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Neutron star and low energy QCDPhys.Rev.C 101 (2020) 035201Phys.Rev.C 102 (2020) 025203
arXiv: 2007.08098
Internal seminar
2nd September 2020, Yonsei University
Kie Sang Jeong in collaboration with
Dyana Duarte, Saul Hernandez, Srimoyee Sen, and Larry McLerran
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Motivation – astrophysical observation
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• 2𝑀⊙ neutron star (PSR J1614-2230, Nature 467 (2010) 1081, P. Demorest et al.)
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• Tidal deformability estimated from GW170817 (LIGO-Virgo)
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Motivation – astrophysical observation
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Outline
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I. Massive neutron stars and quasi-hadron picture• Quantum hadron dynamics and mean-field phenomenology
• Low energy QCD symmetry breaking and sum rules for quasi-baryon states
II. Gravitational wave observation and quarkyonic matter concept• Large 𝑁𝑐 QCD in dense medium
• Excluded-volume model
III. Three-flavor extension of excluded-volume model
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• More observation of 2M⊙ states from a compact binary
• To reach such a massive state, hard enough EOS is required
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Massive states
▪ Tolman-Oppenheimer-Volkof equation
▪ Intrinsic strong repulsive interaction is required
(Nat.Astron.4 (2019) 72 H. T. Cromartie et al.)(Science 340 (2013) 1233232 J. Antoniadis et al.)
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Nuclear symmetry energy
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• For continuous (infinite) matter
• Quasi-particles?
In continuous matter, nuclear potential can be understood as self-energy of quasi-nucleon
Energy dispersion relation can be written with self-energies (Relativistic mean-field theory)
(Near quasi-pole)
Similarly, from equation of state
If one assume linear density dependent potential, the symmetry energy can be easily read off from potential
dependent on (ρ, I)
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Nuclear symmetry energy – stiff or soft?
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• Nuclear phenomenology (RMFT) (Phys. Rept. 410 (2005) 335 V. Baran et al.)
Iso-spin scalar channel (attraction) becomes weaker at dense matter ⇒ stiffly increases
If symmetry energy is stiff, neutron sea will become unstable at dense regime• leads to nn → pΔ- type scattering• iso-spin density becomes lower• subsequently changed hadron yield such as
π−/π+ from final state will become smaller
NLρδ model (Iso-spin dependent interaction)
Σ𝛒
Σδ
NLρδ
NLρ
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Hyperons and EOS
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• Hyperons (S≠0) in medium (In vacuum, MN~940 MeV, MΛ~1115 MeV, MΣ~1190 MeV)
• If hyperon energy is a similar order of nucleon energy? (ρ>ρ0, I=1)
At normal density
Λ is bounded (V ~ -30 MeV) Σ potential is repulsive (V ~ +100 MeV)
(PRC38 (1988) 2700 D. J. Millener et. al.) (PRL89 (2002) 072301 H. Noumi et al.)
New degree of freedom (hyperon) can appear in the nuclear matter→ matter becomes softer → maximum neutron star mass will be bounded around 1.5𝑀⊙
2𝑀⊙ neutron star has been observed (Nature 467 (2010) 1081 P. B. Demorest et al.)→ should we exclude hyperons in neutron star? How to obtain such a stiff EOS?→ related to density evolution of hyperon energy and symmetry energy
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• Classical symmetry
• After quantum correction
QCD symmetry breaking
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Axial charge is not conserved
For ν=1 configuration, classical solution (instanton) can be found as
, → trapped color already breaks symmetry
which resides in closed Wilson loop , (zero curvature on the contour)
(Belavin, Polyakov, Schwartz, Tyupkin, PLB59(1975)85)
flavor color
( )
Left Right
ν=0
ν=1
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• Fermionic zero-modes in ν>0 configuration
• 𝜃 vacuum
QCD symmetry breaking
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,
leads to and ( )
different topological configuration measures different axial charge
→ all possible topological configuration contributes
helicity bases can be correlated via Instantons
For a given gauge configuration breaks chiral symmetry
hadron property can be described via the symmetry breaking pattern
flavor color
ν0 ⇒ … =
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• Interpolating current correlator
• Operator product expansion (Example: 2-quark condensate diagram)
QCD sum rules
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Vacuum expectation value of localized operators
Short distance q>μ
Long distance q
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Outline
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I. Massive neutron stars and quasi-hadron picture• Quantum hadron dynamics and mean-field phenomenology
• Low energy QCD symmetry breaking and sum rules for quasi-baryon states
II. Gravitational wave observation and quarkyonic matter concept• Large 𝑵𝒄 QCD in dense medium
• Single flavor excluded-volume model
III. Three-flavor extension of excluded-volume model
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• Tidal deformability observed from GW170817 (LIGO-Virgo)
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Inspiral of two object (𝑅𝑖>0)
Successive analysis (PRL121.161101 B.P. Abbott et al.)
