Conformal Finite Size Scaling of Conformal Finite Size Scaling of - - PowerPoint PPT Presentation

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conformal Finite Size Scaling of Twelve Fermion Flavors Conformal Finite Size Scaling of Twelve Fermion Flavors

Chik Him Wong Lattice Higgs Collaboration (LHC): Zoltán Fodor $, Kieran Holland ∗, Julius Kuti †, Dániel Nógrádi −, Chik Him Wong †

†: University of California, San Diego *: University of the Pacific $: University of Wuppertal -: Eötvös University

LATTICE 2012

1 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Outline Outline

Background

Conformality Controversy of Nf = 12 SU(3) Fundamental Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with Finite Size Scaling study

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Outstanding Problems Future Plans

2 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conformality Controversy of Nf = 12 SU(3) Fundamental Conformality Controversy of Nf = 12 SU(3) Fundamental

Is Nf = 12 SU(3) Fundamental inside Conformal Window?

• Previously. . .

Likely: T. Appelquist et al, A. Deuzeman et al,A Hasenfratz et al, Y. Aoki et al Not Likely: X.-Y. Jin et al, Z. Fodor et al

Latest findings of our group: Finite Size Scaling Conformal scenario is Inconsistent and Suspect

3 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conformality Controversy of Nf = 12 SU(3) Fundamental Conformality Controversy of Nf = 12 SU(3) Fundamental

Is Nf = 12 SU(3) Fundamental inside Conformal Window?

• Previously. . .

Likely: T. Appelquist et al, A. Deuzeman et al,A Hasenfratz et al, Y. Aoki et al Not Likely: X.-Y. Jin et al, Z. Fodor et al

Latest findings of our group: Finite Size Scaling Conformal scenario is Inconsistent and Suspect

3 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conformality Controversy of Nf = 12 SU(3) Fundamental Conformality Controversy of Nf = 12 SU(3) Fundamental

Is Nf = 12 SU(3) Fundamental inside Conformal Window?

• Previously. . .

Likely: T. Appelquist et al, A. Deuzeman et al,A Hasenfratz et al, Y. Aoki et al Not Likely: X.-Y. Jin et al, Z. Fodor et al

Latest findings of our group: Finite Size Scaling Conformal scenario is Inconsistent and Suspect

3 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conformality Controversy of Nf = 12 SU(3) Fundamental Conformality Controversy of Nf = 12 SU(3) Fundamental

Is Nf = 12 SU(3) Fundamental inside Conformal Window?

• Previously. . .

Likely: T. Appelquist et al, A. Deuzeman et al,A Hasenfratz et al, Y. Aoki et al Not Likely: X.-Y. Jin et al, Z. Fodor et al

Latest findings of our group: Finite Size Scaling Conformal scenario is Inconsistent and Suspect

3 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conformality Controversy of Nf = 12 SU(3) Fundamental Conformality Controversy of Nf = 12 SU(3) Fundamental

Is Nf = 12 SU(3) Fundamental inside Conformal Window?

• Previously. . .

Likely: T. Appelquist et al, A. Deuzeman et al,A Hasenfratz et al, Y. Aoki et al Not Likely: X.-Y. Jin et al, Z. Fodor et al

Latest findings of our group: Finite Size Scaling Conformal scenario is Inconsistent and Suspect

3 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Simulation Setup Simulation Setup

Action: Tree-level Symanzik-Improved gauge action with Staggered Nf = 12 Fundamental fermions HMC algorithm with multiple time scales and Omelyan integrator Autocorrelations monitored by time histories of Fermion condensate,plaquette and correlators β ≡ 6/g2 = 2.20, which is in the Weak Coupling regime

1/8 1/4 1/2 1 2 1.2 1.4 1.6 1.8 2.0 2.2 2.4 a3 ¯ ψψ β . m 0.02, V 124 m 0.01, V 164

we are here intermediate phase

m β

intermediate phase

• ur simulations here
• C. Schroeder et al
• A. Hasenfratz et al

4 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Simulation Setup Simulation Setup

Action: Tree-level Symanzik-Improved gauge action with Staggered Nf = 12 Fundamental fermions HMC algorithm with multiple time scales and Omelyan integrator Autocorrelations monitored by time histories of Fermion condensate,plaquette and correlators β ≡ 6/g2 = 2.20, which is in the Weak Coupling regime

