Hiroshi Ohki KMI, Nagoya University ! Y. Aoki, T. Aoyama, E. - PowerPoint PPT Presentation

Lattice study of the scalar and baryon spectra in many flavor QCD Hiroshi Ohki KMI, Nagoya University ! Y. Aoki, T. Aoyama, E. Bennett, M. Kurachi, T. Maskawa, K. Miura, K.-i. Nagai, E. Rinaldi, A. Shibata, K. Yamawaki, T. Yamazaki (LatKMI collaboration) SCGT15 @SCGT15

Studies in LatKMI for strong coupling gauge theory • Lattice study of the SU(3) gauge theory with Nf fundamental fermions • all calculations are done with same set-up: Highly Improved Staggered Quark (HISQ) type action with Nf=4*n • Nf=(4),8,(12), generic hadron spectrum properties → Y. Aoki (talk, yesterday) • Nf=8 spectrum of Dirac operator and topology → K. Nagai (talk, yesterday) • Nf=8 scalar and baryon for Dark Matter → this talk

Outline • Introduction • Scalar analysis mass & decay constant • Baryon analysis • Summary

Introduction

“Discovery+of+Higgs+boson” • Higgs like particle (125 GeV) has been found at LHC. • Consistent with the Standard Model Higgs. But true nature is so far unknown. • Many candidates for beyond the SM one interesting possibility – (walking) technicolor • “Higgs” = dilaton (pNGB) due to breaking of the approximate scale invariance Nf=8 QCD could be a candidate of walking gauge theory. We find the flavor singlet scalar ( σ ) is as light as pion. It may be identified a techni-dilaton (Higgs in the SM), which is a pseudo-Nambu Goldstone boson. (LatKMI, Phys. Rev. D 89, 111502(R), arXiv: 1403.5000[hep-lat].)

Dilaton decay constant It is important to investigate the decay constant of the flavor singlet scalar as well as mass, which is useful to study LHC phenomena; the techni-dilaton decay constant governs all the scale of couplings between Higgs and other SM particles. F σ : dilaton decay constant z, w g g σ W W = v EW σ (dilaton) g h SM W W F σ z, w b, τ , … g σ ff = (3 − γ ∗ ) v EW g σ ff σ g h SM ff F σ b, τ , … Dilaton effective theory analysis [S. Matsuzaki, K. Yamawaki, PRD86, 039525(2012)]

Lattice calculation of flavor-singlet scalar mass

Flavor singlet scalar from fermion bilinear operator O S ( t ) ≡ ¯ D ( t ) = � O S ( t ) O S (0) � � � O S ( t ) �� O S (0) � ψ i ψ i ( t ) , Staggered fermion case • Scalar interpolating operator can couple to two states of ! ( 1 ⊗ 1 ) & ( γ 4 γ 5 ⊗ ξ 4 ξ 5 ) ! ! C ± (2 t ) ≡ 2 C (2 t ) ± C (2 t + 1) ± C (2 t − 1) ! ! • Flavor singlet scalar can be evaluated with disconnected diagram. C σ (2 t ) = − C + (2 t ) + 2 D + (2 t ) (8 flavor) = 2 × (one staggered fermion)

N f =8 Result � � Same data as [LatKMI PRD2014] and Some updates

Simulation setup L 3 × T N cf [ N st ] m f ! SU(3), Nf=8 • 0.012 42 3 × 56 2300[2] ! 0.015 36 3 × 48 5400[2] • HISQ (staggered) fermion and tree level Symanzik gauge action 0.02 36 3 × 48 5000[1] ! 0.02 30 3 × 40 Volume (= L^3 x T) 8000[1] L =24, T=32 • 0.03 30 3 × 40 16500[1] L =30, T=40 • L =36, T=48 • 0.03 24 3 × 32 36000[2] L =42, T=56 • Bare coupling constant ( ) 0.04 30 3 × 40 12900[3] beta=3.8 • 0.04 24 3 × 32 50000[2] ! bare quark mass 0.04 18 3 × 24 9000[1] mf= 0.012-0.06, • (5 masses) 0.06 24 3 × 32 18000[1] ! • high statistics (more than 2,000 configurations) 0.06 18 3 × 24 9000[1] ! • We use a noise reduction technique for disconnected correlator. (use of Ward-Takahashi identity[Kilcup-Sharpe, ’87, Venkataraman-Kilcup ’97] )

