Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 1

2PI effective action for gauge theories:Symmetry & Renormalization

Urko Reinosa

Centre de Physique Théorique(CNRS - Ecole Polytechnique)

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 2

2PI in a nutshell

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 3

The 2PI effective action

An approximation scheme in Quantum Field Theory:

• Good convergence properties.

• It captures certain non-perturbative aspects.

• Wide range of applications in- and out-of-equilibrium.

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 3

The 2PI effective action

An approximation scheme in Quantum Field Theory:

• Good convergence properties.

• It captures certain non-perturbative aspects.

• Wide range of applications in- and out-of-equilibrium.

How solid are its theoretical grounds?

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 3

The 2PI effective action

An approximation scheme in Quantum Field Theory:

• Good convergence properties.

• It captures certain non-perturbative aspects.

• Wide range of applications in- and out-of-equilibrium.

How solid are its theoretical grounds?

QED in the covariant gauge with gauge fixing parameter ξ

S =

Z

ddx

ψ [i /∂ − e /A−m]ψ −1

4FµνFµν

ff

−1

ξ

Z

ddx (∂µAµ)2

| {z }

Sgf

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 4

The 2PI effective action

The 2PI effective action for Quantum Electrodynamics

Γ2PI[ϕ,G] = S[ϕ] +i

2Str logG−1 +

i

2StrG−1

0 G

+ + +

+ + + + + + . . .

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 4

The 2PI effective action

The 2PI effective action for Quantum Electrodynamics

Γ2PI[ϕ,G] = S[ϕ] +i

2Str logG−1 +

i

2StrG−1

0 G

+ + +

+ + + + + + . . .

It is convenient to introduce a superfield notation

ϕ ≡

0

B

B

@

A

ψ

ψt

1

C

C

A

and G ≡ 〈ϕϕt〉c ≡

0

B

B

@

G Kt K

K F D

Kt −Dt F

1

C

C

A

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 4

The 2PI effective action

The 2PI effective action for Quantum Electrodynamics

Γ2PI[ϕ,G] = S[ϕ] +i

2Str logG−1 +

i

2StrG−1

0 G

+ + +

+ + + + + + . . .

The physics is obtained from a variational principle

Γ[ϕ] = Γ2PI[ϕ, G[ϕ]] where 0 =δΓ2PI

δG

˛

˛

˛

˛

G[ϕ]

A given truncation of Γ2PI[ϕ,G] defines a non-trivial approximation of Γ[ϕ].

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 4

The 2PI effective action

The 2PI effective action for Quantum Electrodynamics

Γ2PI[ϕ,G] = S[ϕ] +i

2Str logG−1 +

i

2StrG−1

0 G

+ + +

+ + + + + + . . .

To which extent does Γ[ϕ] reflect the basic properties of the theory?

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 4

The 2PI effective action

The 2PI effective action for Quantum Electrodynamics

Γ2PI[ϕ,G] = S[ϕ] +i

2Str logG−1 +

i

2StrG−1

0 G

+ + +

+ + + + + + . . .

To which extent does Γ[ϕ] reflect the basic properties of the theory?

Symmetry? Renormalizability?

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 4

The 2PI effective action

The 2PI effective action for Quantum Electrodynamics

Γ2PI[ϕ,G] = S[ϕ] +i

2Str logG−1 +

i

2StrG−1

0 G

+ + +

+ + + + + + . . .

To which extent does Γ[ϕ] reflect the basic properties of the theory?

Symmetry? Renormalizability?

X Ward-Takahashi identities.

? Gauge-fixing independence.

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 4

The 2PI effective action

The 2PI effective action for Quantum Electrodynamics

Γ2PI[ϕ,G] = S[ϕ] +i

2Str logG−1 +

i

2StrG−1

0 G

+ + +

+ + + + + + . . .

To which extent does Γ[ϕ] reflect the basic properties of the theory?

Symmetry? Renormalizability?

X Ward-Takahashi identities. ⇒ Vertices.

? Gauge-fixing independence.

