IR fixed points in SU(3) Gauge Theories

Transcription

1 205/03/03 SCGT5 IR fixed points in SU(3) Gauge Theories Y. Iwasaki U.Tsukuba and KEK

2 In Collaboration with K.-I. Ishikawa(U. Horoshima) Yu Nakayama(Caltech & IPMU) T. Yoshie(U. Tsukuba)

3 Plan of Talk Introduction Phase structure (brief review of our previous works) Scaling relations based on RG Set up Results Interpretation Conclusions

4 Objectives Identify IR fixed points in SU(3) Gauge Theories with Nf fundamental fermions within the conformal window N c f? N c f N f 6 anomalous mass dimension γ? meson propagator on the fixed point in the continuum limit? Strategy Propose a novel RG method based on the scaling behavior of the propagator through the RG analysis with a finite IR cut-off

5 Constructive approach Define gauge theories as the continuum limit of lattice gauge theories N x = N y = N z = N take the limit a->0 and N -> infinity with N t = rn L = an and L t = an t (r aspect ratio) r=4 in this work fixed when L and/or Lt finite => IR cutoff Conformal theories: IR cutoff: an indispensable ingredient in contrast with QCD

6 Constructive approach (2) Important steps. Clarify the phase structure 2. Clarify what kind of phase exists 3. Clarify the boundary of the phases 4. Clarify the location of UV or IR fixed points our earlier works: step. ~ 3. The phase diagram for various number of flavors 7 \le Nf \le 300 The phase diagram for Nf \le 6 A new phase conformal region in addition to the confining region and deconfining region Phys. Rev. Lett. 69(992), 2 Phys. Rev. D69(2004), Phys. Rev. D54(996), 700 Phys.Rev. D87 (203) 7, Phys.Rev. D89 (204) 4503 we intend to perform step 4 in this work

7 as in 2004 m = 0 q N f m = 0 q confinement N deconfinement

8 as in 2004 m q m q N f confinement N deconfinement

9 as in 204 m =0 q m =0 q 0 0 m >0 q m =0 q 0 0

10 Conformal region A new concept conformal theories with an IR cutoff m q Λ IR meson propagators show a power-modified Yukawa-type decay Two sets of Conformal window Large Nf and QCD in high temperature Nf=7 ~ T/Tc =.0 ~2.0: unparticle meson model strongly support the conjecture that the conformal window: 7 N f 6

11 Nf=7 confining K=0.400, mq=0.22 loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) Nf=7 conformal K=0.459, mq=0.045 loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0)

12 Nf=2 deconfining Beta=0.0, K=0.25, Nf=2, 6 3 x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) Nf=2 conformal Beta=0.0, K=0.35, Nf=2, 6 3 x64, PS loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0)

13 Scaling relations from RG RG equation for the propagator at vicinity of IRFP ) 3 2γ (N G(t; g, m q,n,µ)= G(t ; g,m q,n,µ). N at the IRFP g = g = g m q = m q =0 γ = γ. B =0 Simplified expression ion as G(τ,N)=G(t, N) with τ = t/n t. N t = N/s = t/s e.g. Del Debbio, Zwicky 200 RG scale change Scaling relation G(τ; N) = (N N ) 3 2γ G(τ; N ).

14 Scaled effective mass m(t, N) = N N 0 ln G(t, N) G(t +,N). N m(τ,n)= N 0 τ ln G(τ,N) Scaling relation 2 m(τ,n)=m(τ,n ) Stringent condition for the IR fixed point

15 Strategy With given Nf, choose \beta, and tune mq ~ 0.0 Calculate the meson propagator on the lattices with size 8^3x32, 2^3x48 and 6^3x64 Plot the scaled effective mass In general, three points and lines do not coincide repeat this process narrow the region of \beta in such a way that the three approach together finally find the \beta at which three pots and lines coincide within the standard error identify the \beta IR fixed point

16 Stage and Tools SU(3) gauge theories with Nf quarks in the fundamental representation Action: the RG gauge action (called the Iwasaki gauge action) Wilson fermion action Nf = 7, 8, 2, 6 Lattice size: 8^3x32, 2^3x48, 6^3 x 64 Boundary conditions: periodic boundary conditions an anti-periodic boundary conditions (t direction) for fermions Algorithm: Blocked HMC for 2N and RHMC for : Nf=2N + Statistics:,000 +,000 ~ 4000 trajectories Computers: U. Tsukuba: CCS HAPACS; KEK: HITAC 6000

17 Measurement Plaquette Polyakov loop quark mass m q = h0 r 4A 4 PSi 2h0 P PSi meson propagator G H (t) = X x h H (x, t) H (0)i effective mass cosh(m H (t)(t N t /2)) cosh(m H (t)(t + N t /2)) = G H(t) G H (t + ) m H (t) =ln G H (t) G H (t + )

18 N f = 6, β = 0.5, K =0.292 N N traj acc plaq m q () () () () () () () () (5) N f = 2, β =3.0, K =0.405 N N traj acc plaq m q () (2) 0.002() () () 0.002() () (2) 0.004() N f = 8, β =2.4, K =0.47 N N traj acc plaq m q () () 0.007() () () 0.006(3) () (2) (5) N f = 7, β =2.3, K = N N traj acc plaq m q () () 0.007(2) () () (3) () (3) (6)

