Nuclei in holographic QCD
Yoshinori Matsuo
Kyoto University
Based onarXiv:1902.07444

with Koji Hashimoto (Kyoto U.),
Takeshi Morita (Shizuoka U.)

arXiv:2103.03563

with Koji Hashimoto (Kyoto U.)
April 28, 2021@Kyoto U.

Nuclei can be described bymatrix model in holographic QCD
Nuclei = bound states of nucleons = bound states of D-branes
D-branes (baryon vertex)

nucleons in QCD

correspondence

Effective theory

Nuclear matrix model

Nuclei in holographic QCD appear as bound states in matrix model
Nuclei have following properties• Saturation of nucleon number density• Saturation of nuclear binding energy• Nuclear magic number
These properties can be reproduced from nuclear matrix modelPlan of Talk
1. Important properties of nuclei
2. Holographic QCD
3. Nuclei in nuclear matrix model

4. Conclusion

1. Important properties of nuclei

1. Important properties of nuclei
Saturation of nucleon number density
Saturation of nuclear binding energy
Nuclear magic number

Saturation of nucleon number density:
nucleon density is constantNucleon density can be read off from charge distribution
almost constant𝜌 ≃ 𝐴Τ𝑉 ≃ const.

Constant is almost independent of nuclide for large mass number
Nuclear radius 𝑅
𝑅 ≃ 1.2 × 𝐴1Τ3 fm

𝐴: mass number (# of nucleons)Saturation of nuclear binding energy:
binding energy per nucleon is constantNuclear binding energy can be read off from mass defectBinding energy per nucleon is almost constant in large 𝐴

Nuclear magic number:# of protons (neutrons) for stable nuclei
Magic Numbers: 2, 8, 20, 28, 50, 82, 126, ⋯Nuclei at magic number are stable

Magic number is explained bynuclear shell model
Harmonic potential 𝑉𝐻 ∝ 𝑟 2𝜓 ∼ 𝑥 𝑛𝑥 𝑦 𝑛𝑦 𝑧 𝑛𝑧
𝑛012

orbital state 1s

1p2s+1d# of states 1×2

3×2(1+5)×2=128201. Important properties of nuclei
Saturation of nucleon number density
Saturation of nuclear binding energy
Nuclear magic number2. Holographic QCD

2. Holographic QCD
Effective theory is obtained by holography
Sakai-Sugimoto model gives effective theory of QCD(in confinement phase)
Baryons in holographic QCD = D-branes (baryon vertex)
Effective theory of baryon vertex = matrix modelEffective theory is obtained by holography
holography (AdS/CFT, gauge/gravity, fluid gravity, AdS/CMT)
CFT (gauge theory)

Classical solution in AdS

correspondence

Field theory on D-branes

D-branes in SUGRA

Holography is useful to calculate strongly coupled QCD
holographic QCD

Strongly coupled QCDClassical solution in gravity

correspondence

ResultCalculation

Sakai-Sugimoto model gives effective theoryof QCD (in confinement phase)𝒙𝟎𝒙𝟏𝒙𝟐𝒙𝟑𝒙𝟒

D4✓

✓✓✓

✓

D8✓✓

✓✓

𝒙𝟓𝒙𝟔𝒙𝟕𝒙𝟖𝒙𝟗✓✓✓✓✓D8-brane
D8D4

quarksgluonscorrespondence

D4-branegeometry

Sakai-Sugimoto model gives effective theory
of QCD (in confinement phase)𝒙𝟎𝒙𝟏𝒙𝟐𝒙𝟑𝒙𝟒D4

✓

✓

✓

✓

✓

D8✓

✓

✓

✓

D8-brane𝒙𝟓𝒙𝟔𝒙𝟕𝒙𝟖𝒙𝟗

✓

✓

✓

✓

✓

D8-branequarks𝑥4𝑥4

correspondence

D4-branegluonsD4-branegeometry

Sakai-Sugimoto model gives effective theory
of QCD (in confinement phase)
Holographic QCD (Sakai-Sugimoto model)

Strongly coupled QCD

D4: color
D8: flavor

Classical solution in gravity

correspondence

QCD on D4-branesD4-brane geometry

Effective theory of mesons

Field theory on D8-branes

Sakai-Sugimoto model reproduces meson spectra
Baryons in Sakai-Sugimoto model
Baryons

Soliton on D8-branes

D4-branes

Baryons in AdS/CFT correspondence
Baryons = D-branes wrapping on color D-branes
RR-flux coupled withgauge field on baryon vertex
න𝑅×𝑆 𝑛𝐴 ∧ 𝐺 = 𝑁𝑐 න 𝐴𝑅Gauge field RR-flux
RR-flux

