understanding chiral gauge theories using extra dimensions
TRANSCRIPT
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Understanding Chiral Gauge Theories using
Extra Dimensions
Dorota M Grabowska UC Berkeley
Work done with David B. Kaplan: Phys. Rev. Lett. 116 (2016), no. 21 211602
Phys. Rev. D 94, 114504
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Parity ViolationOne of the great surprises of 20th century was discovery of parity violation: LH and RH fermions can carry different gauge charge!
2
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Chiral Gauge Theories
3
Extremely well-motivated
• Strong agreement between observations and predictions
• Ubiquitous in speculative models of BSM physics
On poor theoretical footing
• No (proven) nonperturbative regulator
• No all-orders proof for a perturbative regulator
We have seen exactly one chiral gauge theory: Standard Model
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Chiral Gauge Theories
3
Extremely well-motivated
• Strong agreement between observations and predictions
• Ubiquitous in speculative models of BSM physics
On poor theoretical footing
• No (proven) nonperturbative regulator
• No all-orders proof for a perturbative regulator
We have seen exactly one chiral gauge theory: Standard Model
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Chiral Gauge Theories
3
Extremely well-motivated
• Strong agreement between observations and predictions
• Ubiquitous in speculative models of BSM physics
On poor theoretical footing
• No (proven) nonperturbative regulator
• No all-orders proof for a perturbative regulator
We have seen exactly one chiral gauge theory: Standard Model
Could nonperturbative regulator lead to unexpected phenomena or address some outstanding puzzles?
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Need nonperturbative definition of fermion path integral
• Vector theory: fermions in real representations of gauge group
• Chiral theory: fermions in complex representations
Fermion Path Integrals
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Need nonperturbative definition of fermion path integral
• Vector theory: fermions in real representations of gauge group
• Chiral theory: fermions in complex representations
Fermion Path Integrals
4
Δ(A) is really product of eigenvalues
4
Witten: ‘We often call the fermion path integral a “determinant” or a “Pfaffian,” but this is a term of art.’
Fermion path integral
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
The Problem
5
Need to find eigenvalues of fermion operator
• Vector theory:
• Chiral theory:
LH
RH
LH
RH
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
The Problem
5
Need to find eigenvalues of fermion operator
• Vector theory:
• Chiral theory:
LH
RH
LH
RH
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
The Problem
5
Need to find eigenvalues of fermion operator
• Vector theory:
• Chiral theory:
Ill-defined eigenvalue problem leads to phase ambiguity for product of the eigenvalues of chiral fermions
LH
RH
LH
RH
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
A (Perturbative) Proposal
6
Proposal: Introduce neutral RH spectator fermions*
• Well defined eigenvalue problem
• Overall phase of Δ(A) related to η-invariant*
• Uncertain if amenable to lattice regularization
* Alvarez-Gaume et al, ’86, ’87* Atiyah, Patodi & Singer, ’75, ’75, ’76
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
A (Perturbative) Proposal
6
Proposal: Introduce neutral RH spectator fermions*
• Well defined eigenvalue problem
• Overall phase of Δ(A) related to η-invariant*
• Uncertain if amenable to lattice regularization
* Alvarez-Gaume et al, ’86, ’87* Atiyah, Patodi & Singer, ’75, ’75, ’76
Charged LH Weyl Fermion
Neutral RH Weyl Fermion
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
A (Perturbative) Proposal
6
Proposal: Introduce neutral RH spectator fermions*
• Well defined eigenvalue problem
• Overall phase of Δ(A) related to η-invariant*
• Uncertain if amenable to lattice regularization
* Alvarez-Gaume et al, ’86, ’87* Atiyah, Patodi & Singer, ’75, ’75, ’76
Charged LH Weyl Fermion
Neutral RH Weyl Fermion
{ Are there other reasonable pert. limits for Δ(A)? }
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Ex: Massless electrons in two dimensions
• Fermion charge does not change: U(1)V preserved
• Axial charge does change if Dirac sea infinite: U(1)A violated
7
Anomalies
Right Moving
Left Moving
p
ω
Initial Ground State
Classical symmetries violated by quantum effects
