Modiﬁca(on	  of	  spectrum	  of	  heavy-­‐
light	  mesons	  in	  nuclear	  medium	
Masayasu	  Harada	  (Nagoya	  Univ.)	
at	  “Hadrons	  and	  Hadron	  Interac(ons	  in	  QCD”	  	  	  	  	  	  (YITP,	  Kyoto,	  March	  2,	  2015)	

Based	  on	  ・ D.Suenaga,	  B.-­‐R.He,	  Y.L.Ma	  and	  M.H.,	  	  	  Phys.	  Rev.	  C	  89,	  068201	  (2014);	  	  Phys.	  Rev.	  D	  91,	  036001	  (2015)	  	  INTRODUCTION	☆ Heavy-Light Mesons (Qq- type) Baryons (Qqq)

Q“Light-quark cloud” (Brown Muck)・・・ made of light quarks and gluons
typical energy scale ～ ΛQCD << MQ◎ Heavy mesons ・・・ 3 or 3bar, … 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.

Heavy-light mesons as a probe to thestructure (symmetries) of nuclear matter	uIn this talk, I shall show main results of D.Suenaga,B.-R.He, Y.L.Ma and M.H., Phys. Rev. C 89, 068201(2014); Phys. Rev. D 91, 036001 (2015)• In PRC89, 068201 (2014), we propose to study themass spectrum of the heavy-light mesons to probe
the structure of the spin-isospin correlation in nuclearmatter.
– The spin-isospin correlation generates a mixing among
the heavy-light mesons carrying different spins andisospin such as D+, D0, D*+ and D*0 mesons.
– The structure of the mixing reflects the pattern of thecorrelation, i.e. the remaining symmetry.
– The magnitude of the mass modification provides
information of the strength of the correlation.

Mass degeneracy of heavy-light mesons asa probe to the half-Skyrmion phase	• In D.Suenaga, B.-R.He, Y.L.Ma and M.H.,
PRC89, 068201 (2014), we explored the mass
splitting of heavy-light mesons with chiralpartner structure in nuclear matter.
– We regard [D(JP=0-),D*(1-)] and [D*0(0+),D1(1+)] as
chiral partners.
– We construct the nuclear matter by putting
Skyrmions onto a crystal.
– Our results show that the masses of chiral partners
are degenerated in the half-Skyrmion phase in which
the chiral symmetry is restored globally.

Outline	
1. Introduction
2. Heavy meson effective fields
3. D-D* mixing in Spin-Isospin
correlated Nuclear matter
4. Mass degeneracy in the halfSkyrmion phase
5. Summary

Heavy meson effective fields	

☆ Heavy quark symmetry
・・・ a symmetry of QCD at MQ → ∞ limit◎ velocity super-selection rulegluon

heavy quark

V µ = (1,0,0,0) at rest frame
The velocity of a heavy quark is
not changed by the QCD interaction.

◎ Heavy quark number conservation
No pair production of heavy quarks by QCD interaction.
◎ SU(2) spin symmetry
QCD interaction cannot flip the spin of heavy quarks.

☆ Heavy-Light Mesons (Qq- type)
Q“Light-quark cloud” (Brown Muck)・・・ made of light quarks and gluons
typical energy scale ～ ΛQCD<< MQ

angular momentum
carried by “Brown muck”◎ spin of meson

spin of heavy quark・ MQ → ∞ limit

same Ｊｌ 	Q(↑)

conservation of Jl⇒ classification of hadrons by Jl

Q(↓)

Heavy Meson Multiplet
・・・ degenerate masses

◎ Parity of Mesons	・ Quark model ・・・ Brown muck ⇔ qangular momentum	
spin of light quark	Parity of meson ・・・	◎ classification of mesons	A heavy meson multiplet includes

・	⇒ JP = (0- , 1-) ・・・ ground states(0+ , 1+) ・・・ excited states	

・	⇒ JP = (1+ , 2+)・・・・	states	・・・ excited states	

Heavy meson effective fields☆ Ground states ・・・ Jl =1/2 ; JP = (0- , 1-)
Pseudoscalar meson D ; Vector meson D*
D = ( D 0, D + )D* = ( D*0, D*+ )・ Bi-spinor field; Ψ ・・・ light constiuent quark fieldannihilates heavy mesons (not generate)☆ First excited states ・・・ Jl =1/2 ; JP = (0+ , 1+)
Scalar meson D0* ; Axial-vector meson D1
D 0* = ( D 0* 0, D 0* + )D 1 = ( D 10, D 1+ )

