Chiral symmetry restoration with functional
renormalization group methods
Gergely Fejős
RIKEN
Theoretical Research Division, Nishina Center
Quantum Hadron Physics Laboratory

Progress on J-PARC hadron physics in 2014a
1 December, 2014

Gergely FejősChiral symmetry restoration with functional renormalization gro

Outline
aaa
Motivation

Chiral symmetry in effective meson models

Functional renormalization group flows

Numerical results

Summary

Gergely FejősChiral symmetry restoration with functional renormalization...

Motivation
QCD Lagrangian with quarks and gluons:1 a µνa
L = − GµνG
+ ψ̄i (iγµ D µ − m)ij ψj4Approximate chiral symmetry:a a

ψL → e iT θL ψL ,a a

ψR → e iT θR ψR−→ UL (Nf ) × UR (Nf ) ∼ UV (Nf ) × UA (Nf )−→ depending on the energy, Nf = 2, 3 have relevance

Gergely FejősChiral symmetry restoration with functional renormalization...

Motivation
QCD Lagrangian with quarks and gluons:1 a µνa
L = − GµνG
+ ψ̄i (iγµ D µ − m)ij ψj4Approximate chiral symmetry:a a

ψL → e iT θL ψL ,a a

ψR → e iT θR ψR−→ UL (Nf ) × UR (Nf ) ∼ UV (Nf ) × UA (Nf )−→ depending on the energy, Nf = 2, 3 have relevance
Chiral symmetry is spontaneuously broken in the ground state:< ψ̄i ψi > = < ψ̄i,R ψi,L > + < ψ̄i,L ψi,R > 6= 0−→ < ψ̄i,R ψj,L > ∼ δij ⇒ symmetry broken to UV (Nf )Chiral symmetry restoration? Critical temperature?
Quark mass dependence? Axial anomaly?
Gergely FejősChiral symmetry restoration with functional renormalization...

Motivation−→ Nf = 2 case: 2nd order nature depends on anomaly strenght−→ small anomaly case: subtle, fixed point?
−→ vanishing quark masses: first order, but no direct evidence
Gergely FejősChiral symmetry restoration with functional renormalization...

Chiral symmetry in effective meson models
Lagrangian of the n-flavor low energy strong interaction:
g1
g2
L = ∂µ M∂ µ M † −µ2 Tr (MM † )− 2 [ Tr (MM † )]2 − Tr (MM † )2nn−→ M = T a (s a + iπ a ) [scalar and pseudoscalar mesons]−→ vanishing quark masses−→ no anomaly
Renormalization group analysis: fixed point(s)?

Gergely FejősChiral symmetry restoration with functional renormalization...

Chiral symmetry in effective meson models
Lagrangian of the n-flavor low energy strong interaction:
g1
g2
L = ∂µ M∂ µ M † −µ2 Tr (MM † )− 2 [ Tr (MM † )]2 − Tr (MM † )2nn−→ M = T a (s a + iπ a ) [scalar and pseudoscalar mesons]−→ vanishing quark masses−→ no anomaly
Renormalization group analysis: fixed point(s)?
β functions (-expansion, 1-loop)
n2 + 4 2 4nḡ1 + ḡ1 ḡ2 + ḡ22
3
32n 2
= −ḡ2 + ḡ2 + 2ḡ1 ḡ23

β1 = −ḡ1 +β2−→ in 3d, fixed points: ḡ1 = n23+4 ,
aaaaaaaaaaaaaaaaaaaaaā
g1 = 0,−→ no IR stable fixed point exists!
Gergely Fejősḡ2 = 0ḡ2 = 0[O(2n2 ) W.F.][Gaussian]Chiral symmetry restoration with functional renormalization...

Chiral symmetry in effective meson models
Symmetric
Phase

k

C
ri
al
t ic
Li

Broken
Phase

ne

_2
k

Broken or
Unstable

Inclusion of only the first quartic coupling:−→ O(2n2 ) model with second order transition

Gergely FejősChiral symmetry restoration with functional renormalization...

Chiral symmetry in effective meson models
Symmetric
Phase

k

C
ri
al
t ic
Li

Broken
Phase

ne

_2
k

Broken or
Unstable

Inclusion of only the first quartic coupling:−→ O(2n2 ) model with second order transition
Adding the second coupling: RG trajectories diverge from f.p.
−→ no second order transition−→ indirect evidence of a first order transition
Direct evidence?
−→ Construction of the finite temperature effective potential
Gergely FejősChiral symmetry restoration with functional renormalization...

