カイラル相転移臨界点におけるクォークスペクトル
Quark spectrum near the critical point of chiral transition

根本幸雄（聖マリアンナ医大）
共同研究者：北沢正清（阪大）, 国広悌二（京大）
熱場の量子論とその応用 2013.8.26.

Our previous study on the quark spectrum
How do the fluctuations of the chiral condensate
affect the quark spectrum near Tc?

 model: Nambu-Jona-Lasinio model (2-flavor, chiral limit)
 phase diagram of the chiral transitionchiral restored
2nd order in the low density region
1st order in the high density regionchiral broken

spectrum of the fluctuations of the chiral condensate(Hatsuda-Kunihiro 85)

ωT = 1.1Tcµ=0m

msmσp−msσ ,π-modes

ms (T )mπTc

T(Kitazawa-Kunihiro-YN 07)

quark spectrumfluctuations

quark self-energy: Σ( p0 , p) :quark spectral function:
three-peak structure

ρ+=++…

T = 1.05TC , µ = 0

The scattering off the fluctuations
forms the three-peak structure.

EIm Σ +p0Ep0
Contour ofthe spectral funcion
Re Σ +

red lines:

ω − | p | −ReΣ + =0

This study: FINITE current quark mass
phase diagram of the chiral phase transitioncurrent quark mass: 5.5 MeV
The pseudo-critical line isdetermined from a maximum ofthe spectral
function for p=10 MeV (dynamicchiral susceptibility).

critical point (CP)masses of the sigma, pion, and dynamical quarkµ=0 MeVµ=µCP

Soft mode near CP
The sigma meson has still a non-zero mass at CP,
because the chiral symmetry is explicitly broken.
What is the soft mode at CP?
The soft mode is not the sigma mode,
but appears in the space-like region.
scalar density fluc.
〈𝑞𝑞� 𝑞𝑞 𝑝𝑝, 𝑝𝑝0 𝑞𝑞𝑞𝑞
� 0,0 〉

softneing𝜇𝜇 = 𝜇𝜇𝐶𝐶𝐶𝐶 , 𝑇𝑇 ∼ 𝑇𝑇𝐶𝐶𝐶𝐶

sigma meson, 𝑚𝑚𝜎𝜎 ∼ 2𝑚𝑚𝑞𝑞(Fujii 03, Fujii-Ohtani 04)

pseudo-scalar density fluc.
〈𝑞𝑞𝑖𝑖𝛾𝛾� 5 𝜏𝜏𝜏𝜏 𝑝𝑝, 𝑝𝑝0 𝑞𝑞� 𝑖𝑖𝛾𝛾5 𝜏𝜏𝜏𝜏 0,0 〉NOT softneingpion pole: not shownpion dispersion relations in medium[MeV]

𝜇𝜇 = 𝜇𝜇𝐶𝐶𝐶𝐶 , 𝑇𝑇 ∼ 𝑇𝑇𝐶𝐶𝐶𝐶[MeV]

𝜇𝜇 = 0, 𝑇𝑇 ∼ 𝑇𝑇𝑃𝑃𝑃𝑃 (pseudo-critical)

𝑚𝑚𝑞𝑞 : dynamically generated (constituent) quark mass (mean field)

pion dispersion relation:

𝐸𝐸𝜋𝜋 𝑝𝑝 ≠𝑝𝑝2 + 𝐸𝐸𝜋𝜋 0 2c.f: fermion and gauge boson in HTLQuark spectrum near CP•Quark spectral function 𝜌𝜌± for p=0
ρ ± ( p0 ,0) = −1πImscalar fluc.

self-energy

Σ=1p0 + µ  mq − Σ ± ( p0 ,0)pseudo-scalar fluc.

+soft modespace-likesigma modetime-like (cont.)

ex. pion pole contribution
Λ𝑚𝑚Im Σ+ 0, 𝑝𝑝0 ∼ ∫ 𝑑𝑑𝑑𝑑 1 −𝐸𝐸+ 1−+ 1++ 1+𝑞𝑞

𝑚𝑚

𝐸𝐸𝑞𝑞

𝑚𝑚

𝐸𝐸𝑞𝑞

𝑚𝑚

𝐸𝐸𝑞𝑞

s, ps fluctuations=++…

pi modetime-like(pole or cont.)
pion pole residue

BE dist. Func.

FD dist. Func.