෩Λ(1.4M⊙) < 580, , , 𝑹𝟏.𝟒 = 𝟏𝟏. 𝟗−𝟏.𝟒+𝟏.𝟒 km
Gravitational wave
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• Example based on machinery computations
Possible equation of state
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By polytropes interpolation (Quark Matter 2018, NPA982.36 A. Vuorinen)
▪ Gradual “soft” increment and “stiff” increment (small 𝑣𝑠2 after sudden increment)
▪ The 2nd soft part is constrained by pQCD limit
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• Fast and slow speed
Speed of sound
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(PRD 101 (2020) 054016 Y. Fujimoto et al.)
▪ , , 𝑣𝑠2 < 1 (causality)
▪ If 𝑣𝑠2~1, the unit increment of log scale will be directly proportional to each other
→ requires tremendously large repulsive interaction
▪ At some points, this huge force should be turned off to satisfy the ෩Λ
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• 𝑔𝑇2𝑁𝑐 = 𝑐𝑜𝑛𝑠𝑡., 𝑁𝑐 → ∞ ( Nucl.Phys.B72 (1974) 461 G. ‘t Hooft )
Large 𝑁𝑐 QCD and quarkyonic matter
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▪ Effectively U(𝑁𝑐 → ∞) gauge theory, anomaly free
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Large 𝑁𝑐 QCD and quarkyonic matter
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1/𝑁𝑐 (1/𝑁𝑐)2
▪ Faces (F=L+I ), internal lines (P), vertices (V), quark loops (L), index loops (I )
▪ Coupling factor
( , )
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• 𝑔𝑇2𝑁𝑐 = 𝑐𝑜𝑛𝑠𝑡., 𝑁𝑐 → ∞ ( Nucl.Phys.B160 (1974) 57 E. Witten )
Large 𝑁𝑐 QCD and quarkyonic matter
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▪ Easier counting
→→ ~(1/𝑁𝑐)
2𝑁𝑐 ~ 1/𝑁𝑐
▪ Planar diagram dominates
→
~(1/𝑁𝑐)6(𝑁𝑐)3~(1/𝑁𝑐)
3 ~(1/𝑁𝑐)6(𝑁𝑐)1~(1/𝑁𝑐)
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• 𝑔𝑇2𝑁𝑐 = 𝑐𝑜𝑛𝑠𝑡., 𝑁𝑐 → ∞ (Nucl.Phys.B160 (1974) 57 E. Witten)
Large 𝑁𝑐 QCD and quarkyonic matter
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▪ Planar type meson current correlator dominates → saturated by `confined meson’ state
only type in planar diagram
▪ Baryon ground state
~1
𝑁𝑐
𝑁𝑐(𝑁𝑐 − 1)
2~𝑁𝑐 ~
1
𝑁𝑐
𝑁𝑐(𝑁𝑐 − 1)
2~𝑁𝑐
+𝑶(𝑵𝒄) interaction term in medium
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• 𝑔𝑇2𝑁𝑐 = 𝑐𝑜𝑛𝑠𝑡., 𝑁𝑐 → ∞ (Nucl.Phys.A796 (2007) 83 L. McLerran and R. D. Pisarski )
Large 𝑁𝑐 QCD and quarkyonic matter
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▪ Phase diagram
▪ Quarkyonic matter : EOS in order of 𝑵𝒄
▪ Low energy excitation on the quark Fermi surfaces would be baryons
→ baryon interaction will determine low-T matter properties
(Nonrelativistic expansion)
(Baryon interaction)
perturbative quark matter EOS
dilute gas-quarkyonic window
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• Quark Fermi sea cannot screen gluon
• A lot of rooms for quarks in Λ𝑄𝐶𝐷 ball
Large 𝑁𝑐 theory and Quarkyonic matter
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→ O(1/𝑁𝑐)
Quarks on the Fermi surface → confined as done in vacuum confinement mechanism
In low density regime, quarks are broadly distributed
𝑔𝑇2𝑁𝑐 = 𝑐𝑜𝑛𝑠𝑡.
