JP0050013
Japan Atomic Energy Research Institute (T319-1195
This report is issued irregularly. Inquiries about availability of the reports should be addressed to Research Information Division, Department of Intellectual Resources, Japan Atomic Energy Research Institute, Tokai-mura, Naka-gun, Ibaraki-ken 319-1195, Japan.
© Japan Atomic Energy Research Institute, 1999 JAERI-Conf 99-008
10 &
1999^3^ 11
(1999^7^5
, 1999^3^ n B, 12 s
319-1195 2-4 JAERI-Conf 99-008
Proceedings of the First Symposium on Science of Hadrons under Extreme Conditions
March 11 - 12, 1999, JAERI, Tokai, Japan
(Eds.) Satoshi CHIBA and Toshiki MARUYAMA
Advanced Science Research Center (Tokai Site) Japan Atomic Energy Research Institute Tokai-mura, Naka-gun, Ibaraki-ken (Received July 5, 1999)
The first symposium on Science of Hadrons under Extreme Conditions, organized by the Research Group for Hadron Science, Advanced Science Research Center, was held at Tokai Research Establishment of JAERI on March 11 and 12, 1999. The symposium was devoted for discussions and presentations of research results in wide variety of fields such as observation of X-ray pulsars, theoretical studies of nuclear matter, nuclear struc- ture, low- and high-energy nuclear reactions and QCD. Thirty seven papers on these topics presented at the symposium aroused lively discussions among approximately 50 participants.
Keywords: Proceedings, Hadrons under Extreme Conditions, Neutron Stars, X-ray Pulsars, Nuclear Matter, Nuclear Structure, Nuclear Reactions, QCD
Organizers:S. Chiba, T. Maruyama, T. Kido, Y. Nara (Research Group for Hadron Science, Advanced Science Research Center, JAERI), H. Horiuchi, T. Hatsuda (Kyoto University), A. Ohnishi (Hokkaido University), K. Oyamatsu (Nagoya University/ Aichi-Shukutoku University) JAERI-Conf 99-008
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Baryonic P2 Superfluidity under Charged-Pion Condensation with A Isobar 8 Ryozo Tamagaki (Kyoto Univ.) 4. Landau-Migdal Parameters and Pion Condensation 14 Toshitaka Tatsumi (Kyoto Univ.) 5. Effects of Pion Condensation on Hadronic Matter Including Strangeness 19 Takumi Muto (Chiba Institute of Technology) 6. Kaon Condensation and Protoneutron Stars 23 Masatomi Yasuhira (Kyoto Univ.) 7. A — E Mixing through TT Condensation 28 Akinori Isshiki (Hokkaido Univ.) 8. Alpha Clustering in Dilute Nucleonic Sea 33 Akihiro Tohsaki (Shinshu Univ.) 9. Relativistic Approach to Superfluidity in Nuclear Matter — Constructing Effective Pair Wave Function from Relativistic Mean Field Theory with a Cutoff — 39 Masayuki Matsuzaki (Fukuoka Univ. of Education) 10. Effects of Meson Mass Decrease on Superfluidity in Nuclear Matter 45 Tomonori Tanigawa (Kyushu Univ.) 11. Simulation Study for the Nuclear Matter below the Saturation Density 49 Toshihiko Kido (JAERI) 12. Dynamical Simulation of Expanding Nuclear Matter 58 Shinpei Chikazumi (Tsukuba Univ./JAERI) 13. Giant Quadrupole Resonances in Time Dependent Density Matrix Theory 62 Mitsuru Tohyama (Kyorin Univ.) 14. Nuclear Shape Evolution Starting from Superdeformed State - Role of Two-Body Collision and Rotation - 67 Yu-xin Liu (Ibaraki Univ.) 15. Can We Determine the EOS of Asymmetric Nuclear Matter Using Unstable Nuclei? 71 Kazuhiro Oyamatsu (Nagoya Univ./Aichi-Shukutoku Univ.) 16. Effects of the Equation of State of Asymmetric Nuclear Matter in Nuclear Collisions 78 Akira Ono (Tohoku Univ.) 17. Fragmentation Mechanism Reflecting the Cluster Structure of 19B 84 Hiroki Takemoto (Kyoto Univ. / JAERI) 18. Nuclear Phase Transition Studied with AMD-MF 90 Yasuo Sugawa (Kyoto Univ.) 19. The Analysis of Proton Induced Reactions on Light Nuclei 96 Yuichi Hirata (Hokkaido Univ.) V JAERI-Conf 99-008 20. Calculation of Spin Observables with Semiclassical Distorted Wave (SCDW) Model 101 Kazusuke Ogata (Kyushu Univ.) 21. Superconductivity of Quark Matter and the Phase Diagram 105 Masaharu Iwasaki (Kochi Univ.) 22. Color Superconductivity in Quark Matter 109 Tetsuo Hatsuda (Kyoto Univ.) 23. The Lattice Landau Gauge QCD Simulation and the Confinement Mechanism 117 Sadataka Furui (Teikyo Univ.) 24. Analysis of Spectral Functions in QCD with Maximum Entropy Method 122 Yasuhiro Nakahara (Nagoya Univ.) 25. Molecular Dynamics Simulation of Quark Matter 127 Toshiki Maruyama (JAERI) 26. Mesons Above The Deconfining Transition 133 Takashi Umeda (Hiroshima Univ.) 27. Classification of Diquark Condensates with an Effective Potential 138 Eiji Nakano (Tokyo Metropolitan Univ.) 28. Time Evolution of Chiral Phase Transition at Finite Temperature and Density in the Linear Sigma Model 143 Ken-ichi Sato (Tohoku Univ.) 29. Phase-Shift Analyses of pp Scattering at High Energies and Strong Energy-Dependence of Spin-Orbit Interaction 148 Junichi Nagata (Hiroshima Univ.) 30. Three Dimensional Relativistic Hydrodynamical Model for QGP Gas 154 Chiho Nonaka (Hiroshima Univ.) 31. Elliptic Flow Based on a Relativistic Hydrodynamic Model 159 Tetsufumi Hirano (Waseda Univ.) 32. Baryon Stopping and Strangeness Baryon Production in a Parton Cascade Model 163 Yasushi Nara (JAERI) 33. Analysis of Subthreshold Anti-proton Production at KEK-PS and Anti-proton potential 171 Tomoyuki Maruyama (Riken/Nippon Univ.) 34. Nuclear Equation of State in Heavy-ion Collisions 178 P.K. Sahu (Hokkaido Univ.) 35. Thermal Properties of Nuclear Matter under the Periodic Boundary Condition 184 Naohiko Otuka (Hokkaido Univ.) 36. Themodynamical Properties of Hot and Dense Hadronic Gas using URASiMA 189 Nobuo Sasaki (Hiroshima Univ.) 37. Equation of State of Dense Nuclear Matter with a Variational Method 194 Masatoshi Takano (Waseda Univ.) 38. Relativistic EOS Table for Supernova Explosion and r-process 198 Kohsuke Sumiyoshi (Riken) vi JAERI-Conf 99-008 12 (D AF t^ 1 — JAERI-Conf 99-008 JP0050014 2. X-ray Pulsar Rush in 1998 3-E fttih it* E?i, Brt « 1 Abstract We present recent remarkable topics about discoveries of X-ray pulsars. 1. Pulsations from two Soft Gamma-ray Repeaters These pulsars have enormously strong magnetic field (B~1015 G), thus these are called as "magnetar", new type of X-ray pulsars. 2. New Crab-like pulsars These discoveries lead to suggesting universality of Crab-like pulsars. 3. An X-ray bursting millisecond pulsar This is strong evidence for the recycle theory of generating radio millisecond pulsars. 4. X-ray pulsar rush in the SMC This indicates the younger star formation history in the SMC. Crab-like pulsar (rotation-powered pulsar): - 3 X —fill: 3.1 Soft Gamma-ray Repeater —: magnetar Soft Gamma-ray Repeater (SGR) Gamma-ray Burst (GRB) 3 1. GRB 2. S*f 3. 4 oco 5 *, 3 l; 1998 ¥> 2 oW SGR K X SGR 1806-20 (P^ SGR 1900+14 (P=5.16sec)I2][3](4l 3 x G l^ G - 2 - JAERI-Conf 99-008 U Crab-like pulsar (B-1012"13 G) tltm tbMz.m<»hV>T'1bZ> i £ "magnetar"!5' tlfp^o Crab-like pulsar T'li -Ero< >> Lx t ft 0 . dHte^A-df-ro—g|5£ i^H -Erot > L \ magnetar r ix h fi Anoumalous X- 6 ray Pulsar (AXP)i J P-Pdiagram SGR (0 1 ) AXP m31cSGRt AXP t, SGR *« decay AXP t 0 10-10 0.01 0.1 1 10 100 Period [sec] m 1: P-Pdiagram 10,3' 8 37 |io r -Erot == Lx r / •> / : 3B • I io SGR / • * : 3 * / / • Crab-like pulsar] 10,33 10,3Ji3 ,10*n34* 1n1036 1Q36 1Q37 1Q38 1()3 100 1000 10* 10° 10 Rotatinal Energy Loss [ergs/s] Age [yr] I 2: IB (-Ero() t X (LA-) H 3: SGR t AXP - 3 - JAERI-Conf 99-008 3.2 Crab-like pulsar Cfgjl, Crab-like pulsar 14, WjgffS4f IS (II : Type II SN) (' £ 5 k%7L htl t Crab-like pulsar (2$: 1000 ¥(i»^igc(t5 t #^t»tt50 "fSt, Type II SN f—^ -f-Sftk, *£, 2& 10 i@W Crab-like pulsar ##£1-5 ir7-fi£ix-5o , Crab-like pulsar li^W^rtT'ti^T^3 «SJ (Crab, Vela, PSR B1059-58) ^ of;(J ib(C^:-rif y -y%K PSR B0540-69 Type II SN — Crab-like pulsar tl/ 1997-98 ¥t£a»tf"C$ffc& Crab-like pulsar T'lo) ^t±0. Crab-like pulsar WlSli- ^t.tz$L\Z%ft t lifx.4 Crab-like pulsar ^ Type II SN X*$if&&}l6 t V^5Ift^4, I SN T'fc5 SN 1987A T* 7 • Gll.2-0.3l ]: mseco A. D. 386 c J; 19 • RCW103I8!: mseco kick velocity Sr£ttTfllf4*<*;h/fc(D£:5 5 • N157BI9!: J1W 16 msec, Vg (LMC) SN1987A Crab- like pusar ro^Tilftjgo • Puppis Al10!: j^ffl 75 msec 3.3 X-ray bursting millisecond pulsar («£) 5 ]) S-WHT— (0 1). , (1) il (1-10 msec), (2) it«i (109~10 G) 1. (B~1012 G). 9 2. raix.5 (B~10 G)o 3. 4. (LMXB)O 5. 1998^, J1808-3658 = XTE J1808-359 LMXB - 4 - JAERI-Conf 99-008 3.4 (Small Magellanic Cloud: SMC) l± LMC (0.7-320 sec) 15 Pulsar lc [ IO , SMC 7 10 ¥tW^tl>TV>5o -*, LMXB fi 3.3 108 ~ 109 ¥tSit SMC s SMC LMC ^ Our Galaxy £.<9 j fffi mm (s) mm (s) SMC X-1* 1976 (Rocket) 0.716 0.70876° [16][17] 0.7101436 0.706481' RX J0059.2-7138* 1994 ROSAT 2.7632 2.7632 [17][18] 2E 0050.1-7247* 1995 ROSAT 8.88 — [17][19] AX J0051-722 1997 RXTE, ASCA 91.1 91.1 [20][21] 1WGA J0053.8-7226* ASCA 46.63 46.63 [20][22] XTE J0054-720 1998 RXTE 169.3 — [21] [23] XTE J0055-724* RXTE, BeppoSAX 58.97 59.00 [24][25] AX J0049-729* RXTE, ASCA 74.68 74.68 [20][26] AX J0051-733*' ASCA 323.2 323.2 [23] [27] [28] [29] AX J0058-720* ASCA 280.4 280.4 [27] 1SAX J0103.2-7209* BeppoSAX 345.2 348.9 [25][30][31][32] AX J0105-722*** ASCA 3.34300 3.34300 [33] AX J0049-732" ASCA 9.1321 9.1321 [34] XTE J0111.2-7317* RXTE, CGRO 31 30.9497 [35][36][37] RX J0052.1-7319* 1999 ROSAT 15.3 — [37] [38] RX XTE J0054-720 lliB^ft, RX J0059.2-7138 lii o [a-c]: SMC X-1 14 3 1993 ^ [a], 1995 *p [b], 1998 ^ [c] - 5 - JAERI-Conf 99-008 LMXB HMXB • Our Galax M 4: SMC t Our Galaxy GOO 1000 1600 2000 2800 3000 normalized number • magnetar t V> 5 . • Crab-like pulsar SMC X i^^ i 102 101 death? ^r--+- AXP? 10° ' magnetar(SGR) 4 5 / B=10i -10i G \ io-i radio pulsar \ • ''' ^^ \ millisecond 2 io- 1 pulsar i^,^**^ Crab-like pulsar B=108-109G 3 10- - B=1O12-1Q13G *-—• 1 • 10° 103 106 109 1012 [year] 5: - 6 - JAERI-Conf 99-008 [1] Kouveliotou, C. et al. Nature 393, 235 (1998). [2] Hurley, K. et al. Astrophys. J. 510, Llll (1999). [3] Kouveliotou, C. et al. Astrophys. J. 510, L115 (1999). [4] Murakami, T., Kubo, S., Shibazaki, N., Takeshima, T., Yoshida, A. k Kawai, N. Astrophys. J. 510, L119 (1999). [5] Thompson, C. k Duncan, R. C. Astrophys. J. 473, 322 (1996). [6] van Paradijs, J., Taam, R. E. k van den Heuvel, E. P. J. Astron. Astrophys. 299, L41 (1995). [7] Torii, K., Tsunemi, H., Dotani, T. k Mitsuda, K. Astrophys. J. 489, L145 (1997). [8] Torii, K. et al. Astrophys. J. 494, L207 (1998). [9] Marshall, F. E., Gotthelf, E. V., Zhang, W., Middleditch. J. k Wang, Q. D. Astrophys. J. 499, L179 (1998). [10] Pavlov, G. G., Zavlin, V. E. k Triimper, J. Astrophys. J. 511, L45 (1999). [12] in't Zand, J. J. M., Heise, J., Muller, J. M., Bazzano, A., Cocchi, M., Natalucci, L. k Ubertini, P. Astron. Astrophys. 331, L25 (1998). [13] Wijnands, R. k van der Klis, M. Nature 394, 344 (1998). [14] Yokogawa, J. 1999, Master thesis of Kyoto University [15] Nagase, F. Publ. Astron. Soc, Japan 41, 1 (1989). [16] Lucke, R. et al. Astrophys. J. 206, L25 (1976). [17] Bildsten, L. et al. Astrophys. J. Suppl. 113, 367 (1997). [18] Hughes, J. Astrophys. J. 427, L25 (1994). [19] Israel, G. L. et al. Astrophys. J. 484, L141 (1997). [20] Corbet, R., Marshall, F. E., Ozaki, M. k Ueda, Y. IAU Circ. No. 6803 (1997). [21] Lochner, L. C. IAU Circ. No. 6858 (1998). [22] Buckley, D. A. et al. IAU Circ. No. 6789 (1997). [23] Cowley, A. P. et al. Publ. Astron. Soc. Pacific 109, 21 (1997) . [24] Marshall, F. E. k Lochner, J. C; Santangelo, A. et al. IAU Circ. No. 6818 (1998). [25] Israel, G. L. et al. I A U Circ. No. 6999 (1998). [26] Yokogawa, J. k Koyama, K. IAU Circ. No. 6835 (1998). [27] Yokogawa, J. k Koyama, K. IAU Circ. No. 6853 (1998). [28] Cook, K. IAU Circ. No. 6860 (1998). [29] Schmidtke, P. C. k Cowley, A. P. IAU Circ. No. 6880 (1998). [30] Yokogawa, J. k Koyama, K. IAU Circ. No. 7009 (1998). [31] Hughes, J. P. k Smith. R. C. Astron. J. 107, 1363 (1994). [32] Ye, T. et al. Monthly Notices Roy. Astron. Soc. 275, 1218 (1995). [33] Yokogawa, J. k Koyama, K. IAU Circ. No. 7028 (1998). [34] Imanishi, K., Yokogawa, J. k Koyama, K. IAU Circ. No. 7040 (1998). [35] Chakrabarty, D., Levine, A. M., Clark, G. W. k Takeshima, T.; Wilson, C. A. k Finger, M. H. IAU Circ. No. 7048 (1998). [36] Chakrabarty, D. et al. IAU Circ. No. 7062 (1998). [37] Israel, G. L. et al. IAU Cue. No. 7101 (1999). [38] Lamb, R. C, Prince, T. A., Macomb, D. J. k Finger, M. H. IAU Circ. No. 7081 (1999). - 7 - JAERI-Conf 99-008 JP0050015 3. We study the baryonic 3P2 superfluidity under charged-pion condensation with isobar (A) degrees of freedom. After a remark on motivations of the present study, the outline of theoretical framework is briefly described, typical results of the superfluid critical temperature are shown, and the possibility of coexistence of the superfluid with charged-pion condensation is discussed. Cosmic Laboratory" t 1 3 0.57A( S0),kBTc~ 0.60A( P2) §«ref. 6) ( - 8 - JAERI-Conf 99-008 X\ - N 1 &-%&&%& t , 7T- N X 0 TT0) > (3 - 5)/9oo 7r IM±l-ft, fk^*"^ y v (r)) ~ AQ*m(k0-r) &tZia (Alternating-Layer-Spin; A L S«3tt X.2) OPEP (^X >- V /HICO 1 J^' f co-7 ^/v > (3 - jtli, AL S^ [Mitt] v), r modified #'" cooling" U ^-^li, - 9 - JAERI-Conf 99-008 y/<— (A) 3 l §2. «f*/USW6ffil=fclt-5 P2/< J^->ffl3ftlfr—aiftCDout line- 1 ) m/^tt^V (occupied) (i) f$- (2) (real) Pauli spi = (q±kc,a) ii^< y vp\Pf3-) (3) io u uq ~ ~ -iav. 2) 4l/\* J = 2,mj = ±2 £o< 5(?5T% aligned pairing scheme t - 10- JAERI-Conf 99-008 ^CO scheme "C^feS " - (TV) A) u,d) quark CO spin, isospin , quark spin, isos (AT + 4) # i> 0} X- h Z A-pair/ i. N _ K r lr kc) — NN K'i c BB l 'M r^)radial integral &?&< V 3) pair / 3 V^- (g , (/; irc) F2CO aligned pairing scheme 2 H(9)| , Yn(q) (6) L T# 9 , Tc in 10 K ~ 7.1 x in MeV]o §3 fam : OPECJ-A: steep odd-state repulsive core CO t>cO&t/OPEG-B: gentle odd-state repulsive core CO £>(/)„ CPC- model parameters jyf, ;;2, AvfCoV ^T(i, M^;#X./i ::.O»^ (Landau-Migdal pa- 7) rameter 57J5H&3) ^^-5 •• (i) strong CPC:(flf'=0.5) > (ii) weak CPC(«/'=0.6)D fc*N (iii) A.-C,v - n - JAERI-Conf 99-008 Xm* = MB/MN = 0.8 -+ O.G £M-#J t ^ST A 2) ' p2m HI 1 (a) cO^$|}2^j//> OPEG-B, CPC model fr (i) ~ 0.7 &n%\CTc{if) > Tin ^ *) "5 , gentle repulsive steep repulsive core 1CG-A) (Ci:t It, «t CPG < strong CPC, weak CPC cote^ yiM^{z& t IU 1 1 1 1 . (a ) 1 <"b> ' I m' _ ^ -. ' " . • •^^. T, —{*- — . 0.8 • 0.8 j \^j — 10' id* : 0.7 r A-mixj \"\ /NN-CPC ^^~ -* (K) (K) ^—-^. - 0.6 J- • O.G ^: 8 ,o . • \ with A without A without A no CPC no CPC 1 _J i L_ 1 1 i P'P. 5 4 p/g 5 mi. 'Pa OPEG-B CPC(fuD) i§£o ,^H : CPC W^ (a) strong CPC model <7)#£-( < , * (b) *;.., t»25:iffi^l/j:CPC model CPC; 2 tfC CPC -12- JAERI-Conf 99-008 CPC model (iii) CPC t iz (1) (2) rco-o^ "9, (3) rr(:«^ So fat, j^ cool- ing > References 1) T. Kunihiro, T. Muto, T. Takatsuka, R. Tamagaki and T. Tatsumi, Prog. Theor. Pliys. Supplement No. 112(1993). :: 2) mum. xfflR&nftRifi ^•\t f-s.(Dm±tm{t-mwitmm(Dmmmi, RESCEU W%£ (1998.12.24-26) #^o 3) S. Tsunita, Phys. Reports 292(1998), 1. 4) T. Takatsuka and R. Tamagaki, Prog. Theor. Phys.97(1997), 263. 5) T. Takatsuka and R. Tamagaki, Prog. Theor. Phys.98(1997), 393. 6) T. Takatsuka and R. Tamagaki, Prog. Theor. Phys.Vol. 101, No.5 (1999). 7) T. Tatsumi, Prog. Theor. Phys.68(1982), 1231. -13- JAERI-Conf 99-008 JP0050016 4. Landau-Migdal^ y * - $ t n Wfe Landau-Migdal parameters and Pion condensation (Toshitaka Tatsumi) (De])artment, of Physics, Kyoto University) The possibility of pion condensation, one of the long-standing issues in nuclear physics, is reexamined in the light of the recent experimental data on the giant Gamow-Teller resonance. The experimental result tells that the coupling of nucleon particle-hole states with A isobar-hole states in the spin-isospin channel should be weaker than that priviously believed. It, in turn, implies that nuclear matter has the making of pion condensation at low densities. The possibility and implications of pion condensation in the heavy-ion collisions and neutron stars should be seriously reconsidered. I. INTRODUCTION - 7 -< y * tr^ S Gamow-Teller O(uA = mA - mN) t &oTl^50 Gamow-Teller (A T4V'^-)- %RM®)fe (1.1) Landau- Migdal o ^e-X\ £ (Universality) ISM bWftftZ&%&&., *E-mail address: tatsumi«*ruby.s(;phyH.kyoto-ti.ac.jp - 14- JAERI-Conf 99-008 o.c~0.8 [2,3] t [4] Gamow-Teller II. WHAT IS PION CONDENSATION? tll^ Wltiirflillf ©If (A Tjy<—)- Bose-Einstein (2.1) §1 t°>-- r-ry -Ktc (2.2) (r/ AA (2.3) with ( 11 r( ')rr(0) (2.4) = 1 + + + (UNNUAA - VNA and Lindhanl r r T 4 A ) -15- JAERI-Conf 99-008 III. GAMOW-TELLER GIANT RESONANCE Gamow-Teller *R|^^f^H)fe^^^^r—ft D(u < mK, k = 0) - 0 id ( (3-1) m HoO Landau- (Quenching factor) . r -r y (3-2) U N Gamow-Teller (0) it /7U ^ -3(iV - Z) UNA A (3.3) / - -3(iV - Z) (3.4) 7T JO enching factor) 1- (3.5) fc Q Bloc Landau-Migdal Gamow-Teller *« 1 & ffi© RCNP mNb(p, w), ^ i 6 £ , d^f LT Q = 0.9 ± 0.05 && * [7,8], t dt, Q = 0.5-0.7, (3.5) [9] - 0.585. (3.6) Q = 0.9 0.12 + -16- JAERI-Conf 99-008 IV. CRITICAL DENSITY FOR PION CONDENSATION l D; (kr,u = 0;/;,.) = mi + kf. + U(kr,u = 0;/>r) = 0, (4.1) l OD; /dk\k=kr = o (4.2) 1.6/Jo(m* = 0.7m,//AA = Symmetric nuclear matter Symmetric nuclear matter (Non-Universality; 9 Q=0.7 --- . Q=0.8 3.5 0.7 -- / 8 Q=0.9 - 0.6 / / Q=1 --- 3 0.5 •--,-'- 7 universality 0.4 /--- 6 2.5 • o 5 Q. Q. 2 ,..-••' ...••• - 4 3 1.5 - 2 1 -'•"•••;::--- 0 5 i 1 1 • i ° ) 0.2 0.4 0.6 0.8 1 1 2 ) 0.2 0.4 0.6 0.8 1 1 2 g' Q'&A FIG. i. I&IM££O Q n FIG. 2. = 0.9 Q = 0.9 ±0.05 m* = 0. ex), , Uff, 00 -Cfe t), ^ (2.4) 0 oo)) 1 + f/(f/]}/(f/]} ++ C/A ) (Universality) (4.3) / '"'jry — / <2 / / UAA ~ UNA ~ \!JNN ~ (4.4) — 1 (Universality) , AV -17- JAERI-Conf 99-008 . Landau-Migdal V. SUMMARY AND CONCLUDING REMARKS -4/;0(m* = 0.8m, [io]0 -t ^), Pr ~ 1.3^)(Z = 0) [1] F. Osterfeld, Rev. Mod. Phya. 64, 491 (1992), and references therein. [2] R. Tamagaki et al., Prog. Theor. Phys. Supplement No.112 (1993), and references therein. [3] A.B. Migdal et, al.,Phys. Rep. 192, 179(1990). [4] J. Meyer-ter-Vehn, Phys. Rep. 74, 323 (1981); E. Sliiino et al., Phys. Rev. C 34, 1004 (1986); [5] T. Suzuki, H. Sakai and T. Tatsuini, nucl-th/9901097 [6] H, RESCUE Workshop report (1998) [7] T. Wakasa et al., Phys. Rev. C 55, 2909 (1997); [8] m#, mm. mm^hm, 52 (1997) 44i. [9] T. Suzuki and H. Sakai, to be published. [10] A. Akinal and V.R. Pandharipande, Phys. Rev. C56, 2201(1997). 18- JAERI-Conf 99-008 JP0050017 5. Onset condition of hyperons in charged-pion(?rc)-condensed phase is examined. It is shown that the onset of hyperons shifts to higher baryon number density in the ?rc-condensed phase than in the normal neutron-star matter. Relevance of this result to possible realization of kaon condensation from the ?rc-condensed phase is discussed. I.** ;vr y (Y)- (N) RTIY - Y KN >(S",A---) #ttJ3§ Briickner-Hartree-Fock pj = 2 . K \\\ Gamow-Teller II. TT A. , A-r *email address: [email protected] -19- JAERI-Conf 99-008 A'U (n,p) Hnp = sin Op* • sin sin ^p^ • .25, Z?=0.81, F=0.44 t (2) P ± b, Fermi sea A,E", S°, P $.(D TVYY' s ®.^ (3) B. (3) [5] epot V;: = = n,p, 1) ^case2 tbTSfflTS [5]: A -20- JAERI-Conf 99-008 , (4) , x = pp/pB, $&&o=2.1 fm, 6i = -1.0fm, Bo=-4.3 fm, Bx=0, a=0.5 £^5 [5]o 2 V% potential in neutron-star matter potential in neutron-star matter (MeV) 1500 1000 500 -500 0.4 0.6 0.8 3 pB(fm- ) FIG. 1. III. Fig.3 tFig.4 t case 1,2 \Z /u7l t Ey(p = 0) cDft, 'y(P = 0) , e ->• o (MeV) 2000 1 lM 1 tff —— 4 f 1600 if/ s s X 1200 1200 2° •*" 1000 800 0.2 0.4 0.6 0.2 0.4 0.6 0.8 3 3 pB(fm- ) pB(fm- ) FIG. 3. FIG. 4. -21 - JAERI-Conf 99-008 case 1 =1.8po "C E TTC vs(2) case 2 Tte, S JStT£tf>«A T, CA tfpB=3.0po , easel «t iv. 7T- A* u NN,YN (fib, YY t, (*P) [6,7]o A' AT" Oft , A- References [1] V\L=L — tLX, M. Prakash et al., Phys. Rep.280(1997), 1. [2] M. Baldo et al., Phys.Rev.C58(1998),3688. [3] E.g.,T. Muto, R. Tamagaki and T. Tatsumi, Prog.Theor..Phys. Supplement 112(1993), 159. [4] T. Suzuki, H. Sakai and T. Tatsumi, preprint nucl-th 9901097. [5] S. Balberg and A. Gal, Nucl.Phys.A625(1997),435. [6] D. B. Kaplan and A. E. Nelson, Phys. Lett.B175 (1986),57 ;B179 (1986),409(E). [7] mm i5, B«II^ 1996 ^x -22- JAERI-Conf 99-008 JP0050018 6. e-mail: [email protected] [email protected] abstruct Equation of state with kaon condensate is derived for isentropic and neutrino-trapping matter. Both are important ingredients to study the delayed collapse of protoneutron stars. By solving the TOV equation, we discuss the static properties of protoneutron stars and their implications for the delayed collapse. 1 Introduction K ^m^mmmt, 1987 *Z\Z Kaplan and Nelson K«fc 9 *Jf«| [1] fc [2]. JH *.# supernovae explosion "• black-hole star m (large mass ) hot and dens v-trapped matter 1 "A, low mass neutron star black-hole 1987A T«-^- h u fcBaumgarte -23- JAERI-Conf 99-008 . Prakash et al.[ Thorsson and Ellis[3] t tl consistent 2 Formulation ~ SU(3); G = SU(3)L x Goldstone - exp(2iTa , (pa = { transparent 5. Z Green massive tf*O, - t?D V Goldstone-mode 3 Numerical Results and Discussions tl -24- JAERI-Conf 99-008 S 9 •§>< EOS -:i- h U J £ b y 100 , ...... 400 90 350 80 300 70 • / w - 250 / / - fin $••'' $/ - E v-trapped.// jl | 50 - f f i 200 - h? 40 150 - 30 • I J v-trapping / 100 20 effect / 50 10 ••€/ ( / / N o T=0, v-free U ,5 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 2 3 4 5 6 7 U 2: = 0.4), = 0,50[MeV] S^«t 0 EOS Maxwell construction ^rfflV^T realistic^: EOS tlfc2. i CO Maxwell construction lZ£Z EOS TOV [10], 3[£] K strapped zm 3[£;] iz t& {Y,e =0.4,5 = , i/-free, 5 = 1 or 2 C*5 1 strapping ©3^^!' -25- JAERI-Conf 99-008 1.75 1.8 5=2 \v_ 1.7 1.6 -y\ \ trapped • 5=0 1.4 • N 1.65 V- ^ 1.2 free m • 1 5=0^— 0.8 '•••• H 1.55 i i i 7 6 9 10 11 12 lin13 ; ,14 15 2e+57 21e+57 2.2e+57 2.3e+57 fl[km] 3: v-trapped(Yle = 0A) ,v- dynamical &J , 4-IUffffl VTz Heavy Baryon Limit ^ b $ b £J£3gL, Propagator (' f!0 dynamical simulation K [1] D.B. Kaplan and A.E. Nelson, Phys. Lett. B175 (1986) 57; B179 (1986) 409(E). [2] G. E. Brown and H. A. Bethe, ApJ. 423 (1994) 659. [3] V. Thorsson and P.J. Ellis, Phys. Rev. D55 (1997) 5177. [4] M. Prakash, I. Bombaci, M. Prakash, P. J. Ellis, J. M. Lattimer and R. Knorren, Phys. Rep. 280 (1997) 1. [5] T. Tatsumi and M. Yasuhira, Nucl. Phys., to be published, nud-th/9811067. [6] T. Tatsumi, Prog. Theor. Phys. Suppl. 120 (1995) 111 and references cited therein. [7] T.W. Baumgarte et al, ApJ. 443 (1996) 680. [8] V. Thorsson, M. Prakash, J.M. Lattimer A572 (1994) 693. [9] N.K. Glendenning and J. Schaffner-Bielich, Phys. Rev. Lett. 81 (1998) 4564. -26- JAERI-Conf 99-008 [10] T. Takatsuka, Prog. Theor. Phys. 95 (1996) 901 [11] W. Keil and H.-Th. Janka, A&A. 296 (1995) 145 [12] J.A. Pons, S. Reddy, M. Prakash, J.M. Lattimer and J.A. Miralles astro-ph/9807040 -27- JAERI-Conf 99-008 JP0050019 7. —fe BS Abstract We investigate A E° mixing through neutral pion condensation interior neutron matter,symmetric nuclear matter and hyperonic nuclear matter. Although there is no direct AA7r coupling, the system gains energy as much as nucleon systems through A - E° mixing. It is necessary to understand TT condensation consistently with behavior of I] in A hyper nuclei. -6 t 4#:(^He) 75s under binding L, 3fch Afatfhl X 1 Kt % t^l&it 5 fr^ overbinding t TT° E (A-E induced pion condensation) fit iz>o or, * Tr0 t p , n, A, E ^ Lagrangian^ltti, pnAE - — * 28- JAERI-Conf 99-008 ffl 1: ALS fff*t -V - Cint (1) (2) SU(3) D) ^ 0.3 ~ 0.4 * = 2(1 - tf > ALS fllit [6] iM 1 \ Lowest Harmonics Approximation [6] {C X <{> ~ < (4) m. cos(A;2) (5) T (6) mv dz -29- JAERI-Conf 99-008 n[ = \MN- cos{kz)\ (7) AT t 1 E MA 0 1 MA" o 0 1 i 0 Ms 2 0 I i o if Ufa (8) (H 2) t > schematic i (p:n:A = 1:1:1) Hi5lt*7r ^ energy gain (121 3) I) , TT ffiUJSV^T A AATT **#ft = 0.3 &WI&3frZ% A-E |g (75MeV ^IJt ~ NA © 1/4 energy gain **# f t tfTF $ tL^ (d 4) o t 11.5 % L L to 7T , AH i: AHe Slt* underbinding > A ^ n > tfftfcf h ff L v > 7T JFLt£v>x.J:-9 , realistic 30- JAERI-Conf 99-008 Energy gain per baryon Density dependence of total energy per nucleon 100 N < vA tu ALS-n -100 ALS-p,n ALS-p.n FG-A ----- ALS-[),n,A —•— '•s. •50 -120 1 2 3 2 3 density [ p B / p 0 ] density [ p B / p 0 ] El 4: 7T° Peli=-' &lZ £& Energy gain / 2: Baryon Fermi Gas frb ALS ffif Density dependence of total energy per baryon Mixing ratio E / ( A +1) ~i 1 r FG-p,n,A --•--• ALS - p,n,A -..»-- •50 - 1 2 3 density [ p B / p „ ] 3: p:n:Y=l:l:l (D^V U >!$jftIZJSlf %> n° 15: TT° -31 - JAERI-Conf 99-008 [1] R.Tamagaki, Prog. Theor. Phys. Suppl. Noll2 (1993), Preface. [2] N.K.Glendenning, Phys. Lett. 114B (1982), 392. [3] T.Muto, 4*ffl%£f^f [4] T.Harada and Y.Akaishi, oral presentation at Strangeness Nuclear Physics 99. [5] E.Hiyama et al., oral presentation at Strangeness Nuclear Physics 99. [6] T.Takatsuka, R.Tamagaki and T.Tatsumi, Prog. Theor. Phys. Suppl. 112 (1993), 67. [7] T.Kunihiro, T.Takatsuka, R.Tamagaki and T.Tatsumi, Prog. Theor. Phys. Suppl. 112 (1993), 123. -32- JAERI-Conf 99-008 JP0050020 8. i& Alpha-clustering in dilute nucleonic sea Akihiro TOHSAKI Department of Fine Materials Engineering, Shinshu University, Veda, 386-8567 a-clusters are expected to come out here and there in nucleonic sea owing to energetic benefit as its density is diluted. We propose a precise treatment to elucidate o-clusterized process in nucleonic sea after the breakdown of the uniformness. In order to do this, an infinite number of nucleons are considered by taking account of both the Pauli exclusion principle and effective internucleon forces. This method is called a microscopic approach, which has been successful in an a-cluster structure in light nuclei. In particular, we shed light on overcoming difficulties in a static model within the microscopic framework. This improvement is verified by using the empirical value in Weizaecker's mass formula. §1. Introduction In general, in spite of the divergence of the binding energy of an aggregation of an infinite number of nucleons, each cluster comprising a few nucleons may have a definite value of binding energy. However, the Coulomb interaction between protons can give a logarithmic divergence to the binding energy even of an individual clus- ter. Nevertheless, a residual value which remains after subtracting the divergence is easily evaluated to be considered. It is also a serious problem that the binding energy inevitably includes the spurious energy coming from a static treatment of aggregation. This approaches a zero-point oscillation energy of isolated a-particle in the most dilute sea, which can be estimated only in a dynamical treatment. We show how to overcome such serious difficulties in quantum mechanically precise treatment. In this report, we represent a brief indication how we obtain the binding energy per a cluster, and we give the verification of this approach seeing effects which take place in nucleonic sea interior and surface. §2. Method We briefly recapitulate the static model which has been proposed in ref.(l). Imagine nucleonic sea in the generator coordinate space(GC space) in which treat- ment is equivalent to that of resonating group method2'. As shown in Fig.l. we cut out a certain domain F in nucleonic sea where 4JV nucleons exist. It is possible to contain the surface of sea by this cutting which allows us to estimate surface effects of semi-infinite aggregation of nucleons. The total wave function is written by where A is the antisymmetrizer for all the nucleons with the same spin-isospin state, \p. The spatial wave function, -33- JAERI-Conf 99-008 dimensionless spatial parameter s\ by i(q-S^)2}, (2-2) where the parameter SJ is regarded as the GC of the i-th nucleon. The realistic variable, q, is integrated over infinite space. The parameter S\ is directly related to the spatial generator coordinate, R,- ' = 6S, where b is the size parameter of (0s) h.o. wave function with %u> = h2/(Mb2) ( M is a nucleon mass ). When the labels, p, differ from each other, the GCs are also different due to the spatial distribution of nucleons although they have the same label, i. In Fig.l, the nucleons with labels, i\, Z2, 13 and 14 are placed on the same position, then four nucleons make an a-cluster in nucleonic sea with random arrangement of nucleons as shown in Fig.l. After inserting a microscopic Hamiltonian, 7i, the energy overlapping is written by 1 l 1 < W\H\W > = < 9\9 > [TriTB' ) + TT{(Vd - Ve){B~ ® B' )}}, (2-3) where the matrices are defined as B=(< m) >) , (2-4) (2-6) & - - and 1 / — * e — \ ^ 0 *P \^kl \01 01- / I i [ Z' i j 2 v ' •? ' * / where Vj and Ve have a tensor representation, and the symbol ® means the tensor product. Besides, the matrix element with three-body operator can be estimated by an extended formalism from eq.(2.3). The respective binding energies of nucleons are straightforwardly regarded as corresponding elements of traces in eq.(2.3). The domain F' is given by being slightly expanded from F. If the binding energy of a nucleon in the center of F' is almost unchanged from that in F, we can regard the convergence of the value as its binding energy. Nonetheless, we have two difficulties : that is, 1. how to get the convergence with the Coulomb force which inevitably leads to logarithmic divergence, and 2. how to remove the spurious energy which is included in such a static approach. The first difficulty is eliminated by subtracting the Coulomb direct energy without the Pauli exchange effects. This term and the background coming from negative charges are canceled out each other. The remaining value is just regarded as the Coulomb exchange term which rapidly converges into a definite value. We think that this value should be included in the volume term of the Weizaecker's mass formula -34- JAERI-Conf 99-008 as suppressing the binding energy. The second difficulty is fundamental because it is impossible to treat dynamically an infinite number of nucleons. Therefore, we propose an approximate method to remove the spurious energy. We assume that the a-cluster in a cavity, which is surrounded by the remaining nucleons, moves freely. Then we solve the equation of motion for a-cluster in the cavity in F using the Hill-Wheeler variational method as / < W{S)\H - X\9(S') > /(S')rfS' = 0. (2-8) an The deviation of Yli Ei d Amm may correspond to the spurious energy for the centered a-cluster, thus the spurious free energy is written in E^s in the following section. The weight function, f(S), to be solved expresses characteristics of motion through the transform into a wave function. This method can exactly remove the spurious energy in the rarefied limit of nucleon gas. One of the developments of this treatment is to enable us to investigate intercluster potential in a medium within a microscopic framework. §3. Results and conclusion As a preliminary step, we employ Brink-Bocker No.l force3) as an effective internucleon force which can reproduce the saturation property of nuclear matter. Unfortunately, the aggregation of nucleons always makes a-clusters over wide range of nucleon density because this force is a-clustering favorable. Therefore, we cannot find out the uniform density region in the nucleonic sea. The most likely a-clusterized nucleonic sea is composed of fee ( or ccp ) lattice configuration of an infinite number of a-clusters. The difference between fee and ccp appears only in the structure of surface. In Fig.2, we show the energy quantities of a-clusters which belong to three positions such as the sea interior, the surface of fee and the surface of ccp. Here, we use b = 1.4fm which reproduces the minimum binding energy of an isolating a-particle. We summarize the traits in Fig.2 as follows 1. The Coulomb exchange energy is about —0.22MeV per nucleon which is at- tractive against the Coulomb repulsion. This value seems to be independent of the positions of a-clusters. 