Recent lattice results on heavy flavor probes of QGP Peter Petreczky

2018 Santa Fe Jets and Heavy Flavor Workshop, Santa Fe, New Mexico, January 29-31, 2018

QCD at high temperature (introduction): EoS and susceptibilities Heavy quark interactions at high temperatures: color screening TUMQCD, arXiV:1802.xxxx Charm fluctuations and correlations: open charm hadrons above Tc ? Mukherjee, PP, Sharma, PRD 93 (2016) 014502 Spatial meson correlation functions: open charm mesons and quarkonia Bazavov, Karsch, Maezawa, Mukherjee, PP, PRD91 (2015) 054503 S. Sharma and PP, work in progress Quarkonium correlators and spectral functions from NRQCD S. Kim, PP, A. Rothkopf, PRD91 (2015) 054511 ,work in progress

Equation of state at high temperatures

T [MeV]

p/T4 (2+1+1)-f LQCD

ideal3-loop, HTLhadron gas

(2+1)-f LQCD

0

1

2

3

4

5

6

200 400 600 800 1000 1200 1400 1600 1800 2000

2+1 flavor: Bazavov, PP, Weber, arXiv:1710.05024 2+1+1 flavor (with charm quark): Borsányi et al (BW Coll.), Nature 539 (2016) 69

Charm quark contribution to QCD pressure is significant for T>400 MeV The pressure is well describe by weak coupling calculations for T>1000 MeV

Lattice results are extrapolated to continuum (a=0)

QCD thermodynamics at non-zero chemical potential

Taylor expansion :

hadronic

quark

Taylor expansion coefficients give the susceptibilities, i.e. the fluctuations and correlations of conserved charges, e.g.

information about carriers of the conserved charges ( hadrons or quarks )

probes of deconfinement

p(T, µu, µd, µs, µc)

T 4=

X

ijkl

1

i!j!k!l!�udscijkl

⇣µu

T

⌘i ⇣µd

T

⌘j ⇣µs

T

⌘k ⇣µc

T

⌘l

�abcdijkl = T i+j+k+l @i

@µib

@j

@µjb

@k

@µic

@l

@µld

lnZ(T, V, µa, µb, µc, µd) |µa=µb=µc=µd=0

p(T, µB , µQ, µS , µC)

T 4=

X

ijkl

1

i!j!k!l!�BQSCijkl

⇣µB

T

⌘i ⇣µQ

T

⌘j ⇣µS

T

⌘k ⇣µC

T

⌘l

Quark number fluctuations at high T

Good agreement between lattice and the weak coupling approach for 2nd and 4th order quark number fluctuations

quark number fluctuations

Bazavov et al, PRD88 (2013) 094021, Ding et at, PRD92 (2015) 074043

T [MeV]

rud11

EQCDNo=6

81012

cont

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

250 300 350 400 450 500 550 600 650 700

quark number correlations

T [MeV]

ru4/rideal

4

EQCDr4

u(cont)3-loop HTL

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

300 350 400 450 500 550 600 650 700

ru2/rideal

2

T [MeV] 0.85

0.9

0.95

1

1.05

300 500 700 900

ru2/rideal

2

T [MeV] 0.85

0.9

0.95

1

1.05

300 500 700 900

ru2/rideal

2

T [MeV] 0.85

0.9

0.95

1

1.05

300 500 700 900

ru2/rideal

2

T [MeV] 0.85

0.9

0.95

1

1.05

300 500 700 900

ru2/rideal

2

T [MeV] 0.85

0.9

0.95

1

1.05

300 500 700 900

ru2/rideal

2

T [MeV] 0.85

0.9

0.95

1

1.05

300 500 700 900

ru2/rideal

2

T [MeV] 0.85

0.9

0.95

1

1.05

300 500 700 900

Correlations are large for T<200 MeV but agree with the weak coupling expectations for T>300 MeV

Continuum extrapolated lattice results:

Fluctuation and correlations and deconfinement of charm

Bazavov et al, PLB 737 (2014) 210

mc � T only |C|=1 sector contributes

r13BC/r22

BC

r11BC/r13

BC

1.0

1.5

2.0

2.5

3.0

140 160 180 200 220 240 260 280

T [MeV]

No: 8 6

un-corr.hadrons

non-int.quarks

In the hadronic phase all BC-correlations are the same !

