Lattice computation for the QCD Lambda parameter

Transcription

1 Lattice computation for the QCD Lambda parameter twisted gradient flow scheme for quenched system Issaku Kanamori (Hiroshima Univ.) Workshop on Hadron Physics & QCD July 17, 2017 at Academia Sineca based on arxiv: with Ken-Ichi Ishikawa, Yuko Murakami, Ayaka Nakamura, Masanori Okawa and Ryoichiro Ueno

2 Outline 1. Introduction Λ parameter and Schemes 2. Twisted Gradient Flow Scheme 3. Step scaling and estimate of Λ in TGF scheme 4. Measuring in the Physical Quantities: σ, r 0 5. Converting Λ to MS scheme 6. Summary Feasibility test of the Twisted Gradient Flow Scheme quenched simulation 2/23

3 Introduction: Λ parameter QCD Λ-parameter typical scale in QCD dimensional mutation in the chiral limit/quenched limit, no dimensionful parameters in the theory renormalization scheme dependent perturbative scheme/ non-perturbative scheme 3/23

4 Introduction: Λ parameter QCD Λ-parameter typical scale in QCD dimensional mutation in the chiral limit/quenched limit, no dimensionful parameters in the theory renormalization scheme dependent perturbative scheme/ non-perturbative scheme How to calculate Λ non-perturbatively? 3/23

5 Introduction: Λ parameter QCD Λ-parameter typical scale in QCD dimensional mutation in the chiral limit/quenched limit, no dimensionful parameters in the theory renormalization scheme dependent perturbative scheme/ non-perturbative scheme How to calculate Λ non-perturbatively? Ok, let us use lattice simulation ( first principle calc.) (I am a lattice physicist anyway) 3/23

6 Introduction: Λ parameter QCD Λ-parameter typical scale in QCD dimensional mutation in the chiral limit/quenched limit, no dimensionful parameters in the theory renormalization scheme dependent perturbative scheme/ non-perturbative scheme How to calculate Λ non-perturbatively? Ok, let us use lattice simulation ( first principle calc.) lattice in 3 line: (I am a lattice physicist anyway) spacetime Euclidean lattice as a UV regulator defined non-perturbatively Monte Carlo estimation of path integral (lattice simulation) 3/23

7 Introduction: Schemes Λ parameter for pure SU(N C ) Yang-Mills: Λ = µ ( b 0 g 2 (µ) ) [ ] [ b 1 1 2b 2 0 exp exp 2b 0 g 2 (µ) g(µ) 0 dξ ( β(ξ)+ 1 b 0 ξ 3 b 1 b 2 0ξ µ: scale β(g) = b 0 g 3 b 1 g }{{} 5 +, b 0 = 11 N C, 3 16π 2 ( b 1 = 34 NC ) π 2 scheme indep. g, β(g) and Λ are all renormalization scheme dependent ) ] 4/23

8 Introduction: Schemes Λ parameter for pure SU(N C ) Yang-Mills: Λ = µ ( b 0 g 2 (µ) ) [ ] [ b 1 1 2b 2 0 exp exp 2b 0 g 2 (µ) g(µ) 0 dξ ( β(ξ)+ 1 b 0 ξ 3 b 1 b 2 0ξ µ: scale β(g) = b 0 g 3 b 1 g }{{} 5 +, b 0 = 11 N C, 3 16π 2 ( b 1 = 34 NC ) π 2 scheme indep. g, β(g) and Λ are all renormalization scheme dependent schemes MS scheme: useful for phenomenology, but perturbative scheme Schrödinger functional scheme: popular in lattice, non-perturbative Twisted Gradient Flow scheme: this work, non-perturbative (Gradient Flow scheme: non-perturbative, does not work) ) ] 4/23

9 Recipe to calculate Λ with lattice simulation 1. choose your scheme 2. measure the renormalized coupling with lattice simulation 3. use step scaling to obtain (discrete) β-function 4. get your Λ in lattice unit: Λa or ΛL 5. convert Λ to physical unit 5/23

10 Recipe to calculate Λ with lattice simulation 1. choose your scheme 2. measure the renormalized coupling with lattice simulation 3. use step scaling to obtain (discrete) β-function 4. get your Λ in lattice unit: Λa or ΛL 5. convert Λ to physical unit lattice setup L/a = 12,16,18,24,36 quenched (no dynamical quark) O(1000) O(10000) configuration β = 2N C g 2 0 = (weak coupling regime: β = 2N C g0 2 plaquette action (+ twisted b.c.) = 40.0,60.0,80.0 ) U µ 5/23

