Walking, the dilaton, and complex CFT (II) Walking, the dilaton, and complex CFT (II) Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Implications Chik Him (Ricky) Wong Lattice Higgs Collaboration (L at HC) LATTICE 2019 1 / 18

Outline Outline Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Outline Introduction Implicit Maximum Likelihood Estimate Results and Dilaton Effective Field Theory Implications Implicit Maximum Liklihood Estimate Conclusion 2 / 18

Dilaton Effective Field Theory Dilaton Effective Field Theory Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Sextet model: SU(3) N f = 2 fermions in two-index symmetric Wong representation Outline Introduction Near conformal window ⇒ candidate of walking theory Implicit Maximum Emergent 0 ++ scalar gets lighter as conformal window is Likelihood Estimate approached Results and Implications In the pion mass range of our simulation, the scalar mass is comparable with the pions’ ⇒ χ PT cannot work properly Previous attempt of χ PT analysis: Inconsistent values of f π are obtained from M π and F π =3.20 n f =2 sextet model F NLO "chiral log fit" =3.20 n f =2 sextet model M 2 NLO "chiral log fit" 0.1 0.03 fitted M 2 = 2B m [ 1+ ( 2B m/(n f 16 2 f 2 ) log(2B m/ 3 ) ] 0.09 not fitted =3.20 NLO chiral log fit 2 f 2 ) log(2mB / 2 ) ] F = f [ 1 - (n f 2mB /32 0.025 0.08 4 B = 3.562 0.085 B = 3.34 0.20 0.07 f = 0.0292 0.0042 f = 0.01252 0.0016 0.02 3 = 0.333 0.038 4 = 0.479 0.055 0.06 a 2 M 2 2 /dof= 0.27 F 0.05 2 /dof= 0.31 0.015 0.04 0.03 0.01 0.02 0.005 0.01 FSS inputs: 32 3 64 to 56 3 96 volume range m fit range: 0.0015 - 0.004 0 0 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 m 10 -3 10 -3 a m 3 / 18

Dilaton Effective Field Theory Dilaton Effective Field Theory Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong σ model: masses of σ and π comparable ⇒ Non-linear σ model Outline may work, but not linear Introduction Dilaton hypothesis: The scalar acts as a dilaton from scale Implicit Maximum Likelihood symmetry breaking Estimate Lagrangian of Dilaton Effective Field Theory: Results and Implications L = 1 2 ∂ µ χ ∂ µ χ − V ( χ ) � χ � χ � 2 � y + f 2 + f 2 π m 2 � ∂ µ Σ † ∂ µ Σ � � Σ + Σ † � π π Tr Tr 4 f d 4 f d y = 3 − γ , γ : mass anomalous dimension χ ( x ) = f d e σ ( x ) / f d , σ ( x ) : Dilaton field Σ = e i π a τ a / f π : τ a = Pauli matrices, m 2 π = 2 B π m 4 / 18

Dilaton Effective Field Theory Dilaton Effective Field Theory Walking, the dilaton, and complex CFT (II) In this talk, we fit our data against two representative forms of V Chik Him (Ricky) and compare the corresponding results Wong Deformation of CFT parametrically Outline [e.g. M. Golterman and Y. Shamir,Phys. Rev. D94 (2016) 054502] Introduction Implicit Maximum Likelihood m 2 � 4ln χ � Estimate d χ 4 V d = − 1 16 f 2 f d Results and d Implications 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 ] V σ = m 2 � 2 � χ 2 − f 2 d d 8 f 2 d 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 ∼ χ 4 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] ) 5 / 18

Walking, the Tree Level predictions: dilaton, and complex CFT (II) V-independent scaling relation : Chik Him (Ricky) π F γ − 1 − 2 B π f γ − 1 Wong M 2 m = 0 π π Outline Introduction Expanding: � χ � χ Implicit Maximum � y � y f 2 π m 2 = n f m 2 π f 2 Likelihood � Σ + Σ † � π π + ... Tr Estimate 4 f d 2 f d Results and One can define: Implications W ( χ ) ≡ V ( χ ) − ( n f m 2 π f 2 π / 2 )( χ / f d ) y , W ′ ( F d ) = 0 , W ′′ ( F d ) = M 2 d W ′ d ( χ = F d ) = 0 : B π m ( m d / f π ) − 2 ( f d / f π ) − 2 = 0 F γ + 1 ln ( F π / f π ) − ( 3 − γ ) n f f γ − 1 π π W ′′ d ( χ = F d ) = M 2 d : π ) ( 3 ln ( F π / f π )+ 1 ) ( m d / f π ) 2 − 2 ( M 2 π ) − ( 3 − γ ) ( 2 − γ ) n f ( f d / f π ) − 2 = 0 2 ( F 2 π / M 2 d / M 2 W ′ σ ( χ = F d ) = 0 : B π m ( m d / f π ) − 2 ( f d / f π ) − 2 = 0 F γ + 1 π ) − 2 ( 3 − γ ) n f f γ − 1 ( 1 − f 2 π / F 2 π π W ′′ σ ( χ = F d ) = M 2 d : ( m d / f π ) 2 − 2 M 2 π − ( 3 − γ ) ( 2 − γ ) n f ( f d / f π ) − 2 = 0 � � 3 F 2 π / M 2 π − f 2 π / M 2 d / M 2 π 6 / 18

