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A New Perspective on Chiral Gauge Theories D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Why an interest in chiral gauge theories? • There are strongly coupled χGTs which are thought to exhibit massless composite fermions, etc • There does not exist a nonperturbative regulator • There isn’t an all-orders proof for a perturbative regulator D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Why an interest in chiral gauge theories? • There are strongly coupled χGTs which are thought to exhibit massless composite fermions, etc • There does not exist a nonperturbative regulator • There isn’t an all-orders proof for a perturbative regulator But of paramount importance: The Standard Model is a χGT! Nonperturbative definition • unexpected phenomenology? • answers to outstanding puzzles? D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The Problem: A vector-like gauge theory like QCD consists of Dirac fermions, = Weyl fermions in a real representation of the gauge group. ZV = Z Nf [dA]e SY M Y / det(D mi ) i=1 D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The Problem: A vector-like gauge theory like QCD consists of Dirac fermions, = Weyl fermions in a real representation of the gauge group. ZV = Z Nf [dA]e SY M Y / det(D mi ) i=1 A chiral gauge theory consists of Weyl fermions in a complex representation of the gauge group. Z = Z [dA]e SY M [A] ? D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The Problem: A vector-like gauge theory like QCD consists of Dirac fermions, = Weyl fermions in a real representation of the gauge group. ZV = Z Nf [dA]e SY M Y / det(D mi ) i=1 A chiral gauge theory consists of Weyl fermions in a complex representation of the gauge group. Z = Z [dA]e SY M [A] ? Witten: “We often call the fermion path integral a ‘determinant’ or a ‘Pfaffian’, but this is a term of art.” D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The Problem: A vector-like gauge theory like QCD consists of Dirac fermions, = Weyl fermions in a real representation of the gauge group. ZV = Z Nf [dA]e SY M Y / det(D mi ) i=1 A chiral gauge theory consists of Weyl fermions in a complex representation of the gauge group. Z = Z [dA]e SY M [A] ? Witten: “We often call the fermion path integral a ‘determinant’ or a ‘Pfaffian’, but this is a term of art.” We mean a product of eigenvalues… …but there is no good eigenvalue problem for a chiral theory D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Vector-like gauge theory with Dirac fermions: / = D ✓ 0 Dµ ¯ µ Dµ 0 µ ◆✓ R L ◆ = ✓ R L ◆ D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Vector-like gauge theory with Dirac fermions: / = D ✓ 0 Dµ ¯ µ Dµ 0 µ ◆✓ R L ◆ ✓ = R L ◆ Chiral gauge theory with Weyl fermions: ✓ 0 0 Dµ 0 µ ◆✓ 0 L ◆ = ✓ R 0 ◆ D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Vector-like gauge theory with Dirac fermions: / = D ✓ 0 Dµ ¯ µ Dµ 0 µ ◆✓ R L ◆ ✓ = R L ◆ Chiral gauge theory with Weyl fermions: ✓ 0 0 Dµ 0 Can define: µ ◆✓ R 0 L ◆ = = | | ei✓ ✓ R 0 ◆ R but no unambiguous way to define the phase θ D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Vector-like gauge theory with Dirac fermions: / = D ✓ 0 Dµ ¯ µ Dµ 0 µ ◆✓ R L ◆ ✓ = R L ◆ Chiral gauge theory with Weyl fermions: ✓ 0 0 Dµ 0 Can define: µ ◆✓ R 0 L ◆ = = | | ei✓ ✓ R 0 ◆ R but no unambiguous way to define the phase θ So: [A] = e i [A] q / | det D| D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The fermion integral for a χGT: [A] = e i [A] The phase δ encodes both anomalies and dynamics q / | det D| δ[A] is generally not gauge invariant (eg, when fermion representation is anomalous) D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The fermion integral for a χGT: [A] = e i [A] The phase δ encodes both anomalies and dynamics q / | det D| δ[A] is generally not gauge invariant (eg, when fermion representation is anomalous) Do we know δ≠0 for anomaly-free theory? D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The fermion integral for a χGT: [A] = e i [A] The phase δ encodes both anomalies and dynamics q / | det D| δ[A] is generally not gauge invariant (eg, when fermion representation is anomalous) Do we know δ≠0 for anomaly-free theory? Consider two gauge-anomaly-free SO(10) gauge theories D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The fermion integral for a χGT: [A] = e i [A] The phase δ encodes both anomalies and dynamics q / | det D| δ[A] is generally not gauge invariant (eg, when fermion representation is anomalous) Do we know δ≠0 for anomaly-free theory? Consider two gauge-anomaly-free SO(10) gauge theories • Model A has N x (16 + 16*) LH Weyl fermions - vector theory • gauge invariant fermion condensate expected D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The fermion integral for a χGT: [A] = e i [A] The phase δ encodes both anomalies and dynamics q / | det D| δ[A] is generally not gauge invariant (eg, when fermion representation is anomalous) Do we know δ≠0 for anomaly-free theory? Consider two gauge-anomaly-free SO(10) gauge theories • Model A has N x (16 + 16*) LH Weyl fermions - vector theory • • gauge invariant fermion condensate expected Model B has 2N x 16 LH Weyl fermions - chiral theory • no gauge invariant fermion bilinear condensate possible D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The fermion integral for a χGT: [A] = e i [A] The phase δ encodes both anomalies and dynamics q / | det D| δ[A] is generally not gauge invariant (eg, when fermion representation is anomalous) Do we know δ≠0 for anomaly-free theory? Consider two gauge-anomaly-free SO(10) gauge theories • Model A has N x (16 + 16*) LH Weyl fermions - vector theory • • gauge invariant fermion condensate expected Model B has 2N x 16 LH Weyl fermions - chiral theory • no gauge invariant fermion bilinear condensate possible If δ=0, A & B would have same measure, same glue ball spectra…unlikely! D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The fermion integral for a χGT: [A] = e i [A] q / | det D| Alvarez-Gaume et al. proposal for perturbative definition (1984,1986): ✓ 0 [A] ⌘ det @µ ¯ µ Dµ 0 µ ◆ gauged LH Weyl fermion neutral RH Weyl fermion D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The fermion integral for a χGT: [A] = e i [A] q / | det D| Alvarez-Gaume et al. proposal for perturbative definition (1984,1986): ✓ 0 [A] ⌘ det @µ ¯ µ Dµ 0 µ ◆ gauged LH Weyl fermion neutral RH Weyl fermion Well-defined eigenvalue problem with complex eigenvalues Extra RH fermions decouple Amenable to lattice regularization? D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The key problem in representing chiral symmetry on the lattice (global or gauged) is the anomaly D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The key problem in representing chiral symmetry on the lattice (global or gauged) is the anomaly One way of looking at anomalies: massless electrons in E field, 1+1 dim ω RH ➠ LH E⇒ ➠ p D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The key problem in representing chiral symmetry on the lattice (global or gauged) is the anomaly One way of looking at anomalies: massless electrons in E field, 1+1 dim ω RH ➠ LH E⇒ ➠ p d=(1+1): @µ j5µ qE = ⇡ quantum violation of a classical U(1)A symmetry D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 The key problem in representing chiral symmetry on the lattice (global or gauged) is the anomaly One way of looking at anomalies: massless electrons in E field, 1+1 dim ω RH ➠ LH E⇒ ➠ p d=(1+1): @µ j5µ qE = ⇡ In the continuum, the Dirac sea is filled…but is a Hilbert Hotel which always has room for more quantum violation of a classical U(1)A symmetry D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Not so on the lattice: Can reproduce continuum physics for long wavelength modes… ω ➠ ➠ LH RH E⇒ p D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Not so on the lattice: Can reproduce continuum physics for long wavelength modes… ω ➠ ➠ ➠ LH RH ➠ …but no anomalies in a system with a finite number of degrees of freedom E⇒ p µ @µ j5 =0 D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Not so on the lattice: Can reproduce continuum physics for long wavelength modes… ω …but no anomalies in a system with a finite number of degrees of freedom ➠ ➠ ➠ LH RH E⇒ ➠ p µ @µ j5 =0 anomalous symmetry in the continuum must be explicitly broken symmetry on the lattice D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Wilson fermions reproduce the U(1)A anomaly in QCD: Karsten, Smit 1980 2 ¯ / L= D + m + aD D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Wilson fermions reproduce the U(1)A anomaly in QCD: Karsten, Smit 1980 2 ¯ / L= D + m + aD D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Wilson fermions reproduce the U(1)A anomaly in QCD: Karsten, Smit 1980 2 ¯ / L= D + m + aD • Wilson fermions eliminate doublers by giving them a big mass D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Wilson fermions