11 Twisted Gradient Flow Scheme 6/23

12 Gradient Flow observables are made of smeared field diffusion process via Flow equation : dv µ (n,t) = g 2 S[V] 0 V µ (n,t) dt n,µ t : flow time V µ (n,t): smeared gauge field on link (n,n+ˆµ) no operator renormalization for bosons M. Lüscher JHEP 1008 (2010) 071 How much the fields should be diffused? 8t = c L (L: size of box, c = 0.3) 1/L naturally gives the scale keeps zero mode of gauge field non-analytic term in α S appears, unable to obtain Λ Fodor et. al JHEP 1211(2012) 007 7/23

13 Twisted Gradient Flow A.Ramos JHEP 1411(2014)101 Twisted boundary condition for x-,y-directions (others are periodic) U µ (n+l/aˆν) = Γ ν U µ (n)γ ν ν = x,y Γ x Γ y = ωγ y Γ x, ω = exp(2π/n C ) no zero mode (technical detail) boundary condition can be casted to a phase S = β 2N C n µ ν Z µν(n)u µ (n)u ν (n+ ˆµ)U µ(n+ ˆν)U ν(n) }{{} plaquette action { Z µν (n) = Z νµ (n) ω for µ = x,ν = y and n x = n y = 0 = 1 otherwise coupling through the energy density: clover definition gtgf(1/l) 2 = normalization G µν (n) 2 n,µ ν G µν (n) Im( ) F µν 8/23

14 Step Scaling and Estimate of Λ in TGF Scheme 9/23

15 Step Scaling: idea M.Lüscher et.al NPB359(1991)221 We want to estimate β-function non-perturbatively: fixed g 0 : a is fixed, L L/s, scale changes so g TGF changes fixed g TGF : scale is fixed, but a a/s g TGF (1/L max ) g TGF (s/l max ) g TGF (s/l max ) g TGF (s 2 /L max ) 10/23

16 Step Scaling: discrete beta function change the scale 1/L s/l s = 3 2 > 1 Discrete β-function: B s (gtgf 2 (1/L)) = g2 TGF (s/l) g2 TGF (1/L) log[s 2 ] scaling of the renorm. coupling: u i = u i 1 +B s (u i 1 )log[s 2 ], u 0 = u = gtgf 2 (1/L max) global fit of the discrete β-function on the lattice: Bs LAT (u,a/l) = 5 [ ( c k +d a ) 2 k cont. limit k ]u B L s (u) = 5 c k u k k=2 k=2 c 2 = b 0, c 3 = c 2 2 log[s2 ] b 1 (two-loop) B LAT 3/2 (u,a/l) L/a = cont. limit B 3/2 (u) loop 2-loop u = g TGF u = g TGF 11/23

17 after n steps... scale: 1/L max s n /L max (s n 1) renorm. coupling: u 0 = u = g 2 TGF (1/L max) u n = g 2 TGF(s n /L max ) }{{} Λ µ = ( b 0 g 2 (µ) ) at µ = s n /L max b 1 [ ] 2b 2 0 exp 1 2b 0 g 2 (µ) cl max Λ TGF s n (b 0 u n ) b 1 2b 2 0 exp 1 [ exp ( )] g(µ) 0 dξ β(ξ)+ 1 b 0 b ξ 3 1 b 2 0 }{{ ξ } 1 for g(µ) 1 [ 1 2b 0 u n ] c = 0.3 is for our TGF scheme 12/23

18 Λ TGF in physical unit L max Λ TGF : dimensionless number want to measure with a dimensionful quantity L max Λ TGF knowing L max A phys = Λ TGF L max A phys A phys 13/23

19 how to choose bare parameters to take cont. limit? input β (= 2N C, bare coupling at the cutoff): governs lattice spacing a g0 2 L max /a : (dimension less!) output gtgf 2 (β,l max/a) aa phys (β) (assume no finite volume effect) A phys : hadronic quantity with dim.=1 continuum limit: L max fixed L max /a (and β ) with gtgf 2 (β,l max/a) = u fixed [ ] Lmax L max A phys = lim β, a aa phys Lmax/a gtgf 2 =u Given u and L max /a, we need a corresponding value of β 14/23