D-D* MIXING IN SPIN-ISOSPIN
CORRELATED NUCLEAR MATTER	
based onD.Suenaga, B.-R.He, Y.L.Ma and M.H.,
Phys. Rev. C 89, 068201 (2014)	

Model (heavy-light mesons)	
• Heavy-light mesons made of a heavy c quark and (u, d)quarks1+ γ νV ν ! *;µ# ; (a = 1,2 : isospin index)
Ha =Dγ+iD
γaµa5"$2• Symmetries (at rest frame: Vν = (1,0,0,0) )– heavy-quark spin : SU(2)• representation : Sh = ½– spin of Brown-muck : SU(2)• representation : Jl = ½– isospin : SU(2)• representation : I = ½• Degeneracy– 2×(2Jl+1)×2 = 8 states make (2Jl+1)×2 = 4 HQ pairs• An HQ pair belongs to the representation Sh=1/2. 	

Chiral	  Lagrangian	  for	  pion	
◎ Chiral symmetry breaking
SU(Nf )L×SU(Nf )R → SU(Nf )V◎ Basic Quantity

U=ea2iπ T a/F π→ gL U g †R ;gL,R∈ SU(N f) L,R◎ Lagrangian
2Fπ	µ†
L = 4 tr［ ∂µ U ∂ U ］	i π aTa /Fπ††U = ξ L ξR ; ξR = ξ L = e;
ξR† →gR ξR† h † ( gR ,gL , π );
ξL → h(gR ,gL , π )ξL gL†; h(gR ,gL , π ) ∈ SU(Nf )V()

L =Fπ2 tr !"α ⊥µ α ⊥µ #$ ; α ⊥µ = ∂µξR ⋅ ξR† − ∂µξL ⋅ ξL† / (2i )Interaction to the light sector	
• Lagrangian
L = tr !"H iV µDµ H #$ + g A tr !"H γ µγ 5α ⊥µ H #$()– gA = 0.56 determine from D* → D π decay3• covariant derivativeDµ H =∂µ H − i α||µ H

1α ⊥µ = ∑α ⊥AµTA ; TA = τ A
2
A=1

position dependent pioncondensation	
• Spin-isospin correlation in the nuclear medium will cause theposition dependent pion condensation.
• In the equilibrium case, the pion condensation does notdepend on the time.
!"!"AA
α ⊥0 x = α||0 x = 0!"
A• α||i x does not contribute at the rest frame.

()

()

()

• We are interested in the mass spectrum of the heavy-lightmesons, so that we take the spatial components of theresidual momentum of the heavy-light mesons to be zero.
• Only the space average of α⊥ contributes.	
!"1 AA
3Aα ⊥i = ∫V d x α ⊥i x : α ⊥iA =∂i π +#fπ()

P-wave pion condensation	

Inverse propagator matrix	
# Db[i ,j ] %*i
Lagrangian → Da ,Da Δab% Db* j$(#[i ,j ] % Δ abΔab =% Δi$ ab&((')&j−Δab(Δijab ('Δab = p 0 δabcauses a mixing
among D and D* mesons	Δiab = −i g A α ⊥i A (τ A )abΔiabj = p 0 δab −i g A ε ijk α ⊥k A (τ A )ab

causes a mixing among
different modes of D* mesons	Mass spectrum (pattern 1)	
• α ⊥i A = α δi A motivated by the hedgehog ansatz of skyrmion

Nucleon as Skyrme soliton	Skyrme model

hedgehog ansatz	

αia

2$!
F ' (sinF )
2sinF
a 1
3#&= δi ∫ d x+ F ' cosF +3
1+ cosF &%#" r

Mass spectrum (pattern 1)	
• α ⊥i A = α δi A motivated by the hedgehog ansatz of skyrmion
6detΔ = ( p 0 − g Aα ) ( p 0 + 3g Aα )2Mass "MeV#SU(2)diag [diagonal subgroup of SU(2)spin X SU(2)isospin] exists.	
100

