Toward the understanding of QCD phase structure at finite temperature and density

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Cameron Mills
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Hatsuda3 K. Kanaya Y. akagawa1 H. Ohno H. Saito and T.

1 Toward th undrstanding o QCD phas structur at init tmpratur and dnsity Shinji Ejiri iigata Univrsity HOT-QCD collaboration S. Ejiri1 S. Aoki T. Hatsuda3 K. Kanaya Y. akagawa1 H. Ohno H. Saito and T. Umda5 1 iigata Univ. Univ. o Tsukuba 3 IKE Billd Univ. 5 Hiroshima Univ. Osaka Univ. 1/1/

2 QCD phas structur at high tmpratur and dnsity Lattic QCD Simulations has transition lins T quark-gluon plasma phas Equation o stat Dirct simulation: Impossibl at. HIC LHC SS HIC E-scan J-AC AGS dconinmnt? hadron phas chiral SB? nuclar mattr quarkyonic? color supr conductor? q

3 Histogram mthod roblm o Complx Dtrminant at Boltzmann wight: complx at Mont-Carlo mthod is not applicabl. Distribution unction in Dnsity o stat mthod (Histogram mthod) X: ordr paramtrs total quark numbr avrag plaqutt tc. ( X m T ) DU δ( X-Xˆ ) dtm ( m ) Expctation valus O O 1 DU O dt M ( m ) Z g ( ) ( m T ) Sg ( ) ; 1 Z [ X ] ( ) dx O[ X ] ( X m ) T m T Z complx S ( m T ) dx ( X m T )

4 (β m )-dpndnc o th Distribution unction Distributions o plaqutt (1x1 ilson loop or th standard action) ˆ ( ) 6 sit β ( β m ) DUδ( - ˆ ) dtm ( m ) ( β β m m ) ( β m ) ( β ) m S g (wight actor) 6 ˆ sit β 6sit ( β β ) ( ) ' δ ( ˆ dtm ) ( m ) - dtm δ ( - ˆ ) ( m ) ( β ) ( β ) 6 sit ( β β ) ( ) ' dtm m dtm ( m ) ' Ectiv potntial: V indpndnt o β ( β m ) [ ( βm )] V ( β m ) ( β β m m ) ( ) 6 ( β β ) sit + ( m ) ( m ) dtm dtm

5 Sign problm ( m ) ( m ) dt i dtm X ixd dt dtm ( m ) ( m ) M M X ixd : complx phas o (dtm) Sign problm: I i changs its sign rquntly ( X ) ~ i dtm dtm ( m ) ( m ) X ixd << (statistical rror)

6 Ovrlap problm O 1 1 O ( X ) dx xp( V ( ) ( )) X + O dx Z Z V ( X ) ( X ) is computd rom th histogram. Distribution unction around X whr V ( X ) ( O) is minimizd: important. V must b computd in a wid rang. V ( X ) ( O) V ( X ) ( O) + Out o th rliabl rang rliabl rang I X-dpndnc is larg. rliabl rang

7 Distribution unction in qunchd simulations Ectiv potntial in a wid rang o : rquird. laqutt histogram at K1/mq. Drivativ o V at β sit 5 β points qunchd. dv dv dv/d is adjustd to β5.69 using β β1 6 β sit β d d Ths data ar combind by taking th avrag. ( ) ( ) ( ) 1

8 Distribution unctions or and Expctation valu o olyakov loop and its suscptibility by th rwightuing mthod at. 3 lattic χ κ β * β + 8 K κ β * β + 8 K I () is a Gaussian distribution Th pak position o () (<><>) Th width o () suscptibilitis χ χ I () hav two paks irst ordr transition ~ χ

9 V Z -dpndnc o th ctiv potntial Crossovr ( X T ) ( T ) dx ( X T ) Corrlation gth: short V(X): Quadratic unction V ( X ) Critical point X: ordr paramtrs total quark numbr avrag plaqutt quark dtrminant tc. ( X ) Corrlation gth: long Curvatur: Zro T QG 1 st ordr phas transition hadron CSC? Two phass coxist Doubl wll potntial

10 Quark mass dpndnc o th critical point nd ordr Qunchd 1 st ordr hysical point ms 1 st ordr Crossovr mud hr is th physical point? Extrapolation to init dnsity invstigating th quark mass dpndnc nar Critical point at init dnsity?

