Truncation Effects in the FRG Method Istvn Nndori MTA-DE Particle - - PowerPoint PPT Presentation

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary

Truncation Effects in the FRG Method

István Nándori

MTA-DE Particle Physics Research Group, University of Debrecen

Non-perturbative Methods in Quantum Field Theory, Balatonfüred 2014

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Wetterich RG equation

Functional Renormalization Group Wetterich RG equation k∂kΓk = 1 2Tr k∂kRk Γ(2)

k

+ Rk ! , Rk(p) ≡ p2 r(y), y = p2 k2 regulator: Rk!0(p) = 0, Rk!Λ(p) = ∞, Rk(p → 0) > 0 Approximations Γk[ϕ] = Z

x

 Vk(ϕ) + Zk(ϕ)1 2(∂µϕ)2 + ...

• ,

Vk =

Ncut

X

n=1

gn(k) (2n)! ϕ2n Problems = ⇒ approximated RG flow depends on Rk = ⇒ phase structure depends on truncations

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Regulator functions

Power-law regulator rpow(y) = a yb = ⇒ compatible with the derivative expansion, = ⇒ momentum integral analytic (b = 1, 2) but not UV safe Exponential regulator rexp(y) = a exp [c yb] − 1 = ⇒ compatible with the derivative expansion, = ⇒ UV safe but the momentum integral non-analytic Litim’s regulator ropt(y) = a ✓ 1 yb − 1 ◆ Θ(1 − y) = ⇒ momentum integral analytic and UV safe, = ⇒ confront to the derivative expansion,

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Properties I.

Compactly Supported Smooth (CSS) regulator general CSS regulator r gen

css (y) = exp[cyb 0 /(f − hyb 0 )] − 1

exp[cyb/(f − hyb)] − 1θ(f − hyb)

• I. N., JHEP 04 (2013) 150

normalised CSS regulator (f ≡ 1 and y0 fixed) r norm

css

(y) = exp[ln(2)c] − 1 exp h

ln(2)cyb 1hyb

i − 1 θ(1 − hyb) = 2c − 1 2

c yb 1−hyb − 1

θ(1 − hyb)

• I. N., I. G. Márián, V. Bacsó, PRD 89 (2014) 047701

= ⇒ smooth: compatible with the derivative expansion, = ⇒ compact support: UV safe, = ⇒ momentum integral cannot be performed analytically

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Properties II.

Properties of the general CSS regulator lim

c!0 r gen css = =

yb 1 − yb ✓ 1 yb − 1 ◆ Θ(1 − y), lim

f!1 r gen css = = yb

yb , lim

h!0,c=f r gen css (y) = = exp[yb 0 ] − 1

exp[yb] − 1. Properties of the normalised CSS regulator lim

c!0,h!1 r norm css

= ✓ 1 yb − 1 ◆ θ(1 − y) lim

c!0,h!0 r norm css

= 1 yb lim

c!1,h!0 r norm css

= 1 exp[ln(2)yb] − 1

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Shortest Trajectory in Theory Space

Optimization I. Amplitude (ω = k2V 00

k ) expansion in LPA

k∂kVk = −αdkd Z 1 dy r 0 y1+ d

2

(1 + r)y + ω =

1

X

m=1

2m d a2md (−ω)m1 = ⇒ best convergence: Litim’s regulator (b=1, a=1) = ⇒ confront to the derivative expansion Shortest Trajectory (ST) in the Theory Space

Jan M. Pawlowski, Annals of Physics 322 (2007) 2831

= ⇒ works in any order of the derivative expansion, = ⇒ it gives the Litim regulator in LPA, = ⇒ no explicit r(y) beyond LPA CSS is differentiable, recovers the Litim regulator in LPA BUT according to ST the Litim is NOT the best beyond LPA

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Principle of Minimal Sensitivity

Optimization II. Principle of Minimal Sensitivity (PMS):

• L. Canet, B. Delamotte, D. Mouhanna and J. Vidal, PRD 67 (2003) 065004

r(y) = α

1 ey1

= ⇒ αopt = 6

0.65 0.652 0.654 0.656 0.658 0.66 0.662 0.664 1 2 3 4 5 6 7 8 9 10

ν α

νpms

uk(ρ) u10 Léonie Canet, PRB 71 (2005) 012418

• ptimal choice for the parameters for a given regulator

properties: = ⇒ works in any order of the derivative expansion, = ⇒ regulators cannot be compared directly CSS provides a framework to compare regulators directly.

