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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Walking, the dilaton, and complex CFT (II) Walking, the dilaton, and complex CFT (II)

Chik Him (Ricky) Wong Lattice Higgs Collaboration (LatHC) LATTICE 2019

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Outline Outline

Dilaton Effective Field Theory Implicit Maximum Liklihood Estimate Conclusion

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Dilaton Effective Field Theory Dilaton Effective Field Theory

Sextet model: SU(3) Nf = 2 fermions in two-index symmetric representation Near conformal window ⇒ candidate of walking theory Emergent 0++ scalar gets lighter as conformal window is approached In the pion mass range of our simulation, the scalar mass is comparable with the pions’ ⇒ χPT cannot work properly Previous attempt of χPT analysis: Inconsistent values of fπ are obtained from Mπ and Fπ

1 2 3 4 5 6

m

10-3 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 F =3.20 nf=2 sextet model F NLO "chiral log fit" =3.20 NLO chiral log fit F = f [ 1 - (nf 2mB /32

2 f2) log(2mB / 4 2) ]

B = 3.34 0.20 f = 0.01252 0.0016

4 = 0.479

0.055 2/dof= 0.31 FSS inputs: 323 64 to 56 3 96 volume range m fit range: 0.0015 - 0.004 fitted not fitted

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

a m

10-3 0.005 0.01 0.015 0.02 0.025 0.03

a2 M2 =3.20 nf=2 sextet model M 2 NLO "chiral log fit" M2 = 2B m [ 1+ ( 2B m/(nf16 2 f2) log(2B m/

3) ]

B = 3.562 0.085 f = 0.0292 0.0042

3 = 0.333

0.038 2/dof= 0.27

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Dilaton Effective Field Theory Dilaton Effective Field Theory

σ model: masses of σ and π comparable ⇒ Non-linear σ model may work, but not linear Dilaton hypothesis: The scalar acts as a dilaton from scale symmetry breaking Lagrangian of Dilaton Effective Field Theory:

L =1 2 ∂µχ ∂µχ −V(χ) + f 2

π

4 χ fd 2 Tr

• ∂µ Σ† ∂µΣ
• + f 2

π m2 π

4 χ fd y Tr

• Σ+Σ†

y = 3−γ, γ : mass anomalous dimension χ(x) = fd e σ(x) /fd, σ(x) : Dilaton field Σ = e i πa τa /fπ : τa = Pauli matrices, m2

π = 2Bπm

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Dilaton Effective Field Theory Dilaton Effective Field Theory

In this talk, we fit our data against two representative forms of V and compare the corresponding results

Deformation of CFT parametrically

[e.g. M. Golterman and Y. Shamir,Phys. Rev. D94 (2016) 054502]

Vd = m2

d

16 f 2

d

χ4

• 4ln χ

fd −1

• Linear-σ model inspired potential

[T. Appelquist et al, JHEP 07 (2017) 035; T. Appelquist et al, JHEP 07 (2018) 039;W. D. Goldberger et al, Phys. Rev. Lett. 100 (2008) 111802 ]

Vσ = m2

d

8 f 2

d

• χ2 −f 2

d

2

Previous similar attempt of analysis with our data by another group

[T. Appelquist et al, JHEP 03 (2018) 039] only fitted against the assumption of V ∼ χ4

asymptotically and was not comprehensive We are fitting our own data against the two explicit forms of V We investigate with implicit maximum likelihood analysis ( different from [Z. Fodor et al, PoS LATTICE2018 (2019) 196] )

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Tree Level predictions: V-independent scaling relation : M2

π F γ−1 π

−2Bπ f γ−1

π

m = 0 Expanding: f 2

π m2 π

4 χ fd y Tr

• Σ+Σ†

= nf m2

π f 2 π

2 χ fd y +... One can define: W(χ) ≡ V(χ)−(nf m2

π f 2 π / 2)(χ / fd)y, W′(Fd) = 0, W′′(Fd) = M2 d

W′

d( χ = Fd ) = 0 :

F γ+1

π

ln(Fπ/fπ)−(3−γ) nf f γ−1

π

Bπ m (md/fπ)−2 (fd/fπ)−2 = 0 W′′

d ( χ = Fd ) = M2 d :

2 (F2

π/M2 π) (3 ln(Fπ/fπ)+1) (md/fπ)2 −2 (M2 d/M2 π)− (3−γ) (2−γ) nf (fd/fπ)−2 = 0

W′

σ( χ = Fd ) = 0 :