1/8 1/4 1/2 1 2 1.2 1.4 1.6 1.8 2.0 2.2 2.4 a3 ¯ ψψ β . m 0.02, V 124 m 0.01, V 164

we are here intermediate phase

m β

intermediate phase

• ur simulations here
• C. Schroeder et al
• A. Hasenfratz et al

4 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Simulation Setup Simulation Setup

Action: Tree-level Symanzik-Improved gauge action with Staggered Nf = 12 Fundamental fermions HMC algorithm with multiple time scales and Omelyan integrator Autocorrelations monitored by time histories of Fermion condensate,plaquette and correlators β ≡ 6/g2 = 2.20, which is in the Weak Coupling regime

1/8 1/4 1/2 1 2 1.2 1.4 1.6 1.8 2.0 2.2 2.4 a3 ¯ ψψ β . m 0.02, V 124 m 0.01, V 164

we are here intermediate phase

m β

intermediate phase

• ur simulations here
• C. Schroeder et al
• A. Hasenfratz et al

4 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Simulation Setup Simulation Setup

Action: Tree-level Symanzik-Improved gauge action with Staggered Nf = 12 Fundamental fermions HMC algorithm with multiple time scales and Omelyan integrator Autocorrelations monitored by time histories of Fermion condensate,plaquette and correlators β ≡ 6/g2 = 2.20, which is in the Weak Coupling regime

1/8 1/4 1/2 1 2 1.2 1.4 1.6 1.8 2.0 2.2 2.4 a3 ¯ ψψ β . m 0.02, V 124 m 0.01, V 164

we are here intermediate phase

m β

intermediate phase

• ur simulations here
• C. Schroeder et al
• A. Hasenfratz et al

4 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Simulation Setup Simulation Setup

Lattices used:( ∼ 1000−2000 Trajectories each) L T mq 48 96 0.0100, 0.0150 40 80 0.0100, 0.0150, 0.0200, 0.0250 32 64 0.0020, 0.0040, 0.0060, 0.0080, 0.0100, 0.0150, 0.0200, 0.0250 28 56 0.0150 24 48 0.0020, 0.0040, 0.0060, 0.0080, 0.0100, 0.0150, 0.0200, 0.0250, 0.0300, 0.0325, 0.0350 20 40 0.0020, 0.0040, 0.0060, 0.0080, 0.0100, 0.0150, 0.0200, 0.0250

5 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Spectroscopy Analysis Spectroscopy Analysis

Double Jackknife to obtain good estimate on Covariance Matrix

Generate Jackknife ensembles of Correlators {C(t)}i Fit individual ensemble to obtain mi

0 or Fi π using Covariance Matrix

Ci

x4x′

4 =

N−1

∑

j=1

{meff (x4)i

j −meff (x4)i}{meff (x′ 4)i j −meff (x′ 4)i}

With χ2 defined as: (χ2)i =

t1

∑

x4=t0 t1

∑

x′

4=t0

{mi

0 −meff (x4)i}[(Ci)−1]x4x′

4{mi

0 −meff (x′ 4)i}

(above expressions also apply to Fi

π)

6 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Spectroscopy Analysis Spectroscopy Analysis

Double Jackknife to obtain good estimate on Covariance Matrix

Generate Jackknife ensembles of Correlators {C(t)}i Fit individual ensemble to obtain mi

0 or Fi π using Covariance Matrix

Ci

x4x′

4 =

N−1

∑

j=1

{meff (x4)i

j −meff (x4)i}{meff (x′ 4)i j −meff (x′ 4)i}

With χ2 defined as: (χ2)i =

t1

∑

x4=t0 t1

∑

x′

4=t0

{mi

0 −meff (x4)i}[(Ci)−1]x4x′

4{mi

0 −meff (x′ 4)i}

(above expressions also apply to Fi

π)

6 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Spectroscopy Analysis Spectroscopy Analysis

Double Jackknife to obtain good estimate on Covariance Matrix

Generate Jackknife ensembles of Correlators {C(t)}i Fit individual ensemble to obtain mi

0 or Fi π using Covariance Matrix

Ci

x4x′

4 =

N−1

∑

j=1

{meff (x4)i

j −meff (x4)i}{meff (x′ 4)i j −meff (x′ 4)i}

With χ2 defined as: (χ2)i =

t1

∑

x4=t0 t1

∑

x′

4=t0

{mi

0 −meff (x4)i}[(Ci)−1]x4x′

4{mi

0 −meff (x′ 4)i}

(above expressions also apply to Fi

π)

