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

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

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 1 3 𝑁 3 𝐶 𝑁 2 𝐶 =𝐭𝐚𝐧𝐡 𝟑 𝛍 𝐓 (dotted line) 1 9 𝑁 4 𝐶 𝑁 2 𝐶 =𝟏 1 9 𝑁 3 𝐶 𝑁 1 𝐶 =𝟏 deviation from H.R. Gas STAR Collaboration Phys. Rev. Lett. 112 (2014) 032302 something interesting?

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！

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 0 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

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 =222.5 11 𝑀𝑒𝑉 small 𝛽 → low temperature large 𝛽 → high temperature

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 = 1+ 3 𝜋 2 𝜇 𝐵 3𝑇 2 𝜇 𝐵 3𝑇 + 1 𝜋 2 𝜇 𝐵 3𝑇 3 Q.G. gas (dotted line)

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 ,2.04 𝛽≾1.6 𝜇 𝐵 /𝑇 =3.00 𝛽≾1.5 , 𝜇 𝐵 /𝑇=3.60 𝛽≾1.4 consistent with H.R. gas 𝛽∼2.1 approach to Q.G. gas

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, 074511 This work consistent behavior application range of HR gas becomes small as 𝜇/𝑇 increases measure the H.R. gas / Q.G. gas transition

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

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

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

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 ,2.04 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

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

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)

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.

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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 2 2 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

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) 032302

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. 12 3 ×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

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