Functional renormalization group flows
FRG: follows the idea of Wilsonian renormalization group
Z

R
R 1
Zk [J] = exp(iWk [J]) = Dφe i S[φ]+ Jφ+ 2 φRk φ
Rk : IR regulator function⇒ requriements: 1., scale separation (suppress modes q . k)aaaaaaaaaaaaaaa 2., Rk −→ ∞ as k −→ ∞aaaaaaaaaaaaaaa 3., Rk −→ 0 as k −→ 0

Gergely FejősChiral symmetry restoration with functional renormalization...

Functional renormalization group flows
FRG: follows the idea of Wilsonian renormalization group
Z

R
R 1
Zk [J] = exp(iWk [J]) = Dφe i S[φ]+ Jφ+ 2 φRk φ
Rk : IR regulator function⇒ requriements: 1., scale separation (suppress modes q . k)aaaaaaaaaaaaaaa 2., Rk −→ ∞ as k −→ ∞aaaaaaaaaaaaaaa 3., Rk −→ 0 as k −→ 0
scale-dependent effective action:
ZZ
1
Γk [φ̄] = Wk [J] − J φ̄ −φ̄Rk φ̄
2

Z
R δΓ
R
i S[φ]+ δφ̄k (φ−φ̄)+ 12 (φ̄−φ)Rk (φ̄−φ)iΓk [φ̄]e=Dφe=⇒ Γk≈∞ [φ̄] = S[φ̄],Γk=0 [φ̄] = Γ1PI [φ̄]scale-dependent effective action interpolates between theclassical- and quantum effective action
Gergely FejősChiral symmetry restoration with functional renormalization...

Functional renormalization group flows
The scale-dependent effective action obeys the following flow#"
equation:
1
1∂k Γk = STr (2)∂k R k2
Γk [φ̄] + Rk−→ functional integro-differential equation−→ not solvable, approximation(s) needed

Gergely FejősChiral symmetry restoration with functional renormalization...

Functional renormalization group flows
The scale-dependent effective action obeys the following flow#"
equation:
1
1∂k Γk = STr (2)∂k R k2
Γk [φ̄] + Rk−→ functional integro-differential equation−→ not solvable, approximation(s) needed
Approximations? → derivative expansion!
Z Γk [φ] =Vk [φ] + φ(Zk ∂τ − Ak ∇2 − Wk ∂τ2 )φ + ...
xaaaa⇒ Zk , Ak = 1, Wk = 0 with Vk 6= 0 is reliable (LPA)

Gergely FejősChiral symmetry restoration with functional renormalization...

Functional renormalization group flows
The scale-dependent effective action obeys the following flow#"
equation:
1
1∂k Γk = STr (2)∂k R k2
Γk [φ̄] + Rk−→ functional integro-differential equation−→ not solvable, approximation(s) needed
Approximations? → derivative expansion!
Z Γk [φ] =Vk [φ] + φ(Zk ∂τ − Ak ∇2 − Wk ∂τ2 )φ + ...
xaaaa⇒ Zk , Ak = 1, Wk = 0 with Vk 6= 0 is reliable (LPA)
Finite temperature flow equation for the local potential:∂k Vk =

1
k4 X XT22
26π
ωm + k + µ2i (k)ωmGergely FejősiChiral symmetry restoration with functional renormalization...

Functional renormalization group flows
Symmetry breaking pattern (Wafa-Vitten theorem):=⇒ UL (n) × UR (n) → UV (n)=⇒ < M >= v0 T 0 ∼ 1̂Vk is a function of:=⇒ chiral invariants: I1 = Tr [MM † ]aaaaaaaaaaaaaaaaaaa I2 = Tr [MM † − Tr (MM † )/n)]2
aaaaaaaaaaaaaaaaaaa I3 = Tr [MM † − Tr (MM † )/n)]3
aaaaaaaaaaaaaaaaaaa ...

Gergely FejősChiral symmetry restoration with functional renormalization...