F γ+1

π

(1−f 2

π /F2 π)−2 (3−γ) nf f γ−1 π

Bπ m (md/fπ)−2 (fd/fπ)−2 = 0 W′′

σ( χ = Fd ) = M2 d :

• 3F2

π/M2 π −f 2 π /M2 π

• (md/fπ)2 −2 M2

d/M2 π −(3−γ) (2−γ) nf (fd/fπ)−2 = 0

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Implicit Maximum Likelihood Estimate Implicit Maximum Likelihood Estimate

From numerical data, we estimate Mdata

π

, Fdata

π

, Mdata

d

in the infinite volume limit at each m. Then minimize the following χ2

χ2 = ∑

m

• Mdata

π

(m)−Mπ(m) 2 ∆Mπ (m) +

• Fdata

π

(m)−Fπ(m) 2 ∆Fπ (m) +

• Mdata

d

(m)−Md(m) 2 ∆Md(m) ∆Mπ ,∆Fπ ,∆Md: variances

Subject to the constraints:

V = Vd        M2

π F γ−1 π

−2Bπ f γ−1

π

m = 0 F γ+1

π

ln(Fπ/fπ)−(3−γ) nf f γ−1

π

Bπ m (md/fπ)−2 (fd/fπ)−2 = 0 2 (F2

π/M2 π) (3 ln(Fπ/fπ)+1) (md/fπ)2 −2 (M2 d/M2 π)− (3−γ) (2−γ) nf (fd/fπ)−2 = 0

• r

V = Vσ        M2

π F γ−1 π

−2Bπ f γ−1

π

m = 0 F γ+1

π

(1−f 2

π /F2 π)−2 (3−γ) nf f γ−1 π

Bπ m (md/fπ)−2 (fd/fπ)−2 = 0

• 3F2

π/M2 π −f 2 π /M2 π

• (md/fπ)2 −2 M2

d/M2 π −(3−γ) (2−γ) nf (fd/fπ)−2 = 0

Fitted parameters: fπ, Bπ, γ, md/fπ, fd/fπ

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Implicit Maximum Likelihood Estimate Implicit Maximum Likelihood Estimate

Mdata

π

and Fdata

π

values are defined in infinite volume limit, while simulations are done in finite volumes ⇒ Extrapolation into infinite volume limit is required Ansatz:

Mπ (L) = Mπ (L → ∞)+cM g1(Mπ L,η = Nt/NL), Fπ (L) = Fπ (L → ∞)+cF g1(Mπ L,η = Nt/NL)

30 40 50 60 70 L/a 0.098 0.1 0.102 0.104 0.106 0.108 0.11 a Mπ Nf=2 Sextet, β=3.20, m=0.0015 Mπ(L) = Mπ + cM g1(Mπ L), Fπ(L) = Fπ + cF g1(Mπ L) a Mπ = 0.09838(26), a cM = 0.0120(92) a Fπ = 0.02842(17), a cF = -0.0160(40) χ

2/DoF = 0.595

35 40 45 50 55 60 L/a 0.024 0.0245 0.025 0.0255 0.026 0.0265 0.027 0.0275 0.028 0.0285 a Fπ Nf=2 Sextet, β=3.20, m=0.0015 Mπ(L) = Mπ + cM g1(Mπ L), Fπ(L) = Fπ + cF g1(Mπ L) a Mπ = 0.09838(26), a cM = 0.0120(92) a Fπ = 0.02842(17), a cF = -0.0160(40) χ

2/DoF = 0.595

In the presence of light dilaton Md ∼ Mπ, the particle being exchanged can also be dilaton. However, the combined correction term would still be g1 Md is noisier ⇒ FSS of Md is more difficult, Md at largest volume is used instead

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Implicit Maximum Likelihood Estimate Implicit Maximum Likelihood Estimate

Markov Chain Monte Carlo (MCMC)

Using the means and covariances of {Mπ(m),Fπ(m)}, and the means and variances of Md(m), the posterior distributions of {Mπ,Fπ,Md}(m) are generated with MCMC P({Mπ,Fπ,Md})|m ∼exp(−1/2((((Mπ − ¯ Mπ)2 +(Fπ − ¯ Fπ)2)/Σ(M,F)+(Md − ¯ Md)2/∆Md))|m Markov Chain Monte Carlo algorithms sample a desired distribution by constructing a Markov Chain with such distribution as the equilibrium distribution A simple implementation is the familiar Metropolis-Hastings algorithm using a Gaussian proposal density Using the posterior distributions as input, one can obtain the Implicit Maximum Likelihood estimate of the parameters

