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Gauge invariant regularization forperturbative chiral gauge theory
Yu HamadaKyoto Univ.
2018/8/9 PPP2018@YITP
Based onYH, H. Kawai, arXiv:1705.01317,YH, H. Kawai, K. Sakai, arXiv:1806.00349YH, H. Kawai, K. Sakai, to appear
1 / 23
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Motivation
Standard Model is a chiral gauge theory (CGT)
SU(3)C × SU(2)L × U(1)Y (EW sector)
However, regularization of CGT is difficult!
• No lattice regulator [cf. Grabowska-Kaplan, 2015]• No manifestly gauge-invariant perturbative regulator
because fermion mass term is forbidden by chiral gaugesymmetry...
2 / 23
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Regularization problem for CGT
Eg. Dimensional Regularization
Lreg. = ψ̄(/D(4) + /D(�)
)PLψ,
[/D(�), γ5
]= 0
• �-dimensional kinetic term behaves as “mass term”• Gauge sym. is broken even when anomaly-free theory• Need extra local counter terms to restore gauge sym:
Γ[A] + ∆Γ[A] s.t. δω (Γ[A] + ∆Γ[A]) = 0
• However, the procedure is rather complicated...
Is there a gauge-invariant regularization for CGT?(except for gauge anomaly)
Note: Only for one-loop calculations, the naive prescription{/D(�), γ5
}= 0
can be used.3 / 23
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Our answer
� �5D Domain-Wall fermion with PV regulators
+(4 + �)-D gauge field� �
This regularization is quite useful!4 / 23
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One-loop diagrams
5 / 23
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One-loop diagrams
5 / 23
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Domain-wall fermion [Kaplan, 1992]
SDW =
∫d4x
∫ ∞
−∞ds ψ̄(x, s)
[/∂(4) + γ5∂s −M�(s)
]ψ(x, s)
• �(s) is the sign function (M > 0)→ induce LH massless mode ∝ e−M |s|
• s-direction’s size is infinitely large→ RH massless mode is decoupled• Massive modes form a continuous
spectrum and give rise to IR divergence→ cancel by bosonic field φ(x, s) withconstant mass (−M )
s = 0
+M
�M
M✏(s)
s
6 / 23
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Action
� �S =
∫d4x
∫ds ψ̄(x, s)
[/D(4) + γ5∂s −M�(s)
]ψ(x, s)
+
∫d4x
∫ds φ̄(x, s)
[/D(4) + γ5∂s +M
]φ(x, s)
+1
4g2
∫d4x tr(Fµν(x)Fµν(x))� �
• boson φ will cancel IR div. from ψ• Gauge field is 4-dimensional one Aµ(x):
s-indep. & A5 = 0
• Dirac fermion (boson) are expected to beregularized in a gauge-invariant way
s
7 / 23
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Action in 4D form [Narayanan-Neuberger, 1993]� �S =
∫d4x
∫ds ψ̄(x, s)
[/D(4) + γ5∂s −M�(s)
]ψ(x, s)
+
∫d4x
∫ds φ̄(x, s)
[/D(4) + γ5∂s +M
]φ(x, s)
+1
4g2
∫d4x tr(Fµν(x)Fµν(x))� �
⇓ “mass operators”{M̂ψ ≡ −∂s −M�(s)M̂φ ≡ −∂s +M� �
S =
∫d4x
∫ds ψ̄(x, s)
[/D(4) + M̂ψPL + M̂
†ψPR
]ψ(x, s)
+
∫d4x
∫ds φ̄(x, s)
[/D(4) + M̂φPL + M̂
†φPR
]φ(x, s)
+1
4g2
∫d4x tr(Fµν(x)Fµν(x)),� �
s-space looks like an internal space of 4D spinors ψ, φ8 / 23
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Vacuum polarization diagram
• Propagator
Gψ(p) =(−i/p+ M̂ψ
)1
p2+M̂†ψM̂ψPR +
(−i/p+ M̂†ψ
)1
p2+M̂ψM̂†ψPL
Gφ(p) =−i/p+M̂φPR+M̂†φPL
p2+M̂†φM̂φ
• Vaccum polarization diagram
�kp
k
p0 = p � 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
⇧µ⌫(k) =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
=
∫d4p
(2π)4(Tr[Gψ(p)γµGψ(p
′)γν]− Tr
[Gφ(p)γµGφ(p
′)γν])
• Tr includes the trace over s-space where M̂ψ and M̂φ act.• We will regulate the UV divergence later. 9 / 23
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How to define Tr ?� �Tr[Gψ(p)γµGψ(p
′)γν]− Tr
[Gφ(p)γµGφ(p
′)γν]
� �• Each trace is IR divergent→ First subtract these contents,
and then take the trace over s-space.• However, there is an ambiguity in the choice of the starting
point of the loop.
