Variatonal description of continuum states

Transcription

1 Variatonal description of continuum states A. Kievsky INFN, Sezione di Pisa (Italy) Collaborators Three-body systems in reactions with rare isotopes ECT*, October 2016 M. Viviani, L.E. Marcucci - INFN & University of Pisa L. Girlanda - University of Lecce M. Gattobigio - Institute Non Lineaire de Nice (France) E. Garrido - CSIC - Madrid (Spain) A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

2 Variatonal description of continuum states A. Kievsky INFN, Sezione di Pisa (Italy) Collaborators Three-body systems in reactions with rare isotopes ECT*, October 2016 M. Viviani, L.E. Marcucci - INFN & University of Pisa L. Girlanda - University of Lecce M. Gattobigio - Institute Non Lineaire de Nice (France) E. Garrido - CSIC - Madrid (Spain) A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

3 Why studying scattering states specific motivations from nuclear physics Few bound states exist in very light nuclei Most of the investigations on the nuclear interaction are done using scattering data More than 5000 data in np and pp data are used to construct specific models of the NN interaction However: A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

4 3 H and 4 He Bound States and n d scattering length Potential(NN) Method 3 H[MeV] 4 He[MeV] 2 a nd [fm] AV18 HH FE/FY Bochum FE/FY Lisbon CDBonn HH FE/FY Bochum FE/FY Lisbon NCSM 7.99(1) N3LO-Idaho HH FE/FY Bochum FE/FY Lisbon NCSM 7.852(5) 25.39(1) Potential(NN+NNN) AV18/UIX HH FE/FY Bochum CDBonn/TM HH FE/FY Bochum N3LO-Idaho/N2LO HH NCSM 8.473(5) 28.34(2) Exp ±0.010 A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

5 Why studying scattering states specific motivations from nuclear physics Few bound states exist in very light nuclei Most of the investigations on the nuclear interaction are done using scattering data More than 5000 data in np and pp data are used to construct specific models of the NN interaction However: It seems very difficult to reproduce simultaneously the triton and alpha particle binding energies and the n d scattering length Furthermore: A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

6 dσ/dω [mb/sr] AV18 -no MM V NN θ c.m. [deg] T θ c.m. [deg] A y θ c.m. [deg] E c.m. = 2.0 MeV T θ c.m. [deg] it θ c.m. [deg] 0 T θ c.m. [deg] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

7 AV18/UIX N3LO/N2LO V low-k (CDBonn) E c.m. = 2.0 MeV dσ/dω [mb/sr] θ c.m. [deg] T θ c.m. [deg] A y θ c.m. [deg] T θ c.m. [deg] it θ c.m. [deg] 0 T θ c.m. [deg] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

8 Why studying scattering states specific motivations from nuclear physics Few bound states exist in very light nuclei Most of the investigations on the nuclear interaction are done using scattering data More than 5000 data in np and pp data are used to construct specific models of the NN interaction However: It seems very difficult to reproduce simultaneously the triton and alpha particle binding energies and the n d scattering length Furthermore: Severe discrepancies between experimental data and theoretical predictions can be found in some three- and four-nucleon observables in the very low energy sector A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

9 Actual situation quality of the description: 2N system: χ 2 per datum 1 3N and 4N system: χ 2 per datum >> 1 Some comments: Polarization observables are small. They magnify some problems in the spin structure of the nuclear force. Possibility of using 3N data to determine the 3N force The variational implementation of the scattering description could help Partially this project already started A=3,4: A theoretical lab for the nuclear interaction A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

10 Actual situation quality of the description: 2N system: χ 2 per datum 1 3N and 4N system: χ 2 per datum >> 1 Some comments: Polarization observables are small. They magnify some problems in the spin structure of the nuclear force. Possibility of using 3N data to determine the 3N force The variational implementation of the scattering description could help Partially this project already started A=3,4: A theoretical lab for the nuclear interaction A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

