Charmed meson masses in medium based on effec1ve chiral models

Transcription

1 Charmed meson masses in medium based on effec1ve chiral models Masayasu Harada (Nagoya KEK theory center workshop on Hadron and Nuclear Physics in 217 (KEK, JAPAN, January 9, 217) Based on M. Harada, Y.L. Ma, D. Suenaga, Y. Takeda, arxiv: D.Suenaga, M.Harada, Phys. Rev. D 93, 765 (216). D.Suenga, B.-R.He, Y.L.Ma, M.Harada, Phys. Rev. D 91, 361 (215) D.Suenaga, B.-R.He, Y.L.Ma, M.Harada, Phys. Rev. C 89, 6821 (214) D.Suenaga, S.Yasui, M.Harada, in prepara1on 217/1/9 Hadron and Nuclear Physics in KEK 1

2 1. Introduc,on 217/1/9 Hadron and Nuclear Physics in KEK 2

3 of Hadrons Origin of Mass? of Us = One of the Interes,ng problems of QCD 217/1/9 Hadron and Nuclear Physics in KEK 3

4 Phase diagram of Quark-Gluon system high temperature 1 trillion kelvin Early Universe Chiral Symmetry Restra,on Deconfinement of Quarks Spontaneous Chiral Symmetry Breaking Confinement of Quarks Neutron Star High density 1 million ton/cm 3 217/1/9 Hadron and Nuclear Physics in KEK 4

5 Heavy-light hadrons as probes of vacuum structue Heavy-Light Mesons (Qq type) Baryons (Qqq) - Q Light-quark cloud (Brown Muck) made of light quarks and gluons typical energy scale ~ Λ QCD << M Q Heavy mesons 3 or 3 bar, of SU(3) l Heavy baryons 6, of SU(3) l Flavor representations, which do not exist in the light quark sector, give a new clue to understand the hadron structure. In this talk, I will summarize main points of a series of papers, in which we studied spectra of charmed mesons in nuclear matter. 217/1/9 Hadron and Nuclear Physics in KEK 5

6 Outline 1. Introduc1on 2. D-D* mixing in spin-isospin correlated maeer D.Suenaga, B.-R.He, Y.L.Ma and M.Harada, Phys. Rev. C 89, 6821 (214) 3. Dispersion rela1ons of charmed mesons in spin-isospin correlated maeer D.Suenaga, M.Harada, Physical Review D 93, 765 (216) 4. Effects of par1al chiral symmetry restora1on in nuclear maeer to effec1ve masses of charmed mesons M. Harada, Y.L. Ma, D. Suenaga, Y. Takeda, arxiv: D.Suenaga, B.-R.He, Y.L.Ma and M.Harada, Phys. Rev. D 91, 361 (215) 5. Modifica1on of spectral func1on of charmed mesons in nuclear maeer D.Suenaga, S.Yasui, M.Harada, in prepara1on see Poster by D.Suenaga 6. Summary 217/1/9 Hadron and Nuclear Physics in KEK 6

7 2. D-D* mixing in spin-isospin correlated maeer D.Suenaga, B.-R.He, Y.L.Ma and M.Harada, Phys. Rev. C 89, 6821 (214) 217/1/9 Hadron and Nuclear Physics in KEK 7

8 dual chiral density wave (DCDW) Nakano and Tatsumi showed the existence of the DCDW phase with inhomogeneous chiral condensate (Nakano-Tatsumi, PRD71, 1146 (25)) in the high density maeer. qq x!" ( ) = σ x!" ( ) + iπ 3 x!" ( )τ 3 =φ cos( 2 αx ) + iτ 3 φ sin( 2αx ) An analysis using an effec1ve hadron model in [A.Heinz, F.Giacosa, D.Rischke, NPA 93, 34 (215)] showed that the phase transi1on to the inhomogeneous phase occurs at the density of 2.4 1mes of normal nuclear maeer density. 217/1/9 Hadron and Nuclear Physics in KEK 8

9 position dependent pion condensation Spin-isospin correlation in the nuclear medium will cause the position dependent pion condensation. In the equilibrium case, the pion condensation does not depend on the time. A α ( x!" ) = α A ( x!" ) = We are interested in the mass spectrum of charmed mesons, so that we take the spatial components of the residual momentum of charmed mesons to be zero. Only the space average of α contributes. A α i = d 3 x V α A ( i x!" ) A : α i α A µ = 1 π 3 ( x!" A )= φ sin( 2αx ) leads to α i 217/1/9 f π µ π A +! ; chiral field for pion = 1 f π i π A +# P-wave pion condensation = α δ i 3 δ a3 Hadron and Nuclear Physics in KEK 9