(i)∫ds 〈s|
[GψγµG
′ψγν −GφγµG′φγν
]|s〉
(ii)∫ds 〈s|
[G′ψγνGψγµ −G′φγνGφγµ
]|s〉
• As seen later, the ambiguity vanishes when anomaly-free.• We adopt the symmetric choice (iii) ≡ 1
2(i) +
1
2(ii) for the
moment.
10 / 23
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How to define Tr ?� �Tr[Gψ(p)γµGψ(p
′)γν]− Tr
[Gφ(p)γµGφ(p
′)γν]
� �• Each trace is IR divergent→ First subtract these contents,
and then take the trace over s-space.• However, there is an ambiguity in the choice of the starting
point of the loop.
(i)∫ds 〈s|
[GψγµG
′ψγν −GφγµG′φγν
]|s〉
(ii)∫ds 〈s|
[G′ψγνGψγµ −G′φγνGφγµ
]|s〉
• As seen later, the ambiguity vanishes when anomaly-free.• We adopt the symmetric choice (iii) ≡ 1
2(i) +
1
2(ii) for the
moment.10 / 23
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Computation of Tr :(1)
∫ds 〈s|
[GψγµG
′ψγν −GφγµG′φγν
]|s〉
=
∫ds 〈s|
[tr(i/pγµi/p
′γνPR)
+ tr (γµγνPR)M̂†ψM̂ψ(p2 + M̂†ψM̂ψ)(p′2 + M̂
†ψM̂ψ)
+tr(i/pγµi/p
′γνPL)
+ tr (γµγνPL)M̂ψM̂†ψ(p2 + M̂ψM̂†ψ)(p′2 + M̂ψM̂
†ψ)
−tr(i/pγµi/p
′γν)
+ tr (γµγν)M̂†φM̂φ(p2 + M̂†φM̂φ)(p′2 + M̂
†φM̂φ)
]|s〉
In the 1st and 2nd terms, insert the complete sets of M̂†ψM̂ψ and M̂ψM̂†ψ:
1 = |0〉 〈0|+∫ ∞0
dω |lω〉 〈lω| , 1 =∫ ∞0
dω |rω〉 〈rω|M̂†ψM̂ψ |0〉 = 0, M̂
†ψM̂ψ |lω〉 = (M
2 + ω2) |lω〉M̂ψM̂†ψ |rω〉 = (M
2 + ω2) |rω〉
In the 3rd term, insert the complete set of M̂†φM̂φ:
1 =
∫ ∞0
dω |φω〉 〈φω| , M̂†φM̂φ |φω〉 = (M2 + ω2) |φω〉
Note:[M̂φ,M̂†φ
]= 0
11 / 23
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Computation of Tr : (2)
=
∫ds | 〈s|0〉 |2
tr(i/pγµi/p
′γνPL)
p2p′2
+
∫ds
∫ ∞0
dω
[| 〈s|rω〉 |2
tr(i/pγµi/p
′γνPR)
+ tr (γµγνPR) (M2 + ω2)
(p2 +M2 + ω2)(p′2 +M2 + ω2)
+ | 〈s|lω〉 |2tr(i/pγµi/p
′γνPL)
+ tr (γµγνPL) (M2 + ω2)
(p2 +M2 + ω2)(p′2 +M2 + ω2)
−| 〈s|φω〉tr(i/pγµi/p
′γν)
+ tr (γµγν) (M2 + ω2)
(p2 +M2 + ω2)(p′2 +M2 + ω2)
]
=tr(i/pγµi/p
′γνPL)
p2p′2
+
∫ds
∫ ∞0
dω[[| 〈s|rω〉 |2 + | 〈s|lω〉 |2 − 2| 〈s|φω〉 |2
]=2δ(ω)
tr(i/pγµi/p
′γν)
+ tr (γµγν) (M2 + ω2)
2(p2 +M2 + ω2)(p′2 +M2 + ω2)
+
∫ds
∫ ∞0
dω[| 〈s|rω〉 |2 − | 〈s|lω〉 |2
]= 2M
M2+ω2
tr(i/pγµi/p
′γνγ5)
+ tr (γµγνγ5) (M2 + ω2)
2(p2 +M2 + ω2)(p′2 +M2 + ω2)
12 / 23
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Pauli-Villars regularization
To make the loop UV-finite, we introduce Pauli-Villars fields
�kp
kp
