Running coupling measurements in Technicolor models

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1 measurements in Technicolor models University of the Pacific August Lattice Higgs Collaboration Zoltan Fodor, Julius Kuti, Daniel Nogradi, Chris Schroeder

2 outline context: technicolor running coupling role method test: pure gauge theory first results dynamical fermions conclusions

3 revived technicolor on lattice busy field always same question: N f which theories are conformal? gray: fundamental blue: 2-index antisymmetric red: 2-index symmetric green: adjoint this is a hard question, but answerable, we hope Dietrich&Sannino N c

4 consistency more than 1 signal for conformal/non-conformal behavior p-regime and techni-hadron spectrum epsilon-regime and lowest Dirac eigenvalues ] Julius Kuti Monday finite temperature transitions running coupling (Schrodinger functional, Wilson loops)...

5 RG flow if theory conformal, non-trivial zero of β(g) = (β 1 g 3 16π 2 + β 2 g 5 (16π 2 ) 2 ) β infrared fixed point Caswell-Banks-Zaks g 2 16π 2 = β 1 β 2 g g pert.theory trustworthy if g not too large use lattice: non-perturbative

6 Wilson-loop method continuum R2 g 2 (R/L, L) = k(r/l) 2 R T ln W(R, T, L) T=R lattice g 2 1 ((R + 1/2)/L, L) = k(r/l) (R + 1/2)2 χ(r + 1/2, L), [ ] W(R + 1, T + 1, L)W(R, T, L) χ(r + 1/2, L) = ln W(R + 1, T, L)W(R, T + 1, L) T=R, keep where (R + 1/2)/L fixed: is the Creutz ratio [67], and the renormalizat coupling flows with L also tested by Bilgici et al

7 step-scaling g 2 (L 1, β 1 ) = g 2 (L 2, β 2 ) = g 2 (L 3, β 3 ) tune: g 2 (2L 1, β 1 ) g 2 (2L 2, β 2 ) g 2 (2L 3, β 3 ) measure: extrapolate RG step to continuum g 2 (L) g 2 (2L)

8 test: pure gauge theory 4-dimensional SU(3) 3.2 L = 14 many runs on smaller lattices to tune coupling scheme: r = (R + 1/2)/L = 0.25 g 2 (L) !

9 test: pure gauge theory 0.65 run on doubled lattice 0.6 linear fit data e.g. 2L = 28, β = where coupling tuned g 2 (L = 14, β = 6.99) = 2.1 (R+1/2) 2! r = 0.25 interpolation: 0.25 g 2 (2L = 28, β = 6.99) = 2.82(2) R+1/2

10 continuum extrapolation tune: g 2 (L i, β i ) = doubled lattices linear fit data extrapolation: 1.636(23) 2L = 20, 24, 28, 32 g 2 (2L) 1.7 cut-off effects O(a 2 ) 1.65 i.e. O(1/L 2 ) /(2L) 2

11 connect RG steps 6 iterate procedure tune g 2 (L i, β i ) to previous extrapolation g 2 (2L) g 2 (L)=1.44 g 2 (L)=1.7 g 2 (L)=2.1 g 2 (L)=2.8 range of doubled lattices 2L = 20,..., 44 actually, not that cheap /(2L) 2

12 RG flow simulation: 4 RG steps weak coupling: calculate Wilson loop in pert.theory (with improvement - Julius) g 2 (L) loop 2-loop analytic simulations nice agreement with loop RG flow ln(l/l 0 )

13 why fermions expensive For non-lattice people: pure gauge theory Z = DU exp( S gauge [U]) gauge theory with fermions Z = DU {Det(D[U])} N f exp( S gauge [U]) lattice DU determinant makes computation slow typical U

14 dynamical fermions (ongoing) SU(3), N f = 16 fundamental g loop guaranteed(?) conformal lattice: staggered fermions no step-scaling g 2 (L, β, m) g 2 (L) !=5!=7!=12!=15!=35 consistent with flow to fixed point L similar to Hietanen at al SU(2), N f = 2 adjoint

15 dynamical fermions (ongoing) fundamental SU(3), N f = 12 g loop interesting/controversial lattice: staggered no step-scaling g 2 (L, β, m) g 2 (L) !=6.0!=6.25!=6.5!=7.0!=8.0!=9.0!=12.0!=15.0!=25.0 no sign yet of IRFP L

16 conclusions method works well in pure gauge theory indication that SU(3), N f = 16 fundamental conformal no sign yet of IRFP for SU(3), N f = 12 fundamental faster continuum limit? twisted Polyakov loop method long-term: consistency with other conformal signals