Functional renormalization group flows
Symmetry breaking pattern (Wafa-Vitten theorem):=⇒ UL (n) × UR (n) → UV (n)=⇒ < M >= v0 T 0 ∼ 1̂Vk is a function of:=⇒ chiral invariants: I1 = Tr [MM † ]aaaaaaaaaaaaaaaaaaa I2 = Tr [MM † − Tr (MM † )/n)]2
aaaaaaaaaaaaaaaaaaa I3 = Tr [MM † − Tr (MM † )/n)]3
aaaaaaaaaaaaaaaaaaa ...
Chiral expansion around the < M > configuration:
Vk (I1 , I2 , ...In ) = Uk (I1 ) +

X
{α}

(α)

Ck (I1 )

n
Y
Iiαi
i=2

We derive and solve flow equations for the coefficient(α)
functions Uk and Ck−→ very efficent numerically−→ 1-dimensional grids (not n-dim.)Gergely FejősChiral symmetry restoration with functional renormalization...

Functional renormalization group flows
Flow equations of the coefficients are similary to the
Dyson-Schwinger hierarchy:(0,1,0,..)Uk (I1 ) ←− Ck
Ck(0,1,0...)←− Ck(0,0,1,0...)←− ...

Ck(0,0,1,0...)Gergely Fejős(0,2,0,..), Ck

Chiral symmetry restoration with functional renormalization...

Functional renormalization group flows
Flow equations of the coefficients are similary to the
Dyson-Schwinger hierarchy:(0,1,0,..)Uk (I1 ) ←− Ck
Ck(0,1,0...)←− Ck(0,0,1,0...)←− ...

Ck(0,0,1,0...)(0,2,0,..), Ck

Truncation is necessary−→ we keep only those coefficients that are alreadyaaaanonzero at classical level(0,1,0,0..)Vk ≈ Uk (I1 ) + Ck

(I1 ) · I2

Evaluation of Vk at < M >= v0 T 0−→
I1 |v0 = v02 /2,
I2 |v0 = 0
Gergely FejősChiral symmetry restoration with functional renormalization...

Functional renormalization group flowsn2 − 1
1
k 4T Xn2+ 2+ 2∂k Uk (I1 ) =2 + E26π 2 ω
ωm
ωm + Ea20
ωm + Eσ2πmGergely FejősChiral symmetry restoration with functional renormalization...

Functional renormalization group flows
k 4T Xn2 − 1
1
n2∂k Uk (I1 ) =+ 2+ 2
2 + E26π 2 ω
ωm
ωm + Ea20
ωm + Eσ2πm"4(3Ck + 2I1 Ck0 )2 /n
k 4T X∂k Ck (I1 ) =2 + E 2 )2 (ω 2 + E 2 )6π 2 ω
(ωm
a0
m
σ
m

+

128Ck5 I13 /n
2 + E 2 )3 (ω 2 + E 2 )3
(ωm
π
m
a0


4Ck 4Ck (n2 − 3) + (1 − 4n2 )I1 Ck0 /n+
2 + E 2 )3(ωma0

4 3Ck Ck0 I1 + 4I12 Ck0 + Ck (3Ck − 2Ck00 I12 ) /n+
2 + E 2 )(ω 2 + E 2 )2
(ωm
a0
m
σ
+

64Ck3 I12 (Ck − I1 Ck0 )/n
48Ck2 I12 Ck0−+ ...
2 + E 2 )2 (ω 2 + E 2 )3
2 + E 2 )(ω 2 + E 2 )3
(ωm
(ωm
π
m
a0
π
m
a0
Gergely FejősChiral symmetry restoration with functional renormalization...

Functional renormalization group flows
Assumption of Vk : (form of classical potential)
Vk = µ2k Tr (MM † ) +

g1,k
g2,k
[ Tr (MM † )]2 +Tr (MM † )22nn

We recover the one-loop β-functions:n2 + 4 2
4n2
ḡ1,k + ḡ1,k ḡ2,k + ḡ2,k3
32n 2= −ḡ2,k + ḡ2,k+ 2ḡ1,k ḡ2,k3

β1 = −ḡ1,k +β2Flow of the mass parameter:∂k µ2k = −k 4

(n2 + 1)g1,k + 2ng2,k6(k 2 + µ2k )2Gergely FejősChiral symmetry restoration with functional renormalization...

Functional renormalization group flows
Assumption of Vk : (form of classical potential)
Vk = µ2k Tr (MM † ) +

g1,k
g2,k
[ Tr (MM † )]2 +Tr (MM † )22nn

We recover the one-loop β-functions:n2 + 4 2
4n2
ḡ1,k + ḡ1,k ḡ2,k + ḡ2,k3
32n 2= −ḡ2,k + ḡ2,k+ 2ḡ1,k ḡ2,k3

β1 = −ḡ1,k +β2Flow of the mass parameter:∂k µ2k = −k 4

(n2 + 1)g1,k + 2ng2,k6(k 2 + µ2k )2The functional flow equations contain much more:−→ infinite resummation of n-point couplings
Gergely FejősChiral symmetry restoration with functional renormalization...