20 renorm. coupling as a function of bare coupling Given u and L max /a, we need a corresponding value of β = 2N C g 2 0 We have: u = g 2 TGF (β,l max/a) Fit: g 2 TGF (β,l max/a) = g k=1 c kg 2k 0 g TGF β L/a = 12 L/a = 16 L/a = 18 15/23

21 Hadronic quantities: string tension aa phys = a σ 1. fitting data to obtain (a σ)(β) Allton et.al JHEP 0807 (2008) 021 A. Gonz ález-arroyo and M. Okawa PLB718(2013) choose value of renorm. coupling u ( choose L max ) 3. choose L max /a 4. we know corresponding β = β from g 2 TGF (β,l max /a) = u 5. L max /a(a σ)(β ) = L max σ(β ) for given u and L max /a string tension L max sqrt(σ) L max σ g TGF 2 (1/Lmax ) 16/23

22 Hadronic quantities: Sommer scale r 0 aa phys = a/r 0 Sommer scale: determined from static quark potential, widely used to scale setting in lattice QCD (r 0 0.5fm) 1. fitting data to obtain (a/r 0 )(β) ALPHA collab. NPB535(1998) choose value of renorm. coupling u ( choose L max ) 3. choose L max /a 4. we know corresponding β = β from g 2 TGF (β,l max /a) = u 5. L max /a(a/r 0 )(β ) = L max /r 0 (β ) for given u and L max /a Sommer scale L max /r L max /r g TGF 2 (1/Lmax ) 17/23

23 Converting Λ to MS Scheme Λ MS /A phys =? Λ TGF /A phys 18/23

24 Intermediate scheme: SF MS is a perturbative scheme we need a long, tedious perturbative calculation to convert TGF scheme to MS on going work E. Ibanez Bribian and M. Gracia Perez, PoS LATTICE alternative use intermediate scheme: 19/23

25 Intermediate scheme: SF MS is a perturbative scheme we need a long, tedious perturbative calculation to convert TGF scheme to MS on going work E. Ibanez Bribian and M. Gracia Perez, PoS LATTICE alternative use intermediate scheme: Schrödinger Functional Scheme (SF scheme) widely used in lattice QCD 1/L gives the scale requires a lot of statistics Λ MS Λ SF is known! S.Sint and R.Sommer NPB465(1996)71 19/23

26 Intermediate scheme: SF MS is a perturbative scheme we need a long, tedious perturbative calculation to convert TGF scheme to MS on going work E. Ibanez Bribian and M. Gracia Perez, PoS LATTICE alternative use intermediate scheme: Schrödinger Functional Scheme (SF scheme) widely used in lattice QCD 1/L gives the scale requires a lot of statistics Λ MS Λ SF is known! Λ SF S.Sint and R.Sommer NPB465(1996)71 Now we need: Λ Λ MS = Λ MS Λ SF Λ TGF TGF Λ SF Λ TGF this ratio does not depend on the coupling we simulate matching in perturbative region is enough 19/23

27 Conversion of coupling form TGF to SF Convert the coupling at the same scale 1/L (and β): perturbative relation: gsf 2 (a/l,β) gtgf 2 (a/l,β) = 1+c g(a/l)gtgf 2 (a/l,β)+ Λ SF Λ TGF = exp [ c (0) g 2b 0 ] = (61) with c g (a/l) = c (0) g +c (1) g (a/l) 2 + g SF 2 / gtgf L/a = c g g TGF (a/l) 2 20/23

28 Convining everything... Λ MS A phys = Λ MS Λ SF Λ SF Λ TGF L max Λ TGF L max A phys with A phys = σ,1/r 0 Λ MS σ = 0.517(10) stat. ( +8 7) syst. r 0 Λ MS = 0.593(12) stat. ( ) syst. Λ MS-bar / (σ) Our average Value from [G. S. Bali, K. Schilling, 1993] u r 0 * Λ MS-bar Our average Value from [FLAG2016] systematic errors are from renormalization condition (u ) dep. u 21/23

29 Summary 22/23

30 Summary Twisted Gradient flow scheme to calculate gauge coupling and Λ parameter for SU(3) pure Yang-Mills NEW methodology: feasible! coming soon: more statistics to reduce the error...current error is as large as 20 years ago 23/23

31 Summary Twisted Gradient flow scheme to calculate gauge coupling and Λ parameter for SU(3) pure Yang-Mills NEW methodology: feasible! coming soon: more statistics to reduce the error...current error is as large as 20 years ago Thank you very much. 23/23