F-strings

𝑁𝑐 endpoints of F-strings
on baryon vertex
Corresponds to baryons

Baryons in Sakai-Sugimoto model:
D4-branes wrapping on color D4-branes
𝒙𝟎𝒙𝟏𝒙𝟐𝒙𝟑𝒙𝟒D4✓✓✓✓

✓

D8✓✓✓

✓

Baryon✓BaryonD4-braneColorD4-brane

𝑼𝜽𝟏

𝜽𝟐

𝜽𝟑𝜽𝟒✓

✓✓✓

✓

✓✓✓

✓

Baryon D8-brane
D4-braneEffective theory of baryons = Matrix models
Action for baryons[Hashimoto-Iizuka-Yi,’10]

𝑆 = 𝑆0 + 𝑁𝑐 න𝑑𝑡 tr𝐴𝑡1
11 2 𝛼𝑖ሶ
ሶ𝐼2𝛼𝑖
𝑆0 = න 𝑑𝑡 tr ቈ 𝐷𝑡 𝑋 + 𝐷𝑡 𝑤
ഥ𝐷𝑡 𝑤𝛼𝑖ሶ − 𝑀 𝑤ഥ 𝑤𝛼𝑖ሶ
2221
ሶ𝛽ሶ
𝐼2𝐼𝐼𝐽𝐾𝐽𝐾𝛼𝑖
𝐼+𝐷 + 𝐷 2𝑖𝜖 𝑋 𝑋 + 𝑤ഥ 𝜏 𝛼ሶ 𝑤𝛽𝑖ሶ ቉4𝜆𝑋 𝐼 : D4-D4 scalar𝑤 (𝑤):
ഥ D4-D8 scalar

Position of baryon vertex
Spin, flavor, baryon numberNuclei = bound states of baryons
Nuclei appear in eigenstates of matrix quantum mechanics

2. Holographic QCD
Effective theory is obtained by holography
Sakai-Sugimoto model gives effective theory of QCD(in confinement phase)
Baryons in holographic QCD = D-branes (baryon vertex)
Effective theory of baryon vertex = matrix model3. Nuclei in nuclear matrix model3. Nuclei in nuclear matrix model
Saturation of nuclear number density (Review of [Hashimoto-Morita,’11])
Eigenstates of nuclear matrix model have
similar structure to nuclear shell model
Saturation of nuclear binding energy

magic number

Saturation of nucleon number density[Hashimoto-Morita,’11]

Potential is approximated by harmonic potential
11
1𝐷𝑡 𝑋 𝐼 2 + 2𝜆 𝑋 𝐼 , 𝑋𝐽 2 ≃ න 𝑑𝑡 tr
𝐷𝑡 𝑋 𝐼 2 − 𝑚2 𝑋 𝐼 222216𝐴𝜆3 ≃ 8𝜆𝐴
22𝑚𝑚 ≃𝑋2𝐴
3 𝐴2 − 1𝑟0 =𝑚

𝑆𝑋 = න 𝑑𝑡 tr

Nucleon density

13 𝑘 𝑒 −𝑖𝑘𝑥 ⟨tr exp 𝑖𝑘𝑋⟩ ≃𝜌=න𝑑2𝜋 3Nuclear radius
𝑅2 =1tr 𝑋 2
𝐴

𝐸 = 2 𝑉 = 𝑚2 tr 𝑋 2 = 32𝑚 𝐴2𝑅∼

𝐴/𝑚 ∝ 𝐴1/3

Eigenstates of nuclear matrix model hassimilar structure to nuclear shell model
Action for baryons

𝑋 𝐼 : Position of baryon vertex
𝑎𝑤𝛼𝑖
ሶ : Spin, flavor, baryon number

𝑆 = 𝑆0 + 𝑁𝑐 න𝑑𝑡 tr𝐴𝑡1
11 2 𝛼𝑖ሶ
ሶ𝐼2𝛼𝑖
𝑆0 = න 𝑑𝑡 tr ቈ 𝐷𝑡 𝑋 + 𝐷𝑡 𝑤
ഥ𝐷𝑡 𝑤𝛼𝑖ሶ − 𝑀 𝑤ഥ 𝑤𝛼𝑖ሶ
2221
ሶሶ𝐼 2𝐼
𝐼𝐽𝐾 𝐽 𝐾𝛼𝑖
𝐼 𝛽+𝐷 + 𝐷 2𝑖𝜖 𝑋 𝑋 + 𝑤ഥ 𝜏 𝛼ሶ 𝑤𝛽𝑖ሶ ቉4𝜆Eigenstate of Hamiltonian