3 HQ pairs : triplet of SU(2)diag 	

0!100!200!300!400
0100!Α! "MeV#
200

300

400

1 HQ pair : singlet of SU(2)diag 	Mass spectrum (pattern 2)	
• α ⊥i A = α δi 3δA3

motivated by the chiral density wave
See, e.g.,
A. Heinz, F. Giacosa, and D. H. Rischke, NPA933, 34 (2014);
D. Mller, M. Buballa, and J.Wambach, PLB727, 240 (2013);
T. Kojo, Y. Hidaka, L. McLerran, and R. D. Pisarski, NPA843, 37 (2010);
E. Nakano and T. Tatsumi, Phys. Rev. D 71, 114006 (2005)	

Chiral density wave	
• Several literatures pointed that the chiral field U
has the position dependent expectation value inthe nuclear medium.
• Here I take the following ansatz as an example:!"
U x()

⇔

= cos (2 α z ) − i τ 3 sin (2 α z )!"U x = 1 at vacuum()ξR = ξL† = cos ( α z ) + i τ 3 sin (α z )• This leads toαia = α δi 3δa 3

Mass spectrum (pattern 2)	
• α ⊥i A = α δi 3δA3

motivated by the chiral density wave
4detΔ = ( p 0 − g Aα ) ( p 0 + g Aα )4

U(1)s X U(1)I X Z2 [subgroup of SU(2)spin X SU(2)isospin] exists.	
150

2 HQ pairs :(+,+) and (-,-) related by Z2 	Mass "MeV#100
500!502 HQ pairs :(+,-) and (-,+) related by Z2 	!100!150
0100!Α! "MeV#200

300

400

Inclusion of mass difference at thevacuum	• We include the violation of the heavy quark
symmetry by the mass difference between Dand D* mesons at vacuum.	
• Meson spin denoted by SU(2)J together withthe isospin SU(2)I is conserved at vacuum.
• In the spin-isospin correlated matter, theSU(2)J X SU(2)I is broken to a subgroupdepending on the symmetry structure of thecorrelation.	

Mass "MeV#Inclusion of mass difference at the vacuum(pattern 1) 	• Pattern 1 : SU(2)J X SU(2)I → SU(2)diagØ 6 D* mesons are split into 2 and 4 of SU(2)diag:3⊗2= 2⊕4
Ø 2 D mesons make 2 of SU(2)diag which mixes withanother 2 from D*.
2200

4 states	

2100

2 states	

2000
1900
1800
1700

2 states	

1600
0100!Α! "MeV#
200

300

400

Inclusion of mass difference at the vacuum(pattern 2) 	• Pattern 2 : SU(2)J X SU(2)I → U(1)J X U(1)I X Z2
Ø 8 states are split into 4 pairs of Z2 symmetry.
Ø 2 D mesons provide 1 pair {(0,+), (0,-) }Ø 6 D* provide 3 pairs :Ø {(0,+), (0,-)}, {(+,+), (-,-)}, {(+,-), (-,+)｝.
Ø 2 of {(0,+), (0,-)} are mixed with each other
Mass "MeV#

2200
2100

1 pair	2000

1 pair	

1900

1 pair	

1800

1 pair	

1700
0100!Α! "MeV#
200

300

400

MASS DEGENERACY IN THE HALFSKYRMION PHASE
based onD.Suenaga, B.-R.He, Y.L.Ma and M.H.,
Phys. Rev. D 91, 036001 (2015)	

Nuclear matterfrom Skyrmions on a crystal 	• We put Skyrmions onto the face-centered cubic (FCC)
crystal and regard the Skyrmion matter as nuclear matter.
e.g. M. Kugler and S. Shtrikman, PLB 208, 491 (1988); PRD 40, 3421 (1989).
H. -J. Lee, B. -Y. Park, D. -P. Min, M. Rho and V. Vento, NPA 723, 427 (2003).

z• There appears the “half-Skyrmion phase”,
which is characterized by σ = d 3 x U = 0 ⇔ d 3 x qq = 0∫∫Vσ••density	

B.-Y. Park, H.-J. Lee, V. Vento, J.-I. Kim, D.-P.
Min and M. Rho, NPB. Suppl. 141, 267 (2005).
H. -J. Lee, B. -Y. Park, D. -P. Min, M. Rho and V.
Vento, NPA 723, 427 (2003).

See also•
Y.-L. Ma, M. Harada, H. K. Lee, Y. Oh, B.Y. Park, M. Rho, PRD88, 014016 (2013);
PRD90, 034015 (2014).
•M. Harada and H.K.Lee, Y.-L. Ma and M.
Harada, arXiv:1502.02508.