Numerical results (effective potential)µ2/Λ2 = -0.05, g1 = -0.8, g2 = 1.1, n = 2
0.002µ2/Λ2 = -0.05, g1 = -0.8, g2 = 1.1, n = 3

T/Λ=0
T/Λ=0.3
T/Λ=0.6
T/Λ=0.9
T/Λ=1.2

0.001

T/Λ=0
T/Λ=0.3
T/Λ=0.6
T/Λ=0.9
T/Λ=1.2

0.005

0
0

-0.002

Vk/Λ4

Vk/Λ4

-0.001-0.003-0.0050.00010.0002T/Λ=0.96

8e-05

T/Λ=0.97
0.0001T/Λ=0.98

6e-05-0.004-0.01

T/Λ=0.99
4e-05

0
2e-05

-0.0050-0.0001-2e-05-0.006T/Λ=0.88-0.015T/Λ=0.89-0.0002-4e-05T/Λ=0.90
T/Λ=0.91-6e-05-0.007

0

0

0.05

0.1

0.1

0.15

0.2

0.25

0.3

0.2

0.35

0.4

-0.0003

0.45

0.3

0.4

0.5

0.6

0.7

0

0

0.1

0.2

0.3

0.4

0.5

0.2

0.6

0.4

v0/√
2Λ

0.6

0.8

1

v0/√
2Λ

k = 0 is very demanding to reach numerically−→ the flow was stopped at k/Λ = 0.2−→ at finite k, the potential is not convex
First order transition is observed−→ in the whole range of the parameter space−→ irrespectively of the flavor number nGergely FejősChiral symmetry restoration with functional renormalization...

Numerical results (effective potential)µ2/Λ2 = -0.05, g1 = -0.8, g2 = 1.1, n = 3µ2/Λ2 = -0.05, g1 = -0.8, g2 = 1.1, n = 3

0.003
5e-05
0.002
0.001
0

Vk/Λ4

Vk/Λ4

0
-0.001-5e-05-0.002-0.003-0.0001-0.004k/Λ = 0.6k/Λ = 0.5k/Λ = 0.4k/Λ = 0.3k/Λ = 0.2k/Λ = 0.1

-0.005k/Λ = 0.130k/Λ = 0.100k/Λ = 0.055k/Λ = 0.030k/Λ = 0-0.00015-0.006
0

0.2

0.4

0.6

0.8

v0/√
2Λ

1

0

0.1

0.2

0.3
v0/√
2Λ

0.4

0.5

0.6

When k → 0, the potential gradually becomes convex
The effective potential is not a convenient quantity for
identifying 1st order transitions−→ crit. temp. and discontinuation are defined as limits:
TC = limk→0 TC (k),∆v0 = limk→0 ∆v0 (k)−→ numerics: they can be obtained via extrapolation
Gergely FejősChiral symmetry restoration with functional renormalization...

Numerical results (TC )µ2/Λ2 = -0.05, n = 2µ2/Λ2 = -0.05, n = 3
g1 = -0.2
g1 = -0.4
g1 = -0.6
g1 = -0.8

1.4

g1 = -0.2
g1 = -0.4
g1 = -0.6
g1 = -0.8

1.4

1.2

1.2

1
Tc/Λ

Tc/Λ

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2
0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.4

0.6

0.8

1

1.2

g2
2

1.4

1.6

1.8

g2

2

2

µ /Λ = -0.05, n = 42µ /Λ = -0.1, n = 3
g1 = -0.2
g1 = -0.4
g1 = -0.6
g1 = -0.8

1.4

g1 = -0.2
g1 = -0.4
g1 = -0.6
g1 = -0.8

1.4

1.2
1.2

Tc/Λ

Tc/Λ

1

0.8

1

0.6
0.8
0.4
0.6

0.2
0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

g2

0.8

1

1.2

1.4

1.6

1.8

g2

Gergely FejősChiral symmetry restoration with functional renormalization...

