1. Calculation of high-order cumulants  with canonical ensemble method  in lattice QCD
Contents
Asobu Suzuki
Univ. of Tsukuba
for Zn Collaboration
S. Oka (Rikkyo Univ.)
S. Sakai (Kyoto Univ.)
Y. Taniguchi (Univ. of Tsukuba)
A. Nakamura (Hiroshima Univ.)
R. Fukuda (Tokyo Univ.)
Quark number cumulant
Sign problem
Canonical ensemble method
Winding number
Numerical results
Conclusion

2. Hadronic phase : Hadron Resonance Gas QGP phase : Quark Gluon Gas
Motivation 𝑇 𝜇
QGP phase
critical
point
Hadronic phase
QCD phase diagram
Hadronic phase : Hadron Resonance Gas
QGP phase : Quark Gluon Gas
Hadronic phase　→　Quark-Gluon phase
from the viewpoint of quark number cumulant

3. Quark number cumulant H.R. Gas =𝐭𝐚𝐧𝐡 𝟑 𝛍 𝐓 =𝟏 =𝟏
𝑁 𝑘 𝐶 ≔ 𝜕 𝑘 𝜕 𝜇/𝑇 𝑘 log 𝑍 𝐺.𝐶.
H.R. Gas
𝑁 3 𝐶 𝑁 2 𝐶
=𝐭𝐚𝐧𝐡 𝟑 𝛍 𝐓
(dotted line)
𝑁 4 𝐶 𝑁 2 𝐶 =𝟏
𝑁 3 𝐶 𝑁 1 𝐶 =𝟏
deviation from H.R. Gas
STAR Collaboration
Phys. Rev. Lett. 112 (2014)
something interesting?

4. Canonical ensemble method A.Hasenfratz , D.Toussaint (1992)
𝑍 𝐺.𝐶. 𝑇,𝜇;𝑉 = 𝑛 𝑍 𝑐𝑎𝑛. 𝑇;𝑛,𝑉 𝜉 𝑛 ,𝜉= 𝑒 𝜇 𝑇
Fugacity expansion
Canonical partition function
Equivalent
Grand canonical partition function
Residue theorem
𝑍 𝑐𝑎𝑛. 𝑇;𝑛,𝑉 = 1 2𝜋𝑖 𝐶 𝑑𝜉 𝜉 − 𝑛+1 𝑍 𝐺.𝐶. 𝑇;𝜉,𝑉
pure
imaginary
𝐶:𝜉= 𝑒 𝑖 𝜇 𝑇 , 𝜇 𝑇 ∈[−𝜋,𝜋)
= 1 2𝜋 −𝜋 𝜋 𝑑 𝜇 𝑇 𝑒 −𝑖 𝜇 𝑇 𝑛 𝑍 𝐺.𝐶. 𝑇;𝑖𝜇,𝑉
Fourier transformation！

5. Winding number expansion Li, X. Meng, A. Alexandru,K. F. Liu (2008)
Canonical partition function
Grand canonical partition function
Fourier trans.
𝑍 𝑐𝑎𝑛. 𝑇;𝑛,𝑉 = 1 2𝜋 −𝜋 𝜋 𝑑 𝜇 𝑇 𝑒 −𝑖 𝜇 𝑇 𝑛 𝐷𝑒𝑡 𝐷 𝑖𝜇 𝐷𝑒𝑡 𝐷 𝑔
sign
real , positive
calculate 𝐷𝑒𝑡 𝐷(𝑖𝜇)
at low cost !
𝛾 5 𝐷 𝜇 𝛾 5 =𝐷 − 𝜇 ∗ †
𝐷𝑒𝑡 𝐷 𝜇 =𝐷𝑒𝑡 1−𝜅𝑄 𝜇 = 𝑒 𝑇𝑟 log 1−𝜅𝑄
𝜅 : hopping parameter　　　　
𝑇𝑟 log 1−𝜅𝑄 =− 𝑛 𝜅 𝑛 𝑛 𝑇𝑟 𝑄 𝑛
= 𝑘 𝑊 𝑘 𝑒 𝜇 𝑇 𝑘
𝑘 ; winding number

6. Numerical results − log 𝑍 𝑐𝑎𝑛. 𝑁 𝛽=1.1 − log 𝑍 𝑐𝑎𝑛. 𝑁 𝛽=2.1
Iwasaki gauge action
2-flavor Wilson Clover
Lattice Size : 8 3 ×4 𝜷 𝜿
𝑻/𝑻 𝑪
0.9
0.137
0.644
1.1
0.133
0.673
1.3
0.706
1.5
0.131
0.813
1.6
0.130
1.7
0.129
1.00
1.8
0.126
1.9
0.125
1.68
2.1
0.122
3.45
− log 𝑍 𝑐𝑎𝑛. 𝑁 　𝛽=1.1
− log 𝑍 𝑐𝑎𝑛. 𝑁 　𝛽=2.1
𝑁 𝑘 = 𝑁 𝑁 𝑘 𝑍 𝑐𝑎𝑛. 𝑁 𝑒 𝜇 𝑇 𝑁 𝑁 𝑍 𝑐𝑎𝑛. 𝑁 𝑒 𝜇 𝑇 𝑁
𝑇 𝐶 𝜇=0 = 𝑀𝑒𝑉
small 𝛽 → low temperature
large 𝛽 → high temperature