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 5

2PI and 2PI-resummed vertices

Two possible definitions of the two-point function

i δ2Γ

δϕ2 δϕ1or G−1

12

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 5

2PI and 2PI-resummed vertices

If no truncation, both definitions agree

i δ2Γ

δϕ2 δϕ1= G−1

12

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 5

2PI and 2PI-resummed vertices

In a given truncation, they become different

i δ2Γ

δϕ2 δϕ16= G−1

12

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 5

2PI and 2PI-resummed vertices

In a given truncation, they become different

i δ2Γ

δϕ2 δϕ16= G−1

12

A similar remark applies for higher vertices

i δnΓ

δϕn · · · δϕ16=

δn−2G−112

δϕn · · · δϕ3

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 5

2PI and 2PI-resummed vertices

In a given truncation, they become different

i δ2Γ

δϕ2 δϕ16= G−1

12

A similar remark applies for higher vertices

2PI-resummed verticesi δnΓ

δϕn · · · δϕ16=

δn−2G−112

δϕn · · · δϕ32PI vertices

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 6

2PI and 2PI-resummed vertices

The 2PI-resummed two-point vertex involves the 2PI three-point vertex

δ2Γ

δϕ2δϕ1= (−1)q1 iG−1

0,21 +δ2Γint

δϕ2δϕ1+ G

δG−1

δϕ2Gδ2Γint

δGδϕ1

δ2Γ

δAµδAν= iG

−10,µν + +

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 6

2PI and 2PI-resummed vertices

The 2PI-resummed two-point vertex involves the 2PI three-point vertex

δ2Γ

δϕ2δϕ1= (−1)q1 iG−1

0,21 +δ2Γint

δϕ2δϕ1+ G

δG−1

δϕ2Gδ2Γint

δGδϕ1

δ2Γ

δAµδAν= iG

−10,µν + +

The 2PI-resummed three-point vertex involves the 2PI three- and four-point vertices

δ3Γ

δϕ3δϕ2δϕ1=

δ3Γint

δϕ3δϕ2δϕ1+

„

Gδ2G−1

δϕ3δϕ2G + 2 G

δG−1

δϕ2GδG−1

δϕ3G

«

δ2Γint

δGδϕ1

δ3Γ

δψαδψαδAµ

= −e γµαα + +

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 6

2PI and 2PI-resummed vertices

The 2PI-resummed two-point vertex involves the 2PI three-point vertex

δ2Γ

δϕ2δϕ1= (−1)q1 iG−1

0,21 +δ2Γint

δϕ2δϕ1+ G

δG−1

δϕ2Gδ2Γint

δGδϕ1

δ2Γ

δAµδAν= iG

−10,µν + + div!div!

The 2PI-resummed three-point vertex involves the 2PI three- and four-point vertices

δ3Γ

δϕ3δϕ2δϕ1=

δ3Γint

δϕ3δϕ2δϕ1+

„

Gδ2G−1

δϕ3δϕ2G + 2 G

δG−1

δϕ2GδG−1

δϕ3G

«

δ2Γint

δGδϕ1

δ3Γ

δψαδψαδAµ

= −e γµαα + +div! div!div!

div!

div!

2PI in a nutshell

● 2PI Effective action

● 2PI vertices

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 6

2PI and 2PI-resummed vertices

The 2PI-resummed two-point vertex involves the 2PI three-point vertex

δ2Γ

δϕ2δϕ1= (−1)q1 iG−1

0,21 +δ2Γint

δϕ2δϕ1+ G

δG−1

δϕ2Gδ2Γint

δGδϕ1

δ2Γ

δAµδAν= iG

−10,µν + + div!

The 2PI-resummed three-point vertex involves the 2PI three- and four-point vertices

δ3Γ

δϕ3δϕ2δϕ1=

δ3Γint

δϕ3δϕ2δϕ1+

„

Gδ2G−1

δϕ3δϕ2G + 2 G

δG−1

δϕ2GδG−1

δϕ3G

«

δ2Γint

δGδϕ1

δ3Γ

δψαδψαδAµ

= −e γµαα + +div! div!

