A New Perspective on Chiral Gauge Theories D. B. Kaplan ~ Lattice - - PowerPoint PPT Presentation
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 GTs which are thought to exhibit massless composite fermions, etc There does not exist
composite fermions, etc
composite fermions, etc
The Standard Model is a χGT!
But of paramount importance:
Nonperturbative definition ⇒
puzzles?
A vector-like gauge theory like QCD consists of Dirac fermions, = Weyl fermions in a real representation of the gauge group.
ZV = Z [dA]e−SY M
Nf
Y
i=1
det( / D − mi) The Problem:
A chiral gauge theory consists of Weyl fermions in a complex representation of the gauge group.
Zχ = Z [dA]e−SY M ∆[A]
A vector-like gauge theory like QCD consists of Dirac fermions, = Weyl fermions in a real representation of the gauge group.
ZV = Z [dA]e−SY M
Nf
Y
i=1
det( / D − mi) The Problem:
Witten: “We often call the fermion path integral a ‘determinant’ or a ‘Pfaffian’, but this is a term of art.”
A chiral gauge theory consists of Weyl fermions in a complex representation of the gauge group.
Zχ = Z [dA]e−SY M ∆[A]
A vector-like gauge theory like QCD consists of Dirac fermions, = Weyl fermions in a real representation of the gauge group.
ZV = Z [dA]e−SY M
Nf
Y
i=1
det( / D − mi) The Problem:
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 A chiral gauge theory consists of Weyl fermions in a complex representation of the gauge group.
Zχ = Z [dA]e−SY M ∆[A]
A vector-like gauge theory like QCD consists of Dirac fermions, = Weyl fermions in a real representation of the gauge group.
ZV = Z [dA]e−SY M
Nf
Y
i=1
det( / D − mi) The Problem:
Vector-like gauge theory with Dirac fermions:
/ Dψ = ✓ Dµσµ Dµ¯ σµ ◆ ✓ψR ψL ◆ = λ ✓ψR ψL ◆
Vector-like gauge theory with Dirac fermions:
/ Dψ = ✓ Dµσµ Dµ¯ σµ ◆ ✓ψR ψL ◆ = λ ✓ψR ψL ◆
Chiral gauge theory with Weyl fermions:
✓0 Dµσµ ◆ ✓ 0 ψL ◆ = ✓χR ◆
Vector-like gauge theory with Dirac fermions:
/ Dψ = ✓ Dµσµ Dµ¯ σµ ◆ ✓ψR ψL ◆ = λ ✓ψR ψL ◆
Chiral gauge theory with Weyl fermions:
✓0 Dµσµ ◆ ✓ 0 ψL ◆ = ✓χR ◆
but no unambiguous way to define the phase θ Can define:
χR = |λ| eiθ ψR
Vector-like gauge theory with Dirac fermions:
/ Dψ = ✓ Dµσµ Dµ¯ σµ ◆ ✓ψR ψL ◆ = λ ✓ψR ψL ◆
Chiral gauge theory with Weyl fermions:
✓0 Dµσµ ◆ ✓ 0 ψL ◆ = ✓χR ◆
but no unambiguous way to define the phase θ Can define:
χR = |λ| eiθ ψR
So:
∆[A] = eiδ[A] q | det / D|
∆[A] = eiδ[A] q | det / D|
The fermion integral for a χGT: The phase δ encodes both anomalies and dynamics δ[A] is generally not gauge invariant (eg, when fermion representation is anomalous)
∆[A] = eiδ[A] q | det / D|
The fermion integral for a χGT: The phase δ encodes both anomalies and dynamics δ[A] is generally not gauge invariant (eg, when fermion representation is anomalous) Do we know δ≠0 for anomaly-free theory?
∆[A] = eiδ[A] q | det / D|
The fermion integral for a χGT: The phase δ encodes both anomalies and dynamics 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
∆[A] = eiδ[A] q | det / D|
The fermion integral for a χGT: The phase δ encodes both anomalies and dynamics 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
∆[A] = eiδ[A] q | det / D|
The fermion integral for a χGT: The phase δ encodes both anomalies and dynamics 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
∆[A] = eiδ[A] q | det / D|
The fermion integral for a χGT: The phase δ encodes both anomalies and dynamics 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
If δ=0, A & B would have same measure, same glue ball spectra…unlikely!
