Towards Faddeev-AGS equations in a Coulomb basis in momentum space

Transcription

1 Towards Faddeev-AGS equations in a Coulomb basis in momentum space V. Eremenko 1,5 L. Hlophe 1 Ch. Elster 1 F. M. Nunes 2 I. J. Thompson 3 G. Arbanas 4 J. E. Escher 3 TORUS Collaboration ( 1 Institute of Nuclear & Particle Physics and Dept. of Physics & Astronomy, Ohio University 2 NSCL, Michigan State University 3 Lawrence Livermore National Laboratory 4 Oak Ridge National Laboratory 5 Moscow State University, Moscow, Russia Supported by U.S. Department of Energy, Office of Science of Nuclear Physics V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 1 / 21 Coll

2 Introduction & Motivation Focus on deuteron-induced reactions Deuteron-induced reactions are a point of the theoretical interest To treat experimental results on the exotic nuclei beams with the deuterated targets in inverse kinematics. Low energy E lab from few up to 50 MeV/A. Broad range of involved nuclei From He isotopes up to Pb. Main focus is on (d, p) But the tools must analyze other three-body reaction channels (e.g. deuteron breakup) on the same footing. V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 2 / 21 Coll

3 Introduction & Motivation d + A reaction channels Approximated three-body: elastic scattering, direct deuteron breakup A(d, pn)a without core excitation, direct deuteron stripping (d, p) or (d, n) with nucleon capture without core excitation. Approximated three-body with excitations: direct deuteron breakup and stripping with core excitations, non-elastic direct (d, d ) scattering with core excitations. Many-body problems: core breakup, compound-nucleus reactions... V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 3 / 21 Coll

4 Introduction & Motivation Deuteron-induced reactions as three-body problem Deltuva et al.. // Phys. Rev. C 76, (2007) Neglect the internal degrees of freedom of the nucleus to get 3-body problem. Use Faddeev-AGS approach to treat the problem. To do later: Take into account some target degrees of freedom: collective core excitations, more complicated internal dynamics. V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 4 / 21 Coll

5 Introduction & Motivation Formal considerations about a 3-body system 3-body configurations & Jacobi coordinates A(pn) = d + A p(na) n(ap) Numerical indices: A 1, p 2, n 3. Odd man out configuration notation: a(bc) labeled by a. Denoting the two-body interactions V a V bc, i.e. a is a spectator. V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 5 / 21 Coll

6 Introduction & Motivation 3-body Hamiltonian H = H 0 + V np + U na + v pa, H 0 = q a /2M a + p a /2µ a. Interactions between the particles v pa = V C pa + v S pa, V C pa = Z Aα 2, vpa S = U pa + v cd r pa, V np : NN-interaction potential, e.g. chiral. U na, U pa : phenomenological optical potentials, e.g. CH89. vpa cd : short-range Coulomb interaction, usually, the charged sphere potential. V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 6 / 21 Coll

7 Introduction & Motivation Faddeev approach to a three-body problem φ = 3 a=1 φ a [= φ 1 ] is initial state [A + (pn)]. Faddeev components ψ a 3 Ψ = φ + g0 C (E)Va S Ψ, a=1 Ψ = a g C 0 (E) = ( E H 0 V C pa + iε ) 1, δ a,1 φ 1 + g0 C (E)Va S Ψ = }{{} a ψ a ψ a. Faddeev-type equations ψ 1 = φ 1 + g C 0 (E)V S 1 ψ a a ψ 2 = g C 0 (E)V S 2 ψ a a ψ 3 = g C 0 (E)V S 3 ψ a a ψ 1 = φ 1 + g C 0 (E)tS 1 ψ a a 1 ψ 2 = g C 0 (E)tS 2 ψ a a 2 ψ 3 = g C 0 (E)tS 3 ψ a a 3 V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 7 / 21 Coll

8 Introduction & Motivation Coulomb interaction in momentum space Screening factor (g C 0 Ṽ C = Z 1Z 2 α 2 exp( µr n ). r becomes g 0, Ṽ C goes with the other potentials) Various short-range amplitude corrections required.. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 8 / 21 Coll