2PI in a nutshell

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 7

WT identities

2PI in a nutshell

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 8

2PI Ward-Takahashi identities

At any truncation order, the 2PI effective action is gauge invariant

δα`Γ2PI[ϕ,G] − Sgf [A]

´= 0

under a generalized gauge transformation

δαϕ =

0

BB@

−(1/e)∂α

ie αψ

−ie α ψt

1

CCA

and δαG =

0

BB@

δαG = 0 δαKt δαK

δαK δαF δαD

δαKt −δαDt δαF

1

CCA

2PI in a nutshell

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 8

2PI Ward-Takahashi identities

At any truncation order, the 2PI effective action is gauge invariant

δα`Γ2PI[ϕ,G] − Sgf [A]

´= 0

under a generalized gauge transformation

δαϕ =

0

BB@

−(1/e)∂α

ie αψ

−ie α ψt

1

CCA

and δαG =

0

BB@

δαG = 0 δαKt δαK

δαK δαF δαD

δαKt −δαDt δαF

1

CCA

WT identity for the 2PI-resummed four-photon vertex [UR & J. Serreau, JHEP 0711:097 (2007)]

∂µ δ4Γ

δAµδAνδAρδAσ= 0

WT identity for the 2PI four-photon vertex [UR & J. Serreau, JHEP 0711:097 (2007)]

∂ρ i δ2G−1

µν

δAρδAσ= 0

2PI in a nutshell

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 8

2PI Ward-Takahashi identities

At any truncation order, the 2PI effective action is gauge invariant

δα`Γ2PI[ϕ,G] − Sgf [A]

´= 0

under a generalized gauge transformation

δαϕ =

0

BB@

−(1/e)∂α

ie αψ

−ie α ψt

1

CCA

and δαG =

0

BB@

δαG = 0 δαKt δαK

δαK δαF δαD

δαKt −δαDt δαF

1

CCA

WT identity for the 2PI-resummed two-photon vertex [UR & J. Serreau, JHEP 0711:097 (2007)]

∂µ δ2Γ

δAµδAν= i∂µG−1

0,µν

BUT! 2PI two-photon vertex unconstrained [UR & J. Serreau, JHEP 0711:097 (2007)]

i∂µG−1µν = i∂µG−1

0,µν + O(higher order contributions)

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 9

Renormalization

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 10

Renormalization factors

QED is renormalizable: one can redefine the fields

Aµb ≡ Z

1/23 (d)Aµ

ψb ≡ Z1/22 (d)ψ ψb ≡ Z

1/22 (d)ψ

as well as the parameters of the theory

Z2(d)mb ≡ Z0(d)m Z2(d)Z1/23 (d)eb ≡ Z1(d)e

Z3(d)

ξb=Z4(d)

ξ

in order to remove all UV divergences from the vertices, as d→ 4.

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 11

Exact theory

In terms of renormalized fields, the 2PI effective action gets the additional contribution

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y+ · · ·

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 11

Exact theory

In terms of renormalized fields, the 2PI effective action gets the additional contribution

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y+ · · ·

The 2PI-resummed photon propagator

gets the additional contribution

δ2Γ

δAµδAν= δZ3

`gµν∂

2 − ∂µ∂ν´

+ · · ·

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 11

Exact theory

In terms of renormalized fields, the 2PI effective action gets the additional contribution

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y+ · · ·

The 2PI-resummed photon propagator

gets the additional contribution

δ2Γ

δAµδAν= δZ3

`gµν∂

2 − ∂µ∂ν´

+ · · · The 2PI photon propagator

gets the additional contribution

iG−1µν = δZ3

`gµν∂

2 − ∂µ∂ν´

+ · · ·

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 11

Exact theory

In terms of renormalized fields, the 2PI effective action gets the additional contribution