11 Why studying scattering states using a variational approach General motivations Variational predictions can be improved systematically Implementation using large basis are at present feasible The specific form of the boundary conditions is not always needed The variational principle can be given in terms of integral relations some applications are: Applications to the Hyperspherical Adiabatic expansion Breakup of three charged particles Study of universal aspects of the few-body dynamics A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

12 Why studying scattering states using a variational approach General motivations Variational predictions can be improved systematically Implementation using large basis are at present feasible The specific form of the boundary conditions is not always needed The variational principle can be given in terms of integral relations some applications are: Applications to the Hyperspherical Adiabatic expansion Breakup of three charged particles Study of universal aspects of the few-body dynamics A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

13 Why studying scattering states using a variational approach General motivations Variational predictions can be improved systematically Implementation using large basis are at present feasible The specific form of the boundary conditions is not always needed The variational principle can be given in terms of integral relations some applications are: Applications to the Hyperspherical Adiabatic expansion Breakup of three charged particles Study of universal aspects of the few-body dynamics A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

14 Variational Bounds (1) Two-body system: Bound states (T + V E n )Ψ n = 0 ( 2 m 2 + V E n )Ψ n (r) = 0 For E n < 0 and l = 0, Ψ n (r) = u n (r)/r 4π numerical solution in a grid {r i } u n (r i ), E n E n = < Ψ n H Ψ n > < Ψ n Ψ n > E n E n 10 7 E n expansion in a complete basis φ k (r) Ψ n (r) = N A n k φ k(r) k N A n k < φ k H E φ k >= 0 E n E n and, for N, E n E n k A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

15 Variational Bounds (1) Two-body system: Bound states (T + V E n )Ψ n = 0 ( 2 m 2 + V E n )Ψ n (r) = 0 For E n < 0 and l = 0, Ψ n (r) = u n (r)/r 4π numerical solution in a grid {r i } u n (r i ), E n E n = < Ψ n H Ψ n > < Ψ n Ψ n > E n E n 10 7 E n expansion in a complete basis φ k (r) Ψ n (r) = N A n k φ k(r) k N A n k < φ k H E φ k >= 0 E n E n and, for N, E n E n k A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

16 Variational Bounds (1) Two-body system: Bound states (T + V E n )Ψ n = 0 ( 2 m 2 + V E n )Ψ n (r) = 0 For E n < 0 and l = 0, Ψ n (r) = u n (r)/r 4π numerical solution in a grid {r i } u n (r i ), E n E n = < Ψ n H Ψ n > < Ψ n Ψ n > E n E n 10 7 E n expansion in a complete basis φ k (r) Ψ n (r) = N A n k φ k(r) k N A n k < φ k H E φ k >= 0 E n E n and, for N, E n E n k A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

17 Variational Bounds (2) Two-body system: Scattering states (T + V E)Ψ k = 0 ( 2 m 2 + V E)Ψ k (r) = 0 For k 2 = m 2 E and l = 0, Ψ k (r) = u k (r)/r 4π Asymptotic behavior u(r ) [A sin(kr) + B cos(kr)] / k numerical solution in a grid {r i } u k (r i ), A, B the following integral relations are verified tanδ = B/A m < Ψ 2 k H E F >= B with F = m < Ψ 2 k H E G >= A with G = k sin(kr) 4π kr k cos(kr) 4π kr tanδ = B A tanδ tan δ 10 7 tan δ A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

18 Variational Bounds (2) Two-body system: Scattering states (T + V E)Ψ k = 0 ( 2 m 2 + V E)Ψ k (r) = 0 For k 2 = m 2 E and l = 0, Ψ k (r) = u k (r)/r 4π Asymptotic behavior u(r ) [A sin(kr) + B cos(kr)] / k numerical solution in a grid {r i } u k (r i ), A, B the following integral relations are verified tanδ = B/A m < Ψ 2 k H E F >= B with F = m < Ψ 2 k H E G >= A with G = k sin(kr) 4π kr k cos(kr) 4π kr tanδ = B A tanδ tan δ 10 7 tan δ A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