6 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 1: Fitting scaling function with physical model LM = c1x+cexp(cπx)−1/2e−cπx, x > xcut LM = c0 +cαxα, x < xcut where x ≡ Lm1/1+γ and M = Mπ,Mρ,MN or Fπ Conformal behavior:

M ∼ m1/1+γ as L → ∞ at fixed m M ∼ L−1 as m → 0 Other terms are finite size corrections

cexp(cπx)−1/2e−cπ x : leading exponential correction from wrap-around effect of the lightest Goldstone pion state (cπ = c1 for pions and fitted value becomes cπ of other channels) cαxα : simplest ansatz, could be polynomials

Continuity and first derivative continuity imposed at xcut 5 fit parameters: γ , c1 , cexp , α , xcut

7 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 1: Fitting scaling function with physical model LM = c1x+cexp(cπx)−1/2e−cπx, x > xcut LM = c0 +cαxα, x < xcut where x ≡ Lm1/1+γ and M = Mπ,Mρ,MN or Fπ Conformal behavior:

M ∼ m1/1+γ as L → ∞ at fixed m M ∼ L−1 as m → 0 Other terms are finite size corrections

cexp(cπx)−1/2e−cπ x : leading exponential correction from wrap-around effect of the lightest Goldstone pion state (cπ = c1 for pions and fitted value becomes cπ of other channels) cαxα : simplest ansatz, could be polynomials

Continuity and first derivative continuity imposed at xcut 5 fit parameters: γ , c1 , cexp , α , xcut

7 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 1: Fitting scaling function with physical model LM = c1x+cexp(cπx)−1/2e−cπx, x > xcut LM = c0 +cαxα, x < xcut where x ≡ Lm1/1+γ and M = Mπ,Mρ,MN or Fπ Conformal behavior:

M ∼ m1/1+γ as L → ∞ at fixed m M ∼ L−1 as m → 0 Other terms are finite size corrections

cexp(cπx)−1/2e−cπ x : leading exponential correction from wrap-around effect of the lightest Goldstone pion state (cπ = c1 for pions and fitted value becomes cπ of other channels) cαxα : simplest ansatz, could be polynomials

Continuity and first derivative continuity imposed at xcut 5 fit parameters: γ , c1 , cexp , α , xcut

7 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 1: Fitting scaling function with physical model LM = c1x+cexp(cπx)−1/2e−cπx, x > xcut LM = c0 +cαxα, x < xcut where x ≡ Lm1/1+γ and M = Mπ,Mρ,MN or Fπ Conformal behavior:

M ∼ m1/1+γ as L → ∞ at fixed m M ∼ L−1 as m → 0 Other terms are finite size corrections

cexp(cπx)−1/2e−cπ x : leading exponential correction from wrap-around effect of the lightest Goldstone pion state (cπ = c1 for pions and fitted value becomes cπ of other channels) cαxα : simplest ansatz, could be polynomials

Continuity and first derivative continuity imposed at xcut 5 fit parameters: γ , c1 , cexp , α , xcut

7 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 1: Fitting scaling function with physical model LM = c1x+cexp(cπx)−1/2e−cπx, x > xcut LM = c0 +cαxα, x < xcut where x ≡ Lm1/1+γ and M = Mπ,Mρ,MN or Fπ Conformal behavior:

M ∼ m1/1+γ as L → ∞ at fixed m M ∼ L−1 as m → 0 Other terms are finite size corrections

cexp(cπx)−1/2e−cπ x : leading exponential correction from wrap-around effect of the lightest Goldstone pion state (cπ = c1 for pions and fitted value becomes cπ of other channels) cαxα : simplest ansatz, could be polynomials

Continuity and first derivative continuity imposed at xcut 5 fit parameters: γ , c1 , cexp , α , xcut

7 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 1: Fitting scaling function with physical model LM = c1x+cexp(cπx)−1/2e−cπx, x > xcut LM = c0 +cαxα, x < xcut where x ≡ Lm1/1+γ and M = Mπ,Mρ,MN or Fπ Conformal behavior:

M ∼ m1/1+γ as L → ∞ at fixed m M ∼ L−1 as m → 0 Other terms are finite size corrections

cexp(cπx)−1/2e−cπ x : leading exponential correction from wrap-around effect of the lightest Goldstone pion state (cπ = c1 for pions and fitted value becomes cπ of other channels) cαxα : simplest ansatz, could be polynomials