7. H.R. gas Q.G. gas 𝑁 2 𝐶 / 𝑁 1 𝐶 𝛽 not depend on 𝜷 (solid line)
𝑁 2 𝐶 / 𝑁 1 𝐶 𝛽
H.R. gas
(solid line)
𝑁 2 𝐶 𝑁 𝐶 = 3 coth 𝜇 𝐵 𝑇
not depend
on 𝜷
𝑁 2 𝐶 𝑁 𝐶 = 𝜋 𝜇 𝐵 3𝑇 𝜇 𝐵 3𝑇 + 1 𝜋 𝜇 𝐵 3𝑇 3
Q.G. gas
(dotted line)

8. 𝑁 2 𝐶 / 𝑁 1 𝐶 𝛽 consistent with H.R. gas approach to Q.G. gas
𝑁 2 𝐶 / 𝑁 1 𝐶 𝛽
𝜇 𝐵 /𝑇=0.12 ,0.24 ,1.08 , 𝛽≾1.6
𝜇 𝐵 /𝑇 = 𝛽≾ , 𝜇 𝐵 /𝑇= 𝛽≾1.4
consistent with H.R. gas
𝛽∼2.1
approach to Q.G. gas

9. Comparison with previous work
374.7
767.0
222.5
157.1
143.3
T[MeV]
Christof Gattringer and Hans-Peter Schadler Phys. Rev. D 91,
This work
consistent behavior
application range of HR gas becomes small as 𝜇/𝑇 increases
measure the H.R. gas / Q.G. gas transition

10. artificial phase transition
Function of 𝜇 𝐵 /𝑇
We truncate
the fugacity expansion
deviate from HR gas
approach to QG gas
𝑍 𝐺.𝐶. 𝜇 𝑞 /𝑇 ∼ 𝑛=− 𝑁 𝑐𝑢𝑡 𝑁 𝑐𝑢𝑡 𝑍 𝑐𝑎𝑛. 3𝑛 𝑒 3𝑛 𝜇 𝑞 𝑇
𝑍 𝑐𝑎𝑛. 3𝑛 =0 , 𝑛 > 𝑁 𝑐𝑢𝑡
deviate from QG gas
not depend on 𝑵 𝒄𝒖𝒕
𝑁 2 𝐶 / 𝑁 1 𝐶 𝜇 𝐵 /𝑇 ,𝛽=0.9
HR gas(blue line)
QG gas(red line)
𝑵 𝒄𝒖𝒕 creates
artificial phase transition

11. 𝑁 2 𝐶 / 𝑁 1 𝐶 𝜇 𝐵 /𝑇 at low temperature
HR gas(blue line)
QG gas(red line)
𝜷=𝟏.𝟓
𝜷=𝟎.𝟗
consistent
small 𝝁 𝑩 /𝑻 ∼ H.R. gas
𝝁 𝑩 /𝑻 ∼𝟒 for 𝜷=𝟎.𝟗
𝝁 𝑩 /𝑻 ∼𝟑.𝟓 for 𝜷=𝟏.𝟓
large 𝝁 𝑩 /𝑻 ∼ Q.G. gas
Application range of HR gas becomes small as 𝛽 increases.
circumstantial evidence of phase transition

12. 𝑁 2 𝐶 / 𝑁 1 𝐶 𝜇 𝐵 /𝑇 at high temperature
𝜷=𝟐.𝟏
HR gas(blue line)
QG gas(red line)
𝜷=𝟏.𝟗
approach to QG gas
as 𝜷 increases
𝜷=𝟏.𝟗
volume dependence
𝟖 𝟑 ×𝟒
𝟏𝟐 𝟑 ×𝟒
approach to QG gas
as volume increases
interaction remains

13. Skewness (fig. above) Kurtosis (fig. bottom)
Skewness , Kurtosis 𝑣𝑠. 𝛽
𝑁 3 𝐶 𝑁 1 𝐶
𝑁 4 𝐶 𝑁 2 𝐶
solid line : H.R. gas
dotted line : Q.G. gas
Skewness (fig. above)
Kurtosis (fig. bottom)
𝜇 𝐵 /𝑇=0.12 ,1.08 , 𝛽≾1.6
𝝁 𝑩 /𝑻=𝟐,𝟎𝟒 ,𝜷=𝟏.𝟔
deviate from H.R. gas
negative value
consistent with H.R. gas
as same as 𝑵 𝟐 𝑪 / 𝑵 𝑪

