Check of a new non-perturbative mechanism for elementary fermion - PowerPoint PPT Presentation

The model Lattice action Lattice correlators Outlook Check of a new non-perturbative mechanism for elementary fermion mass generation S. Capitani a ) , P. Dimopoulos b ) , G.M. de Divitiis c ) , R. Frezzotti c ) , M. Garofalo d ) , B. Knippschild e ) , B. Kostrzewa e ) , K. Ottnad e ) , G.C. Rossi b ) c ) , M. Schröck f ) , C. Urbach e ) a ) University of Frankfurt b ) Centro Fermi, Museo Storico della Fisica e Centro Studie Ricerche "Enrico Fermi" c ) University of Rome Tor Vergata, Physics Department and INFN - Sezione di Roma Tor Vergata d ) Higgs Centre for Theoretical Physics, The University of Edinburgh e ) HISKP (Theory), Universitaet Bonn f ) INFN, Sezione di “ Roma Tre ” Check of a new non-perturbative mechanism for elementary fermion mass generation

The model Lattice action Lattice correlators Outlook In [Frezzotti and Rossi Phys. Rev. D92 (2015) 054505] a new non-perturbative mechanism for the elementary particle mass generation was conjectured We are testing this conjecture in a toy model where a fermion doublet Q is coupled to a non-Abelian SU(3) gauge field and a scalar Φ L toy ( Q, A, Φ) = L kin ( Q, A, Φ) + V (Φ) + L W il ( Q, A, Φ) + L Y uk ( Q, Φ) � � • L kin ( Q, A, Φ) = 1 µν + Q L γ µ D µ Q L + Q R γ µ D µ Q R + 1 4 F a µν F a ∂ µ Φ † ∂ µ Φ 2Tr • V (Φ) = µ 2 � � � � �� 2 , + λ 0 0 Φ † Φ Φ † Φ Φ ≡ [ ϕ | − iτ 2 ϕ ∗ ] 2 Tr Tr 4 � � • L W il ( Q, A, Φ) = b 2 ← − ← − D µ Φ † D µ Q L 2 ρ Q L D µ Φ D µ Q R + Q R "Wilson-like" � � Q L Φ Q R + Q R Φ † Q L • L Y uk ( Q, Φ) = η "Yukawa" UV cutoff ∼ b − 1 • Fermion chiral symmetry ˜ χ broken if ( ρ, η ) � = (0 , 0) Check of a new non-perturbative mechanism for elementary fermion mass generation

The model Lattice action Lattice correlators Outlook L toy ( Q, A, Φ) = L kin ( Q, A, Φ) + V (Φ) + L W il ( Q, A, Φ) + L Y uk ( Q, Φ) � � • L kin ( Q, A, Φ) = 1 µν + Q L γ µ D µ Q L + Q R γ µ D µ Q R + 1 4 F a µν F a ∂ µ Φ † ∂ µ Φ 2Tr • V (Φ) = µ 2 � � � � �� 2 , + λ 0 0 Φ † Φ Φ † Φ Φ ≡ [ ϕ | − iτ 2 ϕ ∗ ] 2 Tr Tr 4 • L W il ( Q, A, Φ) = b 2 � � ← − ← − D µ Φ † D µ Q L 2 ρ Q L D µ Φ D µ Q R + Q R "Wilson-like" � � Q L Φ Q R + Q R Φ † Q L • L Y uk ( Q, Φ) = η "Yukawa" � in suitable � Symmetries & power counting = ⇒ Renormalizability UV-regul. Invariant under χ (global) SU(2) L × SU(2) R transformations χ L,R ⊗ (Φ → Ω L,R Φ) ⊗ (Φ † → Φ † Ω † • χ L,R : ˜ L,R )  Q L,R → Ω L,R Q L,R  Ω L,R ∈ SU (2) L,R χ L,R : ˜  Q L,R → Q L,R Ω † L,R Check of a new non-perturbative mechanism for elementary fermion mass generation