Continuity and first derivative continuity imposed at xcut 5 fit parameters: γ , c1 , cexp , α , xcut

7 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 1: Fitting scaling function with physical model LM = c1x+cexp(cπx)−1/2e−cπx, x > xcut LM = c0 +cαxα, x < xcut where x ≡ Lm1/1+γ and M = Mπ,Mρ,MN or Fπ Conformal behavior:

M ∼ m1/1+γ as L → ∞ at fixed m M ∼ L−1 as m → 0 Other terms are finite size corrections

cexp(cπx)−1/2e−cπ x : leading exponential correction from wrap-around effect of the lightest Goldstone pion state (cπ = c1 for pions and fitted value becomes cπ of other channels) cαxα : simplest ansatz, could be polynomials

Continuity and first derivative continuity imposed at xcut 5 fit parameters: γ , c1 , cexp , α , xcut

7 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 1: Fitting scaling function with physical model LM = c1x+cexp(cπx)−1/2e−cπx, x > xcut LM = c0 +cαxα, x < xcut where x ≡ Lm1/1+γ and M = Mπ,Mρ,MN or Fπ Conformal behavior:

M ∼ m1/1+γ as L → ∞ at fixed m M ∼ L−1 as m → 0 Other terms are finite size corrections

cexp(cπx)−1/2e−cπ x : leading exponential correction from wrap-around effect of the lightest Goldstone pion state (cπ = c1 for pions and fitted value becomes cπ of other channels) cαxα : simplest ansatz, could be polynomials

Continuity and first derivative continuity imposed at xcut 5 fit parameters: γ , c1 , cexp , α , xcut

7 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 1: Fitting scaling function with physical model LM = c1x+cexp(cπx)−1/2e−cπx, x > xcut LM = c0 +cαxα, x < xcut where x ≡ Lm1/1+γ and M = Mπ,Mρ,MN or Fπ Conformal behavior:

M ∼ m1/1+γ as L → ∞ at fixed m M ∼ L−1 as m → 0 Other terms are finite size corrections

cexp(cπx)−1/2e−cπ x : leading exponential correction from wrap-around effect of the lightest Goldstone pion state (cπ = c1 for pions and fitted value becomes cπ of other channels) cαxα : simplest ansatz, could be polynomials

Continuity and first derivative continuity imposed at xcut 5 fit parameters: γ , c1 , cexp , α , xcut

7 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting scaling function with physical model Fitting scaling function with physical model

Mπ

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'()'*'+$,-

+

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%!(M! %M(ML %L(#O &%(OM M!(L! ML(KO 8 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting scaling function with physical model Fitting scaling function with physical model

Mρ

! !"# $ $"# % %"# ! % & ' ( $! $% $& $' $(

)*+),)-$./

• ),0l1,2

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%!*&! %&*&( %(*#' H%*'& &!*(! &(*J' 9 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting scaling function with physical model Fitting scaling function with physical model

MN

! !"# $ $"# % %"# & ' $! $% $( $& $' %! %%

)*+),)-$./

• ),01234,5

)361768-9:);<<)))))))))43=23:>61)3?911>:5

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,01234,5+3$)*)@)3>*Au)43/)*5ï$.%)>*A4ï3/)*5))))*)B)*32C ,01234,5+3!)@)3_)*_))))))))))*)D)*32C /-+$@aE)3$E)3>*AE)_E)*32C))))#)7FC)A989->C>8G a+)!"%'')!)!"!%H _)+)("(I)!)!"I' *32C+)$"&()!)!"$% 3$+)J"'#)!)!"#'' r%.K67+)$"(# 3_)+)!"I& 3!)+)$I"% 3/)+)("I%&

%!*(! %(*(' %'*#& I%*&( (!*'! ('*J& 10 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting scaling function with physical model Fitting scaling function with physical model

Fπ

! !"# $ $"# % ! !"# $ $"# % %"# &

'()'*'+$,-

+

'*.//*0 '123425+67'.88'''''/.9:'ï';<=<'1>633?70

' '

*.//*0)1!'@'1_'(_''''''

• +)$@aA'1!A'_A'1_''''''''B'4:C'9656+?C?5D

a)'!"%$B'!'!"!$E _')'!"F#%'!'!"!%G 1!)'ï!"$HE'!'!"!& 1_)'$"$H%'!'!"!#E r%,I24)'$B"&

%!(B! %B(BF %F(&% &%(EB B!(F! BF(HE

Only 4 parameters are kept because 5 parameter fit is not stable The unexpectedly curious behavior of the data set against conformal FSS remains unresolved. Question: What to be expected if conformal, e.g. SU(2) Adjoint?

