Classification of higher-dimensional operators in the Standard Model - PowerPoint PPT Presentation

Classification of higher-dimensional operators in the Standard Model Mateusz Iskrzy´ nski University of Warsaw IMPRS Workshop, Munich 19.07.2010

Higher-dimensional operators in the Standard Model 1 Introduction Effective theories Structure of the Standard Model 2 Reasoning scheme 3 Basis of invariant effective operators 4 Comparison with ”Effective lagrangian analysis of new interactions and flavour conservation” by Buchm¨ uller, Wyler (1986)

Effective theories Standard Model → Extension

Effective theories Standard Model → Extension But how does Extension correct Standard Model interactions in low-energy processes?

Effective theories Standard Model → Extension But how does Extension correct Standard Model interactions in low-energy processes? Appelquist-Carazzone decoupling theorem: � � L = L (4) c (5) O (5) c (6) O (6) SM + 1 + 1 + O ( 1 Λ 3 ) Λ i i Λ 2 i i i i

Effective theories Standard Model → Extension But how does Extension correct Standard Model interactions in low-energy processes? Appelquist-Carazzone decoupling theorem: � � L = L (4) c (5) O (5) c (6) O (6) SM + 1 + 1 + O ( 1 Λ 3 ) Λ i i Λ 2 i i i i What are the operators O (5) and O (6) ? i i

Effective theories Standard Model → Extension But how does Extension correct Standard Model interactions in low-energy processes? Appelquist-Carazzone decoupling theorem: � � L = L (4) c (5) O (5) c (6) O (6) SM + 1 + 1 + O ( 1 Λ 3 ) Λ i i Λ 2 i i i i What are the operators O (5) and O (6) ? i i ◮ Gauge and Lorentz symmetry ◮ Dependencies through EOM

Effective theories Standard Model → Extension But how does Extension correct Standard Model interactions in low-energy processes? Appelquist-Carazzone decoupling theorem: � � L = L (4) c (5) O (5) c (6) O (6) SM + 1 + 1 + O ( 1 Λ 3 ) Λ i i Λ 2 i i i i What are the operators O (5) and O (6) ? i i ◮ Gauge and Lorentz symmetry ◮ Dependencies through EOM Classification given in the paper ”Effective lagrangian analysis of new interactions and flavour conservation” by W. Buchm¨ uller, D. Wyler (1986), but...

Effective theories Standard Model → Extension But how does Extension correct Standard Model interactions in low-energy processes? Appelquist-Carazzone decoupling theorem: � � L = L (4) c (5) O (5) c (6) O (6) SM + 1 + 1 + O ( 1 Λ 3 ) Λ i i Λ 2 i i i i What are the operators O (5) and O (6) ? i i ◮ Gauge and Lorentz symmetry ◮ Dependencies through EOM Classification given in the paper ”Effective lagrangian analysis of new interactions and flavour conservation” by W. Buchm¨ uller, D. Wyler (1986), but... 22 (of 81) operators are redundant and 1 is absent.

SM - gauge group representations structure representation (dimension) hypercharge Field SU(3) SU(2) U(1) G µ 8 1 0 W µ 1 3 0 1 1 0 B µ 1 q 3 2 6 2 u 3 1 3 − 1 3 1 d 3 − 1 l 1 2 2 e 1 1 − 1 1 1 2 ϕ 2

SM - lagrangian density µν G A µν − 1 µν W I µν − 1 L 0 = − 1 4 G A 4 W I 4 B µν B µν + ( D µ ϕ ) † ( D µ ϕ ) + m 2 ϕ † ϕ − 1 2 λ ( ϕ † ϕ ) 2 + i ¯ u � Du + i ¯ l � Dl + i ¯ e � De + i ¯ q � Dq + i ¯ d � Dd + − (¯ l Γ e e ϕ + ¯ q Γ u u ( ˜ ϕ ) + ¯ q Γ d d ϕ + h . c . )

Mass-dimension of fundamental objects in units � = c = 1 Type vector V µ tensor X µν spinor Ψ skalar ϕ 3 ( GeV ) 1 ( GeV ) 2 ( GeV ) 1 Dimension ( GeV ) 2 Object D µ W µν , G µν , B µν q , l , u , d , e ϕ ◮ for SU(3) G A µν = ∂ µ G B ν − ∂ ν G C µ − g s f ABC G B µ G C ν ◮ for SU(2) W I µν = ∂ µ W I ν − ∂ ν W I µ − g ε IJK W J µ W K ν ◮ for U(1) B µν = ∂ µ B ν − ∂ ν B µ