∆[A] = eiδ[A] q | det / D|
The fermion integral for a χGT: Alvarez-Gaume et al. proposal for perturbative definition (1984,1986):
∆[A] ≡ det ✓ Dµσµ ∂µ¯ σµ ◆
gauged LH Weyl fermion neutral RH Weyl fermion
∆[A] = eiδ[A] q | det / D|
The fermion integral for a χGT: Alvarez-Gaume et al. proposal for perturbative definition (1984,1986):
∆[A] ≡ det ✓ Dµσµ ∂µ¯ σµ ◆
gauged LH Weyl fermion neutral RH Weyl fermion Well-defined eigenvalue problem with complex eigenvalues Extra RH fermions decouple Amenable to lattice regularization?
The key problem in representing chiral symmetry on the lattice (global
The key problem in representing chiral symmetry on the lattice (global
One way of looking at anomalies:
p ω
E ⇒
➠ ➠
LH RH
massless electrons in E field, 1+1 dim
The key problem in representing chiral symmetry on the lattice (global
d=(1+1): ∂µjµ
5 = qE
π
quantum violation of a classical U(1)A symmetry One way of looking at anomalies:
p ω
E ⇒
➠ ➠
LH RH
massless electrons in E field, 1+1 dim
The key problem in representing chiral symmetry on the lattice (global
d=(1+1): ∂µjµ
5 = qE
π
quantum violation of a classical U(1)A symmetry In the continuum, the Dirac sea is filled…but is a Hilbert Hotel which always has room for more One way of looking at anomalies:
p ω
E ⇒
➠ ➠
LH RH
massless electrons in E field, 1+1 dim
Not so on the lattice: Can reproduce continuum physics for long wavelength modes…
E ⇒
ω p
➠ ➠
LH RH
Not so on the lattice: Can reproduce continuum physics for long wavelength modes…
E ⇒
ω p
➠ ➠
LH RH
∂µjµ
5 = 0
➠ ➠
…but no anomalies in a system with a finite number of degrees of freedom
Not so on the lattice: Can reproduce continuum physics for long wavelength modes…
E ⇒
ω p
➠ ➠
LH RH
anomalous symmetry in the continuum must be explicitly broken symmetry on the lattice ∂µjµ
5 = 0
➠ ➠
…but no anomalies in a system with a finite number of degrees of freedom
How Wilson fermions reproduce the U(1)A anomaly in QCD:
How Wilson fermions reproduce the U(1)A anomaly in QCD:
How Wilson fermions reproduce the U(1)A anomaly in QCD:
How Wilson fermions reproduce the U(1)A anomaly in QCD:
symmetries
How Wilson fermions reproduce the U(1)A anomaly in QCD:
symmetries
breaking does not decouple & correct anomalous Ward identities are found
How Domain Wall Fermions reproduce the U(1)A anomaly in QCD:
DBK, 1992
extra dimension
LH +Λ
RH fermion mass
How Domain Wall Fermions reproduce the U(1)A anomaly in QCD:
DBK, 1992
extra dimension
LH +Λ
RH fermion mass
How Domain Wall Fermions reproduce the U(1)A anomaly in QCD:
DBK, 1992
extra dimension
LH +Λ
RH fermion mass
symmetry…
How Domain Wall Fermions reproduce the U(1)A anomaly in QCD:
DBK, 1992
extra dimension
LH +Λ
RH fermion mass
symmetry…
except for marginal Chern- Simons current in bulk which allows charges to pass between LH and RH zero modes at mass defects
How Domain Wall Fermions reproduce the U(1)A anomaly in QCD:
DBK, 1992
extra dimension
LH +Λ
RH fermion mass
symmetry…
except for marginal Chern- Simons current in bulk which allows charges to pass between LH and RH zero modes at mass defects
a LH - RH mass term in effective theory since they are physically separated
How Domain Wall Fermions reproduce the U(1)A anomaly in QCD:
DBK, 1992
extra dimension
LH +Λ
RH fermion mass
symmetry…
except for marginal Chern- Simons current in bulk which allows charges to pass between LH and RH zero modes at mass defects
a LH - RH mass term in effective theory since they are physically separated