correlator for Nf=8, beta=3.8, L=36, mf=0.015 0.0001 -C(t) 2D(t) 1e-05 1e-06 1e-07 1e-08 1e-09 0 4 8 12 16 20 t C σ (2 t ) = − C + (2 t ) + 2 D + (2 t )

m σ for Nf=8, beta=3.8, L=36, mf=0.015 (same figure as talk by Y. Aoki, yesterday) 0.5 2D + (t) - C + (t) 2D(t) 0.4 - C - (t) m � 0.3 m σ 0.2 0.1 0 0 4 8 12 16 20 t 2 D + ( t ) − C + ( t ) → A σ e − m σ 2 t D + ( t ) = A σ e − m σ 2 t + A a 0 e − m a 0 2 t → A σ e − m σ 2 t , (if m σ < m a 0 ) (in the continuum limit)

m σ for Nf=8, beta=3.8 (same figure as talk by Y. Aoki, yesterday) � L=42 � L=36 0.5 � L=30 � L=24 � L=18 0.4 � � (PV) m 0.3 0.2 0.1 0 0 0.01 0.02 0.03 0.04 0.05 0.06 m f σ is as light as π and clearly lighter than ρ

Scalar decay constant Preliminary

Two possible decay constants for σ (F σ and Fs) 1. F σ : Dilaton decay constant difficult to calculate � 0 |D µ ( x ) | σ ; p � = iF σ p µ e − ipx D µ : dilatation current can couple to the state of σ . � 0 | ∂ µ D µ (0) | σ ; 0 � = F σ m 2 Partially conserved dilatation current relation (PCDC): σ 2. Fs :scalar decay constant not so difficult N F ¯ We use scalar density operator � O ( x ) = ψ i ψ i ( x ) ! i =1 which can also couple to the state of σ . We denote this matrix element as scalar decay constant ! ! ! (Fs : RG-invariant quantity) We study Fs. We also discuss a relation between F σ and Fs later.

scalar decay constant from 2pt flavor singlet scalar correlator Insert the complete set (|n><n|) Asymptotic behavior (large t) of the scalar 2pt correlator C σ (t) NF: number of flavors V: L^3 A: amplitude

What is relation between Fs and F σ ?

A relation between Fs and F σ through the WT id. (in the continuum theory) the (integrated) WT-identity for dilatation transformation Useful relations (trace anomaly relation) (scale transformation) Taking the zero momentum limit (q → 0), (LHS) is zero. the WT-identity gives

Insert the complete set ! into ! N F and use a scalar density operator � ¯ O = m f ψψ i We obtain (in the dilaton pole dominance approximation) [Ref: Technidilaton (Bando, Matumoto, Yamawaki, PLB 178, 308-312)] ψψ N F m f � ¯ ψψ � F σ = � ∆ ¯ � 2 V Am σ Recall ∆ ¯ ψψ = 3 − γ m

ψψ N F m f � ¯ ψψ � F σ = � ∆ ¯ � 2 V Am σ ∆ ¯ ψψ = 3 − γ m (in the dilaton pole dominance approximation) c.f. PCAC relation The (integrated) chiral WT-identity tells us that using PCAC relation, this leads to π = � 4 m f � ¯ m 2 π F 2 ψψ � (GMOR relation) (in the pion pole dominance approximation)

N f =8 Result

Effective amplitude for Nf=8, beta=3.8 L=30, T=40, mf=0.02 0.001 2D + (t) - C + (t) 2D(t) 0.0001 A 1e-05 1e-06 1e-07 0 4 8 12 16 20 t C = A ( e − mt + e − m ( T − t ) )