reproduce the U(1)A anomaly in QCD: Karsten, Smit 1980 2 ¯ / L= D + m + aD • Wilson fermions eliminate doublers by giving them a big mass • Mass & Wilson terms explicitly break the (global) chiral flavor symmetries D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Wilson fermions reproduce the U(1)A anomaly in QCD: Karsten, Smit 1980 2 ¯ / L= D + m + aD • Wilson fermions eliminate doublers by giving them a big mass • Mass & Wilson terms explicitly break the (global) chiral flavor symmetries • fine tune m to continuum limit…find some chiral symmetry breaking does not decouple & correct anomalous Ward identities are found D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Domain Wall Fermions reproduce the U(1)A anomaly in QCD: DBK, 1992 fermion mass ➠ +Λ LH RH ➠ ordinary dimensions -Λ extra dimension D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Domain Wall Fermions reproduce the U(1)A anomaly in QCD: DBK, 1992 fermion mass ➠ +Λ LH RH ➠ ordinary dimensions -Λ extra dimension D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Domain Wall Fermions reproduce the U(1)A anomaly in QCD: DBK, 1992 fermion mass ➠ +Λ LH RH ➠ ordinary dimensions -Λ • Bulk fermion mass violates chiral symmetry… extra dimension D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Domain Wall Fermions reproduce the U(1)A anomaly in QCD: DBK, 1992 fermion mass ➠ +Λ LH RH ➠ ordinary dimensions -Λ • Bulk fermion mass violates chiral symmetry… • …χSB effects are irrelevant except for marginal ChernSimons current in bulk which allows charges to pass between LH and RH zero modes at mass defects extra dimension D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Domain Wall Fermions reproduce the U(1)A anomaly in QCD: DBK, 1992 fermion mass ➠ +Λ LH RH ➠ ordinary dimensions -Λ • Bulk fermion mass violates chiral symmetry… • …χSB effects are irrelevant except for marginal ChernSimons current in bulk which allows charges to pass between LH and RH zero modes at mass defects • …but cannot radiatively generate a LH - RH mass term in effective theory since they are physically separated extra dimension D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Domain Wall Fermions reproduce the U(1)A anomaly in QCD: DBK, 1992 fermion mass ➠ +Λ LH RH ➠ ordinary dimensions -Λ • Bulk fermion mass violates chiral symmetry… • …χSB effects are irrelevant except for marginal ChernSimons current in bulk which allows charges to pass between LH and RH zero modes at mass defects • …but cannot radiatively generate a LH - RH mass term in effective theory since they are physically separated extra dimension “topological insulator” D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Overlap Fermions reproduce the U(1)A anomaly in QCD: ➠ Neuberger, Narayanan 1993-1998 ➠ LH RH extra dim radius L5 ∞ D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Overlap Fermions reproduce the U(1)A anomaly in QCD: ➠ Neuberger, Narayanan 1993-1998 ➠ LH RH • Overlap=effective 4d theory of DWF in limit L5 ∞: L = ¯D • D satisfies Ginsparg-Wilson equation extra dim radius L5 ∞ D 1 , 5 =a 5 D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 How Overlap Fermions reproduce the U(1)A anomaly in QCD: ➠ Neuberger, Narayanan 1993-1998 ➠ LH RH • Overlap=effective 4d theory of DWF in limit L5 ∞: L = ¯D • D satisfies Ginsparg-Wilson equation extra dim radius L5 ∞ D • Solution (chiral basis): D 1 = ✓ 1 , 5 =a 0 C† ◆ C 0 5 a + 2 ✓ 1 0 D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 0 1 ◆ How Overlap Fermions reproduce the U(1)A anomaly in QCD: ➠ Neuberger, Narayanan 1993-1998 ➠ LH RH • Overlap=effective 4d theory of DWF in limit L5 ∞: L = ¯D • D satisfies Ginsparg-Wilson equation extra dim radius L5 ∞ D • Solution (chiral basis): Chiral symmetric D 1 = ✓ 1 , 5 =a 0 C† ◆ C 0 5 a + 2 ✓ 1 0 D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 0 1 ◆ How Overlap Fermions reproduce the U(1)A anomaly in QCD: ➠ Neuberger, Narayanan 1993-1998 ➠ LH RH • Overlap=effective 4d theory of DWF in limit L5 ∞: L = ¯D • D satisfies Ginsparg-Wilson equation extra dim radius L5 ∞ D • Solution (chiral basis): Chiral symmetric D 1 = ✓ 1 , 5 =a 0 C† ◆ C 0 5 a + 2 ✓ 1 0 Explicit chiral symmetry breaking D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 0 1 ◆ Back to the continuum operator: D = ✓ 0 @µ ¯ µ Dµ 0 µ ◆ Two anomaly issues to address: • global U(1)A anomaly • gauge anomaly D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Back to the continuum operator: D = ✓ 0 @µ ¯ µ Dµ 0 µ ◆ Two anomaly issues to address: • global U(1)A anomaly • gauge anomaly Here: focus on global anomaly: It requires explicit U(1)A chiral symmetry breaking on the lattice: D = ✓ X c† C X ◆ D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Back to the continuum operator: D = ✓ 0 @µ ¯ µ Dµ 0 µ ◆ Two anomaly issues to address: • global U(1)A anomaly • gauge anomaly Here: focus on global anomaly: It requires explicit U(1)A chiral symmetry breaking on the lattice: D = ✓ X c† C X ◆ …but if LH fermion is gauged and RH is neutral, X terms coupling them violate gauge symmetry! D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 D = ✓ X c† C X ◆ Apparently two alternatives for the lattice in order to realize U(1)A anomaly in a chiral gauge theory: D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 D = ✓ X c† C X ◆ Apparently two alternatives for the lattice in order to realize U(1)A anomaly in a chiral gauge theory: 1. Gauge the RH mirror fermions (so X does not violate gauge symmetry)… and then decouple somehow D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 D = ✓ X c† C X ◆ Apparently two alternatives for the lattice in order to realize U(1)A anomaly in a chiral gauge theory: 1. Gauge the RH mirror fermions (so X does not violate gauge symmetry)… and then decouple somehow 2. Break the gauge symmetry explicitly in the mirror fermion couplings D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 1. Gauge the RH mirror fermions (so X does not violate gauge symmetry)… and then decouple somehow D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 1. Gauge the RH mirror fermions (so X does not violate gauge symmetry)… and then decouple somehow Historically numerous attempts to endow mirror fermions with exotic interactions in hopes of decoupling them…many have been shown not to work. Currently several proposed for which there is no evidence either way. D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 1. Gauge the RH mirror fermions (so X does not violate gauge symmetry)… and then decouple somehow Historically numerous attempts to endow mirror fermions with exotic interactions in hopes of decoupling them…many have been shown not to work. Currently several proposed for which there is no evidence either way. 2. Break the gauge symmetry explicitly in the mirror fermion couplings D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 1. Gauge the RH mirror fermions (so X does not violate gauge symmetry)… and then decouple somehow Historically numerous attempts to endow mirror fermions with exotic interactions in hopes of decoupling them…many have been shown not to work. Currently several proposed for which there is no evidence either way. 2. Break the gauge symmetry explicitly in the mirror fermion couplings Some work along these lines that looks OK in perturbation theory Bock, Golterman, Shamir 1998, 2004 D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 1. Gauge the RH mirror fermions (so X does not violate gauge symmetry)… and then decouple somehow Historically numerous attempts to endow mirror fermions with exotic interactions in hopes of decoupling them…many have been shown not to work. Currently several proposed for which there is no evidence either way. 2. Break the gauge symmetry explicitly in the mirror fermion couplings Some work along these lines that looks OK in perturbation theory Bock, Golterman, Shamir 1998, 2004 A natural way to examine the problem is with Domain Wall Fermions… D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 0 @µ ¯ µ Dµ 0 Suggests: localizing gauge fields near one domain wall Requires “Higgs” field at boundary to maintain gauge invariance µ ◆ -Λ ordinary dimensions Motivation: D = ✓ localized gauge fields RH LH Higgs D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Old attempts to use domain wall fermions for chiral gauge theory localized gauge fields +Λ RH ➠ LH ➠ Not compelling…how would theory know to fail when there are gauge anomalies? ordinary dimensions Localize the gauge fields around one of the defects? -Λ Higgs D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Old attempts to use domain wall fermions for chiral gauge theory One finds a Dirac fermion & vector-like gauge theory. Golterman, Jansen,Vink 1993 localized gauge fields RH LH ➠ Never works. +Λ ➠ Not compelling…how would theory know to fail when there are gauge anomalies? ordinary dimensions Localize the gauge fields around one of the defects? -Λ Higgs