The model Lattice action Lattice correlators Outlook • Lagrangian not invariant under purely fermionic transformations Q L → Q L Ω † χ L : Q L → Ω L Q L ˜ Ω L ∈ SU (2) L • They yield the bare WTIs ∂ µ � ˜ µ ( x ) ˆ O (0) � = � ˜ L ˆ Y uk ( x ) ˆ O (0) � − b 2 � O L,i W il ( x ) ˆ J L,i ∆ i O (0) � δ ( x ) − η � O L,i O (0) � � � 2 Q L − b 2 = Q L γ µ τ i τ i D µ Φ † τ i ← − ˜ J L,i 2 ρ Q L 2 Φ D µ Q R − Q R 2 Q L µ � � � � τ i D µ τ i ← − W il = ρ O L,i O L,i Y uk = 2 Φ Q R − hc 2 Φ D µ Q R − hc Q L Q L 2 • Mixing of O L,i W il under renormalization b 2 O L,i J − 1) ∂ µ ˜ J L,i − η ( η ; g 2 s , ρ, λ 0 ) O L,i Y uk + . . . + O ( b 2 ) W il = ( Z ˜ µ • Renormalized WTIs read J ˜ J L,i µ ( x ) ˆ O (0) � = � ˜ ∆ i L ˆ O (0) � δ ( x ) − ( η − η ( η )) � O L,i Y uk ( x ) ˆ O (0) � + . . . + O ( b 2 ) ∂ µ � Z ˜ where the ellipses ( . . . ) stand for possible NP mixing contributions Check of a new non-perturbative mechanism for elementary fermion mass generation

The model Lattice action Lattice correlators Outlook • At the critical η cr ( g 2 s , ρ, λ 0 ) s.t. η cr − η ( η cr ; g 2 s , ρ, λ 0 ) = 0 the WTIs become J ˜ J L,i µ ( x ) ˆ O (0) � η cr = � ˜ ∆ i L ˆ O (0) � η cr δ ( x ) + . . . + O ( b 2 ) ∂ µ � Z ˜ � � χ SSB � In Wigner phase � Φ � = 0 → Wilson-like term uneffective for ˜ J ˜ µ ( x ) ˆ O (0) � η cr = � ˜ L ˆ J L,i ∆ i O (0) � η cr δ ( x ) + O ( b 2 ) ∂ µ � Z ˜ � � � In Nambu-Goldstone � Φ � = v 1 1 2 × 2 expect (conjecture) τ i J ˜ J L,i µ ( x ) ˆ O (0) � η cr = � ˜ ∆ i L ˆ 2 U Q R + hc ] ˆ O (0) � + O ( b 2 ) ∂ µ � Z ˜ O (0) � η cr δ ( x ) + � C 1 Λ s [ Q L The term ∝ C 1 Λ s can exist only in the NG phase where � √ Φ † Φ = ( v + σ + i − → τ − → ( v + σ ) 2 + − → π − → U = Φ / π ) / π  Natural  In Γ NG loc a mass term C 1 Λ s [ Q L U Q R + hc ] � = Yukawa mass  C 1 = O ( α 2 s ) Hierarchy Check of a new non-perturbative mechanism for elementary fermion mass generation

The model Lattice action Lattice correlators Outlook • Intuitive idea of the NP mass generation mechanism O ( b 2 ) NP corrections to ( ˜ χ -preserving) effective vertices combined in loop "diagrams" with O ( b 2 ) ( ˜ χ -breaking) vertices from the Wilson-like term • b − 4 loop divergency = ⇒ O ( b 0 ) C 1 Λ s mass term • Phenomenon occurring even in the quenched approximation Check of a new non-perturbative mechanism for elementary fermion mass generation

The model Lattice action Lattice correlators Outlook • Intuitive idea of the NP mass generation mechanism O ( b 2 ) NP corrections to ( ˜ χ -preserving) effective vertices combined in loop "diagrams" with O ( b 2 ) ( ˜ χ -breaking) vertices from the Wilson-like term • b − 4 loop divergency = ⇒ O ( b 0 ) C 1 Λ s mass term • Phenomenon still occurring in quenched approximation Check of a new non-perturbative mechanism for elementary fermion mass generation

