Fermion bags, topology and index theorems Shailesh Chandrasekharan - PowerPoint PPT Presentation

Fermion bags, topology and index theorems Shailesh Chandrasekharan (Duke University) work done in collaboration with V. Ayyar Lattice 2016, Southampton UK Supported by: US Department of Energy, Nuclear Physics Division

Summary

Summary The concepts of topology and index theorems that arise in the context of QCD, have analogies in simple fermion lattice field theories with staggered fermions, when formulated in the fermion bag approach

Summary The concepts of topology and index theorems that arise in the context of QCD, have analogies in simple fermion lattice field theories with staggered fermions, when formulated in the fermion bag approach This connection gives a more complete perspective on fermion mass generation mechanisms, including a mechanism where fermions acquire a mass through four-fermion condensates instead of fermion bilinear condensates.

Summary The concepts of topology and index theorems that arise in the context of QCD, have analogies in simple fermion lattice field theories with staggered fermions, when formulated in the fermion bag approach This connection gives a more complete perspective on fermion mass generation mechanisms, including a mechanism where fermions acquire a mass through four-fermion condensates instead of fermion bilinear condensates. Can non-Abelian gauge theories also demonstrate this alternate mechanism of fermion mass generation?

QCD Partition Function

QCD Partition Function Partition function of a non-Abelian gauge theory (formal, continuum, finite volume)

QCD Partition Function Partition function of a non-Abelian gauge theory (formal, continuum, finite volume) Z Z [ dA ] e − S G ( A ) [ d ψ d ψ ] e − ψ D ( A ) ψ Z = D ( A ) = ( γ µ ∂ µ − iA µ ) background gauge field weight of the anti-Hermitian operator integration background gauge field depends on the gauge field

Fermion Bag Analogy

Fermion Bag Analogy Partition function of staggered fermion lattice field theories in the fermion bag approach.

Fermion Bag Analogy Partition function of staggered fermion lattice field theories in the fermion bag approach. Z X e − S (B) [ d ψ d ψ ] e − ψ W (B) ψ Z = B anti-Hermitian “fermion bag” matrix depends fermion bag configuration sum over fermion bag weight of a configurations fermion bag configuration

Topology and Index Theorem in QCD

Topology and Index Theorem in QCD Q For every gauge field configuration we can define a topological charge: A

Topology and Index Theorem in QCD Q For every gauge field configuration we can define a topological charge: A Dirac operators satisfy: γ 5 D ( A ) = − D ( A ) γ 5 non-zero modes D ( A ) γ 5 | λ i = � i λ γ 5 | λ i D ( A ) | λ i = i λ | λ i come in pairs zero modes are D ( A ) | z ± i = 0, γ 5 | z ± i = ± | z ± i eigenstates of γ 5 | z ± i modes number of n ± = ( n + − n − ) The index of the Dirac operator : D ( A )

Topology and Index Theorem in QCD Q For every gauge field configuration we can define a topological charge: A Dirac operators satisfy: γ 5 D ( A ) = − D ( A ) γ 5 non-zero modes D ( A ) γ 5 | λ i = � i λ γ 5 | λ i D ( A ) | λ i = i λ | λ i come in pairs zero modes are D ( A ) | z ± i = 0, γ 5 | z ± i = ± | z ± i eigenstates of γ 5 | z ± i modes number of n ± = ( n + − n − ) The index of the Dirac operator : D ( A ) Index Theorem: Q = ( n + − n − )

Topology and Index Theorem in QCD Q For every gauge field configuration we can define a topological charge: A Dirac operators satisfy: γ 5 D ( A ) = − D ( A ) γ 5 non-zero modes D ( A ) γ 5 | λ i = � i λ γ 5 | λ i D ( A ) | λ i = i λ | λ i come in pairs zero modes are D ( A ) | z ± i = 0, γ 5 | z ± i = ± | z ± i eigenstates of γ 5 | z ± i modes number of n ± = ( n + − n − ) The index of the Dirac operator : D ( A ) Index Theorem: Q = ( n + − n − ) D(A) has at least |Q| zero modes

Fermion Bag Analogy

Fermion Bag Analogy For every fermion bag configuration we can define a topological charge: Q B

Fermion Bag Analogy For every fermion bag configuration we can define a topological charge: Q B Fermion bag matrix satisfies: Ξ W ( B ) = − W ( B ) Ξ non-zero modes W ( B ) | λ i = i λ | λ i W ( B ) Ξ | λ i = � i λ Ξ | λ i come in pairs zero modes are W ( B ) | z ± i = 0, Ξ | z ± i = ± | z ± i eigenstates of Ξ | z ± i modes number of n ± = The index of the Dirac operator : ( n + − n − ) W ( B )

Fermion Bag Analogy For every fermion bag configuration we can define a topological charge: Q B Fermion bag matrix satisfies: Ξ W ( B ) = − W ( B ) Ξ non-zero modes W ( B ) | λ i = i λ | λ i W ( B ) Ξ | λ i = � i λ Ξ | λ i come in pairs zero modes are W ( B ) | z ± i = 0, Ξ | z ± i = ± | z ± i eigenstates of Ξ | z ± i modes number of n ± = The index of the Dirac operator : ( n + − n − ) W ( B ) Q = ( n + − n − ) Index Theorem:

Fermion Bag Analogy For every fermion bag configuration we can define a topological charge: Q B Fermion bag matrix satisfies: Ξ W ( B ) = − W ( B ) Ξ non-zero modes W ( B ) | λ i = i λ | λ i W ( B ) Ξ | λ i = � i λ Ξ | λ i come in pairs zero modes are W ( B ) | z ± i = 0, Ξ | z ± i = ± | z ± i eigenstates of Ξ | z ± i modes number of n ± = The index of the Dirac operator : ( n + − n − ) W ( B ) Q = ( n + − n − ) Index Theorem: W(B) has at least |Q| zero modes

Example of a Fermion Bag Approach

Example of a Fermion Bag Approach Consider free massive staggered fermions: S = 1 ⇣ ⌘ X X η x , α ψ x ψ x + α − ψ x + α ψ x + m ψ x ψ x 2 x , α x X ψ x D xy ψ y x , y

Example of a Fermion Bag Approach Consider free massive staggered fermions: S = 1 ⇣ ⌘ X X η x , α ψ x ψ x + α − ψ x + α ψ x + m ψ x ψ x 2 x , α x X ψ x D xy ψ y x , y D is an anti-Hermitian matrix of the form even odd ✓ ◆ 0 C even D = − C T 0 odd

Example of a Fermion Bag Approach Consider free massive staggered fermions: S = 1 ⇣ ⌘ X X η x , α ψ x ψ x + α − ψ x + α ψ x + m ψ x ψ x 2 x , α x X ψ x D xy ψ y x , y D is an anti-Hermitian matrix of the form even odd ✓ 1 even odd ◆ 0 even ✓ ◆ 0 C even Ξ = D = − C T 0 − 1 0 odd odd

Example of a Fermion Bag Approach Consider free massive staggered fermions: S = 1 ⇣ ⌘ X X η x , α ψ x ψ x + α − ψ x + α ψ x + m ψ x ψ x 2 x , α x X ψ x D xy ψ y x , y D is an anti-Hermitian matrix of the form even odd ✓ 1 even odd ◆ 0 even ✓ ◆ 0 C even Ξ = D = − C T 0 − 1 0 odd odd D Ξ = − Ξ D

Z [ d ψ d ψ ] e − S Z = Partition function:

Z [ d ψ d ψ ] e − S Z = Partition function: Femion Bag Approach: Z [ d ψ d ψ ] e − ψ D ψ e − m P x ψ x ψ x Z = Z [ d ψ d ψ ] e − ψ D ψ Y � � = 1 − m ψ x ψ x x Z X [ d ψ d ψ ] e − ψ W ( B ) ψ m k = B

Z [ d ψ d ψ ] e − S Z = Partition function: configuration B Femion Bag Approach: Z [ d ψ d ψ ] e − ψ D ψ e − m P x ψ x ψ x Z = Z [ d ψ d ψ ] e − ψ D ψ Y � � = 1 − m ψ x ψ x x Z X [ d ψ d ψ ] e − ψ W ( B ) ψ m k = B B = B1 + B2 + … free space-time defects fermion bags k = total number of defects

Z [ d ψ d ψ ] e − S Z = Partition function: configuration B Femion Bag Approach: Z [ d ψ d ψ ] e − ψ D ψ e − m P x ψ x ψ x Z = Z [ d ψ d ψ ] e − ψ D ψ Y � � = 1 − m ψ x ψ x x Z X [ d ψ d ψ ] e − ψ W ( B ) ψ m k = B B = B1 + B2 + … free S ( B ) = − k log( m ) Defining space-time defects fermion bags Z X [ d ψ d ψ ] e − ψ W ( B ) ψ Z = exp( − S ( B )) k = total number of defects B

Z [ d ψ d ψ ] e − S Z = Partition function: configuration B Femion Bag Approach: Z [ d ψ d ψ ] e − ψ D ψ e − m P x ψ x ψ x Z = Z [ d ψ d ψ ] e − ψ D ψ Y � � = 1 − m ψ x ψ x x Z X [ d ψ d ψ ] e − ψ W ( B ) ψ m k = B B = B1 + B2 + … free S ( B ) = − k log( m ) Defining space-time defects fermion bags Z X [ d ψ d ψ ] e − ψ W ( B ) ψ Z = exp( − S ( B )) k = total number of defects B even odd even ✓ ◆ 0 C ( B ) anti-Hermitian W ( B ) = − C ( B ) T 0 odd

Topology and index theorem for a fermion bag

Topology and index theorem for a fermion bag Q = n even − n odd Topological charge of a fermion bag

Topology and index theorem for a fermion bag Q = n even − n odd Topological charge of a fermion bag Fermion bag Dirac operator even odd ✓ ◆ 0 C ( B ) even W ( B ) = − C ( B ) T 0 odd

Topology and index theorem for a fermion bag Q = n even − n odd Topological charge of a fermion bag Fermion bag Dirac operator define even odd even odd ✓ ◆ even 1 0 ✓ ◆ 0 C ( B ) even Ξ B = W ( B ) = − C ( B ) T 0 0 − 1 odd odd