D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 New proposal: “localize” gauge fields using gradient flow Dorota Grabowska, D.B.K. • Phys.Rev.Lett. 116 211602 (2016) [arXiv:1511.03649] • work in progress Gradient flow smooths out fields by evolving them classically in an extra dimension via a heat equation ➟t D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 New proposal: “localize” gauge fields using gradient flow Dorota Grabowska, D.B.K. • Phys.Rev.Lett. 116 211602 (2016) [arXiv:1511.03649] • work in progress Gradient flow smooths out fields by evolving them classically in an extra dimension via a heat equation ➟t • • • Gradient flow uses an extra dimension… DWF uses an extra dimension… …maybe they fit together? What could go wrong? D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 New proposal: “localize” gauge fields using gradient flow Dorota Grabowska, D.B.K. • Phys.Rev.Lett. 116 211602 (2016) [arXiv:1511.03649] • work in progress Gradient flow smooths out fields by evolving them classically in an extra dimension via a heat equation ➟t • • • Gradient flow uses an extra dimension… DWF uses an extra dimension… …maybe they fit together? What could go wrong? D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Gradient flow (continuum version): D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 4d world Gradient flow (continuum version): Āµ (x, t) lives in 5d bulk ➟t @ Āµ (x, t) = @t D⌫ F̄µ⌫ Āµ (x, 0) = Aµ (x) covariant flow eq. boundary condition Aµ (x) lives on 4d boundary of 5d world D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 4d world Gradient flow (continuum version): Āµ (x, t) lives in 5d bulk ➟t @ Āµ (x, t) = @t D⌫ F̄µ⌫ Āµ (x, 0) = Aµ (x) covariant flow eq. boundary condition Aµ (x) lives on 4d boundary of 5d world Aµ ⌘ @µ ! + ✏µ⌫ @⌫ @t ! ¯=0, D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 @t ¯ = ⇤ ¯ 4d world Gradient flow (continuum version): Āµ (x, t) lives in 5d bulk ➟t @ Āµ (x, t) = @t D⌫ F̄µ⌫ Āµ (x, 0) = Aµ (x) covariant flow eq. boundary condition Aµ (x) lives on 4d boundary of 5d world Aµ ⌘ @µ ! + ✏µ⌫ @⌫ Evolution in t damps out high momentum modes in physical degree of freedom only @t ! ¯=0, @t ¯ = ⇤ ¯ ¯ (p, t) = (p)e D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 p2 t 4d world Gradient flow (continuum version): Āµ (x, t) lives in 5d bulk ➟t @ Āµ (x, t) = @t D⌫ F̄µ⌫ Āµ (x, 0) = Aµ (x) covariant flow eq. boundary condition Aµ (x) lives on 4d boundary of 5d world Aµ ⌘ @µ ! + ✏µ⌫ @⌫ Evolution in t damps out high momentum modes in physical degree of freedom only @t ! ¯=0, @t ¯ = ⇤ ¯ ¯ (p, t) = (p)e p2 t This will allow λ(p) to be localized near t=0 while maintaining gauge invariance D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 RH ➠ ➠ Combining gradient flow gauge fields with domain wall fermions: LH gradient flow gauge fields D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 t Combining gradient flow gauge fields with domain wall fermions: RH ➠ ➠ • quantum gauge field Aμ(x) lives at defect at s=0 where LH fermions live LH gradient flow gauge fields D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 t Combining gradient flow gauge fields with domain wall fermions: RH ➠ ➠ • quantum gauge field Aμ(x) lives at defect at s=0 where LH fermions live LH gradient flow gauge fields D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 t Combining gradient flow gauge fields with domain wall fermions: • quantum gauge field Aμ(x) lives at defect at s=0 where LH fermions live ➠ RH ➠ • gauge field Aμ(x,s) defined as solution to gradient flow equation with BC: Aμ(x,0)= Aμ(x) LH gradient flow gauge fields D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 t Combining gradient flow gauge fields with domain wall fermions: • quantum gauge field Aμ(x) lives at defect at s=0 where LH fermions live ➠ RH ➠ • gauge field Aμ(x,s) defined as solution to gradient flow equation with BC: Aμ(x,0)= Aμ(x) LH gradient flow gauge fields D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 t Combining gradient flow gauge fields with domain wall fermions: • quantum gauge field Aμ(x) lives at defect at s=0 where LH fermions live ➠ • flow equation is symmetric on both sides of defect RH ➠ • gauge field Aμ(x,s) defined as solution to gradient flow equation with BC: Aμ(x,0)= Aμ(x) LH gradient flow gauge fields D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 t Combining gradient flow gauge fields with domain wall fermions: • quantum gauge field Aμ(x) lives at defect