Study of QCD critical point at high temperature and density by lattice simulations

Transcription

1 Stuy of QCD critical point at high tmpratur an nsity by lattic simulations Shinji Ejiri (Brookhavn ational Laboratory) Canonical partition function an finit nsity phas transition in lattic QCD arxiv:84.7 xqcd, July -, 8

2 QCD thrmoynamics at Havy-ion collision rimnt Important rols of lattic QCD stuy Intrsting proprtis of QCD asurabl in havy-ion collisions Critical point at finit nsity Location of th critical point? roprtis of th critical point? Larg fluctuation in quark numbr? Larg bulk viscosity? RHIC SS haron phas quark-gluon plasma phas AGS nuclar mattr color supr conuctor? q

3 atur of phas transitions Crossovr or First orr First orr phas transition wo phass coxists at c.g. SU() ur gaug thory Distribution function of plaqutt () (laqutt histogram) Gaug action artition function S g 6 sit,,, SU() ur gaug thory QCDAX, RD46, 4657 (99) histogram Histogram: f S ', DU t g ' Existnc of th critical point at : Suggst. S.E., hys.rv.d77, 458(8)

4 Canonical approach Canonical partition function Effctiv potntial as a function of th quark numbr. At th minimum, First orr phas transition: wo phass coxist. C GC,, V C ), ( ln ) ( ln ) ( ), ( ln ) ( ln ) ( V C C ) ( ln ( ) V

5 First orr phas transition lin V ( In th thrmoynamic limit,, ) * ln C (, ) / V ix stat First orr transition

6 Fugacity ansion (Laplac transformation) GC canonical partition function Invrs Laplac transformation ot: prioicity Drivativ of ln Canonical partition function,, / V C I C I GC f GC f S t ( ) g t ( ) DU t () i,, i GC GC GC, i, GC * ln C (, ) I I R Intgral Arbitrary Intgral path,.g., imaginary axis, Sal point

7 Canonical partition function (A. Hasnfratz, D. oussaint, ucl. hys. B7 (99) 59) Intgral along th imaginary axis (=) Glasgow mtho (calculating ignvalus of a matrix moifi from th quark matrix) ( B ) * ( ) ln C (, ) S. Kratochvila, h. Forcran os (LA5) 67 (5). f=4 staggr frmions, lattic 6 4 First orr phas transition: wo stats coxist f=4: First orr for all.

8 Invrs Laplac transformation Sal point approximation (vali for larg V, /V ansion) aylor ansion at th sal point. At low nsity: h sal point an th aylor ansion cofficints can b stimat from ata of aylor ansion aroun =. Sal point approximation (S.E., arxiv:84.7) z f I I I i I GC I GC i I C i i t t (),, t ln f z V Sal point: z R I Intgral Sal point t f n f f ln t! ) ( ln t n n n n n n D V n s V V /

9 Canonical partition function in a sal point approximation C GC Chmical potntial Sal point approximation, t ( z ) i f ln Vz, t () V R z Sal point: ( ) * V ln F i (, ) z C (, ) f ln t R V sal point z R F i (, F i (, ) ) i rwighting factor Similar to th rwighting mtho (sign problm & ovrlap problm) (, )

10 Calculation of th canonical partition function Simulations: Bilfl-Swansa Collab., RD7,5458(5). -flavor p4-improv staggr quarks with m77v 6 x4 lattic Approximation: Sal point approximation (/V ansion) ln t : aylor ansion up to O( 6 ) Distribution function of =f Im[ ln t ] : Gaussian typ.

11 Sal point in complx / plan Fin a sal point z numrically for ach conf. V f ln t z wo problms Sign problm Ovrlap problm

12 chnical problm : Sign problm Complx phas of t (phas) f Im ln t ( ) aylor ansion (Bilfl-Swansa, RD66, 457 ()) Goo finition (staggr quarks: 4 th root trick, /4?) Im f t V D n z z n > /: Sign problm happns. i changs its sign. Gaussian istribution Rsults for p4-improv staggr aylor ansion up to O( 5 ) Dash lin: fit by a Gaussian function ll approximat : O in th rang [-, ]. histogram of

13 Sign problm (S.E., hys.rv.d77, 458(8)) Sign problm happns whn (i) changs its sign frquntly. i F (statistical rror) Assum: Gaussian istribution Sign problm is avoi. Gaussian intgral: i F F i F F, ( F ) 4 ( F F F, F ( F ) i F F F C ) histogram with F ral an positiv (o sign problm)

14 hy Gaussian istribution? aylor ansion:.g. st trm: If nsity corrlation: not long & volum: larg, Cntral limit thorm : Gaussian istribution Vali for larg volum (xcpt on th critical point) Also s Splittorff an Vrbaarschot, arxiv:79.8, chiral prturbation thory For th cas: bcaus Vali for low nsity r Im ln t Im Diagonal lmnt: local nsity oprator f lnt 5! lnt! lnt Im, ), ( 6 O F i ), ( ~ ~ O F 6 ~ O () 4 O

15 chnical problm : Ovrlap problm Rol of th wight factor (F+i) h wight factor has th sam ffct as whn () incras. */ approachs th fr quark gas valu in th high nsity limit for all tmpratur. fr quark gas f fr quark gas fr quark gas

16 chnical problm : Ovrlap problm Dnsity of stat mtho (): plaqutt istribution ) z * ( F i ( ) F i ( ) F i ( ) F ( ) Sam ffct whn changs. ( ) ff for small linar for small linar for small

17 Rwighting for =6g - Effctiv (tmpratur) for ff F sit (Data: f= p4-staggar, m/m.7, =) f S ( ) ', DU t g ' Chang: () () Distribution: S S g S g g( ) Sg( ) 6sit otntial: ln 6 sit ln + = ( incrass) ( () incrass)

18 Ovrlap problm, ulti- rwighting hn th nsity incrass, th position of th importanc sampling changs. Combin all ata by multi- rwighting roblm: Configurations o not covr all rgion of. Calculat only whn <> is nar th paks of th istributions. Frrnbrg-Swnsn, RL6,95(989) F i (, F i (, ) laqutt valu by multi-bta rwighting pak position of th istribution <> at ach )

19 Chmical potntial vs nsity Approximations: aylor ansion: ln t Gaussian istribution: Sal point approximation f= p4-staggr, 6 4 lattic wo stats at th sam q/ First orr transition at /c <.8, q/ >. */ approachs th fr quark gas valu in th high nsity limit for all. Soli lin: multi-b rwighting Dash lin: splin intrpolation Dot-ash lin: th fr gas limit umbr nsity

20 Summary An ffctiv potntial as a function of th quark numbr nsity is iscuss. Approximation: aylor ansion of ln t : up to O( 6 ) Distribution function of =f Im[ ln t ] : Gaussian typ. Sal point approximation (/V ansion) Simulations: -flavor p4-improv staggr quarks with m/m.7 on 6 x4 lattic High limit: / approachs th fr gas valu for all. First orr phas transition for /c <.8, q/ >.. Stuis nar physical quark mass: important. Location of th critical point: snsitiv to quark mass