Poincaré Invariant Calculation of the Three-Body Bound State Energy and Wave Function

Transcription

1 Poincaré Invariant Calculation of the Three-Body Bound State Energy and Wave Function q [fm ] M. R. Hadizadeh, Ch. Elster and W. N. Polyzou log Ψ nr log Ψ r p [fm ] log Ψ INPP, Department of Physics and Astronomy, Ohio University, Department of Physics and Astronomy, The University of Iowa. 4th Joint Meeting of the APS Division of Nuclear Physics and the Physical Society of Japan Waikoloa, HI, 9 Oct. 4

2 Motivation/Goal Why relativistic few-body physics? Few GeV scale few-body scattering with realistic interactions Separate evidence for sub-nucleon DOF from relativistic effects in Few GeV scale cross sections Goals Present Relativistic 3B binding energy and wave function; spinless & using direct 3D integration Net Relativistic 3N binding energy and wave function; spin-isospin dependent & using direct 3D integration Long-term Relativistic 3N scattering at the few GeV scale; spin-isospin dependent & using direct 3D integration M. R. Hadizadeh, Ch. Elster and W. N. Polyzou Poincaré Invariant Calculation of the 3B Bound State...

3 Relativistic Jacobi momenta ( ) k µ i := Λ (P/M ) µ νpi ν k i = p i + P P p i M ω m(p i ) M + M +P p µ jk := Λ (k jk /m jk ) µ νk ν j p µ i : single-particle four momenta p jk = k j + k jk m jk ( k jk k j m jk + m jk +k jk ) ω m (k j ) P µ := 3 i= pµ i : total four momentum of the non-interacting 3B system M = Pµ P µ : invariant mass of the non-interacting 3B system ω m (p i ) = p i k µ jk = kµ j = m + pi : the energy of the i-th particle + k µ k : the four momentum of the pair (jk) in the 3B rest frame m jk = kµ jk k µjk: the two-body invariant mass of the (jk) subsystem The Jacobian of the basis change P, p jk, k i = N (k j, k k ) P, k j, k k / [ ] N (k j, k k ) = (k j, k k ) / ωm (p jk ) + ω m (p jk ) ω m (k j )ω m (k k ) (p jk, k i ) = ω m (k j ) + ω m (k k ) ω m (p jk )ω m (p jk ) M. R. Hadizadeh, Ch. Elster and W. N. Polyzou Poincaré Invariant Calculation of the 3B Bound State...

4 Relativistic Faddeev equations for 3B bound state Faddeev component: ψ i ψ jk,i = (M t M ) T jk (M t ) P ψ i M t = E t + 3m: 3B mass eigenvalue M : non-interacting 3B mass operator P = P P 3 + P 3 P 3 T jk (M t ): boosted B t-matri 3B wave function: Ψ = ψ i + P ψ i Representation in momentum space: p jk, k i p jk, k i ψ i = dp jk dk i dp jk dk i M t M (p jk, k i δ(k i k ( i) T jk pjk, p jk; M t ω m (k i ) ) p jk, k i P p jk, k i p jk, k i ψ i p jk, k i Ψ = p jk, k i ψ i + N (k j, k k ) N (k k, k i ) p ki, k j ψ i + N (k j, k k ) N (k i, k j ) p ij, k k ψ i M. R. Hadizadeh, Ch. Elster and W. N. Polyzou Poincaré Invariant Calculation of the 3B Bound State...

5 Boosted B t-matri The relativistic half-shell t-matri in B c.m. frame can be obtained from non-relativistic one by F (p jk, p jk, k i ) ( ) T jk p jk, p jk ; mjk (p jk ) + k i + i + = ( ) t nr p jk, p jk ; p jk m + i+ = 4m mjk (p jk) + ki + m jk (p jk ) + k i F (pjk, p jk, k i = 5 f m ) p p jk [f m ] jk [f m ] k i ; the slope of F ) We obtain the fully off-shell t-matri T jk (p jk, p jk ; E(k i) from half-shell t-matri by solving the first resolvent equation: T (z j ) = T (z i ) + T (z j ) [g (z j ) g (z i )] T (z i ) for each value of off-shell momentum p jk for all values of B subsystem energy, dependent to Jacobi momentum k i M. R. Hadizadeh, Ch. Elster and W. N. Polyzou Poincaré Invariant Calculation of the 3B Bound State...