at s=0 where LH fermions live ➠ • flow equation is symmetric on both sides of defect RH ➠ • gauge field Aμ(x,s) defined as solution to gradient flow equation with BC: Aμ(x,0)= Aμ(x) LH gradient flow gauge fields D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 t Combining gradient flow gauge fields with domain wall fermions: • quantum gauge field Aμ(x) lives at defect at s=0 where LH fermions live ➠ • flow equation is symmetric on both sides of defect RH ➠ • gauge field Aμ(x,s) defined as solution to gradient flow equation with BC: Aμ(x,0)= Aμ(x) LH • RH mirror fermions behave as if with very soft form factor…”Fluff”…and decouple from gauge bosons gradient flow gauge fields D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 t Combining gradient flow gauge fields with domain wall fermions: • quantum gauge field Aμ(x) lives at defect at s=0 where LH fermions live ➠ • flow equation is symmetric on both sides of defect RH ➠ • gauge field Aμ(x,s) defined as solution to gradient flow equation with BC: Aμ(x,0)= Aμ(x) LH • RH mirror fermions behave as if with very soft form factor…”Fluff”…and decouple from gauge bosons gradient flow gauge fields D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 t Combining gradient flow gauge fields with domain wall fermions: • quantum gauge field Aμ(x) lives at defect at s=0 where LH fermions live ➠ • flow equation is symmetric on both sides of defect RH ➠ • gauge field Aμ(x,s) defined as solution to gradient flow equation with BC: Aμ(x,0)= Aμ(x) LH • RH mirror fermions behave as if with very soft form factor…”Fluff”…and decouple from gauge bosons • gauge invariance maintained gradient flow gauge fields D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 t • Mirror top quark (fluff) • mass = 170 GeV • couples only to radio waves? • lattice gauge theorist D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Decoupling mirror fermions as soft fluff in a gauge invariant way: D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Decoupling mirror fermions as soft fluff in a gauge invariant way: 🙂 Can show that this could only lead to a local 4d quantum field theory if the fermion representation has no gauge anomalies ☹ …but exp(-p2t) form factors are a problem in Minkowski spacetime • • flow doesn’t damp out instantons, which can induce 🤔 gradient interactions with fluff • D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Decoupling mirror fermions as soft fluff in a gauge invariant way: 🙂 Can show that this could only lead to a local 4d quantum field theory if the fermion representation has no gauge anomalies ☹ …but exp(-p2t) form factors are a problem in Minkowski spacetime • • flow doesn’t damp out instantons, which can induce 🤔 gradient interactions with fluff • ➟t D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Decoupling mirror fermions as soft fluff in a gauge invariant way: 🙂 Can show that this could only lead to a local 4d quantum field theory if the fermion representation has no gauge anomalies ☹ …but exp(-p2t) form factors are a problem in Minkowski spacetime • • flow doesn’t damp out instantons, which can induce 🤔 gradient interactions with fluff • ➟t Suggests taking t ➝ ∞ limit first…gradient flow like a projection operator A ➝ A D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Decoupling mirror fermions as soft fluff in a gauge invariant way: 🙂 Can show that this could only lead to a local 4d quantum field theory if the fermion representation has no gauge anomalies ☹ …but exp(-p2t) form factors are a problem in Minkowski spacetime • • flow doesn’t damp out instantons, which can induce 🤔 gradient interactions with fluff • ➟t Suggests taking t ➝ ∞ limit first…gradient flow like a projection operator A ➝ A t ➝ ∞ limit suggests finding an overlap operator for this system D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 n particular Refs. [23–25]. The domain wall theory for ector-like gaugeoperator theories pictured in Fig. 2, correThe overlap forisvector theories: 2 Neuberger, Narayanan 1993-1998 ponding to the lattice action D = 1 + ✏ V 5 X ⇤ ¯ S= [ P r5 + P+ r5 + 5 H] (10) Hw ✏ ⌘ ✏(Hw ) = p Hw2 x,s where 5 Hw = ⇥1 (r µ µ 2 + ⇤ rµ ) ⇤ m ⇤ 1 r r µ µ 2 with µ = 1, . . . , 4, 1± 5 P± = , 2 nd r↵ being the covariant forward derivative, r↵ n = (U↵ (n) n+↵ ˆ n) , D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 (11) (12) (13) n particular Refs. [23–25]. The domain wall theory for ector-like gaugeoperator theories pictured in Fig. 