2. The spurious energies are different from each other for the positions of a-cluster, and — Q.09MeV per nucleon for sea interior and —0.42MeV per nucleon for sea surface. But the difference for surface between fee and ccp is almost negligible. 3. The method of removing the spurious energy works well because of the energy a-cluster can be reproduced in dilute density. Finally we list the bulk properties of nucleonic sea comparing the stable binding energy and empirical values of Weizaecher's mass formula. Here we obtain the surface term bs by bs = (V3ir)UEs, (3-1) where oo -C's = / _, \&i ~ i = sur face -35- JAERI-Conf 99-008 for the static model and CO Es= £ (E^ - E^Lr), (3-3) i=surface for the dynamic model. The incompressibility is estimated by diagonalizing the Hessian matrix for the partial differentials of the second order with respect to the density of nucleonic sea and the size of a-cluster. Table.1 Static Dynamic Ordinary Empirical model model nuclear matter value bv(MeV) 14.1 14.4 15.7 15.6 bs(MeV) 36.6 22.8 17.2 K(MeV) 181 185 184 ~ 300 The bs for the dynamic model is remarkably improved from that for the static model. We anticipate to clarify the a-clusterized process in nucleonic sea by the most realistic internucleon force. The details in the static model appear in Refs(4-6). References 1) A.Tohsaki, Prog. Theor. Phys. Suppl. No.l32(1998),213. 2) D.M.Brink,Proceeding of the International school of physics, 'Enrico Fermi' course 36, Verenna, 1966,ed.C.B\och (Academic Press, New York and London,1966),p247. 3) D.M.Brink and E.Boeker, Nucl. Phys. A91 (1967),1. 4) A.Tohsaki, Prog. Theor. Phys. 88(1992),1119. 5) A.Tohsaki, Prog. Theor. Phys. 90(1993),871. 6) A.Tohsaki, Phys. Rev. Lett. 76(1996),3518. -36- JAERI-Conf 99-008 Fig. 1 -37- JAERI-Conf 99-008 -20 -30 — .CD -40 - 3 -50 - Ej(d) Repacking E&sXd) i:matt"surface ccp-paeking -60 - ccp-packing pf/is)/ ,\ i: matter interior I 2.5 3.0 3.5 4.0 4.5 Fig. 2 -38- JAERI-Conf 99-008 JP00G0021 9. We propose a simple method to reproduce the 'So pairing properties of nuclear matter, which are obtained by a sophisticated model, by introducing a density-independent cutoff into the relativistic mean field model. This applies well to the physically relevant density range. 1 f°° A(k\ = -— / v{p,k) k2dk (1) 8*2 Jo { (P-P) . Ek li, (N-N) iS Brueckner- vpp [1,2,3,4] ic N-N T [6] J; [7]o - [8]o (RMF) z. [9]0 1996 (p-h) t Ltffl^6t»[10, 11, 12, 13] 31 2 4, 15] T' o RMF C J; 5*- -39- JAERI-Conf 99-008 LTDirac-Brueckner-Hartree-Fock (DBHF) t LT, t Itffll^ "ffi>Ptf^W" Hartree- Fock-Bogoliubov (HFB) %\% [17] J; otWt (p-h HFB i:, N-N (wpp(/fc,JfcF) (l-2GeV) RMF id l(b) X(D 3- . Fermi 2 v{k¥,k)<)>{k)k dk (2) t —U^xft [18] Cooper»«t^ "9 ^ Bonn-B ^x> Ac -40- JAERI-Conf 99-008 no- 939 MeV, ma = 550 MeV, rnu = 783 MeV, 136.2 [19] f*>5o ft (7 = 4) t^tt-f-tl® (7 = 2) (C^LTtTo/i0 Fermi Sift* fcF {1) Ac M* =M-^r- (5) ,,2 =_ ; 1- 2 \ ^(Ek - Ek (6) ~ feonn A(fcF)Bonn ; V CBonn Ac o "RMF'\ " , kF =0.2, 0.3, ... , 1.2 1 Ac II 3.48 fm- <0 (A(ifc) > 0) p(A;,A;F) >0 (A(fc) < 0) 1 vpp(k,kF) gl l(b) ^W 2 fm" plateau [8] X Skyrme E 2(a) &tf (b) (i, L «p Bonn-B : J; 0.2 fm"1 1.2 t [20]o ^ ^^ Ac 19 i> A(feF) ifcF < LX b, Ac M* , Bli(b) I: kF ~ 1.2 fm kF < 1.3 fm- [5] (DX\ N-N -41 - JAERI-Conf 99-008 [1] L. N. Cooper, R. L. Milles and A. M. Sessler, Phys. Rev. 114 (1959), 1377. [2] T. Marumori, T. Murota, S. Takagi, H. Tanaka and M. Yasuno, Prog. Theor. Phys. 25 (1961), 1035. [3] M. Baldo, J. Cugnon, A. Lejeune and U. Lombardo, Nucl. Phys. A515 (1990), 409. [4] 0. Elgar0y, L. Engvik, M. Hjorth-Jensen and E. Osnes, Nucl. Phys. A604 (1996), 466 . [5] As a review, T. Takatsuka and R. Tamagaki, Prog. Theor. Phys. Suppl. 112 (1993), 27. [6] G. E. Brown and A. D. Jackson, The Nucleon-Nucleon Interaction (North-Holland, Amsterdam, 1976). [7] H. Kucharek, P. Ring, P. Schuck, R. Bengtsson and M. Girod, Phys. Lett. B216 (1989), 249. [8] S. Takahara, N. Onishi and N. Tajima, Phys. Lett. B331 (1994), 261. [9] H. Kucharek and P. Ring, Z. Phys. A339 (1991), 23. [10] F. B. Guimaraes, B. V. Carlson and T. Frederico, Phys. Rev. C54 (1996), 2385. [11] F. Matera, G. Fabbri and A. Dellafiore, Phys. Rev. C56 (1997), 228. [12] M. Matsuzaki and P. Ring, Proc. of the APCTP Workshop on Astro-Hadron Physics in Honor of Mannque Rho 's 60th Birthday: Properties of Hadrons in Matter (World Scientific, Singapore, in press), [e-print nucl-th/9712060]. [13] M. Matsuzaki, Phys. Rev. C58 (1998), 3407. [14] A. Rummel and P. Ring, preprint 1996 (unpublished). P. Ring, Prog. Part. Nucl. Phys. 37 (1996), 193. [15] M. Matsuzaki and T. Tanigawa, Phys. Lett. B445 (1999), 254. [16] 0. Elgar0y, L. Engvik, M. Hjorth-Jensen and E. Osnes, Phys. Rev. Lett. 77 (1996), 1428. [17] T. Gonzalez-Llarena, J. L. Egido, G. A. Lalazissis and P. Ring, Phys. Lett. B379 (1996), 13. [18] F. V. De Blasio, M. Hjorth-Jensen, 0. Elgar0y, L. Engvik, G. Lazzari, M. Baldo and H.-J. Schulze, Phys. Rev. C56 (1997), 2332. [19] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16 (1986), 1. [20] 0. Elgar0y and M. Hjorth-Jensen, Phys. Rev. C57 (1998), 1174. -42- JAERI-Conf 99-008 Particle-particle interaction 4 6 8 10 12 14 Cutoff dependence d 6 8 10 12 14 Ac (fm-1) 1 1: (a) Fermi ]I*J*fcF = 0.9 fm" 1 (b) CT- Fermi ffi fcF = 0.9 fm " -43- JAERI-Conf 99-008 Gap at Fermi surface (Symmetnc matter) c J (a) Bonn-B • RMF(Ac=3.48 far1) 4 3 * /" ^s. ) (MeV 2 1 \ n 1 1 1 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 *F (fm-1) Coherence length (Symmetric matter) "T1"" I —| V 50 (b) Bonn-B • 1 RMF(Ac=3.48fm- ) — 40 30 20 10 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1 *F (fin- ) J f 2: (a) Fermi mX' -44- JAERI-Conf 99-008 JP0050022 10. 15'o pairing in nuclear matter is investigated by taking the hadron mass decrease into account via the "In-Medium Bonn potential" which was recently proposed by Rapp ct al. The resulting gap is significantly reduced in comparison with the one obtained with the original Bonn-B potential and we ascertain that the meson mass decrease is mainly responsible for this reduction. t itzTy° 1970 ¥ft<7) Chin t Walecka K n tliQHD r, 4""" -45- JAERI-Conf 99-008 ttz QHD <» (RMFT) x. (f Gogny Sftf fj RMFT QHD H «t Kucharek t Ring (I <£ »9 1991 [2]o , NL1 «t tf2 MeV ^P> 4 MeV [3] Brown t Rho H = 1-CA C = 0.15. (2) T\ M,mPtW, Ap, 0.15 [Brown-Rho (BR) p [5]0 'i&hl±7t(D Bonn-B *°T > i/ ^ )^K::: a 2>Z.tX- OBEP ifo-j tZo Z.0) [In-Medium Bonn *°r >' -46- JAERI-Conf 99-008 In-Medium (C=0 15) Bonn-B ' 0.0 04 0.8 1.2 1 6 g] 1: Fermi ilfil-t JfcF © Fermi ffi±t:^^>7 7°0 [I] 1 H In-Medium Bonn potential H «£ *) f# f> tl/j Fermi 7° £ , Fermi Bonn-B *°r -<7) J; -5 b%\<\ Rummel t Ring ^1^ L ^ «t ^ U [3], ^>7 7° t *° r > v «t /j^^> In-Medium Bonn Bonn-B tf-r > i/V )l k In-Medium Bonn *°-f > v tl^In-Medium Bonn ^r > v -V Dirac-Brueckner-Hartree-Fock ia 3 H -47- JAERI-Conf 99-008 In-Medium bare Meson masses Bonn t°7- y v \ ) bare Nucleon mass both scaled by BR-scaling Bonn-B *°-f > v < M^ L, -?• tl [7]0 **3, ir Fermi < ^ El 3: UftM 1, [8] m L £ [1] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 (1986). [2] H. Kucharek and P. Ring, Z. Phys. A339, 23 (1991). [3] A. Rummel and P. Ring, preprint (1996, unpublished). [4] G. K. Brown and M. Rho, Phys. Rev. Lett. 66, 2720 (1991). [5] R. Rapp et «/., Phys. Rev. Lett. 82. 1827 (1999). [6] R, Machleidt, Adv. Nucl. Phys. 19, 189 (1989). [7] M. Matsuzaki and T. Tanigawa, Phys. Lett. B445, 254 (1999). [8] M. Matsuzaki and P. Ring, Proceedings of the APCTP Workshop on Astro-Hadron Physics in Honor of Mannque Rho's 60th Birthday: Properties of Hadrons in Matter (World Scientific, Singapore, in press), [LANL e-Print archive nucl-th/9712060]; M. Matsuzaki, Phys. Rev. C58, 3407 (1998). -48- JAERI-Conf 99-008 JP0050023 11. QMD ICJ: Abstract The infinite nuclear matter that consists of numerous protons and neutrons is described by using periodic boundary conditions. The motion of each nucleon in the fundamental cell is decided by a Molecular Dynamics. The ground states or the excited states of the nuclear matter are simulated. 2. » 0) QMD 7 7s 9 ~tk$L& I [fin] n «t -49- JAERI-Conf 99-008 (INPUT) «t 16 [MeV] E/A = e A'=280 [MeV] t) ^^"P^ L^'>— > v (R.-R,)2 (P.-P,)2 2. &13 (Skyrme M + ftfo + BM + 'sym ^ = E f 3. ^- CPauli, 90, Po, L, a, $, r, [5] -50- JAERI-Conf 99-008 80 . T= 0.0 MeV t 60 L, £. 40 C =115[MeV] 20 p qo=2.5O[fm] po=12O [MeV/c] EOS of Nuclear Matter 8 K=280[MeV] E/A = 16 [MeV] 14.5 [MeV] -20 0.0 0.5 1.0 1.5 2.0 2.5 p/po Real Part of Optical Potential for p-Nucleus elastic scattering 100 ^T n Ixp. HAMA2 U(»)= 75.43 IMeV QMO 50 y v -v jtf^ Experimental data CD & S.Hama el al..Phys.Rev.C41,2737(199 I. o I 1.333 33 U (p=p(p=0)F ) = -80.3-67.80 |M«VIMeV]] ut == -121.9 IMeV] a B = 197.3 [MeV] f K =280 0(MeV| V™ =•258.5 (MeV] -50 r m' = 0.800 m n, = 2.350 [1/lm] V1^' = 375.6 [MeV] Hi = 0.40011/fm] -100 500 1000 1500 2000 b [MeV] -51 JAERI-Conf 99-008 Binding Energy of Finite Nuclei 10 i 0 0) LU CD 50 100 150 200 250 Mass Number A Radius of Finite Nuclei 5 >io L [fm2]) , A 50 100 150 200 250 Mass Number A • • • • ±z ^ i '•A * i • m • (replica) \ * \ * -52- JAERI-Conf 99-008 dH dH • dH dH (i) > 0 exp[-A£/T] ^flt^T^ 1024 128 896 x mr&i^m t x $ £p t /: ft 9= O.l/9o, Z/A = 0.125 3. 1.0 0.3 -53- JAERI-Conf 99-008 -7&*£;ftLTv> *i)K z/A ^ o m 10 Hfili t?ti, Z/A = 0.08 ^hho LfrL£^^, (exp[-r/a]/r, a=10fm) [1] D.G. Ravenhall el a/., Phys. Rev. Lett. 50 (1983) pp.2066-2069. [2] M. Hashimoto et al, Prog.Theo.Phys. 71 (1984) pp.320-326. [3] T. Maruyama et al., Phys. Rev. C57 (1998) pp.655-665. [4] N. Metropolis et al., J. Chem. Phys. 21 (1953) pp.1087-1092 [5] MP^jf i% B^t/S^^ 1998 [6] -54- JAERI-Conf 99-008 p=1.0p( p=0.7p( 30. 30. P=0.5p0 p=0.3p0 30. 30. P=0.2p0 P=0.1p0 30. 8: (N=Z) =0 [MeV] -55- JAERI-Conf 99-008 T=0.5 [MeV] T=0 [MeV] T=3 [MeV] T=1 [MeV] T=8 [MeV] T=5 [MeV] O 0 30.00frnr -77, ^— -56- JAERI-Conf 99-008 Z/A=64/1024 Z/A=128/1024 39.35fm 39.35fm Z/A=192/1024 Z/A=256/1024 20 i p= 1.0 p0 - P= 0.2 p0 _ -10 -20 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 Z/A Z/A (N>Z) hM -57- JP0050024 JAERI-Conf 99-008 12. l/-y 3 -3 (99.4.30) We study the properties of nuclear matter expanding dynamically using QMD model. For this purpose, we developed an extended periodic boundary condition prescription. To calculate EOS of the expanding nuclear matter, the liquid-gas phase transition is discussed. I. l), QMD 1/10 Jt P=1-Opo p=O.1po 9*?-flfi£fijE-f t VRJSCO Multifragmentation ii. P, = AR,- (1) = AD* dt (2) P,- - P, + MVt (3) -58- JAERI-Conf 99-008 FIG. 1. ^-u pa] Ciliit £ fc, I—., FIG. 2. (l) Pi (total) —• Pi (total) ~ (4) (total) TABLE I. Effective interaction parameter set (A'=280 MeV) a (MeV) -121.9 P (MeV) 197.3 T 4/3 Csa (MeV) 25.0 Cex(1)(MeV) -258.5 375.6 in (MeV) 2.35 H2 (MeV) 0.4 VSF 20.68 115.0 PO 120.0 So 2.5 L (fm2) 1.95 -59- JAERI-Conf 99-008 in. <> QMD *i [1] 2^, ^"77 + Kiud + ^surface + ^Coulomb + Vpauli i l yl 27 A 27 A TPij(k) A 27 A p$ L i=1 k=1 j# VCoulomb = j |R»-«-j+R«e./(t)|2 J(*) (5) , QMD Metropolis (6) Po PF (i Fermi .05p0 I* * ^7 t>tli>o tzhltf 60(fm/c) rz 108 t LtvS, m -60- JAERI-Conf 99-008 h=0.40 h=0.30 £ IO > 3 C/2 oU I I I I I I I j- 0 1 0 1 0 1 P/Po P/Po P/Po FIG. 3. h=0.1 FIG. 4. h=0.1 FIG. 5. T=5 MeV T=5MeV^ T- 30MeV lAp0 \Z 2 v^ v' ' P> ^effective - „ • / „ (8) H 4 ^ ^o -^ H 5 5MeV = o.i -i^/i = o.4 IV. Si to KiW^^Etijf EOS EOS *s/c t'i, path li [3] path I) realistic IC 11? multifragmentation [l] T. Maruyama e( at, "Quantum molecular dynamics approach to the nuclear matter below the saturation density," Phys. Rev. C57, 655 (1997) nucl-th/9705039. [2] P. Finocchiaro et al., "Second order phase transitions: From infinite to finite systems," Nucl. Phys. A600, 236 (1996) nucl-th/9512019. [3] M. E. Fisher, Rep. Prog. Phys. 30 (1967) 615; Proc. International School of Physics, Enrico Fermi Course LI, Critical Phenomena, ed. M. S. Green (Academic, New York, 1971). -61 - ill JAERI-Conf 99-008 JP0050025 13. Giant quadrupole resonances in time dependent density matrix theory Mitsuru Tohyama Kyorin University School of Medicine, Mitaka, Tokyo 181-8611, Japan The time-dependent density-matrix theory (TDDM) [1,2] is an extended version of the time-dependent Hartree-Fock theory (TDHF) to include the effects of two-body correlations, and is able to describe dissipations and fluctuations in low-energy nuclear collective motions. Here we report the results of the application of TDDM to giant quadrupole resonances in 40Ca. We first present the equations of motion in TDDM and then explain the method for calculating the strength functions of a single quadrupole giant resonance (GQR) and a double giant quadrupole resonance (DGQR). The TDDM equations determine the time evolution of a one-body density matrix p and a two-body correlation function C2 defined by C*2 = p2 — A\pp\. where A[p/)] is an antisymmetrized product of the one-body density matrices and p2 is a two-body density matrix. In TDDM, p and C2 are expanded with a finite number of single-particle states {ipa}, p(U',t) = j;nM.(t)i(l,0^(lV), (1) c2(m'2',t)= Y, ca0a,p,(t)Mht)Mt,t)rM,t)il>F(2,t), (2) where the numbers denote space, spin and isospin coordinates. The time evolution of p and Ci is determined by the following three coupled equations [1]: ih-^a(i,t) = h(i,t)ipa(ht), (3) ihhaa. = Y^[(afi\v\'y5)C-ysa'p - Ca/3ls(lS\v\a'/3}], (4) /3-yi 1 ihCapa'pi = Baf)aip> + Pafia'f) + Ha0Qipi, (5) where h(l,t) is the mean-field hamiltonian and v the residual interaction. The term Bapa'p' on the right-hand side of Eq.(5) represents the Born terms (the first-order terms of v). The terms Papa'0' and Hafia'p' in Eq.(5) contain CQpa'(}' and represent higher-order particle- particle (hole-hole) and particle-hole type correlations, respectively. Thus full two-body correlations including those induced by the Pauli exclusion principle are taken into account an in the equation of motion for Capaipi. The explicit expressions for Bagai0i, Pa0a'0' d Hapaip> are given in ref. [l]. To solve the coupled equations, we use the same Skyrme interaction for the calculation of the mean-field potential and the correlation function and assume that 40Ca is a completely spin-isospin symmetric system. The part of the mean-field potential which contains the parameter a:0 [3] associated with the spin exchange operator disappears for such a system. Therefore, terms depending on the spin exchange operator are neglected also in the residual interaction. The residual interaction used is of the form [3,4] -62- JAERI-Conf 99-008 3 3 2 v(r - r') = t0S (r - r') + k{ W(r - r') + 5 (v - r')k } *3(r - r')k + \t,p {^f^i 8\v - r'), (6) where k = (Vr — Vr/)/2z acts to the right and k' = (Vr» — Vr)/2i acts to the left. The factor 1/2 on the density dependent term contains the contribution of a rearrangement effect. We use the parameter set of the Skyrme III force [5]. GQR built on the correlated ground state [6] is excited by boosting the single-particle wave functions (f>a(l) with a phase factor corresponding to the quadrupole mode, ikQ Mht = 0) = e U 2 1 2 2 Q(r) = z - -(x + y ). (8) In the limit k —> 0 the strength function, defined by En), (9) is related to the Fourier transformation of the expectation value Q(t) of the transition oper- ator Q [1]: 1 r°° Et S(E) = — Q(t)sm—dt, (10) irkn Jo n where Q(t) is calculated with the one-body density /?(r, t) as Q(t) = (Q) = I Q(r)p(r,t)d3r. (11) In numerical calculations shown below the integration in Eq.(10) is limited to a finite time interval. The excitation of DGQR and the calculation of its strength function are done in a way similar to GQR [7]. We assume that the motion of DGQR is generated by a two-body operator Q2'- |tf(* = 0)) = e^|$o>, (12) 2 2 where Q2 is a two-body part of Q . We use Q2 instead of Q to reduce the excitation of single phonon states. The initial condition for Capaipi is given as Ccpc'p'it = 0) = (tf (* = 0)\at,apa/}aa\V(t = 0)). (13) Since the evaluation of the above equation with the use of the correlated ground state is obscure, we assume that |$0) in Eq.(12) is the Hartree-Fock (HF) ground-state wavefunction. At first order of k, the initial correlation matrix becomes = 2ik{(n\Q\p)(v\Q\ -63- JAERI-Conf 99-008 where p and a refer to unoccupied single-particle states, and \i and v refer to occupied ones. Other elements of the initial correlation matrix vanish at first order of k. Similarly, the non-varnishing initial values of naai become (16) (17) The strength function of DGQR, defined by (18) is given by the Fourier transformation of the expectation value of Q2 as in the case of single phonon states: Et (19) where Q2 is calculated as Q2(t) = (20) af3a'0' The coupled equations Eqs.(3)-(5) are solved using the Is, lp, 2s, Id, 2p, 1/, 3s, 2d and \g single-particle orbits. The 2p, 3s, 2d and \g states are in the continuum. The wave functions of these states are obtained by confining them to a cylinder with length 16fm and radius 8fm. 600 FIG. 1. £2 strength distributions of 40Ca calculated in TDDM (solid line) and TDHF (dotted line). The obtained strength distribution of GQR is shown in Fig. 1. The peak energy and the full-width-at-half-maximum FpwHF of the main peak in TDDM is 16.8MeV and 2.4MeV, respectively. The corresponding values in TDHF are 17.6MeV and 1.4MeV, respectively. -64- JAERI-Conf 99-008 The main peak at E = 16.8MeV may consist of two unresolved components with a similar amount of the E2 strength since the time evolution of the quadrupole moment has a beat pattern. The separation energy of the possible two components estimated from the period of the beat is 1.3MeV. The small separation of the two components is in contrast to GQR in 16O [4]. In the case of 160 the E2 strength is nearly equally split into two peaks separated by 5MeV. A cancellation mechanism between hole damping and particle damping suggested by Bertsch [8] may be responsible for the small splitting of the two components. Experimentally observed E2 strengths in 40Ca are also split into two or three peaks located at E=14 and 18MeV [9] or at E = 12,14, and 17MeV [10] depending on experiments. However, there is a discrepancy between our result and experimental data concerning the width of the main peak; the experimental strength at 17-18MeV shows a broad distribution with a width of about 4MeV, which is much larger than the value calculated in TDDM. The problem as to whether the broad main peak is reproduced by using more appropriate residual interaction and/or including more complicated configurations should be solved by further studies. 700000 600000- 500000- ~> 400000 -•| 300000- g 200000- VI 100000- 0 10 20 30 40 50 60 70 E[MeV] FIG. 2. Strength distribution of DGQR calculated in TDDM (solid line). Dotted line depicts the unperturbed strength distribution. The strength distribution of DGQR calculated in TDDM is shown in Fig.2 (solid line), together with the unperturbed one (dotted line). The peak at E — 39.9MeV corresponds to DGQR. The results in Figs.2 and 1 show that the energy of DGQR is more than twice the mean energy of GQR. As was pointed out in our previous paper [7], it should be noted that Eq.(3) (describing GQR) and Eq.(5) are solved with different truncation schemes of the single-particle space: Eq.(3) is solved in coordinate space and Eq.(5) with the truncated single-particle space, as described above. We made an RPA calculation for GQR in the truncated single-particle space. The strength function of GQR obtained from such an RPA calculation is shown in Fig.3. The RPA calculation in the truncated single-particle space gives the peak energy of 19.9MeV. Thus the peak energy of DGQR shown in Fig.2 is twice the GQR energy obtained from the RPA calculation done with the same truncated single-particle space. -65- JAERI-Conf 99-008 FIG. 3. Strength distribution of GQR calculated in RPA (solid line). Dotted line depicts the unperturbed distri- bution. The above results show that DGQR in TDDM keeps quite well the harmonicity as the double-phonon state of GQR in RPA. This indicates that RPA-type correlations in each single-phonon state dominate various two-body correlations considered in Eq.(5). The fact that DGQR in TDDM has highly harmonic properties is consistent with other theoretical calculations for DGQR [11]- [13]. In summary, the properties of isoscalar giant resonances in 40Ca were studied using TDDM, an extended version of the time-dependent Hartree-Fock theory. The calculations were done in a self-consistent way: The same Skyrme force as used for the calculation of the mean- field potential was used as the residual interaction to induce two-body correlations. It was found that the calculated strength of GQR is split into two components; a high-energy component with most E2 strength and a low-energy component with fractional E2 strength. The spreading width of a main peak was found too small as compared with experimental data. The double phonon state of GQR was also studied in TDDM. It was found that the excitation energy of DGQR is exactly twice the excitation energy of GQR calculated in RPA. [1] M. Gong and M. Tohyama, Z. Phys. A335 (1990) 153. [2] M. Gong, M. Tohyama, and J. Randrup, Z. Phys. A335 (1990) 331. [3] D. Vautherin and D. M. Brink, Phys. Rev. C5 (1972) 626. [4] M. Tohyama, Phys. Rev. C58 (1998) 2603. [5] M. Beiner, H. Flocard, Nguyen van Giai, and P. Quentin, Nucl. Phys. A238 (1975) 29. [6] M. Tohyama, Prog. Theor. Phys. 99 (1998) 109. [7] M. Tohyama, Prog. Theor. Phys. 100 (1998) 1293. [8] G. F. Bertsch, Phys. Lett. B37 (1971) 470. [9] J. Lisantti et al., Phys. Rev. C40 (1989) 211. [10] H. Diesener et al., Phys. Rev. Lett. 72 (1994) 1994. [11] A. Abada and D. Vautherin, Phys. Rev. C 45 (1992) 2205. [12] P. Ring, D. Vretenar, and B. Podobnik, Nucl.Phys. A598 (1996) 107. [13] G. F. Bertsch and H. Feldmeier, Phys. Rev. C56 (1997) 839. 66- JAERI-Conf 99-008 JP0050026 14. m Nuclear Shape Evolution Starting from Superdeformed State - Role of Two-Body Collision and Rotation - Yu-xin Liu1 and Fumihiko Sakata Department of Mathematical Sciences, Ibaraki University, Mito, Ibaraki 310-8512 Abstract: With the nuclear density distribution being simulated by the Boltzmann- Uhling-Uhlenbeck equation and Vlasov equation with several rotational frequencies, the time evolution of the quadrupole moment of nucleus 86Zr starting with superdeformed shape is studied. The contribution of two-body collisions and the effects of collective rotation to the shape evolution is investigated. The numerical results indicate that the two-body collisions play a role of damping on the evolution from a superdeformed shape to a normal deformed one in a case without rotation. In a case of rotation with lower frequency, the two-body collisions accelerate the evolution process. A new role of the collective rotation to enhance the nuclear fission is proposed. Since the spontaneous fission was observed in the 1930's, the nuclear shape evolution has long been a topic of the nuclear physics. After a process opposite to the fission, i.e., a decay out of the seperdeformed (SD) states to the noraml deformed (ND) states, was observed in recent years [1], much attention has been paid to investigate the mechanism of nuclear shape evolution [2]. On the theoretical side, it has been known that the nuclear shape evolution results from the collective motion as well as the single particle ones. Meanwhile different nuclear shapes are known to be the states built on different yrast states. Even though the pairing effect has been realized to be important in the evolution process [3], the contribution of the two-body collisions of the nucleons moving in the mean field to the evolution has not yet been clarified well. In the liquid drop model [4], it has been shown that the rotation always enhances the deformation, and even leads to the nucleus to fission, due to the centrifugal force. However, detailed studies on how the rotation affects the deformation has not yet been explicated well. In this work, we will discuss the role of the two-body collisions in the nuclear shape evolution by using a semiclassical dynamical model, and attempt to shed light on the way of the contribution of rotation. It has been well known that the Boltzmann-Uhling-Uhlenbeck equation (BUU or VUU equation) is regarded as a semiclassical approach for the nuclear dynamics, and has been quite powerful in describing the low and intermediate energy heavy ion collision (see for example Ref. [5] and references therein). Recently, it has also been shown to be successful in simulating the processes of nuclear spontaneous fission [6] and fusion [7]. We take it to simulate the process of the evolution from a superdeformed shape to a normal deformed shape. In the semiclassical scheme of BUU equation, the nuclear dynamics is described by the variation of the single particle phase-space density distribution. The BUU equation reads ^ + -P .Vr/-Vr[/-Vp/ = C*[/], (1) \Jv lit aOn leave from the Department of Physics, Peking University, Beijing 100871, China -67- JAERI-Conf 99-008 where m, r* and p denote the mass, coordinates and momentum of the particle, respectively. U is the potential of the field in which the particle moves. /(£, r, p) is the single particle phase-space density distribution function, which is related to the spatial nucleon density with the relation p(t,r) = J fdp. The Ct[f] is the two-body collision term, which can be given as Ct[f) = --^— Jdp2dp2,dpvv12 - /2)] S(p + p2 - pv - P2>) • (2) When the two-body collision term vanishs, i.e., Ct[f] = 0, the BUU equation reduces to the Vlasov equation ft +|-V,/-V,l/.VP/ = O. (3) Aiming to investigate the dynamical role of the two-body collisions on the nuclear shape evolution, we simulate it by numerically solving the Vlasov equation and the BUU equation. To explore the effects of the collective rotation on the evolution process, we calculate a case with and without a rotation. All the calculations are carried out by using the test particle method [8]. Exploiting the test particle method, the single particle phase-space density can be given as f(t,f,p) = LYiS(f-fi(t))8(p-pi(t)), (4) where r*j and pi are the coordinate and momentum vectors of the i-th test particle. The motion of each test particle is determined by the classical equation of motion ^ = & , ^ = —ViU . With the stiff potential in the scheme of Skyrme interaction, the equation of motion for the test particles is sovled with the fourth order Runge-Kutta algorithm with time step 8t — 0.05 fm/c. The calculations are performed in the lattice with 51 x 51 x 61 mesh points with nt — 2000 for each nucleon. Since the nucleus 86Zr is believed to have quite good SD states and its nucleon number is neither too large nor too small, we take it as an example to be investigated. To simulate the evolution from a SD shape to a ND shape, the position of each test particle is initiated in a ellipsoid with major to minor axis ratio 2:1. In the case with rotation, the rotational frequency is taken to be huQ = 0.79 MeV which is consistent with the experimentally observed rotational frequency at the band head of the yrast SD band of 86Zr [9]. As the test particles are exploited to simulate the nucleons, we have considered the Pauli principle to restrict their motion and collisions, for taking account of the quantum property of the test particles. With the evolution of the density distribution being simulated, the quadrupole 2 86 moment Q^ = J p(r)r Y2odr of the nucleus Zr is obtained. The results of the time evolution of the quadrupole moment in the cases without and with rotation are illustrated in Fig. la and lb respectively. From the figure, one may easily understand that the quadrupole moment descends globally with time. Since the quadrupole moment well describes how large the deformation of the nucleus is, the figure indicates that the superdeformed nuclear shape can decay to a shape with smaller deformation, and even to a normal-deformed shape. Comparing the results with and without two-body collisions in the case without rotation, one may know how the two-body collisions damp the decay rate. Looking over the result in the case of rotation with frequency u0, one may realize that the difference between with and without two-body collisions is not so monotonous and obvious as that in the case without rotation. At the beginging of the decay (t < 200 -68- JAERI-Conf 99-008 fm/c), the two-body collisions play the same role as that in the case without rotation. In the middle of the evolution process (200 fm/c < t < 500 fm/c), the two-body collisions accelerate the decay and thereafter make the nucleus arrive at a stable normal deformation. However, in the case without two-body collisions, the nucleus can decay continuously to a oblate shape. After a period, the nucleus has a tendency to recover its shape in both cases with and without two-body collisions. 