7
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Ex: Massless electrons in two dimensions
• Fermion charge does not change: U(1)V preserved
• Axial charge does change if Dirac sea infinite: U(1)A violated
7
Anomalies
Right Moving
Left Moving
p
ω
Initial Ground State
Classical symmetries violated by quantum effects
7
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Ex: Massless electrons in two dimensions
• Fermion charge does not change: U(1)V preserved
• Axial charge does change if Dirac sea infinite: U(1)A violated
7
Anomalies
Right Moving
Left Moving Electric field on, off
ω
Final Ground State
pp
ω
Initial Ground State
Classical symmetries violated by quantum effects
7
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Ex: Massless electrons in two dimensions
• Fermion charge does not change: U(1)V preserved
• Axial charge does change if Dirac sea infinite: U(1)A violated
7
Anomalies
Right Moving
Left Moving Electric field on, off
ω
Final Ground State
pp
ω
Initial Ground State
Classical symmetries violated by quantum effects
Anomalies require infinite number of degrees of freedom
7
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Need: Finite phase space to tame UV divergences
• Usually done by introducing mass scale
• Gauged chiral symmetries forbid mass scale
Need: Possibility of anomalous chiral symmetries
• Self-consistency requires cancellation of gauge anomalies
• Anomalies require infinite number of degrees of freedom
Regulating Chiral Gauge Theories
8
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Need: Finite phase space to tame UV divergences
• Usually done by introducing mass scale
• Gauged chiral symmetries forbid mass scale
Need: Possibility of anomalous chiral symmetries
• Self-consistency requires cancellation of gauge anomalies
• Anomalies require infinite number of degrees of freedom
Regulating Chiral Gauge Theories
8
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Need: Finite phase space to tame UV divergences
• Usually done by introducing mass scale
• Gauged chiral symmetries forbid mass scale
Need: Possibility of anomalous chiral symmetries
• Self-consistency requires cancellation of gauge anomalies
• Anomalies require infinite number of degrees of freedom
Regulating Chiral Gauge Theories
8
Fundamental tension between taming UV behavior of chiral gauge theories and maintaining gauge invariance
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
No-Go Theorem*
9
*Nielsen & Ninomiya, ‘81
No-Go Theorem: No lattice fermion operator can satisfy all four conditions simultaneously:
1. Periodic and analytic in momentum space
2. Reduces to Dirac operator in continuum limit
3. Invertible everywhere except at zero momentum
4. Anti-commutes with γ5
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
No-Go Theorem*
9
*Nielsen & Ninomiya, ‘81
No-Go Theorem: No lattice fermion operator can satisfy all four conditions simultaneously:
1. Periodic and analytic in momentum space
2. Reduces to Dirac operator in continuum limit
3. Invertible everywhere except at zero momentum
4. Anti-commutes with γ5
}locality of Fourier
transform
single massless Dirac in
continuum
chiral symmetry preserved
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
No-Go Theorem*
9
*Nielsen & Ninomiya, ‘81
Lattice regulated chiral fermions violate at least one condition
No-Go Theorem: No lattice fermion operator can satisfy all four conditions simultaneously:
1. Periodic and analytic in momentum space
2. Reduces to Dirac operator in continuum limit
3. Invertible everywhere except at zero momentum
4. Anti-commutes with γ5
}locality of Fourier
transform
single massless Dirac in
continuum
chiral symmetry preserved
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 10
Requirements
Lattice regulated chiral gauge theories must have:
• Global chiral symmetry with correct U(1)A anomaly
• Decoupled mirror fermions
• Road to failure for anomalous representations
Basic building block is Dirac fermion
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 10
Requirements
Lattice regulated chiral gauge theories must have:
• Global chiral symmetry with correct U(1)A anomaly
• Decoupled mirror fermions
• Road to failure for anomalous representations
Basic building block is Dirac fermion
Some Previous Proposals
Project out mirrors; construct measure
{Lüscher}
Gauge fix; flow to correct continuum limit
{Golterman, Shamir}
Give mass to mirrors {Golterman, Petcher,
Smit, etc}
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 10
Requirements
Lattice regulated chiral gauge theories must have:
• Global chiral symmetry with correct U(1)A anomaly
• Decoupled mirror fermions
• Road to failure for anomalous representations
Basic building block is Dirac fermion
Some Previous Proposals
Project out mirrors; construct measure
{Lüscher}
Gauge fix; flow to correct continuum limit
{Golterman, Shamir}
Give mass to mirrors {Golterman, Petcher,
Smit, etc}
Our proposal is in a slightly different vein
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 11
Global Chiral Symmetries
Domain Wall Fermions*
• Introduce extra (compact) dimension, s
• Fermion mass depends on s
• Massless modes localized on mass defects
• Massive fermions delocalized into the bulk
extra dimension (s)
ordi
nary
dim
ensi
ons
+Λ-Λ
fermion mass
*Kaplan, ‘92
RH
➠
LH
➠
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 12
Callan-Harvey Mechanism*
U(1)A Anomaly
• Gauge fields independent of s
• Bulk fermions carrying charge between mass defects
• Boundary fermions see axial charge appearing/disappearing
• Condensed matter physicists would call this a topological insulator
*Callan & Harvey, ’85 extra dimension (s)
+Λ-Λ
fermion mass
RH
➠
LH
➠
ordi
nary
dim
ensi
ons
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 12
Callan-Harvey Mechanism*
U(1)A Anomaly
• Gauge fields independent of s
• Bulk fermions carrying charge between mass defects
• Boundary fermions see axial charge appearing/disappearing
• Condensed matter physicists would call this a topological insulator
*Callan & Harvey, ’85
Anomaly: U(1)A explicitly violated
extra dimension (s)
+Λ-Λ
fermion mass
RH
➠
LH
➠
ordi
nary
dim
ensi
ons
Chern Simons Current
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 13
Requirements
Lattice regulated chiral gauge theories must have:
• Global chiral symmetry with correct U(1)A anomaly
• Decoupled mirror fermions
• Road to failure for anomalous representations
Basic building block is Dirac fermion
DONE!
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Fluffy Mirror Fermions*
Idea: Localize gauge field around defect via gradient flow
• Gauge field satisfies gradient flow equation in bulk
• RH fermions couple to physical DoF with have soft form factor
• Both LH and RH fermions couple identically to gauge DoF
*DMG & Kaplan ’15
ΛLH RHRH
-Λs = 0 s = Ls = -L
BC:Flow Eq:
Vector Theory
14
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Fluffy Mirror Fermions*
Idea: Localize gauge field around defect via gradient flow
• Gauge field satisfies gradient flow equation in bulk
• RH fermions couple to physical DoF with have soft form factor
• Both LH and RH fermions couple identically to gauge DoF
*DMG & Kaplan ’15
ΛLH RHRH
-Λs = 0 s = Ls = -L
BC:Flow Eq:
Chiral Theory?
14
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Fluffy Mirror Fermions*
Idea: Localize gauge field around defect via gradient flow
• Gauge field satisfies gradient flow equation in bulk
• RH fermions couple to physical DoF with have soft form factor
• Both LH and RH fermions couple identically to gauge DoF
*DMG & Kaplan ’15
ΛLH RHRH
-Λs = 0 s = Ls = -L
BC:Flow Eq:
Integration variable in path integral
Chiral Theory?
14
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 15
Gradient Flow*
Ex: Two-dimensional QED
• Gauge field decomposes into gauge and physical DoF
• Each obey own flow equation
• RH couple with soft form factors*Used in LQCD (Luscher ’10 etc)
High momentum modes damped out
Gauge DoF unaffected
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 15
Gradient Flow*
Ex: Two-dimensional QED
• Gauge field decomposes into gauge and physical DoF
• Each obey own flow equation
• RH couple with soft form factors*Used in LQCD (Luscher ’10 etc)
High momentum modes damped out
Gauge DoF unaffected s/L
ω(x,s)ω(x,0)
λ(x,s)λ(x,0)
00 1
1
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 15
Gradient Flow*
Ex: Two-dimensional QED
• Gauge field decomposes into gauge and physical DoF
• Each obey own flow equation
• RH couple with soft form factors*Used in LQCD (Luscher ’10 etc)
High momentum modes damped out
Gauge DoF unaffected
Allows mirrors to decouple
s/L
ω(x,s)ω(x,0)
λ(x,s)λ(x,0)
00 1
1
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 16
Requirements
Lattice regulated chiral gauge theories must have:
• Global chiral symmetry with correct U(1)A anomaly
• Decoupled mirror fermions
• Road to failure for anomalous representations
Basic building block is Dirac fermion
DONE!DONE?