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Implicit Maximum Likelihood Estimate Implicit Maximum Likelihood Estimate

A sample {Mdraw

π

,Fdraw

π

,Mdraw

d

} from the posterior distribution at each m is drawn, then the following is minimized:

χ2 = ∑

m

• Mdraw

π

(m)−Mπ(m) 2 ∆Mπ (m) +

• Fdraw

π

(m)−Fπ(m) 2 ∆Fπ (m) +

• Mdraw

d

(m)−Md(m) 2 ∆Md(m) ∆Mπ ,∆Fπ ,∆Md: variances of the posterior distribution

Subject to the constraints:

V = Vd        M2

π F γ−1 π

−2Bπ f γ−1

π

m = 0 F γ+1

π

ln(Fπ/fπ)−(3−γ) nf f γ−1

π

Bπ m (md/fπ)−2 (fd/fπ)−2 = 0 2 (F2

π/M2 π) (3 ln(Fπ/fπ)+1) (md/fπ)2 −2 (M2 d/M2 π)− (3−γ) (2−γ) nf (fd/fπ)−2 = 0

• r

V = Vσ        M2

π F γ−1 π

−2Bπ f γ−1

π

m = 0 F γ+1

π

(1−f 2

π /F2 π)−2 (3−γ) nf f γ−1 π

Bπ m (md/fπ)−2 (fd/fπ)−2 = 0

• 3F2

π/M2 π −f 2 π /M2 π

• (md/fπ)2 −2 M2

d/M2 π −(3−γ) (2−γ) nf (fd/fπ)−2 = 0

Fitted parameters: fπ, Bπ, γ, md/fπ, fd/fπ

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Results and Implications Results and Implications

Tree-level Symanzik-Improved gauge action with 2-step ρ = 0.15 stout-smeared Staggered Nf = 2 SU(3) Sextet fermion In order to study the taste breaking effects, we have dataset of β = 3.20,3.25,3.30 As a pilot study, we present here only β = 3.20, using m = 0.0015,0.002,0.003,0.004

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Results and Implications Results and Implications

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

f fit parameter

10-3 50 100 150 200 250 300 350

synthetic MC distribution = 3.20 rep6 (sextet) nf=2 Vd potential f = 0.00300 0.00068

0.012 0.0125 0.013 0.0135 0.014 0.0145 0.015 0.0155 0.016

f fit parameter

50 100 150 200 250 300 350

synthetic MC distribution = 3.20 rep6 (sextet) nf=2 V potential f = 0.01415 0.00056

Vd : fπ is very low ⇒ validity of tree-level calculation is questionable Vσ : fπ Similar to previous χ PT fits

1 2 3 4 5 6

m

10-3 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 F =3.20 nf=2 sextet model F NLO "chiral log fit" =3.20 NLO chiral log fit F = f [ 1 - (nf 2mB /32

2 f2) log(2mB / 4 2) ]

B = 3.34 0.20 f = 0.01252 0.0016

4 = 0.479

0.055 2/dof= 0.31 FSS inputs: 323 64 to 56 3 96 volume range m fit range: 0.0015 - 0.004 fitted not fitted

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

a m

10-3 0.005 0.01 0.015 0.02 0.025 0.03

a2 M2 =3.20 nf=2 sextet model M 2 NLO "chiral log fit" M2 = 2B m [ 1+ ( 2B m/(nf16 2 f2) log(2B m/

3) ]

B = 3.562 0.085 f = 0.0292 0.0042

3 = 0.333

0.038 2/dof= 0.27

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Results and Implications Results and Implications

3.5 4 4.5 5 5.5 6

B parameter

50 100 150 200 250 300 350 400 450 500

sythetic MC distribution = 3.20 rep6 (sextet) nf=2 Vd potential B = 4.57 0.22

3.45 3.5 3.55 3.6 3.65 3.7 3.75

B fit parameter

50 100 150 200 250 300 350

sythetic MC distribution = 3.20 rep6 (sextet) nf=2 V potential B = 3.587 0.043

Vd : Bπ Higher than previous χPT fits Vσ : Bπ Consistent with previous χPT fits

1 2 3 4 5 6

m

10-3 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 F =3.20 nf=2 sextet model F NLO "chiral log fit" =3.20 NLO chiral log fit F = f [ 1 - (nf 2mB /32