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i=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�Ci
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
Mi✏(s)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
• Ci,Mi have to be chosen such that the sum is UV-finite.• Need another condition to eliminate the extra chiral
fermions : ∑
i=1
Ci = 0
13 / 23
-
Result� �Πreg.µν (k)
=
∫d4p
(2π)4
{1
2tr
[i/p
p2γµi/p′
p′2γν
]− 1
2tr
[i/p+M
p2 +M2γµ
i/p′ +M
p′2 +M2γν
]
−∑i
Ci1
2tr
[i/p+Mi
p2 +Mi2γµ
i/p′ +Mi
p′2 +Mi2γν
]+ f(p, p′)tr
[i/p
p2γµi/p′
p′2γνγ5
]}� �f(p, p
′) ≡
∑i=0
Ci
1−√p2 +Mi2
(√p2 +Mi2
√p′2 +Mi2 +Mi
2)
√p′2 +Mi2
(√p2 +Mi2 +
√p′2 +Mi2
)2 Mi
2√p2 +Mi2
+(p↔ p′)
(C0 ≡ 1, M0 ≡M)
Parity-even Regularized via 4D Pauli-Villars fields in agauge-invariant way
Parity-odd Multiplied by a non-local regularization factor
The entire is UV-finite!14 / 23
-
Gauge anomaly
• The parity-odd part reproduces the correct gauge anomaly.
Of course, 4D anomaly is obtained from the triangle andrectangle diagrams.For simplicity, let us consider 2D non-Abelian CGT.
δωSeff [A] = ωa(x) (Dµ 〈Jµ(x)〉)a
→ − 14π�µν ω
a(x)tr [ta∂µAν(x)] (M2 →∞)
2D consistent anomaly!
taγµ
15 / 23
-
How to define Tr? -revisited-
� �Tr[Gψ(p)γµGψ(p
′)γν]− Tr
[Gφ(p)γµGφ(p
′)γν]
� �(i)∫ds 〈s|
[GψγµG
′ψγν −GφγµG′φγν
]|s〉
(ii)∫ds 〈s|
[G′ψγνGψγµ −G′φγνGφγµ
]|s〉
(iii)1
2(i) +
1
2(ii)
(i)→ covariant anomaly: (DµJµ(x))a = −1
2π�µν tr [t
aFµν ]
∴ The choices correspond to the consistent and covariantanomaly!For anomaly-free cases, no ambiguity because both are 0.
16 / 23
-
One-loop diagrams
17 / 23
-
One-loop diagrams
17 / 23
-
Fermion number anomaly
Fermion number anomaly is physical, so it should be definedwithout any ambiguity.→ Define the fermion number current with a damping factor:
〈jU(1)µ (x)〉 ≡ limξ→0
∫ds e−ξ|s|
[〈ψ̄γµψ〉+ 〈φ̄γµφ〉
]+ (PV-fields)
e−ξ|s|γµ
• The IR divergence is regularized by ξ and then canceled.• The regularized current is automatically gauge invariant
without any ambiguity:
〈∂µjU(1)µ (x)〉 =−1
32π2tr[FF̃]
18 / 23
-
Other one-loop diagrams
• Can be regularized by applying the dim. reg. only to thegauge field.
1
4g2
∫d4x tr (Fµν(x))
2 → 14g2
∫d4+�x tr (Fmn(x̃))
2
x̃ = (x(4), x�), m, n = 1, · · · , (4 + �)
19 / 23
-
Gauge boson loop
• The gauge boson’s loops can be regularized as theordinary dimensional regularization.