“topological insulator”
How Overlap Fermions reproduce the U(1)A anomaly in QCD:
Neuberger, Narayanan 1993-1998
RH
LH
extra dim radius L5 ⇒ ∞
How Overlap Fermions reproduce the U(1)A anomaly in QCD:
Neuberger, Narayanan 1993-1998
RH
LH
L = ¯ ψDψ D
= aγ5
extra dim radius L5 ⇒ ∞
How Overlap Fermions reproduce the U(1)A anomaly in QCD:
Neuberger, Narayanan 1993-1998
RH
LH
L = ¯ ψDψ D
= aγ5
extra dim radius L5 ⇒ ∞
D−1 = ✓ 0 C −C† ◆ + a 2 ✓1 1 ◆
How Overlap Fermions reproduce the U(1)A anomaly in QCD:
Neuberger, Narayanan 1993-1998
RH
LH
L = ¯ ψDψ D
= aγ5
extra dim radius L5 ⇒ ∞ Chiral symmetric
D−1 = ✓ 0 C −C† ◆ + a 2 ✓1 1 ◆
How Overlap Fermions reproduce the U(1)A anomaly in QCD:
Neuberger, Narayanan 1993-1998
RH
LH
L = ¯ ψDψ D
= aγ5
Explicit chiral symmetry breaking extra dim radius L5 ⇒ ∞ Chiral symmetric
D−1 = ✓ 0 C −C† ◆ + a 2 ✓1 1 ◆
Back to the continuum operator: Two anomaly issues to address:
Dχ = ✓ Dµσµ ∂µ¯ σµ ◆
Back to the continuum operator: Two anomaly issues to address:
Dχ = ✓ Dµσµ ∂µ¯ σµ ◆
Here: focus on global anomaly: It requires explicit U(1)A chiral symmetry breaking on the lattice:
Dχ = ✓ X C −c† X ◆
Back to the continuum operator: Two anomaly issues to address:
Dχ = ✓ Dµσµ ∂µ¯ σµ ◆
Here: focus on global anomaly: It requires explicit U(1)A chiral symmetry breaking on the lattice:
Dχ = ✓ X C −c† X ◆
…but if LH fermion is gauged and RH is neutral, X terms coupling them violate gauge symmetry!
Dχ = ✓ X C −c† X ◆
Apparently two alternatives for the lattice in order to realize U(1)A anomaly in a chiral gauge theory:
Dχ = ✓ X C −c† X ◆
Apparently two alternatives for the lattice in order to realize U(1)A anomaly in a chiral gauge theory:
not violate gauge symmetry)… and then decouple somehow
Dχ = ✓ X C −c† X ◆
Apparently two alternatives for the lattice in order to realize U(1)A anomaly in a chiral gauge theory:
not violate gauge symmetry)… and then decouple somehow
the mirror fermion couplings
not violate gauge symmetry)… and then decouple somehow
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
way.
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
way.
the mirror fermion couplings
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
way.
the mirror fermion couplings
Bock, Golterman, Shamir 1998, 2004
Some work along these lines that looks OK in perturbation theory
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
way.
the mirror fermion couplings
Bock, Golterman, Shamir 1998, 2004
Some work along these lines that looks OK in perturbation theory A natural way to examine the problem is with Domain Wall Fermions…
Dχ = ✓ Dµσµ ∂µ¯ σµ ◆
Motivation:
RH
LH
localized gauge fields
Suggests: localizing gauge fields near one domain wall Requires “Higgs” field at boundary to maintain gauge invariance Higgs
+Λ
LH
localized gauge fields
Old attempts to use domain wall fermions for chiral gauge theory Localize the gauge fields around one
RH
Higgs Not compelling…how would theory know to fail when there are gauge anomalies?
+Λ
LH
localized gauge fields
Old attempts to use domain wall fermions for chiral gauge theory Localize the gauge fields around one
RH
Never works.
One finds a Dirac fermion & vector-like gauge theory.
Golterman, Jansen, Vink 1993
Higgs Not compelling…how would theory know to fail when there are gauge anomalies?
New proposal: “localize” gauge fields using gradient flow
:
Gradient flow smooths out fields by evolving them classically in an extra dimension via a heat equation
t
Dorota Grabowska, D.B.K.