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Callan-Harvey Mechanism Revisted
17
Bulk fermions do not decouple completely at low energy
• Generate Chern-Simons terms
• Same mechanism is responsible for U(1)A anomaly
• In 3 dimensions, the Chern Simons action is
Fermion Contribution
Pauli Villars ContributionCS only depends on sign
of domain wall mass
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Callan-Harvey Mechanism Revisted
17
Bulk fermions do not decouple completely at low energy
• Generate Chern-Simons terms
• Same mechanism is responsible for U(1)A anomaly
• In 3 dimensions, the Chern Simons action is
Fermion Contribution
Pauli Villars ContributionCS only depends on sign
of domain wall mass
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Chern-Simon Action
18
Ex: Two-dimensional QED
• Effective two point function is nonlocal
• When flow is turned off, Γ vanishes Determines speed of flow
Chern-Simons term is non-zero with flowed gauge fields
18
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Chern-Simon Action
18
Ex: Two-dimensional QED
• Effective two point function is nonlocal
• When flow is turned off, Γ vanishes Determines speed of flow
Chern-Simons term is non-zero with flowed gauge fields
∂μω
18
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Chern-Simon Action
18
Ex: Two-dimensional QED
• Effective two point function is nonlocal
• When flow is turned off, Γ vanishes Determines speed of flow
Chern-Simons term is non-zero with flowed gauge fields
∂μω ∂μλ
18
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Chern-Simon Action
18
Ex: Two-dimensional QED
• Effective two point function is nonlocal
• When flow is turned off, Γ vanishes Determines speed of flow
Chern-Simons term is non-zero with flowed gauge fields
∂μω ∂μλ
18
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Chern-Simon Action
18
Ex: Two-dimensional QED
• Effective two point function is nonlocal
• When flow is turned off, Γ vanishes Determines speed of flow
Effective 2d theory is nonlocal due to Chern-Simons operator
Chern-Simons term is non-zero with flowed gauge fields
∂μω ∂μλ
18
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 19
Anomalies Cancellation
DWF with flowed gauge fields results in nonlocal theory
• Multiple fermion fields give prefactor to Chern-Simons action
• Theory is local if prefactor vanishes
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 19
Anomalies Cancellation
DWF with flowed gauge fields results in nonlocal theory
• Multiple fermion fields give prefactor to Chern-Simons action
• Theory is local if prefactor vanishesFermion Chirality
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 19
Anomalies Cancellation
DWF with flowed gauge fields results in nonlocal theory
• Multiple fermion fields give prefactor to Chern-Simons action
• Theory is local if prefactor vanishes
Chiral fermion representations that satisfy this criteria are gauge anomaly free representations in continuum
Fermion Chirality
Prefactor depends on dimension
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 20
Requirements
Lattice regulated chiral gauge theories must have:
• Global chiral symmetry with correct U(1)A anomaly
• Decoupled mirror fermions
• Road to failure for anomalous representations
Basic building block is Dirac fermion
DONE!DONE?DONE!
Are the mirror fermions truly decoupled???