19 Variational Bounds (2) Two-body system: Scattering states (T + V E)Ψ k = 0 ( 2 m 2 + V E)Ψ k (r) = 0 For k 2 = m 2 E and l = 0, Ψ k (r) = u k (r)/r 4π Asymptotic behavior u(r ) [A sin(kr) + B cos(kr)] / k numerical solution in a grid {r i } u k (r i ), A, B the following integral relations are verified tanδ = B/A m < Ψ 2 k H E F >= B with F = m < Ψ 2 k H E G >= A with G = k sin(kr) 4π kr k cos(kr) 4π kr tanδ = B A tanδ tan δ 10 7 tan δ A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

20 Explicitely the integrals are: k m dr sin(kr)v(r)u k (r) = B k m 2 2 Some details The integrals are short range! 0 0 dr cos(kr)v(r)u k (r) + k ψ(0) = A The irregular solution G can be regularized: k G reg = f reg (r)g = f reg 4π cos(kr) kr [ For example f reg (r) = (1 e γr ), γ 1 r int ] k Ψ(0) = Iγ If (H E)Ψ k = 0, I γ is independent of γ!!! Independence on γ: a check on Ψ k A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

21 Explicitely the integrals are: k m dr sin(kr)v(r)u k (r) = B k m 2 2 Some details The integrals are short range! 0 0 dr cos(kr)v(r)u k (r) + k ψ(0) = A The irregular solution G can be regularized: k G reg = f reg (r)g = f reg 4π cos(kr) kr [ For example f reg (r) = (1 e γr ), γ 1 r int ] k Ψ(0) = Iγ If (H E)Ψ k = 0, I γ is independent of γ!!! Independence on γ: a check on Ψ k A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

22 The Green s Theorem... Normalization condition: < F H E G > < G H E F >= W(F, G) = 1 (H E)Ψ k = 0 Ψ AF + BG < Ψ k H E G > < G H E Ψ k >= A = < Ψ k H E G > < F H E Ψ k > < Ψ k H E F >= B = < Ψ k H E F > The Integral Relations B = < Ψ k H E F > A = < Ψ k H E G > tanδ = B/A A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

23 Variational Bounds (2) - continuation The Rayleight-Ritz Variational Principle E = < Ψ b H Ψ b > < Ψ b Ψ b > or < Ψ b H E Ψ b >= 0 the Kohn Variational Principle (for a single channel) [tan δ] = tanδ < Ψ s H E Ψ s > Asymptotics Ψ b 0 as ρ with ρ 2 ij r 2 ij Ψ s φ 1 φ 2 [AF(y) + BG(y)] as ρ (two-cluster scattering) as ρ : (H E)Ψ b could be different from 0 (for example Ψ b = n (H E)φ 1 φ 2 [AF(y) + BG(y)] = 0 A n φ n ) A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

24 Variational Bounds (2) - continuation The Rayleight-Ritz Variational Principle E = < Ψ b H Ψ b > < Ψ b Ψ b > or < Ψ b H E Ψ b >= 0 the Kohn Variational Principle (for a single channel) [tan δ] = tanδ < Ψ s H E Ψ s > Asymptotics Ψ b 0 as ρ with ρ 2 ij r 2 ij Ψ s φ 1 φ 2 [AF(y) + BG(y)] as ρ (two-cluster scattering) as ρ : (H E)Ψ b could be different from 0 (for example Ψ b = n (H E)φ 1 φ 2 [AF(y) + BG(y)] = 0 A n φ n ) A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

25 Variational Bounds (2) - continuation The Rayleight-Ritz Variational Principle E = < Ψ b H Ψ b > < Ψ b Ψ b > or < Ψ b H E Ψ b >= 0 the Kohn Variational Principle (for a single channel) [tan δ] = tanδ < Ψ s H E Ψ s > Asymptotics Ψ b 0 as ρ with ρ 2 ij r 2 ij Ψ s φ 1 φ 2 [AF(y) + BG(y)] as ρ (two-cluster scattering) as ρ : (H E)Ψ b could be different from 0 (for example Ψ b = n (H E)φ 1 φ 2 [AF(y) + BG(y)] = 0 A n φ n ) A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