Implicit Maximum Likelihood Estimate Implicit Maximum Likelihood Estimate Walking, the dilaton, and complex CFT (II) From numerical data, we estimate M data , F data , M data in the infinite Chik Him (Ricky) π π d Wong volume limit at each m . Then minimize the following χ 2 Outline Introduction � 2 � 2 � 2 � M data � F data � M data ( m ) − M π ( m ) ( m ) − F π ( m ) ( m ) − M d ( m ) χ 2 = ∑ π π d + + Implicit Maximum ∆ M π ( m ) ∆ F π ( m ) ∆ M d ( m ) Likelihood m Estimate ∆ M π , ∆ F π , ∆ M d : variances Results and Implications Subject to the constraints: π F γ − 1 − 2 B π f γ − 1 M 2 m = 0 π π B π m ( m d / f π ) − 2 ( f d / f π ) − 2 = 0 F γ + 1 ln ( F π / f π ) − ( 3 − γ ) n f f γ − 1 V = V d π π π ) ( 3 ln ( F π / f π )+ 1 ) ( m d / f π ) 2 − 2 ( M 2 π ) − ( 3 − γ ) ( 2 − γ ) n f ( f d / f π ) − 2 = 0 2 ( F 2 π / M 2 d / M 2 or π F γ − 1 − 2 B π f γ − 1 M 2 m = 0 π π B π m ( m d / f π ) − 2 ( f d / f π ) − 2 = 0 F γ + 1 π ) − 2 ( 3 − γ ) n f f γ − 1 V = V σ ( 1 − f 2 π / F 2 π π ( m d / f π ) 2 − 2 M 2 π − ( 3 − γ ) ( 2 − γ ) n f ( f d / f π ) − 2 = 0 � 3 F 2 π / M 2 π − f 2 π / M 2 � d / M 2 π Fitted parameters: f π , B π , γ , m d / f π , f d / f π 7 / 18

Implicit Maximum Likelihood Estimate Implicit Maximum Likelihood Estimate Walking, the dilaton, and complex CFT (II) M data and F data values are defined in infinite volume limit, while π π Chik Him (Ricky) simulations are done in finite volumes ⇒ Extrapolation into infinite Wong volume limit is required Outline Ansatz: Introduction M π ( L ) = M π ( L → ∞ )+ cM g 1 ( M π L , η = Nt / NL ) , F π ( L ) = F π ( L → ∞ )+ cF g 1 ( M π L , η = Nt / NL ) Implicit Maximum N f =2 Sextet, β =3.20, m=0.0015 N f =2 Sextet, β =3.20, m=0.0015 Likelihood 0.11 0.0285 Estimate M π (L) = M π + c M g 1 (M π L), 0.028 Results and 0.108 F π (L) = F π + c F g 1 (M π L) Implications 0.0275 a M π = 0.09838(26), a c M = 0.0120(92) 0.106 0.027 a F π = 0.02842(17), a c F = -0.0160(40) 2 /DoF = 0.595 χ 0.0265 a M π a F π 0.104 M π (L) = M π + c M g 1 (M π L), 0.026 F π (L) = F π + c F g 1 (M π L) 0.102 0.0255 a M π = 0.09838(26), a c M = 0.0120(92) a F π = 0.02842(17), a c F = -0.0160(40) 0.025 2 /DoF = 0.595 0.1 χ 0.0245 0.098 0.024 30 40 50 60 70 35 40 45 50 55 60 L/a L/a In the presence of light dilaton M d ∼ M π , the particle being exchanged can also be dilaton. However, the combined correction term would still be g 1 M d is noisier ⇒ FSS of M d is more difficult, M d at largest volume is used instead 8 / 18

Implicit Maximum Likelihood Estimate Implicit Maximum Likelihood Estimate Walking, the dilaton, and complex CFT (II) Chik Him (Ricky) Wong Markov Chain Monte Carlo (MCMC) Outline Using the means and covariances of { M π ( m ) , F π ( m ) } , and the means Introduction and variances of M d ( m ) , the posterior distributions of Implicit Maximum Likelihood { M π , F π , M d } ( m ) are generated with MCMC Estimate Results and M π ) 2 P ( { M π , F π , M d } ) | m ∼ exp ( − 1 / 2 (((( M π − ¯ Implications +( F π − ¯ F π ) 2 ) / Σ ( M , F )+( M d − ¯ M d ) 2 / ∆ M d )) | 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 9 / 18

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