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y+ · · ·

The 2PI-resummed photon propagator

gets the additional contribution

δ2Γ

δAµδAν= δZ3

`gµν∂

2 − ∂µ∂ν´

+ · · · The 2PI photon propagator

gets the additional contribution

iG−1µν = δZ3

`gµν∂

2 − ∂µ∂ν´

+ · · ·δ2Γ

δAµδAν= iG−1

µν

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 11

Approximations

In terms of renormalized fields, the 2PI effective action gets the additional contribution

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y+ · · ·

The 2PI-resummed photon propagator

gets the additional contribution

δ2Γ

δAµδAν= δZ3

`gµν∂

2 − ∂µ∂ν´

+ · · · The 2PI photon propagator

gets the additional contribution

iG−1µν = δZ3

`gµν∂

2 − ∂µ∂ν´

+ · · ·δ2Γ

δAµδAν6= iG−1

µν

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 11

Approximations

In terms of renormalized fields, the 2PI effective action gets the additional contribution

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y+ · · ·

δZ3 6= δZ3

The 2PI-resummed photon propagator

gets the additional contribution

δ2Γ

δAµδAν= δZ3

`gµν∂

2 − ∂µ∂ν´

+ · · · The 2PI photon propagator

gets the additional contribution

iG−1µν = δZ3

`gµν∂

2 − ∂µ∂ν´

+ · · ·δ2Γ

δAµδAν6= iG−1

µν

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 11

Approximations

In terms of renormalized fields, the 2PI effective action gets the additional contribution

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y+ · · ·

+δZL

2

Z

x

h

∂µx∂

νxGµν(x, y)

i

x=y+δM2

2

Z

xGµ

µ(x, x)

δZ3 6= δZ3, δZL, δM2The 2PI-resummed photon propagator

gets the additional contribution

δ2Γ

δAµδAν= δZ3

`gµν∂

2 − ∂µ∂ν´

+ · · ·

The 2PI photon propagator

gets the additional contribution

iG−1µν = δZ3

`gµν∂

2 − ∂µ∂ν´

+ i δZL ∂µ∂ν + i δM2gµν + · · ·δ2Γ

δAµδAν6= iG−1

µν

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 12

Approximations

In terms of renormalized fields, the 2PI effective action gets the additional contribution

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y+ · · ·

+δZL

2

Z

x

h

∂µx∂

νxGµν(x, y)

i

x=y+δM2

2

Z

xGµ

µ(x, x)

All these new contributions do not affect the gauge invariance of the 2PI effective action

δαG = 0 ⇒ δα“

Γ2PI + δΓ2PI − Sgf [A]”

= 0

⇒ new counterterms allowed by symmetry [UR & J. Serreau, in preparation]

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 12

Approximations

In terms of renormalized fields, the 2PI effective action gets the additional contribution

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y+ · · ·

+δZL

2

Z

x

h

∂µx∂

νxGµν(x, y)

i

x=y+δM2

2

Z

xGµ

µ(x, x)

+δg1

8

Z

xGµ

µ(x, x)Gνν(x, x) +

δg2

4

Z

xGµν(x, x)Gµν(x, x)

All these new contributions do not affect the gauge invariance of the 2PI effective action

δαG = 0 ⇒ δα“

Γ2PI + δΓ2PI − Sgf [A]”

= 0

⇒ new counterterms allowed by symmetry [UR & J. Serreau, in preparation]

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 12

Approximations

In terms of renormalized fields, the 2PI effective action gets the additional contribution

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y+ · · ·

+δZL

2

Z

x

h

∂µx∂

νxGµν(x, y)

i

x=y+δM2

2

Z

xGµ

µ(x, x)

+δg1

8

Z

xGµ

µ(x, x)Gνν(x, x) +

δg2

4

Z

xGµν(x, x)Gµν(x, x)

not wanted! :

Z

xAµ(x)Gµν(x, y)Aν(x) ,

Z

xAµ(x)Aµ(x)Gν

ν(x, x)

All these new contributions do not affect the gauge invariance of the 2PI effective action

δαG = 0 ⇒ δα“

Γ2PI + δΓ2PI − Sgf [A]”