10 D-D* mixing P-wave pion condensa1on gives a mixing between D(J P = - ) and D*(J P =1 - ), as well as a mixing between 2 modes of D*. α µ = 1 f π µ π +! L int = ig A M D µ µ? D ig A µ D D causes a mixing between D and D* mesons causes a mixing between different modes of D* mesons 217/1/9 Hadron and Nuclear Physics in KEK 1

11 Mass spectrum in the heavy quark limit A α i = α δ i 3 δ a3 U(1) s X U(1) I X Z 2 [subgroup of SU(2) spin X SU(2) isospin ] exists. Mass MeV Α MeV 4 states (2 HQ pairs) : (+,+) and (-,-) related by Z 2 4 states (2 HQ pairs) : (+,-) and (-,+) related by Z 2 There are 8 states at vacuum, which are degenerated due to the existence of the heavy quark symmetry D + and D, D* + (3 states) and D* (3 states) 217/1/9 Hadron and Nuclear Physics in KEK 11

12 Inclusion of mass difference at the vacuum We include the violation of the heavy quark symmetry by the mass difference between D and D* mesons at vacuum. Meson spin denoted by SU(2) J together with the isospin SU(2) I is conserved at vacuum. In the spin-isospin correlated matter, the SU(2) J X SU(2) I is broken to a subgroup depending on the symmetry structure of the correlation. 217/1/9 Hadron and Nuclear Physics in KEK 12

13 Inclusion of mass difference at the vacuum Mass MeV SU(2) J X SU(2) I U(1) J X U(1) I X Z 2 Ø 8 states are split into 4 pairs of Z 2 symmetry. Ø 2 D mesons provide 1 pair {(,+), (,-) } Ø 6 D* provide 3 pairs : Ø {(,+), (,-)}, {(+,+), (-,-)}, {(+,-), (-,+)}. Ø 2 of {(,+), (,-)} are mixed with each other /1/9 Α MeV 1 pair 1 pair 1 pair 1 pair Hadron and Nuclear Physics in KEK 13

14 3. Dispersion rela1ons of charmed mesons in spin-isospin correlated maeer D.Suenaga, M.Harada, Physical Review D 93, 765 (216) 217/1/9 Hadron and Nuclear Physics in KEK 14

15 Chiral partner structure for charmed mesons M.A.Nowak, M.Rho and I.Zahed, PRD48, 437 (1993) W.A.Bardeen and C.T.Hill, PRD49, 49 (1994) 2 heavy quark mul1plets with J l =1/2 are regarded as the chiral partner: "D( # ),D * (1 ) $ % (((( " D * chiral partner # ( + ),D 1 (1 + ) $ % Mass difference is generated by the chiral condensate, and the value is roughly equal to the cons1tuent quark mass. Experimental value implies that the chiral partner structure seems to work: m ( + ) m ( ) m ( 1 + ) m ( 1 ).43GeV 217/1/9 Hadron and Nuclear Physics in KEK 15

16 An effec1ve Lagrangian We constructed an effec1ve Lagrangian of the rela1vis1c form based on the heavy quark symmetry and the chiral partner structure. The free Lagrangian and the interac1on with σ are expressed as: L free+ µ D@ µ D µ µ D µ µ D µ µ D 1 m 2 m 2 2 m 2 m 2 2 m 2 + m2 2 D D D D heavy quark symemtry heavy quark symemtry Spontaneous chiral symmetry breaking at zero density leads to the vacuum expecta1on value (VEV) of the σ field as <σ> = σ,which causes the mass difference between the chiral partners. Then, the coupling of σ to the heavy mesons are related to the mass difference (an extended Goldberger-Treiman rela1on): G DD = m 2 + m2 2 m2 2,... D D 1 D 1 D chiral symmetry 217/1/9 Hadron and Nuclear Physics in KEK 16

17 2 Klein-Gordon equa1ons at vacuum We determine the dispersions rela1ons of D, D*, D *, D 1 by solving the coupled equa1ons of mo1on. The equa1ons of mo1on at vacuum are ordinary Klein- Gordon equa1ons. 6 µ@ µ + m 2 MD C A + 1 m2 2 The heavy quark symmetry implies mass degeneracy. M D( ) = M D (1 ) ; M D ( + ) = M D1 (1 + ) The chiral symmetry breaking generates the mass differences: ) MD( (+ ) = MD1 (1 ) + MD (1 ) = m C7 B 1 1 D( ) D (1 ) C D ( + ) A = D 1 (1 + ) 216/7/11 RCNP 17

18 Poten1al in the DCDW The poten1al in the matrix form is expressed as V = V iv V iv iv V iv V Each component is given by 1 C A V = m2 2 cos(2fx), V = m2 2 sin(2fx) = 2fgm sin(2fx), =2fgm cos(2fx) 216/7/11 RCNP 18