11 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting scaling function with physical model Fitting scaling function with physical model

Fπ

! !"# $ $"# % ! !"# $ $"# % %"# &

'()'*'+$,-

+

'*.//*0 '123425+67'.88'''''/.9:'ï';<=<'1>633?70

' '

*.//*0)1!'@'1_'(_''''''

• +)$@aA'1!A'_A'1_''''''''B'4:C'9656+?C?5D

a)'!"%$B'!'!"!$E _')'!"F#%'!'!"!%G 1!)'ï!"$HE'!'!"!& 1_)'$"$H%'!'!"!#E r%,I24)'$B"&

%!(B! %B(BF %F(&% &%(EB B!(F! BF(HE

Only 4 parameters are kept because 5 parameter fit is not stable The unexpectedly curious behavior of the data set against conformal FSS remains unresolved. Question: What to be expected if conformal, e.g. SU(2) Adjoint?

11 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting scaling function with physical model Fitting scaling function with physical model

Fπ

! !"# $ $"# % ! !"# $ $"# % %"# &

'()'*'+$,-

+

'*.//*0 '123425+67'.88'''''/.9:'ï';<=<'1>633?70

' '

*.//*0)1!'@'1_'(_''''''

• +)$@aA'1!A'_A'1_''''''''B'4:C'9656+?C?5D

a)'!"%$B'!'!"!$E _')'!"F#%'!'!"!%G 1!)'ï!"$HE'!'!"!& 1_)'$"$H%'!'!"!#E r%,I24)'$B"&

%!(B! %B(BF %F(&% &%(EB B!(F! BF(HE

Only 4 parameters are kept because 5 parameter fit is not stable The unexpectedly curious behavior of the data set against conformal FSS remains unresolved. Question: What to be expected if conformal, e.g. SU(2) Adjoint?

11 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting scaling function with physical model Fitting scaling function with physical model

Fπ

! !"# $ $"# % ! !"# $ $"# % %"# &

'()'*'+$,-

+

'*.//*0 '123425+67'.88'''''/.9:'ï';<=<'1>633?70

' '

*.//*0)1!'@'1_'(_''''''

• +)$@aA'1!A'_A'1_''''''''B'4:C'9656+?C?5D

a)'!"%$B'!'!"!$E _')'!"F#%'!'!"!%G 1!)'ï!"$HE'!'!"!& 1_)'$"$H%'!'!"!#E r%,I24)'$B"&

%!(B! %B(BF %F(&% &%(EB B!(F! BF(HE

Only 4 parameters are kept because 5 parameter fit is not stable The unexpectedly curious behavior of the data set against conformal FSS remains unresolved. Question: What to be expected if conformal, e.g. SU(2) Adjoint?

11 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting scaling function with physical model Fitting scaling function with physical model

Summary: Quantity γ χ2/dof Mπ 0.393(8) 2.83 Mρ 0.300(17) 1.51 MN 0.288(27) 1.45 Fπ 0.214(16) 14.3

The composite particle masses in several quantum number channels can be reasonably fitted with conformal scaling functions BUT γ’s are incompatible across different channels. ⇒ Global conformal FSS fit with identical γ will fail Conformal fit of Fπ unexpectedly failed

12 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting scaling function with physical model Fitting scaling function with physical model

Summary: Quantity γ χ2/dof Mπ 0.393(8) 2.83 Mρ 0.300(17) 1.51 MN 0.288(27) 1.45 Fπ 0.214(16) 14.3

The composite particle masses in several quantum number channels can be reasonably fitted with conformal scaling functions BUT γ’s are incompatible across different channels. ⇒ Global conformal FSS fit with identical γ will fail Conformal fit of Fπ unexpectedly failed

12 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting scaling function with physical model Fitting scaling function with physical model

Summary: Quantity γ χ2/dof Mπ 0.393(8) 2.83 Mρ 0.300(17) 1.51 MN 0.288(27) 1.45 Fπ 0.214(16) 14.3