26 However...(let us take φ 1 = 1, φ 2 = 1) Ψ s = Ψ c (r) + AF(r) + BG(r) (Ψ c 0 as r ) [tan δ] = tan δ < (1/A)Ψ s H E (1/A)Ψ s > (tan δ = B/A) The variation of the functional [tanδ] = 0 implies < Ψ c H E Ψ s >= 0 < G H E Ψ s >= 0 [tan δ] 2nd = tan δ 1st < F H E (1/A)Ψ s > The Green s Theorem... < F H E G > < G H E F >= 1 < Ψ s H E G > < G H E Ψ s >= A = < Ψ s H E G > < F H E Ψ s > < Ψ s H E F >= B 1st but multiplying by A the equation for [tan δ] 2nd B 2nd = B 1st < F H E Ψ s >= < Ψ s H E F > A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

27 However...(let us take φ 1 = 1, φ 2 = 1) Ψ s = Ψ c (r) + AF(r) + BG(r) (Ψ c 0 as r ) [tan δ] = tan δ < (1/A)Ψ s H E (1/A)Ψ s > (tan δ = B/A) The variation of the functional [tanδ] = 0 implies < Ψ c H E Ψ s >= 0 < G H E Ψ s >= 0 [tan δ] 2nd = tan δ 1st < F H E (1/A)Ψ s > The Green s Theorem... < F H E G > < G H E F >= 1 < Ψ s H E G > < G H E Ψ s >= A = < Ψ s H E G > < F H E Ψ s > < Ψ s H E F >= B 1st but multiplying by A the equation for [tan δ] 2nd B 2nd = B 1st < F H E Ψ s >= < Ψ s H E F > A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

28 However...(let us take φ 1 = 1, φ 2 = 1) Ψ s = Ψ c (r) + AF(r) + BG(r) (Ψ c 0 as r ) [tan δ] = tan δ < (1/A)Ψ s H E (1/A)Ψ s > (tan δ = B/A) The variation of the functional [tanδ] = 0 implies < Ψ c H E Ψ s >= 0 < G H E Ψ s >= 0 [tan δ] 2nd = tan δ 1st < F H E (1/A)Ψ s > The Green s Theorem... < F H E G > < G H E F >= 1 < Ψ s H E G > < G H E Ψ s >= A = < Ψ s H E G > < F H E Ψ s > < Ψ s H E F >= B 1st but multiplying by A the equation for [tan δ] 2nd B 2nd = B 1st < F H E Ψ s >= < Ψ s H E F > A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

29 The Integral Relations B 2nd = < Ψ s H E F > A = < Ψ s H E G > [tan δ] 2nd = B 2nd /A The Integral Relations for coupled channels B 2nd ij = < Ψ s i H E F j > A ij = < Ψ s i H E G j > [R] 2nd = A 1 B 2nd (the eigenvalues of R are tanδ) Properties (i) The integral relations have a variational character (ii) The integrals converge even if (H E)Ψ s 0 asymptotically (iii) If (H E)Ψ s = 0 in the interaction region, the result for [tan δ] 2nd is exact, even if the asymptotic behavior of Ψ s is not the physical one A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

30 The Integral Relations B 2nd = < Ψ s H E F > A = < Ψ s H E G > [tan δ] 2nd = B 2nd /A The Integral Relations for coupled channels B 2nd ij = < Ψ s i H E F j > A ij = < Ψ s i H E G j > [R] 2nd = A 1 B 2nd (the eigenvalues of R are tanδ) Properties (i) The integral relations have a variational character (ii) The integrals converge even if (H E)Ψ s 0 asymptotically (iii) If (H E)Ψ s = 0 in the interaction region, the result for [tan δ] 2nd is exact, even if the asymptotic behavior of Ψ s is not the physical one A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