𝐴: Number of baryons𝑎𝜓0 = 𝜖𝑎1⋯𝑎𝐴 𝑤 𝑎1 ⋯ 𝑤 2𝑁𝑓 𝑋 𝐼 𝑤 ⋯ 𝑋𝐽 𝑤 ⋯ 𝑋 𝐾 ⋯ 𝑋 𝐿 𝑤 𝑎𝐴𝑏× ⋯ × [𝜖𝑏1⋯𝑏𝐴 𝑤 𝑏1 ⋯ 𝑤 2𝑁𝑓 𝑋𝑤 ⋯ 𝑋 ⋯ 𝑋𝑤 𝑏𝐴 ] 0
𝑁𝑐 of 𝜖𝑎1⋯𝑎𝐴 𝑤 𝑎1 ⋯ (𝑋 𝐼 ⋯ 𝑋𝐽 )𝑤 𝑎𝐴

Eigenstates of nuclear matrix model has
similar structure to nuclear shell model
Eigenstate of Hamiltonian

𝐴: Number of baryons𝑎𝜓0 = 𝜖𝑎1⋯𝑎𝐴 𝑤 𝑎1 ⋯ 𝑤 2𝑁𝑓 𝑋 𝐼 𝑤 ⋯ 𝑋𝐽 𝑤 ⋯ 𝑋 𝐾 ⋯ 𝑋 𝐿 𝑤 𝑎𝐴𝑏× ⋯ × [𝜖𝑏1⋯𝑏𝐴 𝑤 𝑏1 ⋯ 𝑤 2𝑁𝑓 𝑋𝑤 ⋯ 𝑋 ⋯ 𝑋𝑤 𝑏𝐴 ] 0

𝐴𝑡 : gauge field (baryon 𝑈(𝐴))Non-dynamical field

EOM gives constraints

𝛿𝑆𝛿𝑆00==− 𝑁𝑐 𝕀 = 𝑄𝑈 𝐴 − 𝑁𝑐 𝕀𝛿𝐴𝑡 𝛿𝐴𝑡We impose constraints to physical states
𝑄𝑆𝑈 𝐴 |𝜓〉 = 0

Singlet state of baryon 𝑆𝑈(𝐴)𝑄𝑈 1 𝐵 |𝜓〉 = 𝑁𝑐 𝐴|𝜓〉Baryon (quark) number is 𝑁𝑐 𝐴

Eigenstates of nuclear matrix model has
similar structure to nuclear shell model
Eigenstate of Hamiltonian

𝐴: Number of baryons𝑎𝜓0 = 𝜖𝑎1⋯𝑎𝐴 𝑤 𝑎1 ⋯ 𝑤 2𝑁𝑓 𝑋 𝐼 𝑤 ⋯ 𝑋𝐽 𝑤 ⋯ 𝑋 𝐾 ⋯ 𝑋 𝐿 𝑤 𝑎𝐴𝑏× ⋯ × [𝜖𝑏1⋯𝑏𝐴 𝑤 𝑏1 ⋯ 𝑤 2𝑁𝑓 𝑋𝑤 ⋯ 𝑋 ⋯ 𝑋𝑤 𝑏𝐴 ] 0
𝑁𝑐 of 𝜖𝑎1⋯𝑎𝐴 𝑤 𝑎1 ⋯ (𝑋 𝐼 ⋯ 𝑋𝐽 )𝑤 𝑎𝐴

Anti-symmetric under exchange of baryon indicesPauli exclusion principle
𝑋 𝐼 has effective mass

Orbital energy levels (excited state)magic number
Magic number for 𝑢 (or 𝑑) quarks (𝑁𝑐 = 1 case for simplicity)𝑤 = 𝑢𝛼ሶMagicnumber

Magicnumber

𝐴=1
𝐴=2
𝐴=3
𝐴=4
⋮
𝐴=8𝛼ሶ = 1, 2 (𝑖 = 1)

2 of 𝑢 (spin ↑ and ↓)𝜓0 = 𝜖𝑢↑ |0〉𝜓0 = 𝜖𝑢↑ 𝑢↓ |0〉𝜓0 = 𝜖𝑢↑ 𝑢↓ 𝑢↑ 0 = 0
𝜓0 = 𝜖𝑢𝑢(𝑢𝑋)(𝑢𝑋)|0〉⋮𝜓0 = 𝜖𝑢𝑢 𝑋𝑢 ⋯ 𝑋𝑢 |0〉
6 of 𝑋𝑢