The composite particle masses in several quantum number channels can be reasonably fitted with conformal scaling functions BUT γ’s are incompatible across different channels. ⇒ Global conformal FSS fit with identical γ will fail Conformal fit of Fπ unexpectedly failed

12 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting scaling function with physical model Fitting scaling function with physical model

Summary: Quantity γ χ2/dof Mπ 0.393(8) 2.83 Mρ 0.300(17) 1.51 MN 0.288(27) 1.45 Fπ 0.214(16) 14.3

The composite particle masses in several quantum number channels can be reasonably fitted with conformal scaling functions BUT γ’s are incompatible across different channels. ⇒ Global conformal FSS fit with identical γ will fail Conformal fit of Fπ unexpectedly failed

12 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 2: Fitting general scaling function using B-form spline functions Vary γ and perform Spline Fits at fixed γ’s:

Piecewise Polynomial fitting Non-uniform B-form used 3 knots used 6 cubic spline functions

Obtain γmin that minimizes χ2(γ)/dof The range of γ that reduce/increase χ2/dof by 1 is regarded as the error estimate of γmin

13 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 2: Fitting general scaling function using B-form spline functions Vary γ and perform Spline Fits at fixed γ’s:

Piecewise Polynomial fitting Non-uniform B-form used 3 knots used 6 cubic spline functions

Obtain γmin that minimizes χ2(γ)/dof The range of γ that reduce/increase χ2/dof by 1 is regarded as the error estimate of γmin

13 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 2: Fitting general scaling function using B-form spline functions Vary γ and perform Spline Fits at fixed γ’s:

Piecewise Polynomial fitting Non-uniform B-form used 3 knots used 6 cubic spline functions

Obtain γmin that minimizes χ2(γ)/dof The range of γ that reduce/increase χ2/dof by 1 is regarded as the error estimate of γmin

13 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 2: Fitting general scaling function using B-form spline functions Vary γ and perform Spline Fits at fixed γ’s:

Piecewise Polynomial fitting Non-uniform B-form used 3 knots used 6 cubic spline functions

Obtain γmin that minimizes χ2(γ)/dof The range of γ that reduce/increase χ2/dof by 1 is regarded as the error estimate of γmin

13 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 2: Fitting general scaling function using B-form spline functions Vary γ and perform Spline Fits at fixed γ’s:

Piecewise Polynomial fitting Non-uniform B-form used 3 knots used 6 cubic spline functions

Obtain γmin that minimizes χ2(γ)/dof The range of γ that reduce/increase χ2/dof by 1 is regarded as the error estimate of γmin

13 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 2: Fitting general scaling function using B-form spline functions Vary γ and perform Spline Fits at fixed γ’s:

Piecewise Polynomial fitting Non-uniform B-form used 3 knots used 6 cubic spline functions

Obtain γmin that minimizes χ2(γ)/dof The range of γ that reduce/increase χ2/dof by 1 is regarded as the error estimate of γmin

13 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 2: Fitting general scaling function using B-form spline functions Vary γ and perform Spline Fits at fixed γ’s:

Piecewise Polynomial fitting Non-uniform B-form used 3 knots used 6 cubic spline functions

Obtain γmin that minimizes χ2(γ)/dof The range of γ that reduce/increase χ2/dof by 1 is regarded as the error estimate of γmin

13 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting under conformal hypothesis with Finite Size Scaling study Fitting under conformal hypothesis with Finite Size Scaling study

Approach 2: Fitting general scaling function using B-form spline functions Vary γ and perform Spline Fits at fixed γ’s:

Piecewise Polynomial fitting Non-uniform B-form used 3 knots used 6 cubic spline functions

Obtain γmin that minimizes χ2(γ)/dof The range of γ that reduce/increase χ2/dof by 1 is regarded as the error estimate of γmin

13 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting general scaling function using B-form spline functions Fitting general scaling function using B-form spline functions

Mπ

! !"# $ $"# % %"# & # ' ( ) * $! $$ $% $& $+ $#

,-.,/01 ,/2/3/4 ,5678690:;,<==,8>?,@>?A,BC;>7D,E:BDF,Gï8690,,3H6;FB?67D,C>674

, ,

BC;>7D,8>?,@>?A,Gï8690,IB>7J,&,>7?D97:;,K76?B ',5IE>5,GïBC;>7D,E:BD,8I75?>67B,0:?5ADF,>7,+,>7?D9L:;B 10.$Ma,AD;F,8>-DF,>7,8>?,?6,Gï8690,ED869D,FD?D90>7DF,8960,,r%3a4,0>7>0I0 %*,F:?:,C6>7?B

a.,!"+!#,!,!"!%$ r%NF68.,$"+(

8>?,O,L:;ID,.,!"!'(

%!-+! %+-+) %)-#' &%-'+ +!-)! +)-*' 14 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting general scaling function using B-form spline functions Fitting general scaling function using B-form spline functions

