Hadron interactions from lattice QCD Sinya Aoki University of - PowerPoint PPT Presentation

Hadron interactions from lattice QCD Sinya Aoki University of Tsukuba GGI Workshop “New Frontiers in Lattice Gauge Theory” GGI, Firenze, Italy, September 12, 2012

1. Introduction

How can we extract hadronic interaction from lattice QCD ? � � � Ex. Phenomenological NN potential ) (~40 parameters to fit 5000 phase shift data) One-pion exchange I III � Yiukawa(1935) II I � � Multi-pions II Taketani et al.(1951) � � � � �������� �������� ore Repulsive core III � � �������� ) Jastrow(1951)

Nuclear force is a basis for understanding ... • Ignition of Type II SuperNova • Structure of ordinary and hyper nuclei • Structure of neutron star Λ Neutron matter quark Matter? Can we extract a nuclear force in (lattice) QCD ?

Plan of my talk 1. Introduction 2. Our strategy 3. Example: Nuclear potential 4. Inelastic Scattering (work in progress) 5. Demonstration (as a conclusion)

2. Our Strategy

Our strategy in lattice QCD Full details: Aoki, Hatsuda & Ishii, PTP123(2010)89. Step 1 define (Equal-time) Nambu-Bethe-Salpeter (NBS) Wave function Spin model: Balog et al., 1999/2001 ϕ k ( r ) = � 0 | N ( x + r , 0) N ( x , 0) | NN, W k � � k 2 + m 2 W k = 2 N energy N ( x ) = ε abc q a ( x ) q b ( x ) q c ( x ): local operator Important property sin( kr − l π / 2 + δ l ( k )) r = | r | → ∞ ϕ l k → A l partial wave kr Lin et al., 2001; CP-PACS, 2004/2005 δ l ( k ) scattering phase shift (phase of the S-matrix) in QCD ! cf. Maiani-Testa theorem How can we extract it ? cf. Luescher’s finite volume method

define non-local but energy-independent “potential” as Step 2 µ = m N / 2 � d 3 y U ( x , y ) � k ( y ) reduced mass [ � k − H 0 ] � k ( x ) = � k = k 2 H 0 = −∇ 2 non-local potential 2 µ 2 µ Properties & Remarks 1. Potential itself is NOT an observable. Using this freedom, we can construct a non-local but energy-independent potential as inner product W k ,W k � ≤ W th η − 1 k , k � : inverse of η k , k � = ( ϕ k , ϕ k � ) k , k � � † � [ � k − H 0 ] � k ( x ) � − 1 U ( x , y ) = k � ( y ) ϕ k is linearly independent. k , k � For ∀ W p < W th = 2 m N + m π (threshold energy) � � d 3 y U ( x , y ) � p ( y ) = [ � k − H 0 ] � k ( x ) � − 1 k , k � � k � , p = [ � p − H 0 ] � p ( x ) k , k � Proof of existence (cf. Density Functional Theory) Of course, potential satisfying this is not unique. (Scheme dependence. cf. running coupling) 2. Non-relativistic approximation is NOT used. We just take the specific (equal-time) flame.

Step 3 expand the non-local potential in terms of derivative as U ( x , y ) = V ( x , r ) δ 3 ( x � y ) V ( x , ∇ ) = V 0 ( r ) + V σ ( r )( σ 1 · σ 2 ) + V T ( r ) S 12 + V LS ( r ) L · S + O ( ∇ 2 ) NLO NNLO LO LO LO spins S 12 = 3 tensor operator r 2 ( σ 1 · x )( σ 2 · x ) − ( σ 1 · σ 2 ) local and energy independent coefficient function V A ( x ) (cf. Low Energy Constants(LOC) in Chiral Perturbation Theory)

Step 4 extract the local potential at LO as V LO ( x ) = [ � k − H 0 ] � k ( x ) � k ( x ) Step 5 solve the Schroedinger Eq. in the infinite volume with this potential. phase shifts and biding energy below inelastic threshold (We can calculate the phase shift at all angular momentum.) δ L ( k ) exact by construction δ L ( p � = k ) approximated one by the derivative expansion expansion parameter � ∆ E p W p � W k W th � 2 m N m π We can check a size of errors at LO of the expansion. (See later). We can improve results by extracting higher order terms in the expansion.

