Conformal Window and Correlation Functions in Lattice Conformal - PowerPoint PPT Presentation

Y. Iwasaki U.Tsukuba and KEK 2012/12/04 SCGT2012 Conformal Window and Correlation Functions in Lattice Conformal QCD 2012 年 12 月 5 日水曜日

In Collaboration with K.-I. Ishikawa(U. Horoshima) Yu Nakayama(Caltech & IPMU) T. Yoshie(U. Tsukuba) 2012 年 12 月 5 日水曜日

Plan of Talk on the phase structure of the lattice SU(3) QCD. Thereby clarify the reason why we conjecture that the conformal window is cutoff”. Propose new predictions about the propagator of a meson. We verify that new numerical results satisfy our proposals for Nf=7 and Nf=16. (to be continued) • Briefly review our previous works • Introduce the concept of “conformal theory with IR 2012 年 12 月 5 日水曜日

Plan of Talk (Cont.) of a meson at T/Tc >1 shows the characteristics of “conformal theory with IR cutoff”. • Point out and verify that the propagator 2012 年 12 月 5 日水曜日

STAGES and TOOLS • Lattice gauge theory • one-plaqutte gauge action • Improved RG action: future plan • Wilson fermion action • Lattice size: Nx=Ny=Nz=N; Nt=r N • Lattice spacing: • PCAC quark mass: • G(t): propagators of mesons 2012 年 12 月 5 日水曜日

Strategy for Part 1 in terms of for on the massless line starting from UVFP taking the limit and with constant • Investigate the phase structure • UV fixed point at • Find critical Nf for the Banks-Zaks IRFP • Construct the field theory toward UVFP, 2012 年 12 月 5 日水曜日

Phase Diagram: The finite temperature as N increases. move toward larger beta, and the chiral transition quenched QCD transition phase transition in the Chiral transition on the massless line starting from the UVFP ! ����� # **+*, " $ � # **+*, $ ! !"#$%#&'&#( )&!"#$%#&'&#( � 2012 年 12 月 5 日水曜日

Phase Diagram: ; N larger As N increases, the green line only the green part survives. in the limit N => infinity becomes longer and ! ����� " � ! !"#$%#&'&#( � ! ����� # "! )*) ∞ $ ))*)+ � % !"#$%#&'&#( � 2012 年 12 月 5 日水曜日

Phase Diagram: 1. the massless line from the UVFP hits the bulk transition 2. no massless line in the confining phase at strong coupling region massless quark line only in the deconfining phase Complicated due to lack of chiral symmetry # ����� ��� ! ������ $ " # ����� $ � )&!"#$%#&'&#( ! !"#$%#&'&#( � 2012 年 12 月 5 日水曜日

Phase Diagram: only in the deconfining phase As N increases, massless quark line is, still, # ����� ��� ! ������ $ " # ����� $ � )&!"#$%#&'&#( ! !"#$%#&'&#( � # ����� ��� ! ������ $ " # ����� $ � ! !"#$%#&'&#( � 2012 年 12 月 5 日水曜日

What found (massless line in the confining phase) for • There are no green lines • Conformal window is • Indirect way to conclude this 2012 年 12 月 5 日水曜日

More direct way For small Nf, g* is in strong coupling region Only upper limit for g* ? theories <= this work (End of part 1) • Identify the IR fixed point • Find out characteristics of Conformal 2012 年 12 月 5 日水曜日

Strategy for part 2 lattice theory, keeping L =finite or infinity IR cutoff” in continuous theories • Define a continuous theory by continuum limit of • Introduce the concept “Conformal theories with • Then propose Conformal theories on a lattice 2012 年 12 月 5 日水曜日

Continuum limit case2 • and , keeping constant • case 1: L=infinity IR cut-off ; • case 2: L=finite IR cut-off ; • A huge difference between case1 and 2012 年 12 月 5 日水曜日

Case 1: simulations deconfining phase in strong coupling region is conjectured based on numerical massive region is confining phase conformal region exists only on the massless line No physical quantities with physical dimensions ./ ,- !"#$"2'34 ! ))*)+ " 0&!"#$%#&1 '&#( !"#$%#&'&#( ! ����� " � )*)+ � ���� 2012 年 12 月 5 日水曜日

Case 1: Propagators of mesons When When UV fixed point IR fixed point scale invariant is a scale parameter for the transition region from UV to IR 2012 年 12 月 5 日水曜日

Case 2: Conformal theories with IR cutoff Physical quantities: “Conformal theories with IR cutoff” region: see figure Boundary of the “conformal” region is given by Propagators : ./ ,- ! ))*)+ !"#$"0'12 " !"#$%#&'&#( ! ����� " � )*)+ � ���� 2012 年 12 月 5 日水曜日

Case 2: Conformal theories with IR cutoff (Cont.) and are t-dependent: evolve with RG transformation 2012 年 12 月 5 日水曜日

Conformal theories on the Lattice simulations on a lattice: transition of meson propagators from an exponential damping form to a modified Yukawa-type, that is, an exponential form with power correction • Note: IR cutoff is inherent in numerical • Primary target of part 2 is to verify the 2012 年 12 月 5 日水曜日