𝑍𝑍 𝐸𝐸𝜋𝜋 𝑞𝑞 𝛿𝛿 𝑝𝑝0 − 𝐸𝐸𝑞𝑞 + 𝜇𝜇 − 𝐸𝐸𝜋𝜋 (𝑞𝑞)

1 + 𝑛𝑛 𝐸𝐸𝜋𝜋 (𝑞𝑞) − 𝑓𝑓 𝐸𝐸𝑞𝑞 − 𝜇𝜇

𝑍𝑍 𝐸𝐸𝜋𝜋 𝑞𝑞 𝛿𝛿 𝑝𝑝0 + 𝐸𝐸𝑞𝑞 + 𝜇𝜇 − 𝐸𝐸𝜋𝜋 (𝑞𝑞)𝑛𝑛 𝐸𝐸𝜋𝜋 (𝑞𝑞) + 𝑓𝑓 𝐸𝐸𝑞𝑞 + 𝜇𝜇

𝑍𝑍 𝐸𝐸𝜋𝜋 𝑞𝑞 𝛿𝛿 𝑝𝑝0 − 𝐸𝐸𝑞𝑞 + 𝜇𝜇 + 𝐸𝐸𝜋𝜋 (𝑞𝑞)𝑍𝑍 𝐸𝐸𝜋𝜋 𝑞𝑞 𝛿𝛿 𝑝𝑝0 + 𝐸𝐸𝑞𝑞 + 𝜇𝜇 + 𝐸𝐸𝜋𝜋 (𝑞𝑞)𝑛𝑛 𝐸𝐸𝜋𝜋 (𝑞𝑞) + 𝑓𝑓 𝐸𝐸𝑞𝑞 − 𝜇𝜇

1 + 𝑛𝑛 𝐸𝐸𝜋𝜋 (𝑞𝑞) − 𝑓𝑓 𝐸𝐸𝑞𝑞 + 𝜇𝜇

Quark spectrum near the critical point
quark spectrum:
one peak at 120 MeVshift by coupling withthe soft modethe other peak at 80 MeV
but small residue ～0.01
self-energy
large imaginary around 200 MeV

𝜌𝜌+Re Σ+through the below process(−𝑞𝑞,⃗ 𝑝𝑝0 − 𝐸𝐸𝑞𝑞 )space-like(0, 𝑝𝑝0 )(𝑞𝑞,⃗ 𝐸𝐸𝑞𝑞 )on-shelldivergence at ±80 MeV

van Hove singularitiesIm Σ+

𝜇𝜇 = 𝜇𝜇𝐶𝐶𝐶𝐶 , 𝑇𝑇 ∼ 𝑇𝑇𝐶𝐶𝐶𝐶

residue: 0.01

𝑚𝑚𝑞𝑞 = 185

van Hove singularity
van Hove singularity = divergence of density of states
density of states 𝐷𝐷(𝐸𝐸)
𝐷𝐷 𝐸𝐸 𝑑𝑑𝑑𝑑 =�

𝐸𝐸<𝐸𝐸 𝑝𝑝 <𝐸𝐸+𝑑𝑑𝑑𝑑

𝑑𝑑 3 𝑝𝑝 ∼1𝑑𝑑𝑑𝑑|𝛻𝛻𝑝𝑝 𝐸𝐸 𝑝𝑝 |𝐷𝐷 𝐸𝐸 diverges when 𝐸𝐸 𝑝𝑝 has a maximum.
ex: plasmino in HTL

joint density of states
𝐷𝐷 𝐸𝐸 𝑑𝑑𝑑𝑑 =�

𝐸𝐸<𝐸𝐸𝑓𝑓 −𝐸𝐸𝑖𝑖<𝐸𝐸+𝑑𝑑𝑑𝑑

𝑑𝑑 3 𝑝𝑝 ∼1|𝛻𝛻𝑝𝑝 𝐸𝐸𝑓𝑓 − 𝐸𝐸𝑖𝑖 |𝑑𝑑𝑑𝑑

𝐸𝐸𝑓𝑓 − 𝐸𝐸𝑖𝑖

𝐷𝐷 𝐸𝐸 diverges when 𝐸𝐸𝑓𝑓 (𝑝𝑝) − 𝐸𝐸𝑖𝑖 𝑝𝑝 has a maximum.
present case𝐸𝐸𝑓𝑓𝐸𝐸𝑖𝑖