Quark spin polarization and Spontaneous magnetization in high density quark matter

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1 Quark sin olarization an Sontaneous magnetization in high ensity quark matter Yasuhiko TSUE Kohi Univ. Jaan With J. a Proviênia Univ. e Coimbra Portugal C. Proviênia Univ. e Coimbra Portugal H. Matsuoka Kohi Univ. Jaan M. Yamamura Kansai Univ. Jaan H. Bohr Danish Tehnial Univ. Denmark Prog. Theor. Ex. Phys. 05 Issue D0 Prog. Theor. Ex. Phys. 06 Issue D0

Introution Quark an/or haroni matter at finite ensity -low ensity haroni hase : hiral broken hase -high ensity quark-gluon or olor sueronuting hase : hiral symmetri hase Possibility of quark sin

2 Introution Quark an/or haroni matter at finite ensity -low ensity haroni hase : hiral broken hase -high ensity quark-gluon or olor sueronuting hase : hiral symmetri hase Possibility of quark sin olarization? Sontaneous quark an T. Hatsua We exet that the sin olarization leas to sontaneous magnetization. Consier a ossibility of sontaneous sin olarization in quark matter at high baryon ensity

3 Introution We show By using the NJL moel with tensor-tye four-oint interation between quarks if there exists the tensor-tye four-oint interation quark sin olarization may ours at high ensity quark matter even in hiral symmetri hase quark mass is zero f. seuovetor interation f E.Nakano T.Maruyama an T.Tatsumi PRD sin olarization isaears in hiral symmetri hase ue to quarks being massless S.Maean PTP sontaneous magnetization may ours at high ensity quark matter ue to quark sin olarization an anomalous magneti moment of quark

4 Sin olarize hase --- two-flavor ase Consier high ensity an low temerature quark matter in two-flavor Is the sin olarize hase ours at high ensity? Moel use here is the NJL moel with tensor-tye interation : L i Here Then S T quark sin olarization ours Hereafter μ=ν= are taken into aount. 0

5 Lagrangian ensity uner the mean fiel aroximation : Here ynamial quark mass hiral on. sin olarize tensor onensate Energy eigenvalue : Mean fiel aroximation 5 T S MA M M i L S M M E T

6 Thermoynami otential : Here Thermynami otential 6 T S M T E T T E T E TS N H ex log ex log temerature otential hemial T

7 Thermynami otential ev T.0 ev 0.6eV

8 8 Thermynami otential

9 Phase iagram 9 In the region with low temerature an large hemial otential the sin olarize hase 0 may be realize. The orer of the hase transition is of the seon orer from normal quark matter to the sin olarize hase.

10 0 Sontaneous Magnetization

11 Sontaneous magnetization Consier high ensity an zero temerature quark matter in two-flavor Does the sin olarization leas to The sontaneous magnetization? Tensor-tye four-oint interation in the NJL moel with the anomalous magneti moment of quark : L AMM L i A u A u N 0.97 ; e m.50 ev/t R..elie A.P.Martinez H.P.Rojas an M.Orsaria Phys. Rev. C N N 7

12 Sontaneous magnetization Let us aot the mean fiel aroximation uner the external magneti fiel B : 0 AMM LMA i D AB T By Bx D iqa A A M 0 Q Q f e for f u - quark or e for f - quark Energy eigenvalue E f ~ f ~ f Q f B f B 0 for for

13 Thermoynami otential : Sontaneous magnetization an be erive by the thermoynami relation: Sontaneous magnetization f T u f f f f E B Q B Q f f ~ max min M B M B M u B u T B for 0 for 0 0

14 Aliation : Magneti fiel of hybri omat star - magneti flux ensity Sontaneous magnetization N/A 0 R Mr r R M z R M B r r m r r m B q q z 0 0

15 Summary 5 We have shown tensor-tye four-oint interation between quarks leas to - the sin olarization of quark matter in the region of high baryon ensity an low temerature sontaneous magnetization may our ue to the anomalous magneti moment of quark uner the existene of the quark sin olarization

16 6 Bak u

17 7 Sin olarization versus olor sueronuting hase

18 Sin olarize hase --- two-flavor ase 8 Tensor-tye four-oint interation in the NJL moel : Energy eigenvalue : A A C C A C T S i i i L A C C j j j i E 5 7 5

19 9 Phase iagram

20 0 Possible hases uner the strong external magneti fiel

21 Possible hases uner strong external magneti fiel Consier high ensity an zero temerature quark matter in two-flavor uner the strong external magneti fiel What hases o aear? The ase of i Tensor-tye four-oint interation in the NJL moel with the anomalous magneti moment of quark : eb ev 0 B 0 Only Lowest Lanau Level ontributes m

22 Possible hases uner strong external magneti fiel The ase of ii Pseuovetor-tye four-oint interation in the NJL moel with the anomalous magneti moment of quark : S AMM MA U U M U M D i L 5 5 i i i L P S for for 0 M U B Q E f f

23 Possible hases uner strong external magneti fiel Only Lowest Lanau Level ontributes

24 Origin of the tensor-tye four-oint interation between quarks

25 Two-gluon exhange 5 auge interation : int L QCD g x A x x Two-gluon exhange : fourth-orer erturbation x L int QCD int x y L y z L z u L u g QCD int QCD int QCD Proer two-gluon exhange iagram : an

26 Two-gluon exhange Aroximation to NJL moel : Two-gluon exhange to four-oint interation : y x g im e M g i e g i i y A x A y x im e M i y x ab g y x i g y x i ab b a ij q y x i q ij j i 6