New proposal: “localize” gauge fields using gradient flow
:
Gradient flow smooths out fields by evolving them classically in an extra dimension via a heat equation
t
wrong?
Dorota Grabowska, D.B.K.
New proposal: “localize” gauge fields using gradient flow
:
Gradient flow smooths out fields by evolving them classically in an extra dimension via a heat equation
t
wrong?
Dorota Grabowska, D.B.K.
Gradient flow (continuum version):
Gradient flow (continuum version): 4d world
t
lives on 4d boundary of 5d world lives in 5d bulk
Aµ(x) ∂ ¯ Aµ(x, t) ∂t = −Dν ¯ Fµν ¯ Aµ(x, 0) = Aµ(x) ¯ Aµ(x, t)
covariant flow eq. boundary condition
Gradient flow (continuum version):
∂t¯ ω = 0 , ∂t¯ λ = ⇤¯ λ ⇒ 4d world
t
lives on 4d boundary of 5d world lives in 5d bulk
Aµ(x) ∂ ¯ Aµ(x, t) ∂t = −Dν ¯ Fµν ¯ Aµ(x, 0) = Aµ(x) ¯ Aµ(x, t)
covariant flow eq. boundary condition
Gradient flow (continuum version):
∂t¯ ω = 0 , ∂t¯ λ = ⇤¯ λ ⇒ 4d world
t
lives on 4d boundary of 5d world lives in 5d bulk
Aµ(x) ∂ ¯ Aµ(x, t) ∂t = −Dν ¯ Fµν ¯ Aµ(x, 0) = Aµ(x) ¯ Aµ(x, t)
covariant flow eq. boundary condition
Evolution in t damps out high momentum modes in physical degree of freedom only
¯ λ(p, t) = λ(p)e−p2t
Gradient flow (continuum version):
∂t¯ ω = 0 , ∂t¯ λ = ⇤¯ λ ⇒ 4d world
t
lives on 4d boundary of 5d world lives in 5d bulk
Aµ(x) ∂ ¯ Aµ(x, t) ∂t = −Dν ¯ Fµν ¯ Aµ(x, 0) = Aµ(x) ¯ Aµ(x, t)
covariant flow eq. boundary condition
This will allow λ(p) to be localized near t=0 while maintaining gauge invariance
Evolution in t damps out high momentum modes in physical degree of freedom only
¯ λ(p, t) = λ(p)e−p2t
gradient flow gauge fields
t
Combining gradient flow gauge fields with domain wall fermions: RH LH
gradient flow gauge fields
t
Combining gradient flow gauge fields with domain wall fermions:
defect at s=0 where LH fermions live
RH LH
gradient flow gauge fields
t
Combining gradient flow gauge fields with domain wall fermions:
defect at s=0 where LH fermions live
RH LH
gradient flow gauge fields
t
Combining gradient flow gauge fields with domain wall fermions:
defect at s=0 where LH fermions live
to gradient flow equation with BC: Aμ(x,0)= Aμ(x)
RH LH
gradient flow gauge fields
t
Combining gradient flow gauge fields with domain wall fermions:
defect at s=0 where LH fermions live
to gradient flow equation with BC: Aμ(x,0)= Aμ(x)
RH LH
gradient flow gauge fields
t
Combining gradient flow gauge fields with domain wall fermions:
defect at s=0 where LH fermions live
to gradient flow equation with BC: Aμ(x,0)= Aμ(x)
sides of defect
RH LH
gradient flow gauge fields
t
Combining gradient flow gauge fields with domain wall fermions:
defect at s=0 where LH fermions live
to gradient flow equation with BC: Aμ(x,0)= Aμ(x)
sides of defect
RH LH
gradient flow gauge fields
t
Combining gradient flow gauge fields with domain wall fermions:
defect at s=0 where LH fermions live
to gradient flow equation with BC: Aμ(x,0)= Aμ(x)
sides of defect
very soft form factor…”Fluff”…and decouple from gauge bosons
RH LH
gradient flow gauge fields
t
Combining gradient flow gauge fields with domain wall fermions:
defect at s=0 where LH fermions live
to gradient flow equation with BC: Aμ(x,0)= Aμ(x)
sides of defect
very soft form factor…”Fluff”…and decouple from gauge bosons
RH LH
gradient flow gauge fields
t
Combining gradient flow gauge fields with domain wall fermions:
defect at s=0 where LH fermions live
to gradient flow equation with BC: Aμ(x,0)= Aμ(x)
sides of defect
very soft form factor…”Fluff”…and decouple from gauge bosons
RH LH
(fluff)
radio waves?
