light in-medium resonances from lattice qcdcmarkert/workshop/agenda... · 2012. 3. 15. ·...

Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012

Light In-Medium Resonances from lattice QCD

Heng-Tong Ding

1

Brookhaven National Laboratory

Resonance workshop at UT AustinMarch 5-7, 2012

The U(1)A symmetry at finite temperature

Thermal dilepton rates & electrical conductivity


The symmetries of QCD

2

At the classical level, the symmetries of QCD with Nf flavors of massless fermions:

SU(Nf )L × SU(Nf )R × U(1)V × U(1)A

• Spontaneous SU(Nf)LxSU(Nf)R chiral symmetry breaking

• U(1)A symmetry is violated by axial anomaly

∂µjµ5 =

g2Nf

16π2tr(F̃µνF

µν)

• The U(1)A anomaly is responsible for the η-η’ mass splitting

gives rise to 8 Goldstone bosons: the π, K, η

9th Goldstone boson η’ ？

Witten & Veneziano


The QCD phase diagram

3

mu,d

ms

cross over

2nd orderZ(2)

2nd orderO(4)

Nf=2

Nf=1

2nd orderZ(2)

1st

order

PUREGAUGE

Nf=3physical point

mtris

1st

ordermu,d

ms

cross over

2nd orderZ(2)

2nd orderO(4)

Nf=2

Nf=1

2nd orderZ(2)

1st

order

PUREGAUGE

Nf=3physical point

mtris

1st

ordermu,d

ms

cross over

2nd orderZ(2)

2nd orderO(4)

Nf=2

Nf=1

2nd orderZ(2)

1st

order

PUREGAUGE

Nf=3physical point

mtris

1st

order

Pisarski, Wilczek PRD ‘84,

Nf=2 theory at m=0: a second order obeys O(4) universality class

Ejiri et al., PRD ’09, ...

Alexandrou et al., PRD’99...

At physical quark masses, a cross over is confirmed Bernard et al., PRD ’05, Cheng et al., PRD ’06,

Aoki et al., Nature ’06...

Nf >3 theory at m=0 or ∞: a first order phase transition is expected

Nf=2 theory at m=0 can also be first order if U(1)A is restored Pisarski &Wilczek

Is U(1)A effectively restored at Tc ? Shuryak


U(1)A restoration seen in HIC experiments??!

4

?


Symmetries, Correlators & Susceptibilities

5

• correlation functions of vector (ρ) and axial-vector (a1) degenerate

• Restoration of the chiral symmetry:

χσ = χδ + χdiscχπ

χδ χη� = χπ − χ5,disc

• susceptibilities of isosinglet and isotriplet (pseudo) scalar are equiv.

• correlation functions of scalar and pseudo-scalar degenerate

• Restoration of the U(1)A symmetry:

• susceptibilities of scalar and pseudo-scalar are equiv.

isotriplet isosinglet


chiral symmetry restoration

6

P. Hegde,arXiv:1112.0364

P. Hegde(hotQCD) lattice 2011: Domain wall fermion results on 163x8xLs lattices with mπ≈200 MeV

Tc≈160 MeV


chiral & U(1)A symmetry

7


χπ − χδ = χdisc = χ5,disc

• chiral symmetry is restored if

•Both are restored if

•U(1)A is restored if

χπ − χδ = 0 = χdisc + χ5,disc

χπ − χδ = 0 = χdisc = χ5,disc

•χπ - χδ, χ5,disc ,χdisc are close to each other at T>170 MeV

•χπ - χδ, χ5,disc ,χdisc are non-zero even at T=200 MeV

•U(1)A remains broken at T=200 MeV~1.25Tc


The origin of the U(1)A violation

8


• In most cases the presence of a non-zero χπ - χδ is associated with a non-zero topological charge Q

• χπ - χδ becomes smaller at higher temperature


The connection to Dirac spectrum

9

the order parameter of chiral symmetry

the order parameter of the U(1)A symmetry

• <|Q|top> terms are the zero mode contributions and should vanish in the infinite volume limit

• Any U(1)A as well as chiral symmetry breaking effects should come from the near-zero modes

• Effective U(1)A restoration: Gap above λ=0 in ρ(λ)

limm→0

limV→∞

�ψ̄ψ� = πρ(0)�ψ̄ψ� =� ∞

0dλ ρ(λ,m)

2m

λ2 +m2+

�|Qtop|�mV

χπ − χδ =

� ∞

0dλ ρ(λ,m)

4m2

(λ2 +m2)2+

2�|Qtop|�m2V

,

ρ(λ): density of fermion states with eigenvalue λ


The Dirac spectrum

10

• Volume effects need to be checked to remove nontrivial topological gauge conf.