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary O(1), d=3

Model: O(1), d=3, LPA Γk[ϕ] = Z d3x " 1 2(∂µϕ)2 +

Ncut

X

n=1

gn(k) n! ϕn #

ν ν ν ν ν Critical exponent ν for the 3D O(1) model (regulator in exponential normalization, b parameter fixed) !

• I. G. Márián, U. D. Jentscura, I.N., JPG 41 (2014) 055001

Best: c = 0.001, h = 1, b = 1 = ⇒ Litim limit

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary QED2, d=2

Model: QED2, d=2, LPA Γk[ϕ] = Z d2x 1 2(∂µϕ)2 + 1 2M2

k ϕ2 + uk cos(

√ 4πϕ)

• χc

χc χc χc χc Critical ratio χc for bosonized QED2 (regulator in exponential normalization, b parameter fixed) !

• I. G. Márián, U. D. Jentscura, I.N., JPG 41 (2014) 055001

Best: c = 0.001, h = 1, b = 1 = ⇒ Litim limit

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Wavefunction renormalization

Beyond LPA? → wavefunction renormalization O(N) model, d = 3, wavefunction renormalization Γk[ϕ] = Z d3x 1 2Zk(ϕ)(∂µϕ)2 + g1(k) 2 ϕ2 + g2(k) 4! ϕ4 + ...

• =

⇒ field-dependent Zk(ϕ) sine-Gordon model, d = 2, wavefunction renormalization Γk[ϕ] = Z d2x 1 2zk(∂µϕ)2 + uk cos(ϕ) + ...

• =

⇒ field-independent zk = 1/β2 = ⇒ β2

c = 8π does not depend on Rk

0.0 0.2 0.4 0.6 0.8 1.0

u

5 10 15 20 25 30 35 40

1/ z

• ~

> > > > > > > < < < < < < <

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary Truncations and phase structure

Spontaneous Symmetry Breaking O(N=1) model in d = 1, truncated RG flow, LPA Γk[ϕ] = Z dx 1 2(∂µϕ)2 + g1(k) 2 ϕ2 + g2(k) 4! ϕ4 + ...

• SG model in d = 1, truncated RG flow, beyond LPA

Γk[ϕ] = Z dx 1 2zk(∂µϕ)2 + uk cos(ϕ) + ...

• 1.0
• 0.8
• 0.6
• 0.4
• 0.2

0.0

g1 g2

Ê Gaussian Ê Wilson Fisher Ê Infrared

0.0 0.2 0.4 0.6 0.8 1.0

u

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1/z

power-law regulator, b = 3, d=1

• >

> > > > > > > > > > > > D

> <

• N. Defenu, P

. Mati, I.G. Márián, I. N., A. Trombettoni, in progress

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary SG, d=1

Model: sine-Gordon, d=1, LPA + z Γk[ϕ] = Z dx 1 2zk(∂µϕ)2 + uk cos(ϕ)

• 0.35

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75

D

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

b

Best values Exponential limit Power-law limit CSS regulator, d=1 dimension

• I. N., I. G. Márián, V. Bacsó, PRD 89 (2014) 047701

Best: c → 0 (small but non-zero), b → 1 = ⇒ Litim limit

Functional Renormalization Group CSS regulator Optimization CSS in LPA CSS beyond LPA Summary

Summary Wetterich RG equation k∂kΓk = 1 2Tr k∂kRk Γ(2)

k

+ Rk , Rk(p) ≡ p2 r(y), y = p2 k2 CSS regulator – "unification" of regulator functions r norm

css

(y) = exp[ln(2)c] − 1 exp h

ln(2)cyb 1hyb

i − 1 θ(1−hyb) = 2c − 1 2

c yb 1−hyb − 1

θ(1−hyb) single numerical code for all regulators no problem with the upper bound of the momentum integral regulators can be compared through the PMS Outlook

• ptimization of CSS (best: Litim limit?)