(Figure from PRL122(2019)122701)
~ 1
𝑁𝑐
Λ𝑄𝐶𝐷 ball
𝑘𝐵𝐹
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• Quarks in Λ𝑄𝐶𝐷 ball can broadly distributed (𝑘𝑁𝐹 < Λ𝑄𝐶𝐷)
Broadly distributed quarks
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𝑁𝑐 = 3 example
(Figure from PRL122(2019)122701)
𝒌𝑵𝑭~𝜦𝑸𝑪𝑫
𝒌𝑵𝑭 < 𝜦𝑸𝑪𝑫
Λ𝑄𝐶𝐷
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• Once quark Fermi sea is formed (𝑘𝑁𝐹~Λ𝑄𝐶𝐷)
Fast enough nucleon-like shell
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▪ There is no sudden jump of total baryon number density
▪ If one assumes constituent quark model (𝑚𝑞~Λ𝑄𝐶𝐷), 𝜇N grows as
▪ Total energy density is kept
→ pressure is enhanced (stiff EOS)
(ε, n are kept in similar order)
Detailed explanations are given in many literaturesNucl.Phys.A796.83, arXiv:1904.05080, etc.
𝑘𝑞𝐹
𝑘𝑁𝐹 = 𝑁𝑐𝑘𝑞
𝐹> Λ𝑄𝐶𝐷
𝑘𝑁𝐹 ≃ Λ𝑄𝐶𝐷
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• 𝑁𝑐 → ∞ limit contains intrinsic scale (Λ𝑄𝐶𝐷) and divergence (𝑁𝑐)
• Like Hagedorn model
• If one tries singular μ near critical density 𝑛0 and quark d.o.f.
Phenomenological approaches
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Density distribution measure for high energy state
→ leads to singular entropy
Near the critical temperature, the energy diverges → system prefer to make new degree of freedom
Sudden increment of μ with onset of quarks and recovery of pQCD limit by large number of quarks
γ=0.7
𝑛0~4ρ0
(PRC101 (2020) 035201 K.S.J., L.M., S.S.)
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Hard-core repulsion by “effective size”
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• To be more realistic
• Hard core nature can be embodied by semi-classical size 𝑣0
Lattice QCD study for hadron interaction (HALQCD T. Hatsuda et al.)
Hardcore effective size and excluded volume→ reduced available space (fast nucleons)
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• Considering Pauli exclusion principle ( )
Quarkyonic-like baryon shell structure
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▪ Quark density is determined by minimization of energy density
▪ IR cutoff Λ is introduced to regularize singularity (artifact) near onset of quark
▪ Appearance of quarks makes the stiff increment of chemical potential
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• After tuning hard-core density to match energy density scale
Quarkyonic-like baryon shell structure
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▪ Hardcore repulsive interaction reproduces the EOS deduced from Quarkyonic picture
▪ What would be happening if we consider the EM charge and strangeness?
𝑘𝑞𝐹
𝑘𝑁𝐹 = 𝑁𝑐𝑘𝑞
𝐹> Λ𝑄𝐶𝐷
(PRC101 (2020) 035201 K.S.J., L.M., S.S.)