Hadronic description breaks down just above Tc ⇒  open charn deconfines above Tc

�XY Cnml = Tm+n+l @

n+m+lp(T, µX , µY , µC)/T 4

@µnX@µm

Y @µlC

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

160 180 200 220 240 260 280 300 320 340

T [MeV]

χucmn/χc

2HTLptEQCD

m n: 22133111

Correlations are large for T<200 MeV but agree with the weak coupling expectations for T>300 MeV, e.g.

Quasi-particle model for charm degrees of freedom

Charm dof are good quasi-particles at all T because Mc>>T and Boltzmann approximation holds

pqC/pC

pBC/pC

pMC/pC

0.0

0.2

0.4

0.6

0.8

1.0

T [MeV]

pBC,S=2/pC

pBC,S=1/pC

pMC,S=1/pC

0.0

0.1

0.2

0.3

150 170 190 210 230 250 270 290 310 330

pqC/pC

pBC/pC

pMC/pC

0.0

0.2

0.4

0.6

0.8

1.0

T [MeV]

pBC,S=2/pC

pBC,S=1/pC

pMC,S=1/pC

0.0

0.1

0.2

0.3

150 170 190 210 230 250 270 290 310 330

Partial meson and baryon pressures described by HRG at Tc and dominate the charm pressure then drop gradually, charm quark only dominant dof at T>200 MeV or ε > 6 GeV/fm3

pC(T, µB , µc) = pCq (T ) cosh(µ̂C + µ̂B/3) + pCB(T ) cosh(µ̂C + µ̂B) + pCM (T ) cosh(µ̂C)

µ̂X = µX/T�C2 , �BC

13 , �BC22 ) pCq (T ), p

CM (T ), pCB(T )

Partial pressures drop because hadronic excitations become broad at high temperatures (bound state peaks merge with the continuum) See Jakovác, PRD88 (2013), 065012 Biró, Jakovác, PRD(2014)065012

Vice versa for quarks

Mukherjee, PP, Sharma, PRD 93 (2016) 014502

Free energy of static quark anti-quark and screening

-5

-4

-3

-2

-1

0

0.01 0.020.03 0.06 0.1 0.2 0.3 0.6 1.0

FS(r,T) [GeV]

r [fm]

T [MeV] 0

220018001200

800500320260220180160140

0

0.2

0.4

0.6

0.8

1

1.2

0.01 0.02 0.03 0.06 0.1 0.2 0.3

_QQ_(r,T)

r [fm]

T [MeV] 0

160180220260320500800

120018002200

singlet Q ¯Q free energy, FS(r, T ) e↵ective coupling ↵QQ̄ = r2@FS(r, T )

@r

Continuum extrapolated lattice results:

TUMQCD, arXiv:1802.xxxx

• At short distances, r < 0.4/T the Q ¯Q interaction is like at T = 0

• ↵QQ̄

(r, T ) has the maximum at r ' 0.4/T : ↵QQ̄

(r ' 0.4/T ) = ↵max

QQ̄

• ↵max

QQ̄

> 0.5 ) QGP is strongly coupled for T < 300 MeV

In-medium properties and/or dissolution of mesons are encoded in the spectral functions:

Melting is see as progressive broadening and disappearance of the bound state peaks

Due to analytic continuation spectral functions are related to Euclidean time quarkonium correlators that can be calculated on the lattice

MEM ρ(ω,p,T) 1S charmonium survives to 1.6Tc ??

Umeda et al, EPJ C39S1 (05) 9, Asakawa, Hatsuda, PRL 92 (2004) 01200, Datta, et al, PRD 69 (04) 094507, ...

