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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
U(1)A restoration seen in HIC experiments??!
4
?
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
chiral & U(1)A symmetry
7
P. Hegde,arXiv:1112.0364
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
The origin of the U(1)A violation
8
P. Hegde,arXiv:1112.0364
• 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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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...
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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:
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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) ,
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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)
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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)
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
Electrical conductivity
22
ρii(ω) = 2χqcBWωΓ/2
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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
Heng-Tong Ding Resonance workshop @ UT Austin, May 5, 2012
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