200 400 600 800 1000 400 600 1000 time (fm/c) time (fm/c) Fig.l. Time evolution of the quadrupole moment in the case a) without and b) with a rotation (u> = u>0). The result without two-body collisions is illustrated in solid line and that with collisions in dashed line. 200 400 600 800 1000 400 600 800 1000 time (fm/c) time (fm/c) Fig.2. Time evolution of the quadrupole moment in the case of rotation with a) u> = 2wo and b) u> = 5u;o. The result without two-body collisions is illustrated in solid line and that with collisions in dashed line. To scrutinize the effect of rotation, we have also evaluated the time evolution of the quadrupole moment for the cases of rotation with frequency u = 2u>0 and 5u>0. The obtained results are shown in Fig. 2a and 2b, respectively. The figure shows that there exists generally shape oscillation in the evolution process. When the rotational frequency is u: = 2a>o, the case without two-body collisions shows a larger oscillating period and a larger amplitude than the case with two-body collisions. When the rotational frequency is LJ = 5u>o, the case without two-body collisions has a smaller oscillating period as well as a smaller amplitude than the case with two-body collisions. During the first decay period (the time for the Q^ to decay from the initial value to its first minimum), the figure also shows that the difference between numerical results with and without two-body collisions decreases when the rotational frequency increases. It indicates that the role of the two-body collisions is weakened as the rotational frequency -69- JAERI-Conf 99-008 increases. Examining the figures more carefully, one may recognize that, on top of the global changing behavior, the quadrupole moment Q^ exhibits an shorter oscillating property. It manifests an intrinsic shape oscillation (like the giant resonance) resulting from the inherent motion of the nucleus. From the results without rotation, one may estimate the period of the intrinsic oscillation to be about Tjn ~ 160 fm/c. Accoeding to the results with different rotational frequencies, the amplitude of the intrinsic oscillation decreases when the rotational frequency increases. As a consequence, the amplitude of the global oscillation is enhanced drastically. This implies that the global oscillation cauised by both the collective rotation and the intrinsic oscillation may couple with each other. As the frequencies of two kind oscillations come to close with each other, a resonance may appear so that the nucleus has very large quadrupole moment. In the case with rotational frequency u = 5u>o, the numerical result on the density distribution indicates an hourglass-shape with a very thin neck when the quadrupole moment reachs its maximum. It may provide us with a clue on the dynamical mechanism of collective rotation to enhance the nuclear fission: the rotation may couple with the intrinsic motion, and eventually the fission takes place when the frequencies of two kind oscillations reach the conditions of resonance. In summary, we have simulated the time evolution of the quadrupole moment starting with the superdeformd shape. The calculated results indicate a gradual change of the nuclear state from a superformed shape to a normal deformed shape. In the case without rotation, the two-body collisions play a role to delay the evolution from a superdeformed shape to a normal deformed shape. In the case with rotation, it is neither so monotonous nor obvious as that in the case without rotation. It may be too early to deduce a conclusive role of two-body collisions in the case of rotation. However, one may state two important points: a) with increasing the rotational frequency, the role of two-body collisions gets less important, b) besides the centrifugal force effects of rotational motion, the fission may be enhanced by a resonance caused by the collective rotation and the intrinsic motion. Many sophisticated probelms, such as how the effect of the two-body collisions changes with the increasing of rotational frequency, remain to be clarified. Acknowledgement: This work is supported in part by Japan Society for Promotion of Science. References [1] See for example, T. L. Khoo et al, Phys. Rev. Lett. 76, 1583 (1996); A. P. Lopez-Martens et al, Phys. Lett. B 386, 188 (1996). [2] for example, H. A. Weindenmuller, P. von Brentano and B. R. Barrett, Phys. Rev. Lett. 81, 3603 (1998); S. Aberg, Phys. Rev. Lett. 82, 299 (1999). [3] T. Dossing et al, Phys. Rev. Lett. 75, 1276 (1995). [4] S. Cohen, F. Plasil and W. J. Swiatecki, Ann. Phys. (N.Y.) 82, 557 (1974). [5] A. Bonasera, F. Gulminelli and J. Molitoris, Phys. Rep. 243, 1 (1994). [6] A. Bonasera and A. Iwamoto, Phys. Rev. Lett. 78, 187 (1997). [7] V. N. Kondratyev and A. Iwamoto, Phys. Lett. 423, 1 (1998). [8] C. Y. Wong, Phys. Rev. C 25, 1461 (1982). [9] D. G. Sarantites et al, Phys. Rev. C 57, Rl (1998). -70- JAERI-Conf 99-008 JP0050027 15. Can we determine the EOS of asymmetric nuclear matter using unstable nuclei? K. Oyamatsu(Nagoya Univ.),A I. Tanihata, Y. Sugahara, K. Sumiyoshi (RIKEN) H. Toki (RCNP) (RCNP) Abstract This paper shows that nuclear radii and neutron skins do directly reflect the saturation density of asymmetric nuclear matter. The proton distributions in a nucleus have been found to be remarkably independent of the equation of state (EOS) of the asymmetric matter. It is the neutron distributions that are dependent on the EOS. Macroscopic model calculations have been performed over the entire range of the nuclear chart based on two popular phenomenological, but distinctively different, EOS : the SHI parameter set for the non- relativistic Skyrme Hartree-Fock theory and the TM1 parameter set in the relativistic mean field theory. The saturation density for a small proton fraction remains almost the same as the normal nuclear matter density for the SHI EOS, but it becomes significantly small for the TM1 EOS. The key EOS parameters used to describe the saturation density are the density derivative of the symmetry energy and the incompressibility of symmetric nuclear matter, while the saturation energy is written using the symmetry energy alone as a good approximation. We conclude that a systematic experimental study of heavy unstable nuclei would enable us to determine the EOS of asymmetric nuclear matter at around the normal nuclear matter density with a fixed proton fraction down to approximately 0.3. Therefore, the answer to the title is yes. RI H- A present address: Aichi-Shukutoku University -71 JAERI-Conf 99-008 RI io"-i"ioo15 3 g/cm 1. Miff CD RI If—A (3xl014 3. 2. + i: 0.4-0.5 tt 0.03 t RI D -72- JAERI-Conf 99-008 TMI , sm t sni (U o o c Yp = 0-0.5, T = 0 - 100 MeV, (1) i5 p=o-io g/cm\ 0.00 0.05 0.10 0.15 0.20 n (fm'3) 30, i r (TMI, sni) (Y = P 0.00 0.05 0.10 0.15 0.20 TMI n (fm"3) sm •e Yp=0, 0.05, 0.1, TMI -73 - JAERI-Conf 99-008 © > TMI , sni RI tf- -5 2&*£ 4. TMI, sm (0 2) )Z\t, TM1 t •.••'" • • ao-j,= .....-•'- : •.',.-..r!'. :••.-.••:•. • .• -U"::O ^ : •••:%.'!:•!•."• ! t> (Wf C 3, TMK SO 4. , TMK sni ^ (02) -74- JAERI-Conf 99-008 , TMI sni x , TMI (SIII) 5 5dtt Na TMU SHI (Df 0 0.5 n). (1) Skin data(A) 0.4-j Skin data(B) n CD ys(n)> S(n) 0.3 J 0.2- 0- -0.1- -0.2^ -0.3- -0.4 19 -.23 25 27 29 31 33 YP Mass number of Na isotopes 0 5. Na nT : w (n)t n=n X &, TMK SIII s 0 Taylor -~l r JAERI-Conf 99-008 JS (2) io n0 (3) o)* So= S(n0 (10) K$, AK nClt 2 *Ks - -Qny "o (4) n = n0 YP>O.3 dS (5) n = n0' (Yp>l/3) dn (6) c\ , Po , Mm , RI tf- Yp 6. (7) RI tf- (8) TJ! 1/ (9) -76- JAERI-Conf 99-008 , RI \z- [1] K. Oyamatsu, I. Tanihata, Y. Sugahara, K. Sumiyoshi and H. Toki : Can we study the equation of state of asymmetric nuclear matter using unstable nuclei?, Nucl. Phys. A634, 3-14, 1998. [2] H. Shen, H. Toki, K. Oyamatsu and K. Sumiyoshi :Relativistic Equation of State of Nuclear Matter for Supernova and Neutron Star, Nucl. Phys. A637, 435-450, 1998. [3] H. Shen, H. Toki, K. Oyamatsu and K. Sumiyoshi : Relativistic Equation of State of Nuclear Matter for Supernova Explosion, Prog. Theor. Phys. 100, 1013- 1031, 1998. -77- JP0050028 JAERI-Conf 99-008 60Ca + 60Ca and 197Au + 197Au collisions are studied with an extended ver- sion Antisymmetrized Molecular Dynamics (AMD-V), in order to investigate whether the reaction observables carry the information of the equation of state of the asymmetric nuclear matter. DiToro© (AMD-V) . AMD-VTU, U, AMDttt, = detfexpj-^r, - (1) 78- JAERI-Conf 99-008 a, =pt, P+, it, n K, AMDTtt, , AMD * TDHF K: . TDHFtt, [4, 5]. )K> Fig T2 t*D©«»^SSfc»|g1-<5#«lffiSfPffl*#AT*. Figure 1 (±) tt, S^© Gogny ^7 [6] (C^^f-5 *)©T*S. Gogny J]\$, ftS AMD-^ AMD-V OfWTftt, »^t*«t)^oTVi*. Gogny (a = 1/3) (2) -^r, Fig. 1 (T) . DiToro© . Gogny[C(p0)] Gogny :, Gogny 2 ;WS-tt445MeV, (r )^ = 3.84 fin, 3.59 fm T**. Sfe, Gogny[C(p0)] 2 -tt 470 MeV, (r 1 fm, (r )^rge - 3.54 fm £fco fc. -**, Gogny[C(p0)] Gogny 2.1 - 35 MeV fc AMD-V = 0 t b = 5 fm = 0 * 6 = 5 fm Figure 2 \Z, tf- , Gogny [C(po)j -79 JAERI-Conf 99-008 Gogny force Gogny force with C(p0) 7=0.00 02 60 eo eoCa + eoc Ca + Ca E/A = 35 MeV ' a E/A = 35 MeV 7=0.25 p b = 0 fm p b = Ofm 7=0.50 —-^ ~~~ ^. n Gogny n Gogny[C(Po)) 7=0.75 -— 0 15 \ 1 = 30 tm/o ' 0.1b t = 30 tm/c 7=1.00 & — 20 \ > o" 0 1 \ 10 p( 0 \ O.Of 0 05 0 t- • n -10 0 2 4 f. 8 10 12 z[1m] Gogny z[fm] •20 0 0.05 0.1 0.15 02 0.25 0.3 0.35 0.4 0.45 0.5 :a + °°Ca E/A = 35 MeV a + ^Ca E/A = 35 MeV -•Si p — b = Ofm p b = 0 fm n Gogny n Gogny(C(po)] I = 60 fm/c I = 60 fm/c 7=0.00 7=0.25 7=0.50 :a + "°Ca E/A = 35 MeV E/A = 35 MeV p — b = Ofm p — b = 0fm n Sogny n Gogny|C(po)] I = 90 tm/c t = 90 fm/c Gogny, C(p0) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 p[fm 3] Figure 1: Gogny j] (±) £ z[tm] Gogny[C(p0)] tl (T) fc Figure 2: 35 MeV/nucleon © 60Ca Gogny , Gogny [C(p0)] , Gogny . ^IIB, Fig. 3 T 0.45 MeV, Gogny [C(p0)] 1.18 MeV ^cfc, Fig.4 JC, tkf&zn (iV/Z) £^T. AMD-V 300 fm/c , 7 5 ^ > . Gogny Gogny 6, ttffh 80- JAERI-Conf 99-008 Gogny force Gogny force with C(po) 60, 60 60 Ca + ^ E/A = 35 MeV Ca + Ca E/A = 35 MeV b.= Q...fm... b...= .0...ta... Gogny Gogny[C(p0)] Before Decay ••«•• v^ Before Decay ••*•• After Decay -B- After Decay -B- Figure 3: > (D Gogny force Gogny force with C(po) 60 60 °Ca + »Ca E/A = 35 MeV Ca + Ca E/A = 35 MeV H b = 0fm b = 0fm Gogny Gogny[C(Po)] hi 3 Before Decay Before Decay After Decay H After Decay 10 15 20 10 15 20 z Z 22 60 60 6 0 6 2.0 .. Ca..+. Ca E/A.=.3.5.Me.V...... Ca.*.. .°Ca E/A.= 3.5..Me.Y... b = 0fm b = 0fm Gogny 1.8 1.6 1.4 1 4 1.2 1.2 10 1.0 Before Decay 0.8 0.8 Before Decay After Decay After Decay 0.6 10 15 20 10 15 20 Z Z 60 Figure 4: 35 MeV/nculeon (D Ca (6 = 0) > (T) . - -81 - JAERI-Conf 99-008 . -mzz 35 , 50 MeV/nucleon C H t He i: *^bfc*«, ^0© 35 MeV/nucleon , £^ Gogny [C(po)]tl 2.2 l97Au - 150 MeV/u . AMD-Vttt, Fig. 5 «, Gogny . Gogny , Fig. 6 fc 20 MeV Gogny tJ Gogny [C(p0)] Figure 5: ^CT"^fcD - 150 MeV TCD 197Au + 197Au -82- JAERI-Conf 99-008 Au + Au E = 150 MeV Au + Au E = 150 MeV 120 120 b< 1 fm b< 1 fm 100 • a> 60 !2 60 • LJJ LU m 40 * Exp 2=1) • . Exp(Z=1) • . Exp 2=2) Exp (Z=2) • AMD-V Z=0) o AMD-V (Z=0) o 20 'Gogny AMD-V Z=1) • ' •Gogny[C(Po)] AMD-V (Z=1) • ' AMD-V 2=2) AMD-V (Z=2) o 2 3 A Figure 6: ~- 150 MeV X , AMD-V 3 AMD-V , 35 References [1] M. Colonna, M. Di Toro, G. Fabbri and S. Maccarone, Phys. Rev. C57, 1410 (1998). [2] A. Ono, H. Horiuchi, Toshiki Maruyama and A. Ohnishi, Phys. Rev. Lett. 68, 2898 (1992); A. Ono, H. Horiuchi, Toshiki Maruyama and A. Ohnishi, Prog. Theor. Phys. 87, 1185 (1992). [3] J. Aichelin, Phys. Rep. 202, 233 (1991). [4] A. Ono and H. Horiuchi, Phys. Rev. C53, 2958 (1996). [5] A. Ono, Phys. Rev. C59, 853 (1999). [6] J. Decharge and D. Gogny, Phys. Rev. C21, 1568 (1980). QO JAERI-Conf 99-008 JP0050029 17. 19B Abstruct Clustering structure of neutron dripline nucleus 19B which was predicted theoritically is investigated by studying the fragmentation reaction of 19B. We compare 19B fragmentation with 13B fragmentation in 19B + 14N and13 B + 14N reactions by using antisymmetrized molecular dynamics, where 13B has no clustering feature in its structure. We find that the cluster structure of the 19B nucleus is reflected in its fragmentation as the simultaneous production of He and Li isotopes. Furthermore we investigate the dependence of the cluster decay of 19B on the incident energy, and find that the cluster structure of19 B in its ground state is more reflected in lower incident-energy reactions. § 1. mm B 7A VY~7\±W±^ § 2. JS^^b^^tt*^ (AMD) (AMD) -eii,4) yi-^^^wSiSMa |*) li- (1) , = ( — J exp -v [ r - — J + -z- (3) present address : B*H^^W^/?f %MSMm^^ >?- MV8." V n -84- JAERI-Conf 99-008 is {z} ijZ^^ and c.c, Co- = ^ (4) i* ^ = 0.16 fm~2 fc U fz Gogny #£fflvifco f ^, B T ^ V h - ^ H To = 8.9 MeV -fif, g (6) R,- t R,- t, AMD •;Jga7&*lR0 Atih firv»4 § 3. B {Z} (± and ' (8) ^ < 0, if /i < 0 (9) -85- JAERI-Conf 99-008 -5 • -5 0 5-5 0 5-5 5 -5 0 5 z[fm] 15 17 19 I. B 7 4 V )- --/ft z-x , B, B, BO {W} 3.6 3.4 • Rr.m.s. Of B isotope 3.2 • AMD EXP 3 2.8 2.6 - LJJ 2.4 •X t I 1 \ - - 2.2 - 2 10 12 14 16 18 10 12 14 16 18 2C Mass Number A Mass Number A 2. B 13B, I5B, I7B, 19 B) , B 7^ V h-7' in 2 ii B AMD ij, /g Q AMD #tcx B H 1 f/fU; B 7^ Vh- 4. 19B B r-f v 13B 19B -86- JAERI-Conf 99-008 10" Charge distributions • Charge distributions in coincidence with Li isotopes 13g 14 10u 3 10 t-: 19B + 14 ° 10" V,, - ^ \\ : : E/A = 35 MeV/nucleon \ E/A = 35 MeV/nucleon V 10u 10"' 01 23456789 1 2 3 Atomic Number Z Atomic Number Z 3 3. 13B m) t Li Li 7 4 v Y - fc^mZ> 7 ??> V r- 3 C 13B 14 N at 35 MeV/nucleon 19B ^f>4^E$tLS He t Li 13B 19 B w^v^^ Li 7A v h - ^ 13B , Li H 74 7 h - 40 COU^fi He Li "b HZ t v^ 7 7 y 19 B 0 3 tC tt\ ^f>^ 19B H 747 h- 13B <, <4oTV>J» , drip line 19 B C 7 4 7 h — I9 19B o He t Li ^o ^ 7 x * - ^ci;, 19B iz& 'El 3 <0 B 74 v t- - r% B 74 v h - -87- JAERI-Conf 99-008 104 1.4 Charge distributions Charge distributionin 1.2 - in coincidence with Li isotope J 13 1 10 B + 4N •+ 1 - 19B +1 4N — 0.8 E/A = 100 MeV/nucleon- 1 102 • X ^ 0.6 0.4 - E/A = 100 MeV/nucleon V 0.2 10u 0 12 3 4 5 6 Atomic Number Z Atomic Number Z 4. 13B 4N at 100 MeV/nucleon KfolZtS Li it £ 7 7 7 > y v f>KUi/'<, 0 4 1C 13B 19B (Hit) + 14N at 100 MeV/nucleon BU&KiS !^0) t Li 74 7 h-: — ^)? 35 MeV/nucleon 4t, 13B i5 i t/ 19B #• ?> ^SS; $ tl/c He t Li -*? 100 MeV/nucleon S T*^ < tt h t % h tltt 5 tt 19B + 14N at 25 (flfelft) fci y~35 MeV/nucleon (£0) t Li 7 4 V -^ 25 MeV/nucleon t j& < ^r ^ t, He t Li 7 4 7 h - 7° , 19B C 7 4 7 h ~ N -88- JAERI-Conf 99-008 10° 10" Charge distributions in coincidence with Li isotope E 10 • D 19 14 in 19B + 14N reactions in B + N reactions at 25 MeV/nucleon --<—- at 25 MeV/nucleon -+—\ at 35 MeV/nucleon -•— at 35 MeV/nucleon 101 10° 0 12 3 4 5 6 7 1 2 3 Mass Number A Atomic Number Z 5. 19B + 14N at 25 MeV/nucleon {% § 5. (AMD) 13B io 19B + 14N at 35 MeV/nucleon KfclZ&tf Z, 13B t 19 B 19B t\ 19B ^ 19 B Jcfc Ji 19B Coulomb breakup 19 » 19B ^^yX^ B 1) M. Seya, M. Kohno, and S. Nagata, Prog. Theor. Phys. 65 (1981) 204. 2) Y. Kanada-En'yo and H. Horiuchi, Phys. Rev. C52 (1995) 647. 3) H. Takemoto, H. Horiuchi, and A. Ono, Prog. Theor. Phys. 101 (1999) 101. 4) A. Ono, H. Horiuchi, T. Maruyama, and A. Ohnishi, Prog. Theor. Phys. 87 (1992) 1185. -89- JAERI-Conf 99-008 JP0050030 18. AMD-MFZ m^t Nuclear phase transition studied with AMD-MF Y. Sugawa and H. Horiuchi Department of Physics, Kyoto University, Kyoto 606-01, Japan (April 28, 1999) Liquid-gas phase transition of finite nucleus is studied by means of microscopic reaction theory, AMD-MF. Thermodynamic variables such as temperature and pressure and their relationship to the excitation energy of the system are obtained by calculating the time evolution of hot system in a potential well, We see clearly the existence of three regions in the calculated caloric curve; namely liquid-dominant, plateau and gas regions. The transition of the system from liquid-dominant phase to gas-phase begins with the cracking of hot liquid nucleus and disintegration into fragments. Property of plateau region strongly depends on the pressure of the system at equilibrium. Gas phase is well reproduced by van der Waals equation. I. INTRODUCTION The idea of nuclear phase transition and the possibility of its observation have been one of the major concerns in nuclear physics. Its origin is the similarity of inter-nucleon force to the inter-molecular force, the combination of repulsive and attractive forces. Recently, the experimental result presented by ALADIN collaboration [1] called much interest in this field. The extracted caloric curve consists of three regions, which is thought to indicate the liquid gas phase transition. However, several difficulties in those experiments such as the mass dependence of points in the curve make it difficult to conclude that the caloric curve indicates the occurrence of liquid-gas phase transition or not. One of the difficulties underlying in analysis of this kind of heavy ion collision is in the observation. We can only get the fragment from excited nucleus and we don't know the detailed information of collision dynamics: how the excitation energy of and the temperature of transient colliding system is reflected to the observables. From this point of view, we believe it is important to deal with the heavy ion collision directly within microscopic molecular dynamics framework and to have the detailed knowledge of excited nucleus. At the same time, it is significant to make it clear how a finite nucleus is excited and change its property from Fermi gas model-like liquid to classical gas. We explored the thermodynamical property of finite nuclear system within AMD(antisymmetrized molecular dynam- ics framework). Here we will show the thermodynamical variables and the dependence of liquid gas nuclear phase transition on the volume of the system. II. FRAMEWORK AND CALCULATIONS Here we briefly explain the framework of AMD and AMD-MF. We refer to [2,3] for the detailed information. In AMD, the wave function of the total nuclear system is represented by one Slater determinant, ^-)]. (1) Each nucleon wave function is the multiplication of spatial part that is Gaussian wave packet and spin-isospin part that is constant in time. The time development of the system is obtained by using the time-dependent variational principle. 9{H) (2) '^-wrr dz, d2 . The equation (2) is the equation of motion that determines the time development of the complex vectors Z. It is shown [3] that if we only follow the time evolution of the central value of wave packets Z, thermodynamical -90- JAERI-Conf 99-008 property such as nuclear temperature is not correctly reflected to the kinetic energy of nucleons that are emitted from excited nucleus, Therefore, we apply the momentum fluctuation procedure [3] at the emission. We call this extension AMD-MF(momentum fluctuation). Within AMD-MF, we put 36 nucleons into a container potential which is oo for r > rwau and 0 for r < rwau. As rwau, we take 5,6,...,12 fm and total excitation energies of the system are taken from E*/A = 2MeV to E* jA = 38MeV. For each pair of rwau and E'/A, we calculated up to about 15000 fm/c to calculate the property of equilibrated system. Cluster of the system at each time step is defined from the chain cluster method(nucleon i and j are connected if |Rj - Rj| < 2.5fm.) The heaviest cluster in the system at the time step is referred as 'liquid' nucleus in the followings. Unconnected nucleons are regarded as 'gas' nucleons and the temperature of the system at the moment is defined from their kinetic energies, |T = j^— Yli=i" K*- Pressure is defined from the momentum given to the wall from the nucleons when they hit the container wall. One have to keep it in mind that these physical variables are defined at each time step and they always have fluctuations around the central values. III. CALORIC CURVES In fig.l, we show the caloric curves for the three cases of the radius of the system, rwan = 5,8 and 12. 18 16 ,6 Fermi gas (a=12) 14 .'A 12 Fermi gas (a=8) 10 o a ' 8 • rwa 6 A rWaii =8fm 4 °rwaii=12fm 2 oo0oo 10 15 , 20 25 30 35 40 E/A (MeV) 2 FIG. 1. Caloric curves for the case of rwan = 5,8 and 12. Thick curves are the caloric curve for Fermi-gas model E'/A = T /a with a = 8 and a = 12. Dotted line is that of ideal gas. In this figure, we can see that the liquid-gas phase transition takes place in all case of volumes. In the low excitation energy, caloric curves is similar to that of Fermi gas model expectation for all case of rwa|j. In this energy region, there are one heavy excited nucleus and several gas nucleons in the system. Therefore, the thermodynamic property is ruled by that of heavy nucleus(Fermi gas-like). For much higher energies, the caloric curves follow that of ideal gas. In this energy region, the system is considered to be in gas phase that is described by the freely moving nucleons. There are somewhat flat regions midst of these two phases in all cases of the radius. In this region, temperature doesn't rise considerably so we call this plateau. From these observations, we can say that the system experiences the liquid-gas phase transition in all cases of volumes. When we look at the caloric curves more precisely, we find there are falls of the temperature just at the end of liquid-dominant region, when the excitation energy grows. The reason for this fall should be attributed to the sudden increase of the number of clusters in the system. In fig.2, we show the corresponding change in the number of clusters for the case of rwan = 8fm When the excitation energy is low (E*/A < 9MeV), number of cluster gradually increases. However, at around lOMeV/A, it shows a sudden rise. This excitation energy corresponds to the point where the temperature suddenly falls in the caloric curve. At this energy, system enters the plateau region. -91 - JAERI-Conf 99-008 15 20 25 E*/A[MeV] FIG. 2. Average number of clusters in the system for each point of the caloric curve. The curve experiences sudden increase of the number (see text.) IV. CRACKING—MULTIPLICITY DISTRIBUTIONS This understanding can also be certified from the change of multiplicity distributions(Fig. 3). 3 10 20 30 0.5 D oE»A=3 OE7A=3 n 0.4 AE7A=10 0.4 AE7A=11 OEM=Z2 • E7A=18 ft 0.3 0.3 • 0.2 0.2 • 0.1 0.1 • 0 15 15 = 10 rwall -5fm __ 10 ^wall 8fm 5 — location and ' 5 —~ location and —— width of peaks width of peaks — n 0 10 20 30 10 20 30 Mass number Mass number FIG. 3. Change in the multiplicity distribution in the case of rwa.n = 5and8fm. Multiplicity peaks for the liquid nucleus are fitted by Gaussian(Upper panels.) Diversity of the center and width of the Gaussian are shown in the lower panels. In the upper panel, three typical multiplicity distribution are shown for each case of radius. The peak in the distribution disappears at the excitation energy that corresponds to the end of plateau region The lower panel shows the change of the location of such peaks and the width. In the case of rwan = 8fm, sudden jump we mentioned above is clear. For lower excitation energy, the mass number of liquid cluster gradually decreases up to certain excitation energy. At E*/A = lOMeV, the mass number suddenly decreases. Therefore, the way the system changes from liquid dominant phase to plateau region can be depicted as follows: As the excitation energy rises, liquid nucleus becomes excited and it emits nucleons. At certain energy, liquid nucleus can't hold excitation energy any more and the energy put into the system is only poured into the separation energy of disintegrated clusters and emitted nucleons and their kinetic energies. Therefore, the number of cluster suddenly increases at this excitation energy. We call this feature 'cracking', which is mentioned in [4]. On the other hand, in the case of rwa|| = 5fm, there is no such a jump in the lower panel of Fig.3. The reason for this lies in the radius of the system. In this case, there is not enough room for hot nucleus to disintegrate or emit too much nucleons. Therefore, in this volume (rwan = 5fm), cracking is gradual. -92- JAERI-Conf 99-008 V. PLATEAU REGION—CORRELATION BETWEEN THE HEAVIEST AND THE SECOND HEAVIEST CLUSTER We next show the property of the nuclear phase transition by looking at the favored cluster configuration during the rise of excitation energy. In Fig. 4, the correlation between the heaviest cluster and the second heaviest cluster that exist during the time development at several excitation energies are shown. a hMvtaM duiMr 11. !l =5(m r rwai =8fm wal) =12f m FIG. 4. Correlation between the heaviest cluster and the second heaviest cluster at each excitation energy in the case of fwaii = 5,8andl2 fm. When the system is in the smallest volume (rwan = 5fm in Fig.4), there are various sizes of clusters as the 1st and 2nd heaviest clusters. It means the transitional phase from the liquid-dominant to the gas phase consists of clusters with various sizes. Therefore, the transition is somewhat violent as the boiling water. When the system is in the middle volume (rwan = 8fm in Fig.4), clusters with A > 2 are hardly formed as the second heaviest cluster and the equilibration is established between the heaviest nucleus and the mixture of gas nucleons and deuterons in the plateau region. This description is closer to the naive picture of phase transition where there are only liquid nucleus and gas nucleons in the mixed phase. When the system is in the largest volume (rwaii = 12fm in Fig.4, during the plateau, many a particles are pro- duced in the space outside of the liquid nucleus. In this volume, the plateau temperature is very low (around 1 MeV/nucleon); therefore, the emitted nucleons don't have much kinetic energy and are easily recombined to form heavier cluster in the 'gas' space. From these figures, one finds that the transitional phase between the liquid-dominant phase and gas-phase is not simple mixture of these two, but it gives qualitatively different pathways from liquid phase to gas phase depending on the volume of the system. Our figures are drawn by changing the total excitation energy and keeping the volume constant. The nuclear system that is formed during the nuclear collision has more complex feature, since the volume changes during the time evolution. Therefore, the experimental verification of the nuclear phase transition might require much more delicate treatment. -93- JAERI-Conf 99-008 FIG. 5. Relationship between temperature and pressure. Squares are calculated points at each excitation energy. Thick lines are caloric curves for ideal gas. Thin lines are van der Waals fits. VI. GAS PHASE In Fig.5, the relationship of these temperature and pressure is shown. Each square represents the averaged value of p and T for the entire time development under fixed value of excitation energy and radius. Solid line starting from the origin is the relationship of p and T at the same density in the ideal gas case and the other line is obtained by the best fit of the two parameters, a and 6, in van der Waals equation: (5) where v is volume per nucleon, v = V/A (6) The characteristics common for all the cases of radius are that the pressure can be expressed as a linear function of temperature when the system is in high excitation and that it ceases to follow the line at certain lower temperatures. In the case of large volume, squares corresponding to high temperature are close to both lines of ideal gas and van der Waals gas. For the smaller volume, the gas squares are not close to the ideal gas any more but they are still well fitted by van der Waals equation. When one looks at Table I, the parameter a constantly increases from negative value to positive value when the volume decreases. Since a represents the strength of the attractive part of van der Waals force, the nuclear force of the gas phase works repulsively when the volume is big, and turns to attractive for the smaller volume. Parameter b corresponds to the volume (per gas nucleon) of the effective repulsive core where particles can't go inside. In our calculation, not only the repulsive part of nuclear force but also the momentum fluctuation procedure acts repulsively. It is because succeeding absorption of nucleon and emission of another nucleon by a cluster is equivalent as a reflection collision between cluster and nucleon. TABLE I. Parameters a and b of van der Waals equation that is fitted to the calculated relation between pressure and temperature of the gas phase 7"wall 12 11 10 9 8 7 6 5 a (MeV/fnT*) -172 -109 -51.0 -19.9 25.9 50.4 93.4 102 6 (fin") 5.42 5.75 7.33 5.80 5.68 5.13 5.95 5.17 -94- JAERI-Conf 99-008 In Table I the best-fit values of the parameter b are also shown. Although it looks like the value of b doesn't change so much for different value of rwan, we should not take the values of b for large rwan so seriously. The reason is as follows. For large rwau, v is much larger than the value of b given in Table I. Since b is contained in the van der Waals equation only in the form of (v — b), even if we make the choice of 6 = 0 it fits still for the case of large rwau. On the other hand, for small rwan the values of b given in Table I should be regarded as being determined uniquely. If the value of b is 5.8 fm3, it means that the volume which gas nucleons should avoid amounts to 185.6 fm3, which corresponds to the volume of the nucleus of mass number A — 29 — 37 with normal density. The fact that the gas nucleon obeys van der Waals equation in all case of rwan justifies our definition of temperature, since the kinetic energy part of the internal energy of van der Waals gas is the same as that of ideal gas and is independent on parameters a and b. VII. DISCUSSION AND SUMMARY By means of the microscopic theory AMD-MF, we studied the change in the behavior of nuclear system when its excitation energy and volume are varied. Transitional stage that connects liquid-dominant phase and gaseous phase is shown not to be the mere mixture of the gas nucleus and liquid nucleus but to be complicated compounds of clusters with various sizes. This consequence qualitatively agrees with the result which is predicted in the statistical multi-fragmentation studies [4,5]. The most important point is that in the present study, the Fermi-gas like property of the liquid nucleus, van der Waals like behavior of gas nucleons, the existence of cracking and the property of plateau region are all obtained from microscopic calculations without any assumptions. This feature makes it possible to explore the collision dynamics with AMD. In AMD-MF (and its superset, AMD-V [6]), quantum statistics is automatically taken into account and one can safely compare the calculation with the experimental results. We don't need any assumption on the reaction stage like equilibration, reaction geometry or switching from dynamical to quantum statistical stage as statistical framework and quantum molecular dynamics do. With AMD-MF we will be able to check whether the agreement of the isotopic temperature to other temperatures in this ideal condition. Our study revealed the importance to treat the volume of the equilibrated system explicitly, since the volume deter- mines the property of the plateau region even qualitatively. The temperature of plateau is dependent on the volume and hence so is the way the liquid-dominant system turns into totally gaseous phase. From the observation that the volume is the key parameter to determine the behavior of the transition, we realize the importance of the estimation, in the experimental condition, of the equivalent volume in which the system can be regarded as equilibrated. Each point on experimental caloric curve may correspond to different effective volume and might be a representative of different path of the phase transition. [1] J. Pochodzalla, T. Mohlenkamp, T. Rubehn, A. Schuttauf, A. Worner, E. Zude, M. Begemann-Blaich, T. Blaich, C. Gross, H. Emling, A. Ferrero, G. Imme, I. Iori, G.J. Kunde, W.D. Kunze, V. Lindenstruth, U. Lynen, A. Moroni, W.F.J. Muller, B. Ocker, G. Raciti, H. Sann, C. Schwarz, W. Seidel, V. Serfling, J. Stroth, A. Trzcinski, W. Trautmann, A. Tucholski, G. Verde, B. Zwieglinski, Phys. Rev. Lett. 75, 1040 (1995). [2] A. Ono, H. Horiuchi, T. Maruyama and A. Ohnishi, Prog. Theor. Phys. 87, 1185 (1992). [3] A. Ono and H. Horiuchi, Phys. Rev. C53, 845 (1996), Phys. Rev. C53, 2341 (1996) [4] J. P. Bondorf, R. Donangelo, I. N. Mishustin, C..J. Pethick, H. Schulz, K. Sneppen, Nucl. Phys. A443, 321 (1985). [5] D. H. E. Gross, Z. Xiao-ze, and X. Shu-yan Phys. Rev. Lett. 56, 1544 (1986). [6] A. Ono and H. Horiuchi, Phys. Rev. C53, 2958 (1996). -95- JAERI-Conf 99-008 JP0050031 19. -The analysis of proton induced reactions on light nuclei- Department of Physics, Faculty of Science, y Hokkaido University Sapporo 060, Japan Abstract We investigate quantum fluctuations effects on the fragment mass and angu- lar distributions. We find that the quantum fluctuations enable us to describe dynamical fragmentation and improve the desctiption of the fragment angular distribution in the intermediate energy proton induced reaction on 12C nuclei. Introduction - lif 7-AltSS<7)£ & L 197 238 Z [1, 2, 3, 4, 5]o ftl^Cli, P(11.5GeV)+ Au, U [8, 9], p(12GeV) +197Au [10], p(14.6GeV)+197Au [11], p(28GeV) +197Au,U [12]) ^(D Fig. lCD^fflii, Brookheaven AGS Mf?T% p(14.6GeV)+197Au fltz Z=5 ~ 9 f multiplicity ^?3 \>X±. 