Download A New Perspective on Chiral Gauge Theories

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