Size Dependence of Critical mass When the lattice size is increased The critical mass is decreased If we keep the quark mass in the region Yukawa-type disappears Have to carefully choose the parameters to find the “Conformal region” 2012 年 12 月 5 日水曜日

Numerical Simulations RHMC for 1 : Nf=2N + 1 U. Tsukuba: CCS HAPACS; KEK: HITAC 16000 • Algorithm: Blocked HMC for 2N and • Computers: • Nf=7, 16 and (Nf=2, 6) • Lattice size: • Statistics: 1,000 +1000 trajectories 2012 年 12 月 5 日水曜日

Parameters of Simulations 0.055 0.125 0.126 0.127 0.13 0.1315 0.13322 mq 0.25 0.22 0.19 0.1 0.003 Nf16 mH(96) 0.30 0.27 0.32 mH(64) 0.54 0.54 0.49 0.43 0.38 0.38 mH(32) K 0.74 Masses are preliminary ! 0.006 Nf7 K 0.1400 0.1446 0.1452 0.1459 0.1472 mq 0.22 0.084 0.062 0.045 mH(96) 0.74 0.66 0.33 0.33 0.20 mH(64) 0.68 0.46 0.42 0.41 0.41 mH(32) 0.74 2012 年 12 月 5 日水曜日

From now on, let me show you examples of Yukawa-type propagators for Nf=7 at beta=6.0 and Nf=16 at beta=11.5 detailed analyses will come later 2012 年 12 月 5 日水曜日

Nf7: mq=0.045: example of Yukawa-type Beta=6.0, K=0.1459, Nf=7, 16 3 x64, PS-channel Beta=6.0, K=0.1459, Nf=7, 16 3 x64, V-channel 1 1 loc(t)-loc(0) loc(t)-loc(0) 0.9 0.9 loc(t)-dsmr(0) loc(t)-dsmr(0) loc(t)-dwal(0) loc(t)-dwal(0) 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1459, Nf=7, 16 3 x64, PS-channel Beta=6.0, K=0.1459, Nf=7, 16 3 x64, V-channel 0.53 0.53 loc(t)-loc(0) loc(t)-loc(0) 0.52 0.52 loc(t)-dsmr(0) loc(t)-dsmr(0) 0.51 loc(t)-dwal(0) 0.51 loc(t)-dwal(0) 0.5 0.5 0.49 0.49 0.48 0.48 0.47 0.47 0.46 0.46 0.45 0.45 0.44 0.44 0.43 0.43 0.42 0.42 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 2012 年 12 月 5 日水曜日

Nf=7: mq=0.22; example of exp!!-damp Beta=6.0, K=0.1400, Nf=7, 16 3 x64, PS-channel Beta=6.0, K=0.1400, Nf=7, 16 3 x64, V-channel 1 1 loc(t)-loc(0) loc(t)-loc(0) 0.9 0.9 loc(t)-dsmr(0) loc(t)-dsmr(0) loc(t)-dwal(0) loc(t)-dwal(0) 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1400, Nf=7, 16 3 x64, PS-channel Beta=6.0, K=0.1400, Nf=7, 16 3 x64, V-channel 0.75 0.75 loc(t)-loc(0) loc(t)-loc(0) 0.74 0.74 loc(t)-dsmr(0) loc(t)-dsmr(0) loc(t)-dwal(0) loc(t)-dwal(0) 0.73 0.73 0.72 0.72 0.71 0.71 0.7 0.7 0.69 0.69 0.68 0.68 0.67 0.67 0.66 0.66 0.65 0.65 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 2012 年 12 月 5 日水曜日

Nf=7: mq=0.045 Yukawa-type fit[15:31] Beta=6.0, K=0.1459, Nf=7, 16 3 x64, PS-channel 0.5 loc(t)-loc(0) fit[15:31] 0.49 0.48 0.47 0.46 0.45 0.44 0.43 0.42 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 2012 年 12 月 5 日水曜日

Nf16: mq=0.055; example of Yukawa-type Beta=11.5, K=0.1315, Nf=16, 16 3 x64, PS-channel Beta=11.5, K=0.1315, Nf=16, 16 3 x64, V-channel 1 1 loc(t)-loc(0) loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dsmr(0) 0.9 0.9 loc(t)-dwal(0) loc(t)-dwal(0) 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.1315, Nf=16, 16 3 x64, PS-channel Beta=11.5, K=0.1315, Nf=16, 16 3 x64, V-channel 0.51 0.51 loc(t)-loc(0) loc(t)-loc(0) 0.5 0.5 loc(t)-dsmr(0) loc(t)-dsmr(0) loc(t)-dwal(0) loc(t)-dwal(0) 0.49 0.49 0.48 0.48 0.47 0.47 0.46 0.46 0.45 0.45 0.44 0.44 0.43 0.43 0.42 0.42 0.41 0.41 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 2012 年 12 月 5 日水曜日