238 ttz. Fig. 2(i, P(0.8GeV ~ 11.5GeV)+ U X**>&•&, Ki, lilOGeV^x^;l/^-p+197Au,238UR£.-e|i«IJ^tlTV^ IMF 9 W Dynamics HJ^S* 4 XL ^> - • E-mail: [email protected] , Fax: +81-11-706-4926 -96- JAERI-Conf 99-008 &ZbizX ^ y y - (55MeV,100MeV) 2 Model G, 17, 18] {;*o*< AMD-QL t^^m [18] *) T&.&Ztltz-lfrf-mSlW&n Slater ^# AMD mfrfiMR [13] H, f§L^=S:^tf LXy fc g 7r*y 12 p(45 MeV)+ C ; [18]o «^ 200fm/c 3 Results - m-J-m p(55 MeV,100 MeV)+12C ^ Fig. yv :, o • o A V Error Bar ri*oV^/ ^ LT V> £ o gjO h|^^ AMD+*fcftJW±IflM t/i AMD-QL AMD+IBtiK±lfIM-e S^ X < U AMD AMD-QL tOi , AMD-QL h m^i-X • -97- JAERI-Conf 99-008 [15] p(100 MeV)+"C - y 3 , Fig. 7 Fig. Fig. fc H scale i97 238 P+ Au, u z IMF 4.5 Exp INC + SMM init s = 11.5 GeV =5 0-10 0 MeV 4 E|MF'AMF =3 0-5.0 MeV =3.0-5.0 MeV 3.5 =12-3 0 MeV = 1 2-3.0 MeV [rar y u 3 irbi l J, P a 2.5 T .= 3.0 GeV 2 w $ 3- 1.5 a 1 X3 T3 0 100 200 100 200 = 0.8 GeV 0Lab (de9) Figure 1: £BI:p(14.6GeV)+197Au ^J^ . fltz IMF(Z=5 ~ 9,7V/MF > 3) Figure 2: p+238U H^Tif b fltz :, IMF 48 Sc [9] -98- JAERI-Conf 99-008 10" AMD(55 MeV] AMD (100 MeV) 10° 101 10u if) AMD-QL ( 55 MeV ) AMD-QL (100 MeV) CO o O 10' 10 J _u 2 4 6 8 10 12 2 4 6 8 10 12 Fragment Mass Figure 3: 12C *m Ltz 55MeV,100MeV 35 10 30 AMD+Casc A= 11 AMD+Casc A=7 AMD+Casc A=10 8 AMD+Casc A=6 25 ExpA=11 •• e- AMD+Casc A=9 ExpA=10 1—•«- 6 Exp A=7 20 Exp A=6 Exp A=9 15 4 10 2 5 jo I 35 10 30 AMD-QL+Casc A= 11 a AMD-QL+Casc A=7 P AMD-QL+Casc A=10 8 AMD-QL+Casc A=6 25 ExpA=11 •• Q '• e AMD-QL+Casc A=9 •o ExpA=10 '--«--' 6 Exp A=7 20 Exp A=6 15 4 Exp A=9 10 5 0 -1 -0.5 0.5 -1 -0.5 0 0.5 cos cos ^ Figure 4: 12C Ltz lOOMeV ft Z>7 7 -99- JAERI-Conf 99-008 References [1] S. Furihata, S. Iwai, A. Ono, and H. Horiuchi, JAERI-Conf 95-012, 124. [2] H. Takemoto, H. Horiuchi, and A. Ono, Phys. Rev. C57 (1998), 811. [3] Y. Tosaka, A. Ono, and H. Horiuchi, JAERI-Conf 98-012, 27. [4] K. Niita et al, Phys. Rev. C52 (1995), 2620. [5] S. Chiba et al, Phys. Rev. 54 (1996), 285. [6] T. Maruyama and K. Niita, Prog. Theor. Phys. 97 (1997) 579. [7] C. T. Roche, R. G. Clark, G. J. Mathews, and V. E. Viola, Jr., Phys. Rev. C14 (1976), 410. [8] J. A. Urbon, S. B. Kaufman, D. J. Hederson, and E. P. Steinberg, Phys. Rev. C21 (1980), 1048. [9] D.R. Fortney and N.T. Porile, Phys. Lett. 76B (1978), 553; [10] K.H. Tanaka et al, Nucl. Phys. A583 (1995) 581. [11] W.-c. Hsi et al, Phys. Rev. C58 (1998), R13. [12] L. P. Remsberg and D. G. Perry, Phys. Rev. Lett. 35 (1975), 361; [13] A. Ono, H. Horiuchi, T. Maruyama, and A. Ohnishi, Prog. Theor. Phys. 87 (1992), 1185; Phys. Rev. Lett. 68 (1992), 2898; Phys. Rev. C47 (1993), 2652. [14] F.Puhlhofer, Nucl. Phys. A280 (1977), 267. [15] A. Ono and H. Horiuchi, Phys. Rev. C53 (1996), 2958. [16] A. Ohnishi and J. Randrup, Phys. Rev. Lett.75 (1995), 596. [17] A. Ohnishi and J. Randrup, Ann. Phys. 253 (1997), 279. [18] Y. Hirata, Y. Nara, A. Ohnishi, T. Harada, and J.Randrup, Submitted to Prog. Theor. Phys. -100- JAERI-Conf 99-008 JP0050032 20. Abstract The semiclassical distorted wave (SCDW) model is applied to the calculation of the depolar- izations in iSNi(p,p'x) at 80 MeV, 90Zr(p, p'x) and 90Zr(p, nx) at 160 MeV by one- and two-step processes. The result for 58Ni(p,p'x) is compared with experimental data. The calculated spin flips in (p,p'x) and (p, nx) on 90Zr are analyzed in terms of the effects of in-medium modification of the N — N interaction and the contributions of individual components of the effective interaction. 1 Introduction ^&W&mmi (Semi-Classical Distorted Wave model; SCDW) [1, 2, 3, 4] hVftf jaswBSiE*t«*-cv>s. scDwii DWBA SCDW 2 SCDW ^*ftffl ±ikc(R) s/2 Xc(R ± s/2) £ Xc(R)e , f = i, / (1) s/2 <0-t;K7) + -CS.ffl^«: [AM • JfcfcH3m=-W^#fM&» kc(R) la LSCA WffiH, gfS(:itLt^7.Ul/ 5 #;*fiS! (Local-density Fermi Gas model; LFG) 2 ^i^Il = f h ' fdR| X,(R)| •-'| X<(R)| // dkadkl} x - 8{ki} - ka + k/(R) - k,(R)) ^(^(k'i ~ k'i) + Qali - J) (2) -101- JAERI-Conf 99-008 Qaj3 liEiSo Q . itz mit m{ co fc Dortmans & Amos (DA) [5, 6] <7> G > NN - Tr(TTt) SNN = 11(1 - (4) , a- \z 3 t: mi £ fc^ifc n IS'/E L ^. jfiift [2] ^'£. 80MeV AW^O 58Ni (p,p'x) [7, 8] t [7] [8] -6. ti: (free space If #) t I (in-medium It^) ti T z DNN t •^(I 160MeV 90 Zr (p, (p,nx) 1, (p,nj) hti, tfmti (DA CO G t V'l2 = Vb - T2 + V'T(S) 5I2 + V'TT(S) (5) 102- JAERI-Conf 99-008 160MeV AI+SOMeVftftO 90Zr (p,ni) 5ID^*-6. ft# <*>£*, 41 My S*, 58Ni (p.p'j-) £ 90Zr (p,A) S^ (p.nj:) \Zt$LT*>t\%*ft<3tztZ.h, (p,nx ,S [9, 10, 11] L/HI^ftT.fcb-f, #<7)J0°-ew^|t [12, 13, 14] [1] Y.L. Luo and M. Kawai, Phys. Lptt,. B235, 211 (1990); Phys. Rev. C 43, 2367 (1991). [2] M. Kawai and H.A. Weidenmiiller, Phys. Rev. C 45, 1856 (1992). [3] Y. Watanabe and M. Kawai, Nucl. Phys. A560, 43 (1993). [4] Y. Watanabe et al., Phys. Rev. C 59, (1999), in press. [5] P.J. Dortmans and K. Amos, Phys. Rev. C 49, 1309 (1994). [6] H.V. von Gerainb et al., Phys. Rev. C 44, 73 (1991). [7] H. Sakai et al., J. Phys. 55, 622 (1986). [8] T. Takahashi, Master thesis, University of Kyoto, 1989 (unpublished). [9] H. Sakai et al., J. Phys. 55, 624 (1986). [10] O. Hausser, et al., Phys. Rev. G 43, 230 (1991). [11] J. Lisantti et al., Phys. Rev. C 44, R1233 (1991). [12] II. Sakai et al., Phys. Rev. C 35, 1280 (1987). [13] T. Wakasa et al., Phys. Rev. C 55, 2909 (1997). [14] T. Wakasa et al.. Phys. Lett. B426, 257 (1998). -103- JAERI-Conf 99-008 01 10 0.8 80MeV HNi(p,p') M 80 MeV Ni(p,p') 0.6 Ep'=52.5MeV 10°° r--O r Ep'=62.5MeV " -01 , 1+2STEP 0.4 X! 10 ; ISTEP 2STEP 0.2 -02 | O Sakai et al. O 10 \ l+2step (in-medium) , 58 m 1: Ni (p,p'x), Ep = 80 MeV 0.9 0.9 -. i 0.8 —"— l+2step (free space) 0.8 160MeV Zr(p,n) : 0.7 —a - - K- - j step (free space) 0.7 En=80MeV ) j ,_ —«— l+2step (free space) © 0.6 Istep (in-med.) 0.6 i | --«-. istep (free space) 0.5 at i 0.5 -~.j r l+2step (in-med.) N 0.4 K »-.. ' ' ' zatio n 0.4 . _ ^-*» - I- >steP (in-med.) a •u - m 0.3 it. 0.3 Im "3 0.2 -m ?~ 0.2 t Q 0.1 Depo l 0.1 0 161»MeV "Zr(p,p') 0 1 Ep'=80MeV ^HM^I -0.1 -0.1 -0.2 -0.2 ,,I,,,,!,,,,!,,,,I, 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 Emission angle (deg.) Emission angle (deg.) 2: 160MeV Alt 80MeV 90Zr {p,p'x) (p,nx) ff (p, nx). 0.6 0.6 0.5 0.5 in-medium 0.4 0.4 --*-- c_st a 0.3 0.3 ••«-• C_st-T_t c Istep part £ 0.2 - - -#-'**«..-j..^V't •• - - -i- - - c Istep part e 2step part 0.2 2step part "a Total 'a Total C/3 0.1 V3 0.1 I | 0 0 •••>•••§••• -0.1 -0.1 ! 160MeV MZr(p,n) En=80MeV 160MeV"Zr(p,n) En=80MeV -0.2 -0.2 ,,!,,,,I 0 20 40 60 80 100 120 140 160 20 40 60 80 100 120 140 160 Emission angle (deg.) Emission angle (deg.) El 3 9nZr (p,nx), lOOMeV AW8OM0V X t* free-space itW-, in-medium Htt£- -104- JAERI-Conf 99-008 JP0050033 21. Abstract Superconductivity of cjuark matter at finite temperature is investigated within a mean field (BCS) approximation. The energy gap is calculated at various densities mid tem- peratures and the phase diagram is dniwned. We also discuss the relation between the spin-structure of the Cooper pjiir and the superconductivity. l \tCth\z -C1C1 9 8 Landau -PSbftfc. V0GE= Y ^ (^ BCS L -C 0BCS Siitt (2) - Ho (3) «t (4) -105- JAERI-Conf 99-008 and eli2 = e2>3 = c3,i = *(*) = ( +*'* V (5) m2 - M = e* - iite—%fr(r)=z -t^^H ,Nambu h) — Ek A* \ / h) + £k — Afc \ , 2 2 I l"-'n ~ ^t 2 22 F(k) = G(fc)ll2 = -Ak(k 0 -ee k- 5. S(p) = 0. (9) "CE (p) $r ff #-f Sit r it L rjfiiH+Sl^^^icj ©jffMfe (Higashijima - Miransky iff ^-^f"F(fc)f''D^(p-A;). (10) "5, J; 9 3 4TT /• d fc ^(efce.,, + k • p)A(fc) A(p) = Tj Mq) g ^ft? t # g P^^^^O. ^ g i3W>#£1-fttt*, ^ (Dtewfttf «t A(p) = y ^3tTj]rF(k)ta;B)rD^(p- A,-). (12) -106- JAERI-Conf 99-008 4TT f (13) P < i^ u.-cfes 3. (14) 11 » v 3 6 - lOMeV A = 400MeV vl = 1-5A2. < tezt'bZ < (A;F oc th lc L r i ^, L, m 3. 0 10 20 P (fm'3) -107 JAERI-Conf 99-008 ,T ~ t r 0 ±{91*3 r 1 60 40 • f-°20 - • • 1 1 0 • • • 0 50 100 150 0J; ->J— • l 0 MeV ,T T4 TT 11 -108- JAERI-Conf 99-008 JP0050034 22. iff&JS* *- *&R M% KUNS-1573, April, 1999 Color Superconductivity in Quark Matter* T. HATSUDA Physics Department, Kyoto University, Kyoto 606-8502, Japan E-mail: [email protected] Abstract We have studied color superconductivity (CSC) in high density quark matter with two flavors on the basis of a model hamiltonian with Debye-screened gluon propagator. We found that the most attractive quark interaction of this hamiltonian is in the Jp = 0+ — 0~ channel with color anti-symmetric and flavor anti-symmetric representation. We also found that there is an attraction in the JF = 1+ — 1~ channel although the strength is somewhat weaker. Depending on the strength of as, the magnitude of the gap can be as large as 50-100 MeV. Even at extremely high baryon density pg ~ '20po, the gap still exists with the size of 10-20 MeV. Open problems related to the physics of CSC are also discussed. I. INTRODUCTION At extremely high baryon density (PB), hadronic matter is believed to undergo a phase transition to quark matter where quarks and gluons are deconfined. Simple percolation picture suggests that the critical baryon density pc is somewhere between 3 to 10 times the 3 nuclear matter density p0 = 0.17fm~ . When ps ^> pc, the system can be approximated as a degenerate fermi system with a short-range interaction, because of the color Debye screening and the asymptotic freedom of QCD. This was first pointed out by Collins and Perry [1]. The typical fermi momentum of the quark in u — d quark matter is estimated as 3 2 pF = 460 (580) MeV for pB = 5 (10) p0 through the relation pB = 2p F/3ir = pQ/3 with pQ being the quark number density. Even if the system has relatively weak interaction at high density, there is always an instability of the fermi surface as far as there exists an attractive channel in a quark pair near the fermi surface. This is the celebrated color superconductivity (CSC) in quark mat- ter originally proposed by Bailin and Love [2]. In CSC, the attraction between quarks is *Talk presented at the 1st Workshop on Hadron Science under Extreme Conditions, (JAERI, Tokai, March 11-12, 1999). -109- JAERI-Conf 99-008 brought solely by gluons, while in the standard BCS superconductivity in metals, the attrac- tion is brought as a result of the competition between phonon-attraction and the Coulomb repulsion. Treatment of CSC starting from a microscopic model of QCD has been first discussed by Iwasaki and Iwado [3]. They studied CSC with the Debye-screened gluon propagator together with running coupling constant in Higashijima-Miransky approximation. Although they studied one-flavor case only, their treatment of CSC on the basis of the color Bogoliubov transformation should be considered as the state-of-art calculation. Later, several groups has analyzed the phase structure for two and three flavors using much more simplified 4-fermi models [4]. In this talk, I will closely follow the seminal works by Iwasaki and Iwado and generalize their approach to two-flavors. Details can be seen in the Master thesis of R. Horie [5] who did actual calculations. II. BASIC QUESTIONS AND ANSWERS The Basic questions I would like to raise in my talk are Ql: What are the most attractive channels (MACs) in the q — q pair in color-flavor-spin space ? Q2: How large the CSC gap could be at high density? Q3: Are there any critical density pcrn above which the CSC turns back to the normal fermi system ? The basic tools I will take for studying these questions are the same with Iwasaki and Iwado: namely, the Debye screened gluon exchange with Higashijima-Miransky running coupling, and the color Bogoliubov transformation. The differences are (i) we take color SUC(3) with flavor SUj(2), and (ii) we fully utilize the Fierz transformation to extract the MAC. Let me first present answers to these questions and explain the reasons later. Al: The most attractive channel is the color anti-symmetric, flavor anti-symmetric and Lorentz scalar (Jp = 0+) or pseudo-scalar (Jp = 0~) one. In terms of the order parameter, it reads 6 r (^> = £^6» ((g )-C(l,75)rf>, (1) where (a,/?,7) is the color index, (a, b) is the flavor index and C = i^f2^0 being the charge conjugation operator. In the vector channel, there is also an attraction, but the gap is smaller than the scalar ones. A2: The gap at the fermi surface can be as large as 50-100 MeV due to rather strong attraction by gluons. A3: We did not find any sign of the existence of pcrit. In fact, as far as there is an attraction between quarks, the gap has finite value no matter how the attraction is weak at extremely high baryon density. -110- JAERI-Conf 99-008 III. WHAT ARE THE MOST ATTRACTIVE CHANNELS? The gluonic interaction between quarks and its Fierz transform can be written as l i i a V = -Jd xjd y [q(x)rt q(x)} asD(x,y) [q(y)ft°q(y)i (2) T T [q (y)Tiq(x)] asD(x,y) [q (x)Tiq{y)], (3) where D(x,y) is the screened gluon propagator in the Feynman gauge, and ta = Xa/2 with A" being the color Gell-Mann matrix. F; summarizes all the color-spin-flavor structure after the Fierz transform. After a simple algebra, one can extract the channels where the overall coefficient a; becomes negative as shown in Table 1. Lorentz color flavor at scalar (JF = 0+, C75), AS AS AS + 1 -4/3 +1/2 -2/3 p-scalar (JF = 0~,C), AS AS AS + 1 -4/3 + 1/2 -2/3 vector (JF = l-,(7775), AS AS AS + 1/2 -4/3 +1/2 -1/3 p-vector (J** = l+,Cj), S AS S + 1/2 -4/3 + 1/2 -1/3 Table 1. Fierz coefficients. AS (S) means anti-symmetric (symmetric) combination. For example, AS : ud — du (S : uu, uu + dd, dd) for flavor quantum number. Table 1 shows that MACs corresponding to large |a;| are the scalar and pseudo-scalar channels. The vector and pseudo-vector channels have coefficients 2 times smaller than MACs, thus the gaps in these channels are reduced. Just for completeness, the relation between the Lorentz structure and its non-relativistic reduction are shown in Table 2. Since we are working in almost massless quarks at high density, such non-relativistic reduction does not make much realistic sense. Nevertheless, it would be helpful to get intuition on the spin and angular-momentum structure of the Cooper pair in the non-relativistic limit. channel Lorentz symmetry non-relativistic reduction scalar (O+) anti-sym. 5 = 0 (anti-sym.), L = 0 (sym.), P = + p-scalar (O~) c anti-sym. S = 1 (sym.), L = 1 (anti-sym.), P = — vector (1~) C7V anti-sym. 5 = 1 (sym.), L = 1 (anti-sym.), P = — p-vector (1+) sym. S = 1 (sym.), 1 = 0 (sym.), P = + Table 2. Lorentz structure of the Cooper pair and its non-relativistic reduction. S (L) is the total spin (angular momentum) of the pair, and P is the parity. Ill- JAERI-Conf 99-008 IV. HOW LARGE THE CSC GAP COULD BE ? In the momentum space, the screened gluon propagator in the static limit reads 2 2 2 q + m D q where Pi{Pj) is the projection operator to the longitudinal (transverse) direction. Only the longitudinal/electric part of D^v is Debye screened as is well-known in perturbative theory [7]; 4 2 mD = -as(q )pF. (5) 7T Also we use the running coupling in Higashijima-Miransky approximation [6] following Iwasaki [3] with the infrared (IR) finite form of as: 2 where q = p — k, g^ax = rnax.jp , P}, and qc is a phenomenological IR regulator scale and 2 A corresponds to AQCD- We take A = 400 MeV and q\ = 1.5A following ref. [3]. The gap equation has a general form: ^3k. (7) Here F(p, k) is a kernel which contains the information of the gluon propagator and the running coupling. E(k) is the quark energy relative to its chemical potential. For the color anti-symmetric case, there appears one gapless and two gapful states: (8) Solving the gap equation numerically, one finds Fig.l for the momentum dependence of the gap. The solid (dashed) line corresponds to 0+ — 0~ (1+ — 1~) channel. In the figure, the baryon density (quark fermi momentum) is taken to be 3.5 po (410 MeV). In the 0+ — 0~ channel, the gap is 50-100 MeV for wide range of the quark momentum. Even in the 1+ — 1~ channel, 20-40 MeV is obtained for the gap. One should note, however, that the absolute value of the gap is rather sensitive to the magnitude of as. We believe that the choice of A = 0.4 GeV in [3], which we have also taken here, overestimates the realistic value of as by factor 2 at 1 GeV scale [8]. V. TRANSITION TO NORMAL PHASE AT HIGH DENSITY? Suppose we use a fixed value of as instead of the running coupling. Then, only the intrinsic mass scale in the theory becomes pp, and the dimensionful quantity such as the gap at the fermi surface A(p = pp) should be strictly proportional to pp. Namely, as the density increases, the gap also increases and will never vanish at high pg. On the other -112- JAERI-Conf 99-008 hand, if we use the running coupling which has extra dimensionful parameters such as qc and A, it is by no means trivial whether A(pp) increases or decreases as pp increases. Explicit solution of the gap equation shows, however, that A(PF) decreases very slowly as the density increases, which is plotted in Fig.2 by the solid line. For comparison, the case for fixed as with (without) the Debye screening is shown by the dashed (dash-dotted) line. Because of the asymptotic freedom, the attraction between quarks become weak as the density increases. This is the reason why gap decreases slowly as the density increases. To see whether there is a critical density above which CSC turns back to the normal phase, we have done the calculation up to ps ~ 20/9O and found no evidence of such phase transition (see the right figure in Fig.2). No matter how weak the coupling is, the gap is expected to be finite as far as there exists attractions, and our calculation confirms this. Also our result does not agree to ref. [3] in which the existence of pcrit is claimed. VI. SUMMARY AND FUTURE PROBLEMS We have studied CSC in two flavors starting from a hamiltonian with Debye-screened gluon propagator. This is a direct generalization of the method by Iwasaki and Iwado [3] who studied the case of one-flavor. We found that most attractive channel in this hamiltonian is in Jp = 0+ — 0~ with color anti-symmetric and flavor anti-symmetric representation. Also we found that there is an attraction in Jp = 1+ — 1~ channel although the strength is somewhat weaker. Depending on the strength of as, the magnitude of the gap can be as large as 50-100 MeV. Even at extremely high density PB ~ 20/90, the gap still exists with the size of 10-20 MeV. There are many directions we should explore starting from these results [11]. 1. Although we take massless quarks throughout this study, quarks are known to acquire plasmino mass in the medium [7]. At finite baryon density, it is O(y/a~spF) which can be the same order of magnitude with pF and is not negligible. The calculation taking into account the plasmino mass as well as its dispersion relation should be done. 2. Since as is not really small at the scale pF — 300 — 1000 MeV, the weak coupling treatment a la BCS may not be adequate to study CSC. This can be seen already in our numerical result in Fig.2 where the Fermi surface is completely distorted and cannot be identified from the momentum dependence of the gap. The strong coupling treatment through bosonization may be more appropriate for quantitative study of CSC. 3. Since color symmetry is broken in CSC, there should be the Higgs mechanism which leads to massive gluons. Then, it, in turn, affects the attraction mechanism of the quark pair. Therefore, one needs to develop a self-consistent scheme to treat both quarks and gluons simultaneously to study CSC in a consistent manner. The situation here is quite different from the superconductivity in metals, where the attraction between electrons is induced by phonons, and once Cooper pair is formed and symmetry is broken, the Nambu-Goldstone mode is absorbed by "photon" and not by "phonon". 4. CSC automatically implies the standard electric superconductivity since quarks have electric charges. Therefore, the standard Meissner effect under the magnetic field should take place. For example, if CSC is formed in the core of the neutron star, the magnetic -113- JAERI-Conf 99-008 field is expelled from that region (see, however, the recent paper [9]). More interesting thing may happen when the external magnetic field becomes stronger. Because of the magnetic interaction, quark-spin alignment occurs and the 0+ pairing will be broken. However, the quark attraction for the parallel spin exists e.g. in the vector channels. Therefore, as the strength of the magnetic field is increased, a new phase transition from "0+ CSC" to e.g. "1+ - I" CSC" may occur. 5. For even stronger magnetic field, we need to take into account the dynamical breaking of chiral symmetry together with CSC. In fact, it is known that the magnetic field induces the chiral symmetry breaking due to the dimensional reduction [10]. The interplay of the CSC and the chiral symmetry breaking under strong magnetic field with high chemical potential is an interesting physics to be studied. Also, the recent observation of the the compact stars having super strong magnetic field may have close relevance to this problem. ACKNOWLEDGEMENT Most of the results presented in this talk are based on the work with R. Horie to whom I am grateful. I also thank M. Iwasaki, who is a pioneer of this subject, for fruitful discussions, and to K. Itakura for suggestions and discussions on the physics of CSC as well as CSC under strong magnetic field. -114- JAERI-Conf 99-008 FIGURES 120 100 80 0) 60 Q_ CO en 40 - 20 - 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 momentum(GeV) FIG. 1. CSC gap A(p) as a function of the quark momentum p. The solid (dashed) line corresponds to the 0+ — 0~ (1+ — 1~) channel. The baryon density ps is taken to be 3.5 po which corresponds to pp = 410 MeV. 140 density p(fm'" density p(fm'' FIG. 2. CSC gap on the fermi surface A(PF) as a function of the quark density p = 3pB for 0 < PB < 4/90 (left figure) and for 0 < PB < 20po (right figure). The solid line shows the result with the running coupling. The dashed (dash-dotted) line is a result of a fixed as with (without) the Debye-screening. -115- JAERI-Conf 99-008 REFERENCES [1] J. C. Collins and M. J. Perry, Phys. Rev. Lett. 34 (1975) 1353. [2] D. Bailin and A. Love, Nucl. Phys. B190 (1981) 175; ibid. 751; Phys. Rep. 107 (1984) 325. [3] M. Iwasaki and T. Iwado, Phys. Lett. B350 163; Prog. Theor. Phys. 94 (1995) 1073. M. Iwasaki, Prog. Theor. Phys. 96 (1996) 1043; ibid. 98 (1998) 461; ibid. 101 (1999) 19. [4] R. Rapp, T. Schafer, E. V. Shuryak and M. Velkovsky, Phys. Rev. Lett. 81 (1998) 53. M. Alford, K. Rajagopal and F. Wilczek, Phys. Lett. B422 (1998) 247; Nucl. Phys. 537 (1999) 443. J. Berges and K. Rajagopal, Nucl. Phys. B538 (1999) 215. T. Schafer and F. Wilczek, Phys. Lett. B450 (1999) 325. N. Evans, S. D. H. Hsu and M. Schwetz, Phys. Lett. B449 (1999) 281. K. Langfeld and M. Rho, hep-ph/9811227. D. T. Son, hep-ph/9812287. M. Alford, J. Berges and K. Rajagopal, hep-ph/9903502. [5] R. Horie, On color superconductivity in high density quark matter, Master thesis sub- mitted to Phys. Dep., Kyoto Univ. (Feb., 1999). [6] K. Higashijima, Prog. Theor. Phys. Suppl. 104 (1991) 1. [7] J. H. Kapusta, Finite Temperature Field Theory, (Cambridge Univ. Press, New York, 1989). M. Le Bellac, Thermal Field Theory, (Cambridge Monographs, New York, 1996). [8] Review of Particle Properties, Euor. Phys. J. C3 (1998) 1. [9] D. Blaschke, D. M. Sedrakian and K. M. Shahabasyan, astro-ph/9904395. [10] See e.g., K. G. Klimenko, hep-ph/9809218, hep-ph/9809323. V. P. Gysynin, V. A. Miransky and I. A. Shvkovy, hep-th/9811079. V. A. Miransky, hep-th/9511224. C. N. Leung, hep-th/9806208. [11] H. Abuki, K. Itakura and T. Hatsuda, under investigation. -116- JAERI-Conf 99-008 JP0050035 23. Landau^- vfe^QCD > ^ a. V - v a > £ fflt&tb * iJ - X A The Lattice Landau Gauge QCD Simulation and the Confinement Mechanism Hideo Nakajima1 Department of Information science, Utsunomiya University Sadataka Furui 2 School of Science and Engineering, Teikyo University Abstract Numerical results of the gluon propagator and the ghost propagator in quenched lattice Landau gauge QCD simulation are reported. We observe that the gluon propa- gator is infrared finite and the ghost propagator is infrared divergent. The divergence of the ghost propagator is less singular than k~4. 1 The lattice Landau gauge QCD simulation The Landau gauge fixing, the restriction on the gauge field such that dA = 0, specifies a region of the configuration space of gauge fields. The region such that the Faddeev-Popov determinant — dD^A) is positive is called the Gribov regionfl]. The Gribov region 0 is denned from the variation of the square norm of the gauge field [1, 2, 3, 4] : A\\Ag\\2 = -(dA\t) + (t\ - dV\e)/2 + ••• (1) where g = ee . (2) The configuration is however not unique and, on the lattice, the minimal Landau gauge, i.e. restriction to the fundamental modular region is attempted[6, 7]. In the minimal Landau gauge[2], the fundamental modular region A is defined as the global minimum of ||J43||2 for each gauge orbit (A9) Gribov and Zwanziger[l, 5] showed that the restriction yields the gluon propagator which is free from infrared divergence and speculated that the singularity of the ghost propagator 1 Y' is enhanced from — to —. k2 k4 A solution of the Dyson-Schwinger equation in the Landau gauge[8] suggests that the gluon propagator is infrared finite and the ghost propagator is infrared divergent Dc{k) oc 1 (£2)1+0.92 ' We perform the lattice QCD simulation in /3 = 5 and 5.5 on 83 x 16 lattice in quenched Axi>x approximation, using the new definition of the gauge field : e = Ux U £ 517(3), A£M = — AriM, Tr Ax The restriction to the fundamental modular region was attempted by the smeared gauge fixing method[6] which was applied after the Landau gauge fixing in our particular method. [email protected] 2Speaker at the workshop, e-mail [email protected] -117- JAERI-Conf 99-008 We observed that the smeared gauge transformation appreciably reduces the norm in about 15% of the samples. Although there are about 8% exceptional samples where the norm is increased, the differences in the norms are less than 1% in all the 100 samples. We fixed remaining global gauge such that the zero mode of the gauge field component in the (1,1,1,1) direction is diagonalized and ordered, and observed that the expectation value of the gauge field is small but significantly nonvanishing.[7] The Polyakov loop after the global gauge fixing has the diagonal part expressed as U = eA Aa+j4 Ag. The expectation values of the diagonal components after the global gauge fixing is in the form diag(eia, 1, e~'a). It is a consequence of the diagonalization of the zero-mode of the gauge field which was suggested by the idea of the Abelian projection: i.e. if the U(l) part of the compact group SU(N) or SO(3) is left unbroken, one can minimise the off-diagonal component of the gauge fields and identify the magnetic charge. When the global gauge transformation is not done, the expectation values of {A3), (A8) are consistent to 0. t-direction z-direction diagonal 0.022(23) 0.028(26) 0.102(27) 3 1 A , A* \ -0.003(21) -0.003(23) 0.003(24) -0.019(20) -0.026(25) -0.105(28) Table 1: The diagonal part of the Polyakov loop SU(3) algebra, after the smeared and global gauge fixing. 2 The gluon propagator We measured the connected part of the gluon propagator : D(c(nk)) = I (3) In the case of (3 — 5.5, we show the results without the subtraction of the zero-mode and after the smeared and global gauge fixing and with subtraction. The scale is denned from the one loop renormalization group equation, i _I^ 2 2 2 2 ahiattice — e A>9 ((30g ) "°, where A/a«ice = AJV/5/28.81 for nj = 0. We take AMS = lOOMeV but for the data of (3 = 6[10], we take AMs = 0.128GeV according to their de- termination. Using the parameterisation of [11] based on the truncated Dyson-Schwinger equation, we obtained the complex masses of the gluon as in Table 2. 3 The ghost propagator The ghost propagator is denned as (4) -118- JAERI-Conf 99-008 Figure 1: The gluon propagator without Figure 2: The gluon propagator with global subtraction. (3 = 5.5, 83 x 16. and smeared gauge fixing and subtraction. (3 = 5.5, 83 x 16. size Mi a-VGeV MJGeV ref /? = 5,83 x 16 0.75(15)a"1 0.48 0.36 this work /? = 5.5,83 xl6 0.71(15)a-1 0.84 0.59 this work /? = 6,323 x 64 0.32a"1 1.882 0.60 (Leinweber et al.) Table 2: Complex mass of the gluon. a 1 is defined either from = OAGeV or 0.12SGeV. Fit of the off-axis diagonal propagator data. where a, b specify the color. The Faddeev-Popov operator is M[U] = -(d • D[A]) = -(D[A] • d). (5) X x The ghost propagator M. ^J\ — (Mo — Mi) is calculated perturbatively by using the Green function of the Poisson equation MQ1 = (—d2)'1 whose zero-mode is eliminated. f (6) where Ad is the lattice adjoint operation[7]. The matrix elements between color eigenstates 6 a and |A x0) ab -xo) dx = (7) are measured and obserbed that G°£(k) fitting by (ii) is better than (i) 1 . The simulation data shows that the off-diagonal element is consistent to zero but in the infrared region there are significant fluctuations. -119- JAERI-Conf 99-008 (3 = 5 0 = 5.5 (3 = 5 0 = 5.5 a b a b z K z K A 0.392(3) 1.2026(3) 0.773(9) 1.08(1) A 1.655(2) 0.1184(5) 1.582(5) 0.1293(12) A9 0.333(3) 1.2060(3) 0.683(6) 1.09(1) A3 1.597(2) 0.1045(5) 1.579(4) 0.1289(10) Table 3: Parameters of the ghost propagator using A without global gauge fixing and with global gauge fixing A9. The left table is for the fit option i and the right table is for the fit option ii. Figure 3: The ghost propagator after the Figure 4: The ghost propagator after the 3 3 global gauge fixing Gfg{k) (3 = 5.5, 8 x 16. global gauge nxmg GfJk) (3 = 5.5, 8 x 16. The fitted curve is 1.619/A;2*11137. 