Mρ

! !"# $ $"# % %"# & ' ( ) $! $$ $% $* $+ $# $& $'

,-.,/01 ,/2l3/4 ,5678690:;,<==,8>?,@>?A,BC;>7D,E:BDF,Gï8690,,35HA6%,5A:77D;4

, ,

,BC;>7D,8>?,@>?A,Gï8690,IB>7J,*,>7?D97:;,K76?B &,5IE>5,GïBC;>7D,E:BD,8I75?>67B,0:?5ADF,>7,+,>7?D9L:;B 10.$Ma,AD;F,8>-DF,>7,8>?,?6,Gï8690,ED869D,FD?D90>7DF,8960,,r%3a4,0>7>0I0 %),F:?:,C6>7?B

a.,!"*$#,!,!"!'# r%NF68.,$"!%

8>?,O,L:;ID,.,!"++

%!-+! %+-+( %(-#& *%-&+ +!-(! +(-)& 15 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting general scaling function using B-form spline functions Fitting general scaling function using B-form spline functions

MN

! !"# $ $"# % %"# $! $# %! %#

&'(&)*+

*

&),-./0)1 &/2-324*56&378&978:&;<67-=&>ï324*&&0/?./6=2-&/:5--=61

& &

&;<67-=&378&978:&>ï324*&.;7-@&#&A-28; B&/.C7/&>ï;<67-=&C5;=&3.-/872-&*58/:=D&7-&E&7-8=4F56; B&378&<545*=8=4;&978:&+*($Ga&:=6D&37'=D&7-&378 %H&D585&<27-8;

a(&!"II&!&!"$I r%JD23(&!"KK

378&L&F56.=&(&!"KK

%!'E! %E'EM %M'#B I%'BE E!'M! EM'HB 16 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting general scaling function using B-form spline functions Fitting general scaling function using B-form spline functions

Fπ

! !"# $ $"# % %"# ! !"% !"& !"' !"( $ $"% $"& $"' $"( %

)*+),-.

• ),//0,1

234536-78)/99)5:;)<:;=)>?8:4@)A7>@B)Cï536-))0//):4)DEFE)2=744@81

) )

)>?8:4@)5:;)<:;=)Cï536-)G>:4H)I):4;@6478)J43;> %K)B7;7)?3:4;>

a+)!"%I)!)!"!%)L r%MB35)+)("!#

%!*&! %&*&( %(*#' I%*'& &!*(! &(*K' 17 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting general scaling function using B-form spline functions Fitting general scaling function using B-form spline functions

Summary Quantity γ χ2/dof Mπ 0.405(21) 1.47 Mρ 0.315(75) 1.02 MN 0.33(13) 0.77 Fπ 0.23(2) 8.05 Mπ improved as expected Tension of γ across channels decreased, but still problematic Fit to Fπ remains unacceptable

18 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting general scaling function using B-form spline functions Fitting general scaling function using B-form spline functions

Summary Quantity γ χ2/dof Mπ 0.405(21) 1.47 Mρ 0.315(75) 1.02 MN 0.33(13) 0.77 Fπ 0.23(2) 8.05 Mπ improved as expected Tension of γ across channels decreased, but still problematic Fit to Fπ remains unacceptable

18 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting general scaling function using B-form spline functions Fitting general scaling function using B-form spline functions

Summary Quantity γ χ2/dof Mπ 0.405(21) 1.47 Mρ 0.315(75) 1.02 MN 0.33(13) 0.77 Fπ 0.23(2) 8.05 Mπ improved as expected Tension of γ across channels decreased, but still problematic Fit to Fπ remains unacceptable

18 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Fitting general scaling function using B-form spline functions Fitting general scaling function using B-form spline functions

Summary Quantity γ χ2/dof Mπ 0.405(21) 1.47 Mρ 0.315(75) 1.02 MN 0.33(13) 0.77 Fπ 0.23(2) 8.05 Mπ improved as expected Tension of γ across channels decreased, but still problematic Fit to Fπ remains unacceptable

18 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conclusion Conclusion

The extended study of the spectroscopy of Nf = 12 Fundamental shows problems in the conformal scenario, consistent with our previous findings Outstanding Problem:

Scaling violation effect may fake scaling with inconsistent gamma values, but unlikely to be the only effect ⇒ Leading scaling violation effects has to be analyzed Can the strongly squeezed wave function effects in chirally broken scenario explain the good scaling form in separate quantum number channels?

