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Inflection point as a characteristic of the QCD critical point

Inflection point as a characteristic of the QCD critical point. Mingmei Xu ( 许明梅 ) Yuanfang Wu ( 吴元芳 ) Central China Normal University ( 华中师范大学 ). arXiv:1307.2285[nucl-th]. 10 th workshop on QCD phase transition and relativistic heavy-ion physics, Chengdu, 9 Aug., 2013. 1) Introduction.

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Inflection point as a characteristic of the QCD critical point

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  1. Inflection point as a characteristic of the QCD critical point Mingmei Xu (许明梅) Yuanfang Wu (吴元芳) Central China Normal University (华中师范大学) arXiv:1307.2285[nucl-th] 10th workshop on QCD phase transition and relativistic heavy-ion physics, Chengdu, 9 Aug., 2013

  2. 1) Introduction Search for the critical point (CP)

  3. High cumulants as promising observables • Sensitive to the correlation length • Related to the susceptibility M. A. Stephanov, PRL 102, 032301 (2009); S. Ejiri, F. Karsch and K. Redlich, PLB 633, 275 (2006); M. Cheng et al, PRD 79, 074505 (2009); ( conserved charges) F. Karsch, K. Redlich, PLB 695, 136 (2011);

  4. Mostly the signals are characterized by the sign or the change of the sign of various cumulants. • The third order cumulants change signs passing across the phase boundary. • The fourth order cumulants (kurtosis) should be negative when the CP is approached on the crossover side and will change signs in the critical region. • The sixth order cumulants and the eighth order cumulants oscillate in the crossover region. M. Asakawa, S. Ejiri and M. Kitazawa, PRL 103, 262301 (2009); M. A. Stephanov, PRL 107, 052301 (2011); R. V. Gavai and S. Gupta, PLB 696, 459 (2011); B. Friman et al, EPJC 71, 1694 (2011); V. Skokov et al, arXiv:hep-ph/1108.1300; M. Cheng et al, PRD 79, 074505 (2009); C. Schmidt, Prog. Theor. Phys. Suppl. 186, 563 (2010);

  5. The cumulants measure the derivatives of the EoS M. Asakawa, S. Ejiri and M. Kitazawa, PRL 103, 262301 (2009); V. Koch, EMMI rapid reaction task force meeting 2013; nth cumulant (n-1)th derivative of EoS What feature of the EoS results in the sign change behavior of its high order derivatives?

  6. The counting law: the nth order derivative changes the sign n-1 times in the neighborhood of the inflection point. 2) Inflection point and the oscillations of the high order derivatives What is an inflection point? A point on a curve at which the sign of the concavity changes (the second derivative changes sign).

  7. Inflection point of the van der Waals equation of fluid 3) Inflection point of EoS and the critical point Critical point is exactly the inflection point in the critical isotherm.

  8. Inflection point of the magnetization in the spin models In the 3d Ising (O(1)), O(2) and O(4) model without external magnetic field, it is found that all higher cumulant ratios change dramatically the sign near Tc. Xue Pan, Lizhu Chen, Xiaosong Chen and Yuanfang Wu, arXiv:1306.0335; An inflection point is obviously seen in M.

  9. In an universality analysis at \mu_B=0 and PQM model calculations at \mu_B≥0, the high cumulants, , , , show oscillating behavior close to the pseudo-critical temperature (the temperature of crossover). • Inflection point of the baryon number density of QCD matter The critical behavior can also be observed during a crossover at \mu_B=0 due to the remnants of O(4) criticality on the crossover line. B. Friman et al, EPJC 71, 1694 (2011); V. Skokov et al, arXiv:hep-ph/1108.1300;

  10. Inflection points indeed exist in the curves of the quark number density n_q(\mu_q) for T=Tc and T>Tc, as shown in an earlier paper. B-J Schaefer et al, PRD 75, 085015 (2007);

  11. To summarize, an inflection point of EoS is a characteristic of the CP and is closely related to the oscillating behavior of high cumulants. Physically, the second order phase transition passing through the CP is always accompanied by the numerical change of the order parameter. Why an inflection point? order parameter

  12. An appearance of the inflection point in the EoS of crossover does not contradict the statement above. The inflection point in the EoS of crossover may be due to the close location with the CP. The peak structure of the usual susceptibility survived along the crossover line, which induces the phrase “critical region” as mentioned in references.

  13. Application: according to the O(4) universal scaling analysis. B. Friman et al, EPJC 71, 1694 (2011); The universal analysis for mu_q/T=0:

  14. The PQM model at mu_q/T=0: The lattice at mu_q/T=0: The counting law about the times of sign change works well.

  15. 4) Conclusion • The critical behavior of high cumulants, in particular, the sign change, is closely related to the appearance of inflection point. • The number of times of sign change of high cumulants near the critical temperature is determined by the order of cumulants, i.e., the order of derivative of EoS at the inflection point. • The second, forth, sixth and eighth cumulants of baryon number at zero chemical potential from current lattice QCD, an O(4) universal scaling analysis as well as PQM model calculations, show consistent sign changes as we predicted from the order of the cumulants. • We demonstrate that the character of inflection point of EoS is as visible as the sign change of high cumulants. • We propose that the EoS, or the mean of baryon number density, should be measured together with its high cumulants.

  16. Thanks!

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