F σ for Nf=8, beta=3.8 ψψ N F m f � ¯ ψψ � F σ = � ∆ ¯ � 2 V Am σ 0.075 L=42 L=36 Preliminary L=30 with � ¯ ψψ � � � ¯ ψψ � 0 L=24 L=18 chiral limit 0.05 F σ ∆ ¯ ψψ Chiral extrapolation fit 0.025 Blue (mf=0.012-0.03) F σ = c 0 + c 1 m f 0 Black (mf=0.015-0.04) 0 0.01 0.02 0.03 0.04 0.05 m f F σ F σ = c 0 + c 1 m f + c 2 m 2 in the chiral limit ∼ 1 . 5 ∆ ¯ f ψψ ∼ 3 F π with assumption of γ ∼ 1 , ( ∆ ¯ ψψ = 3 − γ ∼ 2) c.f. Another estimate via the scalar mass in the dilaton ChPT (DChPT). DChPT: m 2 σ ∼ d 0 + d 1 m 2 π F σ √ � N F = 2 2 d 1 = (1 + γ ) ∆ ¯ ∼ N F F 2 F π ψψ π ∼ 1 4 F 2 σ

Technibaryon Dark Matter

Technibaryon • The lightest baryon is stable due to the technibaryon number conservation ! • Good candidate of the dark matter (DM) ! • Boson or fermion? (depend on the #TC) our case: DM is fermion (#TC=3). ! • Direct detection of the dark matter is possible.

DM effective theory Technibaryon(B) interacts with quark(q), gluon in standard model µ ν G aµ ν + Bi ∂ µ γ ν B O µ ν + · · · L eff = c ¯ qq + c ¯ BBG a M ¯ 1 BB ¯ One of the dominant contributions in spin-independent interactions comes from the microscopic Higgs (technidilaton σ ) mediated process (below diagram) B : DM B : DM Technibaryon-scalar effective y ¯ BB σ coupling σ nucleon-scalar effective coupling y ¯ nn σ Nucleon Nucleon

(Techni)baryon Chiral perturbation theory with dilaton leading order of BChPT L = ¯ B ( i γ µ ∂ µ − m B + g A 2 γ 5 γ µ u µ ) B u µ = i u † ( ∂ µ − i r µ ) u − u ( ∂ µ − i l µ ) u † � � U = u 2 = e 2 π i/F π L = ¯ B ( i γ µ ∂ µ − e σ /F σ m B + g A 2 γ 5 γ µ u µ ) B χ = e σ /F σ The dilaton-baryon effective coupling (leading order) is uniquely determined as y ¯ BB σ = m B /F σ

DM Direct detection B B Spin-independent cross section with nucleus σ SI ( χ , N ) = M 2 σ π ( Zf p + ( A − Z ) f n ) 2 R Nucleon Nucleon = (3 − γ ∗ ) v EW g σ ff Note: Yukawa coupling is different from the SM : g h SM ff F σ | i f ( N ) ⌘ h N | m q ¯ qq | N i /m N Nucleon sigma term in QCD T q Nucleon matrix element non-perturbatively determined by lattice QCD calculation Lattice calculation for both nucleon and technibaryon interactions

An illustrative example of DM cross section 1e-38 F σ = 250 [GeV] 1e-40 F σ = 1 [TeV] 1e-42 F σ = 3 [TeV] 1e-44 1e-46 1e-48 [GeV] m B 100 1000 10000 Sample input values m σ = 125 [GeV] f ( p ) f ( n ) γ = 1 0.019(5) 0.013(3) T u T u f ( n ) f ( p ) 0.040(9) 0.027(6) T d T d f ( n ) f ( p ) 0.009(22) 0.009(22) T s T s Lattice calculation of the nucleon sigma term (fTq) Ref [R.D. Young, and A. W. Thomas,’10, HO et al. JLQCD ’13, ]

LatKMI result Baryon mass in Nf=8 QCD 0.8 0.6 m B M N 0.4 quad 0.012-0.04 linear 0.012-0.03 0.2 0 0 0.01 0.02 0.03 0.04 0.05 m f