Gradient flow running coupling: SU(2) with 6 fundamental flavors - PowerPoint PPT Presentation

Gradient flow running coupling: SU(2) with 6 fundamental flavors Viljami Leino Kari Rummukainen Joni Suorsa Kimmo Tuominen University of Helsinki and Helsinki Institute of Physics 28.07.2016 Lattice 2016, Southampton

Motivation • Nearly • Previous studies at N f =6 conformal theories inconclusive 2 3 4 can have walking behavior • 4-loop MS IRFP g 2 ∼ 30 needed by technicolor • SU(2) with 8 massless fer- 0.5 mions has a fixed point 1 1.08 12 8 0.25 1.06 1.04 GF GF , 2) /g 2 1.02 β 0 1.00 σ ( g 2 continuum 0.98 -0.25 16-32 2-loop 0.96 4-loop MS 0.94 4 0 1 2 3 4 5 6 7 8 9 g 2 -0.5 GF 0 1 2 3 4 5 6 g 1 V. Leino et al. Lattice 2015 (hep-lat/1511.03563 ) , 2 T. Karavirta et al. JHEP 1205 (2012) 003 (hep-lat/1111.4104) , 3 T. Appelquist et al. Phys. Rev. Lett. 112, 111601 (2014) (hep-lat/1311.4889) 4 M. Hayakawa et al. Phys. Rev. D 88, 094504 (2013) (hep-lat/1307.6997) 1 / 14

Model • HEX-smeared 1 Wilson-clover action • Schrödinger functional • Use trivial (Dirichlet) boundaries (no background field) • Used to reach zero mass (Tune the κ cr at L = 24) • Allows the measurement of mass anomalous dimension • Lattice sizes: 8,12,16,18,20,24,30,(36) • Use step scaling step s = 3 / 2 ( 8-12, 12-18, 16-24, 20-30) • Can compare to s = 2 at 8-16 and 12-24 • β between 8 and 0.5 • We run into bulk phase transition at β < 0 . 5 • Smaller lattices ∼ 80 000 trajectories, larger ∼ 15 000 1 S. Capitani, S. Durr and C. Hoelbling, JHEP 0611 (2006) 028 2 / 14

Gradient Flow • Use the gradient flow 1 2 GF = t 2 g 2 N � E ( t + τ 0 a 2 ) � • Flow can be evolved using both Wilson plaquette (W) and Lüscher-Weisz (LW) actions • Energy can be measured with both clover and plaquette definitions • We use LW and clover unless otherwise specified √ • Fix flow time t to L by setting scale: c = 8 t / L = 0 . 3 • Use τ 0 correction to tune down the a 2 effects 3 • Measuring also the topological charge: 1 � ǫ µναβ G a µν ( x ; t ) G a Q = αβ ( x ; t ) 32 π 2 x 1 M. Luscher and P. Weisz , JHEP 1102 (2011) 051 (hep-th/1101.0963) , 2 P. Fritzsch and A. Ramos , JHEP 1310 (2013) 008 (hep-lat/1301.4388) , 3 A. Cheng, A. Hasenfratz, Y. Liu, G. Petropoulos and D. Schaich. JHEP 1405 (2014) 137 (hep-lat/1404.0984) 3 / 14

Measured couplings 26 β =8 24 β =6 β =5 22 β =4 β =3 20 β =2 18 β =1.7 β =1.5 16 β =1.3 14 β =1.1 2 g β =1 12 β =0.9 10 β =0.8 β =0.75 8 β =0.7 6 β =0.65 β =0.6 4 β =0.55 β =0.53 2 β =0.5 0 8 12 16 20 24 28 32 36 L/a 4 / 14

Topology β =0.53 β =0.53 15 15 10 10 5 5 0 0 -5 -5 -10 -10 β =0.7 β =0.7 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 β =2 β =2 0.0015 0.0015 0.001 0.001 0.0005 0.0005 0 0 -0.0005 -0.0005 -0.001 -0.001 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 L-W evolved flow Wilson evolved flow • Topology frozen at small couplings, becomes unfrozen at largest couplings • LW evolved flow fluctuates more • Don’t use configurations that are frozen to nonzero values • Projecting δ Q , 0 1 could work, but for N f = 8 effects were small 1 P. Fritzsch et al. PoS Lattice 2013 , 461 (2014) (hep-lat/1311.7304) 5 / 14

Step scaling function • Interpolate couplings using a rational function, m = 7, n = 2 1 + � m i = 1 a i g 2 i g 2 GF ( g 2 0 , L / a , t ) = g 2 0 . 0 1 + � n j = 1 b j g 2 j 0 • Estimate systematic errors by changing the fit parameters • Step scaling function: � GF ( g 0 , s L Σ( u , s , a / L ) = g 2 � a ) , σ ( u , s ) = a / L → 0 Σ( u , s , a / L ) lim � � g 2 GF ( g 0 , L a )= u • Extrapolate to continuum limit: Σ( u , s , a / L ) = σ ( u , s ) + c ( u )( L a ) − 2 • Fix τ 0 to minimize a 2 effects 6 / 14