14. circumstantial evidence of phase transition
Kurtosis 𝑣𝑠. 𝜇 𝐵 /𝑇 ,𝛽=1.6
consistent with H.R. gas
approach to Q.G. gas
singularity
circumstantial evidence of phase transition

15. more high-order cumulant 𝛽=1.6
𝑁 5 𝐶 / 𝑁 1 𝐶 𝜇 𝐵 /𝑇
𝑁 6 𝐶 / 𝑁 2 𝐶 𝜇 𝐵 /𝑇
≠𝟖𝟏 𝝁 𝑩 /𝑻~𝟎
≠𝟖𝟏 𝝁 𝑩 /𝑻~𝟎
𝑁 5 𝐶 𝑁 𝐶 = 𝑁 6 𝐶 𝑁 2 𝐶 =81
H.R. gas
(blue line)
2nd ,3rd ,4th : consistent with H.R. gas
5th ,6th : inconsistent with H.R. gas
(suggestion)

16. Conclusion Canonical ensemble method Hadron Resonance gas model
Sign problem become mild.
We can see as a function of 𝜇 𝐵 /𝑇 .
𝑁 𝑐𝑢𝑡 makes an artificial phase transition!
Hadron Resonance gas model
Less than 4th-order → HR gas is good model.
It may be difficult to adapt HR gas for 5th and 6th.
Quark Gluon gas model
large 𝛽 or 𝜇/𝑇 → approach to Q.G. gas
Circumstantial evidence of phase transition
We measured singular behavior for 𝛽=1.6 .
We saw HR gas / QG gas transition.

17. back up

18. Grand Canonical ensemble
Quark number cumulant
𝑁 𝑘 𝐶 ≔ 𝜕𝐺 𝜃 𝜕 𝜃 𝑘 𝜃=0 𝐺 𝜃 ≔ log ∫𝑑𝑛 𝑒 𝜃𝑛 𝑃(𝑛)
Distribution
𝑃(𝑛)
characterize
𝑁 1 𝐶 =𝜇
𝑁 2 𝐶 = 𝜎 2
𝑁 𝑘 𝐶 =0 , 𝑓𝑜𝑟 𝑘>2
𝑃 𝑛 = 1 2𝜋 𝜎 𝑒 − 𝑛−𝜇 𝜎 2
2 𝜎 2
ex. Gaussian 𝜇
non-Gaussian
Grand Canonical ensemble
𝑃 𝑛 ≔ 𝑒 𝜇 𝑇 𝑛 𝑍 𝑐𝑎𝑛. (𝑛) 𝑍 𝐺.𝐶. (𝜇)
system
heat bath
𝑁 𝑘 𝐶 = 𝜕 𝑘 𝜕 𝜇/𝑇 𝑘 log 𝑍 𝐺.𝐶.
energy
particle

19. around phase transition line finite density lattice QCD
Quark number cumulant
𝑁 𝑘 𝐶 = 𝜕 𝑘 𝜕 𝜇/𝑇 𝑘 log 𝑍 𝐺.𝐶. 𝑁 2 𝐶 = 𝛿 𝑁 2 𝑁 3 𝐶 = 𝛿 𝑁 3 𝑁 4 𝐶 = 𝛿 𝑁 4 −3 𝛿 𝑁 2 2
2nd
4th
3rd
𝑁 3 𝐶 𝑁 2 𝐶
around phase transition line
𝑁 4 𝐶 𝑁 2 𝐶
𝑁 𝑘
𝑁 𝑘 𝐶
expectation
value
𝑁 3 𝐶 𝑁 1 𝐶
sign problem
finite density lattice QCD
STAR Collaboration
Phys. Rev. Lett. 112 (2014)

20. Volume dependence 8 3 ×4 𝑣𝑠. 12 3 ×4 𝑁 2 𝐶 / 𝑁 1 𝐶 𝜇 𝐵 /𝑇 𝛽=0.9
8 3 ×4 𝑣𝑠 ×4
𝑁 2 𝐶 / 𝑁 1 𝐶 𝜇 𝐵 /𝑇
𝛽=0.9
𝑁 2 𝐶 / 𝑁 1 𝐶 𝜇 𝐵 /𝑇
𝛽=1.5

21. How to determine 𝑁 𝑐𝑢𝑡 ? d’Alembert 𝑛 𝑎 𝑛 lim 𝑛→∞ 𝑎 𝑛+1 𝑎 𝑛 <1
𝑍 𝐺.𝐶. (𝜇)= 𝑛 𝑍 𝑐𝑎𝑛. 𝑛 𝑒 𝜇𝑛/𝑇
𝑍 𝑐𝑎𝑛. 𝑛+1 𝑍 𝑐𝑎𝑛. 𝑛 𝑒 𝜇 𝑇 <1