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 Nf = 10 flavors IR conformal or chirally broken? There was/is some controversy about Nf = 12 . . . . . . Nf = 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? Nf = 10 interesting on its own

0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 ( g2(sL) - g2(L) ) / log(s2) g2(L) approaching the conformal window Nf = 4 c = 3/10 s = 3/2 Nf = 8 c = 3/10 s = 3/2 Nf = 12 c = 1/5 s = 2

Last year: Nf = 3 sextet, Nf = 14 fund (both conformal, 1711.00130) This conference: Kieran Holland: Nf = 13 fund (conformal)

What, why and how? Nf = 10 interesting on its own

0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 ( g2(sL) - g2(L) ) / log(s2) g2(L) approaching the conformal window fund Nf = 4 c = 3/10 s = 3/2 fund Nf = 8 c = 3/10 s = 3/2 sextet Nf = 2 c = 7/20 s = 3/2 fund Nf = 12 c = 1/5 s = 2

Last year: Nf = 3 sextet, Nf = 14 fund (both conformal, 1711.00130) This conference: Kieran Holland: Nf = 13 fund (conformal)

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

What, why and how? Results in literature - domain wall

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7 8 9 ( g2(sL) - g2(L) ) / log(s2) g2(L) c = 3/10 s = 2 H,R,W T-W C 5-loop 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 < g2(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

trajectory

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

λ2

min

0.01 0.02 0.03 0.04 0.05 0.06 0.07

Nf = 10 244 smallest eigenvalue

β = 2.90 Remez bound β = 4.00 Remez bound

Infrared regulator 1/L acts similarly to m in large volumes → stable algorithm

Rooting - taste breaking in Dirac eigenvalues

k

1 2 3 4 5 6 7 8

λk

0.085 0.09 0.095 0.1 0.105 0.11

Nf = 10 244 lowest eigenvalues

β = 2.90

k

1 2 3 4 5 6 7 8

λk

0.195 0.2 0.205 0.21 0.215 0.22

Nf = 10 244 lowest eigenvalues

β = 4.00

Lowest 8 eigenvalues First (low β): doublets, then (high β): quartets

Rooting - taste breaking in Dirac eigenvalues

1 2 3 4 5 6 7

a2/L2

10-3 0.5 1 1.5 2 2.5 3

normalized doublet splitting

10-3

nf=10 doublet splitting s1 g 2 = 5.5

/ = c0 + c1 a2/L2 + c2 a4/L4

2/dof = Inf Q = 0

c0 = 4.26e-05 0.00017 c1 = 1.57 0.51 c2 = -367 3.3e+02

1 2 3 4 5 6 7

a2/L2

10-3 0.5 1 1.5 2 2.5 3

normalized doublet splitting

10-3

nf=10 doublet splitting s2 g 2 = 5.5

/ = c0 + c1 a2/L2 + c2 a4/L4

2/dof = 0.17 Q = 0.84

c0 = 3.49e-05 7.6e-05 c1 = 1.54 0.19 c2 = -214 90

1 2 3 4 5 6 7

a2/L2

10-3 0.5 1 1.5 2 2.5 3

normalized doublet splitting

10-3

nf=10 doublet splitting s3 g 2 = 5.5

/ = c0 + c1 a2/L2 + c2 a4/L4

2/dof = 0.3 Q = 0.74

c0 = 1.12e-05 7.2e-05 c1 = 1.53 0.18 c2 = -204 84

1 2 3 4 5 6 7

a2/L2

10-3 0.5 1 1.5 2 2.5 3

normalized doublet splitting

10-3

nf=10 doublet splitting s4 g 2 = 5.5

/ = c0 + c1 a2/L2 + c2 a4/L4

2/dof = 0.26 Q = 0.61

c0 = -1.3e-05 0.00014 c1 = 1.68 0.41 c2 = -328 2.6e+02

Fix g2(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

β

2.8 3 3.2 3.4 3.6 3.8 4

g2(L)

3 4 5 6 7 8 9 10

124 → 244

χ2/dof= 0.4 and 1.7

124 244

Nf = 10 SSC c = 0.25 interpolation

β

2.8 3 3.2 3.4 3.6 3.8 4

g2(L)

3 4 5 6 7 8 9 10

164 → 324

χ2/dof= 0.92 and 0.56

164 324

Nf = 10 SSC c = 0.25 interpolation

β

2.8 3 3.2 3.4 3.6 3.8 4

g2(L)

