Critical statics and dynamics of QCD CEP

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1 Ciical saics and dynamics of QCD CEP Eiji Nakano Dep. Physics, Kochi Univesiy, and ExeMe Mae GSI,Gemany w/ B. Fiman, V. Skokov, K. Redlich, and B-J. Schaefe

2 Conen: 1. Moivaion. Basics on ciical phenomena 3. Fncional Renomalizaion Gop 4. Chial effecive heoy and Isenopes aond CEP 5. Ciical dynamics 6. Smmay

3 1, Moivaion Ciical End Poin (CEP) exiss o no? 0 A( Hadon) B( QGP) A B( CEP) CEP: a coninos phase ansiion poin (3D-Ising nivesaliy)

4 ~ nl ng

5 Relaivisic Heavy Ion Collision Ealy ime hemalizaion ~1fm/c isenopic evolion (Hydodynamics) [ collecive moion and had pobe ]

6 Seach fo QCD CEP in fe expeimen; GSI/CBM exp. S / N ~ cons.

7 Why is CEP expeced o appea? In MFA, Chial ansiion along empeae axis (high T expansion): V one loop ~ a 4 a b 4 (1 #log[ T c ] #log[ ]) c T 0 whee ~ 0 Finie chemical poenial hampes his cancelaion a fis ode ansiion somewhee a finie densiy Ode paamee fo CEP: ~ CEP

8 Low enegy flcaions and non-linea ineacion develops vey mch nea CEP (IR poblem) Non-pebaive eamen, sch as laice QCD. Poenial V 4 4 MFA+flcaions someime beaks LETs. Fncional RG

10 , Basics on ciical phenomena (Ising model) Kadanoff s block spin agmen Coelaion lengh ξ a phase ansiion poin 1/ ~ ~, ~, ~, ~ h C h C C T T T ), ( ), ( h hb b F b h F s d s b: e.g., Block spin size in ni of a Hypohesis of geneal homogeneiy :

11 Z D e D D e D e Z k d L k k d L k T H E E / T log H Z T A geneal effecive heoy wih UV coff : Oiginal heoy and low enegy effecive heoy Wilson s enomalizaion gop mehod (Z sym. heoy)

12 RG ansfomaion = seps wih a paamee b 1 UV k / b 1) Inegae he momenm shell in loop coecions / b k / b / b IR 0 ) Change he lengh scale fo all vaiables (zoom o) ~1/ k 1/ bk Repea 1) and ) = RG ansfomaion flows in,

13 Non-ivial fixed poin (a ciical poin ): Flow eqaions :(below ciical dimension ) Scaling dimensions (fom Lineaized RG) 4 d ) ( 36 ) ( * * O O ) ( ) ( 1 ) (,0) ( ), ( ) ( / 3 ~ b F b h F O s d s

14 Physical qaniy: A b A Close o F.P., and fo lage b no oo lage n fo LRG: a1 C a1 A A b O( / / {a}, { b n a1 a1 A {a} * * {a a} { a};a, a};a ), b n a~ / {Ca1, * C a1;a 0 Scaling woks only asympoically! a { b a n a 1 *,a a, b,a n 1/ a 3 a~, Sepaae agmens ino elevan and ielevan ones. n imes RG ~ a};a a1 ~ a};a * * a( a1) 0

15 Wilson s RG + epsilon expansion povides only nivesal popeies scaling dimensions of elevan vaiables, and h scaling elaions among exponens No non-nivesal qaniies de o asympoic adjacence o FP Exapolaion (non-pebaion eamen) o 3D, njsified in geneal. Assme Boel smmabiliy? d 4 d 3 Fncional FG woks a d=3! becase hisis non-pebaive fomalism.

16 3, Fncional Renomalizaoin Gop Effecive Aveage Acion Fncional Z in Eclidean space: Geneaing fncional fo conneced Geen fncions: Effecive acion (Legende ansfom):

17 Scale dependen fncional: Flcaions wih q < k ae sppessed by a scale dependen mass em: R k (q) Scale dependen effecive acion: 0 k q

18 Flow eqaion:

19 Flow eqaion looks like a one-loop fom, say wih fll -poin fncion. Flow eqaion fo -poin fncion involves 4- and 6- poin fncions... Tip1: Deivaion of flow eqaion mch mch easie fom PI fomalism!