• The gap seen at 180 and 200 MeV is due to zero mode contributionsLin, arXiv:1111.0988


JLQCD results with fixed topology

11

• Effective U(1)A restoration is observed in the Nf=2 QCD at T≈1.15Tc

Cossu, PoS(lattice 2011)188

overlap fermions

163x8

Tc≈180 MeV


JLQCD results with fixed topology

12

• Effective U(1)A restoration is observed in the Nf=2 QCD at T≈1.15Tc

Cossu, PoS(lattice 2011)188

overlap fermions

163x8

Tc≈180 MeV

• pion mass unknown, uncertainties on the volume...


Motivation: PHENIX results on dilepton rates

13

• the low mass, the low pt PHENIX puzzle (c.f. Bratkovskaya et al, Dusling et al)

dNl+l−

dωd3p= Cem

α2em

6π3

ρV (ω, �p, T )

(ω2 − �p2)(eω/T − 1)Cem = e2

nf�

f=1

Q2f• dilepton rates:


Vector correlation & spectral functions

q(0)

q(0)

ΓH Γ†H

q(x)

q(x)

14

ρ(ω, �p) = D+(ω, �p)−D−(ω, �p) = 2 ImDR(ω, �p)

Spectral function

Jµ(τ, �x) ≡ q̄(τ, �x)γµq(τ, �x)

Gµν(τ, �p) =

�d3x�Jµ(τ, �x)J†

ν(0,�0)� ei�p·�x

Euclidean correlation function

GH(τ, �p, T ) =

� ∞

0

dω

2πρH(ω, �p, T )

cosh(ω(τ − 1/2T ))

sinh(ω/2T ), H = 00, ii, V .

Spectral representation

D+(t, �x) = D−(t+ iβ, �x)G(τ, �p) =

�d3x e−i�p·�x D+(−iτ, �x) ,


Prior information on spectral functions

free vector spectral function (in the infinite temperature limit)

ρfreeii (ω) = 2πT 2ωδ(ω) +3

2πω2 tanh(

ω

4T)

vector spectral function at T<∞

-functionδ✦ in ρii is smeared out

-functionδ✦ in ρ00 is protected

ρfree00 (ω) = −2πT 2ωδ(ω)

-functionsδ cancel in ρV (ω) ≡ ρ00(ω) + ρii(ω)✦s

possible form: Breit-Wigner (BW) form + modified continuum

ρ00(ω, T ) = −2πχqωδ(ω)

ρii(ω, T ) = χqcBWωΓ

ω2 + (Γ/2)2+

3

2π

�1 +

αs

π

�ω2 tanh(

ω

4T)

3-4 parameters: (χq), cBW ,Γ,αs

15


previous lattice results on electrical conductivity

16

Electrical conductivity:σ

T=

Cem

6limω→0

ρii(ω)

ωT

The emission rate of soft photons:

Quite different results from previous lattice calculations:

G. Aarts et al., PRL 99(2007)

σ/T � (0.4± 0.1)Cem

S. Gupta PLB 597(2004)57

σ/T � 7Cem

staggered fermions used ρeven and ρodd need to be distinguished

Nτ = 8− 14, Nσ ≤ 44 Nτ = 16, 24, Nσ = 64

limω→0

ωdRγ

d3p= lim

ω→0Cem

αem

4π2

ρ(ω = |�p|, T )eω/T − 1

=3

2π2σ(T )Tαem


previous lattice results on thermal dilepton rate

17

F. Karsch, E. Laermann, P. Petreczky, S. Stickan and I. Wetzorke, Phys.Lett. B530 (2002) 147-152

• 643x16 lattices, finite volume & lattice cutoff effects?

• Nτ not sufficiently large to extract spectral function

Wilson fermions

• ρ(ω) should be linear in ω at small ω, not captured by the MEM analysis

• Integral Kernel needs to be redefined in MEM to explore low frequency region G. Aarts et al, PRL ’07


Vector correlation functions on large & fine lattices

SU(3) gauge configurations at T/Tc≃1.45 lattice size N3

σ ×Nτ with Nσ = 32-128 & Nτ=16,24,32,48

Non-perturbatively clover O(a) improved Wilson fermionsQuark masses close to chiral limit κ � κc

volume dependence

cut-off dep. & continuum extrapolation

close to continuum18


Continuum extrapolation

19

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

0 0.1 0.2 0.3 0.4 0.5

T

T2GV( T)/[ qGVfree( T)]

1283x161283x241283x321283x48

contfit [0.2:0.5]

• Increase of GV (τT )/GfreeV (τT ) with τT is obvious

• The rise with τT indicates that vector spectral function in the low frequency region is different from the free case

• Motivation for the Breit-Wigner type ansatz fitting


Estimate of electrical conductivity

20

k = 0.0465(30) , �Γ =2 .235(75) , 2cBW �χq/�Γ = 1.098(27)

0

1

2

3

4

5

0 2 4 6 8 10

BW+continuumfree

ii( )/ T

/T

σ

T=

Cem

6limω→0

ρii(ω)

ωT

= (0.37± 0.01)Cem

=Cem

3

2cBW �χq

�Γ

�ρii(�ω) =2cBW �χq

�Γ2�ω(�Γ/2)2

�ω2 + (�Γ/2)2+

3

2π(1 + k) �ω2 tanh(

�ω4)


Estimate of electrical conductivity

20

k = 0.0465(30) , �Γ =2 .235(75) , 2cBW �χq/�Γ = 1.098(27)

0

1

2

3

4

5

0 2 4 6 8 10

BW+continuumfree

ii( )/ T

/T

σ

T=

Cem

6limω→0

ρii(ω)

ωT

= (0.37± 0.01)Cem

=Cem

3

2cBW �χq

�Γ

(accidentally) close to Aarts’ result!