Chiral partner structure of Heavylight mesons	u To Probe the half-Skyrmion phase, we use the chiralparner structure of heavy-light mesons.
u Ground states ・・・ Jl =1/2 ; JP = (0- , 1-)Ø Pseudoscalar meson D ; Vector meson D*Ø D = ( D0, D+ , Ds)

D* = ( D*0, D*+ , Ds*)chiral partner	
M.A.Nowak, M.Rho and I.Zahed, PRD48, 4370 (1993);
W. A. Bardeen and C. T. Hill, PRD49, 409 (1994)u First excited states ・・・ Jl =1/2 ; JP = (0+ , 1+)
Ø Scalar meson D0* ; Axial-vector meson D1
Ø D0* = ( D0*0, D0*+ , Ds0*)

D1 = ( D10, D1+ , Ds1 )Heavy meson effective fields☆ Ground states ・・・ Jl =1/2 ; JP = (0- , 1-)
Pseudoscalar meson D ; Vector meson D*
D = ( D 0, D + )D* = ( D*0, D*+ )・ Bi-spinor field; Ψ ・・・ light constiuent quark fieldannihilates heavy mesons (not generate)☆ First excited states ・・・ Jl =1/2 ; JP = (0+ , 1+)
Scalar meson D0* ; Axial-vector meson D1
D 0* = ( D 0* 0, D 0* + )D 1 = ( D 10, D 1+ )

Heavy meson Lagrangian	◎ chiral doubler fields for heavy mesons	

gL,R∈ SU(2) L,R◎ chiral field for pion	 U =e 2i π /fπ = ξ 2 →gL U gR†☆ model Lagrangian	ΔΜ term generates themass difference between(D, D*) and (D0*, D1).

Mideum effects to the heavy meson
chiral doublet fields	
☆ Lagrangian written in the parity eigenstates	•We are interested in the mass spectrum of the heavy-light mesons at the!"rest frame, so that we take the space average.
3e.g. ∂iU = ∫ d x ∂iU x†††• We can show ∂iU − ∂iU= ∂iUU − ∂iU U = 0

so that only ΔM term contributes.
•U +U † → 0 in the half-Skyrmion phase,so that G doublet and H doublet are degeneratewith each other.

V()Summary	• We proposed to study the mass spectrum ofheavy-light mesons to probe the structure of thespin-isospin correlation in the nuclear medium,
which is expected to occur in the skyrmion crystal
or the chiral density wave phase.
• Our result shows that the structure of the mixing
among D and D* mesons reflects the pattern ofthe correlation, i.e. the remaining symmetry.
• Furthermore, the magnitude of the massmodification provides information of the strength ofthe correlation.

Summary 2	
• We explore the mass splitting of heavy-light mesons
with chiral partner structure in nuclear matter.
• We regard [D(JP=0-),D*(1-)] and [D*0(0+),D1(1+)] as
chiral partners.
• The nuclear matter is constructed by puttingSkyrmions onto a crystal.
• The masses of chiral partners are degenerated in thehalf-Skyrmion phase in which the chiral symmetry isrestored globally.
• Our results show that the medium modified mass
splitting of H and G doublets can be used as a probeof the existence of the half-Skyrmion phase.

Remarks	
•[B. Friman, J. Phys. G 30, S895 (2004)] and [C. Sasaki, Phys. Rev.
D 90, 114007 (2014)] pointed that the degeneracy occurs when thechiral symmetry is restored. In the half-Skyrmion phase, although
the chiral symmetry is not restored, the masses are degenerated
since the space average of the quark condensate vanishes.

d 3x
σ =∫
U = 0 ⇔ ∫ d 3 x qq = 0
V• In the present analysis, we included only the pion effect. The effectsof the heavier resonances, such as ρ, ω, σ and so on which areessential for nuclear matter properties, on the heavy-light meson
spectrum will be reported elsewhere.
• In the present analysis, we studied heavy meson mass defined at
the rest frame to probe the global structure of the nuclear matter. It
would be interesting to see the dispersion relation of the heavy
meson to probe the local structure of the matter, i.e., the potential
provided by the nuclear matter. (D. Suenaga, Y.-L. Ma and M.
Harada, work in progress.)