11 Distribution unction in th havy quark rgion Hopping paramtr xpansion ( K ) ( ) dt M dt M (HOT-QCD hys.v.d8 55(11)) t 3 t ( 88 K + 1 K ( cosh( T ) + i sinh( T ) ) + ) sit study th proprtis o (X) in th havy quark rgion. rorming qunchd simulations + wighting. ind th critical surac. Standard ilson quark action + plaqutt gaug action S g 6sitβ lattic siz: 3 5 simulation points; β : plaqutt +ii : olyakov loop dt M ( ) 1 s I phas

12 Ectiv potntial nar th qunchd limit() HOT-QCD hys.v.d8 55(11) at phas transition point Qunchd Simulation (mq K) K~1/mq or larg mq Critical point or : Kcp.658(3)(8) T c m π. V ( β m ) [ ( βm )] V ( β m ) ( β β m m ) dtm: Hopping paramtr xpansion ( ) 6 ( β β ) + irst ordr transition at K changs to crossovr at K >. sit ( m ) ( m ) dtm dtm

13 Ordr o th phas transition olyakov loop distribution Ectiv potntial o on th psudo-critical lin at Th psudo-critical lin is dtrmind by χ pak. χ Doubl-wll at small K irst ordr transition Singl-wll at larg K Crossovr Critical point : κ. 1 5

olyakov loop distribution in th complx plan 6 κ. 5. 1 () 5 κ κ 1. 1 I I I κ 1.

14 olyakov loop distribution in th complx plan 6 κ () 5 κ κ 1. 1 I I I κ κ. 1 5 κ I I critical point I on βpc masurd by th olyakov loop suscptibility.

15 Distribution unction o at init dnsity Hopping paramtr xpansion ( β K ) ( β ) Adopting ( β K ) DU δ( ˆ ) dtm ( κ) xp Ectiv potntial: β β + 8 K V ˆ 6 ( ) sit ( ( ) ) ˆ t 3 t 6 β β + 88 K 1 K cosh( T ) [ ˆ ] sit s + i ; β K ( ( ) ) t 3 t K sinh T ˆ 1 s ( ; β K ) ( ; β K ) I V t 3 t ( β K ) V ( β ) 1 K cosh( T ) V ( β K ) has-qunchd part i ; β s K has avrag i ; β K V is V () whn w rplac t 3 t (at V ( ) ( ) ) β K V β 1 s K K K t t cosh ( T )

16 Sign problm: I changs its sign Cumulant xpansion Odd trms vanish rom a symmtry undr ( ) Sourc o th complx phas I th cumulant xpansion convrgs o sign problm. + + C C C C i i i 3! 1 3! 1 xp Avoiding th sign problm (SE hys.v.d77158(8) HOT-QCD hys.v.d8 158(1)) rror) (statistical ixd << i + C C C C cumulants <..>: xpctation valus ixd and. i : complx phas M Im dt ( ) I s T K t t sinh 1 3

17 Convrgnc in th larg volum (V) limit Th cumulant xpansion is good in th ollowing situations. I th phas is givn by o corrlation btwn x. atios o cumulants do not chang in th larg V limit. Convrgnc proprty is indpndnt o V although th phas luctuation bcoms largr as V incrass. Th application rang o can b masurd on a small lattic. hn th distribution unction o is prctly Gaussian th avrag o th phas is giv by th scond ordr x x n C n n i n i! xp x n C n x n x i i i n i x x x! xp ) ( ~ V O x C n x C n C i 1 xp

18 β * 5.69 K ( μ) sinh( μ T ). Cumulant xpansion i 1 C K + 1! C 1 6! ( μ) sinh( μ T ). 5 6 C + K ( μ) sinh( μ T ). 1 Th ct rom highr ordr trms is small nar th critical point o phas-qunchd part. K t t ( ) K ( μ) cosh( μ T ) K ( μ) sinh( μ T ) cp t cp cp > ~.