9 Introduction & Motivation Coulomb interaction in momentum space Screening factor (g C 0 Ṽ C = Z 1Z 2 α 2 exp( µr n ). r It works for dp scattering becomes g 0, Ṽ C goes with the other potentials) Various short-range amplitude corrections required. Deltuva // Phys. Rev. C 80, (2009). E p(lab) = 9 MeV. The most advanced screening techniques to the date. AV18 and AN18+UIX potentials. V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 8 / 21 Coll

10 Introduction & Motivation A(d, p)b reaction with screened Coulomb interaction It works for the nuclei up to 48 Ca 20 (Deltuva) Deltuva. // Phys. Rev. C 80, (2009). Upadhyay, Deltuva, Nunes. // Phys. Rev. C 85, (2012). The charge wall Discrepancy between the results of CDCC & Faddeev calculations becomes unreasonably large as the nucleus charge climbs up. Faddeev-AGS method hits the charge wall. V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes,eqs. I. J.inThompson, a Coulomb G. basis Arbanas, J. E. Escher[1mm] TORUS 9 / 21 Coll

11 Introduction & Motivation Faddeev approach on the way to the highly-charged nuclei Total recall Ṽ C = Z 1Z 2 α 2 exp( µr n ) r g 0 = [E H 0 + iε] 1 (H 0 = q a /2M a + p a /2µ a ). V C = Z 1Z 2 α 2 r g0 C = [, E H 0 + iε VpA C ] 1 In terms of the g 0 and g C 0 proper basis states (simplified notation): g 0 = φ φ E H 0 + iε, gc 0 = η = Z A e 2 µ p /k. ψk,η C ψk,η C E H 0 + iε, V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 10 / 21 Coll

12 Coulomb basis in momentum space Coulomb function in momentum space 3D: p ψ C k,η = ψ C k,η (p) = 1 2π 2 lim γ +0 { [ d p 2 + (γ ik) 2] } iη dγ [γ 2 + (p k) 2 ] 1+iη. Guth and Mullin. // Phys. Rev. 83, 667 (1951). In partial waves: ψl,k,η C (p) = 2πeπη/2 pk lim γ +0 d dγ { [p 2 (k + iγ) 2 2pk ] iη } Q iη l (ζ) (ζ 2 1) iη/2, η = Z 1 Z 2 α 2 µ/k; ζ = (k 2 + p 2 + γ 2 )/2kp. Dolinskii and Mukhamedzhanov. // Sov. Journ. of Nucl. Phys., vol. 3, No. 2, p. 180 (1966). V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 11 / 21 Coll

13 Coulomb basis in momentum space The two representations The regular representation (p k): ψl,k,η C (p) = q(pk) l [ 4πηe πη/2 (p 2 + k 2 ) 1+l+iη lim p 2 (k + iγ) 2] 1+iη γ +0 [ ] ( Γ(1 + l + iη) 2 + l + iη 2F 1, 1 + l + iη 4k 2 p 2 ), l + 3/2; (1/2) l (p 2 + k 2 ) 2. The pole-proximity representation (p k): [ ] (p + k) ψl,k,η C 2 l (p) = i2π p exp( πη/2 + iσ l) lim 4pk 2 γ +0 [ Γ(1 + iη)e iσ l (p + k) 1+iη )] (p k)2 Im (p k + iγ) 1+iη 2F 1 ( l, l iη, 1 iη; (p + k) 2. V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 12 / 21 Coll

14 Coulomb basis in momentum space Coulomb function in momentum space (l = 0 plot) 10 2 Re ψ C l,k,η (p) [fm3 ] η = 1 η = 2 η = 3 k = 1.5 fm p [fm -1 ] Eremenko et al. // Comp. Phys. Comm. 187, 195 (2015). V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 13 / 21 Coll

15 Coulomb basis in momentum space The two-body t-matrix elements in Coulomb basis t C a,l (k,k,e) { ψ C }} { g0 C (E)t S (l)a = l,k,η ψ C t S ψ C l,k,η (l)a l,k,η ψl,k,η C, E H 0 + iε t C a,l (k, k, E) = dp dp ψl,k C,η (p ) t S a,l (p, p, E) ψl,k,η C (p). Some indices omitted for the simplisity. Pinch singularity in the elastic channel: E = 2µk 2 = 2µk 2. Since γ +0, singularities of ψ C l,k,η and ψl,k,η C are pinching the integration contour. +iγ k = k p, p iγ V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 14 / 21 Coll