4 Discussion and Outlook We observed that at /? = 5 the gluon propagator in the infrared region is suppressed[7]. At (3 = 5.5 the suppression is less significant, but it is infrared finite and the effective mass of the gluon is about 600MeV, consistent to the Mandula and Ogilvie[9], which are manifestations of the confinement. These results are qualitatively the same as in the SU(2) lattice simulation[12]. The ghost propagator is infrared divergent, but its divergence is less singular than k~4. We observe in the infrared region, significant fluctuations in the color space, which can be regarded as another manifestation of the confinement in the Landau gauge. Concerning the expectation value of the gauge field, which we observed in the lattice Landau gauge, there is an interesting conjecture on the quark confinement[13, 14]. The Hamiltonian of QCD describing the interaction of the light u and d quarks with the gauge fields contains the mass term M and the potential V = -iguj^A^u - (8) When (M) is small, (M + V) < 0 can be realized and a pair of u,d quarks from the condensate will convert qq pairs created by a collision into a meson pairs. A lattice simulation is necessary. The color confinement problem in general was extensively analyzed in the BRS formula- tion of the continuum gauge theory[15]. A sufficient condition of the color confinement given -120- JAERI-Conf 99-008 by them is that u% defined by the two-point function of the Fadeev Popov ghost fields c(x),c(y) and Au(y), x c)b(y)\0)dx = (g^ - ^)<(p2) (9) / satisfies ul = —S%. The meaurement of w£ in the lattice Landau gauge is under way. We have encouraging results for the study of the color confinement mechanism. The authors thank Dr. Kikukawa and Dr. Hatsuda for suggestion of checking the Kugo- Ojima criterion, and S.F. thanks organizers of the workshop for stimulating discussion. This work was supported by High Energy Accelerator Research Organization, as KEK Supercomputer Project (Project No.98-34). References [1] V.N. Gribov, Nucl. Phys. B139 (1978) 1. [2] G. Dell'Antonio, D. Zwanziger, Commun. Math. Phys. 138 (1991) 291. [3] T. Maskawa, H. Nakajima, Prog. Theor. Phys. 60 (1978) 1526, Prog. Theor. Phys. 63 (1980) 641. [4] M. A. Semenov-Tyan-Shanskii, V. A. Franke, Zapiski Nauchnykh Seminarov, Leningrad- skage Otdelleniya Mathemat. Instituta im Steklov ANSSSR 120 (1982) 159. [5] D. Zwanziger, Nucl. Phys. B364 (1991) 127, Nucl. Phys. B412 (1994) 657. [6] J.E. Hetrick and P.H. de Forcrand, Nucl. Phys B (Proc. Suppl.)63A-C,(1998) 838. [7] H.Nakajima and S. Furui, Lattice 98 contribution, Bouldar(1998), hep- lat/9809080,9809081; Confinement III proceedings, June 1998, Jefferson Lab, NewPort. hep-lat/ 98090 78 [8] von Smekal, A. Hauck, R. Alkofer, Ann. Phys.267(1998) 1, hep-ph/9707327. [9] J.E. Mandula and M. Ogilvie, Phys. Lett. B185 (1987) 127. [10] D.B. Leinweber, J.I Skullerud, A.G. Williams and C. Parrinello, Phys. Rev.D58 031501, hep-lat/9803015; D.B. Leinweber, J.I Skullerud and A.G. Williams, hep-lat/9811027; [11] M. Stingl, Z. Phys. A353 (1996) 423. [12] A. Cucchieri, Phys. Lett. B422 (1998) 233, hep-lat/9709015. [13] V.N. Gribov, Physica Scripta T15 (1987) 164. [14] K. Cahill and G. Herling, hep-lat/9809149 [15] T. Kugo and I. Ojima, Prog. Theor. Phys. Supp. 66 (1979) 1. -121- JAERI-Conf 99-008 JP0050036 24. Abstract QCD spectral functions of hadrons in the pseudo-scalar and vector channels are extracted from lattice Monte Carlo data of the imaginary time Green's functions. The maximum entropy method works well for this purpose, and the resonance and continuum structures in the spectra are successfully obtained in addition to the ground state peaks. QCD fc QCD .R = a{e+e~ —• hadrons)/a(e+e 'fj,~) oc p(y/s) dN(e+e~production at T) + 1 , r = 0) = /" (1) ,r ^ 0) = (2) (flSH) MEMtt, -122- JAERI-Conf 99-008 (ill-posed Pv[A\DH) = Lattice Data £ I>, X'S* h^Wft* A, 5fclfc«fcfl|*8£ # Lattice 2 *fe C a Covariance Matrix T\ Lattice Pr[A\H] S*fflWt;»flfiTSfcJ&f^ Default Model () (1 Pr [ [1]. ) log T4 = 7/ V f m(u) Z.CD S[A] Hl> hnfcT-iWSn i4(w) © Default Model m(w) Lattice Data*^V^-nttPr[A| £>//•] « Pr ;:£5. BayesO^Stt, Lattice = eQ Q = a5 - L PT[A\DH}= I' Vx[Aa\DH]da= IPr[A\aDH}Pi[a\D H)da ^1 /weflM KWffip (Q[a,i]+ I^VVQ^) (4) ) J L\Ot) \ Z J SA = A — A "C> A \a>-£> (x (C-fel/^X Q %tHck.\Z.~$•£> HO. -123- JAERI-Conf 99-008 h;HI**#*fcJ&fciJ5.S»«I^MfflMH*<0 Lattice Data >l:tt Wilson = 0.429GCV-1 O(x) = d(xbs«(y) (5) O(x) = d(x)7Mu(») (6) MEM > Lattice Data© {r|a < T < 12a} (7)12 Lattice Data »3 MEM C. MEM*«#S»"r*-6 7 (Mock Data) tr^fLT MEM C x x r) J ^Tdu: , Hf^CO Lattice Data i^^, Lattice Data \Z. MEM , (0 2)0«t3C&ofc. Default ModellC 2 ilttt, P(LJ) = m0LJ QCD ^ o;2 fc Jt09f S r m tt#^ QCD H fcfrfcofc (0 3). : t*, -124- JAERI-Conf 99-008 2 4 0) [GeV] 0.001, a < r < 12a) „ : MEM (C = 0.0001,a < T < 24a),, Aa> = lOMeVo Mock Data [1] See e.g., J. Skilling, in Maximum Entropy and Bayesian Methods, ed. J. Skilling (Kluwer, London, 1989), pp.45-52; S. F. Gull, ibid, pp.53-71. [2] See the review, M. Jarrell and J. E. Gubematis, Phys. Rep. 269, 133 (1996). -125- JAERI-Conf 99-008 75 (a) PS channel : 50 Q. 25 0 30 (b) V channel i Q. 10 0 0 4 co [GeV] m 2: (a) tei^QCD tlSttSX^ hJPBiS: (Pseudo Scalar Channel)» p(u) = A(u>)/u2o H* m = 0.0834(GeV), Jftijl : m = 0.141, ^i^ : m = 0.215» Aw = lOMeV, TO0 = 2.0» 2 (b) &T QCD tI*5tt^X^^7 h;H|» (Vector Channel),, p(w) = A(w)/u 0 W$k : m 0.0834(GeV), ^i^ : m = 0.141, ^H : m = 0.215o Aw = lOMeV, m0 = 0.86o 2 4 co [GeV] 3: Wit Vector Channel, «; = 0.1557 (l -126- JAERI-Conf 99-008 JP0050037 25. * Abstract We propose a microscopic simulation for quark many-body system based on a molecular dynam- ics. Using confinement potential, one-gluon exchange potential and meson exchange potentials, we can construct color-singlet nucleons, nuclei and also an infinite nuclear/quark matter. Statistical feature and the dynamical change between confinement and deconfinement phases are studied with this molecular dynamics simulation. [2] irt% , FMD[5] L * ~ 1 o -127 JAERI-Conf 99-008 = M')Xi, (2) = exp[-(r-R,)2/2L-z-P,r], (3) cos a,- e~*P' cos ft Xi = | sin a,-e+''^ cos ft \ . (4) sin 6>, eiVi U x.-li*7- SU(3) Pi, a,-, Pi, 9i, ifi] £ = (*|iftl-^|*) (5) .-R,- + hpi cos 2a,- cos2 0, - ft^i sin2 ^] - H (7) M A = 2hsin 2a,- cos20,-5^' 2^sin^in^,-- cos 6i6i dtpi' 1 g^ cos 2a,- 2 ^ sin 2a,- cos2 0,- 3/?,- 2ftsin2ai cos2 ft dtpi' ' ^ 1 dH cos 2a,- dH ~ 2^ sin ft cos ft d6i + 2^sin 2a,- cos2 ft <9a~' 5^ fi), (16) e Ks(r) = -<*s -^-, (17) VL(r) = Lir, (18) VM(r) = Cff—j—+ CW-^—, (19) Aa = Gell-Mann matrices. (20) -128- JAERI-Conf 99-008 ttz, VL(r) y Y t7%Atl&o ^ < )V h -7 >fH 1 -^h it, 1 spurious 2 1 o m= 320 MeV, L = 0.25 fm , as = 1.0, p = 0.5 [far ], = 2800 [MeV/fm], rcut = 3 [fm], = -8.94/9, (ia = 550 [MeV], Cw = 24.5/9, //w = 782 [MeV]. (3 7 -iftt (White) - h White (21) -* t White t^ f |Ri-Rj| < ^cluster (»,> = 1,2,3), 3 (22) < I. -129- JAERI-Conf 99-008 .. ,, . 1 ... 1 ... 0.78 p0 . 0.8 \ 1.17 p0- 1.68 p0 : ; 2.28 po . M A lift •Q 0.6 : \\ —• 4.11 p0 - - 0.4 • \ \\ 0.2 n n 200 400 600 200 400 600 - [MeV/q] - [MeV/q] 1. 0 2. h boost -v3 -130- JAERI-Conf 99-008 A=66(q=198) T ' ' ' 1 • • • 1 ' ' •" rmsr=3.08 fm Eblnd=15MeV/q (Without Fermi motion) 0 10 20 30 40 [fm/c] H3. 66 -:? 198$^ (A=66)+(A=66) (A=66)+(A=66) PCM=0.842 10 t =0.0 fni/< 10 t -ttO fm/c 1.5 fif fe * 0 r 10 t £ • • t T 0 -10 -in 10 o-#" P=28.0fm/c 10 t =28.0 trh/c Lt -10 ••/?: -10 -10 10 -10 0 10 h 0 5. . fih !§J -131- JAERI-Conf 99-008 * - [1] J. Harris and B. Miiller, Ann. Rev. Nucl. Part. Sci. 46 (1996) 71; W. Greiner and D. Rischke, Phys. Rep. 264 (1996) 183; C. Y. Wong, Introduction to High-Energy Heavy-Ion Physics, World Scientific Publ. Singapore, 1994. [2] G. Baym, Nucl. Phys. A590 (1995) 233. [3] J. Aichelin and H. Stocker, Phys. Lett. B 176 (1986) 14; J. Aichelin, Phys. Rep. 202 (1991) 233; D. H. Boal and J. N. Glosli, Phys. Rev. C 38 (1988) 1870; T. Maruyama, A. Ohnishi and H. Horiuchi, Phys. Rev. C42 (1990), 386. [4] A. Ono, H. Horiuchi, T. Maruyama and A. Ohnishi, Phys. Rev. Lett 68 (1992) 2898, Prog. Theor. Phys. 87 (1992) 1185, Phys. Rev. C 47 (1993) 2652. [5] H. Feldmeier, Nucl. Phys. A515 (1990) 147. [6] T. Maruyama, K. Niita, K. Oyamatsu, T. Maruyama, S. Chiba and A. Iwamoto, Phys. Rev. C57 (1998) 655. [7] T. Hatsuda, Prog. Theor. Phys. 70 (1983) 1685. [8] K. Saito, K. Tsushima and A.W. Thomas, nucl-th/9901084; and references therein. -132- JP0050038 JAERI-Conf 99-008 26. 3Effl £&#>*B^ttj£"C<7);< v Mesons Above The Deconfining Transition QCD-TARO Collaboration Ph. de Forcrand1, M. Garcia Perez2, T. Hashimoto3, S. Hioki4, H. Matsufuru5'6, 0. Miyamura6, A. Nakamura7, I.-O. Stamatescu5'8, T. Takaishi9 and T. Umeda6 XSCSC, ETH-Zurich, CH-8092 Zurich, Switzerland 2Dept. Fisica Teorica, Universidad Autonoma de Madrid, E-28049 Madrid, Spain ^Department of Applied Physics, Faculty of Engineering, Fukui University, Fukui 910-8507, Japan 4Department of Physics, Tezukayama University, Nara 631-8501, Japan 5 Institut fur Theoretische Physik, Univ. Heidelberg D-69120 Heidelberg, Germany 6Department of Physics, Hiroshima University, Higashi-Hiroshima 739-8526, Japan 7Res. Inst. for Information Science and Education, Hiroshima University, Higashi-Hiroshima 739-8521, Japan 8FEST, Schmeilweg 5, D-69118 Heidelberg, Germany 9Hiroshima University of Economics, Hiroshima 731-01, Japan Abstract: We analyze temporal and spatial meson leads to systematic errors [3]. Moreover, having the correlators in quenched lattice QCD at T > 0. Above t-propagators at only a few points makes it difficult Tc we find different masses and (spatial) "screening to characterize the unknown structure in the corre- masses", signals of plasma formation, and indication sponding channels. To obtain a fine ^-discretization of persisting "mesonic" excitations. and thus detailed t-correlators, while avoiding pro- hibitively large lattices (we need large spatial size With increasing temperature we expect the physical to avoid finite size effects, typically la ~ 3/T), we picture of QCD to change according to a phase tran- use different lattice spacings in space and in time, sition where chiral symmetry restoration and decon- aa/aT = £ > 1. For this we employ anisotropic Yang- finement may simultaneously occur. For model inde- Mills and fermionic actions [4]: pendent non-perturbative results one attempts lat- tice Monte Carlo studies [1]. Since in the Euclidean ( 7 0 1G (see, e.g., [2]), physics appears different, depending = -^-9W9, — = 2(7710 , (2) on whether we probe the space ("a": x) or time ("T": t) direction: the string tension, e.g., measured "h.c." ) (3) from -133- JAERI-Conf 99-008 ture of the hadrons (see, e.g. [5] for reviews). Two the above lattices T ~ 0,0.93Tc, 1.15TC and 1.5TC 1 "extreme" pictures are frequently used for the inter- and aT ~ 0.044 fm = (4.5 GeV)"" . We calculate cor- mediate and the high T regimes, respectively: the relators for 3 quark masses, with f ~ 5.3(1). The weakly interacting meson gas, where we expect the various parameters are given in the Table. Details mesons to become effective resonance modes with a small mass shift and width due to the interac- tion; and the quark-gluon plasma (QGP), where the mp, TTly mesons should eventually disappear and at very high 0.081 4.05 0.178(1) 0.196(1) 0.379 1.289 T perturbative effects should dominate. These gen- 0.084 3.89 0.149(1) 0.175(1) 0.380 1.277 uine temperature effects should be reflected in the 0.086 3.78 0.134(1) 0.164(1) 0.380 1.263 low energy structure in the mesonic channels. But this structure cannot be observed directly, due to the inherently coarse energy resolution 1//T = T. Our Table 1: Simulation parameters used at every T and strategy is the following: we fix at T = 0 a mesonic meson masses at T ~ 0 (in units a"1). The source source which gives a large (almost 100%) projection parameters a, p eq. (5) are extracted from the T ~ 0 onto the ground state. Then we use this source to wave function. determine the changes induced by the temperature on the ground state. For T > Tc we do not have a will be given in a forthcoming paper [7]. We use pe- good justification to use that source as representa- riodic (anti-periodic) boundary conditions in the spa- tive of the meson but we assume that it still projects tial (temporal) directions and gauge-fix to Coulomb onto the dominant low energy structure in the spec- gauge. We investigate correlators of the form: tral function. This is a reasonable procedure if the mesons interact weakly with other hadron-like modes in the thermal bath and the changes in the correlators are small. Large changes will signal the breakdown t (Tr [S( ,0;z, t) M755 (y ,0;z + x, OTSTM]) (4) of this weakly interacting gas picture and there we yi 7 2 S p shall try to compare our observations with the QGP «>i,2(y) ~ (y) {point); tuii2(y) ~ exp(-ay ) (exp.) (5) picture. with On a periodic lattice the contribution of a pole in 5 the quark propagator and JM = {75,7ii liTiTs} the mesonic spectral function to the t-propagator is for M = {Ps, V, S, A} (pseudoscalar, vector, scalar cosh(m(t — NT/2)) (this m is therefore called "pole- and axial-vector, respectively). We use point and mass"). A broad structure or the admixture of ex- smeared (exponential) quark sources and point sink. cited states leads to a superposition of such terms. We fix the exponential (exp) source taking the param- Fitting a given t-propagator by cosh(m(t)(t- NT/2)) eters a, pin (5) from the observed dependence on x of at pairs of points t, t + 1 defines an "effective mass" the temporal Ps correlator with point-point source at m(t) which is constant if the spectral function has T ~ 0 (see Table). The results of a variational anal- only one, narrow peak. We shall simply speak about ysis using point-point, point-exp and exp-exp sources m(t) as "mass": it connects directly to the (pole) indicate that the latter ansatz projects practically en- mass of the mesons below Tc, while above Tc it will tirely onto the ground state at T = 0. This is well help analyze the dominant low energy structure in the seen from the effective mass in Fig. 1. frame of our strategy above. By contrast, we shall Therefore we use throughout the exp-exp sources speak of "screening mass" (m(a)) when extracted a in the Table, according to our strategy for defining from spatial propagators. mS > is different from the hadron operators at high T [8]. All masses are given T > 0 mass (the propagation in the space directions in units a"1, i.e. we plot mass X a,-. Errors are represents a T = 0 problem with finite size effects). statistical only. 3 We use lattices of 12 X NT with NT = 72,20,16 a) Effective masses. In Fig. 2 we show the effective and 12 at /3 = 5.68, yG = 4 , no.conf = 60. We mass m(t) of the Ps and V time-propagators at T ~ find [6] Tc at NT slightly above 18, which fixes for 0.93Tc, 1.15TC and 1.5TC. -134- JAERI-Conf 99-008 0.30 K,= 0.086 o point-point i point-exp 1.0 0.25 o exp-exp 0.8 c $ 0.20 .2 | 0.6 ve f I 5 0.4 0.15 0 6 12 18 24 Figure 1: Effective pseudoscfeiar mass m(t) in units of a"1 v0.1s 0t, for various sources at T ^ 0 . 0.24 Figure 3: Ps wave functions (Set-B, Ka = 0.086, exp- N t=20 N_t=16 N_t=12 0.22 KO =0.086 exp source) normalized at r = 0 vs quark separation r O Ps at t = 2,4 and 6, using full and free propagators. Also 0.20 A V * a s •-• Ps(free) plotted is the initial distribution of separations as is *•* V (free) E 0.18 given by the source, / (Pyw(y + r)w(y). T ~ 0.93Tc at *^as amis (left) and T ~ 1.5TC (right). | 0.16 0.14 O" ^ffl QI 01 ID 35 0.12 tions normalized at x = 0, Gp,(x,t)/GpB(0,t), at several t for T ~ 0.93Tc (which is very similar to 0.10 2 4 6 2 4 T ~ 0) and T ~ 1.5TC at our lightest quark mass {na = 0.086) and for the free quark case {mqav — 0.1, Figure 2: From left to right, effective mass m(t) in 7F = * = 5.3). units of a.-1 at T ~ 0.93, 1.15 and 1.5T (open sym- C Our exp-exp source appears somewhat too broad bols) vs t. Also shown are the effective masses from at T ~ 0.93T : the quarks go nearer each other while the same correlators calculated using free quarks. c propagating in t. Interestingly enough, this is also Ka - 0.086. the case at T ~ 1.5TC: the spatial distribution shrinks and stabilizes, indicating that even at this high tem- We notice practically no change from T = 0 (Fig. perature there is a tendency for quark and anti-quark 1) to 0.93rc, while above Tc clear changes develop: to stay together. This is in clear contrast to the free m(t) depends strongly on t, it increases significantly, quark case which never shows such a behaviour re- and the Ps and V reverse their positions. Because gardless of the source (in Fig. 3 we use the exp-exp of the large changes at high T we compare here with source and mo = 0.1; for heavy free quarks we ex- the unbound quark picture of the high T regime of pect a "wave function" similar to the source at all t). QGP. For this we calculate mesonic correlators us- Hence the only effect of the temperature on the wave ing, with the same source, free quark propagators 5o function is to make it slightly broader. The same instead of 5 in (4), with jjr — f = 5.3, illustratively, holds also for the other mesons at all quark masses. mo — m^o- = mqaT£ = 0.1. Generally above Tc, c) Masses and screening masses. We fit the Ps, V, m{t) shows stronger i-dependence, which means that S and A time-correlators to single hyperbolic func- the spectral function selected by the exp-exp source tions at the largest 3 t-values [9]. Above Tc we assume no longer has just one, narrow contribution, as for that these masses characterize the dominant low en- T -135 JAERI-Conf 99-008 in the mesonic channels above Tc, which would also 0.4 KO =0.086 | O mPs explain the variation of m(t). On the other hand, A m * v the behaviour of the wave functions obtained from 0.3 • D ms CO SLLJ the 4-point correlators suggests that there can be low n a V m A energy excitations in the mesonic channels above Tc, \ 0.2 • m /^ A P6 remnants of the mesons below Tc. They would be 0.1 characterized by a mass giving the location of the corresponding bump in the spectral function. The 0.0 variation of m(t) with t and with the source would then indicate a resonance width increasing with T, 0.4 - ***. g O mPs A mv although it may simply reflect the uncertainty in our treatment of the low energy states [14]. Remember 0.3 M D ms CO 2 CO that our source is not chosen arbitrarily but such as a) m a E 0.2 • m l°V^ " to reproduce a "pure" meson source at T = 0; at high s T, however, it may allow admixture of other contri- v 0.1 butions, and m becomes increasingly ambiguous. We m 0.0 • .5 . • ...... see chiral symmetry restoration above Tc both in the 0.00 0.02 0.04 0.06 0.08 0.10 0.12 masses and in the screening masses, with the latter 1/N t increasing faster than the former and remaining be- low the free gas limit at T ~ 1.5TC. The exact amount Figure 4: Temperature dependence of the mass m of splitting among the channels and the precise ratio (open symbols) and screening mass m^ (full sym- between m and tnt"' may, however, be affected also 1 bols) in units a" , Kg. = 0.086 (upper plot) and by uncertainties in our £ calibration. Finally, note in the chiral limit (lower plot). The vertical gray that this is a quenched simulation, with incomplete lines indicate Tc. The data correspond to T ~ dynamics. 0, 0.93TC) 1.15TC and 1.5TC. A possible physical picture is this: Mesonic ex- citations subsist above Tc (up to at least 1.5TC) as source for all T (a gauge invariant extended source unstable modes (resonances), in interaction with un- leads to similar results). The results for m and m^ bound quarks and gluons. Our results are consis- at KO = 0.086 are shown in Fig. 4 (m, and to a smaller tent with this, but there may be also other pos- extent also m^ may be overestimated). sibilities (cf [15, 16], cf [5] and references therein). We extrapolate m and mS"' in l//c = 2TUQ + 8 to E.g., in a study of meson propagators including dy- the chiral limit from the 3 quark masses analyzed [10]. namical quarks but without wavefunction informa- Up to Tc screening masses and masses remain similar, tion [16], one found masses and (spatial) screening masses oc T above T and indication for QGP with while above Tc the former become much larger than c the latter, both at finite quark mass and in the chiral "deconfined, but strongly interacting quarks and glu- limit - see Fig. 4 [11]. In the high T regime of QGP ons". The complex, non-perturbative structure of a 1 QGP (already indicated by equation of state stud- one expects m\ > ~ 2TT/NT ~ 0.4(0.5) in units of a" ies up to far above Tc [17]) is also confirmed by our for T ~ 1.15TC(1.5TC), respectively, to be compared with the values in Fig. 4 of ~ 0.3(0.4). analysis of general mesonic correlators. From our In conclusion, in this quenched QCD analysis the study however, the detailed low energy structure of changes of the meson properties with temperature the mesonic channels appears to present further in- teresting, yet unsolved aspects. appear to be small below Tc, while above Tc they become important and rapid, but not abrupt. Here Further work is needed to remove the uncertainties we observe apparently opposing features: On the one still affecting our analysis. This concerns particularly hand, the behaviour of the propagators, in particu- the £ calibration and the question of the definition of lar the change in the ordering of the mass splittings hadron operators at high T, which appear to have could be accounted for by free quark propagation [13] been the major deficiencies, besides the smaller lat- -136- JAERI-Conf 99-008 tices, affecting earlier results [18]. We shall also try to J. Berges, D.U. Jungnickel and C. Wetterich, hep- extract information directly about the spectral func- ph/9705474. tions [19]. [12] G. Boyd, Sourendu Gupta and F. Karsch, Nucl. Phys. B385 (1992) 481. Acknowledgments: We thank JSPS, DFG and the European Network "Finite Temperature Phase Tran- [13] We mean 5b with some effective mq induced by the in- sitions in Particle Physics" for support. H.M. thanks teraction. The larger m(t) in Pa channel as compared T. Kunihiro and I.O.S. thanks F. Karsch for inter- with V is a peculiarity of the free Wilson quarks (3). esting discussions. We thank F. Karsch for reading Its counterpart for staggered fermions is an oscillat- ing behaviour, quoted in [16] also as indicator of free the manuscript. We are indebted to two anonymous quark propagation. referees for useful comments. The calculations have been done on AP1000 at Fujitsu Parallel Comp. Res. [14] We remark that using free quark propagators the Facilities and Intel Paragon at INSAM, Hiroshima source dependence is much stronger. Univ. [15] C. DeTar and J. Kogut, Phys. Rev. D36 (1987) 2828; C. DeTar, Phys. Rev. D37 (1988) 2328. References [16] G. Boyd, Sourendu Gupta, F. Karsch, E. Laermann, Z. f. Physik C 64 (1994) 331. [1] For a recent review see: E. Laermann, Nucl. Phys. B (Proc.SuppI.) 63A-C (1998) 114. [17] G. Boyd et al., Nucl. Phys. B 469 (1996) 419. [2] N.P. Landsman and Ch.G. van Weert, Phys. Rep. 145 [18] T. Hashimoto, A. Nakamura and I.-O. Stamatescu, (1987) 141. Nucl. Phys. B400 (1993) 267. [3] J. Engels, F. Karsch and H. Satz, Nucl. Phys. [19] QCD-TARO: Ph. de Forcrand et al., Nucl. Phys. B B205[FS5] (1982) 239. (Proc.SuppI.) 83 (1998) 460. [4] F. Karsch, Nucl. Phys. B205[FS5] (1982) 285. [5] H. Meyer-Ortmanns, Rev. Mod. Phys. 68 (1996) 473; H. Satz, hep-ph/9706342. [6] From the peak of the Polyakov loop susceptibility; see QCD-TARO: M. Fujisaki et al., Nucl. Phys. B (Proc.SuppI.) 53 (1997) 426. [7] QCD-TARO: Ph. de Forcrand et al., in preparation. [8] Source dependence has various origins. At fixed physi- cal distance, say, taT — 5oT, we find a variation in m(t) between point-exp, exp-exp and wall source of ~ 20% below Tc and ~ 25% above Tc. This source depen- dence disappears exponentially with t at low T, see Fig. 1. By choosing an optimal source, we try to min- imize that part of the m(t) variation which is present at T = 0. [9] For the 5-meson we only use the connected correlator. [10] We assume that the P$ extrapolates in mass squared for T < Tc. Above Tc the /c dependence is very weak. See [7]. [11] A similar behaviour is obtained in T. Hatsuda and T. Kunihiro, Phys. Rep. 247 (1994) 221 from an NJL effective model; an increasing mK is also obtained in -137- JP0050039 JAERI-Conf 99-008 27. 9 * - ? ffl tf^ A colour superconducting phase is expected to occer in a high density quark matter. A general effective potential for this phase is built up to the fourth order of the order parameter on the basis of colour symmetry invariance . The stability conditions for three typical charactaristic phases and symmetry breaking patterns in the colour symmetry are discussed. T V efef it11h 7 1.1 i: M lt-H8L\Z&mWce$> 0 > 3 tf 3 -^W £ M = UXV^e*4 (1.1) 0 0 \ U, V 1.2 M -^ GMG -138- JAERI-Conf 99-008 Vi = TrM*M = TrU*XVlUXV1 = TrWXW*X, (W = V*U) (1.2) 2 V2 = TrM*M = TrX (1.3) 2 V3 = Tr(M*M) = TrWXW*XWXW*X (1.4) f 2 4 VA = Tr(M M) = TrX (1.5) t 3 V5 = 7VM MM*M = TrW*XVKX (1.6) 1 l :i VQ = TrM^MM^M = TrW^XW X (1.7) W = TrMfMM*Ml = TrW*X2WtX2 (1.8) 2 2 K8 = [Tr Af*Af] = (7'rWXW*X) (1.9) K, = [TrAftAf]2 = (TrX2)2 (1.10) 10 = (TrM*M)(TVMtM) = (Tr^2)(rrWXVK*X) (1.11) Veff = miVi + 772216 + W3V3 + 7724 V4 - V6) + m7l/7 + ?7i8V8 + m9V9 + mioVw (1.12) f fc, (6J|Jtp[) (S3M^) w2o<7)ffi^^^wii:^3©6{;5WST-§*o io fiflv fr ve!I = veff{w,x) , ft/Mi (§/Mi) w ' -139- JAERI-Conf 99-008 2.1 &T\ ftfflti* Mo VM )V Veff (DMHi xux2,x3 £ W eSU(3) = v*wxviei+ v \t tfr^i/vfr Veff = Veff(W,X) V = iz ws0 t xs Wo + 6W = W0(l (2.1) = (xl0 hj i = 1,2,3 (2.2) a A : 1^, zero-mode(-g-tf Goldstone-mode) t 2.2 , l/e M e// (±mi (rn-s + m\ ± 2m5 + 7717) (xf + x\ + x\) 2 2 2 (m$ + mg ± mw)(x + x , + xj) (2.3) 2.3 M sQ Mo Wo /I 0 0\ 'Aa 0 0\ I As 0 0 \ Ms0 = 0 10 0 0 0 As 0 (2.4) Vo 0 1) 0 0 Aa) V 0 Q As) -(mi +m ) 2 (2.5) 2(7713 + 7«4 + 2m5 -f 7777) + 6(70-8 + ing 4- v -v m )2 2 (2.6) 4(7713 + "14 + 2m5 + 7777 + 3(mg + 7719 a at G(€ S£/c(3)) \Zft LT Mo (±> Mo + i(yaA M0 + MoyaA ) a a< yaA M0 + Mo2/aA = 0 ^fj^ -140- JAERI-Conf 99-008 = Ma ; O = o li&KK/h 50(3) W suc(s) - 50(3) m10 (6) t 2.4 KMWJ: Ma0 Mo ; 0 0\ (Aa 0 0\ (° -Aa o^ Ma0 = Wa{ 1 0 0 A 0 = Aa 0 0 (2.7) 0 a 0 0 1/ \o 0 Vo 0 -(-mi +m2) Aa = (2.8) 4(m8 > v Vc//(Ma0) = (2.9) , Ma0 = , SUC(Z) -» 5£/(2) &ft (3) 2.5 ti t^; '0 0 -1\ 0 0\ 0 0 0\ 0 1 0 0 0 0 0 0 0 (2.10) 10 0 / 0 0 0/ Aas 0 0 t Vas0 V 2(m4+mfl)' 4(m4+m9) ^, 5t/c(3) (3 © 6) 141- JAERI-Conf 99-008 2.6 Veff(Mo) Veff{M0) (6) t (3) WfflH$t:-(± Ve//(Ma0) > Ve//(M,0)> (3 0 6) t (6) "Cfi Veff(Ms0) > Veff{Mas0), (306) t (3) "Cfi Veff(Ma0) > Veff(Mas0) t^i& 3 • tttb e ^ff^^lo < l = M a/3 ia2ai, ox -J^ 5t/c(3) ® 50(3) ^'^ Wg^B'Sriaife^fe^-t* - t > *^!»^Sr t tiif ?>- ^t X7)3fA^Higgs [1] R.Rapp,T.Schafer,F,Shuryak and M.Velkovsky, Phys.Rev.Lett.81(1998)53. [2] M.Iwasaki and T.Iwado, Physics Letters B 350(1995)163. [3] M.Alford,K.R.ajagopal and F.Wilczek, Physics Letter B422(1998)247. [4] L.P.Gor'kov, Sov.Phys.-JETP,9(1959)1364. -142- JAERI-Conf 99-008 JP0050040 28. There are various approaches to nonequilibrium system. We use the projection operator method investigated by F.Shibata and N.Hashitsume on the linear sigma model at finite temperature and density. We derive a differential equation of the time evolution for the order parameter and pion number density in chiral phase transition. QCD(Quamtum Chro- modynamics) QCD Itii y -i y . RHIC tivistic Heavy Ion Collider) tdf tiftl-5MIS£ QCD [1] v [4] i-t- , v (c70 -> v + V + gov/3]i/j ~ +(M2 h + (ml + ^V) - ) - M2]a2 + \[{ml (1) 2 m0 m o u (i condensate *), = ,/M2 143- JAERI-Conf 99-008 \z , = e^PiLO(0) + e'Lt / at Jo (2) Liouville $&=?••?&Zo «D«3ttSg (t = 0) fcftSt SafffiSf*^J± Pto< = Ps ® PE TrE[pEA] 7^ - if- t ffig "t S g ftffi (System) £> B ffi^ 5 )V Y =. 7 >&%. Us t W% (Environment) ^+M^], (3) +Nonlinear Terms, (5) , (6) Hoop co^4 (0 1 W^4) co**#itL/>:*^i:^oTv^o = P... = TrB[pB...],pfi = exp[-(3(HE - »NE)\ T* »J, ^ = 1/T "CT lijBS, &a ttz, HR((3,n;t) li'MMMik (retarded function) X-$)Z> -144- JAERI-Conf 99-008 n V * * Tadpole diagram n ooooSelfenergy diagram o (5) 2 tod 2 1. /i + S (/3,/i;M)W) = M , (7) 2. u(M2 + ^u2) - A = 0, (8) o fi S'Qd(/3, /x; M, U) 4: m2 ^-OMiE t Ltiv\ -f <7)ffiiE4-#S L/i fc (7)4- M2 t LTGap^fl [3] m 2 (i Fermion ^^v^-g- (0(4) jj^^v 7?1I) W, fiJt T = 0,180[MeV] (C 3s = 0-C (CTO> =0, d(ao)/dt = O t -> XR{/3,p, oo), T{/3,ii;t) ->r(ft/i;oo)) ^t*,v^/co = 0[MeV]) IC^V^Tu = 93[MeV],MCT = 550[MeV],Mff = 140[MeV] 2 3 2 (A = 98,/x = -120 x 10 [MeV ]), t L/co T = 180[MeV] t?Jiv = 65[MeV],Mff = 410[MeV],M^ = 170[MeV] fc& 4 a^ /'? 7 /• - 9 ~nm$Mitl~Wl Wk (Environment) n^W^Lit^Wi^^ - t 3 (Kl:;140[MeV]) -145- JAERI-Conf 99-008 (nn{t)) 3 T = 0[MeV] -C (n,) = 0[l/fm ] t Lfco vfco T = 180[MeV] Af (9) [1] M.Gell-Mann and M.Levy.Nuovo.Cim.Vol 16.Num 4.1729(1960) J.Schwinger,Ann.Phys.2.407(1958) B.W.Lee,Chiral Dynamics(Gordon and Breach,1972) [2] N.Hashitsume ,F.Shibata and M.Shingu, J.Stati.Physl7,155(1977) F.Shibata and N.Hashitsume,J.Phys.Soc44,1435(1978) [3] T.koide ,M.Maruyama .F.Takagi, Prog.Theor.Phys.l01373(1999) [4] C.Greiner and B.Muller, Phys.Rev.D55,1026(1997) [5] -146- JAERI-Conf 99-008 130 120 T = 0 [MeV] 110 T= 180 [MeV] 100 90 80 70 60 50 40 30 20 10 0 0 3 4 5 6 8 10 Time [1/fm] 2: 0.45 T = 0 [MeV] T= 180 [MeV] 0.05 0 3 4 5 6 8 10 Time [1/fm] 3: -147- JAERI-Conf 99-008 JP0050041 29. Phase-Shift Analyses of pp Scattering at High Energies and Strong Energy-Dependence of Spin-Orbit Interaction Junichi Nagata1, Kazuo Harada2 and Msanori Matsuda3 1 Venture Business Laboratory, Hiroshima University, Higashi-Hiroshima 739-8527, Japan 2 Faculty of Food Culture, Kurashiki-Sakuyo University, Tamashima-Nagao 3515, Kurashiki, 710- 0292, Japan 3Division of Materials Science,Faculty of Integrated Arts and Sciences Hiroshima University,Higashi-Hiroshima 739, Japan Abstract A possible weak first-order phase transition of sub-nuclear medeium at Tc=100 ~ 150 MeV is indicated by means of the phase-shift analyses of pp scattering at PL=1 ~ 12 GeV/c. A phenomeno- logical model on the analogy of BCS-theory is proposed for a scenario of the phase transition. §1. Introduction We have performed the phase-shift analyses (PSA) of elastic pp scattering at PL=1 ~ 12 GeV/c so far[l, 2]. It has been found that the spin-orbit phase-shift of p-wave has a strong eenrgy dependence, where it decreases rapidly at PL=3 ~ 6 GeV/c and tends to zero at higer energies. As regards of such a strong energy-dependence, we suggest a " shrinkage " of spin-orbit interaction and give its threshold radius. The implications of the indecated shrinkage of spin-orbit force in the weak first- order phase transition of sub-nuclear medium at Tc= 100 ~> 150 MeV are discussed. In order to write a scenario of the weak first-order transition from Quark-Gluon phase to Hadron phase, we propose a phenomenological model on the analogy of the BCS-theory of the Fermi liquid. §2. Phase-shift analyses of pp scattering at 1~ 12 GeG/c and spin-orbit interaction. In order to do the partial wave analyses of N-N scattering data in this energy region, we have proposed a study of nuclear interaction in which the scattering amplitudes are determined on the basis of the following correspondence principle. This method is a relativistic extension of Taketani's way[3] of studying nuclear force. Principle I The peripheral part of the amplitude of nucleon-nucleon scattering in the outer region of the distance r ;> 2.5 fm is provided with the one-pion-exchange(OPE) contribution. Principle II In the region 2.5 ^ r ^ 1.0 fm, the scattering amplitude is evaluated by the modi- fied one-boson-exchange(OBE) contributions^]. The modification of the OBE-amplitude of nucleon- nucleon scattering is 9 9 * 9 A g _ g f in (1) m2-t m2 l A2 - t/nJ ' K ' where t is the squared momentum transfer of nucleons and m the observed mass of the exchanged boson, and g, A and n are parameters peculiar to boson. The full amplitude of nucleon-nucleon scattering is represented by M = £ [fj(6j,vj)} + £ {fj(6j(OBE),nj)} + MOBE(J > Jx). (2) J 148- JAERI-Conf 99-008 e Here 6^, 6^ and 6 LS are the central, tensor and spin-orbit phase-shifts, respectively. The recently obtained solutions of Sj^g and S^g at 3, 6 and 12 GeV/c are shown in Fig. 1 together with the other solutions. The energy-dependence of spin-orbit phase-shifts indicates that there exists a repulsive spin-orbit interaction which is so strong as to be superior to the attractive one due to the one vector- boson exchange, and the repulsion of such an interaction becomes prominent at the range r < 0.5 fm. The scattering matrix M in the two-proton spin space is determined by a PSA of p-p scattering at each energy. In the eikonal model, the scattering matrix has an impact parameter representation in terms of the eikonal x(b) as follows: M(q) = Up/2) J[l - exp(ix(6))] exp(-iq • b)d% (4) where fa is the impact parameter, q is the momentum transfer q = p' — p, and p and p' are the incident and scattered momenta in the cm. system, respectively. The eikonal is numerically evaluated by Fourier-Bessel transformation of the scattering matrix M(q) of the PSA solutions. [5] The obtained spin-orbit components of eikonal XLS at 3, 6 and 12 GeV/c s are shown in Fig. 2, respectively. Because Rexz,s(6) ' proportional to — f*££ VLs{r)dz, where VLS{T) is the spin-orbit part of a local optical potential and r — (fa2 + z2)1/2, Re XLS presents the information on the 6-dependence of the spin-orbit interaction. The solutions of Rexz,s at 3, 6 and 12 GeV/c agree among them in the outer region b ^ 1.0 fm, which represent the force due to peripheral mesonic contribution. In the region of 0.5~1.0 fm, Rexz,s exhibits the nonstatic effect due to the modified one-boson-exchange interaction. In the inner region of b £> 0.5 fm, Rexx-S gives new indication on an existence of a short-range repulsive force with a strong energy dependence. It is, however, to be noted that the spin-orbit component phase shifts have a tendency to be zero progressively with energy. These found results can be never understood by the usual hadron physics. Our only possible explanation about them is that a phase transition takes place in the incident energy region of proton PL= 3.5~12 GeV/c.