31 The Integral Relations B 2nd = < Ψ s H E F > A = < Ψ s H E G > [tan δ] 2nd = B 2nd /A The Integral Relations for coupled channels B 2nd ij = < Ψ s i H E F j > A ij = < Ψ s i H E G j > [R] 2nd = A 1 B 2nd (the eigenvalues of R are tanδ) Properties (i) The integral relations have a variational character (ii) The integrals converge even if (H E)Ψ s 0 asymptotically (iii) If (H E)Ψ s = 0 in the interaction region, the result for [tan δ] 2nd is exact, even if the asymptotic behavior of Ψ s is not the physical one A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

32 Example I: Scattering from bound state solutions V(r) = V 0 e r 2 /b 2 2 /m = MeV fm 2, NN system: bound state Expansion in a complete basis: (V 0 = 51.5 MeV, b = 1.6 fm) Ψ b (r) = E 2B = MeV M A n φ n (r) n=0 φ n (r) = L (2) n (z) e z/2 (z = βr) H nn =< φ n H φ n >, N nn =< φ n φ n >= δ nn A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

33 E [MeV] M=40 E 0 = MeV E 1 E 2 E β [fm -1 ] M Ψ b (r) = A b nφ n (r), < Ψ b H Ψ b >= E 0 n=0 M Ψ i (r) = A i n φ n(r), < Ψ i H Ψ i >= E i, i = 1, 2, 3,... n=0 A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

34 B = < Ψ i H E i F i > F i = sin k i r/k i r A = < Ψ i H E i G i > G i = f reg (r)[cos k i r/k i r] B/A = [tanδ] 2nd k 2 i = E i /( 2 /m) M E 0 [MeV] E 1 [MeV] tanδ 2nd tan δ E 2 [MeV] tanδ 2nd tan δ E 3 [MeV] tanδ 2nd tan δ A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

35 B = < Ψ i H E i F i > F i = sin k i r/k i r A = < Ψ i H E i G i > G i = f reg (r)[cos k i r/k i r] B/A = [tanδ] 2nd k 2 i = E i /( 2 /m) M E 0 [MeV] E 1 [MeV] tanδ 2nd tan δ E 2 [MeV] tanδ 2nd tan δ E 3 [MeV] tanδ 2nd tan δ A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

36 2 Polynomial M=40 exact 1 Φ(r) r [fm] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

37 Example II: Two-body scattering with Coulomb potential V(r) = V 0 e r 2 /b 2 + e2 r W(r) = V 0 e r 2 /b 2 +[e (r/rsc)n ] e2 r Ψ k F c k (r) + tan δc k Gc k (r) Ψ (n,rsc) k F k (r)+tan[δ (n,rsc) k ]G k (r) B = < Ψ (n,rsc) k T + V E Fk c > A =< Ψ (n,rsc) k T + V E Gk c > [tan δ] 2nd = B/A [tanδ] 2nd n=2 n=3 n=4 n=1-1 c tanδ k n=5 n= r sc [fm] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

38 Applications to the three-nucleon system The three-nucleon bound state in the HH formalism = A α,k m m,α, K > Ψ 1/2+ T z m,α,k < ρ,ω m,α, K >= L (5) m (βρ) e βρ/2 [ Y Lα [K α] (Ω) χ S α ] The linear coefficients are the solutions of A α,k m < m,α, K H E m,α, K >= 0 m,α,k The 3 H bound state (in MeV) 1/2 + ζtα T z Method AV18 CDBonn N3LO AV18+UR CDBonn+TM N3LO+N2LO HH FE Bochum FE Lisbon NCSM 7.99(1) 7.852(5) 8.473(5) A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

