [ tT 2 ≤ 1/(8 N t 2 ) ~ 0.0009, 0.0003 for N t =12, 20] => SF t X well reproduces the results of conventional integral method.

SF t X method based on GF H. Suzuki, PTEP 2013, 083B03 (2013) [E: 2015, 079201] Making use of the finiteness of the GF, H. Suzuki developed a general method to correctly calculate any renormalized observables non-perturbatively on the lattice . Small Flow- t ime eXpansion (SF t X) method physical obs. we want GF finite, continuum continuum physically well-defined t =0 t >0 t → 0 a → 0 a → 0 t → 0 by linear window lattice lattice evaluate their values non-perturbatively t =0, a >0 t >0, a >0 GF construct corresponding operators on the lattice Because we can construct a lattice operator directly from the continuum operator, this method is applicable also to observables whose base symmetry is broken on the lattice (Poincaré inv., chiral sym. , etc.) ➯ energy-momentum tensor ➯ QCD with Wilson-type quarks, to cope with the problems due to chiral violation. When we can identify a proper window, we may exchange the order of two extrapolations.

[0] Introduction [1] N F = 2+1 QCD with slightly heavy u,d and ≈ physical s quarks [1A] Issue of renormalization-scale in N F = 2+1 QCD with slightly heavy u,d --- an improvement of the SF t X method --- [1B] 2-loop matching coefficients in N F = 2+1 QCD with slightly heavy u,d [2] N F = 2+1 QCD with physical u,d,s quarks --- a status report --- [3] Summary

[1] N F = 2+1 QCD with slightly heavy u,d and ≈ physical s quarks Taniguchi-Ejiri-Iwami-KK-Kitazawa-Suzuki-Umeda-Wakabayashi, Phys.Rev. D 96, 014509 (2017) Taniguchi-KK-Suzuki-Umeda, Phys.Rev. D 95, 054502 (2017)