20 Typical fom of C-off fncion Rk(p) Peak a k conibes o flow evolion!

21 Easy o exend involving Femions: Opimized c-off fncions (analyic inegaion a finie empeae and densiy):

22 So fa nohing has been done. Js an exac ansfomaion fom inegal o diffeenial fom in evalaing fncional Z. We need an appoximaion o make flow eqaion be of a closed fom. Thee ae some appoximaions, we employ `` deivaive expansion + LPA 1 ( ) k Uk ( ) Zk ( ) O 4 ( ) This is siable fo ciical phenomena whee long-ange physics does mae. This is a pojecion of fncional space ono leading ode of deivaive expansion wiho spoiling non-pebaive nae of FRG. ciical exponens and hei scaling elaions ae epodced vey well. good even away fom ciical poin (good descipion of non-nivesal popeies) easy o handle!

23 4, Chial effecive heoy and Isenopes aond CEP Possesses he CEP and he same nivesaliy class as QCD CEP (3D Ising model)!

24 Deivaive expansion fo QM model: Finie empeae Conibions fom Bosonic and Femionic pas. /

25 Flow eqaion fo hemodynamic poenial of QM model a finie T and μ : whee Bose and Femi disibion fncions:

26 Gid mehod: Bonday condiion a c-off scale: U k ( ) 1 m 1 4 Scale evolion by flow eqaion 4 c U( Φ) Fo T<Tc Φ U k0 ( ) Convex fncion! 0 Λ Flow k Paamees ae fixed o povide low enegy physics: Pion decay cons. fπ and meson masses.

27 Taylo expansion aond minimm (mch simple han Gid mehod): Flow eqaions fo bea fncions.:

28 s Isenopes : cons. n

29 Fom Fncional RG appoach o a Chial model CEP Invese of Sigma mass ~ sscepibiliy

30 Pevios woks on Isenopes Mean-Field Appoximaion (MFA) o effecive heoies Linea Sigma model, Scavenios eal, PRC01 Linea Sigma model wih Polyakov loop, Kahaa and Tominen, RPD08 MFA shows `Kink' behavio along he cossove line. Tip: The kink is an aifac fom 1 s ode-ansiion behind cossove egion.

31 PNJL (MFA), Fkshima 09

32 Focsing effec, by Nonaka and Asakawa PRD04 Consced Enopy (Bayon nmbe) Densiy: 3D Ising Univesal fncion wih Amplide Paamees, which inepolaes Hadon and Qak-Glon enopies. e.g., AniPoon/Poon aio vs P, Asakawa eal RRL08, Lo eal PLB09.

34 Laice QCD (Taylo exp. abo μ/t ), Ejii eal PRD06 Smooh? (Kink = laice aifac) B had o see if he CEP exiss.

35 5, Ciical dynamics Ciical dynamics of QCD CEP = ha of Liqid-gas ciical poin (Son & Sephanov `04) Review on Liqid-gas ciical dynamics (Siggia-Hohenbeg-Halpen 77) Relaion o isenopic evolion

36 Sochasic eqaion of moion (Langevin eq.) A ( k) i Mode-mode coplings : Slow modes {Ode paamee, conseved vaiables} : Kineic (anspo) coefficiens Lij ( k) H H({ A i }) : e.g., Ginzbg-Landa Hamilonian A i A i Li ( k) ~ e Whie noise (jsified fo long ime scale) The EoM descibes how modes elax o hei eqilibim configlaions.

37 A simple case of Langevin eqaion fo a scala heoy d H d wih In Gassian case ( 0 ) d k k k d ( ) (0) H ( ) 3 k 0 fo non -conseved OP ( 0k k * fo conseved OP) 1 k c ( ) k k 0 k 0 k a a k Z 1 ( ) k Gk 0 Ciical limi: k 1 Dynamical ciical exponen: Z (4) * If hee ae mode-mode coplings, hey may enomalize λ o be singla.

