Gauge invariant regularization for perturbative chiral ... › ~ppp.ws › PPP2018 › slides › hamada.pdfGauge invariant regularization for perturbative chiral gauge theory Yu Hamada

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

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

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

Our answer

� �5D Domain-Wall fermion with PV regulators

+(4 + �)-D gauge field� �

This regularization is quite useful!4 / 23

One-loop diagrams

5 / 23

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

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

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

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

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

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


]|s〉

(ii)∫ds 〈s|


]|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

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

Pauli-Villars regularization

To make the loop UV-finite, we introduce Pauli-Villars fields

�kp

kp

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X

i=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AAACkXichVG7SgNBFD2u73d8FIJNMEQEJUxEMFoFbQQbo8YEYpDdzSQO2Re7k4AGf0BbxcJKwUL8Bxsbf8DCTxDLCDYW3mwWRIPxDrNz7pl77ty7V3MM4UnGXjqUzq7unt6+/oHBoeGR0dDY+J5nV1ydp3XbsN2spnrcEBZPSyENnnVcrpqawTNaeb1xn6ly1xO2tSuPHJ431ZIlikJXJVG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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


]|s〉

(ii)∫ds 〈s|


]|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

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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Documents

Gauge invariant regularization for perturbative chiral ... › ~ppp.ws › PPP2018 › slides › hamada.pdfGauge invariant regularization for perturbative chiral gauge theory Yu Hamada