[2][6] The threshold radius (Rth) for the " shrinkage" of the spin-orbit force is about 0.5 fm, which almost agrees with the bag-radius predicted by the MIT bag model[7]. The critical temperature for the transition from QGP to HP was estimated as 150 ~ 200 MeV by using the Gibbs condition, provided that the transition is approximately same to that from the massless quark-gluon gas to the massless pion gas. [8] The energy density of fire ball at this critical 3 point is ec= 0.9 - 2.5 GeV/fm . In the next, we calculate the energy density of plasma in high energy collision of two nuclei in the cm. system by means of Landau's fire ball model. Both of two nuclei are supposed to have the same atomic mass A. The incident energy of one nucleon is given by £Cm = W/(2A), where W is the incident energy of nucleus. The incident two nuclei are thin disks owing to the Lorenz contraction, the thickness of which is 2/2/7. Here R is the nuclear radius: R = 1.2Al/3[fm\ and 7 is the Lorenz factor: 'y=Ecm/rn,N- Two nuclei are closed in the volume occupied by two disks at the moment of collision. The energy density of this fire ball is given by 2 W IAR = E m (5) nR2 / 7 2?r x 1.23mjv " V ' If this is equal to the energy density of quark-gluon plasma at the critical point, i.e., e = ec, the incident energy per nucleon in the cm. system is evaluated as i^cm = 3~5 GeV, which corresponds to the critical temperature Tc= 150~200 MeV. The momentum region PL= 3.5~12 GeV/c, where the catastrophic transition of spin-orbit force is found, corresponds to the energy region £cm= 1.5~2.5 GeV, which is equivalent to Tc= 100~150 MeV. §3. A phenomenological model for a scenario of the weak lst-order phase transition We propose a phenomenological model [9] of the phase transition with the QCD effects on the analogy of the BCS-theory of the Fermi liquid in order to clarify the mechanism of the weak lst-order transition from QGP to HP. In analogy with the BCS theory of the Fermi-liquid, the total Hamiltonian of the quark Cooper- pair in the momentum representation is written as pa p a /) -149- JAERI-Conf 99-008 2 2 where £p = (p — pp ) /2m, aj,a and apa axe the creation and annihilation operators of quark with the momentum p and the color freedom a (red, blue and green), respectively, and pF is the Fermi momentum. The symbol £]' represents the summation of the momentum which is taken only in the neighborhood of the Fermi-surface. A is the order parameter which is defined by v p where G is a direct interaction constant or a form factor and V is a volume of system. We take into account the following; (1) The force between two quarks with the same color is repulsive. Then, the order parameters Arr,Abj, and Agg are zero. (2) The two quarks with different colors combine and form the qq-pair. The order parameters Arfc, A;,g and Agr take the finite values. The third quark interacts with the qq-pair to create nucleon. We write a wave function of the 3-quarks system as tanh where THJQ is an averaged energy of the gluon exchanged between the Cooper-paired quarks. Np is the state density of the quark liquid on the Fermi surface, which is given by 3 n 3/0 where n and p are the number- and the mass-densities of the quark liquid, respectively, and t Pp are the Fermi energy and momentum of the quark liquid, respectively. The gap equation presents a relation between the temperature and the order parameter. When the temperature is smaller than the critical temperature Tc, the qq-pair wave function 2. The Fermi momentum of the quark liquid at T ~ Tc is evaluated by Heisenberg uncertainty relation as Pp — h/L, where L = 2Rth = 1.0/m and Rth is the threshold radius for the shrinkage of spin-orbit force given in the above section. 3. The interaction between the Cooper-paired quarks is so strong that it is nonrelativistically represented in a good approximation. The direct interaction constant G in the gap equation is denned by the volume integral of the conventional potential between the Cooper-paired quarks : the one-gluon exchange and confining string (OGC) potential, [10] Here b is the universal confinement strength taken as 0.18 GeV2 and a is the running coupling constant [7] given by -150- JAERI-Conf 99-008 47r n. = 9 ln(l + l/(Ar)2) with A = 0.172 GeV and b = 0.18GeV2. 4. The gluon energy hujo needed to produce the quark Cooper-pair is independent of T. 5. The order parameter A is nearly constant for T < Tc/ 2, so that A(T)~A(T = 0). 6. The value of A rapidly tends to be zero for T > Tc/ 2. 7. The quark density p in the fire ball is given by p — 2(2mu + m^/v, where mu and rrid are the u- and d-quark masses, respectively, and v is a volume of the fire ball. Here mu — rrid = m is assumed for simplicity. The calculated relations of the internal distance of Cooper-paired quarks R with Tc and A given by the gap equation for OGC-potential are given in Fig. 3(a) and Fig. 3(6), respectively. It is interesting to estimate the hadron masses based on our model. In analogy with the Weizsacker mass formula for nucleus, the pion and the nucleon masses in BCS-phenomenology will be given by M = ni x mq + n-z x A, (13) where n\ — (Number of constituent quarks), ri2 = n\— (Number of paired quarks), A is the excitation energy of unpaired quark from the paired quarks, and mq is the mass of stripped quark at T ~ Tc. • in the case of pion, ni—0, n MN = 3 x 60MeV + 750MeV = 930MeV. • the observed mass of u- or d-quark (m#) is predicted as follows, 3 x 60MeV + 750MeV = 3m# m* = 930MeV/3 = 310MeV. 4. Concluding remarks We considered for the analogy of the dynamical parameters between the present model and the BCS-theory to be more important than the scheme of the relativistic field theory. Such a model should be permitted in a phase transition of the "strong" interaction by the reason mentioned in the above section. In the case of the first-order phase transition, it is meaningless to reinterpret the hadron dynamics by means of the QCD. By the present model of the phase transition from nucleon phase to quark gluon phase, the shrinkage of spin orbit force is interpreted as follows; o The threshold radius of shrinkage (~ 0.5 fm) is a half of Pippard length of the qq-pair wave function. o When a distance between two nucleons is shortened and amounts to almost the same as the Pippard length of the qq-pair, the qq-pair composing nucleon starts to disintegrate. o The critical temperature of the disintegration is about 100 ~ 150 MeV, which corresponds to the c.m.s. energy of N-N scattering, £OT = 1.5 ~ 2.5 GeV and the laboratory momentum Pi, = 3.5 ~ 12 GeV/c. The liquid model permits us to write a scenario for a possible mechanism of the first-order phase transition from the quark-gluon system to the hadron system as follows; 1. In the high temperature region of T > Tc, the chiral symmetry of the quark-gluon system is completely realized, and no quark condensation occurs. 2. When the temperature of the quark-gluon system goes down and comes near the critical temper- ature (T ^, Tc), the chiral symmetry is weakly broken to result in the massive quark (mq ~ 60 MeV). 3. In the temperature region of T < Tc, the chiral symmetry is completely broken in the current- quark system, the quark condensation occurs and the massive quarks constitute the Cooper pair, i.e. the hadrons, where Tc = 100 ~ 150 MeV and ftw0 = 10 ~ 50 MeV. -151- JAERI-Conf 99-008 References [1] J. Nagata, H. Yoshino and M. Matsuda, Prog. Theor. Phys. 93(1995) 559; 95(1996) 691. [2] M. Matsuda, J. Nagata, H. Yoshino K. Harada and S. Ohara, Prog. Theor. Phys. 93(1995) 1059. [3] J. Iwadare, S. Otsuki, R. Tamagaki and W. Watari, Prog. Theor. Phys. Suppl. No.3 (1956) 32; M. Taketani, S. Nakamura and M. Sasaki, ibid. 169. [4] M. Kawasaki, Y. Susuki and M. Yonezawa, Prog. Theor. Phys. 47(1972) 589. [5] M. Matsuda, H. Suemitsu and M. Yonezawa, Prog. Theor. Phys. 77(1987) 497. [6] M. Matsuda, Proceedings of XIII International Conf. Few-Body Problems Phys.,Flinders Univ. FLAS-R-216(1992),118. [7] C. E. Carlson, T. H. Hansson and C. Peterson, Phys. Rev. D27(1983) 1556. [8] E.V. Shuryak, Phys. Rev. 61(1980) 71. [9] K. Harada and M. Matsuda, Hiroshima Univ. Preprint HUFLAS-HP97-1, (contributed paper to Memorial Sym. for Prof. K. Nishikawa's retirement, May 1997); Proceedings of 15th International Conference of Few-Body Phys. July 1997, Groningen, (to be published); Soryushiron-Kenkyu (mimeographed circular in Japanese) Vol.96, No.4(1998) 139. [10] R. K. Bhaduri, "Models of the Nucleon" Addison-Wesley Publishing Company,Inc.(1988) 40 1 • Our solutions 30 - • o Y. Higuchirta/. _ P -wave F -wave V Y. Higuchi el al. • N. Hoshizaki era/. (L—1) (L—3) SAID(VPI) 20 c- 3. Bystricky et al. _ D J. Bystricky etal. u -a 10 - a \ - o • V , ^- . -10 • i i i i 1 1 2 3 456789 3 4 5 6 7 8 9 10 10 PL (GeV/c) PL (GeV/c) Figure 1: The spin-orbit phase-shifts of P and F-waves. The dots show our obtained solutions and the solid lines the interpolated ones by Spline function. 152 JAERI-Conf 99-008 1 3GeV/c 6GeV/c 12GeV/c 0.1 X 0.01 04 0.001 I I I 0.00 0.25 0.50 0.75 1.00 0.0001 0.0 05 1.5 Figure 2: The real part of the spin-orbit eikonal Rexxs(6) obtained from the PSAs at 3, 6 and 12 GeV/c. 1 1 ' 1 1 •T- | . | - ' 1 m =30 q — mq =30 300 P mq =35 ^0.6 mq=35 (a) \ (b) \\ mq =60 mq=60 I mq =70 ---mq=70 2 200h lo.4 I £ - 2 100 k 13 u OD .1.1 , 1 °0.0 1 . 1 1 U 0.0.8 I.O 1.2 1.4 0.8 1.0 1.2 1.4 Internal Distance of qq Pair [fm] Internal Distance of qq Pair [fm] Figure 3: (a)The relations between the critical temperature (Tc) and the internal distance of Cooper- Paired quarks (R) given by the gap equation for OGC- potential, (b) The relations between the order parameter(D) and the internal distance of Cooper-Paired quarks (R). -153- JAERI-Conf 99-008 JP0050042 30. Three dimensional relativistic hydrodynamical model for QGP gas C. Nonaka1, S. Muroya*2, O. Miyamura3 Dep. of Physics, Hiroshima Univ., Higashi-hiroshima, Hiroshima, 739, Japan Tokuyama Women's College, Tokuyama, Yamaguchi, 745, Japan* Abstract We numerically solved fully (3+l)-dimensional relativistic hydrodynamical equation coupled with the baryon number conservation law without spatial symmetry. We discuss the effect of transverse expansion based on the deviation our numerical result from Bjorken's scaling solution. We analyze the space-time evolution of the QGP gas in the case of non cylindrical initial conditions. 1 Introduction The various kinds of collective flow phenomena such as directed flow, elliptic flow and radial flow has been observed in recent experiments at AGS [1] and SPS [2]. It is a matter of interest that such flow are results of hydrodynamical motion of hadronic fluid. Our first trial to tackle the problem is to develop (3+l)-dimensional hydrodynamical model. Assuming the local thermal equilibrium for hot and dense fire ball produced in ultra relativistic nuclear collisions, we analyze the evolution of the fire ball based on the (3+l)-dimensional hydrodynamical model. The hydrodynamical model for Quark-Gluon Plasma (QGP) fluid has already been discussed in many papers since Bjorken first introduced the simple scaling model based on (1+1)-dimensional expansion picture [3]. For simplicity, cylindrical symmetry is assumed in usual hydrodynamical [4, 5, 6, 7] analysis, but this assumption disables us from discussing the anisotropic collective flow. In this paper, in order to investigate collective flow not only in the central collisions but also in the non-central ones we numerically solve the (3+l)-dimensional relativistic hydrodynamical equation coupled with the baryon number conservation law. 2 The relativistic hydro dynamical model The relativistic hydrodynamical equation for perfect fluid is given as v dllT" = 0, (1) where TM" is energy momentum tensor, - vv"). (2) Here, t is energy density, P is pressure, metric tensor is q*" = diag.(l, —1,-1,-1) and local velocity is M (/ = (1, vx, vy,vz)/f respectively. In order to take account of the finite baryon number density, we must solve the baryon number conservation law, d»{nB(T,n)U"} = 0} (3) ttl also, where rts(T,^) being baryon number density. Through the time like component of Eq.(l), Ul/dtiT ' = 0, Eq.(3) and thermodynamical relation, e+P — TS+fms, we can obtain the conservation law of entropy density current, S* = 5(/M, (4) 'E-mailmonakaObutsuri.sci. hiroshima-u.ac.jp 2 E-mail:[email protected] 3E-mail:miyamura@fusion. aci.hiroshima-u.ac.jp -154- JAERI-Conf 99-008 Our numerical algorithm solving the hydrodynamical equation is based on the entropy conservation law Eq.(4). In order to solve the hydrodynamical equation, the equation of state is needed. Though we consider the QGP gas and the hadron gas(excluded volume model [9]) for the realistic model equation of states, in this paper we adopt the QGP gas for numerical simplicity. The QGP gas model of massless Nf flavor quarks is given by, '-^^V + ^ + sM- <5> where the number of flavor being Nf = 3 and chemical potential for quarks being /iq = fi/S. 3 The numerical calculation We solve the (3+l)-dimensional hydrodynamical equation without symmetrical conditions by using an algorithm in which lattice points of volume element is moved along local velocity and the entropy con- servation law Eq.(4) is adopted explicitly. D. H. Rischke et al. discuss relativistic hydrodynamics in (3+l)-dimensional situation and collective behavior by using Eulerian hydrodynamics [8]. Our numeri- cal calculation is explained briefly as follows: In the first step the coordinates xm = Xm(t,i,j,k) (m = 1,2,3) of lattice points at time t + At are replaced by Xm(t + AM, j.fc) = Xm(t,i, j,k) + "^'''i'Sto (6) In the determination of lattice points in Eq.(6), the coordinates move in parallel with TIBU^, SU*. In the next step the local velocity is determined by, m m t v (t + At,i,j,k) = v (t,i,j,k) + dtv (i,j,k,t)At (7) 3 where d^v^ obtained from Eq.(l), Eq.(3) is used. In the final step the temperature and chemical potential of lattice points is calculated by using Eq.(3), Eq.(4). 4 Comparison with Bjorken's solution Comparing our numerical solutions with Bjorken's scaling solution vz = z/t, we can easily evaluate the effect of the transverse flow. Based on Bjorken's scaling solution and Eq.(4) entropy density is given as, 5(r) = S(r0)^, (8) where proper time T, T = \/t2 — z2. In order to make comparison clear, in this section the velocity of our model in the longitudinal direction is fixed to the Bjorken's scaling solution. We put the initial temperature distribution and chemical potential distribution respectively as follows: } (9) -155- JAERI-Conf 99-008 x [fm] Figure 1: the comparison with Bjorken's Figure 2: the difference between our so- scaling solution lution and Bjorken's scaling solution by changing XQ, yo from 1.0 fm to 6.0 fm Nucl. Phys. A638 ('98) 357c 180 z= o I fm ] > 0) 120 60 200 400 600 800 1000 M [ MeV] -4 Figure 4: The solid line stands for freeze- Figure 3: the non cylindrical initial condi- out which we assume from chemical freeze- tion of temperature distribution at z = 0 out (the dashed line) and thermal freeze-out fm (the dotted line). where xo = yo = zo = 1.0 fm, ZB = 0.7 fm. ox — ay — az = 1.0 fm, OB = 0.7 fm, 7b = 200 MeV, /Jo = 210 MeV and the initial transverse velocity is set to 0. We focus to the volume elements at (x, y, z) = (0,0.0) for the comparison of our numerical solution with Bjorken's scaling solution. Figure 1 shows that our numerical calculation is coincident with Bjorken's scaling solution up to r = 3.0 fm. After T = 3.0 fm the difference between them increases with the proper time because of the transverse flow. We define a characteristic time TB at which instance the difference between two models becomes larger than 0.1 % . Figure 2 indicates that TB is almost proportional to the initial xo, j/o during 1.0 fni to 0.0 fm, and Bjorken's scaling solution seems to be a proper solution for a large system. 5 Non cylindrical initial conditions In this section we investigate the space-time evolution of the hydrodynamical flow with non cylindrical initial temperature distribution and chemical distribution. Other parameters are put as the previous section. Figure 3 shows the initial temperature distribution in b = 0.8 fni and H — 1.0 fm. We evaluate the effect, of the flow in particle distributions by giving initial conditions like this, though this condition is not realistic to analyze the experimental data. We use Cooper-Frye formula [10] for particle emission from hadronic fluid. 1 (ID (2*)» - 156- JAERI-Conf 99-008 t = 3.0 fm y [fm] •W\ I //• [fm] \\\ Figure 5: the space-time evolution of the flow at z = 0 fm (b = 0.8 fm) 1.1 1.0 ---- b=0 [fin] 0.9 100 200 300 PT [GeV] 300 azimuthal angle ( 6 Summary We solved (3+l)-dimensional relativistic hydrodynamical equation without cylindrical symmetry condi- tions by Lagrangian hydrodynamics. We discussed the effect of the transverse flow and confirmed that Bjorken's scaling solution is a proper solution in a large system by making a comparison with numerical calculation. The effect of the flow to the particle distributions was also investigated. The influence of flow is large at = 90°, 270°. This is because the freeze-out hypersurface is large at -157- JAERI-Conf 99-008 to adopt the output from the event generator as initial conditions and to use the equation of states including phase transition from the QGP phase to the hadron phase. Investigating the collective flow in experimental data is our next task. References [1] J. P. Wessels: E877, Nucl. Phys. A638(1998), 69c; H. Liu : E895, Nucl. Phys. A638(1998), 451c; S. A. Voloshin: E877, Nucl. Phys. A638(1998), 455c [2] H. Appelshauser et al. : NA49, Phys. Rev. Lett. 80(1998), 4136; M. M. Aggarwal et al.: WA98, nucle-ex/9807004 [3] J. D. Bjorken, Phys. Rev. D27 (1983), 140. [4] T. Ishii and S. Muroya, Phys. Rev. D46(1992), 5156 [5] J. Sollfrank, P. Huovinen, M. Kataja, P. V. Ruuskanen, M. Prakash, R. Venugopalan, Phys. Rev. C55(1997), 392 [6] C. M. Hung and E. Shuryak, Phys. Rev. C57(1998), 1891 [7] B. R. Schlei and D. Strottman, Phys. Rev. C59(1999), R9 [8] D. H. Rischke, S. Bernard, J .A .Maruhn, Nucl. Phys. A595(1995), 346; D. H. Rischke, Nucl. Phys. A610(1996), 88c [9] D. H. Rischke, M. I. Gorenstein, H. Stocker, and W. Greiner, Z. Phys. C51(1991), 485 [10] F. Cooper and G. Frye, Phys. Rev. D10(1974), 186 [11] U. Heinz, Nucl. Phys. A638(1998), 357c -158- JAERI-Conf 99-008 JP0050043 31. #mst&imimMK&<3< Elliptic Flow Based on a Relativistic Hydrodynamic Model Tetsufumi Hirano Department of Physics, Waseda University, Tokyo 169-8555, Japan Based on the (3+l)-dimensional hydrodynamic assuming cylindrical symmetry [10, 11] by specifying model, the space-time evolution of hot and dense the model EoS and we obtain the space-time depen- nuclear matter produced in non-central relativistic dent thermodynamical variables and the four velocity. heavy-ion collisions is discussed. The elliptic flow pa- We use the following models of the EoS with a phase rameter V2 is obtained by Fourier analysis of the az- transition. Hagedorn's statistical bootstrap model [12] imuthal distribution of pions and protons which are with Hagedorn temperature Tu = 155 MeV is em- emitted from the freeze-out hypersurface. As a func- ployed for the hadronic phase. We directly use the in- tion of rapidity, the pion and proton elliptic flow pa- tegral representation of the solution of the bootstrap rameters both have a peak at midrapidity. equation [13] instead of using the very famous hadronic mass spectrum, exp(m/2H), which is the asymptotic solution of this equation. It is well known that this One of the main goals in relativistic heavy-ion model has a limited temperature range-, i.e., the energy physics is the creation of a quark-gluon plasma (QGP) density and pressure diverge at Tn- This singularity, and the determination of its equation of state (EoS) however, disappears when an exclude volume approx- [1]. It is therefore very important to study collective imation [14] (with a Bag constant B* =230 MeV) flow in non-central collisions, such as directed or el- is associated with the Hagedorn model. In the QGP liptic flow [2]. Recently experimental data concerning phase, we use massless free u, d and s-quarks and the collective flow in semi-central collisions at SPS energies gluon gas model for simplicity. The two equations of has been reported [3, 4, 5]. This data should be anal- state are matched by imposing Gibbs' condition for ysed using various models. Some groups [6, 7, 8] have phase equilibrium. Consequently we obtain a first or- used their microscopic transport models to analyse the der phase transition model which has a critical tem- collective flow obtained by the NA49 Collaboration [3]. perature 2c — 159 MeV and a mixed phase pressure In this paper we investigate collective flow, especially of Pmix = 70.9 MeV/fm3 at zero baryon density. elliptic flow, in terms of a relativistic hydrodynamic model. We mention our numerical algorithm for the rela- In non-central collisions elliptic flow arises due to tivistic hydrodynamic model. It is known that the the fact that the spatial overlap region of two colliding Piecewise Parabolic Method (PPM) [15] is very ro- nuclei in the transverse plane has an "almond shape". bust scheme for the non-relativistic gas equation with That is, the hydrodynamical flow becomes larger along a shock wave. We have extended the PPM scheme the short axis than along the long axis because the pressure gradient is larger in that direction. Therefore this spatial anisotropy causes the nuclear matter to also have momentum anisotropy. Consequently, the azimuthal distribution may carry information about the pressure of the nuclear matter produced in the early stage of the heavy-ion collisions [9]. The relativistic hydrodynamical equations for a per- fect fluid represent energy-momentum conservation 2'b (1) Reaction Plane T"" = (E + PJi^u" - (2) Transverse Plane and baryon density conservation Figure 1: Schematic view of the initial geometry in the center of mass system. The left figure shows the (3) reaction plane and the right the transverse plane. The (4) initial condition is in the region with slanting lines, b M where E, P, ne and u are, respectively, the energy is the impact parameter vector. rp and rt are respec- density, pressure, baryon density and local four ve- tively the distances from the center of the projectile locity. We numerically solve these equations without and the target nucleus in the transverse plane. -159- JAERI-Conf 99-008 of Eulerian hydrodynamics to the relativistic hydro- dynamical equation. Note that this is a higher order i-to=o.» t-MMI.0, Z=0 extension of the piecewise linear method [16]. Assuming non-central Pb+Pb collisions at SPS en- ergy, we choose very simple formulas for the initial con- dition at the initial (or passage) time to = 2ro/(~fv) ~ 1.4 fm (ro, 7 and v are, respectively, the nuclear ra- dius, Lorentz factor and the velocity of a spectator in the center of mass system) E{x,y,z) = Ei(z)0(So - z)9(z + zo)p{rp)p{rt), (5) nB{x,y,z) = nBi(z)6(z0 - z)0{z + zo)p(rp)p(rt),(6) vz(x,y,z) = VQ tanh(z/z0) x 0{z0 - z)9(z + zo)p(rp)p(rt), (7) where 8(z) is the step function, p(r) is the Woods- Saxon parameterization in the transverse direction, 1 p(r) = (8) I »• E\(z) is Bjorken's solution [17] and the z dependence of the baryon density TIBI{Z) is taken from Ref. [18] -i Ei(z) = Eo (9) to TlBl(z) = KXO.17^-5 . (10) See also Fig. 1. We have employed Bjorken's longi- tudinal solution just as an initial condition. This is in contrast to Ref. [9, 19], in which Bjorken's boost- invariant solution was used as an assumption and the hydrodynamical equation was numerically solved only in the transverse plane. At relativistic energies the Lorentz-contracted spec- tators leave the interaction region after ~ 1 fm, we therefore assume the hydrodynamical description is valid only in the overlap region and neglect the interac- tion between the spectators and the fluid. Therefore we can say that our model gives a good description only in the vicinity of the midrapidity region and fails to reproduce directed flowa t present. It may be possi- i ... ble to treat this problem if we use a hadronic cascade model for both spectators and particles emitted from the freeze-out hypersurface, together with the hydro- dynamic model. There are four initial (and adjustable) parameters in our hydrodynamic model: 1) the energy density at z = 0, Eo = 2500 MeV/fm3, 2) the factor in the baryon density distribution K = 2.5, 3) the initial longitudi- Figure 2: Time evolution of pressure and baryon flow nal factor e = 0.9 and 4) the "diffuseness parameter" in the transverse plane. Left: The pressure contours. Right: The baryon flow velocity vector Sr = 0.3 fm. In the present analysis we select these values 'by hand', i.e., we guess them. These param- eters, however, should be chosen so as to reproduce -160- JAERI-Conf 99-008 the experimental data for the (pseudo-)rapidity and structure corresponding to the mixed phase with the the transverse momentum distribution. To make our same pressure ~ 70 MeV/fm3, and the initial pressure analysis more quantitative, we need this experimental gradient gives the baryons transverse flow. The QGP data. We would like the experimental group to ana- phase disappears at t = to + 1.0 fm and after that the lyze the centrality dependence of the hadron spectra, mixed phase occupies the central region. There is still especially, the (pseudo-)rapidity distribution. For this no transverse flow near the origin due to the absence reason we wish to emphasize that our numerical results of a pressure gradient. At about t = to + 5.0 fm all the presented below are only preliminary. nuclear matter initially in the QGP phase has gone Figure 2 shows our numerical results for the tempo- through the phase transition and is in the hadronic ral behavior of the pressure (left column) and the bary- phase. We can see from these figures that the shape onic flow (right column) at z = 0 in the non-central of the nuclear matter is changing from almond (top) Pb+Pb collision with impact parameter b = 7 fm at to round (bottom), and the elliptic flow reduces the SPS energy. Initially almost all matter in this plane is initial geometric deformation. in the QGP phase and there is no transverse flow any- The numerical results of the hydrodynamical simu- where by definition. At t — to+0.5 fm we see the shell lation give us the momentum distribution through the Cooper-Frye formula [20] with freeze-out temperature T( = 140 MeV. The elliptic flow parameter ti2, as a pion 0 Ptj \pt d Before calculating V2 in non-central collisions with impact parameter 6 = 7 fm, we checked the numerical error in our hydrodynamic model in central collisions. Since there is no special direction in the transverse plane for head-on collisions, ideally the elliptic flow vanishes in the infinite particle limit. Performing the Figure 3: Rapidity dependence of elliptic flow for pion. numerical simulation with b = 0 fm, we obtain the Four curves correspond to the different transverse mo- value of V2 as less than lO^1 percent, therefore we mentum regions. The midrapidity is 2.92. can safely neglect the numerical error. Note that the numerical error in the energy and baryon density con- servation of the fluid is less than one percent in our I25 proton D.5 N *» peak at midrapidity. This seems to be in contrast with \ t \ \ the experimental data obtained by the NA49 Collab- 10 \ \ \ • oration [3]. Their data appears to be slightly peaked at medium-high rapidity. 5 Our results for V2 for protons are shown in Fig. 4. We see the same behavior as for the pion case. We 0 ^^ . obtain a larger Vi for protons than for pions because 0 1 3 we are integrating over a larger transverse momentum y region. Since the initial parameters in our hydrody- namic model have been chosen by hand, we would like Figure 4: Rapidity dependence of elliptic flow for pro- readers to not take these results quantitatively. ton. Three curves correspond to the different trans- In summary, we reported our preliminary analysis verse momentum regions. Note that the integral re- of elliptic flow in non-central heavy-ion collisions using gion of transverse momentum is larger than for pions. the hydrodynamic model. We numerically simulated 161- JAERI-Conf 99-008 the hydrodynamic model without assuming cylindrical [12] R. Hagedorn, Suppl. Nuovo. Cim. 3 (1965) 147; symmetry or Bjorken's boost-invariant solution, us- see also R. Hagedorn and J. Rafelski, in Statis- ing the extended version of the Piecewise Parabolic tical Mechanics of Quarks and Hadrons, edited Method which is known as a robust scheme for the by H. Satz (1981) p. 237, North Holland, Ams- non-relativistic gas equation with a shock wave. We terdam; R. Hagedorn, in Hot Hadronic Matter, presented the temporal behavior of high temperature Theory and Experiment, edited by J. Letessier, and high density nuclear matter produced in Pb+Pb H. H. Gutbrod and J. Rafelski (1995) p. 13, collisions with b — 7 fm at SPS energy. Our prelimi- Plenum Press, New York. nary results showed that the elliptic flow parameter V2 has a peak at midrapidity for both pions and protons [13] R. Hagedorn and J. Rafelski, Commun. Math. and increases with transverse momentum. Since there Phys. 83 (1982) 563. are some ambiguities in the initial parameters of our [14] J. I. Kapusta and K. A. Olive, Nucl. Phys. A408 hydrodynamical model, we should fix these parameter (1983) 478. using experimental data for the rapidity distribution in non-central collisions. If we regard the hydrodynam- [15] P. Colella and P. R. Woodward, J. Comput. Phys. ical model as a predictive one, we can choose initial 54 (1984) 174. parameters using results from a parton cascade model, such as VNI [22]. The study of these issues is a future [16] V. Schneider et al., J. Comput. Phys. 105 (1993) work. 92. The author is much indebted to Prof. I. Ohba, Prof. [17] J. D. Bjorken, Phys. Rev. D27 (1983) 140. H. Nakazato, Dr. Y. Yamanaka and Prof. S. Muroya for their helpful comments, and to Dr. H. Naka- [18] J. Sollfrank et al., Phys. Rev. C55 (1997) 392. mura and Dr. C. Nonaka for many interesting discus- sions. The numerical calculations were performed on [19] D. Teaney and E. V. Shuryak, nucl-th/9904006. workstations of the Waseda Univ. high-energy physics [20] F. Cooper and G. Frye, Phys. Rev. D10 (1974) group. 