19 Poten1al by σ and π mesons in the DCDW D mesons get medium correc1ons in the DCDW through the exchange of σ and π mesons. D D D * σ D * D * D* π π D D These effects are included in the equa1ons of mo1on as a poten1al obtained by replacing the σ and π fields in the Lagrangian as (x) = cos(2fx); 3 (x) = sin(2fx) Here, x is one of the 3 direc1ons of the space, f is a frequency of the density wave, and φ is a strength of the condensate 217/1/9 Hadron and Nuclear Physics in KEK 19 σ

20 E [GeV] Dispersion rela1ons with no D*Dπ interac1on 4 modes appear from (D, D * ) sector (2 par1cles at vacuum) s E 2 = m (qx +2f) 2 + qx 2 + k 2 2 y + kz 2 ± 1 [(q x +2f) 2 2 qx] m2 2 2 s E 2 = m q 2 2 x +(q x 2f) 2 + ky 2 + kz 2 ± 1 [qx 2 2 (q x 2f) 2 ] 2 +4 m2 2 f = 4 MeV (k x = k y = ) free D * q x [MeV] gray curve: free D Group velocity : v x = de v x = f E v x = 2 41 ± f E 2 41 ± dq x q x =k y =k z = 4f 2 q16f ( m 2 ) f 2 q16f ( m 2 ) > 3 5 < For -f < q x < or < q x < f, v x - q x Minimum energy at q x = f or q x = -f Only 4 modes instead of infinite modes for usual Bloch s treatment. No energy gaps. 217/1/9 Hadron and Nuclear Physics in KEK 2 2

21 Effects of chiral symmetry restora1on We consider the effects of chiral symmetry restora1on by reducing φ: qq x!" ( ) = σ x!" ( ) + iπ a x!" ( )τ a =φ cos 2 f x ( ) + iτ 3 φ sin 2f x ( ) f = 2 MeV ; =1 =.6 217/1/9 Hadron and Nuclear Physics in KEK 21 =.3 As the value of φ becomes small, the red and green curves move up, and the orange and blue curves move down. For φ =, there are only 2 modes which have a degenerate dispersion rela1on, reflec1ng the chiral symmetry restora1on. Note that 2 modes disappear since the eigenvectors vanish. =

22 Dispersion rela1ons with D*Dπ interac1on included All of (D, D*, D *, D 1 ) mix. We can solve the coupled EoM analy1cally. 8 modes appear, which is compared with 4 modes at vacuum q [MeV] f = 2 MeV (g=.5) For all the modes, v x - q x. Minimum energy is realized at q x = f or q x =-f. 217/1/9 Hadron and Nuclear Physics in KEK f = 4 MeV (g=.5) Level crossing occurs between red doeed and orange solid curves The spliung between a solid curve and a doeed curve with the same color is caused by the mixing between D and D* mesons as well as the mixing between D * and D 1 mesons.

23 4. Effects of par,al chiral symmetry restora,on in nuclear maker to effec,ve masses of charmed mesons M. Harada, Y.L. Ma, D. Suenaga, Y. Takeda, arxiv: D.Suenaga, B.-R.He, Y.L.Ma and M.Harada, Phys. Rev. D 91, 361 (215) 217/1/9 Hadron and Nuclear Physics in KEK 23

24 Par1al chiral symmetry restora1on in nuclear maeer It is expected that the chiral symmetry is par1ally restored in nuclear maeer: e.g. in linear density approxima1on f f =1 N m 2 f 2 B 1 An experiment showed the decreasing of f π in maeer K.Suzuki et al., PRL 92, 7232 (24) 2 apple f ( B ) B / f '.64 at B = 217/1/9 Hadron and Nuclear Physics in KEK 24

25 Masses of charmed mesons in nuclear maeer D D D σ D ω Effec1ve masses for D(J P = - ), D(J P = + ) in nuclear maeer 217/1/9 m (e ) D( ) = m 1 h i 2 M f m (e ) D(+) = m + 1 h i 2 M f + g!dd h! i + g!dd h! i Effec1ve masses for an1-charmed mesons m (e ) D( ) = m 1 2 M h i f m (e ) D(+) = m M h i f g!dd h! i g!dd h! i Hadron and Nuclear Physics in KEK 25

26 Effec1ve masses in linear density approxima1on h! i = g!nn m 2! B, h i h i =1 N m 2 f 2 (e ) md( ) = m 1 h i 2 M + g f!dd h! i B m (e ) D(+) = m M h i f + g!dd h! i m (e ) D( ) = m 1 2 M h i f m (e ) D(+) = m M h i f g!dd h! i g!dd h! i An example g!dd =3.7, g!nn =6.23, N = 45 MeV B / g!dd g!nn < D(+) D(+) D(+) D(+) D( ) D( ) D( ) D( ) 217/1/9 Hadron and Nuclear Physics in KEK g!dd g!nn > Increasing or decreasing of pseudo-scalar D(-) meson mass only is not enough for measuring the par1al chiral symmetry restora1on. 26