38 Model H (Liqid-Gas ciical poin) : choice of slow modes and Hamilonian Slow modes ; (k) A i n : Ode paamee(paicle densiy) J : Tansvese momenm e : Enegy densiy Symmey+Deivaive expansion(p o maginal ems) H H S d 1 d x J T C J H H T T a linea combinaion of 0 1 Specific hea e and n T S The ohe modes ae decopled a long-disance limi.

39 In case of Liqid-Gas ciical poin, wha is he slowes ode-paamee? e n ~ e C 0 n 0 p ~ p 0 n Ts 1 C p seves as ``mass of OP, which vanished a ciical poin as he same manne of densiy sscepibiliy. Liqid- Vapo Ising model The mos singla appoach o ciical poin is he isenope!

40 PB ], [ J mmc Η g J Η J J Η g Η Eqaion of moion fo Liqid-gas:

41 Dynamical enomalizaion gop (Hohenbeg-Halpein) Basic idea : RG ansfomaion Fixed poins of EoM Saic case :mass :copling Dynamical case : mass, :copling Tanspo coefficiens E :enegy - diffsion cons. :shea viscosiy cons. : OP elaxaion ae : blk viscosiy cons. k Z Z : Dynamical ciical exp. of he slowes mode, obained fom Fixed poin.

42 Wilson s RG + epsilon expansion ( d 4 ) Flow eqaion fom Kbo fomla and EoM (o leading ode of ε): J T f j J T fo 3-poin veex Fom above 3 flow eqaions, only fixed poins scaling laws

43 Isenopic evolion Model H : Liqid- Gas ciical poin 1 C p C p ~ `Mass vanishes= enopy flcaion has he lages ciicaliy! * finie * f * j finie Z 4 x ' 3 a d 3 No scaling Scaling law: x y ' y ~, ~ x, y x 18 O( ) 19 1 O( ) 19 OP flcaion ansfes momenm! J Noe: ansvese momenm neve eaches a finie FP no scaling (Z= mch fase!)

44 As fo QCD CEP, densiy and chial OP mix de o finie mass and chemical po. (Fjii 04). B, densiy dominaes as he slowes mode a long-ime-lengh limi, wheeas chial OP is a slave mode (Son-Sephanov 04). Isenopic appoach? Model H and QCD CEP Lacey e al. Helim gas-liqid ciical poin (Model H)

45 6, Smmay FRG o Chial Effecive Theoy and QCD CEP Model-dependen-Regla pa dominaes S and N Model H dynamical nivesaliy (if isenopic evolion)

46 Backps

47 ) / ( ] [ 4 ] [ ] [ ) / ( ] [ 0 ] / [ d / 4 d h d d T H h x T H i i d Scaling dimension (in lengh) of bae fncional

48 ), ( ), ( b b 0 *) *, ( *) *, ( 1, 1, 1, sysem eigen i i i b v Mv M, ielevan 1 elevan 1, 1 i i i v, *) *,, ( i X i i b b i i X dimension of Scaling : ) 1 ( ~ 1) ( ~ 1 1 Evalaion of ciical exponens:scaling fields and dimensions Coninos limi b 1 Lineaize RG eqaion abo F.P.: *, * Fixed poins:

49 Physical qaniy: A b A Close o F.P., and fo lage b no oo lage n fo Linea RG: a1 C a1 O( A A b / / {a}, { b n a1 a1 A {a} * * {a a} { a};a, a};a ), b n a~ / {Ca1, * C a1;a 0 Scaling woks only asympoically! a { b a n a 1 *,a a, b,a n 1/ a 3 a~, Sepaae agmens ino elevan and ielevan ones. n imes RG ~ a};a a1 ~ a};a * * a( a1) 0

50 1 1/ * 1/ * ~, whee,, n n n Δ ν wih b C C b b Fo insance, scaling of coelaion lengh a h=0 is given by In his case, hee ae only independen exponens fo (, h) and he ohes ae nde consains; scaling elaions: ') ( ' ~, ~ Domain Coelaion lengh

51 Bosonic pa: Masbaa sm, analyic hanks o opimized c-off fncion

52 Femionic pa: Masbaa sm, analyic hanks o opimized c-off fncion

53 Fll acion Oiginal acion (classical/mf)

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation

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