D(⌧, p, T ) =

Z 1

0d!⇢(!, p, T )

cosh(! · (⌧ � 1/(2T )))

sinh(!/(2T ))

Meson correlators and spectral functions

D(⌧, p, T ) =

Zd

3xe

ipxhJ(x,�i⌧)J(0, 0)iT

⇢(!, p, T ) =1

2⇡Im

Z 1

�1dte

i!t

Zd

3xe

ipxh[J(x, t), J(0, 0)]iT

If there is no T-dependence in the spectral function, D(τ,T)/Drec(τ,T)=1

temperature dependene of D(τ,T)

Datta, Karsch, P.P., Wetzorke, PRD 69 (2004) 094507

Temperature dependence of temporal charmonium correlators

D(⌧, T ) =

Z 1

0d!⇢(!, T )

cosh(! · (⌧ � 1/(2T )))

sinh(!/(2T ))

Drec(⌧, T ) =

Z 1

0d!⇢(!, T = 0)

cosh(! · (⌧ � 1/(2T )))

sinh(!/(2T ))

Limited sensitivity to medium effects

D0(⌧, T )/D0rec(⌧, T )D(⌧, T )/Drec(⌧, T )

Spatial correlation functions can be calculated for arbitrarily large separations z → ∞

but related to the same spectral functions

Low T limit : High T limit :

Temporal meson correlator only avalaible for τT < ½ and thus may not be very sensitive to In-medium modifications of the spectral functions; also require large Nτ (difficult in full QCD) Spatial correlators can be studied for arbitrarily large separations and thus are more sensitive to the changes in the meson spectral functions; do not require large Nτ (easy in full QCD).

Lattice calculations: spatial meson correlators in 2+1 flavor QCD for ssbar, scbar and ccbar sectors using 483x12 lattices and highly improved staggered quark (HISQ) action (HotQCD) , physical ms and mπ=161 MeV.

Temporal vs spatial meson correlators

G(z, T ) = 2

Z 1

�1dpeipz

Z 1

0d!

⇢(!, p, T )

!

G(z, T ) =

Z 1/T

0d⌧

ZdxdyhJ(x,�i⌧)J(0, 0)iT , G(z ! 1, T ) = Ae

�mscr(T )z

mscr(T ) ' 2q

m2q + (⇡T )2

Temperature dependence of spatial meson correlators

Medium modifications of meson correlators increase with T, but decrease with heavy quark content; larger for 1P charmonium state than for 1S charmonium state

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0.0 0.5 1.0 1.5

z [fm]

GO(z,T)/GO(z,0), cc-, 0++

T [MeV]149171197220248

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5

z [fm]

GNO(z,T)/GNO(z,0), ss-, 1−−

T [MeV]149171197220248 0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5

z [fm]

GNO(z,T)/GNO(z,0), sc-, 1−

T [MeV]149171197220248

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5

z [fm]

GNO(z,T)/GNO(z,0), cc-, 1−−

T [MeV]149171197220248

� D⇤s

J/

�c0

Temperature dependence of meson screening masses

0.5

1

1.5

2

2.5

3

3.5

150 200 250 300 350 400 450 500

M [GeV]

T [MeV]

ss

0--

0++

1--

1++

1.8 2

2.2 2.4 2.6 2.8

3 3.2 3.4 3.6 3.8

150 200 250 300 350 400 450 500

M [GeV]

T [MeV]

sc

0-

0+

1-

1+

2.8

3

3.2

3.4

3.6

3.8

4

4.2

150 200 250 300 350 400 450 500

M [GeV]

T [MeV]

cc

0--

0++

1--

1++

Qualitatively similar behavior of the screening masses for ssbar, scbar and ccbar sectors Screening Masses of opposite parity mesons become degenerate at high T (restoration of chiral and axial symmetry) Screening masses are close to the free limit 2 (mq

2+(π T)2 ) ½ at T>200 MeV, T>250 MeV, T>300 MeV for ssbar, scbar and ccbar sectors, respectively.

