Remarks on 2PI formalisms and the fRG Functional Renormalization - - PowerPoint PPT Presentation

Remarks on 2PI formalisms and the fRG

Jean-Paul Blaizot, IPhT-Saclay

Heidelberg, March 10, 2017

Based on work done in collaboration with

• J. Pawlowski and U. Reinosa

A paper to appear… soon !

Functional Renormalization

— from quantum gravity and dark energy to ultracold atoms and condensed matter

Motivations

Ω[G] = 1

2Tr logG−1 − 1 2TrΣG + Φ[G]

∂κ ˆ Γκ[φ] = 1

2

R

q ∂κRκ(q)Gκ(q, −q; φ)

Study (controlable ?) non perturbative methods in many-body physics and field theory

G−1

κ (q, −q; φ) = Γ(2) κ (q, −q; φ) + Rκ(q)

Thermodynamic potential as a functional of the propagator Flow of the effective action

Two exact formulae These formulae are useful mostly for the approximations that they suggest One can use one formalism to shed light on the other (this talk)

S [ϕ] = R ddx n 1

2 (∂ϕ(x))2 + m2 2 ϕ2(x) + λ 4! ϕ4(x)

• Present discussion limited to scalar field theory

(can be generalized)

Some representative recent related works (not limited to scalar field) JPB, J. Pawlowski, U. Reinosa (2010)

• M. Carrington et al (2014)
• N. Dupuis (2013)
• V. Meden et al (2016)

Ω[G] = 1

2Tr logG−1 − 1 2TrΣG + Φ[G]

Σ(p) = 2 δΦ

δG(p)

Σ[G] = G−1 − G−1

δΩ[G] δG

• G0 = 0

Φ[G]

Basics of 2PI formalisms (1)

Luttinger-Ward functional Self-energy Self-consistency condition Stationarity property

I(q, p) = 2 δΣ(p)

δG(q) = 4 δ2Φ δG(q)δG(p) = I(p, q)

Γ(4)(q, p) = I(q, p) − 1

2

R

l Γ(4)(q, l)G2(l) I(l, p)

Basics of 2PI formalisms (2)

Irreducible kernel Bethe-Salpeter equation

∂κ ˆ Γκ[φ] = 1

2

R

q ∂κRκ(q)Gκ(q, −q; φ)

G−1

κ (q, −q; φ) = Γ(2) κ (q, −q; φ) + Rκ(q)

Basics of functional RG

Flow equation (Wetterich) Infinite hierarchy of coupled flow equations for the n-point functions Equation for the 2-point function

∂κΓ(2)κ(p) = − 1

2

R

q ∂κRκ(q)G2 κ(q) Γ(4) κ (q, p)

And so on…..

G−1

κ (p) = p2 + m2 + Σκ(p) + Rκ(p)

∂κΣκ(p) = 2 R

q ∂κGκ(q) δ2Φ[G] δG(q)δG(p)

• Gκ = 1

2

R

q ∂κGκ(q) Iκ(q, p)

Equation for the 2-point function (or self-energy)

The theory in the presence of Rκ(q)

All formal relations between n-point functions hold for any κ One can then take derivatives w.r.t. κ …. thereby obtaining flow equations This is NOT quite the usual flow equation

Γ(4)

κ (q, p) = Iκ(q, p) − 1 2

R

l Γ(4) κ (q, l)G2 κ(l) Iκ(l, p)

∂κΣκ(p) = − 1

2

R

q ∂κRκ(q)G2 κ(q) Γ(4) κ (q, p) The exact flow equation for the 2-point function

Solving the Bethe-Salpeter equation to get

Γ(4)(q, p)

and using this equation to eliminate Iκ(q, p) we are left with

A possible truncation scheme (1)

Truncate the Luttinger-Ward functional (keeping selected skeletons) Obtain the kernel

I(q, p) = 4

δ2Φ δG(q)δG(p)

Then solve the coupled equations

Γ(4)