2, correThe overlap forisvector theories: 2 Neuberger, Narayanan 1993-1998 ponding to the lattice action D = 1 + ✏ V 5 X ⇤ ¯ S= [ P r5 + P+ r5 + 5 H] (10) Hw ✏ ⌘ ✏(Hw ) = p Hw2 x,s where 5 Hw = ⇥1 (r µ µ 2 + ⇤ rµ ) ⇤ m ⇤ 1 r r µ µ 2 with µ = 1, . . . , 4, DV 1 , 1= ± a 55 P = , ± : s e i t 2 r e p ✓ ◆ pro 1 0 Dµ µ nd r↵ being the covariant forward derivative, lim DV = a!0 r↵ n 5 am = (U↵ (n) Dµ ¯ µ 0 n+↵ ˆ n) , D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 (11) (12) (13) ✏(Hw ) arises as 1 Tn lim n!1 1 + T n where T is the transfer matrix, T =e Hw D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 ✏(Hw ) arises as 1 Tn lim n!1 1 + T n Hw To compute overlap operator for DWF with flowed gauge field, need only replace n T ! n/2 n/2 T? T LH gauge field A gauge field A D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 RH ➠ T =e ➠ where T is the transfer matrix, ✏ ⌘ ✏(Hw [A]) ✏? ⌘ ✏(Hw [A? ]) ➠ An opera H is the Hamiltonian in Eq. (10). This procedure Can construct a gauge invariant overlap operator ds (DG, to DBK, thetomain result of this paper, appear): 1 satisfies the D = 1 + 5 1 (1 ✏? ) (1 ✏) . (22) ✏ ✏? + 1 or equivalen LH We now demonstrate key properties of the chiral overoperator D . where we ha . A. ➠ PROPERTIES OF THE CHIRAL OVERLAP OPERATOR D RH For the chir The continuum limit of D gauge field A This operat The first thing to show is that D has the expected ✓ tinuum limit Eq. (8), up to possibly an overallgauge nor-field A 2 X = 16 ization. The operator ✏(H) has the continuum exD. B. Kaplan ~ Lattice 2016 ~ 30/7/16 ✏ ⌘ ✏(Hw [A]) ✏? ⌘ ✏(Hw [A? ]) ➠ An opera H is the Hamiltonian in Eq. (10). This procedure Can construct a gauge invariant overlap operator ds (DG, to DBK, thetomain result of this paper, appear): 1 satisfies the D = 1 + 5 1 (1 ✏? ) (1 ✏) . (22) ✏ ✏? + 1 or equivalen Obeys Ginsparg-Wilson eq. - U(1)A LH We •now demonstrate key properties of the chiral overHas continuum limit: operator D . where we ha . ➠ PROPERTIES OF THE CHIRAL OVERLAP OPERATOR D RH For the chir • A. The continuum limit of D gauge field A This operat The first thing to show is that D has the expected ✓ tinuum limit Eq. (8), up to possibly an overallgauge nor-field A 2 X = 16 ization. The operator ✏(H) has the continuum exD. B. Kaplan ~ Lattice 2016 ~ 30/7/16 ✏ ⌘ ✏(Hw [A]) ✏? ⌘ ✏(Hw [A? ]) ➠ An opera H is the Hamiltonian in Eq. (10). This procedure Can construct a gauge invariant overlap operator ds (DG, to DBK, thetomain result of this paper, appear): 1 satisfies the D = 1 + 5 1 (1 ✏? ) (1 ✏) . (22) ✏ ✏? + 1 or equivalen Obeys Ginsparg-Wilson eq. - U(1)A LH We •now demonstrate key properties of the chiral overHas continuum limit: operator D . where we ha . ➠ PROPERTIES OF THE CHIRAL OVERLAP OPERATOR D RH For the chir • ✓ ◆ 0 µ Dµ (A) D ! 0 limit of D µ Dµ (A ?) A. ¯The continuum gauge field A This operat The first thing to show is that D has the expected ✓ tinuum limit Eq. (8), up to possibly an overallgauge nor-field A 2 X = 16 ization. The operator ✏(H) has the continuum exD. B. Kaplan ~ Lattice 2016 ~ 30/7/16 1 satisfies the D = 1 + 5 1 (1 ✏? ) (1 ✏) . (22) p ✏ ✏? + 1 a l r or equivalen e v o l a r ✓ ◆ hi t or C a 0 r µ Dµ (A) e p o D ! ¯µOF Dµ (A 0 ?) PROPERTIES THE CHIRAL OVERLAP For the chira OPERATOR D We now demonstrate key properties of the chiral overoperator D . A. where we ha The continuum limit of D he first thing to show is that D has the expected inuum limit Eq. (8), up to possibly an overall norization. The operator ✏(H) has the continuum exsion ✓ ◆ D. B. Kaplan/~ Lattice 2016 ~ 30/7/16 This operato ✓ 2 X = 16 ✓ = 16 1 satisfies the D = 1 + 5 1 (1 ✏? ) (1 ✏) . (22) p ✏ ✏? + 1 a l r or equivalen e v o l a r ✓ ◆ hi t or C a 0 r µ Dµ (A) e p o D ! ¯µOF Dµ (A 0 ?) PROPERTIES THE CHIRAL OVERLAP For the chira OPERATOR D Looks like a LH Weyl fermion interacting with gauge field A, We now demonstrate key properties of the chiral overoperator D . where we ha • A. The continuum limit of D he first thing to show is that D has the expected inuum limit Eq. (8), up to possibly an overall norization. The operator ✏(H) has the continuum exsion ✓ ◆ D. B. Kaplan/~ Lattice 2016 ~ 30/7/16 This operato ✓ 2 X = 16 ✓ = 16 1 satisfies the D = 1 + 5 1 (1 ✏? ) (1 ✏) . (22) p ✏ ✏? + 1 a l r or equivalen e v o l a r ✓ ◆ hi t or C a 0 r µ Dµ (A) e p o D ! ¯µOF Dµ (A 0 ?) PROPERTIES THE CHIRAL OVERLAP For the chira OPERATOR D Looks like a LH Weyl fermion interacting with gauge field A, We now demonstrate keyinteracting