Lattice studies of the conformal window

Transcription

1 Lattice studies of the conformal window Luigi Del Debbio University of Edinburgh Zeuthen - November 2010 T H E U N I V E R S I T Y O F E D I N B U R G H Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

2 Outline 1 Introduction 2 RG flow 3 Scaling laws 4 Schrödinger functional 5 Outlook Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

3 Introduction Technicolor models ETC M ETC??? flavor SM TC 3 <ΨΨ>=Λ TC Λ TC TeV gauge Λ QCD GeV Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

4 Introduction Flavor constraints for TC ETC: mass term for the SM fermions, FCNC L 1 M ΨΨ 1 ETC 2 ETC ψψ, ψψ M ψψ ETC 2 running of ΨΨ : (near) conformal IR behaviour ( ΨΨ ETC = ΨΨ METC ) dµ TC exp Λ TC µ γ(µ) [holdom, lane, appelquist, luty, sannino, chivukula,...] Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

5 Introduction Infrared fixed points [Banks & Zaks 82] β(ḡ) = µ d dµ ḡ(µ) = β0ḡ 3 β 1ḡ 5 + O(ḡ 7 ) loop 2-loop β 0 > 0 = asymptotic freedom, n f < nf af β(ḡ) < 0 = confining theories β(ḡ ) = 0 = IR fixed point conformal window: IR zero of the β function β(g) g n c f < n f < n af f loop 2-loop 0.00 (4π) 2 β 0 (4π) 4 β 1 SU(3), n f = 3 fund 9 64 SU(3), n f = 12 fund 3-50 SU(2), n f = 2 adj 2-40 β(g) g Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

6 Introduction The scales of an IRFP g g* Λ IR Λ U µ asymptotic freedom at high-energy scale Λ UV scale invariance at large distances = no massive spectrum large-distance dynamics: anomalous dimensions at the IRFP nonperturbative IRFP: strong coupling, large anomalous dimensions scale invariance explicitly broken by a mass term/finite volume Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

7 Introduction The conformal window SU(N) Phase Diagram Fund γ =1 γ =2 Ladder Ryttov & Sannino 07 Dietrich & Sannino 07 Sannino & Tuominen 04 2A 2S Adj Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

8 RG flow RG transformation Action with a cutoff µ: S(g i ) = i g iµ 4 d i O i RG transformation: µ µ = µ/b at constant physics g i = R i (g) flow in the (infinite-dimensional) parameter space. Fixed point: In the vicinity of the fixed point: g i = R i (g ) δg i = R i g k v i = b y i v i δg k y i characterize the fixed point y i > 0 relevant parameters Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

9 RG flow RG flow for QCD-like theories YM theory - g lattice bare coupling, g space of irrelevant couplings g g Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

10 RG flow RG flow for conformal theories flow in the m = 0 plane - assuming only the mass is relevant in the IR g g Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

11 RG flow RG flow for conformal theories flow in the presence of a mass deformation am? (g,g ) Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

12 Scaling laws Meson spectrum [luty 09, degrand09, ldd&zwicky 10] C H (t; g, m, µ) = d 3 x H(x, t)h(0) F 2 H e M H t RG transformation: Z H (µ) 2 C H (t; g, m, µ) = Z H (µ ) 2 C H (t; g, m, µ ) C H (t; g, m, µ) = b 2γ H C H (t; b yg g, b ym m, µ ) naive dimensional analysis: C H (t; b yg g, b ym m, µ ) = b 2d H C H (tb 1 ; b yg g, b ym m, µ) choosing b such that b ym m = 1: C H (t; g, m, µ) = m 2 H /y m F(tm 1/ym ), H = d H + γ H M H m 1/(1+γ ) Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

13 Scaling laws Scaling of matrix elements [ldd&zwicky 10] RG transformation: T φ1 Oφ 2 (g, m, µ) = φ 1 O φ 2 g,m,µ T φ1 Oφ 2 (g, m, µ) = b γ O T φ1 Oφ 2 (g, m, µ ) = b γ O T φ1 Oφ 2 (b yg g, b ym m, µ/b) naive dimensional analysis: T φ1 Oφ 2 (b yg g, b ym m, µ/b) = b d O +d 1 +d 2 T φ1 Oφ 2 (b ym m, µ) choosing b such that b ym m = 1 T φ1 Oφ 2 (g, m, µ) m ( O +d 1 +d 2 )/y m Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

14 Scaling laws Scaling from the trace anomaly trace anomaly θ α α = β 2 G 2 + n f (1 + γ m)m qq matrix elements of the energy-momentum tensor H(p) θ αβ H(p) = 2p αp β hence we can write the trace at the fixed point 2MH 2 = n f (1 + γ m) H(p) m qq H(p) using the scaling law for the matrix element M H m 1/ym Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

15 Scaling laws Conformal spectrum In the presence of a mass deformation (and infinite volume) no parametric separation between the PS and the other channels all masses/decay constants scale like m 1/ym (but M F (0)/L!!) very different spectrum from a QCD-like theory [Miransky 94] Λ had m PS Λ YM m PS QCD-like conformal Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

16 SU(2) adjoint Exploring the chiral regime of MWT [catterall et al 07, ldd et al 08/10, rummukainen et al 08] MC simulations with n f = 2 Wilson fermions Catterall et al. Hietanen et al. Del Debbio et al. 2.6 β m 0 Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

17 SU(2) adjoint Chiral limit 0.20 β = 2.25 a m c = (1) a m x8 3 24x x x a m 0 Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

