Fate of a recent conformal fixed point and -function in the SU (3) - PowerPoint PPT Presentation

Fate of a recent conformal fixed point and β -function in the SU (3) BSM gauge theory with ten massless flavors Daniel Nogradi in collaboration with Zoltan Fodor, Kieran Holland Julius Kuti, Chik Him Wong

What, why and how? SU (3) gauge theory with N f = 10 flavors IR conformal or chirally broken? There was/is some controversy about N f = 12 . . . . . . N f = 10 should be simpler Relevant for model building: conformal walking: 4+6 model, tune masses → walking if 10 flavors conformal, if chirally broken: usual walking

What, why and how? N f = 10 interesting on its own approaching the conformal window 2.5 N f = 4 c = 3/10 s = 3/2 N f = 8 c = 3/10 s = 3/2 N f = 12 c = 1/5 s = 2 2 ( g 2 (sL) - g 2 (L) ) / log(s 2 ) 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 g 2 (L) Last year: N f = 3 sextet, N f = 14 fund (both conformal, 1711.00130) This conference: Kieran Holland: N f = 13 fund (conformal)

What, why and how? N f = 10 interesting on its own approaching the conformal window 2.5 fund N f = 4 c = 3/10 s = 3/2 fund N f = 8 c = 3/10 s = 3/2 sextet N f = 2 c = 7/20 s = 3/2 2 ( g 2 (sL) - g 2 (L) ) / log(s 2 ) fund N f = 12 c = 1/5 s = 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 g 2 (L) Last year: N f = 3 sextet, N f = 14 fund (both conformal, 1711.00130) This conference: Kieran Holland: N f = 13 fund (conformal)

What, why and how? Calculate N f = 10 running coupling, β -function in continuum Periodic finite volume gradient flow scheme Step scaling, L → 2 L , discrete β -function

What, why and how? Results in literature - domain wall c = 3/10 s = 2 1.2 H,R,W T-W C 1 ( g 2 (sL) - g 2 (L) ) / log(s 2 ) 5-loop 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 g 2 (L) Hasenfratz, Rebbi, Witzel: 1710.11578, 8 → 16 , 10 → 20 , 12 → 24 Chiu: PoS LATTICE2016 (2017) 228, 8 → 16 , 10 → 20 , 12 → 24 , 16 → 32 Discrepancy for 4 . 5 < g 2 ( L ) < 6 . 0

Outline • Numerical setup • Rooting, taste breaking, etc • Continuum extrapolation • Comparison with literature • Conclusion and outlook

Numerical setup • Tree-level improved Symanzik gauge action • Periodic gauge field • 4-step stout-improved rooted staggered fermions ( ̺ = 0 . 12) • Anti-periodic in all directions • m = 0 • 12 → 24, 16 → 32, 18 → 36, 20 → 40, 24 → 48

Rooting - eigenvalue gap - Remez algorithm N f = 10 24 4 smallest eigenvalue 0.07 β = 2.90 Remez bound β = 4.00 0.06 Remez bound 0.05 0.04 min λ 2 0.03 0.02 0.01 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 trajectory Infrared regulator 1 /L acts similarly to m in large volumes → stable algorithm

Rooting - taste breaking in Dirac eigenvalues N f = 10 24 4 lowest eigenvalues N f = 10 24 4 lowest eigenvalues 0.11 0.22 β = 2.90 β = 4.00 0.105 0.215 0.1 0.21 λ k λ k 0.095 0.205 0.09 0.2 0.085 0.195 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 k k Lowest 8 eigenvalues First (low β ): doublets, then (high β ): quartets