propertieswith of the chiral • RH Weyl fermion gauge fieldoverA operator D . where we ha • A. The continuum limit of D he first thing to show is that D has the expected inuum limit Eq. (8), up to possibly an overall norization. The operator ✏(H) has the continuum exsion ✓ ◆ D. B. Kaplan/~ Lattice 2016 ~ 30/7/16 This operato ✓ 2 X = 16 ✓ = 16 1 satisfies the D = 1 + 5 1 (1 ✏? ) (1 ✏) . (22) p ✏ ✏? + 1 a l r or equivalen e v o l a r ✓ ◆ hi t or C a 0 r µ Dµ (A) e p o D ! ¯µOF Dµ (A 0 ?) PROPERTIES THE CHIRAL OVERLAP For the chira OPERATOR D Looks like a LH Weyl fermion interacting with gauge field A, We now demonstrate keyinteracting propertieswith of the chiral • RH Weyl fermion gauge fieldoverA operator D . where • With lattice Wilson flow A ➝A , A will be pure gauge (no we ha stable instantons) • A. The continuum limit of D he first thing to show is that D has the expected inuum limit Eq. (8), up to possibly an overall norization. The operator ✏(H) has the continuum exsion ✓ ◆ D. B. Kaplan/~ Lattice 2016 ~ 30/7/16 This operato ✓ 2 X = 16 ✓ = 16 1 satisfies the D = 1 + 5 1 (1 ✏? ) (1 ✏) . (22) p ✏ ✏? + 1 a l r or equivalen e v o l a r ✓ ◆ hi t or C a 0 r µ Dµ (A) e p o D ! ¯µOF Dµ (A 0 ?) PROPERTIES THE CHIRAL OVERLAP For the chira OPERATOR D Looks like a LH Weyl fermion interacting with gauge field A, We now demonstrate keyinteracting propertieswith of the chiral • RH Weyl fermion gauge fieldoverA operator D . where • With lattice Wilson flow A ➝A , A will be pure gauge (no we ha stable instantons) • A.could Theignore continuum limit D gauge invariance by Or: derivation andof break setting A =0…? This operato he first thing to show is that D has the expected ✓ inuum limit Eq. (8), up to possibly an overall nor2 X = 16 ization. The operator ✏(H) has the continuum ex✓ sion = 16 ✓ ◆ D. B. Kaplan/~ Lattice 2016 ~ 30/7/16 • 1 satisfies the D = 1 + 5 1 (1 ✏? ) (1 ✏) . (22) p ✏ ✏? + 1 a l r or equivalen e v o l a r ✓ ◆ hi t or C a 0 r µ Dµ (A) e p o D ! ¯µOF Dµ (A 0 ?) PROPERTIES THE CHIRAL OVERLAP For the chira OPERATOR D Looks like a LH Weyl fermion interacting with gauge field A, We now demonstrate keyinteracting propertieswith of the chiral • RH Weyl fermion gauge fieldoverA operator D . where • With lattice Wilson flow A ➝A , A will be pure gauge (no we ha stable instantons) • A.could Theignore continuum limit D gauge invariance by Or: derivation andof break setting A =0…? This operato he first thing to show is that D has the expected ✓ inuum limit Eq. (8), up to possibly an overall nor2 Looks like non-interacting RH fermion? X = 16 ization. The operator ✏(H) has the continuum ex✓ sion = 16 ✓ ◆ D. B. Kaplan/~ Lattice 2016 ~ 30/7/16 • Lots of open questions…but the explicit form of the chiral overlap operator allows for experimentation. D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Lots of open questions…but the explicit form of the chiral overlap operator allows for experimentation. Need to understand: D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Lots of open questions…but the explicit form of the chiral overlap operator allows for experimentation. Need to understand: • How the gauge invariant and gauge variant forms differ; D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Lots of open questions…but the explicit form of the chiral overlap operator allows for experimentation. Need to understand: • How the gauge invariant and gauge variant forms differ; • how it can fail if fermion representation has a gauge anomaly; D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Lots of open questions…but the explicit form of the chiral overlap operator allows for experimentation. Need to understand: • How the gauge invariant and gauge variant forms differ; • how it can fail if fermion representation has a gauge anomaly; • whether it can reproduce known results for a vector-like theory D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 Lots of open questions…but the explicit form of the chiral overlap operator allows for experimentation. Need to understand: • How the gauge invariant and gauge variant forms differ; • how it can fail if fermion representation has a gauge anomaly; • whether it can reproduce known results for a vector-like theory • whether the U(1) chiral gauge theory constructed this way has a connection to Lüscher’s implicit GW construction… D. B. Kaplan ~ Lattice 2016 ~ 30/7/16 …and if these ideas don’t work, try something else! D. B. Kaplan ~ Lattice 2016 ~ 30/7/16