3 4 5 6 7 8 9 10

184 → 364

χ2/dof= 0.97 and 0.094

184 364

Nf = 10 SSC c = 0.25 interpolation

β

2.8 3 3.2 3.4 3.6 3.8 4

g2(L)

3 4 5 6 7 8 9 10

204 → 404

χ2/dof= 1.4 and 0.69

204 404

Nf = 10 SSC c = 0.25 interpolation

Continuum extrapolation

β

2.8 3 3.2 3.4 3.6 3.8 4

g2(L)

3 4 5 6 7 8 9 10

244 → 484

χ2/dof= 1.7 and 0.9

244 484

Nf = 10 SSC c = 0.25 interpolation

Continuum extrapolation

a2/L2

×10-3 1 2 3 4 5 6 7 8 9 10

( g2(sL) - g2(L) )/log(s2)

• 0.2
• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7

continuum WSC: 0.386 ± 0.061 continuum SSC: 0.45 ± 0.046 g2 = 5, c = 0.30, s = 2 Nf = 10 beta function

Continuum extrapolation

a2/L2

×10-3 1 2 3 4 5 6 7 8 9 10

( g2(sL) - g2(L) )/log(s2)

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

continuum WSC: 0.575 ± 0.043 continuum SSC: 0.646 ± 0.04 g2 = 6, c = 0.30, s = 2 Nf = 10 beta function

Continuum extrapolation

a2/L2

×10-3 1 2 3 4 5 6 7 8 9 10

( g2(sL) - g2(L) )/log(s2)

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

continuum WSC: 0.727 ± 0.037 continuum SSC: 0.795 ± 0.042 g2 = 7, c = 0.30, s = 2 Nf = 10 beta function

Final result from 12 → 24, 16 → 32, 18 → 36, 20 → 40, 24 → 48

g2(L)

4.5 5 5.5 6 6.5 7 7.5 8 8.5

( g2(sL) - g2(L) )/log(s2)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Nf = 10 beta function

c = 1/4 s = 2

Final result from 12 → 24, 16 → 32, 18 → 36, 20 → 40, 24 → 48

g2(L)

4.5 5 5.5 6 6.5 7 7.5 8 8.5

( g2(sL) - g2(L) )/log(s2)

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Nf = 10 beta function c = 0.30

c = 3/10 s = 2

Final result

0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 ( g2(sL) - g2(L) ) / log(s2) g2(L) approaching the conformal window Nf = 4 c = 3/10 s = 3/2 Nf = 8 c = 3/10 s = 3/2 Nf = 12 c = 1/5 s = 2

Final result

0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 ( g2(sL) - g2(L) ) / log(s2) g2(L) approaching the conformal window Nf = 4 c = 3/10 s = 3/2 Nf = 8 c = 3/10 s = 3/2 Nf = 10 c = 3/10 s = 2 Nf = 12 c = 1/5 s = 2

Final result

0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 ( g2(sL) - g2(L) ) / log(s2) g2(L) approaching the conformal window fund Nf = 4 c = 3/10 s = 3/2 fund Nf = 8 c = 3/10 s = 3/2 fund Nf = 10 c = 3/10 s = 2 sextet Nf = 2 c = 7/20 s = 3/2 fund Nf = 12 c = 1/5 s = 2

Comparison with existing literature

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7 8 9 ( g2(sL) - g2(L) ) / log(s2) g2(L) c = 3/10 s = 2 H,R,W T-W C this work 5-loop

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 < g2(L) Only Gaussian UV fixed point, governs continuum limit, g0 → 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 g2

∗ and is

positive for 0 < g2(L) < g2

∗

Gaussian UV fixed point still there (only these 2) Non-trivial RG flow between UV fixed point g2 = 0 and IR fixed point g2 = g2

∗ as L = 0, . . . , ∞

In particular, running is via dimensionless quantity ΛL Small ΛL: g2(L) ∼

1 log

• 1

ΛL

• Large ΛL: g2(L) ∼ g2

∗ − const (ΛL)α

Non-universality? For 0 < g2(L) < g2

∗ the volume L is finite in physical units (Λ)

At finite L, i.e. g2(L) < g2

∗, 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 g2(L) < g2

∗ exactly the same story as outside conformal window

Non-universality? Other example: T 3 × R Hamiltonian formulation, g2(L) < g2

∗

• Non-trivial finite masses Mi = Ci/L
• Well-defined ratios Cij = Mi/Mj = Ci/Cj
• Lattice corrections: O(a2)
• Continuum limit g0 → 0 or β → ∞
• Same as outside conformal window

