[PPT] - U(1) axial symmetry at high temperature Sinya AOKI for JLQCD PowerPoint Presentation

SLIDE 1

U(1) axial symmetry at high temperature

Sinya AOKI for JLQCD Collaboration

Yukawa Institute for Theoretical Physics, Kyoto University 研究会「有限温度密度系の物理と格子QCDシミュレーション」

2015 September 5, CCS, University of Tsukuba, Tsukuba

SLIDE 2

1. Introduction

Symmetries of QCD at high temperature Restoration of non-singlet chiral symmetry Theoretical questions

2. Eigenvalue distribution of Dirac operator
1. U(1)_A symmetry at high T ?

relation ?

U(1)B ⊗ SU(Nf)L ⊗ SU(Nf)R

ρ(λ) λ: eigenvalue of Dirac operator

SLIDE 3

Eigenvalue density Banks-Casher relation if chiral symmetry is restored. ρ(λ) =

n

ρn λn n! lim

m→0 ¯

ψψ = πρ(0) ρ(0) = ρ0 = 0 Anomalous U(1)A symmetry is fully restored. If ρ(λ) has a gap ρ(λ) λ gap (See later.)

SLIDE 4

Susceptibility

σ meson: 1 ⊗ 1

π meson: γ5 ⊗ τ a

δ meson: 1 ⊗ τ a

η meson: γ5 ⊗ 1

chiral SU(2) chiral SU(2)

U(1)A U(1)A

χA

Γ = 1

V

d4xM A

Γ (x)M A Γ (0)

U(1)A susceptibilities χπ−η ≡ χπ − χη χπ−δ ≡ χπ − χδ χσ−η ≡ χσ − χη If U(1)A is recovered, χσ−η = χπ−δ = χπ−η = 0.

Nf = 2

SLIDE 5

2. Previous Theoretical Investigation

S.A, H. Fukaya, Y. Taniguchi, “Chiral symmetry restoration, eigenvalue density of Dirac operator and axial U(1) anomaly at finite temperature”,

Phys. Rev D86(2012)114512.

SLIDE 6

Set up

Lattice regularization with Overlap fermion, 2-flavor Eigenvalue spectrum

λA

n + ¯

λA

n = aR¯

λA

n λA n

1/Ra

2/Ra

−1/Ra 1/Ra

x y

zero modes(chiral) doublers(chiral)

Exact “chiral” symmetry but explicit U(1)_A anomaly form Ginsparg-Wilson relation

Dγ5 + γ5D = aDRγ5D A: gauge configuration

D(A)φA

n = λA n φA n

D(A)γ5φA

n = ¯

λA

n γ5φA n

complex pair

SLIDE 7

Propagator Measure

zero modes(chiral) doublers(chiral) bulk modes(non-chiral)

fm = 1 + Rma 2

Pm(A) = eSY M(A)(−m)NfN A

R+L

2 Ra NfN A

D

λA

n >0

Z2

m¯

λA

n λA n + m2

S(x, y) =

n

φn(x)φ†

n(y)

fmλn − m + γ5φn(x)φ†

n(y)γ5

fm¯ λn − m

−

NR+L

k=1

1 mφk(x)φ†

k(y) + ND

K=1

Ra 2 φK(x)φ†

K(y)

Z2

m = 1 − (ma)2 R2

4

positive definite and even function of m = 0 for even Nf

N_f=2 in this talk.

# of zero modes # of doublers

SLIDE 8

Chiral symmetry is restored

lim

m→0δaOn1,n2,n3,n4m = 0 On1,n2,n3,n4 = (P a)n1(Sa)n2(P 0)n3(S0)n4 Sa =

d4x Sa(x),

P a =

d4x P a(x)

scalar pseudo-scalar

chiral rotation at N_f=2

δaSb = 2δabP 0, δaP b = −2δabS0 δaS0 = 2P a, δaP 0 = −2Sa

SLIDE 9

Some assumptions

non-singlet chiral symmetry is restored. Assumption 1 Assumption 2

if O(A) is m-independent O(A)m = f(m2)

f(x) is analytic at x = 0

Note that this does not hold if the chiral symmetry is spontaneously broken. Ex.

lim

V →∞

1 V Q(A)2m = m Σ Nf + O(m2)

topological charge

A: gauge configuration

(Too strong. We should loosen this condition.)