27 Two-gluon exhange Aroximation to NJL moel : 0 8 g q a a a T T T NJL eff M M T T g x L L u z y x g L x 7

28 8 Interlay between sin olarization an olor sueronutivity at zero temerature

29 Interlay between sin olarization an olor sueronutivity Lagrangian ensity with -flavor olor sueronutivity Mean fiel aroximation i i L f k f k T i L 0 C

30 0 Interlay between sin olarization an olor sueronutivity Hamiltonian formalism Hamiltonian for quark an antiquark where heliity isosin olor

31 Mean iel Aroximation for olorsueronuting ga Δ an sin olarization Mean fiel aroximation f f e e k k a ositive energy states a~ negative Quark matter ositive energy artiles are retaine energy states

32 Interlay between sin olarization an olor sueronutivity BCS state for ositive energy artiles where an ' " ' ' " ' ' ' ' " ' ' " ' ' ' 0 0 ~ 0 e S a a K a a K S a K K K K ~ ~ ~ ~ ~ sin sin K K K K

33 Interlay between sin olarization an olor sueronutivity Variational equations etermine θk Thermoynami otential M f V f V N H V

34 Interlay between sin olarization an olor sueronutivity 0 H M N 0 f f V Namely a equation

35 Numerial results for interlay between sin olarization an SC 5 Ouation number for SC hase ev ev Usual utoff Λ=0.6 ev is vali for this alulation

36 6 Numerial results for interlay between sin olarization an SC Pressure vs hemial otential Parameters use here

37 Numerial results for interlay between sin olarization an SC 7 Thermoynami otential

38 8 Qualitative unerstaning qualitative feature of thermoynami otential

39 Relation to fermi surfae In sin olarize hase fermi surafae has following form : torusmajour raius : minor raius: μ volume of torus: Qualitative unerstaning qualitative feature of thermoynami otential 6 N N N N 0 :normal quark matter :sin olarize hase 9

40 0 Numerial results for sin olarization Relation to fermi surfae

41 Interlay between sin olarization an olor-flavor loke hase at zero temerature

42 Stability of sin olarize hase --- three-flavor ase High ensity an low temerature quark matter in threeflavor: olor-flavor loke CL hase may exist Is the sin olarize hase survives at high ensity against CL hase?

43 Interlay between sin olarization an olor sueronutivity Lagrangian ensity with -flavor olor sueronutivity Mean fiel aroximation f k a C C f k a f k f k i i i L 5 5 f k a C ak k a ak f k a C ak M f f k k k f k k M T M M T i h i L i L L L i L *

44 Interlay between sin olarization an olor sueronutivity Hamiltonian formalism Hamiltonian for quark an antiquark where olor flavor heliity s u SP CL SP CL V V N H V V V V N H H ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ' " ' " ' ' ' ' ' ' ' ' ' ' 0 8 0

45 Mean iel Aroximation for olorsueronuting ga Δ without sin olarization =0 Mean fiel aroximation for quasi-artile oerators s u ' " ' ' ' ' ' " " ; " a a a a a a CL CL V V V N H H ; ; ; ; 9 ; ; ; ; 0 Thermoynami otential for =0 CL V V H V

46 Mean iel Aroximation for sin olarize ga without CL onensate Δ =0 s u Thermoynami otential for Δ=0 8 V s u 0 or a equations for Φ 0 CL Φ SP 6

47 7 Numerial results for interlay between sin olarization an CL Pressure vs hemial otential Parameters use here

48 8 Orer of hase transition seon orer erturbation on CL hase with reset to SP term Hamiltonian uner onsieration H H CL H SP H SP Here H SP is regare as erturbation term irst orer erturbation = 0 Seon orer erturbation E orr E i ; groun H E i i 0 E H i state energy i ; intermeia te exite state 0 i E ; exite state energy

49 9 Orer of hase transition seon orer erturbation on CL hase with reset to SP term Thermoynami otential oeffiients of an 8 are always ositive 0 0 V E orr is loal minimum 8 8 mu / ev + / 8 + /

50 50 Orer of hase transition seon orer erturbation on CL hase with reset to SP term Thus an 0 stable 0 8 Then the hase transition may be the first orer

51 Heliity of eah flavor for u >0 <0 s <0 Total heliity sin is zero beause is satisfie. Then is obtaine. Eah an total heliity 8 s u an s s u u S S S 5

52 5 Effetive otential aroah

53 Moel NJL moel 5 Nambu-Jona-Lasinio moel with tensor-tye interation At high baryon ensity hiral symmetry is restore 0 : quarks are massless an then S.Maean PTP : seuovetor onensate is zero 5 ue to quark being massless

54 Moel tensor interation 5 Tensor interation is retaine : L L kin L T Here Then 0 quark sin olarization ours Hereafter μ=ν= are taken into aount.

55 Effetive otential for sin olarization 55 enerating funtional Z Inserting auxiliary fiel k k Then finally : single-artile energy

56 Effetive otential for sin olarization 56 Effetive otential : V[] At finite ensity introue hemial otential : Then finally L L 0 where

57 Effetive otential for sin olarization 57 Effetive otential : V[] Thus Thermoynami relation : ressure : quark number ensity q

58 Effetive otential for sin olarization 58 Thermoynami relation : ressure : quark number ensity q

59 Numerial results for sin olarization 59 Effetive otential : V[] Parameters use here 0 ev with vaume olarization.ev with utoff 0.6eV

60 Numerial results for sin olarization 60 Pressure vs hemial otential or baryon number ensity Critial ensity fm

Quark spin polarization and spontaneous magnetization in high density quark matter

Quark spin polarization and spontaneous magnetization in high density quark matter Quark sin olarization and sontaneous magnetization in high density quark matter Physics Division, Faculty of Science, Kochi University, Kochi 780-8520, Jaan E-mail: tsue@kochi-u.ac.j João da Providência