= 0

⇒ new counterterms allowed by symmetry [UR & J. Serreau, in preparation]

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 13

Four photon leg subgraphs

The 2PI photon propagator G−1µν contains subgraphs involving four photons legs

G−1µν = · · · +

G0

+ · · ·

δ

δGρσ0

“

G−1µν

”

= · · · +

0G

+ · · · ∝ Vµν,ρσ

V fulfills a Bethe-Salpeter equation whose kernel Λ is obtained from the truncation of Γ2PI

Vµν,ρσ = Λµν,ρσ −1

2Λµν,µ′ν′Gµ′ρ′Gν′σ′ Vρ′σ′,ρσ

Similar structures occur in the case of scalar theories [Knoll, van Hees ; Blaizot, Iancu, Reinosa]

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 13

Four photon leg subgraphs

The 2PI photon propagator G−1µν contains subgraphs involving four photons legs

G−1µν = · · · +

G0

+ · · ·

δ

δGρσ0

“

G−1µν

”

= · · · +

0G

+ · · · ∝ Vµν,ρσ

A similar analysis applies to the 2PI-resummed photon propagator δ2Γ/δAµδAν

δ

δGρσ0

„δ2Γ

δAµδAν

«

∝δ2G−1

ρσ

δAµδAν

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 14

Renormalized 2PI effective action

A similar analysis on other 2PI and 2PI-resummed vertices leads to

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+ δZ2

Z

xψ(x)i /∂xψ(x) −mδZ0

Z

xψ(x)ψ(x)

− e δZ1

Z

xψ(x) /A(x)ψ(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y

+δZL

2

Z

x

h

∂µx∂

νxGµν(x, y)

i

x=y+δM2

2

Z

xGµ

µ(x, x)

+δg1

8

Z

xGµ

µ(x, x)Gνν(x, x) +

δg2

4

Z

xGµν(x, x)Gµν(x, x)

− δZ2

Z

xtr [i /∂xD(x, y)]x=y + δm

Z

xtrD(x, x)

+ e δZ1

Z

xtr [ /A(x)D(x, x)] − e δZ1

Z

x

ˆψ(x)K(x, x) + K(x, x)ψ(x)

˜

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 14

Renormalized 2PI effective action

A similar analysis on other 2PI and 2PI-resummed vertices leads to

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+ δZ2

Z

xψ(x)i /∂xψ(x) −mδZ0

Z

xψ(x)ψ(x)

− e δZ1

Z

xψ(x) /A(x)ψ(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y

+δZL

2

Z

x

h

∂µx∂

νxGµν(x, y)

i

x=y+δM2

2

Z

xGµ

µ(x, x)

+δg1

8

Z

xGµ

µ(x, x)Gνν(x, x) +

δg2

4

Z

xGµν(x, x)Gµν(x, x)

− δZ2

Z

xtr [i /∂xD(x, y)]x=y +mδZ0

Z

xtrD(x, x)

+ e δZ1

Z

xtr [ /A(x)D(x, x)] − e δZ1

Z

x

ˆψ(x)K(x, x) + K(x, x)ψ(x)

˜

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 14

Renormalized 2PI effective action

A similar analysis on other 2PI and 2PI-resummed vertices leads to

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+ δZ2

Z

xψ(x)i /∂xψ(x) −mδZ0

Z

xψ(x)ψ(x)

− e δZ1

Z

xψ(x) /A(x)ψ(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y

+δZL

2

Z

x

h

∂µx∂

νxGµν(x, y)

i

x=y+δM2

2

Z

xGµ

µ(x, x)

+δg1

8

Z

xGµ

µ(x, x)Gνν(x, x) +

δg2

4

Z

xGµν(x, x)Gµν(x, x)

− δZ2

Z

xtr [i /∂xD(x, y)]x=y +mδZ0

Z

xtrD(x, x)

+ e δZ1

Z

xtr [ /A(x)D(x, x)] − e δZ1

Z

x

ˆψ(x)K(x, x) + K(x, x)ψ(x)