18 SU(2) adjoint Pseudoscalar mass β = a m ps x8 3 24x x x a m Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

19 SU(2) adjoint Pseudoscalar mass β = a m ps 2 /m x8 3 24x x x a m Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

20 SU(2) adjoint Pseudoscalar mass lattice 12 3 lattice 16 3 lattice 24 3 lattice fit m PS /m a m log m PS = 1 log m + C = 0.85 log m + C 1 + γ Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

21 SU(2) adjoint Vector mass β = x8 3 24x x x24 3 m v /m ps a m Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

22 SU(2) adjoint Gluonic sector x x16 3 fit with fixed σ fit with 1/R 3 term fit with variable σ V(R) am=0.12, am PS = R Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

23 SU(2) adjoint Gluonic sector - 2 a M PS meson mass 0 ++ glueball mass 2 ++ glueball mass σ 1/2 (σ = string tension) values at m PCAC = a m PCAC Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

24 SU(2) adjoint Gluonic sector m PS / σ a m PS Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

25 Scaling laws Finite-size scaling RG transformation naive dimensional analysis C H (t; ˆm, L, µ) = b 2γ H C H (t; ˆm, L, µ ), C H (t; ˆm, L, µ) = b 2(d H +γ H ) C H (b 1 t; b ym ˆm, b 1 L, µ). choosing b such that b 1 L = L 0 ( ) 2 H ( L t C H (t; ˆm, L, µ) = C H ; x L/L 0 L 0 where the scaling variable is x = L ym m. f (x) 0 when x 0 m scaling in the thermodynamic limit M H = L 1 f (x), f (x) x 1/ym, as x. 1 µl ym 0 ), L 0, µ, Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

26 SU(2) adjoint Finite-size scaling 3 γ * =0.1 γ * =0.2 L= 8 L=12 L=16 LF PS γ * =0.4 γ * =1.0 LF PS x x Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

27 Scaling laws Scaling of the chiral condensate RG analysis where qq m η qq relation to the eigenvalue density η qq = qq y m = 3 γ 1 + γ qq = 2 µ/m 0 dx ρ(mx) 1 + x 2 + A(m), This in turn implies: qq m 0 m η qq ρ(λ) λ 0 λ η qq. η qq QCD like = 0, η qq mcgt > 0, Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

28 SU(2) adjoint scaling of the eigenvalues ν(m, m) = +Λ Λ ν(λ, 0) = C(Λ g) η qq+1 dλ ρ(λ), Λ = M 2 m 2 mode number distribution h2st Entries 18 mode number distribution h2st Entries Mean RMS χ / ndf / Mean RMS χ / ndf / 2 80 Prob p e+04 ± 9.964e Prob p e+05 ± 2.411e+05 p ± p ± p ± p ± Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

29 Schrödinger functional Renormalized coupling Schrödinger functional [ALPHA collaboration] Z[η] = e Γ[η] = DUDψD ψ exp( S[U, ψ, ψ]) L L 3 Dirichlet boundary conditions at t = 0, L, dependent on η k ḡ 2 (L) = Γ η, ḡ 2 = g0 2 + O(g0 4 ) η=0 Lattice step scaling function Σ(u, a/l) = ḡ 2 (bl) ḡ 2 (L)=u,m=0 Step scaling function σ(u) = lim a 0 Σ(u, a/l) fixed point: σ(u ) = u Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

30 Schrödinger functional Renormalized mass Definition of the renormalized mass µ(a R ) µ = 2 mp R (A R ) µ(x) = Z A ψ(x)γµγ 5ψ(x) (P R )(x) = Z P ψ(x)γ5ψ(x) Lattice step scaling function Step scaling function 3f1 Z P (g 0, L/a) = c f P (L/2) Σ P (u, a/l) = Z P(g 0, bl/a) Z P (g 0, L/a), σ P (u) = lim a 0 Σ P (u, a/l) ḡ 2 (L) = u Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

31 Schrödinger functional Scheme-dependence existence of the IRFP is scheme-independent β(ḡ) = µ d dµ ḡ(µ) change of scheme: ḡ = Φ(ḡ), Φ (ḡ) > 0 universality of the first two coefficients β (ḡ ) = ḡ Φ(ḡ)β(ḡ) β-function away from IRFP is not universal! m = mf (ḡ) γ = γ + β ḡ log F Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

32 Schrödinger functional SU(2) adjoint n f = 2 Wilson, Schrödinger functional coupling [Hietanen et al 08, Bursa et al 09] loop 2-loop Statistical 1.06 σ(u)/u u (L 1, L 2) = 1/g 2 (L 2) 1/g 2 (L 1) Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

33 Results SU(2) adjoint n f = 2 Wilson, Schrödinger functional mass [Bursa et al 09] loop Statistical Error σ P (4/3,u) u log σ P (u, b) = γ log b = 0.05 < γ < 0.56 Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34

34 Outlook Towards phenomenology tools have been developed and tested - keep looking for better tools preliminary results need to be put on solid ground benchmark codes/analysis control systematics: small masses & large volumes improved actions all methods have systematics more confident when different approaches yield consistent results anomalous dimensions are small SU(2) adjoint evidence for conformality deformations away from conformality n f < n c f mass deformation 4fermi interactions lattice results to be taken into account for model building talk to phenomenologists! Luigi Del Debbio (University of Edinburgh) Lattice studies of the conformal window Zeuthen - November / 34