Rooting - taste breaking in Dirac eigenvalues nf=10 doublet splitting s1 g 2 = 5.5 nf=10 doublet splitting s2 g 2 = 5.5 10 -3 10 -3 3 3 2.5 2.5 normalized doublet splitting normalized doublet splitting 2 /dof = Inf Q = 0 2 /dof = 0.17 Q = 0.84 2 2 / = c 0 + c 1 a 2 /L 2 + c 2 a 4 /L 4 / = c 0 + c 1 a 2 /L 2 + c 2 a 4 /L 4 1.5 1.5 c 0 = 4.26e-05 0.00017 c 0 = 3.49e-05 7.6e-05 c 1 = 1.57 0.51 c 1 = 1.54 0.19 1 1 c 2 = -367 3.3e+02 c 2 = -214 90 0.5 0.5 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 a 2 /L 2 10 -3 a 2 /L 2 10 -3 nf=10 doublet splitting s3 g 2 = 5.5 nf=10 doublet splitting s4 g 2 = 5.5 10 -3 10 -3 3 3 2.5 2.5 normalized doublet splitting normalized doublet splitting 2 /dof = 0.3 Q = 0.74 2 /dof = 0.26 Q = 0.61 2 2 / = c 0 + c 1 a 2 /L 2 + c 2 a 4 /L 4 / = c 0 + c 1 a 2 /L 2 + c 2 a 4 /L 4 1.5 1.5 c 0 = 1.12e-05 7.2e-05 c 0 = -1.3e-05 0.00014 c 1 = 1.53 0.18 c 1 = 1.68 0.41 1 1 c 2 = -204 84 c 2 = -328 2.6e+02 0.5 0.5 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 a 2 /L 2 10 -3 a 2 /L 2 10 -3 Fix g 2 ( L ) = 5 . 5, taste breaking disappears in the continuum

Continuum extrapolation • Interpolate by polynomials (rather than tune) • Larger c : smaller cut-off effects, larger stat errors (knew this already) • Take c = 1 / 4 , 3 / 10 and s = 2 (also s = 3 / 2) • Check consistency of SSC and WSC discretizations

Continuum extrapolation N f = 10 SSC c = 0.25 interpolation N f = 10 SSC c = 0.25 interpolation 10 10 9 9 8 8 16 4 12 4 7 7 32 4 g 2 (L) g 2 (L) 24 4 6 6 5 5 12 4 → 24 4 16 4 → 32 4 4 4 χ 2 /dof= 0.4 and 1.7 χ 2 /dof= 0.92 and 0.56 3 3 2.8 3 3.2 3.4 3.6 3.8 4 2.8 3 3.2 3.4 3.6 3.8 4 β β N f = 10 SSC c = 0.25 interpolation N f = 10 SSC c = 0.25 interpolation 10 10 9 9 8 8 20 4 18 4 40 4 36 4 7 7 g 2 (L) g 2 (L) 6 6 5 5 18 4 → 36 4 20 4 → 40 4 4 4 χ 2 /dof= 0.97 and 0.094 χ 2 /dof= 1.4 and 0.69 3 3 2.8 3 3.2 3.4 3.6 3.8 4 2.8 3 3.2 3.4 3.6 3.8 4 β β

Continuum extrapolation N f = 10 SSC c = 0.25 interpolation 10 9 24 4 8 48 4 7 g 2 (L) 6 5 24 4 → 48 4 4 χ 2 /dof= 1.7 and 0.9 3 2.8 3 3.2 3.4 3.6 3.8 4 β

Continuum extrapolation N f = 10 beta function 0.7 continuum SSC: 0.45 ± 0.046 0.6 0.5 ( g 2 (sL) - g 2 (L) )/log(s 2 ) 0.4 continuum WSC: 0.386 ± 0.061 0.3 0.2 0.1 g 2 = 5, c = 0.30, s = 2 0 -0.1 -0.2 0 1 2 3 4 5 6 7 8 9 10 a 2 /L 2 × 10 -3

Continuum extrapolation N f = 10 beta function 0.8 0.7 continuum SSC: 0.646 ± 0.04 0.6 ( g 2 (sL) - g 2 (L) )/log(s 2 ) 0.5 0.4 continuum WSC: 0.575 ± 0.043 0.3 0.2 0.1 g 2 = 6, c = 0.30, s = 2 0 -0.1 0 1 2 3 4 5 6 7 8 9 10 a 2 /L 2 × 10 -3

Continuum extrapolation N f = 10 beta function 0.9 0.8 continuum SSC: 0.795 ± 0.042 0.7 ( g 2 (sL) - g 2 (L) )/log(s 2 ) 0.6 0.5 0.4 continuum WSC: 0.727 ± 0.037 0.3 0.2 0.1 g 2 = 7, c = 0.30, s = 2 0 -0.1 0 1 2 3 4 5 6 7 8 9 10 a 2 /L 2 × 10 -3