Numerical results (discontinuity)µ2/Λ2 = -0.05, n = 2µ2/Λ2 = -0.05, n = 3
g1 = -0.2
g1 = -0.4
g1 = -0.6
g1 = -0.8

0.45
0.4

0.4

0.35

0.35

0.3

0.3
v0*/Λ

v0*/Λ

g1 = -0.2
g1 = -0.4
g1 = -0.6
g1 = -0.8

0.45

0.25

0.25

0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0.4

0.6

0.8

1

1.2

1.4

0.4

0.6

0.8

1

g2
2

1.2

1.4

g2

2

2

µ /Λ = -0.05, n = 42µ /Λ = -0.1, n = 3
g1 = -0.2
g1 = -0.4
g1 = -0.6
g1 = -0.8

0.45

g1 = -0.2
g1 = -0.4
g1 = -0.6
g1 = -0.8

0.45

0.4
0.4
0.35
0.35
v0*/Λ

v0*/Λ

0.3

0.25

0.2

0.3

0.25

0.15
0.2
0.1
0.15
0.4

0.6

0.8

1

1.2

1.4

g2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

g2

Gergely FejősChiral symmetry restoration with functional renormalization...

Numerical results (large-n)Large-n scalings of functions:
Uk (I1 ) = n2 uk (i1 ),Ck (I1 ) = ck (i1 )/n,I1 = n2 i1

n-dependence of TC :µ2/Λ2 = -0.05, g1 = -0.4, g2 = 1.0
0.615

0.61

0.605

Tc/Λ

0.6

0.595

0.59

0.585

0.58

aaaaa

0.575
2

3

4

5

6

7

8

9

10

n

Less than 3% difference between n = 3 and n = ∞−→ large-n expansion is quite robust
Gergely FejősChiral symmetry restoration with functional renormalization...

Numerical resultsµ2/Λ2 = -0.05, g1 = -0.8, g2 = 1.1, n = 3
0.36

0.34

0.32

Ck

0.3

0.28

0.26
k/Λ = 0.6k/Λ = 0.5k/Λ = 0.4k/Λ = 0.3k/Λ = 0.2

0.24

0.22
0

0.2

0.4

0.6

0.8

1

v0/√
2Λ

Approximating Ck (I1 ) ≈ const. is crude−→ the function develops a structure as k → 0

Gergely FejősChiral symmetry restoration with functional renormalization...

Numerical results−→ our method: no anomaly included−→ for Nf = 2, 3 we obtain first order transitions−→ if anomaly disappears at TC ⇒ Columbia plot has to change
Gergely FejősChiral symmetry restoration with functional renormalization...

Anomaly
Anomaly term in the Lagrangian:
LUA (1) = c(det M + det M † )−→ changes the masses and spoils chiral symmetry∂k Vk =

k4 X X
1T
2 + k 2 + µ2 (k)6π 2 ω
ωmi
mi−→ Vk 6= Vk (I1 , I2 , ...)

Gergely FejősChiral symmetry restoration with functional renormalization...

Anomaly
Anomaly term in the Lagrangian:
LUA (1) = c(det M + det M † )−→ changes the masses and spoils chiral symmetry∂k Vk =

k4 X X
1T
2 + k 2 + µ2 (k)6π 2 ω
ωmi
mi−→ Vk 6= Vk (I1 , I2 , ...)Way out: expand the r.h.s. in terms of the anomaly coefficientThis procedure is compatible with the Ansatz
Vk = Uk (I1 ) + Ck (I1 ) · I2 + ck (I1 )(det M + det M † )−→ obtain the flow and T -dependence of ck (I1 ) coeff.
Gergely FejősChiral symmetry restoration with functional renormalization...

Finite quark masses

Finite quark masses are realized as explicit symmetry breaking
terms:
Lh = Tr [H(M + M † )] ≡ h0 s 0 + h8 s 8−→ these couplings do not change the flow equations at all−→ they do not have an RG-flow−→ only effect: shift the value of the effective potential
Implementation is easy and straightforward
Work is under progress...

Gergely FejősChiral symmetry restoration with functional renormalization...

Conclusions
Analysis of the U(n) × U(n) meson model−→ no anomaly, zero quark masses−→ top left and bottom left regions of the Columbia plot
Functional renormalization group method−→ local potential approximation−→ chiral invariant expansion
Calculation of the effective potential−→ convexity−→ TC = limk→0 TC (k), ∆v0 = limk→0 ∆v0 (k)Only first order transitions have been observed,
irrespectively of n−→ if the anomaly is recovered around TC ,aaaano second order transition appears!
−→ Columbia plot changes
Gergely FejősChiral symmetry restoration with functional renormalization...