Reasoning scheme 1. Description in terms of matter fields ϕ , ψ , field strength tensors X µν and covariant derivatives D µ . Dimensional analysis. 2. Gauge and Lorentz symmetry. 3. Reduction of the set of operators using algebraic properties and SM EOM:

Reasoning scheme 1. Description in terms of matter fields ϕ , ψ , field strength tensors X µν and covariant derivatives D µ . Dimensional analysis. e.g. dim-6 expressions containing both fermionic and bosonic fields: ψψ XD , ψψ X ϕ , ψψϕϕϕ , ψψϕϕ D , ψψϕ DD , ψψ DDD 2. Gauge and Lorentz symmetry. 3. Reduction of the set of operators using algebraic properties and SM EOM:

Reasoning scheme 1. Description in terms of matter fields ϕ , ψ , field strength tensors X µν and covariant derivatives D µ . Dimensional analysis. 2. Gauge and Lorentz symmetry. 3. Reduction of the set of operators using algebraic properties and SM EOM:

Reasoning scheme 1. Description in terms of matter fields ϕ , ψ , field strength tensors X µν and covariant derivatives D µ . Dimensional analysis. 2. Gauge and Lorentz symmetry. e.g. ψψϕ DD : many possible choices of ψ - the only singlet in ˆ 2 SU (2) ⊗ ˆ 2 SU (2) hypercharge conservation ( q † εϕ ∗ ) u , ( q † ϕ ) d , ( l † ϕ ) e , + h . c . 3. Reduction of the set of operators using algebraic properties and SM EOM:

Reasoning scheme 1. Description in terms of matter fields ϕ , ψ , field strength tensors X µν and covariant derivatives D µ . Dimensional analysis. 2. Gauge and Lorentz symmetry. e.g. ψψϕ DD : many possible choices of ψ - the only singlet in ˆ 2 SU (2) ⊗ ˆ 2 SU (2) hypercharge conservation ( q † εϕ ∗ ) u , ( q † ϕ ) d , ( l † ϕ ) e , + h . c . Lorentz structure contains 2 singlets: ( 1 2 , 0) ⊗ ( 1 2 , 0) ⊗ ( 1 2 , 1 2 ) ⊗ ( 1 2 , 1 2 ) = (0 , 0) ⊕ (0 , 0) ⊕ (1 , 0) ⊕ (2 , 0) ⊕ (1 , 1) ⊕ (0 , 1) ⊕ (1 , 1) ⊕ (2 , 1) 3. Reduction of the set of operators using algebraic properties and SM EOM:

Reasoning scheme 1. Description in terms of matter fields ϕ , ψ , field strength tensors X µν and covariant derivatives D µ . Dimensional analysis. 2. Gauge and Lorentz symmetry. e.g. ψψϕ DD : many possible choices of ψ - the only singlet in ˆ 2 SU (2) ⊗ ˆ 2 SU (2) hypercharge conservation ( q † εϕ ∗ ) u , ( q † ϕ ) d , ( l † ϕ ) e , + h . c . 2 independent Lorentz invariants (for each): ¯ ¯ ψ L ψ R ϕ D µ D µ ψ L σ µν ψ R ϕ D µ D ν 3. Reduction of the set of operators using algebraic properties and SM EOM:

Reasoning scheme 1. Description in terms of matter fields ϕ , ψ , field strength tensors X µν and covariant derivatives D µ . Dimensional analysis. 2. Gauge and Lorentz symmetry. 3. Reduction of the set of operators using algebraic properties and SM EOM: We have (omitting full div) the following operators: ( ¯ ψ L σ µν ψ R )( D µ D ν ϕ ) (1) ( ¯ ψ L σ µν D µ D ν ψ R ) ϕ (2) ( ¯ ψ L σ µν D µ ψ R )( D ν ϕ ) (3) ( ¯ ψ L D µ D µ ψ R ) ϕ (4) ( ¯ ψ L D µ ψ R )( D µ ϕ ) (5) ( ¯ ψ L ψ R )( D µ D µ ϕ ) (6)