Non-universality? Only g2(L = ∞) = g2

∗ is tricky (but not needed for g2(L) < g2 ∗)

both on T 4 and T 3 × R Staggered, Wilson, domain wall, overlap, etc, MUST give the same result for g2(L) < g2

∗

If not, they would disagree in QCD too

Non-universality? Proposal: fix g2(L) = 5.7 What is (g2(sL) − g2(L))/ log(s2) ? All 3 results give positive β-function What happens for g2(L) > 5.7 is irrelevant There MUST be agreement once continuum limit is carefully/correctly done via g0 → 0 or β → ∞ All 3 groups should agree (before going to higher g2(L))

Non-zero domain wall residual mass Finite 5th dimension: H,W,R: mostly Ls = 12, T-W C: Ls = 16 Bare mass zero, but residual mass non-zero Most severe: large g2(L) ∼ 6 − 7 Finite mass effect: g2(β, L/a, am) = g2(β, L/a) + ∆(L/a) Introduce: x = g2(L), y = (g2(sL) − g2(L))/ log(s2) Mass effect in x direction: ∆x = ∆(L/a) Mass effect in y direction: ∆y = ∆(sL/a)−∆(L/a)

log(s2)

Volume-independent mass-dependence completely cancels, remain- ing effect: volume-dependent mass-dependence → ∆y expected to be small

Non-zero domain wall residual mass (cartoon sketch)

0.2 0.4 0.6 0.8 1 1.2 1 2 3 4 5 6 7 8 9 ( g2(sL) - g2(L) ) / log(s2) g2(L) m = 0 m > 0

Is the shifted b(g2)

• m>0 curve above or below b(g2)
• m=0 ?

Depends on db(g2)

dg2

• m=0

more or less than ∆y

∆x = ∆(sL/a)−∆(L/a) ∆(L/a) log(s2)

H,W,R: m > 0 is always above m = 0 ???

Conclusions and outlook

• No IR fixed point for 5 < g2(L) < 8
• Potential reasons for disagreements: too small lattice volumes

and/or too large residual mass for H,W,R and T-W C

• Even with IRFP: g2(L) < g2

∗ is universal (in usual sense)

• 3-way discrepancy should be resolved, e.g. at g2(L) = 5.7
• Would be good: low energy observables in p-regime
• Hadron/glueball spectrum, chiral condensate, etc.
• Running coupling in p-regime

Thank you for your attention!

Backup slides

Results

• 1
• 0.5

0.5 1 1.5 3 4 5 6 7 8 9 10 ( g2(sL) - g2(L) ) / log(s2) g2(L) c = 1/4 WSC s = 3/2 12->18 16->24 20->30 24->36 32->48

Results

• 1
• 0.5

0.5 1 1.5 3 4 5 6 7 8 9 10 ( g2(sL) - g2(L) ) / log(s2) g2(L) c = 1/4 SSC s = 3/2 12->18 16->24 20->30 24->36 32->48

Results

• 1
• 0.5

0.5 1 1.5 3 4 5 6 7 8 9 10 ( g2(sL) - g2(L) ) / log(s2) g2(L) c = 1/4 WSC s = 2 12->24 16->32 18->36 20->40 24->48

Results

• 1
• 0.5

0.5 1 1.5 3 4 5 6 7 8 9 10 ( g2(sL) - g2(L) ) / log(s2) g2(L) c = 1/4 SSC s = 2 12->24 16->32 18->36 20->40 24->48

Results

• 1
• 0.5

0.5 1 1.5 3 4 5 6 7 8 9 10 ( g2(sL) - g2(L) ) / log(s2) g2(L) c = 3/10 WSC s = 3/2 12->18 16->24 20->30 24->36 32->48

Results

• 1
• 0.5

0.5 1 1.5 3 4 5 6 7 8 9 10 ( g2(sL) - g2(L) ) / log(s2) g2(L) c = 3/10 SSC s = 3/2 12->18 16->24 20->30 24->36 32->48

Results

• 1
• 0.5

0.5 1 1.5 3 4 5 6 7 8 9 10 ( g2(sL) - g2(L) ) / log(s2) g2(L) c = 3/10 WSC s = 2 12->24 16->32 18->36 20->40 24->48

Results

• 1
• 0.5

0.5 1 1.5 3 4 5 6 7 8 9 10 ( g2(sL) - g2(L) ) / log(s2) g2(L) c = 3/10 SSC s = 2 12->24 16->32 18->36 20->40 24->48