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Chiral phase transition of Nf=2+1 QCD with the HISQ action

Heng-Tong Ding

Brookhaven National Lab & Columbia University

1

Lattice conference at MainzJuly 29, 2013

in collaboration with A. Bazavov, F. Karsch, Y. Maezawa, S. Mukherjee and P. Petreczky

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QCD phase diagram at mu=0

2

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

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

mc

How large is the chiral phase transition Tc ?

?

?

columbia plot:

How large is the influence of scaling regimes to the physical world ?

Nf=2+1 theory: at m=0 or ∞ has a first order phase transition

Intermediate quark mass region an analytic cross over is expected

At physical quark masses, a cross over is confirmed

Critical lines of second order transition Nf=2: O(4) universality class Nf=3: Ising universality class

Pisarski, Wilczek PRD ‘84,

Karsch, Laermann, Schmidt PLB ’04,...

Ejiri et al., PRD ’09, ...

Alexandrou et al., PRD’99...

Bernard et al., PRD ’05, Cheng et al., PRD ’06, Aoki et al., Nature ’06...

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O(N) spin models and Nf=2 QCD

3

QCD at low energies can be described effectively by O(N) symmetric spin models

• SU(2)L x SU(2)R is isomorphic to O(4)

• external field H corresponds to quark mass m

• order parameter “magnetization” Σ=<σ>

• O(4) fields: σ=qq, π=qɣ5tiq, and η=qɣ5q, δ=qtiq--

This description is valid both below and in the vicinity of the chiral phase transition region

- -

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chiral phase transition and universal scaling

4

Behavior of the free energy close to critical lines

f(m,T)=h1+1/δ fs(z) + freg(m,T), z=t/h1/βδ

fs(z): universal scaling function, O(N) etc.h: external field, t: reduced temperature, β,δ: universal critical exponents

Magnetic Equation of State (MEoS):

M = -∂fs(t,h)/∂h =h1/δ fG(z) fχ(z)=h1/δ(ml/ms)1-1/δ∂M/∂h0

h= 1h0

mlms

T-Tc

Tc

1t0

t=

/15

0.00

0.50

1.00

1.50

2.00

-5 -4 -3 -2 -1 0 1 2 3 4 5

t/h1/

Mb/h1/

O(2)

all masses

ml/ms=2/51/5

1/101/201/401/80

Recent O(N) universal scaling studies

5

BNL-Bielefeld PRD ‘09

the scaling window depends on discretization schemes: standard v.s. improved staggered fermions

Nτ=4, p4fat3

0.1

1.0

0.1 0.2 0.5 1.0 2.0 4.0t/h1/

Mb/h1/

O(2)

(2+1)-f, impr: 1/801/401/20

2-f, std: 0.0080.01335

0.0267

f(m,T)=h1+1/δ fs(z) + freg(m,T), z=t/h1/βδ

scaling violations seen at ml/ms > 1/10 using p4 action on Nt=4 lattices

Mb=ms<ψψ>/T4

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Recent O(N) universal scaling studies

6

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

t/h1/

f

O(2)

O(4)

mq/ms=1/801/401/20

Nτ=4, p4fat3

135140145150155160165170175180185190195

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Tc [MeV]

N-2

Physical ml/ms

Combined continuum extrapolationHISQ/tree: quadratic in N-2

Asqtad: linear in N-2

HISQ/treeAsqtad

no fitting

HotQCD, PRD ’11BNL-Bielefeld PRD ‘09

combined fitting to condensates & susceptibilities

Reasonably good prediction of chiral susceptibilities using parameters obtained from the scaling fit to chiral condensates

Useful tool to determine the critical temperature, chiral curvature etc.

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Nf=2+1QCD

7

mu,d

ms

cross over

2nd orderO(4)

Nf=2

Nf=1

2nd orderZ(2)

1st

order

PUREGAUGE

Nf=3physical point

mu,d

ms

cross over

2nd orderO(4)

Nf=2

Nf=1

2nd orderZ(2)

1st

order

PUREGAUGE

Nf=3physical point

mu,d

ms

cross over

2nd orderO(4)

Nf=2

Nf=1

2nd orderZ(2)

1st

order

PUREGAUGE

Nf=3physical point

mu,d

ms

cross over

2nd orderO(4)

Nf=2

Nf=1

2nd orderZ(2)

1st

order

PUREGAUGE

Nf=3physical point

2nd order O(4) scaling regime may have more influence on the physical world ?

The chiral first order phase transition region shrinks with better improved staggered fermions mπ ≈ 290 MeV —> mπ ≲ 45 MeVcc

HTD, xQCD 2012, arXiv:1302.5740

• Fix the strange quark mass to be physical

• Decrease the light quark mass approaching to the chiral limit

• Simulations on Nt=6 lattices using Highly Improved Staggered Quarks with 5 different quark masses corresponding to mπ from 160 MeV down to 80 MeV

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volume dependence at physical pion mass

8

• volume effects are small in 3 largest volume

• mπL > 4 is ensured in the following other datasets

483x6 with mπ=80 MeV, 403x6 with mπ=90 MeV, 323x6 with mπ=110 MeV, 243x6 with mπ=160 MeV

0

5

10

15

20

25

120 130 140 150 160 170T [MeV]

< - >l/T3 123x6

163x6203x6243x6323x6

0

10

20

30

40

50

60

70

120 130 140 150 160 170T [MeV]

dis/T2

123x6163x6203x6243x6323x6

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volume dependence at mπ=80 MeV

9

• Mild volume dependence is seen from chiral condensates

• No evidence of linear volume scaling as signatures of first order phase transition