2 f2) log(2mB / 4 2) ]

B = 3.34 0.20 f = 0.01252 0.0016

4 = 0.479

0.055 2/dof= 0.31 FSS inputs: 323 64 to 56 3 96 volume range m fit range: 0.0015 - 0.004 fitted not fitted

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

a m

10-3 0.005 0.01 0.015 0.02 0.025 0.03

a2 M2 =3.20 nf=2 sextet model M 2 NLO "chiral log fit" M2 = 2B m [ 1+ ( 2B m/(nf16 2 f2) log(2B m/

3) ]

B = 3.562 0.085 f = 0.0292 0.0042

3 = 0.333

0.038 2/dof= 0.27

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Results and Implications Results and Implications

1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2

fit parameter

50 100 150 200 250 300 350

sythetic MC distribution = 3.20 rep6 (sextet) nf=2 Vd potential = 1.150 0.011

1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2

fit parameter

50 100 150 200 250 300 350

sythetic MC distribution = 3.20 rep6 (sextet) nf=2 V potential = 1.151 0.011

Both Vd and Vσ give the same value of γ Further study needed: Which scale does this γ correspond to?

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Results and Implications Results and Implications

0.7 0.8 0.9 1 1.1 1.2 1.3

md/f fit parameter

50 100 150 200 250 300

sythetic MC distribution = 3.20 rep6 (sextet) nf=2 Vd potential md/f = 1.005 0.100

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

md/f parameter

50 100 150 200 250 300 350

sythetic MC distribution = 3.20 rep6 (sextet) nf=2 V potential md/f = 2.36 0.21

md/fπ is more sensitive to the Md(m) input which is of less control Vd : md/fπ dramatically lower than previous extrapolated results Vσ : md/fπ Closer to previous estimations

0.01 0.02 0.03 0.04 0.05 Mπ

2

2 4 6 8 10 12 14 M / Fπ a0 π f0 β=3.25 0.01 0.02 0.03 0.04 0.05 Mπ

2

2 4 6 8 10 12 14 M / Fπ 0.5 1 1.5 2 2.5 3 M / TeV N a1 ρ β=3.20 Decreasing Mπ Decreasing Mπ

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Results and Implications Results and Implications

2 2.5 3 3.5 4

fd/f parameter

50 100 150 200 250 300

sythetic MC distribution = 3.20 rep6 (sextet) nf=2 Vd potential fd/f = 3.15 0.29

2.5 3 3.5 4 4.5

fd/f fit parameter

50 100 150 200 250 300 350

sythetic MC distribution = 3.20 rep6 (sextet) nf=2 V potential fd/f = 3.26 0.31

Both Vd and Vσ give similar fd/fπ ∼ 3 >> 1 ⇒ phenomenological difficulties for potential BSM applications

[e.g. J. Ellis and T. You JHEP06(2012)140] 16 / 18

Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Results and Implications Results and Implications

Similar situation for Fund Nf = 8 model, at β = 4.8 (Data are taken from LSD collaboration [T. Appelquist et al, Phys. Rev. D 99, 014509]) Vd Vσ fπ 0.000947(24) 0.01082(36) Bπ 1.833(64) 2.413(30) γ 0.891(11) 0.897(10) md/fπ 1.821(73) 5.41(13) fd/fπ 3.424(86) 3.315(81)

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Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications

Conclusion Conclusion

Sextet Nf = 2, β = 3.20 Fund Nf = 8, β = 4.8 Vd Vσ Vd Vσ fπ 0.00300(68) 0.01415(56) 0.000947(24) 0.01082(36) Bπ 4.57(22) 3.587(43) 1.833(64) 2.413(30) γ 1.150(11) 1.151(11) 0.891(11) 0.897(10) md/fπ 1.005(100) 2.36(21) 1.821(73) 5.41(13) fd/fπ 3.15(29) 3.26(31) 3.424(86) 3.315(81) The sextet model and fund Nf = 8 model are analyzed under the dilaton hypothesis with two typical dilaton potentials The linear-σ model inspired potential form Vσ seems to fit the data well, while Vd does not seem to work fd/fπ and md/fπ are particularly important but more sensitive to the noisy Md inputs A full analysis in which Md inputs are better controlled and taste breaking effects are taken into account with different β’s, would shed new light on crucial properties of the models.

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