6 Permutation Operator in Relativistic Basis p jk, k i P p jk, k i = p jk, k i p ki, k j + p jk, k i p ij, k k { ( = N(k i, k i ) δ 3 p jk k i ) ( k ic(k i, k i ) δ 3 p jk + k i + ) k i C(k i, k i ) ( +δ 3 p jk + k i + ) ( k ic(k i, k i ) δ 3 p jk k i ) } k i C(k i, k i ) where N(k i, k i ) = N ( k i k i, k i ) N ( k i k i, k i) C(k i, k i) = + ω m (k i ) ω m( k i + k i ) ω m (k i ) + ω m( k i + k i ) + ( ω m (k i ) + ω m( k i + k i ) ) k i In the limit: m p jk, k i N C M. R. Hadizadeh, Ch. Elster and W. N. Polyzou Poincaré Invariant Calculation of the 3B Bound State...

7 Matri elements of Jacobian N & coefficient C N (k cos(), k sin(), ) k = f m π/ k = f m π/ k = 5 f m π/ k = f m π/ N varies in the range: C varies in a wider range: C(k cos(), k sin(), ) k = f m π/..9.8 k = f m π/ k = 5 f m π/..8 k = f m π/ M. R. Hadizadeh, Ch. Elster and W. N. Polyzou Poincaré Invariant Calculation of the 3B Bound State...

8 The relativistic (r) and non-relativistic (nr) 3B binding energies calculated with the MT-V potential E nr [MeV] E r [MeV] E r E nr [%] E nr E appro E r E r [%] E r [MeV] E appro [MeV] N = C = F = G nr N = C = F =, G nr W. Glöckle et al. PRC33 PW epansion truncated to s-wave E r E nr E nr =.7 % Approimations: N = : ignore relativistic Jacobian C = : relativistic Jacobi momenta non-relativistic Jacobi momenta F = : relativistic half-shell t-matri non-relativistic half-shell t-matri G nr : relativistic free propagator non-relativistic free propagator [%] Eappr Er Er 3 N= C= F= C= N= G nr F= G nr M. R. Hadizadeh, Ch. Elster and W. N. Polyzou Poincaré Invariant Calculation of the 3B Bound State...

9 Relativistic 3B Wave Function ki [f m ] log Ψ r (a) log Ψ (c) (b) (d) p jk [f m ] (a) & (c): k i p jk (b) & (d): k i p jk The largest relativistic effect appear at large values of the momentum of spectator particle k i, due to the Fermi motion. M. R. Hadizadeh, Ch. Elster and W. N. Polyzou Poincaré Invariant Calculation of the 3B Bound State...

10 Momentum Distribution Functions n(k i ) = 8π k i n(p jk ) = 8π p jk n(p jk ), n(k i ) [fm] dp jk p jk dk i k i n r (k i ) n nr (k i ) n r (p jk )/ n nr (p jk )/ p jk, k i [fm - ] d pjk Ψ (p jk, k i, pjk ) d pjk Ψ (p jk, k i, pjk ) n(p jk ) [fm] n(k i ) [fm] r - nr r C= - nr r N= - nr r C=N= - nr (a) r - nr r C= - nr r N= - nr r C=N= - nr (b) p jk, k i [fm - ] M. R. Hadizadeh, Ch. Elster and W. N. Polyzou Poincaré Invariant Calculation of the 3B Bound State...

11 Summary Accurate solution of the relativistic 3B bound state in spinless 3D representation Calculation of the relativistic 3B wave function for the first time The relativistic effect in the 3B binding energy is small, about 3.3 % and decreases the binding energy Relativistic effect in the 3B bound state arises from 4 terms, by order: Relativistic Transition Matri Relativistic Free Propagator 3 Relativistic Jacobian 4 Relativistic Jacobi Momenta Combination of all four ingredients leads to a small correction, while individually the corrections do not have to be small. So, we should keep them all to get the correct relativistic effect. arxiv:49.65 Net step: Spin M. R. Hadizadeh, Ch. Elster and W. N. Polyzou Poincaré Invariant Calculation of the 3B Bound State...