Fixing τ 0 6.0 14 13 12 5.8 11 10 5.6 9 8 σ ( g 2 , 2) σ ( g 2 , 2) 5.4 7 6 5 5.2 4 τ 0 = 0 3 5.0 τ 0 = 0 . 05 2 τ 0 = 0 . 1 1 4.8 0 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 ( a/L ) 2 ( a/L ) 2 u = 5 τ optimal vs. τ 0 = 0 • Drop the smallest lattice from continuum extrapolation • Estimate: τ optimal = 0 . 012 log ( 1 + 20 g 2 ) (Preliminary) • Logarithm makes sure the τ 0 doesn’t grow too large 7 / 14

Different discretizations c = 0 . 3 , τ 0 = 0 2.2 13 12 2.0 11 1.8 10 σ ( g 2 , 2) σ ( g 2 , 2) 1.6 9 8 1.4 W Clover W Clover 7 W Plaq W Plaq 1.2 LW Clover LW Clover 6 LW Plaq LW Plaq 1.0 5 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 ( a/L ) 2 ( a/L ) 2 u = 2 u = 11 • Plaquette and Clover agree on continuum limit, plaquette has stronger discretization effects • LW and W diverge slightly on large couplings, W has stronger discretization effects 8 / 14

Step scaling on the lattice c = 0 . 3 1.15 1.2 1.15 1.1 1.1 1.05 1.05 1 1 0.95 0.95 2 2 2 ,2)/g 2 ,2)/g 0.9 0.9 σ (g σ (g 0.85 0.85 L=8 0.8 L=12 L=8 0.75 0.8 L=16 L=12 L=20 0.7 L=18 L=24 0.75 2-loop 2-loop 0.65 4-loop 4-loop 0.7 3-loop 0.6 3-loop 0.65 0.55 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 2 2 g g s = 3 / 2 s = 2 9 / 14

Continuum limit 1.15 1.15 1.10 1.10 1.05 1.05 GF GF GF , 2) /g 2 GF , 2) /g 2 1.00 1.00 σ ( g 2 σ ( g 2 continuum continuum 0.95 20-30.0 0.95 16-24.0 2-loop 2-loop 3-loop MS 3-loop MS 0.90 0.90 4-loop MS 4-loop MS 0 5 10 15 20 0 5 10 15 20 g 2 g 2 GF GF 12 − 30 8 − 24 • L20-30 has less statistics than L8-12 • L8-12 behaves oddly on strong coupling and L8 was not used when defining τ 0 10 / 14

Effects of parameters 1.15 1.15 1.10 1.10 1.05 1.05 GF GF GF , 2) /g 2 GF , 2) /g 2 1.00 1.00 σ ( g 2 σ ( g 2 LW τ 0 No τ 0 0.95 W 0.95 2-loop 2-loop 3-loop MS 3-loop MS 0.90 0.90 4-loop MS 4-loop MS 0 5 10 15 20 0 5 10 15 20 g 2 g 2 GF GF 1.15 1.15 1.10 1.10 1.05 1.05 GF GF GF , 2) /g 2 GF , 2) /g 2 1.00 1.00 σ ( g 2 σ ( g 2 Clover continuum c=0.4 0.95 Plaq 0.95 20-30.0 2-loop 2-loop 3-loop MS 3-loop MS 0.90 0.90 4-loop MS 4-loop MS 0 5 10 15 20 0 5 10 15 20 g 2 g 2 GF GF 11 / 14

Mass anomalous dimension • Schrödinger functional pseudoscalar density renormalization constant allows calculation of γ 1 • Interpolate Z P with Z P = 1 + � 5 i = 1 a i g 2 i 0 • Near fixed point approximate as γ ∗ √ Nf 1 L ) = Z P ( g 0 , sL a ) � Z P ( g 0 , L Σ P ( u , a � a ) = a ) , � f p ( 1 L Z P ( g 0 , L a ) � g 2 GF = u 2 γ ∗ = − log σ P ( g 2 ) a → 0 Σ P ( g 2 , a σ P ( g 2 ) = lim L ) , log s 1 S. Capitani, M. Luscher, R. Sommer and H. Witting Nucl. Phys. B 544 (1999) (hep-lat/9810063) 12 / 14

γ ∗ 0.55 0.40 L=8 0.5 L=12 0.35 L=16 0.45 L=20 0.30 0.4 0.35 0.25 0.3 0.20 γ ∗ γ 0.25 0.15 0.2 0.10 0.15 continuum 20-30 0.1 0.05 Perturbative 0.05 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 0 0 2 4 6 8 10 12 14 16 18 20 22 24 g 2 2 GF g 0.40 1.2 0.35 1.0 0.30 0.8 0.25 0.20 γ ∗ Z P 0.6 0.15 0.4 0.10 continuum 20-30 0.2 0.05 Perturbative L=30 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 0.0 0 2 4 6 8 10 g 2 GF g 2 13 / 14 0

Conclusions • Finite volume GF step scaling works at strong coupling • This choice of action, boundaries and smearing allows us to reach relatively small β before running into a bulk phase transition • Topological freezing mostly problem only on certain range of β ’s • N f = 6 seems to approach a IRFP around g 2 ∼ 15 • Check also the posters: • γ with spectral density method – Joni Suorsa • Spectrum of N f = 2 , 4 , 6 , 8 – Sara Tähtinen 14 / 14