�ρii(�ω) =2cBW �χq

�Γ2�ω(�Γ/2)2

�ω2 + (�Γ/2)2+

3

2π(1 + k) �ω2 tanh(

�ω4)


Breit-Wigner + truncated continuum Ansatz

21

ρii(ω) = 2χqcBWωΓ/2

ω2 + (Γ/2)2+

3

2π(1 + k) ω2 tanh(

ω

4T) Θ(ω0,∆ω)

0

1

2

3

4

5

6

0 2 4 6 8 10

ii( )/ T

/T

/T=0.5

0/T=00.51.01.5

1.75cont

delay the onset (ω0) of the continuum partΘ(ω0,∆ω) =�1 + e(ω

20−ω2)/ω∆ω

�−1

0

1

2

3

4

5

6

0 2 4 6 8 10

ii( )/ T

/T

0/T=1.5

/T=00.1

0.250.5

cont

• Rise of BW peaks compensate for the cut from continuum parts

• Fits become worse with increasing and/or increasing ω0 ∆ω


Breit-Wigner + truncated continuum Ansatz

21


ω2 + (Γ/2)2+

3

2π(1 + k) ω2 tanh(

ω

4T) Θ(ω0,∆ω)

0

1

2

3

4

5

6

0 2 4 6 8 10

ii( )/ T

/T

/T=0.5

0/T=00.51.01.5

1.75cont

delay the onset (ω0) of the continuum partΘ(ω0,∆ω) =�1 + e(ω

20−ω2)/ω∆ω

�−1

18

18.5

19

19.5

20

0 0.5 1 1.5 2

G(2)ii /G

(0)ii

0/T

/T=0.5

datafit

MEMfree

• Spectral function should not deviate much from free behavior for ω/T � (2− 4)

•Ratio of thermal moments reacts sensitive to the truncation of the cont. part


Electrical conductivity

22


ω2 + (Γ/2)2+

3

2π(1 + k) ω2 tanh(

ω

4T) Θ(ω0,∆ω)

Θ(ω0,∆ω) =�1 + e(ω

20−ω2)/ω∆ω

�−1

0

1

2

3

4

5

0 2 4 6 8 10

0/T=0, /T=00/T=1.5, /T=0.5

HTLfree

ii( )/ T

/T

1/3 � 1

Cem

σ

T� 1 at T � 1.45 Tc

electrical conductivity

HTL conductivity is divergent at w~0

Soft photon emission rate

limω→0

ωdRγ

d3p= (0.0004 − 0.0013)T 2

c � (1− 3) · 10−5 GeV2 at T � 1.45 Tc

T~1.5 Tc~

Hard thermal loop (HTL) :Braaten &.Pisarski, NP B337 (1990) 569


Thermal dilepton rates

23

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0 2 4 6 8 10

dNl+l-/d d3pp=0

/T

BW+continuum: 0/T=0, /T=00/T=1.5, /T=0.5

HTLBorn

dNl+l−

dωd3p= Cem

α2em

6π3

ρV (ω, �p, T )

(ω2 − �p2)(eω/T − 1)

HTD, Francis, Kaczmarek,Karsch, Laermann, Soeldner,Phys.Rev. D83 (2011) 034504

0

1

2

3

4

5

0 2 4 6 8 10

0/T=0, /T=00/T=1.5, /T=0.5

HTLfree

ii( )/ T

/T

Hard thermal loop (HTL): Braaten &.Pisarski, NP B337 (1990) 569

• thermal dilepton rate approaches leading order Born rate at w/T > 4~

• enhancement at small w/T


Summary & Outlook

24

• Dilepton rate is enhanced at small ω/T and approaches leading order Born rate at ω/T ≳ 4 at 1.45 Tc

• Electrical conductivity 1/3 � 1

Cem

σ

T� 1 at T � 1.45 Tc

‣ The U(1)A symmetry at finite temperature

• The calculation of vector correlation functions at different T is underway

• Many progress have been made by HotQCD & JLQCD collaborations

• Studies with larger volumes and at different temperatures are in progress

‣ Thermal Dilepton rates & electrical conductivity

light in-medium resonances from lattice qcdcmarkert/workshop/agenda... · 2012. 3. 15. ·...

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