27 h i h i =1 Effective Mass Difference MeV Par1al chiral symmetry restora1on N m 2 f 2 B m (e ) D( ) = m 1 2 M h i f m (e ) D(+) = m D(+) D( ) Ρ B Ρ M h i f g!dd h! i + g!dd h! i m (e ) D( ) = m 1 2 M h i f m (e ) D(+) = m D( )+ D( ) M h i f g!dd h! i g!dd h! i D(+) + D(+) In addi1on to study the mass difference of chiral partners, taking average of par1cle and an1-par1cle will give a clue for par1al chiral symmetry restora1on. Threshold energy for produc1on of D and an1-d meson pair in medium is larger than vacuum reflec1ng the par1al chiral symmetry restora1on. 217/1/9 Hadron and Nuclear Physics in KEK 27

28 Nuclear maeer from a parity doublet model In [Y. Motohiro, Y.Kim, M.Harada, Phys. Rev. C 92, 2521 (215)], we constructed a mean field model of Walecka type, using a parity doublet model for nucleon, which reproduces the proper1es of nuclear maeer at normal nuclear maeer density. Here we use the model to determine the mean fields of sigma and omega. g! DD < Ρ Ρ g! DD > D(+) D(+) D(+) D(+) D( ) D( ) D( ) D( ) m (e ) D( ) = m 1 h i 2 M + g f!dd h! i m (e ) D(+) = m /1/9 Hadron and Nuclear Physics in KEK 28 M h i f m (e ) D( ) = m 1 2 M h i f m (e ) D(+) = m Density dependence is quite similar to the one in the linear density approxima1on. M h i f + g!dd h! i g!dd h! i g!dd h! i

29 Nuclear maeer from Skyrme crystal Y. L. Ma, M. Harada, H.K. Lee, Y. Oh, B. Y. Park, M. Rho, PRD88 (213) 1416 Y. L. Ma, M. Harada, H.K. Lee, Y. Oh, B. Y. Park, M. Rho, PRD9 (214) 3415 M. Harada, H.K. Lee, Y.-L. Ma, M. Rho, PRD91, (215) 9611 d 3 x σ D.Suenga, B.R.He, Y.L.Ma. M.Harada, PRD 91, 361 (215) In the Skyrme crystal model, there exists the half-skyrmion phase, where the space average of the sigma condensate vanishes. Effective Mass MeV g! DD < 217/1/9 Hadron and Nuclear Physics in KEK 29 σ g! DD > D(+) D(+) D( ) D( ) D(+) D(+) D( ) D( ) Ρ Ρ Chiral partners are degenerate with each other in the half-skyrmion phase. Proper1es in the normal nuclear maeer are similar to the other cases. density

30 5. Modifica,on of spectral func,on of charmed mesons in nuclear maker D.Suenaga, S.Yasui, M.Harada, in prepara1on 217/1/9 Hadron and Nuclear Physics in KEK 3

31 Inclusion of fluctua1on modes We include the fluctua1on modes of σ and π to study the spectral func1on of charmed mesons. 217/1/9 Hadron and Nuclear Physics in KEK 31

32 Modifica1on of masses - Mass of D and D are not changed from mean field level or : D or D meson mass with mean field or : D or D meson mass with one loops Mass [MeV] [/fm3 ] 217/1/9 Hadron and Nuclear Physics in KEK 32

33 Spectral func1on 1 ρ B /ρ = ρ 2 B /ρ =.41 ρ B /ρ = q [MeV] Landau damping Posi1on moves to higher energy region. Hight is enhanced 2 Threshold enhancement Posi1on moves to higher energy region. Hight is enhanced. 3 Decay of D* Dπ Collisional broadening. Posi1on moves to lower energy region. q [MeV] /1/9 Hadron and Nuclear Physics in KEK 33 q [MeV]

34 6. Summary We proposed to study the mass spectrum of heavy-light mesons to probe the symmetry structure of nuclear maeer. Existence of spin-isospin correla1on generates a mixing among D and D* mesons. In addi1on to study the mass difference of chiral partners, taking average of par1cle and an1-par1cle will give a clue for par1al chiral symmetry restora1on. Threshold energy for produc1on of D and an1-d meson pair in medium is larger than vacuum reflec1ng the par1al chiral symmetry restora1on. Spectral func1on is also modified by the effect of par1al chiral symmetry restora1on. Our results imply that studying charmed meson spectrum will give clues to understand the chiral symmetry structure of nuclear maeer. 217/1/9 Hadron and Nuclear Physics in KEK 34

35 217/1/9 Hadron and Nuclear Physics in KEK 35