˜

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 15

Renormalized gauge invariance

The unrenormalized 2PI effective action is gauge invariant

δα“

Γ2PI − Sgf [A]”

= 0

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 15

Renormalized gauge invariance

The unrenormalized 2PI effective action is gauge invariant

δα“

Γ2PI − Sgf [A]”

= 0

The renormalized 2PI effective action is a priori not gauge invariant

δα“

Γ2PI + δΓ2PI − Sgf [A]”

=

„

Z2 − Z2 − (Z1 − Z1)Z2

Z1

«

δα

Z

xψ(x)i /∂xψ(x)

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 15

Renormalized gauge invariance

The unrenormalized 2PI effective action is gauge invariant

δα“

Γ2PI − Sgf [A]”

= 0

The renormalized 2PI effective action is a priori not gauge invariant

δα“

Γ2PI + δΓ2PI − Sgf [A]”

=

„

Z2 − Z2 − (Z1 − Z1)Z2

Z1

«

δα

Z

xψ(x)i /∂xψ(x)

WT identities + Renormalization conditions

„

Z2 − Z2 − (Z1 − Z1)Z2

Z1

«

= 0

It is then possible to define a gauge-invariant renormalization scheme.

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 16

Renormalization conditions

13 counterterms need 13 renormalization conditions!

δΓ2PI =δZ3

2

Z

xAµ(x)

“

gµν∂2x − ∂µ

x∂νx

”

Aν(x)

+ δZ2

Z

xψ(x)i /∂xψ(x) −mδZ0

Z

xψ(x)ψ(x)

− e δZ1

Z

xψ(x) /A(x)ψ(x)

+δZ3

2

Z

x

h“

gµν∂2x − ∂µ

x∂νx

”

Gµν(x, y)i

x=y

+δZL

2

Z

x

h

∂µx∂

νxGµν(x, y)

i

x=y+δM2

2

Z

xGµ

µ(x, x)

+δg1

8

Z

xGµ

µ(x, x)Gνν(x, x) +

δg2

4

Z

xGµν(x, x)Gµν(x, x)

− δZ2

Z

xtr [i /∂xD(x, y)]x=y +mδZ0

Z

xtrD(x, x)

+ e δZ1

Z

xtr [ /A(x)D(x, x)] − e δZ1

Z

x

ˆψ(x)K(x, x) + K(x, x)ψ(x)

˜

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 17

Renormalization conditions

Green counterterms (δZ1 here) fix the parameters of the theory

e γµαα = δZ1e γ

µαα +

δ3Γ

δAµδψαδψα

˛˛˛˛⋆

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 17

Renormalization conditions

Green counterterms (δZ1 here) fix the parameters of the theory

e γµαα = δZ1e γ

µαα +

δ3Γ

δAµδψαδψα

˛˛˛˛⋆

Blue counterterms (δZ1 and δZ1 here) restore unicity of vertices at the renormalization point

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 17

Renormalization conditions

Green counterterms (δZ1 here) fix the parameters of the theory

e γµαα = δZ1e γ

µαα +

δ3Γ

δAµδψαδψα

˛˛˛˛⋆

Blue counterterms (δZ1 and δZ1 here) restore unicity of vertices at the renormalization point

iδG−1αα

δAµ6=

iδG−1αµ

δψα6=

δ3Γ

δAµδψαδψα

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 17

Renormalization conditions

Green counterterms (δZ1 here) fix the parameters of the theory

e γµαα = δZ1e γ

µαα +

δ3Γ

δAµδψαδψα

˛˛˛˛⋆

Blue counterterms (δZ1 and δZ1 here) restore unicity of vertices at the renormalization point