19 Results Nf=6 Perturbation: beta function up to two loops RG scheme independent β =.5 On the other hand, β = β 2 + c 2 and β RG = β MS 0.3 β one plaquette = β MS +3.. higher order contribution will be large for one-plaquette action may expect \beta_rg ~.2

20 Nf=6 Effective mass: Nf=6; beta=0.0, K=0.294 Effective mass: Nf=6; beta=.5, K= M(t) 0.9 M(t) t/n t t/n t Effective mass: Nf=6; beta=0.5, K= Effective mass: Nf=6; beta=0.5, K= M(t) 0.9 M(t) t/n t t/n t

21 Nf=6 Effective mass: Nf=6; beta=0.0, K=0.294 Effective mass: Nf=6; beta=.5, K= M(t) 0.9 M(t) t/n t Effective mass: Nf=6; beta=0.5, K= t/n Effective mass: Nf=6; beta=0.5, t K= M(t) 0.9 M(t) t/n t t/n t

22 Nf=2, 8, 7 Effective mass: Nf=2; beta=3.0, K=0.405 Effective mass: Nf=08; beta=2.4, K= M(t) 0.9 M(t) t/n t t/n t Effective mass: Nf=07; beta=2.3, K= M(t) t/n t

23 The location of IR fixed points N f = 6: β = 0.5 ± 0.5 N f = 2: β =3.0 ± 0. N f =8: β =2.4 ± 0. N f = 7: β =2.3 ± 0.05 The conformal window 7 N f 6

24 Continuum limit of propagators at IRFP continuum limit of scaled effective mass is given by the limit N --> infinity Even up to N=6, the limit is almost realized for \tau \ge 0.. As N becomes larger, it will be realized for \tau \le 0. Note the limit depends on the aspect ratio and boundary conditions, but not on L= N a Note that local-local propagators are not local observables, due to the summation over the space coordinates

25 Scaling relation for propagators G(τ; N) = (N N ) 3 2γ G(τ; N ). G(t) Propagator: Nf=8; beta=2.4, k= t/n t

26 Effective mass for Nf=6, 2, 8, 7 Effective mass:all Nf 0.95 M(t) t/nt

27 Local analysis of propagators G(t) =c(t) exp( m(t) t) t α(t) parametrization using data at three points useful for seeing the characteristics

28 local mass Beta=0.5, K=0.292, Nf=6, 6 3 x64, PS-channel (loc(t)-loc 0.8 win_size=3 0.7 win_size=5 Fit Mass Local Mass Beta=2.4, K=0.47, Nf=8, 6 3 x64, PS-channel (loc(t)-loc(0 0.8 win_size=3 0.7 win_size= t of Fit Range [t,t+win_size-] t of Fit Range [t,t+win_size-] Fit Mass Local Mass Beta=3.0, K=0.405, Nf=2, 6 3 x64, PS-channel (loc(t)-loc( 0.8 win_size=3 0.7 win_size= Beta=2.3, K=0.4877, Nf=7, 6 3 x64, PS-channel (loc(t)-loc( 0.8 win_size=3 0.7 win_size= t of Fit Range [t,t+win_size-] t of Fit Range [t,t+win_size-]

29 local exponent Nf=6, Beta=0.5, K=0.292 win_size=3 win_size= Nf=2, Beta=3.0, K=0.405 win_size=3 win_size=5 Local Exponent 2.5 Local Exponent Nf=8, Beta=2.4, K=0.47 win_size=3 win_size= Nf=7, Beta=2.3, K= win_size=3 win_size=5 Local exponent 2.5 Local exponent

30 Nf= Polyakov loop; Nf=6, beta=0.5, K=0.292 almost free particle in the Z(3) twisted vacuum imaginary part Beta=0.5, K=0.292, Nf=6, 6 3 x64, PS-channel (loc(t)-loc 0.8 win_size=3 0.7 win_size= real part free quark (/3, /3, /3); mass and alpha Fit Mass m PS t of Fit Range [t,t+win_size-] t Nf=6, Beta=0.5, K=0.292 win_size=3 win_size= free quark (/3, /3, /3); mass and alpha Local Exponent Exponent t

31 Nf=7, 8 meson unparticle model*

32 Correspondence between two sets γ =2.0 γ γ ~.3 =0.0 meson unparticle free fermion Z(3) twisted vacuum

33 Conclusions (cont.) two scaling relations are derived scaling of scaled effective masses provides a stringent test of IRFP able to identify the location of IRFP for Nf=7, 8, 2 and 6. established the conformal window continuum limit of propagators at IRFP is derived It depends on the aspect ratio and boundary conditions, not L=N a

34 Conclusions Nf=6 is similar to free fermions in the Z(3) twisted vacuum Nf=7 and 8 are consistent with meson unparticle model there is a nice correspondence between large Nf and high temperature. A lot of things should be done Larger N and high statistics estimate γ by several methods

35 Thank you!