theory if the fermion representation has no gauge anomalies
spacetime
interactions with fluff
theory if the fermion representation has no gauge anomalies
spacetime
interactions with fluff
t
theory if the fermion representation has no gauge anomalies
spacetime
interactions with fluff
t
Suggests taking t ➝ ∞ limit first…gradient flow like a projection
theory if the fermion representation has no gauge anomalies
spacetime
interactions with fluff
t
Suggests taking t ➝ ∞ limit first…gradient flow like a projection
t ➝ ∞ limit suggests finding an overlap operator for this system
Neuberger, Narayanan 1993-1998
DV = 1 + 5✏ ✏ ≡ ✏(Hw) = Hw p H2
w
2µ(rµ + r∗ µ) 1 2rµr∗ µ m
w
Neuberger, Narayanan 1993-1998
DV = 1 + 5✏ ✏ ≡ ✏(Hw) = Hw p H2
w
2µ(rµ + r∗ µ) 1 2rµr∗ µ m
w
= aγ5
V
lim
a→0 DV =
1 am ✓ Dµσµ Dµ¯ σµ ◆
properties:
✏(Hw)
lim
n→∞
1 − T n 1 + T n
T = e−Hw
✏(Hw)
lim
n→∞
1 − T n 1 + T n
T = e−Hw
gauge field A gauge field A
RH LH
T n → T n/2
?
T n/2
RH LH
gauge field A gauge field A
(DG, DBK, to appear):
1 − (1 − ✏?) 1 ✏ ✏? + 1 (1 − ✏)
✏ ≡ ✏(Hw[A])
RH LH
gauge field A gauge field A
(DG, DBK, to appear):
1 − (1 − ✏?) 1 ✏ ✏? + 1 (1 − ✏)
✏ ≡ ✏(Hw[A])
RH LH
gauge field A gauge field A
(DG, DBK, to appear):
1 − (1 − ✏?) 1 ✏ ✏? + 1 (1 − ✏)
✏ ≡ ✏(Hw[A])
D → ✓ σµDµ(A) ¯ σµDµ(A?) ◆
D → ✓ σµDµ(A) ¯ σµDµ(A?) ◆
1 − (1 − ✏?) 1 ✏ ✏? + 1 (1 − ✏)
D → ✓ σµDµ(A) ¯ σµDµ(A?) ◆
1 − (1 − ✏?) 1 ✏ ✏? + 1 (1 − ✏)
Chiral overlap
D → ✓ σµDµ(A) ¯ σµDµ(A?) ◆
1 − (1 − ✏?) 1 ✏ ✏? + 1 (1 − ✏)
Chiral overlap
D → ✓ σµDµ(A) ¯ σµDµ(A?) ◆
1 − (1 − ✏?) 1 ✏ ✏? + 1 (1 − ✏)
stable instantons)
Chiral overlap
D → ✓ σµDµ(A) ¯ σµDµ(A?) ◆
1 − (1 − ✏?) 1 ✏ ✏? + 1 (1 − ✏)
stable instantons)
setting A=0…?
Chiral overlap
Looks like non-interacting RH fermion?
D → ✓ σµDµ(A) ¯ σµDµ(A?) ◆
1 − (1 − ✏?) 1 ✏ ✏? + 1 (1 − ✏)
stable instantons)
setting A=0…?
Chiral overlap
Lots of open questions…but the explicit form of the chiral overlap
Lots of open questions…but the explicit form of the chiral overlap
Need to understand:
Lots of open questions…but the explicit form of the chiral overlap
Need to understand:
Lots of open questions…but the explicit form of the chiral overlap
Need to understand:
anomaly;
Lots of open questions…but the explicit form of the chiral overlap
Need to understand:
anomaly;
theory
Lots of open questions…but the explicit form of the chiral overlap
Need to understand:
anomaly;
theory
has a connection to Lüscher’s implicit GW construction…
…and if these ideas don’t work, try something else!