19 Ect rom th complx phas actor olyakov loop ctiv potntial or ach at th psudo-critical (β K). Solid lins: complx phas omittd i.. Dashd lins: complx phas is stimatd with ( μ T ) sinh sinh K t cosh ( T ) ( T ) cosh( T ) t 3 ( 1 s K t sinh( T ) ) I V ( ) V ( ) V ( ) + 1 i :ixd :ixd Th ct rom th complx phas actor is vry small xcpt nar.

d8 55(11)) 88 sit dt dt M ( K ) ( ) t t ( 88 K + 1 K + ) M 3 sit s K t ( ) K ( ) cosh( T ) t cp cp t 3 μud μ t t s (

20 Critical lin in +1-lavor init dnsity QCD Th ct rom th complx phas is vry small or th dtrmination o Kcp. +1 ( dt M ( Kud )) dt M ( K s ) 3 ( dt M ( ) ) at : Kcp.658(3)(8) (HOT-QCD hys.v.d8 55(11)) 88 sit dt dt M ( K ) ( ) t t ( 88 K + 1 K + ) M 3 sit s K t ( ) K ( ) cosh( T ) t cp cp t 3 μud μ t t s ( K + K ) + 1 K cosh + K cosh + ud s s ud T s T Th critical lin is dscribd by K t ud Critical lin or uds μ cosh T ud + K s t μ cosh T s K t cp( ) Critical lin or ud s

21 Distribution unction in th light quark rgion HOT-QCD Collaboration in prparation (akagawa t al. arxiv: ) rorm phas qunchd simulations Add th ct o th complx phas by th rwighting. Calculat th probability distribution unction. Goal Th critical point Th quation o stat rssur Enrgy dnsity Quark numbr dnsity Quark numbr suscptibility Spd o sound tc.

22 robability distribution unction by phas qunchd simulation prorm phas qunchd simulations with th wight: ( ' ' β m ) DU δ( ' ) δ( ' ) dtm ( m ) dtm DU δ S g ( m ) ˆ ˆ ( ) ( ˆ ) δ( ˆ ) i ' ' dtm ( m ) S g S g i ' ' ( ) xpctation valu with ixd : plaqutt sit ( ' ' β m ) dt dt M M histogram ( ) ( ) Im dt M Distribution unction o th phas qunchd. ( ) ( ) ( ) ˆ DU ˆ ˆ 6 β ' ' δ ' δ ' dtm sit

23 -dpndnc o th ctiv potntial Curvatur o th ctiv potntial Crossovr [ ( β) ] Critical point [ ( ) ] β [ ] i + T QG 1 st ordr phas transition [ ( ) ] β phas ct Curvatur: Zro [ ] i hadron CSC + phas ct Curvatur: gativ

24 Curvatur o th ctiv potntial I th distribution is Gaussian sit sit ( ) xp ( ) 6 sit 6sit ( ) xp πχ χ πχ χ χ sit ( ) 6 χ sit ( ) ( ) ( ) 6sit ( ) ( ) χ χ sit at th pak o th distribution

25 Complx phas distribution should not din th complx phas in th rang rom -π to π. hn th distribution o q is prctly Gaussian th avrag o th complx phas is giv by th scond ordr (varianc) i 1 xp C () () o inormation C π Gaussian distribution Th cumulant xpansion is good. din th phas ( ) Th rang o is rom - to. π dt M Im dt M ( ) T dt ( ) Im ( T ) π π M d T

26 Distribution o th complx phas ll approximatd by a Gaussian unction. Convrgnc o th cumulant xpansion: good. 1 xp i 1 1 at th pak o in ach simulation β

27 Simulations 8 3 lattic. 8 m π m ρ Simulation point in th (β /T) -lavor QCD Iwasaki gaug + clovr ilson quark action andom nois mthod is usd. ak o () or ach