16 Coulomb basis in momentum space If t has a separable representation t S l (k, k, E) = zy t C l (k, k, E) = zy u l,z (k ) λ l,zy (E) u l,y (k), u C l,z (k ) λ l,zy (E) u C l,y (k), dp u C l,z(k p 2 dp p ) = 2π 2 u l,z(p )ψl,k C,η (p ), u C l,y(k) 2 = 2π 2 u l,y(p) ψl,k,η(p). C Two independent integrals over p and p. Cauchy s theorem. No pinch singularity! k +iγ p V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 15 / 21 Coll

17 Coulomb basis in momentum space Gel fand-shilov regularization for complex form-factors u C l (k) b>0 a<0 f(y) dy y 1+iη b a dy J a (y), (e.g. f(y) = y 2 + y + 1). Subtract as many terms of Laurent expansion of f(y) around the integrand s special point y = 0 as needed to split the integral and get the regular term, plus the analytically calculated terms. Re J a,b,c (y) (a) (b) y (c) Upadhyay et al. // Phys. Rev. C 90, (2014). V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 16 / 21 Coll

18 Coulomb basis in momentum space Results: form-factors in Coulomb basis u C l,1 (k) = dp p 2 2π 2 u l,1(p)ψ C l,k,η (p) n + 12 C l = 0 l = 2 l = 4 Re u l,1 [fm 2 ] l = 0 l = 3 l = 6 l = 0 l = 4 l = 8 n + 48 Ca n Pb Re u l,1 C [fm 2 ] Upadhyay et al. // Phys. Rev. C 90, (2014) k [fm -1 ] V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 17 / 21 Coll

19 Coulomb basis in momentum space Singularity contribution for n + 12 C C 0.02 Re u 0,1 C [fm 2 ] u 0,1 C (k) [full] = 10-1 fm -1 = 10-2 fm -1 = 10-3 fm -1 = 10-6 fm -1 k p k [fm -1 ] Upadhyay et al. // Phys. Rev. C 90, (2014). V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 18 / 21 Coll

20 Coulomb basis in momentum space Singularity contribution for n Pb Re u 0,1 C [fm 2 ] Pb u 0,1 C (k) [full] = 10-1 fm -1 = 10-2 fm -1 = 10-3 fm -1 = 10-6 fm k [fm -1 ] Upadhyay et al. // Phys. Rev. C 90, (2014). k p V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 19 / 21 Coll

21 Faddeev-AGS eqs. in Coulomb basis Faddeev-AGS equations in Coulomb basis (in progress) Published abstract formalism Faddeev-AGS equations in Coulomb basis for real two-body t-matrices and spinless particles. Mukhamedzhanov et al. // Phys. Rev. C 86, (2012). To do: Take spin degrees of freedom into account. Deal with the 3-body singularity. Generalize the equations for the complex potentials. Take care about other implementation details, develop the algorithms and codes to solve the equations. Objective: Calculate scattering observables and make the code available as open source. V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 20 / 21 Coll

22 Summary & Outlook Summary & Outlook Faddeev formalism is the theoretical tool to study deuteron-induced reactions. This formalism treats all possible three-body channels on the same footage. Momentum space is preferable due to the boundary conditions. Coulomb interaction can be treated properly by using the Coulomb basis in momentum space. Pinch singularity is avoided by choosing the two-body interactions in separable form. Mathematics and machinery are developed to compute Coulomb functions and matrix elements in Coulomb basis in momentum space. Work is in progress to cast the Faddeev-AGS equations in the Coulomb basis taking into account spin degrees of freedom and 3-body singularity in order to solve the equations numerically. V. Eremenko, L. Hlophe, Ch. Elster, F. To M. FAGS Nunes, eqs. I. J. inthompson, a CoulombG. basis Arbanas, J. E. Escher[1mm] TORUS 21 / 21 Coll