Reasoning scheme 1. Description in terms of matter fields ϕ , ψ , field strength tensors X µν and covariant derivatives D µ . Dimensional analysis. 2. Gauge and Lorentz symmetry. 3. Reduction of the set of operators using algebraic properties and SM EOM: We can reduce: ( ¯ 2 ( ¯ ψ L σ µν ψ R )( D µ D ν ϕ ) = 1 ψ L σ µν ψ R )([ D µ , D ν ] ϕ ) ψ L σ µν ψ R )( ig W µν + ig ′ B µν ) ϕ ∼ ψψ X ϕ 2 ( ¯ = 1

Reduction scheme

Bosonic invariant operators ϕ 6 ϕ 4 DD XXX ϕϕ XX ε IJK W I ν W J ρ W K µ ( ϕ † ϕ ) 3 ϕ † T I ϕ W I µν B µν ( ϕ † ϕ )( D µ ϕ ) † ( D µ ϕ ) µ ν ρ ε IJK � µν ˜ W I µν W J νδ W K δ ϕ † T I ϕ W I B µν [ ϕ † ( D µ ϕ )][( D µ ϕ ) † ϕ ] µ f ABCc G A ν µ G B ρ ν G C µ ϕ † ϕ W I µν W I µν ρ f ABC ˜ G A ν µ G B δ ν G C µ µν ˜ ϕ † ϕ W I W I µν δ ϕ † ϕ G A µν G A µν µν ˜ ϕ † ϕ G A G A µν ϕ † ϕ B µν B µν ϕ † ϕ B µν ˜ B µν

Invariant operators with 2 fermions ψψϕϕ D ψψϕϕϕ ψψ X ϕ ¯ q γ µ q )( ϕ † D µ ϕ ) ϕ † q )]( ϕ † ϕ ) d σ µν λ A ( ϕ † q ) G A (¯ [¯ u ( ˜ µν [¯ q γ µ T I q )( ϕ † T I D µ ϕ ) d ( ϕ † q )]( ϕ † ϕ ) u σ µν λ A ( ˜ ϕ † q ) G A (¯ ¯ µν ¯ u γ µ d )( ϕ † D µ ˜ e ( ϕ † l )]( ϕ † ϕ ) d σ µν T I ( ϕ † q ) W I (¯ ϕ ) [¯ µν u γ µ u )( ϕ † D µ ϕ ) u σ µν T I ( ˜ ϕ † q ) W I (¯ ¯ µν (¯ d γ µ d )( ϕ † D µ ϕ ) e σ µν T I ( ϕ † l ) W I ¯ µν e γ µ e )( ϕ † D µ ϕ ) u σ µν ( ˜ ϕ † q ) B µν (¯ ¯ (¯ ¯ l γ µ l )( ϕ † D µ ϕ ) d σ µν ( ϕ † q ) B µν ( ϕ † l ) γ µ (¯ e σ µν ( ϕ † l ) B µν lD µ ϕ ) ¯

Fermionic operators ¯ LL ¯ RR ¯ ¯ LL RR (¯ l p 1 γ µ l p 2 )(¯ l p 3 γ µ l p 4 ) e γ µ e ) (¯ e γ µ e )(¯ q p 3 γ µ q p 4 ) u p 3 γ µ u p 4 ) (¯ q p 1 γ µ q p 2 )(¯ (¯ u p 1 γ µ u p 2 )(¯ (¯ d p 1 γ µ d p 2 )(¯ q p 1 γ µ T I q p 2 )(¯ q p 3 γ µ T I q p 4 ) d p 3 γ µ d p 4 ) (¯ q p 1 γ µ q p 2 )(¯ l p 3 γ µ l p 4 ) e γ µ e ) (¯ (¯ u γ µ u )(¯ q p 1 γ µ T I q p 2 )(¯ (¯ l p 3 γ µ T I l p 4 ) e γ µ e ) (¯ d γ µ d )(¯ u p 1 γ µ u p 2 )(¯ d p 3 γ µ d p 4 ) (¯ u p 1 γ µ T A u p 2 )(¯ d p 3 T A d p 4 ) (¯