39 Applications to the three-nucleon system The three-nucleon bound state in the HH formalism = A α,k m m,α, K > Ψ 1/2+ T z m,α,k < ρ,ω m,α, K >= L (5) m (βρ) e βρ/2 [ Y Lα [K α] (Ω) χ S α ] The linear coefficients are the solutions of A α,k m < m,α, K H E m,α, K >= 0 m,α,k The 3 H bound state (in MeV) 1/2 + ζtα T z Method AV18 CDBonn N3LO AV18+UR CDBonn+TM N3LO+N2LO HH FE Bochum FE Lisbon NCSM 7.99(1) 7.852(5) 8.473(5) A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

40 2 + 1 three-nucleon scattering states in the HH formalism F Jπ,T z LS Ψ Jπ,T z LS [ = S = Ψ Jπ,T z C [φ d χ 1 2 Ψ Jπ,T z C + A Jπ,T z LS,LS F Jπ,T z LS + B Jπ,T z LS,L S G Jπ,T z L S L S = m,α,k ] ] S F L (ky) J A α,k m m,α, K > G Jπ,T z LS [ = S [φ d χ 1 2 The scattering matrix R = A 1 B is obtained from the KVP. For J = 1/2 ± R is 2 2, for J = 3/2 ±,..., R is 3 3. ] S G γ L (ky) ] benchmark for p d phases using AV14: PRC63, (2001) E lab = 1 MeV E lab = 2 MeV E lab = 3 MeV J π (2S+1) L J HH FEcs HH FEcs HH FEcs 4 D 1/ S 2 1/ η 1/ P 1/ P 2 1/ ǫ 1/ A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43 J

41 2 + 1 three-nucleon scattering states in the HH formalism F Jπ,T z LS Ψ Jπ,T z LS [ = S = Ψ Jπ,T z C [φ d χ 1 2 Ψ Jπ,T z C + A Jπ,T z LS,LS F Jπ,T z LS + B Jπ,T z LS,L S G Jπ,T z L S L S = m,α,k ] ] S F L (ky) J A α,k m m,α, K > G Jπ,T z LS [ = S [φ d χ 1 2 The scattering matrix R = A 1 B is obtained from the KVP. For J = 1/2 ± R is 2 2, for J = 3/2 ±,..., R is 3 3. ] S G γ L (ky) ] benchmark for p d phases using AV14: PRC63, (2001) E lab = 1 MeV E lab = 2 MeV E lab = 3 MeV J π (2S+1) L J HH FEcs HH FEcs HH FEcs 4 D 1/ S 2 1/ η 1/ P 1/ P 2 1/ ǫ 1/ A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43 J

42 Benchmark calculations for scattering states N3LO (n 3 H and p 3 He) E n(mev) 1 S 0 3 P 0 3 S 1 3 D 1 ǫ P 1 3 P 1 ǫ AGS HH FY AGS HH FY AGS HH FY E p(mev) 1 S 0 3 P 0 3 S 1 3 D 1 ǫ P 1 3 P 1 ǫ AGS HH FY AGS HH FY AGS HH FY A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

43 Scattering states from bound like states Ψ Jπ,T z n = X m,α,k < ρ, Ω m, α, K >= L (5) m (βρ) e βρ/2 X m,α,k A α,k m,n m, α, K > h Y Lα [K α] (Ω) χ S α i A α,k m,n < m, α, K H E n m, α, K >= 0 J π ζtα T z A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

44 Scattering states from bound like states Ψ Jπ,T z n = X m,α,k < ρ, Ω m, α, K >= L (5) m (βρ) e βρ/2 X m,α,k A α,k m,n m, α, K > h Y Lα [K α] (Ω) χ S α i A α,k m,n < m, α, K H E n m, α, K >= 0 J π ζtα T z 0 J=1/2 + 0 J=1/2 - E n [MeV] E d =-2.226MeV E d = MeV E n [MeV] -4 AV14, m=24 AV14, m=24-6 E 0 = MeV β [fm -1 ] β [fm -1 ] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