2

4

6

8

10

12

140 142 144 146 148 150 152 154 156 158 160T [MeV]

< - >l/T3

m =80 MeV323x6483x6

40

60

80

100

120

140

140 142 144 146 148 150 152 154 156 158 160T [MeV]

dis/T2 m =80 MeV

323x6483x6

• Volume scaling analysis needs to understand the volume effects

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chiral condensates & susceptibilities

10

• chiral condensates decrease with increasing temperature and decreasing quark mass

• peak heights of chiral susceptibilities increase and peak locations shift to lower temperatures with decreasing quark mass

2 4 6 8

10 12 14 16 18 20 22

120 130 140 150 160 170

<-

> l/T

3

T [MeV]

m =160 MeVm =140 MeVm =110 MeVm = 90 MeVm = 80 MeV

20

40

60

80

100

120

140

120 130 140 150 160 170

dis,

l/T2

T [MeV]

m =160 MeVm =140 MeVm =110 MeVm = 90 MeVm = 80 MeV

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O(N) scaling behavior

11

For large negative values of z

fG(z) � f−∞(z)G = (−z)β

�1 + c2 β (−z)−βδ/2

�

M = h1/δfG(z) � h1/δf−∞(z)G = (−t)β

�1 + c2β(−t)−βδ/2

√h�

z=t/h1/βδ

For large positive values of z

fG(z) ∼ Rχ z−β(δ−1)

M = h1/δfG(z) ∼ Rχ t−β(δ−1) h

susceptibility of the order parameter ~ 1/sqrt(h)

susceptibility of the order parameter is independent of h

Engels et al., PLB 514(2001)299

Engels et al., NPB 675(2003)533

contribution of Goldstone modes to the order parameter M is enclosed in the scaling function in the low temperature

h= 1h0

mlms

T-Tc

Tc

1t0

t=

/15

20

40

60

80

100

120

140

120 130 140 150 160 170

dis,

l/T2

T [MeV]

m =160 MeVm =140 MeVm =110 MeVm = 90 MeVm = 80 MeV

disconnected chiral sus. at low and high temperatures

12

χl,disc(ml/ms)1/2

•At very low temperature, the disconnected susceptibilities scale as square root of quark mass

•At T~170 MeV, the disconnected susceptibilities seem to be independent on quark mass: a likely indication of U(1)A symmetry breaking

Chris Schroeder’s talk on U(1)A from DWF, 15:30 today

2

4

6

8

10

12

14

120 130 140 150 160 170T [MeV]

m =160 MeVm =140 MeVm =110 MeVm = 90 MeVm = 80 MeV

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scaling and scaling violation of the chiral condensate

13

• scaling violation of chiral condensates seen with mπ ≥110 MeV (ml/ms≥1/40) using the HISQ action on Nt=6 lattices

M = -∂fs(t,h)/∂h =h1/δ fG(z)

•The right plot is generated using the fitting parameters obtained from the fit to the two lightest quark mass shown in the left plot

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1 0 1 2 3

z=t/h1/

M/h1/

z=t/h1/

M/h1/

z=t/h1/

M/h1/

z=t/h1/

M/h1/ ml/ms=1/60ml/ms=1/80

O(2)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1 0 1 2 3

z=t/h1/

M/h1/

z=t/h1/

M/h1/

z=t/h1/

M/h1/ ml/ms=1/20ml/ms=1/27ml/ms=1/40ml/ms=1/60ml/ms=1/80

O(2)

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scaling and scaling violation of the chiral condensate

13

• scaling violation of chiral condensates seen with mπ ≥110 MeV (ml/ms≥1/40) using the HISQ action on Nt=6 lattices

M = -∂fs(t,h)/∂h =h1/δ fG(z)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1 0 1 2 3

z=t/h1/

M/h1/

z=t/h1/

M/h1/

z=t/h1/

M/h1/ ml/ms=1/20ml/ms=1/27ml/ms=1/40ml/ms=1/60ml/ms=1/80

O(2)

0.00

0.50

1.00

1.50

2.00

-5 -4 -3 -2 -1 0 1 2 3 4 5

t/h1/

Mb/h1/

O(2)

all masses

ml/ms=2/51/5

1/101/201/401/80

Nτ=4, p4fat3

Navigation: mπ=160 MeV ~ ml/ms=1/20, mπ=80 MeV ~ ml/ms=1/80

• the scaling window shrinks compared to the results obtained using the p4 action on Nt=4 lattices

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fit to chiral condensates and resulting sus.

14

freg = (a0 + a1(T-Tc)/Tc ) ml/ms , χreg = ∂freg/∂(ml/ms)

fG(z) = (M - freg )/h1/δ fχ(z)=(χM - χreg)h0/h1/δ-1

• After including the regular terms, the chiral condensates can be described by the O(2) scaling function fG(z)

• The susceptibilities can be reasonably reproduced using the fitting parameters obtained from the fit to the chiral condensate

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1 0 1 2 3

z=t/h1/

(M - freg )/h1/ m =160 MeVm =140 MeVm =110 MeVm = 90 MeVm = 80 MeV

O(2)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-2 -1 0 1 2 3

z=t/h1/

( M- reg)h0/h(1/ -1) m =160 MeVm =110 MeV

O2

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Summary

15

• We study the chiral observables on Nt=6 lattices using the HISQ action with mπ =160,140,110, 90 and 80 MeV

•No direct evidence of a first order phase transition in current pion mass window is found

• Regular terms need to be included to extract information on the singular structure

• The scaling window shrinks in the HISQ results compared to that in the p4 results