- -

-

Temperature dependence of meson screening masses (cont’d)

-120-100

-80-60-40-20

0 20 40

140 150 160 170 180 190 200T [MeV]

6M(T) [MeV]cc

1++

1- -

-120-100

-80-60-40-20

0 20 40 60 80

140 150 160 170 180T [MeV]

6M(T) [MeV]

sc

1+

1-

-200-150-100

-50 0

50 100 150 200

140 150 160 170 180T [MeV]

6M(T) [MeV]

ss

1++

1- -

•  At low T changes in the meson screening Masses ΔM=Mscr(T)-M T=0 are indicative of the changes in meson binding energies •  ΔM is significant already below Tc •  Above the transition temperature the changes in ΔM are comparable to the meson binding energy and except for 1S charmonium (sequential melting)

-

-

-

Bottomonium screening masses

S. Sharma, PP, work in progress

9300

9600

9900

10200

10500

10800

11100

300 400 500 600 700 800

T [MeV]

M [MeV]

rb1rb0

Ydb

bb̄

• For T < 400 MeV ⌥(1S) and ⌘b(1S) screening masses are close to their

vacuum value, �b1(1P ) mass shows large shift

• For T > 500 MeV all the screening masses qualitatively follow the free

theory expectation ) all bottomonia are melted

Why NRQCD ?

Quarkonia to a fairly good approximation are non-relativistic bound state

EFT approach: integrate the physics at scale of the heavy quark mass

NRQCD is the EFT at scale << MQ Heavy quark fields are non-relativistic Pauli spinors:

Advantages: •  the large quark is not a problem for lattice calculations, lattice study of bottomonium is feasible (usually a MQ << 1, which is challenging) •  The structure of the spectral function is simpler => more sensitivity to the bound state properties •  Quarkonium correlators are not periodic and can be studied at larger time extent (=1/T) => more sensitivity to bound state properties

LNRQCD = †✓D⌧ � D2

i

2MQ

◆ +�†

✓D⌧ +

D2i

2MQ

◆�+ ...+ 1

4F2µ⌫ + q̄�µDµq

pQ ⇠ MQv ⌧ MQ

NRQCD on the Lattice

(lattices cannot be too fine)

Quark propagators are obtained as initial value problem:

The energy levels in NRQCD are related to meson masses by a constant lattice spacing dependent shift, e.g.

well behaved if naMQ < 3Davies, Thacker, PRD 45 (1992) 915

Thacker, Lepage, PRD43 (1991) 196

M⌥(1S) = E⌥(1S) + Cshift(a)

Light d.o.f (gluons, u,d,s quarks) are represented by gauge configurationsfrom HotQCD, ms = mphys

s , mu,d = ms/20 $ m⇡ = 161 MeVT > 0: 483 ⇥ 12 lattices, Tc = 159 MeV, the temperature is variedby varying a $ � = 10/g2 Bazavov et al, PRD85 (2012) 054503

Inverse lattice spacing provides a natural UV cuto↵

for NRQCD provided a�1 2MQ

) 140MeV T 407MeV2.759 � aMb � 0.954 (ok if n = 2, 4)

0.757 � aMc � 0.427 (ok if n � 8)

D(⌧) =X

x

hO(x, ⌧)SQ

(x, ⌧)O†(0, 0)S†Q

(x, ⌧)iT

, O(3S1;x, ⌧) = �i

, O(3P1;x, ⌧) = �i

�j

��j

�i

SQ(x, ⌧ + a) = U

†4 (1�

p

2

2MQ�⌧)SQ(x, ⌧), �⌧ = a/n

Bayesian Reconstruction of spectral functions

P [D|⇢I] = exp(�L)

Burnier Rothkopf, PRL 111 (2013) 182003

Discretize the integral

using Bayesian approach, i.e. maximizing

Likelihood:

no restriction on the search space no flat directions

Prior probability:

Different from MEM !

⇢land find

Bottomonium spectral functions at T=0

Well resolved ⌥ ground state peak

Acceptable resolution for �b state

E⌥ + Cshift(a) = 9.46030 GeV

Define the NRQCD energy shift Cshift(a) by fixing the ⌥ peak to PDG

) prediction for mass of other states: ⌘b, �b0, �b1, hb

But excited states, ⌥

0, ⌥

00, �0

b cannot be resolved well

Charmonium spectral functions at T=0

Well resolved J/ peak

Only the position of the �c1 statecan be reliably determined

Excited states cannot be determined, artifacts at �! > 3 GeV

Define the NRQCD energy shift Cshift(a) by fixing the J/ peak to PDG

EJ/ + Cshift(a) = 3.097 GeV

) prediction for mass of other states: ⌘c, �c0, �c1, hc

0.0001

0.001

0.01

0.1

1

1.5 2 2.5 3 3.5 4

�J/�

(�)