Bottom line:

Conformal scenario remains inconsistent and suspect in our analysis The issues are not settled yet. More definitive analyses are needed and ongoing

19 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conclusion Conclusion

The extended study of the spectroscopy of Nf = 12 Fundamental shows problems in the conformal scenario, consistent with our previous findings Outstanding Problem:

Scaling violation effect may fake scaling with inconsistent gamma values, but unlikely to be the only effect ⇒ Leading scaling violation effects has to be analyzed Can the strongly squeezed wave function effects in chirally broken scenario explain the good scaling form in separate quantum number channels?

Bottom line:

Conformal scenario remains inconsistent and suspect in our analysis The issues are not settled yet. More definitive analyses are needed and ongoing

19 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conclusion Conclusion

The extended study of the spectroscopy of Nf = 12 Fundamental shows problems in the conformal scenario, consistent with our previous findings Outstanding Problem:

Scaling violation effect may fake scaling with inconsistent gamma values, but unlikely to be the only effect ⇒ Leading scaling violation effects has to be analyzed Can the strongly squeezed wave function effects in chirally broken scenario explain the good scaling form in separate quantum number channels?

Bottom line:

Conformal scenario remains inconsistent and suspect in our analysis The issues are not settled yet. More definitive analyses are needed and ongoing

19 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conclusion Conclusion

The extended study of the spectroscopy of Nf = 12 Fundamental shows problems in the conformal scenario, consistent with our previous findings Outstanding Problem:

Scaling violation effect may fake scaling with inconsistent gamma values, but unlikely to be the only effect ⇒ Leading scaling violation effects has to be analyzed Can the strongly squeezed wave function effects in chirally broken scenario explain the good scaling form in separate quantum number channels?

Bottom line:

Conformal scenario remains inconsistent and suspect in our analysis The issues are not settled yet. More definitive analyses are needed and ongoing

19 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conclusion Conclusion

The extended study of the spectroscopy of Nf = 12 Fundamental shows problems in the conformal scenario, consistent with our previous findings Outstanding Problem:

Scaling violation effect may fake scaling with inconsistent gamma values, but unlikely to be the only effect ⇒ Leading scaling violation effects has to be analyzed Can the strongly squeezed wave function effects in chirally broken scenario explain the good scaling form in separate quantum number channels?

Bottom line:

Conformal scenario remains inconsistent and suspect in our analysis The issues are not settled yet. More definitive analyses are needed and ongoing

19 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conclusion Conclusion

The extended study of the spectroscopy of Nf = 12 Fundamental shows problems in the conformal scenario, consistent with our previous findings Outstanding Problem:

Scaling violation effect may fake scaling with inconsistent gamma values, but unlikely to be the only effect ⇒ Leading scaling violation effects has to be analyzed Can the strongly squeezed wave function effects in chirally broken scenario explain the good scaling form in separate quantum number channels?

Bottom line:

Conformal scenario remains inconsistent and suspect in our analysis The issues are not settled yet. More definitive analyses are needed and ongoing

19 / 19

Conformal Finite Size Scaling of Twelve Fermion Flavors Chik Him Wong Outline Background

Conformality Controversy Simulation Setup

Spectroscopy Analysis Fitting under conformal hypothesis with FSS

Fitting scaling function with physical model Fitting general scaling function using B-form spline functions

Conclusion

Conclusion Conclusion

The extended study of the spectroscopy of Nf = 12 Fundamental shows problems in the conformal scenario, consistent with our previous findings Outstanding Problem:

Scaling violation effect may fake scaling with inconsistent gamma values, but unlikely to be the only effect ⇒ Leading scaling violation effects has to be analyzed Can the strongly squeezed wave function effects in chirally broken scenario explain the good scaling form in separate quantum number channels?

Bottom line:

Conformal scenario remains inconsistent and suspect in our analysis The issues are not settled yet. More definitive analyses are needed and ongoing

19 / 19