Koji Hashimoto (RIKEN, Mathematical physics lab) 7 Aug 2010 @SI 2010 string and cosmology Nuclear Matrix Model 　Koji Hashimoto (RIKEN, Mathematical physics lab) arXiv/1003.4988 N.Iizuka, P.Yi, KH arXiv/1005.4412, 1006.3612 N.Iizuka, KH arXiv/0809.3141,0910.2303 KH arXiv/0901.4449 T.Sakai, S.Sugimoto, KH arXiv/0911.1035 N.Iizuka, T.Nakatsukasa, KH

“Realistic” M(atrix)-theory ?

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

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?

“Nucleons” in string theory

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

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

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

“Realistic” M(atrix)-theory ? arXiv/1003.4988 N.Iizuka, P.Yi, KH arXiv/1005.4412 N.Iizuka, KH Nuclear Matrix Model arXiv/1006.3612 N.Iizuka, KH “Realistic” M(atrix)-theory ?

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

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

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]

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-,…

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

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, 0901.4449] 3-flavor: again, universally repulsive [Iizuka, S.Aoki, KH in progress] 1/r2 agrees with perturbative QCD [Aoki, Balog, Weisz, 1002.0977]

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

Conclusions

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

“Realistic” M(atrix)-theory ?