�� [GeV]

3P1 T�0 Nmeas=400 n=6

�=6.664 324 �=6.740 484 �=6.800 324

�=6.880 484 �=6.950 324 �=7.030 484

0.001

0.01

0.1

1

10

100

1 1.5 2 2.5 3 3.5

�J/�

(�)

�� [GeV]

3S1 T�0 Nmeas=400 n=6

�=6.664 324 �=6.740 484 �=6.800 324 �=6.880 484 �=6.950 324 �=7.030 484 �=7.150 483x64�=7.280 483x64

Temperature dependence of the bottomonium correlators

) hints for sequential melting pattern: stronger medium modification

of �b1 spectral function than for ⌥ spectral function

D(⌧,T )D(⌧,T=0)

D(⌧,T )D(⌧,T=0)

0.99

0.995

1

1.005

1.01

1.015

1.02

0 0.2 0.4 0.6 0.8 1 1.2

3S1

n=2 and n=4

D� (

T>0)

/D� (

T�0)

� [fm]

T=140MeVT=160MeVT=184MeV

T=249MeVT=273MeVT=333MeV

T=407MeV

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

0 0.2 0.4 0.6 0.8 1 1.2

3P1

n=2 and n=4

D� b

(T>0

)/D� b

(T�

0)� [fm]

T=140MeVT=160MeVT=184MeV

T=249MeVT=273MeVT=333MeV

T=407MeV

change in ⌥ correlator < 2%

change in �b1 correlator < 7%

Temperature dependence of the charmonium correlators

change in J/ correlator < 5%

change in �c1 correlator < 12%

) hints for sequential melting pattern:

changes in the J/ correlator are about the same as in the �b correlator

(same size); changes in the �c correlators are factor of two larger

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

0 0.2 0.4 0.6 0.8 1 1.2

3P1 T>0 Nmeas=400 n=8

D� c

(T>0

)/D� c

(T�

0)� [fm]

T=140MeVT=151MeVT=160MeVT=172MeVT=184MeVT=198MeVT=221MeVT=249MeV

0.99

1

1.01

1.02

1.03

1.04

1.05

0 0.2 0.4 0.6 0.8 1 1.2

3S1 T>0 Nmeas=400 n=8

DJ/� (

T>0)

/DJ/� (

T�0)

� [fm]

T=140MeVT=151MeVT=160MeVT=172MeVT=184MeVT=198MeVT=221MeVT=249MeV

D(⌧,T )D(⌧,T=0)

D(⌧,T )D(⌧,T=0)

Reconstructing Spectral Functions at T>0

Two main problems:

1) ⌧ < 1/T ) limited temporal extent at high T2) relatively small number of time slices (N⌧ = 12 in our study)

Study these e↵ects at T = 0 by using only the first 12 data points:

Decreasing ⌧max

= 1/T leads to broadening of the bound state peak

(to be taken into account in comparison T = 0 and T > 0 spectral functions)

Bottomonium Spectral Functions at T>0

Compare T = 0, T > 0 and free spectral functions reconstructed

using the same systematics ( ⌧max

= 1/T and Ndata

= 12)

Both ⌥ and �b survive up to temperature T > 249 MeV

Onia masses at T>0

ΔM140MeV=)5MeV

NRQCD Bayes T�0NRQCD Bayes T�0 Truncated

PDG T=0

NRQCD Bayes T>0

��� ��� ��� ��� ��� ���

���

���

��

���

��

���

����

��[���

]

�b (3P1) @ T>0, n=4

NRQCD Bayes T>0

PDG T=0

������ ��� ��� ��� ��� �����

��

��

��

� [ ��]

���

��[���

]

�(3S1) @ T>0, n=4PRELIMINARY

No T>06boundstate signal

ΔM273MeV=)87(4)MeV

ΔM251MeV=)87(54)MeV

ΔM140MeV=)17(1)MeV

NRQCD Bayes T�0NRQCD Bayes T�0 Truncated

PDG T>0

��� ��� ��� ��� ��� ���

���

��

���

��

��

�

���

��[���

]