20 / 23
-
Fermion self-energy
p
(k, k�)
p′ = p − k
• The �-dimensional momentum k� is not conserved at thevertices because fermion’s momentum is 4-dimensional.
− tata∫
d4+�k
(2π)4+�1
k2 + (k�)2γµGψ(p− k)γµ
= −tata∫
d4k
(2π)4Γ(1− �/2)
(4π)�/21
(k2)1−�/2γµGψ(p− k)γµ
= −tataΓ(−�/2)16π2
[i/p+ 4
(M̂ψPL + M̂†ψPR
)]+ (finite terms)
• Regularized via the parameter �21 / 23
-
Vertex correction
(k, k�)
(l, l�)p p+ k
• Can be regularized similarly
= −iΓ(1− �/2)(4π)�/2
∫d4l
(2π)4tatbta
(l2)1−�/2γµGψ(p+ k − l)γνGψ(p− l)γµ
22 / 23
-
Summary
• We propose a gauge-invariant regularization forperturbative chiral gauge theory.
• 5D domain-wall fermion with PV regulators can describethe 4D chiral fermion’s loop in a gauge-invariant way.
• The expression for the gauge anomaly has an ambiguity,which vanishes for anomaly-free cases.
• Fermion number anomaly can be obtained in thegauge-invariant form.
• The other loop diagrams can be regularized by the(4 + �)-dimensional gauge field.
• (Regularized UV divergences can be renormalized.)
Please use this regularization !23 / 23
-
Backup Slides
23 / 23
-
Anomaly inflow
• Effective theory of DW fermion= Chern-Simons action in bulk + LH and RH chiral fermions
• Gauge current flows from domain wall to anti domain wallthrough bulk
• We decoupled RH mode by L→∞→ Flowing out of gauge current seems to be anomaly!• This current is sensitive to boundary condition at s =∞
(=IR regulator)
s = 0 s = L
CS currentLH mode RH mode
24 / 23
-
Renormalizability (One-loop) (1)
• Vacuum polarization (with external line)
∫d4k
(2π)4d�k�(2π)�
d�k′�(2π)�
Aµ(k, k�)Aν(−k, k′�)Πµν(k)
−→div. term
∫d4x Aµ(x, x� = 0)Aν(x, x� = 0) δµνc logM
2
• The (regularized) UV divergence is renormalized by acounter term on the brane x� = 0.
25 / 23
-
Renormalizability (One-loop) (2)
• Vertex correction (with external line)
• This UV divergence is also renormalized by a counter termon the brane x� = 0.
26 / 23
-
Renormalizability (Two-loop)
• BPHZ scheme can be applied except that loopsub-diagrams with fermion internal lines should bereplaced by counter terms on the brane.
27 / 23
-
Gauge invariance
• Fermion loop induces 4D gauge-invariant term:∫d4x log(∂2/µ2)tr (Fµν(x, 0))
2 , (µ, ν = 1, · · · , 4)
This term is also invariant under the following (4 + �)-D gaugetransformation:
Am(x, x�)→ e−iω(x,x�) [Am(x, x�) + ∂m] eiω(x,x�)
(m,n = 1, · · · , 4 + �)
• Therefore, (4 + �)-D gauge invariance is preserved.
28 / 23
-
Unitarity
The net effect of the mixed dimension is the following:
For external lineNon-conservation of the �-dimensional momentum of Am→ affect unitarity!
For internal lineChange the power of 4D propagators :
1
k2→ 1
(k2)1−�/2
→ does not!
Therefore, unitarity would be restored in the limit of �→ 0.
29 / 23
-
Pauli-Villars
• diagramに現れる全ての発散を打ち消すように条件を課して、 を決める
• UV発散のdiagramに対して、発散を打ち消すために複数個のPauli-Villars場を導入
+ C1M1 M2
C2+ · · ·M
+
M +X
i
CiMi = 0,
...
1 +X
i
Ci = 0,
M2 +X
i
Ci(Mi)2 = 0,
Ci, Mi
Zddp
(2⇡)d
"1
p2 + M2+X
i
Ci1
p2 + (Mi)2
#
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例)
• 繰り込んだ後、最後に の極限をとってdecoupleさせるMi ! 1
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