κ (q, p) = Iκ(q, p) − 1 2

R

l Γ(4) κ (q, l)G2 κ(l) Iκ(l, p)

∂κΣκ(p) = − 1

2

R

q ∂κRκ(q)G2 κ(q) Γ(4) κ (q, p)

• NB. i) The solution is independent of the choice of the "regulator"

ii) Not only a truncation of fRG, but an alternative to solving the 2PI equations

A possible truncation scheme (2)

Instead of solving the Bethe-Salpeter eqn., write a flow equation for the 4-point function

∂κΓ(4)

κ (p, q) = ∂κIκ(p, q)

− 1 2 Z

l

Γ(4)

κ (p, l) ∂κG2 κ(l) Γ(4) κ (l, q)

− 1 2 Z

l

∂κIκ(p, l) G2

κ(l) Γ(4) κ (l, q)

− 1 2 Z

l

Γ(4)

κ (p, l) G2 κ(l) ∂κIκ(l, q)

+ 1 4 Z

l

Z

s

Γ(4)

κ (p, l) G2 κ(l) ∂κIκ(l, s) G2 κ(s) Γ(4) κ (s, q).

• NB. This equation is NOT the "usual" flow equation for the 4-point function

= +

Γ(2)

κ (p) ≡ Zκp2 + m2 κ,

Zκ ∼ ln κ, m2

κ ∼ κ2

Γ(4)

κ

∼ ln κ, Γ(n>4)

κ

∼ κ4−n,

Renormalization issues

Standard lore in fRg: things become "simple" at the "cutoff scale" One expects of course similar features in the 2PI truncation… …but working out the "details" turned out to be tricky Not a priori obvious that the integrals are finite

∂κΓ(4)

κ (p, q) = ∂κIκ(p, q)

− 1 2 Z

l

Γ(4)

κ (p, l) ∂κG2 κ(l) Γ(4) κ (l, q)

− 1 2 Z

l

∂κIκ(p, l) G2

κ(l) Γ(4) κ (l, q)

− 1 2 Z

l

Γ(4)

κ (p, l) G2 κ(l) ∂κIκ(l, q)

+ 1 4 Z

l

Z

s

Γ(4)

κ (p, l) G2 κ(l) ∂κIκ(l, s) G2 κ(s) Γ(4) κ (s, q).

Divergences, and subdivergences…..

In addition to counterterms needed to renormalise the kernel I an infinite number of counterterms are needed to renormalise the BS equation…. Consider the loop expansion of the 4-point function

1 ΓBS

(0)

= 1 λ + λBS

(0)

+ 1 2 Z

q

G2(q), δλBS

(0) = λ

aλ 1 − aλ, a ≡ 1 2 Z

q

G2(q)

δm2 m2 = aλ 1 − aλ = δλ λ

A simple example (1)

Φ[G] Σ(p) = 2 δΦ

δG(p)

Iκ(q, p)

¯ Σ = δm2 + λ + δλBS

(0)

2 Z

q

G(q)

, G(q) = 1 q2 + m2 + ¯ Σ

Standard 2PI renormalization Gap equation BS equation Counterterms

Φ[G] Σ(p) = 2 δΦ

δG(p)

Iκ(q, p)

A simple example (2)

∂κΓκ = 1 2Γ2

κ

Z

q

∂κG2

κ(q).

m2

κ = m2 = m2 Λ + ΓΛ 2

R

q

n Gκ(q) − GΛ(q) + (m2

κ − m2 Λ)G2 Λ(q)

• ∂κm2

κ = − 1 2Γκ

R

q(∂κRκ)G2 κ(q) The two equations to be solved Solution Elimination of "subdivergences " is automatically taken care of by the coupled flow equations

Conclusions

• Two non perturbative methods were compared
• Approximation schemes exist where they completely match
• The comparison help to clarify some renormalisation issues in

non perturbative schemes, such as 2PI

• Truncating the fRG flow equations with 2PI relations may be

useful in some applications