J/�(3S1) @ T>0, n=8

NRQCD Bayes T>0

PDG T=0

��� ��� ��� ��� ��� ���

���

��

���

��

��

�

���

��[���

]

J/�(3S1) @ T>0, n=8

NRQCD Bayes T�0NRQCD Bayes T�0 Truncated

PDG T>0

��� ��� ��� ��� ���

���

���

���

���

���

[�� ]

���

��[���

]

J/�(3S1) @ T>0, n=8PRELIMINARY

Kim, PP, Rothkopf, arXiv:1704.05221

Onia masses from the peak positions:

Shifts in the peak location is smaller at T>0 than in the vacuum for the same temporal extent è the actual onia masses decrease with increasing temperature

Summary

•  Charm correlations and fluctuations carry information about charm hadrons: 1) For T<Tc fluctuations and correlations are described by hadron resonance gas 2) For T>1.3Tc fluctuations and correlations are described by charm quark gas 3) In-medium open heavy flavor bound states may exist for Tc < T < 1.3 Tc

⇒  sQGP, confirmed by study of static quark anti-quark free energy •  Spatial meson correlators and NRQCD correlators are sensitive to the temperature to the changes in the spectral functions and are consistent with sequential melting picture: where mesons more heavy quarks dissolve at higher temperatures and 1S onia survive till higher temperature than 1P onia

•  Need a link between spatial meson propagators and charm fluctuations to establish the existence and nature of open charm hadrons above Tc

Td(J/ ) ' 250MeV, Td(�b) ' 270MeV, Td(⌥) > 407MeV

High T (T>250 MeV) :

Low T: correlations are large ( bound states ?)

Back-up:

pqC/pC

pBC/pC

pMC/pC

0.0

0.2

0.4

0.6

0.8

1.0

T [MeV]

pBC,S=2/pC

pBC,S=1/pC

pMC,S=1/pC

0.0

0.1

0.2

0.3

150 170 190 210 230 250 270 290 310 330

Strange – charm hadrons:

T [MeV]

SQ(T) LONLO, µ=(1-4)/ T

NNLOlattice

0

0.1

0.2

0.3

0.4

0.5

1000 1500 2000 2500 3000 3500 4000 4500 5000

Does the quasi-particle model makes sense ?

4 non-trivial constraints on the model provided by :

�BC31 ,�SC

31 ,�BSC121 ,�BSC

211

T [MeV]

c1/pC

c2/pCc3/pC

c4/pC

-1.0

-0.5

0.0

0.5

1.0

150 170 190 210 230 250 270 290 310 330

Diquark pressure is zero !

Models with charm quark only: correlations from an effective mass

mc = mc(T, µC , µS , µB)

Taylor expand the effective mass in chemical potential c n ⇒ Un-natural fine tuning of the expansion coefficients

How Well NQRCD Works for Bottomonium ?

��� ��� ��� ��� ��� ��� �� �

���

���

���

��

������[�� ]�b(1S) from Spectra

��� ��� ��� ��� ��� ��� �� �

���

����

���

������[�� ]�b(1P) from Spectra

NQRCD

PDG

PDG

NQRCD

NRQCD can reproduce the hyperfine splitting in bottomonium with accuracy < 20-40 MeV depending on the lattice spacing

NRQCD can reproduce the 1P-1S splitting in bottomonium with accuracy < 15 MeV

ηb

χb1

How Well NQRCD Works for Charmonium ?

NRQCD can reproduce the hyperfine splitting in charmonium with an accuracy < 40 MeV

2.950

2.955

2.960

2.965

2.970

2.975

2.980

2.985

2.990

2.995

3.000

6.6 6.7 6.8 6.9 7 7.1 7.2 7.3 7.4

1S0 T�0 Nmeas=400 n=8

m� c

cal

ibra

ted

[GeV

]

�

PDGNRQCD

3.4

3.45

3.5

3.55

3.6

3.65

6.6 6.7 6.8 6.9 7 7.1 7.2 7.3 7.4

3P1 T�0 Nmeas=400 n=8m� c

Bay

es c

alib

rate

d [G

eV]

�

PDGNRQCD

NRQCD can reproduce the 1P-1S splitting in charmonium well for lattice spacing a > 0.08fm