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.)

• Tidal deformability estimated from GW170817 (LIGO-Virgo)

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Motivation – astrophysical observation

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

• 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.)

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)

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

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

• 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

• 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 ⇒ … =

• 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

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

• 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

• 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

• 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

• 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

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

( , )

• 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

• 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

• 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

• 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

• 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

• 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.)

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)

• 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

• 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.)

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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

Hard-soft evolution of EOS

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Strongly

correla

ted

Weakly

correla

ted

• 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

• 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.

• 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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