The model Lattice action Lattice correlators Outlook � d 4 x L toy Chose a cheap lattice regularization of � � u First NP study of a theory with gauge, Ψ= & ( ϕ 0 , � ϕ ) d "Naive" fermions (good for quenched approximation only) � � S lat = b 4 � L Y M kin [ U ] + L sca kin (Φ) + V (Φ) + Ψ D lat [ U, Φ]Ψ x L Y M kin [ U ] : SU( 3 ) plaquette action � � � � �� 2 µ 2 L sca kin (Φ) + V (Φ) = 1 2 tr [Φ † ( − ∂ ∗ Φ † Φ + λ 0 Φ † Φ µ ∂ µ )Φ] + 2 tr tr 0 4 1 + iϕ j τ j F ( x ) ≡ [ ϕ 0 1 1 + iγ 5 τ j ϕ j ]( x ) where Φ = ϕ 0 1 ( D lat [ U, Φ]Ψ)( x ) = γ µ � ∇ µ Ψ( x ) + ηF ( x )Ψ( x ) − b 2 ρ 1 2 F ( x ) � ∇ µ � ∇ µ Ψ( x ) + � � − b 2 ρ 1 ( ∂ µ F )( x ) U µ ( x ) � µ ) � µ ) + ( ∂ ∗ µ F )( x ) U † ∇ µ Ψ( x + ˆ µ ( x − ˆ ∇ µ Ψ( x − ˆ µ ) 4 • Yukawa ( d = 4 ) term ∝ η , Wilson-like ( d = 6 ) term ∝ ρ • Unquenched studies will require DW or Overlap fermions Check of a new non-perturbative mechanism for elementary fermion mass generation

The model Lattice action Lattice correlators Outlook � � u Quenched model with 2 flavours × 16 doublers d • χ L ⊗ χ R classical symmetry in Ψ basis: Ψ L,R = 1 2 (1 ± γ 5 )Ψ Ψ L → Ψ L Ω † χ L : Ψ L → Ω L Ψ L Φ → Ω L Φ Ω L ∈ SU (2) L � �� � � χ L,R � �� � Ψ R → Ψ R Ω † Φ → ΦΩ † χ R : Ψ R → Ω R Ψ R Ω R ∈ SU (2) R R • � χ L ⊗ � χ R classical symmetry at η = 0 , any ρ , all doublers (at classical level the Wilson-like term irrelevant) • Doubling symmetry group, D ξ , ξ = ( ξ 1 , ξ 2 , ξ 3 , ξ 4 ) , ξ i = 0 , 1 D ξ : Ψ( x ) → ( − 1) x · ξ M ξ Ψ( x ) Ψ( x ) → Ψ( x ) M † ξ ( − 1) x · ξ µ ξ µ ϕ i ( x ) M ξ = ( iγ 5 γ 1 ) ξ 1 . . . ( iγ 5 γ 4 ) ξ 4 � D ξ : ϕ 0 → ϕ 0 , ϕ i ( x ) → ( − 1) • Doublers at p µ = π b ξ µ Check of a new non-perturbative mechanism for elementary fermion mass generation

The model Lattice action Lattice correlators Outlook • Only symmetric derivatives � ∇ on fermions in the lattice action = ⇒ Doubling Symmetry = ⇒ Power counting arguments work (even in the presence of doublers) • Remark: this would not be the case if there were forward/backward ∇ / ∇ ∗ lattice derivatives acting on fermions = ⇒ no Doubling Symmetry • The renormalization of the toy model on the lattice is analogous to the one of the continuum model [Frezzotti and Rossi, Phys. Rev. D92 (2015) 054505], as it is based on power counting and symmetries Check of a new non-perturbative mechanism for elementary fermion mass generation

The model Lattice action Lattice correlators Outlook • Symmetries of the lattice toy model SU( 3 ) Gauge χ L ⊗ χ R (previously defined) P , T , C , CPF 2  Φ( x ) → Φ † ( x P ) , x P ≡ ( − � x, x 4 )  Ψ( x ) → ¯ ¯ P : Ψ( x ) → γ 4 Ψ( x P ) , Ψ( x P ) γ 4  k ( x p − ˆ U k ( x ) → U † U 4 ( x ) → U 4 ( x p ) , k )  Φ( x ) → Φ T ( x )  Ψ T ( x ) , Ψ( x ) → − Ψ T ( x ) iγ 4 γ 2 Ψ( x ) → iγ 4 γ 2 ¯ ¯ C :  U µ ( x ) → U ∗ µ ( x )  Φ( x ) → Φ † ( x T ) , x T ≡ ( � x, − x 4 )  Ψ T ( x ) , Ψ( x ) → Ψ T ( x ) γ 4 γ 5 Ψ( x ) → γ 5 γ 4 ¯ ¯ T :  U 4 ( x ) → U † 4 ( x T − ˆ 0) , U k ( x ) → U k ( x T ) � Φ( x ) → τ 2 Φ( x ) τ 2 F 2 : Ψ( x ) → − i ¯ ¯ Ψ( x ) → iτ 2 Ψ( x ) , Ψ( x ) τ 2 Check of a new non-perturbative mechanism for elementary fermion mass generation