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Outline
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I. Massive neutron stars and quasi-hadron system• Quantum hadron dynamics and mean-field phenomenology
• Low energy QCD symmetry breaking and sum rules for quasi-baryon states
II. Gravitational wave observation and quarkyonic matter concept• Large 𝑁𝑐 QCD in dense medium
• Excluded-volume model
III. Three-flavor extension of excluded-volume model
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• Effective sizes and Pauli principle
• Quark measure and energy density
Three-flavor excluded-volume model
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𝑘𝐹𝑏𝑖
α determines relative hard-core size of Λ hyperon
Non-perturbative cutoff scale Λ𝑄
Explicit shell-like distribution in phase space and electro-magnetic charge
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• Baryon number conservation
• EM charge neutrality and weak-equilibrium
• Minimization of energy density
Electromagnetic charge and strangeness
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Tilde represents unit in baryon number
Three-flavor extension of dynamical equilibrium constraints
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• Baryon chemical potential (with )
• Quark chemical potential
Effectively three-flavor matter
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▪ ω𝑖 dependent term can have non-zero contribution even if 𝑛𝑖 = 0→ reduced available space enhances the chemical potential 𝜇𝑏𝑖
▪ If ω𝑖 ≃ ω𝑛,𝑝 and 𝑚𝑖 > 𝑚𝑛,𝑝→ the heavier state (such as Δ isobar) is hard to appear even at high density regime
▪ quark Fermi sea supports the shell-like distribution of baryons→ 𝜇 ෨𝑄𝑖 get enhanced by if 𝑄𝑖 support a huge amount of baryon shell distribution
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• Strongly correlated assumption
Structure of the shell-like distribution
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▪ If d quarks are saturated first, and u quarks follow
▪ Confined u quark momenta should be bounded around the confined d quark momenta
▪ All confined quarks have momentum in the similar order 𝑘𝑄𝑖 ≃ 𝑘𝑏/3
→ one may suppose
▪ Lower boundary of the shell-like distribution
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• Weakly correlated assumption
Structure of the shell-like distribution
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▪ If d quarks are saturated first, and u quarks follow
▪ Confined u quark momenta can be distributed away from the confined d quark momenta
▪ u quarks can have momentum 𝑘𝑢 ≤ 𝑘𝑏/3
▪ Error function can be assumed for the weight
▪ Lower boundary of the shell-like distribution
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• Strongly correlated assumption with α>0 and α0) leads to 𝑛𝛬 = 0
▪ If there are baryon shells supported by the quark Fermi sea, the corresponding quark Fermi sea becomes flavor symmetric
→ asymmetric configuration causes huge enhancement of 𝜇𝑏𝑖→ dynamically unfavored
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• Weakly correlated assumption with α>0 and α0) leads to 𝑛𝛬 = 0
▪ Even if there are baryon shells supported by the quark Fermi sea, the asymmetric configuration of the quark Fermi sea is appearing
→ asymmetric configuration does not cause that huge enhancement of 𝜇𝑏𝑖→ dynamically allowed
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Hard-soft evolution of EOS
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Strongly
correla
ted
Weakly
correla
ted
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• With aid of Urbana IX forces (𝑛0 = 5𝜌0)
Mass-radius relation of stable neutron star
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▪ Urbana IX model (for neutron rich matter)
(chosen by requiring minimal Maxwell construction interval)
▪ 𝑴𝒎𝒂𝒙 = 𝟐. 𝟎𝟑𝑴⊙, 𝑹𝟏.𝟒 = 𝟏𝟐. 𝟓 𝒌𝒎, 𝑹𝟐.𝟎𝟑 = 𝟏𝟏. 𝟒 𝒌𝒎
▪ 𝒏෩𝑸 = 𝟎. 𝟐𝟔 𝒏𝑩, 𝜺෩𝑸 = 𝟎. 𝟐𝟕 𝜺𝒕𝒐𝒕 at the core of maximum mass state
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• Chiral effective theory and pQCD limits
Some similar studies
38Nature Phys. 16 (2020) 907 E. Annala et al.
PhysRevC.101.034904 A. Motornenko et al.
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• Neutron star puzzle may be solved via quarkyonic matter concept
• Repulsive hard-core and excluded-volume model contains dynamically saturated quark Fermi sea and shell-like distributions
• Strangeness does not cause any problem!
(actually, it is needed to reach the massive state)
• What is the true nature of the non-perturbative state on the quark Fermi surface?
Summary
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