28 Curvatur o th ctiv potntial - Th curvatur or dcrass as incrass.

29 Ect rom th complx phas apidly changs around th psudo-critical point.

30 Critical point at init zro curvatur: xpctd at a larg.

31 Curvatur o th ctiv potntial ithout th complx phas ct ( ) ( ) χ ( ) sit χ sit

32 has avrag nd ordr cumulant i β

33 Curvatur o th ctiv potntial Th ct o th phas incrudd. zro curvatur Critical point

34 ak position o () Th slops ar zro at th pak o (). ( ) ( ) ( ) ( ) ( ) i sit i + + β β + β + β β 6 ( ) ( ) ( ) β β ( ) ( ) ( ) ( ) i i + + β + β β I ths trms ar cancld ( ) ( ) ) (const. β β (β ) can b computd by simulations around (β). ) (

35 QCD phas diagram ( ) i β m ( β ) m phas-qunchd QCD Τ Τ init-dnsity QCD i pion condnsd phas mπ/ color suprconductor phas?

36 Summary studid th quark mass and chmical potntial dpndnc o th natur o QCD phas transition. Th shap o th probability distribution unction changs as a unction o th quark mass and chmical potntial. To avoid th sign problm th mthod basd on th cumulant xpansion o is usul. Our rsults by phas qunchd simulations suggst th xistnc o th critical point at high dnsity. To ind th critical point at init dnsity urthr studis in light quark rgion ar important applying this mthod.

37 Backup

38 Complx phas Gaussian distribution Th cumulant xpansion is good. din th phas Th rang o is rom - to. At th sam tim w calculat as a unction o ( ) ( ) Im dt dt dt dt M M M M ( ) T dt ( ) Im ( T ) ( ) T dt ( ) ( T ) Th rwighting actor is also computd M M d T d T C ( ) dt dt M M ( ) T dt ( ) T ( T ) M d T

39 Distribution unction or and ˆ 6 ( ) sit ( β κ) DU δ( ˆ ) δ( ˆ ) dtm ( κ) ( β K ) ( β ) δ ( ) M ( ) ( ) ( K ) -ˆ β β δ -ˆ 6 dt sit δ ( -ˆ ) δ( -ˆ ) dtm ( ) ( β K ) ( β K ) 6 sit ( β β ) ˆ dtm ( K ) dtm ( ) V Ectiv potntial Hopping paramtr xpansion ( β κ) V ( β ) 6( β β ) paramtrs in V: V is th sam as V () whn 1 paramtr in : t 3 t ( + 88 K ) 1 K cosh( T ) ( β κ) V V has-qunchd part K t ( ; β κ) ( ; β κ) i sit s ( ( ) ) t 3 t K sinh T ˆ 1 s * β + 8 K β K ( T ) t cosh sinh K K t t cosh ( T ) t t ( T ) K cosh( T ) tanh( T ) < K cosh( T ) i I

40 Ordr o phas transitions and Distribution unction ( β κ) DU δ( ) δ( ) dtm ( κ) V ( ; β κ) ( ; β κ) dv d dv d ak position o : ˆ ˆ ( ) 6sit Lins o zro drivativs or irst ordr crossovr 1 intrsction irst ordr transition 3 intrsctions

is nglctd ( ) t 3 t + 88 K sit 1 s K cosh( T ) dv d d 3 d 1 t t s K cosh T constant

41 V dv Drivativs o V in trms o and has-qunchd part: whn ( β κ) V ( β ) 6( β β ) ( β K ) dv ( β) dv 6( β β ) ( β K ) dv ( β) + K ( ) sit 88 d d masurd at κ dv d constant shit i is nglctd ( ) t 3 t + 88 K sit 1 s K cosh( T ) dv d d 3 d 1 t t s K cosh T constant shit dv Contour lins o and at (βκ) (β) corrspond to d dv d th lins o th zro drivativs at (βκ).

42 dv dv d dv d lins o and in th () plan ( β) * 6 ( β β + 8 K ) ( β β ) 6 sit d sit dv d ( β ) t 3 1 s K t cosh T dv dv d d K3.E-5 K.E-5 K1.E-5 K. b* dv d Small K: lins o : S-shap irst ordr Larg K: lins o dv : straight lin crossovr d

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

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