δZ1e γµαα +

iδG−1αα

δAµ

˛˛˛˛˛⋆

= δZ1e γµαα +

iδG−1αµ

δψα

˛˛˛˛˛⋆

= δZ1 e γµαα +

δ3Γ

δAµδψαδψα

˛˛˛˛⋆

2PI in a nutshell

WT identities

Renormalization

● Renormalization factors I

● Renormalization factors II

● Renormalized invariance

● Renormalization conditions

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 17

Renormalization conditions

Green counterterms (δZ1 here) fix the parameters of the theory

e γµαα = δZ1e γ

µαα +

δ3Γ

δAµδψαδψα

˛˛˛˛⋆

Blue counterterms (δZ1 and δZ1 here) restore unicity of vertices at the renormalization point

δZ1e γµαα +

iδG−1αα

δAµ

˛˛˛˛˛⋆

= δZ1e γµαα +

iδG−1αµ

δψα

˛˛˛˛˛⋆

= δZ1 e γµαα +

δ3Γ

δAµδψαδψα

˛˛˛˛⋆

Red counterterms restore the transversality of γ-vertices at the renormalization point

kµˆiG−1

µν + δM2gµν + δZLkµkν˜

⋆= kµ δ2Γ

δAµδAν

˛˛˛˛⋆

= kµiG−10,µν

2PI in a nutshell

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 18

Gauge-fixing

2PI in a nutshell

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 19

Gauge-fixing dependence

Example: Consider the pressure of the system

P2PI = −Trh

ln D−1(ξ) +D−10 D(ξ)

i

+1

2Tr

h

ln G−1(ξ) +G−10 (ξ) G(ξ)

i

+ + + . . .

At a given order of approximation, there is a residual gauge-fixing dependence

At order e2,d

dξP2PI = O(e4)

At order e4,d

dξP2PI = O(e6), and so on

How to systematically get rid of the residual gauge fixing dependence?

2PI in a nutshell

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 19

Gauge-fixing dependence

Example: Consider the pressure of the system

P2PI = −Trh

ln D−1(ξ) +D−10 D(ξ)

i

+1

2Tr

h

ln G−1(ξ) +G−10 (ξ) G(ξ)

i

+ + + . . .

No one knows so far! But there are a few things one can try:

• consider additional (non-systematic) approximations [Blaizot, Iancu, Rebhan];

• use the residual gauge fixing dependence as an indication of the error;

• try to find minimum sensitivity gauges [Borsanyi, Reinosa].

2PI in a nutshell

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 20

Two-loop result (1/2)

In the range of convergence ξ ∈ [0, 2], the ξ-dependence is not dramatically big:

comparable to the µ-dependence in the range µ ∈ [πT, 4πT ]

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 0.5 1 1.5 2 2.5

p/p i

deal

e

ξ=0ξ=1ξ=2

e2 order

UR & Sz. Borsanyi, PLB 661 (2008)

2PI in a nutshell

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 21

Two-loop result (2/2)

Minimum sensitivity obtained for ξ = 0 (Landau gauge):

µ dependence minimal for ξ = 0

0.91

0.915

0.92

0.925

0.93

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

p/p i

deaa

l

ξ

e=2

µ=πTµ=2πTµ=4πT

-13

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-3 -2 -1 0 1

log(

p/p ξ

=0-

1)

log(ξ)

e=1e=2

0.9

0.95

0 1 2

p/p i

deal

ξ

e=2

P2PI(ξ) − P2PI(0) ∼ α(e, µ) ξ2

2PI in a nutshell

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 22

Summary

2PI in a nutshell

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 23

Summary

X Truncations of the 2PI effective action for QED are gauge invariant.

X The corresponding 2PI vertices are renormalizable:

→ Duplicated counterterms: (δZ3, δZ3), . . .

→ Additional counterterms: δg1, δg2, δZL and δM2.

X The renormalization procedure is consistent:

→ All these new features are allowed by the symmetry.

→ No new parameters in the game.

X Systematic application of 2PI techniques to abelian gauge theories:

→ Numerical study of gauge-fixing dependence.

→ Landau gauge seems less sensitive to error.

2PI in a nutshell

WT identities

Renormalization

Gauge-fixing

Summary

Urko Reinosa – 2009 2PI effective action for gauge theories: Symmetry & Renormalization – p. 24

Happy Birthday!

& Thank You!