186. Although there is a well-known problem in this formula when it is applied to the space-like References freeze-out hypersurface, we use this formula for simplicity. [1] See, for example, Quark Matter '97, Nucl. Phys. [21] P. Danielewicz, Phys. Rev. C51 (1995) 716. A638 (1998). [2] For a review, see J.-Y. Ollitrault, Nucl. Phys. [22] See, for example, K. Geiger, Phys. Rev. D46 A638 (1998) 195c. (1992) 4965. [3] H. Appelshauser et al. (NA49 Collaboration), Phys. Rev. Lett. 80 (1998) 4136. [4] S. Nishimura et al. (WA98 Collaboration), Nucl. Phys. A638 (1998) 459c. [5] F. Ceretto et al. (CERES Collaboration), Nucl. Phys. A638 (1998) 467c. [6] H. Liu, S. Panitkin and N. Xu, Phys. Rev. C59 (1999) 348. [7] H. Heiselberg and A.-M. Levy, nucl-th/9812034. [8] S. Soff et al., nucl-th/9903061. [9] J.-Y. Ollitrault, Phys. Rev. D46 (1992) 229. [10] D. H. Rischke et al., Nucl. Phys. A595 (1995) 346. [11] Note that to my knowledge there is only one scheme to simulate the non-central heavy-ion col- lisions which uses Lagrangian hydrodynamics: C. Nonaka et al., these proceedings. -162- JP0050044 JAERI-Conf 99-008 32. /*- h yft7s>r~- YW Baryon Stopping and Strangeness Baryon Production in a Parton Cascade Model Yasushi Nara Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Naka, Ibaraki 319-11, Japan A parton cascade model which is based on pQCD incorporating hard partonic scattering and dy- namical hadronization scheme describes the space-time evolution of parton/hadron system produced by ultra-relativistic nuclear collisions. Hadron yield, baryon stopping and transverse momentum dis- tribution are calculated and compared with experimental data at SPS energies. Using new version of parton cascade code VNI in which baryonic cluster formation is implemented, we calculate the net baryon number distributions and A yield. It is found that baryon stopping behavior at SPS energies is well accounted for within the parton cascade picture. As a consequence of the production of the baryon (u and d quark) rich parton matter, parton coalescence naturally explains the enhanced yield of A particle which has been observed in experiment. I. INTRODUCTION Heavy ion experiments at BNL-AGS and CERN-SPS have been performed motivating by the possible creation of QCD phase transition and vast body of systematic data such as proton, pion strangeness particles distributions, HBT correlation, flow, dileptons and J/rp distributions have been accumulated including mass dependence and their excitation functions [1-3]. Data from forthcoming experiment at BNL-RHIC will soon become available. In this work, I discuss the baryon stopping and A yield at SPS energies. Strong stopping of nuclei has been reported both at AGS and at SPS energies [4,5]. Baryon stopping power can be understood within a hadronic models if we consider multiple scattering of nucleon using reasonable pp energy loss [6]. For example, within string based models [7,34,10], baryon stopping behavior at SPS energies is well explained by introducing diquark breaking mechanism in which diquark sitting at the end of the string breaks. Diquark breaking leads to large rapidity shifts of the baryon. Constituent quark scattering within a formation time [34,36] has to be considered in order to generate Glauber type multiple collision at initial stage of nuclear collisions in microscopic transport models which describe full space-time evolution of particles. Since strangeness enhancement in heavy ion collisions relative to pp collision has been discussed as a QGP signal [8,9], the measurements of (anti-)strange particles have been proceeding. For example, enhancement of (multi)strange baryons and anti-baryons have been recently reported for Pb+Pb collisions at SPS energies [17,18]. Several microscopic models have been proposed to explain the data. String fusion mechanism such as rope formation in RQMD [34] is able to creat larger number of ss and diquark-antidiquark pair by the larger string tension. Capella [12] has explained strangeness enhancement by considering sea quark strings. LUCIAE [13] has proposed that the s quark suppression factor increases with energy, centrality and mass of the colliding system. They use the variant string tension for the multigluon string based on the idea that string tension increases as a result of the gluon kinks. HIJING/BB model [11], which is based on string phenomenology and pQCD, introduces the baryon junction exchange as well as baryon junction-antijunction (JJ) loops to explain enhanced hyperon and antihyperon yield within a hadronic scenario. Fast hadronization models for the constituent quark plasma (CQP) using quark coalescence have been proposed [14-16] to study hadronization process and particle species. Event generators based on perturbative QCD (pQCD) are proposed such as HIJING (Heavy Ion Jet Interaction Generator) [20,21], VNI (Vincent Le Cucurullo Con Giginello) [22], in order to describe ultra-relativistic heavy ion collisions. VNI can follow the space-time history of partons and hadrons. Parameters of both models have been fixed from e+e~, pp and pp data. It is found that parton cascade model of VNI describes well the main features of heavy-ion collisions even at SPS energies [31-33]. The aim of this paper is to investigate both the baryon strong stopping and the strange particle enhancement observed at SPS energies using the QCD based space-time model of parton cascade VNI. Can parton cascade describe correct baryon stopping at SPS? What about at RHIC? Can the enhanced strange baryon yield observed in experiment be explained within a parton cascade picture? So far, only two parton cluster (mesonic cluster) formations are included in the Monte-Carlo event generator VNI. Namely, old version of VNI implicitly assumed the baryon free region at mid-rapidity during the formation of hadrons. I have implemented the baryonic cluster in order to be able to answer those questions. The article is organized in the following way. In section II, I summarize the main component of parton cascade model VNI and its extensions. In section HI, I first compare some VNI results to SPS data and discuss the baryon stopping and A yield. In section,IV, I draw conclusions. 163- JAERI-Conf 99-008 II. PARTON CASCADE MODEL In this section, I present the main features of the parton cascade model of VNI. Relativistic transport equations for partons based on QCD [28-30] are basic equations which are solved on the computer in parton cascade model VNI. The hadronization mechanism is described in terms of dynamical parton-hadron conversion model of Ellis and Geiger [23-27]. The main features in the Monte Carlo procedure are summarized as follows. 1) The initial longitudinal momenta of the partons are sampled according to the measured nucleon structure function f(x, Q%) with initial resolution scale Qo. We take GRV94LO (Lowest order fit) [19] for the nucleon structure function. The primordial transverse momenta of partons are generated according to the Gaussian distribution with mean value of px = 0.42GeV. The individual nucleons are assigned positions according to a Fermi distribution for nuclei and the positions of partons are distributed around the centers of their mother nucleons with an exponential distribution with a mean square radius of 0.81fm. 2)With the above construction of the initial state, the parton cascading development proceeds. Parton scattering are simulated using closest distance approach method in which parton-parton two-body collision will take place if their impact parameter becomes less than \J g + g->C1+C2,g+g->C + g,g + g->C + g + g, (3) q + q-+Ci+C2,q + q->C + g, (4) q + g-*C + q,q + g^>C + g + q. (5) In this work, if diquarks are formed with the above formation probability, I introduce baryonic cluster formation as qq + q-*c, (6) qq + q^C, (7) 9192 + 93-•9193 + 92, (8) 9i92 + g -+ 919293 + 93- (9) Note that by introducing those cluster formation processes, I do not introduce any new parameters into the model. 4) Beam clusters are formed from primary partons (remnant partons) which do not interact during the evolution even though they travel in the overlapping region of nuclei. They may be considered as the coherent relics of the original hadron wavefunctions and should have had soft interactions. Those underlying soft interactions are simulated by the beam cluster decay into hadrons in VNI because additional possibility that several parton pairs undergo soft interactions. This may give a non-negligible contribution to the 'underlying event structure' even at the collider energies. The primary partons are grouped together to form a massive beam cluster with its four-momentum give by the sum of the parton momenta and its position given by the 3-vector mean of the partons' positions. III. RESULTS First, particle spectra calculated by VNI are compared with experimental data at SPS energies and then calculated parton distributions both at SPS and RHIC energies are presented in this section. -164- JAERI-Conf 99-008 A. COMPARISON WITH SPS DATA Since current version of VNI differs from older version, we have checked the previously calculated results. First, I have calculated small system of S+S collision at SPS energies using the new version of VNI. Most important different point from previous version is the inclusion of baryonic cluster formation. S(200AGeV) + S Veto trigger S(200AGeV) + s NA44 io6 VNI: bK2.4fm ' ! 5%^^ P io4 1 —^ _ ^TBB —I Sio° , • I , 1 , 1 2 0.0 0.5 1.0 2 Px(GeV/c) m^-mass (GeV/c ) FIG. 1. VNI calculations of the transverse momentum dis- FIG. 2. VNI calculations of the transverse momentum dis- tributions of net protons (upper) and negative charged parti- tributions of protons (2.4 < y < 2.8), kaons (2.5 < y < 3.4) cles (ir~, K~,p) (lower) for S + S collision at 200GeV/c with and pions (3.0 < y < 4.1) for central 10% S + S collision at centrality 2%. Experimental data are taken from NA35. Ref SPS energy (b < 2.4fm). Experimental data from Ref. [40]. [39]. I compare the data [39] on net proton and negative charged particle transverse distributions in Fig 1 and proton, kaon and pion in Fig 2 for S+S collision at 200AGeV/c. New version of VNI improves the results that previously presented in Ref [31]. The agreement is very good for all particles. In the previous calculations, proton transverse momentum distribution was too steep comparing to the data. Secondary hadron-hadron interactions are not included in this calculations. We can see from Figures no effect of such hadron-hadron interactions for S+S collisions at 200AGeV. Note that hadronic models also reproduce these data (for example see Ref. [35,6]). It has been argued [40] that Mass dependence of the slope parameters shows evidence of collective transverse flow from expansion of the system in heavy-ion collisions. Hadronic transport model of RQMD explains the origin of collective transverse flow by the hadronic rescattering. While within a parton cascade model, collective transverse flow is created already by the parton phase .i.e. parton cascading. -165- JAERI-Conf 99-008 S(200AGeV) + S b<1.08fm S(200AGeV) + S b<1.08fm 2 4 rapidity y 40 h' - " h"(7I",K •P) hard : 30 ft 20 - - / 10 - n V \\ -2 0 2 4 6 y FIG. 3. Comparison of the rapidity distributions of net FIG. 4. The rapidity distributions of net A and Kaons for proton and negative particles for S +S collision at SPS en- S+S collision at SPS energy. Data from Ref. [42] for A and ergy between experimental data [39] and VNI calculations Ref. [41] for kaons. The baryon stopping problem is one of the important element in nucleus-nucleus collisions. Older version of VNI implicitly assumed baryon free region at midrapidity because baryonic cluster formation is not included. We can now discuss the baryon stopping problem with VNI. Figure 3 (upper panel) shows the rapidity distribution of net protons at SPS for S+S collisions from VNI together with NA35 experimental data [39]. It is seen that midrapidity protons come from mainly parton cluster decay. Semi-hard scattering of partons account for the baryon stopping at SPS within the parton cascade model. Absolute particle yield in VNI calculation is also good and contribution from parton cluster is presented in the lower panel of figure 3. In Fig. 4, VNI calculations of net A and kaon rapidity distributions are compared to the data [42] and [42]. Particle lying midrapidity again mainly come from parton clusters. However, beam rapidity region, A's come from beam cluster decay. I conclude that VNI describes main features of data at SPS energies for S+S collisions including strangeness productions. -166- JAERI-Conf 99-008 VNI4.20, Pb+Pb at SPS b<1.0fm Pb(158AGeV) + Pb b<3.2fm 80 1.5 • vni: nel [MOlOn HarS A vni: negative NA49 central 5% ' > D NA49 i.o a. 66, 0.0 rapidity y FIG. 5. Comparison of the rapidity distributions of net FIG. 6. The average transverse momentum for net pro- proton for Pb + Pb collision at SPS energy between experi- tons and negative particles as a function of rapidity for mental data [5] and VNI calculations Pb+Pb collisions at SPS energy. Data from Ref. [5]. Next I am going to discuss massive system of Pb+Pb collisions. Net proton rapidity distribution calculated by VNI is compared to experimental data of NA49 [5] in Fig. 5. Similar prediction can be seen but midrapidity protons (2.5 < y < 3.5) come from parton clusters more significantly than those of S+S system. In Pb+Pb collisions, the role of the hard scatterings among partons become more important than small systems. In Fig. 6, VNI results of the mean transverse momenta < p± > for net protons and charged particles as a function of rapidity are compared to the NA49 data [5]. Mean transverse momenta of charged particle are in good agreement with data, while for net protons, VNI underestimates the mean transverse momenta for all rapidity bins. This might be improved by including the final state interaction among hadrons. Pb(158AGeV) + Pb b<3.2 fm 30 10 AandSu A and 2° ^Z O NA49 20 10 0 O1' • ' • •-— 0 2 4 6 y FIG. 7. VNI calculations for the rapidity distributions of A and A for Pb + Pb collision at SPS energy Data for (anti-)A distributions from [43]. An enhanced production of strange particles has long been considered to contain important information on parton phase produced in high energy heavy-ion collisions. For example, if baryon rich plasma is created, A (uds) is likely to form rather than antikaon (su). Hadronic transport models can not explain the enhancement of strange baryons without introducing some collective mechanism like rope formation. We see in Fig. 7 that parton cascade explain the A yield as well as anti-A yield for Pb+Pb collisions at SPS energy. -167- JAERI-Conf 99-008 B. PARTON DYNAMICS FROM PARTON CASCADE Let me see the dynamics of partons from parton cascade model. In Fig.8 and Fig.9, The rapidity distributions of net baryon number dNq/dy — dNq/dy calculated from partons are plotted at both SPS and RHIC energies. At mid-rapidity, VNI predicts the net baryon number of 50 at SPS and 20 at RHIC. VNI: Pb+Pb at SPS b=0.0fm VNI: Au+Au at RHIC b=0.0fm 150 100 Net baryon number q-q Net baryon number q-q All parton All parton produced parton produced parton 100 oil"shell parton off shell parton 50 50 0 0 0 -5 0 5 rapidity rapidity FIG. 8. The rapidity distributions of net baryon number FIG. 9. The rapidity distributions of net baryon number (? ~ ?) from parton density produced from VNI for Pb + Pb (q — q) from parton density produced from VNI for Au + Au collision at SPS energy. collision at RHIC energy. VNI, Pb+Pb at SPS b=2.0fm VNI4.20, Au+Au at RHIC b=0.0fm 5000 - 300 s- a-quark 3-quark : / u-quark; u-quark ( •••• s-quark. s-quark 200 — " 100 I 0 c 7 i i i" 1 2 1 time (fm/c) time (fm/c) Rs=2N(ss)/( N(uu)+N(d3)) Rs=2N(ss)/( N(uu)+N(dS)) 1 2 3 1 2 time (fm/c) time (fm/c) FIG. 10. Time evolution of timelike partons (upper panel) FIG. 11. Time evolution of timelike partons (upper panel) and strange fraction (lower panel) produced from VNI for Pb and strange fraction (lower panel) produced from VNI for Au + Pb collision at SPS energy. + Au collision at RHIC energy. Time development of produced partons (timelike) (u + u, d + d ,s + s and g) are shown in Fig. 10 and Fig. 11 at -168- JAERI-Conf 99-008 w ere SPS and RHIC energies as well as the strangeness ratio defined by R, = jy/u)4.^/aY+jv(d)+Af(J>' ^ N(q) is the number of partons. At SPS, semi-hard scattering of partons produces strangeness ratio R, = 10%. This value is smaller than that of final strangeness yield. This indicates that hadronization itself creates many strange pair ss and this contribution is not negligible. IV. SUMMARY In summary, a microscopic transport model VNI which is based on QCD can explain many data at SPS energies. VNI calculations indicate that baryon stopping is explained by semi-hard scattering among partons and their cluster formation. As a consequence, A and A yield at mid-rapidity is accounted for by the baryonic cluster formation. Most particles come from beam-cluster decay at beam-rapidity region in VNI. Good agreement of VNI results with data for the small system of S+S does not mean that QGP state is created in S+S collisions, because there is not large enough volume and energy density for the parton phase in S+S collisions, However, as shown in Ref. [22], large energy density which is beyond the value predicted by lattice calculations is achieved in Pb+Pb collisions at SPS energies. In this work, we consider only two or three parton coalescence, but in dense parton matter produced in Pb+Pb collisions, this assumption might be broken down. Inverse processes like hadron conversion to parton such as C —• qq are also ignored which might become important at higher colliding energies. In this article, it is shown that parton cascade picture reproduces both the baryon stopping and enhanced A yield at SPS energies. However, hadronic models explain these data. In order to get clear evidence of parton matter which might be created, multi-strangeness particle and their anit-particle yield should be considered [11]. Especially, hadronic models can not explain [11,13] anomalous enhancement of fi reported recently. Systematic study for the strange particles (A, H, fi ,A, H, Cl) using a parton cascade model is interesting. Recently, slope parameters as a function of particle mass has been reported [2,3] and non strange particles show strong evidence of collective flow, however, the slopes of multistrangeness particles are similar to that of protons. RQMD calculation [37] has shown that this observations indicate the early formation of multistrangeness particle and their decouple from the system rather early due to their small cross sections. On the other hand, VNI shows that collective flow is already created at the partonic phase [45]. 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IS Using the RBUU approach we calculate the subthreshold production of antiprotons in the p- and d-nucleus reacitions done by the KEK-ps collaboration. Then we attempt to determine the depth of the anti-proton potential at the normal nuclear density from experimental data of these reacions. by introducing the JINR [1], BEVELAC [2], SIS [3] ^T*S [4] 1/1000, [5], QMD [6] [7]. ZL - U^x)) - (M - t/,(z))M*) = 0 (1) tztzL, C/c = -C/s (2) -171- JAERI-Conf 99-008 —h U - M* M* = M - [/,, (3) 0.7 RH jfiiK [8]. , mm G /\° U ^^ - * RH 0 . RMFT t BUU c RBUU -7° [9] M*/M = 0.65-0.7 • ifJV— Off $f^? # ^ T & ^ -172- JAERI-Conf 99-008 & 0, 3.5GeV/u KEK-ps < , - 3.5GeV KEK-ps , NL6 - B£ = 16MeV, -%%o Ua(N) = csUs(N) (4) t/s(iV) =0 (easel). f/s(iV) = 100 MeV (case II) = 3.5GeV/u -173- JAERI-Conf 99-008 i.5Gev/c - £in = 3.5GeV/u lOOMeV (5) 2-3 Ref.sugaya [1] A.A. Baladin et al., JETP Lett. 48 (1988) 137. [2] J.B. Carrol et al, Phys. Rev. Lett. 48 (1989) 1829; A. Shor et al., Phys. Rev. Lett. 63 (1989) 2192. [3] A. Schroeter et al., Nucl. Phys. A553 (1993) 775c. [4] A. Shor et al., Nucl. Phys. A514 (1990) 717. [5] G. Batko et al., Phys. Lett. B256 (1991) 331. -174- JAERI-Conf 99-008 [6] S.W. Huang, G.Q. Li, T. Maruyama and A. Fassler, Nucl. Phys. A547 (1992) 657. [7] B. D. Serot and J. D. Walecka, The relativistic Nuclear Many Body Problem. In J. W. Negele and E. Voigt, editors, Adv.Nucl.Phys.Vol.16, page 1, Plenum Press, 1986, and references therein. [8] K. Soutome, T. Maruyama, K. Saito, Nucl. Phys. A507 (1990) 731. [9] S. Teis et al., Phys. Rev. C50 (1994) 388. [10] G.Q. Li wt al, Phys. Rev. C49 (1994) 1139. [11] K. Weber, et al., Nucl. Phys. A539 (1992) 713. [12] T. Maruyama et al., Nucl. Phys. A573 (1994) 653. [13] Y. Sugaya et al., Nucl. Phys. A634 (1998) 115. -175- JAERI-Conf 99-008 p + Cu -»p + X d + Cu ->p + X 1 1 1 I 1 1 1 ' 1 i i i i i i i - : # 5.0 GeV I • • 5.0 GeV 0 -Jr # 10° r •~-^. -. 10' V • 4.0 GeV \ : X. ; i I j- X 10"1 \ 4.0 GeV ^ 10-1 y — f m : ..-•''' T3 3.5 GeV '. a. LU 2 10" k -2 : -- • case 1 : "\; \ .5 GeV " "^ \/\ \ : Sugaya et al. . ; i 10,-3 1 1 1 1 1 1 1 1 1 \ " 10-3 111111111 i i i i i i i 1.0 1.5 2.0 2.5 1.0 1.5 2.0 2.5 p (MeV/c) p (MeV/c) — 5.0, 4.0, 31 case L -176- JAERI-Conf 99-008 103 In n 1 I r 2 • 10 =- i l 1 • r • case 1 n l i 11 1 case II • I • Sugaya et al • o - u 10 k=" 1 . 1 i 1 "= 3 4 Ein (GeV/u) fzo &tLt&iltL\tJ£njen case 11 case II -177- JAERI-Conf 99-008 JP0050046 34. m4 Nuclear Equation of State in Heavy- ion Collisions P. K. Sahu and A. Ohnishi Division of Physics Graduate School of Science Hokkaido University Sapporo 060 - 0810, Japan Abstract We extend the baryon flow {Px/A{y)) calculation in the relativistic transport model at SIS (0.25 ~ 2AGeV) to AGS (<20.0AGeV) energies for Au + Au colli- sions. We can reproduce the flow data of the EoS and E877 collaborations and the nucleon optical potential upto 1 GeV, by reducing the strength of the vector poten- tial moderately at high relative momentum. This reduction leads to a softening of the nuclear matter equation of state at high density. 1 Introduction The nuclear equation of state (EOS) at above nuclear matter density (p > 3po) has been a great interest from the theoretical as well as experimental point of view [1] - [8]. Theo- retically, the nuclear EOS plays a crucial role in nuclear physics as well as in astrophysics, such as the maximum mass of neutron stars and dynamics of supernova explosions. Exper- imentally, baryon sidewards flow observables[9] and subthreshold particle productions [10] are mainly determined by the nuclear EOS. Especially, the baryon sidewards flow is the most promising observable to determine the nuclear EOS. To describe the heavy-ion collision data at energies starting from the SIS at GSI to the SPS at CERN, relativistic transport models have been used extensively [11] - [15]. Among them, the Relativistic Boltzmann-Uehling-Uhlenbeck (RBUU) approach is one of the most successful models. It is based on the relativistic mean field (RMF) theory, which is applicable to various nuclear structure as well as neutron star studies in a reliable manner [16]. Thus it is possible to refine the mean field part of RBUU by incorporating these knowledges. Recently, newly measured flow data have been reported [7, 9] for heavy- ion reactions at AGS energies (< 12 AGeV) for Au + Au system, which further provides the nature of nuclear forces and hence the nuclear EOS around that energy regime. The purpose of flow study is to extract the information of EOS at high densities by us- ing the mean field potential at normal nuclear density and the hadron-hadron elementary cross section data. The simple versions of RMF assume that the scalar and vector fields are represented by point-like meson-baryon coupling. This coupling leads to a linearly growing Schrodinger-equivalent nucleon optical potential in nuclear matter as a function of kinetic energy, which naturally explains the energy dependence of nucleon optical po- tential at low energies (< 200 MeV). However, simple RMF does not describe the nucleon optical potential at higher energies, where the optical potential deviates from a linear function and seems to saturate. In order to avoid this unphysical behavior described in RMF, recent sophisticated RBUU approaches invoke explicit momentum dependence of the coupling constant, i.e., the form factor of meson-baryon coupling. In our earlier work [17], we showed that the scalar and vector self energies for nucleons with momentum and density dependence are the key quantities that decide the nature of nuclear EOS. In this work, we extend our model by fixing the scalar and vector self energies with moderate momentum dependence by describing the Schrodinger equivalent potential up to the nucleon kinetic energy of IGeV and then perform our systematic study of Au + Au collisions around/beyond the AGS energy regime in comparison to the recent -178- JAERI-Conf 99-008 experimental data on the collective flow of baryons. We then extract the information of the nuclear EOS. We organize our work as follows: First we describe briefly the relativistic transport ap- proach with known constraints on the momentum dependence of the scalar and vector self energies. Then we compare the calculated flow with the experimental data systematically. Finally, we conclude with summary and discussions. 2 The transport model In the present calculation we perform the theoretical analysis along the line of a relativistic transport approach which is based on a coupled set of covariant transport equations for the phase-space distributions fh(%,p) of a hadron h [13, 18], i.e. {(n, h(x,P) - h(x,p)fh2(x,P2)fh3(x,p3)fh4(x,p4)} + (12 -> 3) + (1 -> 23) . (1) In Eq. (1) Uj?(x,p) and U£{x,p) denote the real part of the scalar and vector hadron self + energies, respectively, while [G G]i2->34<$r(n + n2 — n3 — II4) is the transition rate for the process 1 + 2 —> 3 + 4. Though in quantum many-body systems the transition rate is partly off-shell - as indicated by the index T of the ^-function - we use the semi-classical on-shell limit Y —> 0 since this approximation is found to describe reasonably well hadronic spectra in a wide dynamical regime. The hadron production(absorption) processes, 12 —> 3(1 —> 23), are described by the decay(formation) of resonances and strings. The hadron 1 2 quasi-particle properties in (1) are defined via the mass-shell constraint <$(n/JIT' — M^ ) [18] with effective masses and momenta given by M*h(x,p) = Mh + U*(x,p) , p) , (2) while the phase-space factors fk(x,p) = l-fh{x,p), (3) for fermions account for Pauli-blocking and //, = 1 for bosons. The transport approach (1) is fully specified by f/f (x,p) and U£(x,p) (fi — 0,1,2,3), which determine the mean- field propagation of the hadrons, and the r.h.s. describes the scattering and hadron production/absorption rates. The model inputs to the transport model are the nuclear mean fields U" and Us, which are related to the nuclear incompressibility K at density p0 as well as to the momentum dependence of the mean fields[l2, 13, 15, 17, 19]. In the RBUU approach - due to covari- ance - the scalar and vector mean fields have to be explicitly momentum dependent [18] for a proper description of the Schrodinger-equivalent optical potential [20] defined by 2 U,ep(Ekin) = US + UO + ~{U S - U%) + ^Ekin (4) as a function of the nucleon kinetic energy Ekm with respect to the nuclear matter rest frame. However, above Ekin = lGeV the Schrodinger-equivalent optical potential is not -179- JAERI-Conf 99-008 well known experimentally, such that the flow data from the GSI/EoS Collaboration could provide further constraints also on this quantity. In our model we use a similar Lagrangian density as considered in our earlier calcu- lation [17] for the description of nuclear matter, which has been used in the RBUU be- fore [21]. This Lagrangian contains nonlinear self-interactions of the scalar field U(a) = 2 3 A \rn\a + \Ba + \Ca where the parameters ma, B, C are calculated by fitting the sat- uration density, binding energy, effective nucleon mass as well as the incompressibility at nuclear matter density (cf. NL3 parameter set [21]). In our computations we use the energy density [17] for calculating the scalar and vec- tor potentials as a function of density. Momentum dependent potentials, furthermore, are obtained by fitting the Schrodinger equivalent potential (4) according to Dirac phe- nomenology for intermediate energy proton-nucleus scattering [20]. The scalar and vector form factors at the vertices in [13] are given by Al+P2 with As = 0.8 — 0.9 GeV and Av = 0.9 — 1.0 GeV, respectively to get a good fit to the data. The Schrodinger equivalent potential (4) is shown in Fig. l(a) as a function of the nucleon kinetic energy with respect to the nuclear matter at rest in comparison to the data from Hama et al. [20]. The solid line in Fig. 1, RBUU is for a momentum dependence with the form factors (5). The increase of the Schrodinger equivalent potential upto Ekin = lGeV fits quite well, then the potential decreases and remains almost constant at very high kinetic energies. For the transition rate in the collision term in the transport model we employ in- medium cross sections as in Ref. [21, 22] that are parameterized in line with the corre- sponding experimental data for ,/s < 2.6 GeV. For higher invariant collision energies y/s, we adopt the Lund string formation and fragmentation model [23] as in the HSD transport approach [13] which has been used for the description of nucleus-nucleus collisions from SIS to SPS energies. In the present relativistic transport approach as described in our ear- lier work [17] we explicitely propagate nucleons and A's as well as all baryon resonances up to a mass of 2.2 GeV with their isospin degrees of freedom [24, 25]. Furthermore, 7r, 7/, p, u, K, K and a mesons are propagated, too, where the a is a short lived effective resonance that describes s-wave nir scattering. For more details we refer the reader to Ref. [1, 24] and Ref. [13], respectively. 