𝐴=9

𝜓0 = 𝜖𝑢𝑢 𝑋𝑢 ⋯ 𝑋𝑢 (𝑋𝑢) 0 = 0
6 of 𝑋𝑢

𝐴=9

𝜓0 = 𝜖𝑢𝑢 𝑋𝑢 ⋯ 𝑋𝑢 (𝑋𝑋𝑢) 0
6 of 𝑋𝑢

Additional energy of 𝑋 𝐼

𝜓0 = 𝜖𝑢𝑢 𝑋𝑢 |0〉
𝑋𝐼

𝐼 = 1, 2, 32 × 3 = 6 of 𝑋𝑢

No correction at leading order of large 𝐴Potential is approximated by harmonic potential
𝑆𝑋 = න 𝑑𝑡 tr𝑚2 ≃1
11𝐷𝑡 𝑋 𝐼 2 + 2𝜆 𝑋 𝐼 , 𝑋𝐽 2 ≃ න 𝑑𝑡 tr
𝐷𝑡 𝑋 𝐼 2 − 𝑚2 𝑋 𝐼 222216𝐴𝜆𝑟
2tr
𝑋
3 𝐴2 − 1

𝐸 = 2 𝑉 = 𝑚2 tr 𝑋 2Harmonic oscillator with 𝑁𝑋 excitations
3 2𝐸 = 𝑚 𝑁𝑋 + 𝐴 − 123𝑁𝑋 ≃2𝑚3 = 8𝜆𝑟 𝐴 + 25/3 34/3 𝜆𝑟 𝐴1/3

7/3

𝐴4/3

Binding energy of fermions can beread off from density distribution
Multi-fermion ground state in given potential 𝑉(𝑟)WKB approximation
𝐶 −1/2 𝑖 ‫)𝑟(𝑝 𝑟𝑑 ׬‬𝜓= 𝑝𝑒𝑌𝑙𝑚𝑟
Quantization conditionන𝑑𝑟 𝑝(𝑟) = 𝜋𝑛Fermion density2𝜌 𝑟 = 2 𝐸𝑓 − 𝑉 𝑟
3𝜋3/2

Saturation of nuclear binding energy
Effective potential from nucleon density𝜌 𝑟 =𝑉 𝑟 = 𝐸𝑓 −32/3 𝐴2/3
4/3

22/3 𝑟0

𝑟02 − 𝑟 2 1/3

Saturation of nuclear binding energy
Effective potential from nucleon density𝜌 𝑟 =𝑉 𝑟 = 𝐸𝑓 −32/3 𝐴2/3
4/3

22/3 𝑟0

𝑟02 − 𝑟 2 1/3

Binding energy (we take 𝐸𝑓 = 0)
𝐵 = 4𝜋 න𝑟 2 𝜌 𝑟 𝑉 𝑟 𝑑𝑟 + 2𝜋 න𝑟 3 𝑉 ′ 𝑟 𝑑𝑟
Potential energy

Kinetic term from virial theorem

Saturation of binding energy
38/3 Γ 76
𝐵38/3 𝜋= 2/3− 2/3𝐴2𝜋Γ 232 5Γ 23 Γ 56

1/3𝜆𝑟

Numerical estimation
Mass of nucleons and Δ

Energy of 𝑤 excitations

4𝜆𝐸 = (𝑁𝑐 + 2𝑁𝑓 )𝑀 + 2 𝐼 𝐼 + 1
𝑀Harmonic oscillator

Input

𝑀𝑁 = 939 MeVNuclear radius
𝑅 = 2.4 × 𝐴1/3 fm
Binding energy per nucleon
𝐵= 9.7 MeV
𝐴First order perturbation 𝜆 𝑤 4

𝑀Δ = 1232 MeVExperiments
𝑅 = 1.2 × 𝐴1/3 fm
𝐵= 8 MeV𝐴

3. Nuclei in nuclear matrix modelSaturation of nuclear number density (Review of [Hashimoto-Morita,’11])
Eigenstates of nuclear matrix model have
similar structure to nuclear shell model
Saturation of nuclear binding energy

magic number

4. Conclusion

Nuclei can be described bymatrix model in holographic QCD
Nuclei = bound states of nucleons = bound states of D-branes
D-branes (baryon vertex)

nucleons in QCD

correspondence

Effective theory

Nuclear matrix model

Nuclei in holographic QCD appear as bound states in matrix model
Nuclei have following properties• Saturation of nucleon number density• Saturation of nuclear binding energy• Nuclear magic number
These properties can be reproduced from nuclear matrix modelThank you