Dilaton Chiral Perturbation Theory - Determining Mass and Decay Constant of Technidilaton on the Lattice

The Lagrangian is constructed uniquely (up to total derivatives) by the requirement that the action S ⁡ [ φ ⁡ ( x ) ] S[\varphi(x)] = = ∫ d 4 ​ x ​ ℒ ​ [ φ ⁡ ( x ) ] \int d^{4}x\,{\cal L}[\varphi(x)] be invariant under the scale (and also chiral) transformation δ ​ φ ​ ( x ) \delta\varphi(x) = = ( d φ + x μ ∂ μ ) φ ( x ) (d_{\varphi}+x_{\mu}\partial^{\mu})\varphi(x) , δ ​ x μ = − x μ \delta x_{\mu}=-x_{\mu} . This implies that δ ​ S ​ [ φ ⁡ ( x ) ] = ∫ ( δ ​ d 4 ​ x ​ ℒ + d 4 ​ x ​ δ ​ ℒ ) = ∫ d 4 ​ x ​ ( − 4 ​ ℒ + d ℒ ​ ℒ ) = 0 \delta S[\varphi(x)]=\int(\delta d^{4}x\,{\cal L}+d^{4}x\,\delta{\cal L})=\int d^{4}x(-4{\cal L}+d_{\cal L}{\cal L})=0 , namely the scale dimension of ℒ {\cal L} must be 4, d ℒ d_{\cal L} = = 4 4 . Eq. ( 2

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The scale-WT identity for the operator 𝒪 {\cal O} reads 0 0 = = lim q → 0 q μ ​ ∫ d 4 ​ x ​ e − i ​ q ​ x ​ ⟨ 0 | T ⁡ ( D μ ​ ( x ) ​ 𝒪 ​ ( 0 ) ) | 0 ⟩ \lim_{q\rightarrow 0}\,q_{\mu}\int d^{4}xe^{-iqx}\langle 0|T(D^{\mu}(x)\,{\cal O}(0))|0\rangle = = ⟨ 0 | [ i ​ Q D , 𝒪 ] | 0 ⟩ \langle 0|[iQ_{D},{\cal O}]|0\rangle + + i ​ ∫ d 4 ​ x ​ ⟨ 0 | T ⁡ ( ∂ μ D μ ​ ( x ) ​ 𝒪 ​ ( 0 ) ) | 0 ⟩ i\int d^{4}x\langle 0|T(\partial_{\mu}D^{\mu}(x)\,{\cal O}(0))|0\rangle = = d 𝒪 ​ ⟨ 𝒪 ⟩ − F ϕ ⋅ ⟨ ϕ | 𝒪 | 0 ⟩ d_{\cal O}\,\langle{\cal O}\rangle-F_{\phi}\cdot\langle\phi|{\cal O}|0\rangle , where [ i ​ Q D , 𝒪 ] = d 𝒪 ​ 𝒪 [iQ_{D},{\cal O}]=d_{\cal O}\,{\cal O} , with Q D = ∫ d 3 ​ x ​ D 0 ​ ( x ) Q_{D}=\int d^{3}xD^{0}(x) , and d 𝒪 d_{\cal O} is the scale dimension of the operator 𝒪 {\cal O} . Note that the second term comes from the ϕ \phi pole contribution. Thus we have d 𝒪 ​ ⟨ 0 | 𝒪 | 0 ⟩ = F ϕ ⋅ ⟨ 0 | 𝒪 | ϕ ⟩ d_{\cal O}\langle 0|{\cal O}|0\rangle=F_{\phi}\cdot\langle 0|{\cal O}|\phi\rangle

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The coefficient of the scale-invariant term χ 4 \chi^{4} is uniquely determined as in Eq.( 4

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Note that ⟨ 0 | θ μ μ | ϕ ⟩ m f ≠ 0 \langle 0|\theta_{\mu}^{\mu}|\phi\rangle_{m_{f}\neq 0} = = ⟨ 0 | ∂ μ D μ | ϕ ⟩ m f ≠ 0 \langle 0|\partial_{\mu}D^{\mu}|\phi\rangle_{m_{f}\neq 0} = = − F ϕ ​ M ϕ 2 -F_{\phi}M_{\phi}^{2} , ⟨ 0 | β NP ​ ( α ) 4 ​ α ​ G μ ​ ν 2 | ϕ ⟩ \langle 0|\frac{\beta_{\rm NP}(\alpha)}{4\alpha}G_{\mu\nu}^{2}|\phi\rangle = = ⟨ 0 | ∂ μ D μ | ϕ ⟩ m f = 0 \langle 0|\partial_{\mu}D^{\mu}|\phi\rangle_{m_{f}=0} = = − F ϕ ​ m ϕ 2 -F_{\phi}m_{\phi}^{2} , while the second term on the RHS of Eq.( 6

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Note that r r has no explicit N f N_{f} -independence, since F ϕ 2 F_{\phi}^{2} is associated with the flavor-singlet operator having a sum of N f N_{f} -flavor contributions, namely F ϕ 2 F_{\phi}^{2} is proportional to N f N_{f} , while F π 2 F_{\pi}^{2} is not

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