3 Comparison to experimental data and related ap- proaches We now use the same parameter sets as for the Schrodinger equivalent potential in Fig. l(a) in our flow calculations for nucleus - nucleus collisions. The calculations are performed for the impact parameter b = 6/m for Au + Au systems, since for this impact parameter we get the maximum flow which corresponds to the multiplicity bins M3 and MA as defined by the Plastic Ball collaboration [26]. We have calculated the flow by fitting a linear plus cubic term in normalized rapidity for Au + Au systems at all energies. In Fig. l(b) the flow F is displayed in comparison with the data from Refs. [7, 9, 27] for Au + Au systems. The solid line (RBUU) is obtained with the scalar and vector self energies having explicit momentum dependence, Eq. (5). The dotted line corresponds to cascade calculations for a reference. We observe that the solid line (RBUU cf. Fig. l(a)) -180- JAERI-Conf 99-008 is in good agreement with flow data [27] at AGS energy regime, although at SIS energies it overestimates a little due to strong repulsive force at around 1 AGeV, which we have discussed elaborately in our previous work[17]. We note here is that the flow rises up to 2AGeV and decreases above lAGeV for Au + Au systems and then decreases at a higher beam energy (> 2AGeV). This can be realised in the following way: the repulsive force due to the vector mean field must decrease slowly and remain constant at very high beam energy such that the Lorentz force on the particles generated by the vector field almost vanishes in the initial phase of the collision. In subsequent collisions, which are more important in the Au + Au case due to its size, the kinetic energy of the particles moving relative to the local rest frame is then in a range where the Schrb'dinger equivalent potential is determined by the experimental data [20]. We thus conclude that to explain the flow data up to < IQAGeV one needs a considerable vector potential at low energy and that one has to reduce the vector mean field at high beam energy. In other words, there is only a weak repulsive force at high relative momenta and high densities. Finally, we show in Fig. 2 the EOS associated with the momentum dependent RBUU (dashed line) that describes the flow data (cf. Fig l(b)) and corresponds to a nuclear incompressibility K ~ 340 MeV. The energy per nucleon is shown in comparison to the standard NL3 parameterization [21] (solid line). The vector part for RBUU is substantially lower at high baryon density as compared to the NL3 parameter set and as a result the EOS is soft with respect to NL3, which is slightly stiffer than our earlier proposed value [17]. 4 Summary In this work we have calculated the baryon flow in the energy range up to 20 AGeV in a relativistic transport model for Au + Au collisions. We found that in order to describe the flow data [27, 28] upto AGS energies properly, which shows a gradual decrease in the energy range of 2 - 11 AGeV, the strength of the vector potential has to be reduced moderately in the RBUU model at high relative momenta and/or densities. Otherwise, too much flow is generated in the early stages of the reaction and cannot be reduced at later phases where the Schrodinger equivalent potential is experimentally known and constrains the parameterizations of the explicit momentum dependence of U" and Us in Eq. (1). This information is essential for both theoretical as well as for experimental point of view to extend these flow studies at higher than AGS energies. We conclude from these calculation that the favorable nuclear EOS at high density is neither very soft nor very hard. This conclusion support the idea of astrophysical point of view such as neutron stars as well as supernova explosion. I would like to acknowledge the support from the JSPS, Japan. References [1] W. Cassing and E. L. Bratkovskaya, Phys. Rep. 308 (1999) 65. [2] H. Stocker and W. Greiner, Phys. Rep. 137 (1986) 277. [3] H. H. Gutbrod, A. M. Poskanzer and H. G. Ritter, Rep. Prog. Phys. 52 (1989) 1267. [4] C. Gale, G. M. Welke, M. Prakash, S. J. Lee, and S. Das Gupta, Phys. Rev. C 41 (1990) 1545. -181 JAERI-Conf 99-008 [5] J. Zhang, S. Das Gupta and C. Gale, Phys. Rev. C 50 (1994) 1617. [6] M. D. Partlan et al., Phys. Rev. Lett. 75 (1995) 2100. [7] N. Herrmann et al., Nucl. Phys. A 610 (1996) 49c. [8] J. Chance et al., Phys. Rev. Lett. 78 (1997) 2535. [9] W. Reisdorf and H. G. Ritter, Collective flow in Heavy-Ion collisions, to be published in Ann. Rev. Nucl. Sci. (1998). [10] W. Cassing, V. Metag, U. Mosel, and K. Niita, Phys. Rep. 188 (1990) 363; W. Cassing and U. Mosel, Prog. Part. Nucl. Phys. 25 (1990) 235. [11] Q. Pan and P. Danielewicz, Phys. Rev. Lett. 70 (1993) 2062. [12] T. Maruyama, W. Cassing, U. Mosel, S. Teis and K. Weber, Nucl. Phys. A 573 (1994) 653. [13] W. Ehehalt and W. Cassing, Nucl. Phys. A 602 (1996) 449. [14] S.A. Bass et al., to be published in Prog. Part. Nucl. Phys. (1998) . [15] G. Q. Li, G. E. Brown, C. H. Lee and C. M. Ko, nucl-th/9702023 (1997); nucl- th/9703040 (1997). [16] S. K. Ghosh, S. C. Phatak and P. K. Sahu, Z. Phys. A352, 457. [17] P. K. Sahu, A. Hombach, W. Cassing, M. Effenberger and U. Mosel, Nucl. Phys. A 640 (1998) 493 and references therein. [18] K. Weber, B. Blattel, W. Cassing, H. C. Donges, A. Lang, T. Maruyama and U. Mosel, Nucl. Phys. A 552 (1993) 571. [19] C. Gale, G. Bertsch and S. Das Gupta, Phys. Rev. C 35 (1987) 1666. [20] S. Hama, B. C. Clark, E. D. Cooper, H. S. Sherif and R. L. Mercer, Phys. Rev. C 41 (1990) 2737. [21] A. Lang, B. Blattel, W. Cassing, V. Koch, U. Mosel and K. Weber, Z. Phys. A 340 (1991) 207. [22] V. Koch, B. Blattel, W. Cassing and U. Mosel, Nucl. Phys. A 532 (1991) 715. [23] B. Nilsson-Almqvist and E. Stenlund, Comp. Phys. Comm. 43 (1987) 387; B. An- derson, G. Gustafson and Hong Pi, Z. Phys. C 57 (1993) 485. [24] S. Teis, W. Cassing, M. Effenberger, A. Hombach, U. Mosel and Gy. Wolf, Z. Phys. A 356 (1997) 421. [25] D. M. Manley and E. M. Saleski, Phys. Rev. D 45 (1992) 4002. [26] K. G. R. Doss et al., Phys. Rev. Lett. 57 (1987) 302. [27] N. N. Ajitanand et al., Nucl. Phys. A 638 (1998) 451c. [28] P. Braun-Munzinger and J. Stachel, Nucl. Phys. A 638 (1998) 3c. -182 JAERI-Conf 99-008 CD LU CO .O CD 1 10 100 Beam Energy (AGeV) Fig.l(a) The Schrodinger equivalent potential (4) as a function of the nucleon kinetic energy Ekin- The solid curve (RBUU) results from the momentum dependence discussed in the text. The data points are from Hama et al. [20]. (b)The flow F{y) versus the beam energy per nucleon for Au + Au collisions at b = 6 fm from our RBUU calculations. The solid line results for the parameter set RBUU, the dashed line for Cascade calculation. The data points are from the FOPI and EoS Collaborations [7, 27]. 4.5 Fig.2 The dashed line shows the equation of state for the parameter set RBUU in comparison to NL3 (solid line). The related incompressibilities are given in the parenthesis in MeV. -183- JP0050047 JAERI-Conf 99-008 35. Thermal properties of nuclear matter under the periodic boundary condition Naohiko Otuka * and Akira Ohnishi Division of Physics, Graduate School of Science, Hokkaido University, Sapporo 060-0810, JAPAN (June 18, 1999) We present the thermal properties of nuclear matter under the periodic boundary condition by the use of our hadronic nucleus-nucleus cascade model (HANDEL) which is developed to treat relativistic heavy-ion collisions from BNL-AGS to CERN-SPS. We first show some results of p — p scattering calculation in our new version which is improved in order to treat isospin ratio and multiplicity more accurately. We then display the results of calculation of nuclear matter with baryon density pb = 0.77 fm3 at some energy densities. Time evolution of particle abundance and temperature are shown. I. INTRODUCTION are not taken into account. This point is important when we discuss the statistical property of nuclear matter using microscopic transport calculation. The main aim of relativistic heavy ion collision study In this report we discuss the statistical property of nu- is acquiring the knowledge of the equation of state of nu- clear matter by the use of hadronic nucleus-nucleus cas- clear matter under extreme conditions. Experimentally cade model (HANDEL) which is developed for the calcu- the temperatures and chemical potentials at the freeze- lation of relativistic heavy-ion collisions from BNL-AGS out are discussed based on the data at BNL-AGS [1] and to CERN-SPS energies and includes multi-particle pro- CERN-SPS [2]. When we study thermodynamical char- duction. First we introduce our model and then we show acter of nuclear matter from heavy ion collisions, it is the results of calculation of nuclear matter under periodic important whether the matter formed during collision boundary conditions. reaches equilibrium or not. The fact that the transverse momentum distributions of proton and pion from the ex- periment are to be well fitted with the Boltzmann distri- II. MODEL bution with flow effects shows that the matter approx- imately reaches thermal equilibrium at the freeze-out. In our previous model (referred to as Model-B in [9]), Transverse momentum distributions are studied also by particle production is realized through using microscopic transport calculations such as JAM [3], RBUU [4], RQMD [5], UrQMD [6], ARC [7], ART [8] at (1) decay of baryonic resonances A(1232), AT*(1440) AGS energies. They can reproduce transverse spectra, and A^*(1535) which are generated by although these models has different assumptions for par- ticle production from each other. NN <-+ NR, AA, AAf*(1440), (1) In RQMD and UrQMD baryon-baryon inelastic scat- tering can produce baryonic resonances whose masses are where R denotes A(1232), W(1440) or N*(1535), below 2GeV, then many particles are produced through and the decay of baryonic resonances such as Ri —> mi + R2 + ..., i?2 —+..., where Ri and m; denote a baryonic (2) direct particle production, resonance and a meson, respectively. In addition to these processes, particles are also produced through string de- NN -> NNp, NNu>, AATT, (2) cays when the invariant mass of two incidental nucleons is much larger than the 1-pion production threshold. Mean- while only several low-lying baryonic resonances are in- NN -> AAp, iV*(1440)Aw. (3) cluded in ARC and ART. In these models, multi-particle production are realized through direct particle produc- The maximum number of pions from nucleon-nucleon tion as Ni + N2 —* Bi + #2 + mi + m2..., where N{ and collision is three in the resonance model (via NN —• Bi denote a nucleon and a baryon, respectively. The de- A(1232)JV(1440)) and six in the direct production(via tailed balance is usually violated in treating string decays NN -> A(1232)JV(1440)w). This description of particle and direct particle production because inverse processes production which is an extension of the model used in "e-mail: [email protected] 184- JAERI-Conf 99-008 ART can reproduce the average multiplicity of charged When we calculate heavy ion collisions or nuclear mat- pions in p — p collision (experimental value is 3.43 [12]). ter, we also include baryon-meson inelastic scattering , However this previous model has some defects. First, direct particle production violates the detailed valance TrJV ^ A(1232), JV*(1440), iV*(1535), (5) because our model does not contain multi-particle col- lision processes. Secondly this model may reveal incor- rect property for isospin thermalization due to the large r)N -> 7V*(1535), (6) omega production rate. The omega production is intro- duced somewhat artificially to earn the pion yield and to carry a large part of multi-particle production channels KN *-* KY,K*Y, (7) at high energies. However some particle production, for example pp —> pp7r+7r+7r+7r~7r~7r~, cannot be treated by and meson-meson inelastic scattering: the latter process in the reactions of Eq.(3) because the final state of this process must contain at least one IT0 . (8) Thirdly the particle yield is insufficient at more energetic collisions than AGS energy. as elementary processes in our model, where K = To improve these points we incorporate new frame- (K°,K+) and Y = (A,E). work of particle production into the current model. In this new model we use the same cross sections as previ- ous ones for the channel which contain less than 4 pions in exit channel (corresponding to the channels in Eqs.(l) and (2)). t 10'* 10"2 1 10 1 10 (WGeV/r] fsbl^V'tl p,.b[GtV/cl FIG. 2. Inelastic cross sections of p — p collision. The solid FIG. 1. Average multiplicity of charged pions in p — p in- lines denote the results of our calculation, while crosses denote elastic scattering calculated in our model for n~ (solid) and experimental data taken from Ref. [14] 7r+(dashed). Experimental data for 7r~(open squares with dashed error bars) and 7r (solid squares with dotted error bars) are taken from Ref. [13]. III. RESULTS OF CALCULATIONS UNDER PERIODIC BOUNDARY CONDITION The remainder of cross section is distributed to multi- pion production In the present work, we have made simulation calcu- NN —> NNmnrn, NN —> NNnnnnn, (4) lation of nuclear matter at 4 different energies (e=l,17, 1.56, 1.87, 2.24 GeV/fm3). The box is 8-fm cube, and We set an upper limit of pion number in the exit channel contains 158 protons and 236 neutrons (baryon density to 20. The ratio of cross sections for each multiplicity 3 pb=0.77/fm ). The isospin ratios and energy densities to total remainder is determined from the result of JAM correspond to those of collision system of Au+Au at in- which is the transport model based on HIJING [10] and cidental projectile momenta piab=4.0, 8.0, 12.0, 18.0 A PYTHIA [11] and also includes hadron-hadron scatter- GeV/c. ing in order to treat final state interactions of hadronic gas. As presented in Fig. 1 our new model yields suit- able number of charged pions both at BNL-AGS energies (~ lOGeV/c) and at CERN-SPS energies(~ 200GeV/c). A. Direct particle production and detailed balance The calculated inelastic cross sections for several chan- nels which are shown in Fig. 2 by solid points are also in As mentioned in the previous section our current model good agreement with experimental data. improves multiplicity and isospin ratio of pion at high -185- JAERI-Conf 99-008 energies, while the violation of detailed valance still re- FIG. 4. Time evolution of particle abundance of me- mains due to the existence of direct multi-particle pro- son(dotted line), nucleon(solid line) and baryonic reso- nance(dashed line) with KY —> nN process in nuclear mat- duction. In order to estimate the effects of this violation, 3 we check the time evolution of collision number ratio be- ter at baryon density pb=0.77/fm , and the energy density is e=1.87GeV/fm3. Asterisks, pluses and crosses are the results tween binary collision shown in Eq.(l) and direct particle without KY —> -nN, respectively. Factor 2 is multiplied to the production shown in Eqs. (2) and (4). results of meson and nucleon. In Fig.3, we show the collision number ratio of multi- The particle abundance of meson, nucleon and bary- particle production to binary collision in p — p inelastic onic resonance given in the calculations with and with- scattering. The relative numbers of multi-particle pro- out strangeness pair annihilation (the inverse process in duction are prominent for more energetic gas and at ear- Eq.(7) are displayed in Fig.4. lier stage (t < 5 fm/c) and diminish with time evolution. The sum of nucleon and baryonic resonance (these However they do not disappear after the elapse of sev- do not include hyperons) decrease monotonically when eral tens of fm/c. This means that the inverse processes we do not take account of the inverse process shown in of multi-particle scattering should not be neglected even Eq.(7). The role of this inverse process on chemical equi- after relaxation. libration is clearly seen in this figure, even if the process e=1.17GeV/fm, e=1.56GeV/lm, «=1.B7GeV/fm, e=2.24 GeV/fm3 B. Relaxation to Thermal and Chemical equilibrium We discuss temperature and particle abundance at re- laxation. First relaxations of temperature are shown in Fig.5 by symbols. In this calculation temperature is defined as the ratio of pressure to the number density of hadrons, 15 26 (9) Time(fm/c) Ph 2V Ph 2Vph f^ FIG. 3. Collision number ratio of multi-particle production to binary collision in p — p inelastic scattering in nuclear mat- where the summation is taken for all hadrons in a box. ter with baryon density ph = 0.77/fra3. Energy densities are TVh and V are the number of hadrons in a box and the e =1.17GeV/fm3(solid line), 1.56GeV/fm3(long dashed line). volume of the box, respectively, while the pressure P is 1.87GeV/fm3(short dashed line) and 2.24GeV/fm3(dotted determined from an average of diagonal component of line). energy-momentum tensor Txx and Tyy per unit volume. The importance of detailed balance can be understood in another way. Our model includes hyperon production process from TTN scattering, irN —* KY. To keep de- tailed balance the inverse process of the above scattering KY —> nN should be also included. 1 Mftson' NuclMn Baryon Ftoionanca Mason x 2 *TT"« i»*«- Tlmrtfm/f) FIG. 5. Time evolution of temperature in nuclear mat- ter with baryon density pb=0-77/fm3. Energy densities are ('j^^—.Baiyon Raton* nc* e=1.17(pluses), 1.56(crosses), 1.87(asterisks) and 2.24(open squares) GeV/fm3), respectively. Lines are results of fitting '^^^ to single exponential functions. 20 40 60 80 100 120 140 160 1B0 200 As one can see, these temperatures visibly fluctuate TlmKfm/c) around certain temperatures even in the late stage. We 186- JAERI-Conf 99-008 extrapolate a temperature T^ by fitting the time depen- resonance particles in this statistical treatment. dence of the temperature to a single exponential func- Abundances of A and £ from our calculation show a tion. We show temperature at relaxation T^ in Fig.6 as reasonable agreement with the statistical model. For the a function of energy density. particle abundance of nucleons and baryonic resonances, The results of transport calculation give too high tem- however, there is a large discrepancy between the results peratures to meet statistical calculations at the same en- in our calculation and the statistical model. Statistical ergy densities. This caloric curve exhibits a striking con- model predicts that the number of baryonic resonance is trast to the result of URASiMA [15] in which tempera- superior to that of normal nucleon, while this relation is ture is lower than that in a statistical model at a given reversed in our calculation. energy density and increasing of temperature is strongly suppressed at some temperature around 100 MeV (taking account of pion only). IV. DISCUSSION AND OUTLOOK Here we discuss probable reasons for the discrepancy from the results of transport calculation to that of the statistical model in the particle ratio (Fig.7) and tem- perature (Fig.5). Using our model, in the earlier stage many mesons are produced through direct particle pro- duction which have larger cross sections at higher ener- gies. If we include the inverse processes of direct particle production in the reactions shown in Eqs.(2) and (4), Transport Calculations non-excited but energetic nucleon pair is produced in Statistical Calculation - Statistical Calculation (Only N A") - Statistical Calculation (only N.x) - many-body scattering such as BiB2mlmi.. —» NN. In a later stage, hence, more baryonic resonance production remain than in a current treatment and may cause to Energy Density (GcV/fra5) FIG. 6. Caloric curve for nuclear matter. Dotted points are reverse the particle abundance of nucleons and baryonic the results of transport calculation, while lines denote the re- resonances. If we take account this point, our current sults of statistical calculation for gases with several sets of com- calculation should estimate lower kinetic energy and the ponent. Baryon density is pb=0.77/fm3. temperature than those in the statistical model. Once we stand for this consideration, however, the caloric curve shown in Fig.6 is incomprehensible. In order to understand these results consistently, we have to study (1) the transport calculation at lower densities where statistical mechanics without residual interactions is expected to give a more reliable value, (2) comparison between the results in the statistical model and in the transport calculation which includes only binary scatter- ing in nucleon-nucleon scattering so that one can keep detailed balance strictly, and (3) the effects of a width of unstable particles on the statistical property in statistical calculations. Tlme(fm/c) FIG. 7. Time evolution of particle abundance of nu- cleon(solid line), baryonic resonance(dashed line), A(dotted [1] L. Ahle et al.(E802 Collaboration), Phys. Rev. C59, 2173 3 line) and S(dashed-dotted line) at pb = 0.77 GeV/fm and (1999). e=1.87GeV/fm3. Straight lines show the results of a statistical [2] H. Appelshauser et o/.(NA49 Collaboration), Phys. Rev. model. Lett. 82, 2471 (1999). [3] Y. Nara, in this proceedings. [4] P. K. Sahu and A. Ohnishi, in this proceedings. In Fig. 7 we display the time evolution of particle abun- [5] H. Sorge, H. Stoker and W. Greiner, Ann. Phys.(N.Y.) dance. Horizontal straight lines represent the results of a 192, 286 (1989). statistical model which takes account Fermi-Dirac stat- [6] A. Dumitru, M. Bleicher, S. A. Bass, C. Spieles, L. Neise, ics for fermions and Bose-Einstein statics for bosons as H. Stoker and W. Greiner, Phys. Rev. C57, 6 (1998). well as the baryon number and strangeness conservation. [7] Y. Pang, T. J. Schlagel and S. H. Kahana, Phys. Rev. However, we neglected the effects of residual interactions, Lett. 68, 2743 (1992). such as exclusive volume effects, and finite life-times of [8] B. A. Li and C. M. Ko, Phys. Rev. C52, 2037 (1995). -187- JAERI-Conf 99-008 [9] Y. Nara, N. Otuka, A. Ohnishi and T. Maruyama, Prog. Bussiere. Lett. Nuov. Cim. 6, 121 (1973). Theor. Phys. Suppl. 129, 33 (1997). [14] Total Cross-sections for the Reactions of High Energy [10] X. N. Wang, Phys. Rep. 280, 287 (1997), X. N. Wang Particles, edited by A. Baldini, V. Flaminio, W. G. Moor- and M. Gyulassy, Comp. Phys. Comm. 83, 307 (1994). head and D. R. O. Morrison, (Springer-Verlag, Berlin, [11] T. Sjostrand, Comp. Phys. Comm. 82, 74 (1994). 1988). [12] V. Blobel et al., Nucl. Phys. B69, 454 (1974). [15] N. Sasaki and O. Miyamura, Prog. Theor. Phys. Suppl. [13] M. Antiunucci, A. Bertin, P. Capiluppi, M. D'Agostino- 129, 39 (1997). Bruno, A. M. Rossi, G. Vannini, G. Giacomelli and A. -188 JP0050048 JAERI-Conf 99-008 36. URASiMA £ffl Thermodynamical Properties of Hot and Dense Hadronic Gas using URASiMA abstract In this work, we investigate the time evolution and the thermalization of the infinite system of hot hadron gas using URASiMA, which is based on the two body hadron- hadron collisions. Our results show that the system reaches the stationary state which almost conserves the detailed balance. The temperature of this system is defined by the slopes of Boltzmann distributions. The information about EOS will be studied in our future plan. 1 Introduction UrQMD TIZ, String Si5 \Z Summary ~£ $) -5 189- JAERI-Conf 99-008 2 N + N-^N + N,N + N^N + A(N*), N + A(N*) ^ N + N, N + N -> AA, AA -+ NiV (A(1232) ©*), 7T + N -> A(^V), TT + TT -> p^), A(A^*) -»• N + 7r(/>, a) ^ 7?©, URASiMA fc URASiMA H ^Ex^(MCM) 1 fm/c t E802 Si -f- Al 1 4.6 GeV/c central . Extended De- tailed Balance [9] [10] (T). AGS-E802 -190- JAERI-Conf 99-008 -5 e= 0.9375 GeV/fm3 ?g£H 0.3125(20.0), 0.625(40.0), 0.9375(60.0), 1.25(80.0) [GeV/fm3} - [GeV] o 200 400 T, time [fm/c] M 2: e= 0.9375 GeV/fm3 ,N,p !o.4 - a DECAY S- elastic 10.2 j...\i NN -> NR NN -> RR 30 NR -> N N RR -> NN 200 400 f0.2 200 400 time [fm/c] I time [fm/c] multi-particle production Q. mmtmz, 5; -191- JAERI-Conf 99-008 = {N + N R) -(TT + N -+ R, n + 7T -.. /j(a) E= 0.9375 GeV/fm3 —250 CO 100 time [fm/c] 4: * 4 Summary URASiMA , UrQMD -192- JAERI-Conf 99-008 [I] M. Belkacemet. al, Phys. Rev. C58(1998) 1727 [2] N. Sasaki and 0. Miyamura, Prog. Theor. Phys. Supplement 129(1997) 39 [3] K. Kinoshita, A.Minaka and H. Sumiyoshi, Prog. Theor. Phys. 61(1979) 165 [4] K. Kinoshita, A.Minaka, H. Sumiyoshi, and F. Takagi, Prog. Theor. Phys. Supple- ment 97A(1980) [5] M. Fuki, The Study for High Energy Nucleus-Nucleus Interactions with 3- dimensional Monte Carlo simulation, Doctor Thesis(1986) [6] S. Date, K. Kumagai, O. Miyamura, H. Sumiyoshi and X. Z. Zhang, JPSJ 64(1995) 766 [7] S. Teis et. al., Z. Phys. A356(1997) 421 [8] V. Dmitrev et. al., Nucl. Phys. A459(1986) 503 [9] Gy. Wolf et. al., Nucl. Phys. A545(1992) 139 [10] Bao-An Li, Nucl. Phys. A552(1993) 605 [II] E-802 Collaboration, T. Abbott et. al., Phys. Rev. C50(1994) 1024 -193- JP0050049 JAERI-Conf 99-008 37. Abstract A new variational method is proposed with its application to infinite zero-temperature nuclear matter. In this variational method, approximate energy expressions expressed explicitly with variational functions are constructed, and the Euler-Lagrange equations are derived analytically from them. This variational method has also been applied to liquid3 He giving fairly good results. In the case of nuclear matter, however, calculated energies are much lower than the experimental value, mainly caused by undesirable long tails of noncentral distribution functions. Thus, an effective theory is proposed by adding a density-dependent correction term to the energy expressions to suppress their long tails. This effective theory includes one adjustable parameter whose value is determined so as to reproduce the experimental saturation point of symmetric nuclear matter. The EOS's of nuclear matter calculated by the effective theory with the Hamada-Johnston potential and Paris potential are rather soft, but the study of neutron stars with use of these EOS's gives reasonable results. mm }t Xffl V> h *l*ft£W*£#& t L T\ Fermi Hypernetted Chain (FHNC) ffi"^*&0 ^O&^Tti, lasKov/MWtfrW&i&'x.LtztSi'&ff)^ 5 K> h 2. spin -194- JAERI-Conf 99-008 (2-lb) spin (2-lc) spin N fc? V tf> WC Oi/>T triplet tkM (s=l) ttz\t singlet RIs (J = 0) V V , LJ ty •tit, -e^i/i - t ) + Fso(r)Vso(r) Vr'dr, (2-2) 1 dFF/r) dr m (2n-l (2-3) Jo (s = 0,1) (intrinsically-central distribution functions) (Dressed tensor correlation function) , gm(r) (Dressed spin-orbit correlation function) (i, fc(Di 2 W = fc,W + 8 s [gT(r)] FFl(r) + - s [gso(r)f FqFs(.r), (2-4a) 2 2 = 16 ^irW^r) gr(r) - [gT(r)] FFl(r)} - \ [gso(r)] FqFl{r), (2-4b) -195- JAERI-Conf 99-008 2 Fso(r) = -24 [gT(r)] FF1(r)+ j (2-4C) (11=1,2) exp(i*r,) = 1 + S,(*) + So(*) > 0, (2-5a) = 1 +T S,(*)-S0(*)>0, (2-5b) >=P [F/r)-F,(oo)]exp(*r) , Ref.4) long range 3. range J tV^ l± D(r0; /•) = exp (3-1) iftfit* i 0 7) Hamada-Johnston (HJ) v- v -V ;K Parishf" >"> -196- JAERI-Conf 99-008 soft Paris#r softcore £l £ # X ^ 300 _ . 1 1 1 1 1 1 1 ' ' 250 1 Pai-is 4 200 \ 1 150 : HJ •/- ^ 100 : v/. 50 A? Avi4; 0 „—- 1 1 I 0.0 0.5 1.0 1.5 P [fm~3] 1) V. R. Pandharipande and R. B. Wiringa, Rev. Mod. Phys. A242 (1975), 389. 2) M. Takano and M. Yamada, Prog. Theor. Phys. 91 (1994), 1149. M. Takano, T. Kaneko and M. Yamada, Prog. Theor. Phys. 97 (1997), 569. 3) M. Viviani, E. Buendia, S. Fantoni and S. Rosati, Phys. Rev. B38 (1988), 4523. 4) M. Takano and M. Yamada, Prog. Theor. Phys. 100 (1998), 745. 5) M. Takano and M. Yamada, Prog. Theor. Phys. 88 (1992), 1131. 6) T. Hamada and I. D. Johbston, Nucl. Phys. 34 (1962), 382. T. Hamada, Y. Nakamura and R. Tamagaki, Prog. Theor. Phys. 33 (1965), 769. 7) R. B. Wiringa, R. A. Smith and T. L. Ainsworth, Phys. Rev. C29 (1984), 1207. 8) M. Lacombe, B. Loiseau, J. M. Richard, R. Vinh Mau, J. Cote, P. Pires and R. de Tourreil, Phys. Rev. C21 (1980), 861. -197 JP0050050 JAERI-Conf 99-008 38. fSM Relativistic EOS table for supernova explosion and r-process \ HongShen2, ', Nankai Univ.2, \ R^RCNP4, KEK5, Abstract We provide the table of equation of state (EOS) for supernova explosion andr-process in the relativistic many body approach. We construct the relativistic mean field framework based on the relativistic Brueckner-Hartree-Fock theory. We apply the constructed RMF framework to the ground state properties of many nuclei in the nuclear chart. The agreement with experimental data is excellent and the predicition is being compared with future experiments in radioactive nuclear beam facilities such as the RI Beam Factory in RIKEN. We apply the same RMF framework, which is checked by the data of unstable nuclei in neutron-rich environment, to the derivation of the equation of state of dense matter in neutron stars and supernovae. We complete the EOS table that covers a wide range of density, composition and temperature in supernova explosion. This table enables us to perform full numerical simulations of supernova explosion starting from gravitational core collapse. Hydrodynamical simulations of core collapse are successfully performed as examples of astrophyical applications of the EOS table. 1 ) h =- X A is (t h ±mmco— t t -198- JAERI-Conf 99-008 ^ «t 2) (RBHF) (RMF) t t V 7 r 3) tf u y. -v 3 -199- JAERI-Conf 99-008 Lt, - va >ofl|TNfc£ ^i- h 'J 7° h Z. t UJB3 h t it LT v -is 3 Jt) tf< 4) * - 9 \zx -200- JAERI-Conf 99-008 &%iBM%*faitot LT Reference [1] K. Sumiyoshi, H. Kuwabara and H. Toki, Nucl. Phys. A581 (1995)725. [2] D. Hirata, K. Sumiyoshi, I. Tanihata, Y. Sugahara, T. Tachibana and H. Toki, Nucl. Phys. A616 (1997) 438. [3] H. Shen, H. Toki, K. Oyamatsu and K. Sumiyoshi, Nucl. Phys. A637 (1998) 435. [4] H. Shen, H. Toki, K. Oyamatsu and K. Sumiyoshi, Prog. Theor. Phys. ]00 (1998) 1013. [5] K. Sumiyoshi, H. Suzuki and H. Toki, A&A 303 (1995) 475. [6] K. Sumiyoshi, H. Suzuki and S. Yamada, in preparation. -201- JAERI-Conf 99-008 8 10 SL 10 CO 10 0.16 0.17 0.18 0.19 Time [sec] Fig. 1 Trajectory of mass elements during gravitational core collapse in hydrodynamical calculations with the relativistic EOS table. -202- (si) s i Sli S2 S5 m fa 12 ^ n 12 ^ ft s m 7}. B#, H min, h, d 10" i E 1S ft * r 7 kg 1O P s 2 * s ij -j Y to 1, L 10' 7 T T y T A h y t 10' *• A" G m y 10' M SJt -y IV t' K M i1 +" ^ H eV 103 D k ft /I/ mol u + s 2 S * y x" 7 cd 10 h 10' X da fi ft 7 'y T y rad leV=1.60218xl0-"J it <* ft X •y'T y sr 1 u= 1.66054x10-" kg 10"' T •y d io-2 y x c io-3 'J m S3 io-' ^? 'f ? D V io-' X / n 12^ 12 m IO- f P 1S V Hz s ' Id ^ 10 7 x A h f 2 IO-" T a i -1- y N m-kg/s * y ¥ x h D - A A 2 (I i , it. Pa N/m '< — y b r-ju ¥-, tt*. * •y' J N-m ^-t - ;u bar (a) X m, )K fct 7 h W J/s A' vu Gal i. a 1-5(3 raisswffiiu H5«, HK m%« , U — • y C As Ci K»«fii 1985ipTiJ N( = 105dyn) kgf lbf s. MPa( = 10bar) kgf/cm* atm mmHg(Torr) lbf/inz(psi) 1 0.101972 0.224809 1 10.1972 9.86923 7.50062 x 10s 145.038 9.80665 1 2.20462 fi 0.0980665 1 0.967841 735.559 14.2233 4.44822 0.453592 1 0.101325 1.03323 1 760 14.6959 ffi It lPa-s(N-s/m!)=10P(.f7X)(g/(cm-s)) 1.33322 x 10"' 1.35951 x 10-3 1.31579 x 103 1 1.93368 x 10"2 1 m2/s= 10'StC X h - ^ x ) (cmVs) 6.89476 x 10'' 7.03070 x 10-2 6.80460 x 102 51.7149 1 X J( = 10'erg) kgf'm kW> h caKItWi) Btu ft • lbf eV leal = 4.18605 J(ItMft) 1 0.101972 2.77778x10"' 0.238889 9.47813x10' 0.737562 6.24150x10" = 4.184J (jBiftfO 1 9,80665 1 2.72407 x 10-' 2.34270 9.29487 x 10 3 7.23301 6.12082x10" = 4.1855 J (15 °C) it 3.6x10' 3.67098 x105 1 8.59999x10* 3412.13 2.65522 x 10' 2.24694 xlO25 = 4.1868 JC 4.18605 0.426858 1.16279 x 10"' 1 3.96759 xlO-3 3.08747 2.61272x10" 21 m 1055.06 107.586 2.93072x10-' 252.042 1 778.172 6.58515 xlO = 75 kgf-m/s 3 1.35582 0.138255 3.76616x10"' 0.323890 1.28506 x 10 1 8.46233x10'" = 735.499 W 1.60218 x 10" 1.63377 x 10"2° 4.45050 x 10"26 3.82743 x 10M 1.51857x10"" 1.18171 x 10"" 1 Bq Ci Gy rad C/kg Sv rem ftt 1 2.70270 x 10'" 1 100 3876 1 100 3.7 x 10'° 1 0.01 1 2.58 x 10' 1 0.01 1 ti