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A potential including Heaviside function in 1+1 dimensional Landau hydrodynamics

A potential including Heaviside function in 1+1 dimensional Landau hydrodynamics. Takuya Mizoguchi* *Toba National College of Maritime Technology, Japan. Contents Introduction Landau model and three solutions Data analyses of hadrons (RHIC p , K) Explanation of net-proton Summary.

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A potential including Heaviside function in 1+1 dimensional Landau hydrodynamics

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  1. A potential including Heaviside function in 1+1 dimensional Landau hydrodynamics Takuya Mizoguchi* *Toba National College of Maritime Technology, Japan • Contents • Introduction • Landau model and three solutions • Data analyses of hadrons (RHIC p, K) • Explanation of net-proton • Summary H. Miyazawa**, M. Biyajima**, M. Ide** **Shinshu University, Japan

  2. Introduction • 1+1 dimensional hydrodynamics proposed by Landau • Landau’s solution (1953) • A boost non-invariant solution by Srivastava et al. (1993) • Our solution including Heviside function (2008, 9) • Three solutions cannot explain the net-proton at RHIC • New approach (2009) • Preliminary results on net-proton

  3. Perfect fluid

  4. (1+1) dimension

  5. Landau's solutionL. D. Landau, Izv. Akad. Nauk Ser. Fiz. 17, 51 (1953)

  6. Solution by Srivastava et al.D. K. Srivastava et al., Annals Phys. 228, 104 (1993)

  7. w w= ycs w= -ycs Bessel function ycs

  8. Our analytical solution with the Heaviside functionMizoguchi, Miyazawa, Biyajima, Eur. Phys. J. A 40, 99 (2009) (Cf. D. G. Duffy, “Green‘s functions with applications”, Ivar Stakgold, "Boundary Value Problems of Mathematical Physics“)

  9. e= 3 5 20

  10. Competition!! • T. Mizoguchi and M. Biyajima, Genshikaku Kenkyu (in Japanese), Vol. 52 Suppl. 3 (Feb.2008) 61. • Our talk in Annual Meeting for Physical Society of Japan (Mar. 2008, Kinki Univ. (Osaka))

  11. The same expression as our solutionBeuf, Peschanski, Saridakis, Phys.Rev.C78 (2008) 064909

  12. Comparison of three analytical solutions

  13. Potencial c(y, w) and Contour map

  14. Trajectory of Bjorken (x/t=const.) Bjorken Srivastava Ours (e = 2.5) t = x t = x t = 1 t = 1 Bjorken: boost invariant solution. (Cf. J.D. Bjorken, Phys. Rev. D27 (1983) 140.)

  15. Cooling law at y = 0 t0: proper time at (y, w) = (0, 0). Bjorken: T3t/T03t0= const. (cs2 = 1/3)

  16. Rapidity distribution of hadrons

  17. Data analyses • Parameter fitting by least-squares (CERN MINUIT is used) • Values to input : Tf, mB • Free parameters: wf, cs2(<=1/3), c, e Temperature and baryon chemical Potential (cf. Andronic et al., Nucl. Phys. A772 (2006) 167)

  18. Analysesof charged p and K dataMizoguchi, Miyazawa, Biyajima, Eur. Phys. J. A 40, 99 (2009) RHIC p- (200 GeV) RHIC p+ (200 GeV) Au+Au Landau Srivastava Ours RHIC K- (200 GeV) RHIC K+ (200 GeV)

  19. Comparison of parameter wf wｆ （Srivastava) < wｆ （Ours) < wｆ （Landau) We cannot determine which temperature is right.

  20. Attention! We cannot determine the value uniquely except for Landau’s solution. • Solution of Slivastava et al. depends on y0 • Our solution doesn’t depends on y0

  21. Contribution of the derivative term of H(Q) RHIC p - (200 GeV) RHIC p+ (200 GeV) H-term H’-term H’’-term e=3.4 e=1.7 If e is large, contributions of the derivative terms are small.

  22. Preliminary worksAnalysesofnet-proton data Parameter fitting by means of our solution • Our solution cannot explain the characteristic peak of RHIC net-proton data. • Thus we consider another approach. RHIC(200 GeV) RHIC(62 GeV) Derivative terms ofHe

  23. Remember the famous book!!Morse and Feshbach, ``Methods of Theoretical Physics'', Chapter 7 (1953) See also, Masoliver, Weiss, ``Finite-velocity diffusion '', Eur. J. Phys. 17 (1996)190

  24. Analysesofnet-proton data at AGS and SPS Gaussian I0-term I1/p-term AGS(5 GeV) SPS(17 GeV)

  25. Analysesofnet-proton data at RHIC RHIC(62 GeV) Gaussian I0-term I1/p-term RHIC(200 GeV)

  26. Parameters and initial Temperature upper limit Existence of missing proton !! Estimation of initial Temperature

  27. G. Wolschin, Phys. Lett. B 569 (2003) 67. For fitting our solution to with this form, it is necessary to impose another conditions.

  28. Summary • We consider the (1+1)-dimensional hydrodynamics, and derive the solution including the Heaviside function. • Our solution explains the data p and K distributions fairly well. Preliminary work • Since our solution doesn't explain the characteristic peak at large y of net-proton distribution, we have considered a new analytic solution. • The new solution explains the data of net-proton fairly well, except for data at 200 GeV.

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