Final result from 12 → 24 , 16 → 32 , 18 → 36 , 20 → 40 , 24 → 48 N f = 10 beta function 1 0.9 0.8 ( g 2 (sL) - g 2 (L) )/log(s 2 ) 0.7 0.6 0.5 0.4 0.3 4.5 5 5.5 6 6.5 7 7.5 8 8.5 g 2 (L) c = 1 / 4 s = 2

Final result from 12 → 24 , 16 → 32 , 18 → 36 , 20 → 40 , 24 → 48 N f = 10 beta function c = 0.30 1 0.9 0.8 ( g 2 (sL) - g 2 (L) )/log(s 2 ) 0.7 0.6 0.5 0.4 0.3 4.5 5 5.5 6 6.5 7 7.5 8 8.5 g 2 (L) c = 3 / 10 s = 2

Final result approaching the conformal window 2.5 N f = 4 c = 3/10 s = 3/2 N f = 8 c = 3/10 s = 3/2 N f = 12 c = 1/5 s = 2 2 ( g 2 (sL) - g 2 (L) ) / log(s 2 ) 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 g 2 (L)

Final result approaching the conformal window 2.5 N f = 4 c = 3/10 s = 3/2 N f = 8 c = 3/10 s = 3/2 N f = 10 c = 3/10 s = 2 2 ( g 2 (sL) - g 2 (L) ) / log(s 2 ) N f = 12 c = 1/5 s = 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 g 2 (L)

Final result approaching the conformal window 2.5 fund N f = 4 c = 3/10 s = 3/2 fund N f = 8 c = 3/10 s = 3/2 fund N f = 10 c = 3/10 s = 2 2 ( g 2 (sL) - g 2 (L) ) / log(s 2 ) sextet N f = 2 c = 7/20 s = 3/2 fund N f = 12 c = 1/5 s = 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 g 2 (L)

Comparison with existing literature c = 3/10 s = 2 1.2 H,R,W T-W C 1 ( g 2 (sL) - g 2 (L) ) / log(s 2 ) this work 5-loop 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 g 2 (L)

Why the disagreement? • Domain wall - too small volumes? • Domain wall - residual mass non-zero? • Non-universality (???)

Non-universality? Outside conformal window: β -function positive for all 0 < g 2 ( L ) Only Gaussian UV fixed point, governs continuum limit, g 0 → 0 Bare perturbation theory (i.e. perturbation theory on cut-off scale) reliable close to continuum Various discretizations can be judged to be in the right universality class by perturbation theory Anything = continuum + O ( a ) is okay (dimension, symmetries, locality) Staggered, Wilson, domain wall, overlap, etc. all okay

Non-universality? Inside conformal window: β -function has simple zero at g 2 ∗ and is positive for 0 < g 2 ( L ) < g 2 ∗ Gaussian UV fixed point still there (only these 2) Non-trivial RG flow between UV fixed point g 2 = 0 and IR fixed point g 2 = g 2 ∗ as L = 0 , . . . , ∞ In particular, running is via dimensionless quantity Λ L 1 Small Λ L : g 2 ( L ) ∼ Large Λ L : g 2 ( L ) ∼ g 2 ∗ − const (Λ L ) α � � 1 log Λ L

Non-universality? For 0 < g 2 ( L ) < g 2 ∗ the volume L is finite in physical units (Λ) At finite L , i.e. g 2 ( L ) < g 2 ∗ , continuum limit is governed by Gaus- sian UV fixed point Bare perturbation theory still reliable close to the continuum limit Various discretizations can be judged to be in the right universality class by perturbation theory For g 2 ( L ) < g 2 ∗ exactly the same story as outside conformal window

Non-universality? Other example: T 3 × R Hamiltonian formulation, g 2 ( L ) < g 2 ∗ • Non-trivial finite masses M i = C i /L • Well-defined ratios C ij = M i /M j = C i /C j • Lattice corrections: O ( a 2 ) • Continuum limit g 0 → 0 or β → ∞ • Same as outside conformal window