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Proposal
Recall: Defining chiral gauge theories requires defining Δ(A) in an unambiguous way
21
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Proposal
Recall: Defining chiral gauge theories requires defining Δ(A) in an unambiguous way
DWF
21
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Proposal
Recall: Defining chiral gauge theories requires defining Δ(A) in an unambiguous way
Pauli Villars
21
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Proposal
Recall: Defining chiral gauge theories requires defining Δ(A) in an unambiguous way
Product over species
21
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Proposal
Recall: Defining chiral gauge theories requires defining Δ(A) in an unambiguous way
5d Dirac operator w/ flowed fields
21
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Proposal
Recall: Defining chiral gauge theories requires defining Δ(A) in an unambiguous way
21
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 22
Decoupling the Fluff
Proposal: Decouple mirror fermions in gauge-invariant manner using soft form factors
• The Good: Theory can only be local for anomaly-free representations
• The Bad: Exponential form factors are problematic in Minkowski space
• The (Potentially) Ugly: Gradient flow does not damp out instanton configurations
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D.M. Grabowska BAPTS Spring 2017 03/03/2017 23
Here Be Dragons
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Decoupling Fluff: The Bad
24
Problem: Exponentially soft form factor violates unitarity under analytic continuation to Minkowski space
Solution: Take extra dimension to be infinite*
• Gradient flow acts like a projector operator
• Doing so results in the manifestly 2d or 4d overlap operator that is amenable to lattice simulations*
* DMG & Kaplan, ‘16* Narayanan and Neuberger, ‘95
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Decoupling Fluff: The Bad
24
Problem: Exponentially soft form factor violates unitarity under analytic continuation to Minkowski space
Solution: Take extra dimension to be infinite*
• Gradient flow acts like a projector operator
• Doing so results in the manifestly 2d or 4d overlap operator that is amenable to lattice simulations*
* DMG & Kaplan, ‘16* Narayanan and Neuberger, ‘95
Field LH Fermion Sees
Field RH Fermion Sees
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Decoupling Fluff: The (Potentially) Ugly
25
Continuum flow equation has multiple attractive fixed points, A*
• Gauge Degree of Freedom to maintain gauge invariance
• Topological configurations like instantons
• May result in non-extensive contributions to the action
Mirrors couple to topology if flow equation is continuous
What are the ramifications of these couplings?
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Decoupling Fluff: The (Potentially) Ugly
26
If mirror fermions do not decouple, they are physical states and not just regularization artifacts
• Strong CP problem: massless mirrors make θ unphysical
• Similarly nonstandard interactions with gravity (Ricci flow)
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Decoupling Fluff: The (Potentially) Ugly
26
If mirror fermions do not decouple, they are physical states and not just regularization artifacts
• Strong CP problem: massless mirrors make θ unphysical
• Similarly nonstandard interactions with gravity (Ricci flow)
LH fermion’s kinetic term
RH fermion kinetic term
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Decoupling Fluff: The (Potentially) Ugly
26
If mirror fermions do not decouple, they are physical states and not just regularization artifacts
• Strong CP problem: massless mirrors make θ unphysical
• Similarly nonstandard interactions with gravity (Ricci flow)
LH fermion’s kinetic term
RH fermion kinetic term
Fundamentally different continuum limit than one would expect from perturbative chiral gauge theories
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Decoupling Fluff: The Good
27
“Road to failure” for anomalous theories is based on non-locality
• Extra dim. theory is gauge invariant for all representations
• Bulk fermions do not completely decouple at low energy scales, resulting in (nonlocal) Cher-Simons current
Question 1: Does the theory correctly reproduce fermion number violating processes?
Question 2: Can this setup be used as a toolkit for the behavior of nonlocal quantum field theories?
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D.M. Grabowska BAPTS Spring 2017 03/03/2017
Summary
28
New proposal combines domain wall fermions with gradient flow
• Extra dimension allows for naturally light fermions
• Gradient flow decouples mirrors with soft form factors
• Road to failure for anomalous representations is non-locality
Chiral gauge theories are extremely well-motived but on poor theoretical footing
![Page 65: Understanding Chiral Gauge Theories using Extra Dimensions](https://reader030.vdocument.in/reader030/viewer/2022012605/619a788c2cc72b4de24de3ef/html5/thumbnails/65.jpg)
D.M. Grabowska BAPTS Spring 2017 03/03/2017
Summary
28
New proposal combines domain wall fermions with gradient flow
• Extra dimension allows for naturally light fermions
• Gradient flow decouples mirrors with soft form factors
• Road to failure for anomalous representations is non-locality
Chiral gauge theories are extremely well-motived but on poor theoretical footing
Many questions, both formal and phenomenological, remain