1. Koji Hashimoto (RIKEN, Mathematical physics lab)
7 Aug 2010 string and cosmology
Nuclear Matrix Model
　Koji Hashimoto
(RIKEN, Mathematical physics lab)
arXiv/ N.Iizuka, P.Yi, KH
arXiv/ , N.Iizuka, KH
arXiv/ , KH
arXiv/ T.Sakai, S.Sugimoto, KH
arXiv/ N.Iizuka, T.Nakatsukasa, KH

2. “Realistic” M(atrix)-theory ?

3. How can we describe multi-nucleon (=nucleus) system?
Nuclear physics
How can we describe multi-nucleon (=nucleus) system?
Serious problem
Elementary particles : quarks and gluons in QCD
Nucleon : bound state of 3 quarks via strong coupling
Nuclear properties from strongly coupled QCD?
Our solution
QCD
+ AdS/CFT
Matrix quantum mechanics for baryons,
“Nuclear Matrix Model”
Proper limit
Nuclear force, baryon spectra

4. Formulating a concrete question
Nuclear force
Importance of repulsive core
Short range :
　　　　Strong repulsion
Long range :
　　　　Pion exchange
- Nucleon scattering
- Nuclear density saturation
- Supernova explosion
Nucleon
distance
EoS at high density?
- Neutron star
Core structure?
- QCD phase diagram
Phases at high density?
Effective description of multi-nucleon system?

5. “Nucleons” in string theory

6. Nc units of 5-form flux penetrating S5
Nucleon in AdS5/CFT4 D5 S5
Consider Nc D3-branes.
Baryon
Nc F1s
Infinitely
massive
quarks
AdS5
N=4 SYM
Baryon = D5-brane wrapping S5
[Witten(98)]
CS term in
D5-brane action :
Nc units of 5-form flux penetrating S5

7. Nucleon in N=2 SQCD S3 D5
Add Nf flavor D7-branes. D7 D7
Baryon
Nc F1 D3
Nc F1
AdS5
Finite mass quarks
Baryon with a finite mass
N=2 SQCD

8. Nucleon in non-SUSY QCD
Witten’s setup for pure YM S4 S4
Sakai-Sugimoto model D8 D4 D8 D4
Nc F1
Nc F1 D8
SUSY broken by boundary
conditions for gauginos
Non-SUSY geometry
The baryon vertices are absorbed into the flavor D8
 Baryon = D4 in D8
D8 viewpoint : instanton, D4 viewpoint : ADHM

9. “Realistic” M(atrix)-theory ?
arXiv/ N.Iizuka, P.Yi, KH
arXiv/ N.Iizuka, KH
Nuclear Matrix Model
arXiv/ N.Iizuka, KH
“Realistic” M(atrix)-theory ?

10. 1d Matrix quantum mechanics with CS term
Nuclear matrix model S4
k D4-brane action
= Effective theory
for k baryons D4 D8
Nc F1
D4-D4 strings
 k x k matrix
Nc F1
D4-D8 strings
 k x Nf matrix
RR flux penetrating S4
 CS term
S4 direction unnecessary
 dimensionally reduced to 1d
1d Matrix quantum mechanics with CS term

11. We derive “Matrix model for multi-baryons”
Quantum mechanics with bosonic matrix variables.
: k x k Hermitian matrix
Eigenvalues of  location of k baryons
: k x k auxiliary Hermitian matrix
: k x Nf complex matrix
Size, spin/isospin of baryon
Effective description of k baryon system
in large Nc QCD with Nf flavors at strong coupling

12. After integrating out auxiliary fields hamiltonian is
k=1 : Single nucleon
After integrating out auxiliary fields hamiltonian is
Minimizing large term gives
Classical minimum :
Size of “soliton” [Hong,Rho,Yee,Yi] [Hata,Sakai,Sugimoto,Yamato]

13. Quantization  Baryon mass spectrum
Similar to but different from instanton quantization
[Hata, Sakai, Sugimoto, Yamato]
: Harmonic oscillator
: Harmonic oscillator + SU(2) rotation (spin / isospin)
 Spectrum formula :
:inputs
940+,1359+,1359-,1778+,1778-,…
Experiments :
940+,1440+,1535-,1710+,1655-,…

14. Fermionic nucleon from Aharonov-Bohm effect
Odd / even Nc  fermion / boson
(Chiral soliton model : Wess-Zumino term = )
[Witten]
Target space rotation SO(3) acts on and
Constant  “Charged particle” at
in constant magnetic field
 AB phase
4d WZ term ~ 5d CS term ~ RR background ~ 1d CS term

15. k=2 : Nuclear force, repulsive core
Minimizing large term (nucleon separation )
Hamiltonian evaluated with baryon wave functions is
This is positive, and scales as 1/r2  Universal repulsive core
Similar to the soliton approach [Sakai, Sugimoto, KH, ]
3-flavor: again, universally repulsive [Iizuka, S.Aoki, KH in progress]
1/r2 agrees with perturbative QCD [Aoki, Balog, Weisz, ]

16. k=3 : three-body nuclear force
Quite important in nuclear physics, unrevealed for 50 yrs p n n
Triton
He-3
nucleus p p n
Neutron
star n n n
spin : averaged
All positive, consistent with expectations from various data

17. Conclusions

18. How can we describe multi-nucleon (=nucleus) system?
Nuclear physics
How can we describe multi-nucleon (=nucleus) system?
Serious problem
Elementary particles : quarks and gluons in QCD
Nucleon : bound state of 3 quarks via strong coupling
Nuclear properties from strongly coupled QCD?
Our solution
QCD
+ AdS/CFT
Matrix quantum mechanics for baryons,
“Nuclear Matrix Model”
Proper limit
Nuclear force, baryon spectra

19. “Realistic” M(atrix)-theory ?