SLIDE 10

ρA(λ) ≡ lim

V →∞

1 V

n

δ

λ −
¯

λA

n λA n

Assumption 3

eigenvalues density can be expanded as More precisely, configurations whose eigenvalue density can not be expanded at the origin are “measure zero” in the configuration space.

=

∞

n=0

ρA

n

λn n!

at = 0 ( < )

At finite lattice spacing, integrals over all eigenvalues are convergent, since

|λ| ≤ 2 Ra

(We should remove this assumption and use more general forms in future investigations.)

SLIDE 11

Analysis (some examples)

S0 V = − 1 V

n
F(λn)

fmλn − m + F(¯ λn) fm¯ λn − m

+ N A

R+L

V m

I1 = 1 Zm ΛR dλ ρA(λ)g0(λ2) 2mR λ2 + m2

R

= πρA

0 + O(m)

lim

m→0 lim V →∞

S0m V = 0

V → ∞

ρA

0 m = O(m2)

source of m singularity

lim

V →∞

NR+L V

m

= O(m2) lim

m→0 ¯

ψψm = 0

SLIDE 12

χη−δ = 1 V P 2

0 S2 a

lim

m→0 χη−δ = 0

χη−δ = Nf

1

m2V {2NR+L − NfQ(A)2} + 1 Zm I1 mR + I2

m

Q(A) = N A

R − N A L

I2 = 2 Zm ΛR dλ ρA(λ) m2

R − λ2g0(λ2)gm

(λ2 + m2

R)2

, gm = 1 Z2

m

1 + m2

2Λ2

R

topological charge

=0

I1 mR + I2 = ρA πm m + 2 ΛR

+ 2ρA

1 + O(m),

lim

m→0

N 2

f Q(A)2m

m2V = 2 lim

m→0ρA 1 m

ρA

0 m = O(m2)

repeat these analysis for higher susceptibilities. from others

SLIDE 13

Final Results lim

m→0ρA(λ)m = lim m→0ρA 3 m

|λ|3 3! + O(λ4)

No constraints to higher ρA

n m

ρA

3 m = 0 even for ”free” theory.

lim

V →∞

1 V k (N A

R+L)km = 0,

lim

V →∞

1 V k Q(A)2km = 0

ρA

0 m = 0

SLIDE 14

Consequences

Singlet susceptibility at high T This, however, does not mean U(1)_A symmetry is recovered at high T. is necessary but NOT “sufficient” for the recovery of U(1)_A .

lim

m→0 χπ−η = 0

lim

V →0 χπ−η = lim m→0 lim V →∞

N 2

f

m2V Q(A)2m = 0

”mπ = mη”

SLIDE 15

More general Singlet WT identities

J0O + δ0Om = O(m)

anomaly(measure) singlet rotation

We can show for

where k is the smallest integer which makes the V → ∞ limit finite.

lim

m→0 lim V →∞

1 V k δ0Om = 0

On1,n2,n3,n4 = (P a)n1(Sa)n2(P 0)n3(S0)n4

O =

lim

V →∞

1 V k J0Om = lim

V →∞

Q(A)2 mV O(V 0)

m

= 0

Breaking of U(1)_A symmetry is invisible for these “bulk quantities”. S0 ∼ O(V ), P a, Sa, P 0 ∼ O(V 1/2)

SU(2)L ⊗ SU(2)R ⊗ Z4

SLIDE 16

Remarks

Important conditions Large volume limit chiral limit lattice chiral symmetry m → 0 V → ∞ Ginsparg-Wilson relation

Dγ5 + γ5D = aDRγ5D

Fractional power for the eigenvalue density

ρA(λ) cAλγ, γ > 0

non-singlet chiral symmetry is recovered.

γ ≤ 2 is excluded. γ > 2

consistent with the integer case (n > 2) Universal treatment ? (future investigations)

SLIDE 17

3. Recent Numerical Results
A. Tomiya et al. (JLQCD), Lat2015
G. Cossu et al. (JLQCD), Lat2015

SLIDE 18

Eigenvalue densities

ρ(λ) = lim

V →∞

1 V

n

δ(λ − λn)

Cossu et al. (JLQCD), Overlap

Phys. Rev. D87 (2013) 114514

Gap seems to open at smaller quark mass. Tc 180 MeV

SLIDE 19