Nucleon-nucleon interaction

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1 Nucleon-nucleon interaction Shell structure in nuclei and lots more to be explained on the basis of how nucleons interact with each other in free space QCD Lattice calculations Effective field theory Exchange of lowest bosonic states Phenomenology Realistic NN interactions: describe NN scattering data up to pion production threshold plus deuteron properties Note: extra energy scale from confinement of nucleons

2 (Effective) central force N.Ishii, S.Aoki, T.Hatsuda, Phys.Rev.Lett.99,000( 07). The following diagram is contained. This leads to the one pion exchange at the large spatial separation. Repulsive core: MeV Attractive pocket: about 30 MeV Ishii talk at 009 Oak Ridge workshop Both of these are smaller than we expect. This is answered by the quark mass dependence.

3 Quark mass dependence of the central force: S 0 () m π =380MeV: Nconf=034 [8 exceptional configurations have been removed] () m π =59MeV: Nconf=000 (3) m π =73MeV: Nconf=000 Strong quark mass dependence is found. In the light quark mass region, the repulsive core grows rapidly. Ishii talk at 009 Oak Ridge workshop attractive pocket is enhanced mildly. The lattice QCD calculation at light quark mass region is quite important.

4 Two-body interactions and matrix elements To determine ˆV = αβγδ (αβ V γδ)a αa β a δa γ we need a basis and calculate (αβ V γδ) for given interaction Simplest type: spin-independent & local (also for spinless bosons) with (r r V r 3 r 4 )=(Rr V R r ) = δ(r R ) r V r = δ(r R )δ(r r )V (r) R = (r + r ) r = r r

5 Nucleon-nucleon interaction Yukawa 935 short-range interaction requires exchange of massive particle mass of particle e µr V Y (r) =V 0 µr µ c = mc mesons are the bosonic excitations of the QCD vacuum many quantum numbers; most important: pion T=, 0 - lowest mass! So one encounters also spin and isospin dependence V spin = V σ (r)σ σ V isospin = V τ (r)τ τ V s i = V στ (r)σ σ τ τ

6 Pauli spin matrices represent Spin and isospin matrix elements σ σ 4 s s Use Then S = s + s s s = ( S s s ) So coupled states are required S M S σ σ SM S = (S (S + ) 3) δ S,S δ MS,M S Same for isospin T M T τ τ TM T = (T (T + ) 3) δ T,T δ MT,M T

7 Required for NN scattering Realistic NN interaction τ τ σ σ σ σ τ τ S S τ τ L S L S τ τ L L τ τ L σ σ L σ σ τ τ (L S) (L S) τ τ plus radial dependence Tensor force Short-range interaction suggests use of angular momentum basis Angular momentum algebra Spherical tensor algebra S (ˆr) = 3 (σ ˆr)(σ ˆr) σ σ Often calculations are done in momentum space

8 Operator content pion exchange Decomposition of static pion exchange V π (Q, 0) = 3 f πnn µ π 3σ Qc σ Qc σ σ Q c c µ π + Q c τ τ + 3 f πnn c σ σ c µ π + Q c τ τ 3 f πnn µ π σ σ τ τ First term: tensor force Rewrite S ( ˆQ) = 3 σ ˆQ σ ˆQ σ σ = 4π [ [σ σ ] Y ] 0 0 = 4π µ ( µ µ 0 0) [σ σ ] µ Y, µ( ˆQ) Can couple states with different orbital angular momentum but total spin must be, also [σ σ ] µ = m m ( m m µ) (σ ) m (σ ) m Responsible for quadrupole moment of the deuteron Second term: Yukawa Remaining: delta-function

9 Momentum space Transform to total and relative momentum basis or wave vectors Use to find Yukawa (p p V p 3 p 4 )=(Pp V P p )=δ P,P p V p k V k = V Helps for Coulomb exp {iq r} =4π lm k V k = 4π V k V Y k = 4π V 0 V µ d 3 r exp {i(k k) r}v (r) i l Y lm(ˆr)y lm ( ˆq)j l (qr) dr r j 0 (qr)v (r) with q = k k µ +(k k) k V C k = 4π V q q e when k k (k k)

10 Partial wave basis Requires matrix elements of the form klm L V k L M L = dˆk LM L ˆk dˆk ˆk L M L k V (r) k For Yukawa write and use with Legendre functions yields µ +k +k kk cos θ kk l = l=0 k V Y (r) k = 4π V m= l = klm L V k L M L = δ L,L l=0 V 0 µ kk µ +k +k kk cos θ kk (l + ) Q l ( µ + k + k kk 4π Q l ( µ + k + k kk ) Q 0 (z) = ( ) z + ln z Q (z) = z ( ) z + ln z ( ) Q (z) = 3z z + ln 3 4 z z δ ML,M L (4π) V 0 Vµkk Y lm(ˆk)y lm ( ˆk ) Q L ( µ + k + k ) kk ) P l (cos θ kk )

11 Example Reid soft-core interaction (968) solid S 0 no bound state 00 dashed 3 S deuteron?? V(r) (MeV) 0 note similarity to atom-atom interaction! r (fm)

12 Two-particle states and interactions Pauli principle has important effect on possible states Free particles plane waves Eigenstates of T = p m Use box normalization Nucleons notation (isospin) p s = m s t = m t pm s m t Use successive basis transformations for two-nucleon states to survey angular momentum restrictions Total spin & isospin; CM and relative momentum; orbital angular momentum relative motion; total angular momentum

13 Start with Antisymmetric two-nucleon states p m s m t ; p m s m t = { p m s m t p m s m t p m s m t p m s m t } = SM S TM T {( m s m s SM S )( m t m t TM T ) p p SM S TM T ) ( m s m s SM S )( m t m t TM T ) p p SM S TM T )} then P = p + p p = (p p ) and use p = LM L plm L LM L ˆp = LM L plm L Y LM L ( ˆp) p = LM L plm L LM L p = LM L plm L ( ) L Y LM L ( ˆp) Y LM L ( p) =Y LM L (π θ p, φ p + π) =( ) L Y LM L ( ˆp) as well as ( m s m s SM S )=( ) + S ( m s m s SM S ) ( m t m t TM T ) = ( ) + T ( m t m t TM T )

14 Summarize Antisymmetry constraints for two nucleons p m s m t ; p m s m t = ( m s m s SM S )( m t m t TM T ) YLM L ( ˆp) SM S TM T LM L = [ ( ) L+S+T ] P p LM L SM S TM T ) ( m s m s SM S )( m t m t TM T ) Y LM L ( ˆp) L + S + T SM S TM T LM L JM J (L M L SM S JM J ) [ ( ) L+S+T ] P p (LS)JM J TM T ) must be odd! Notation T=0 T= 3 S 3 D P 3 D... S 0 3 P 0 3 P 3 P 3 F D

15 Phase shifts Dynamic Static Nucleon correlations

16 Phase shifts Nucleon correlations