This procedure gives a new method to extract phase shift from QCD. HAL QCD method (by-pass Maiani-Testa theorem, using space correlation) Sinya Aoki (U. Tsukuba) HAL QCD Collaboration Bruno Charron* (U. Tokyo) Takumi Doi (Riken) Tetsuo Hatsuda (Riken/U. Tokyo) Yoichi Ikeda (TIT) Takashi Inoue (Nihon U.) Noriyoshi Ishii (U. Tsukuba) Keiko Murano (Riken) Hidekatsu Nemura (U. Tsukuba) Kenji Sasaki (U. Tsukuba) Masanori Yamada* (U. Tsukuba) *PhD Students Our strategy Potentials from Nuclear Physics Neutron stars lattice QCD with these potentials Supernova explosion

3. Example:Nuclear potential

Extraction of NBS wave function NBS wave function Potential � ϕ k ( r ) = � 0 | N ( x + r , 0) N ( x , 0) | NN, W k � d 3 y U ( x , y ) � k ( y ) [ � k − H 0 ] � k ( x ) = 4-pt Correlation function source for NN F ( r , t − t 0 ) = � 0 | T { N ( x + r , t ) N ( x , t ) }J ( t 0 ) | 0 � complete set for NN � F ( r , t − t 0 ) = � 0 | T { N ( x + r , t ) N ( x , t ) } | 2 N, W n , s 1 , s 2 �� 2 N, W n , s 1 , s 2 |J ( t 0 ) | 0 � + · · · n,s 1 ,s 2 A n,s 1 ,s 2 ϕ W n ( r ) e − W n ( t − t 0 ) , � = A n,s 1 ,s 2 = � 2 N, W n , s 1 , s 2 |J (0) | 0 � . n,s 1 ,s 2 − → ∞ ground state saturation at large t ( t − t 0 ) →∞ F ( r , t − t 0 ) = A 0 ϕ W 0 ( r ) e − W 0 ( t − t 0 ) + O ( e − W n � =0 ( t − t 0 ) ) lim NBS wave function This is a standard method in lattice QCD and was employed for our first calculation.

Ishii et al. (HALQCD), PLB712(2012) 437 Improved method R ( r , t ) ≡ F ( r , t ) / ( e − m N t ) 2 = � A n ϕ W n ( r ) e − ∆ W n t normalized 4-pt Correlation function n ∆ W n = W n − 2 m N = k 2 − ( ∆ W n ) 2 n 4 m N m N ∂ 2 � � 1 − ∂ ∂ tR ( r , t ) = H 0 + U − R ( r , t ) 4 m N ∂ t 2 potential Leading Order ∂ 2 � � 1 − H 0 − ∂ � d 3 r ′ U ( r , r ′ ) R ( r ′ , t ) = V C ( r ) R ( r , t ) + · · · ∂ t + R ( r , t ) = 4 m N ∂ t 2 total 1st 2nd 3rd 40 30 20 3rd term(relativistic correction) V C (r) [MeV] is negligible. 10 0 -10 -20 total 1st term -30 2nd term 3rd term -40 0 0.5 1 1.5 2 2.5 r [fm] Ground state saturation is no more required ! (advantage over finite volume method.)

NN potential 2+1 flavor QCD, spin-singlet potential (in preparation) a=0.09fm, L=2.9fm phenomenological potential m π ≃ 700 MeV 40 30 1 S 0 20 V C (r) [MeV] 10 0 -10 -20 -30 -40 0 0.5 1 1.5 2 2.5 r [fm] Qualitative features of NN potential are reproduced ! (1)attractions at medium and long distances (2)repulsion at short distance(repulsive core) 1st paper(quenched QCD): Ishii-Aoki-Hatsuda, PRL90(2007)0022001 This paper has been selected as one of 21 papers in Nature Research Highlights 2007.

phase shift NN potential 60 exp 1 S 0 lattice 50 40 a exp ( 1 S 0 ) = 23 . 7 fm 0 30 � [deg] 20 10 a 0 ( 1 S 0 ) = 1 . 6(1 . 1) fm 0 -10 -20 0 50 100 150 200 250 300 350 E lab [MeV] It has a reasonable shape. The strength is weaker due to the heavier quark mass. Need calculations at physical quark mass.

Convergence of velocity expansion If the higher order terms are large, LO potentials determined from NBS wave functions at different energy become different.(cf. LOC of ChPT). K. Murano, N. Ishii, S. Aoki, T. Hatsuda m π ≃ 0 . 53 GeV Numerical check in quenched QCD PTP 125 (2011)1225. a=0.137fm ������ PBC (E � 0 MeV) ���������������������� � � APBC (E � 46 MeV) �

����������������������������������������������� ���������������������� ������ ���������������������������� ���������� ������ � � � � ��������������������������������������������������� � � � Higher order terms turn out to be very small at low energy in HAL scheme. � ��������������������������������������������������������� � ��������������������������� ��������������������������������������� Need to be checked at lighter pion mass in 2+1 flavor QCD. � ��������������������������������������������������� �� ���������� Note: convergence of the velocity expansion can be checked within this method. (cf. convergence of ChPT, convergence of perturbative QCD)

4. Inelastic scattering (work in progress)

Inelastic scattering 1. Particle production Ex. NN → NN, NN + π , NN + 2 π , · · · , NN + K ¯ K, · · · , NN + N ¯ N, · · · 2. Particle exchanges ΣΣ 2386 MeV Ex. ΛΛ → ΛΛ , N Ξ , ΣΣ 129 MeV N Ξ 2257 MeV 25 MeV ΛΛ 2232 MeV