45 Occupation probabilities J = 1/2 + (β = 2.5 fm 1 ) Ψ 1/2+ T z (E 0 ) P S = 90.96%, P P = 0.08%, P D = 8.96% Ψ 1/2+ T z (E 1 ) P S = 94.04%, P P = 0.00%, P D = 5.96% Ψ 1/2+ T z (E 2 ) P S = 1.22%, P P = 2.72%, P D = 96.06% Ψ 1/2+ T z (E 3 ) P S = 94.20%, P P = 0.00%, P D = 5.80% Ψ 1/2+ T z (E 2 ) P S = 1.21%, P P = 2.70%, P D = 96.09% Occupation probabilities J = 1/2 (β = 2.0 fm 1 ) Ψ 1/2 T z (E 1 ) P 1/2 P = 3.36%, P 3/2 P = 94.87%, P D = 1.77% Ψ 1/2 T z (E 2 ) P 1/2 P = 93.80%, P 3/2 P = 3.42%, P D = 2.78% Ψ 1/2 T z (E 3 ) P 1/2 P = 3.47%, P 3/2 P = 94.71%, P D = 1.82% Ψ 1/2 T z (E 4 ) P 1/2 P = 93.42%, P 3/2 P = 3.86%, P D = 2.72% Ψ 1/2 T z (E 5 ) P 1/2 P = 4.32%, P 3/2 P = 93.87%, P D = 2.01% Ψ 1/2 T z (E 6 ) P 1/2 P = 92.72%, P 3/2 P = 4.77%, P D = 2.51% A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

46 Occupation probabilities J = 1/2 + (β = 2.5 fm 1 ) Ψ 1/2+ T z (E 0 ) P S = 90.96%, P P = 0.08%, P D = 8.96% Ψ 1/2+ T z (E 1 ) P S = 94.04%, P P = 0.00%, P D = 5.96% Ψ 1/2+ T z (E 2 ) P S = 1.22%, P P = 2.72%, P D = 96.06% Ψ 1/2+ T z (E 3 ) P S = 94.20%, P P = 0.00%, P D = 5.80% Ψ 1/2+ T z (E 2 ) P S = 1.21%, P P = 2.70%, P D = 96.09% Occupation probabilities J = 1/2 (β = 2.0 fm 1 ) Ψ 1/2 T z (E 1 ) P 1/2 P = 3.36%, P 3/2 P = 94.87%, P D = 1.77% Ψ 1/2 T z (E 2 ) P 1/2 P = 93.80%, P 3/2 P = 3.42%, P D = 2.78% Ψ 1/2 T z (E 3 ) P 1/2 P = 3.47%, P 3/2 P = 94.71%, P D = 1.82% Ψ 1/2 T z (E 4 ) P 1/2 P = 93.42%, P 3/2 P = 3.86%, P D = 2.72% Ψ 1/2 T z (E 5 ) P 1/2 P = 4.32%, P 3/2 P = 93.87%, P D = 2.01% Ψ 1/2 T z (E 6 ) P 1/2 P = 92.72%, P 3/2 P = 4.77%, P D = 2.51% A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

47 Application of the Integral Relations [ Bij 2nd = < Ψ s i H E n F j > F j S [ A ij = < Ψ s i H E n G j > G j S [R] 2nd = A 1 B 2nd [φ d χ 1 2 [φ d χ 1 2 ] ] Sj F Lj (k n y) ] ] Sj G γ L j (k n y) J J 0 J=1/2 + 0 J=1/ E d =-2.226MeV -2 E d = MeV E n [MeV] -4 E n [MeV] E 0 = MeV β [fm -1 ] β [fm -1 ] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

48 J=1/ E lab =1 MeV -2 E lab = 2 MeV -3.5 E lab =3 MeV 4 D1/2 4 D1/2 4 D1/2 δ [deg] δ [deg] S1/ S1/ S1/ mix.par. [deg] 1 ε 1/2+ benchmark 1 ε 1/2+ 1 ε 1/ γ [fm -1 ] γ [fm -1 ] γ [fm -1 ] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

49 J=1/ E lab =1 MeV -6 E lab = 2 MeV -7 2 P1/2 2 P1/2 E lab =3 MeV 2 P1/2 δ [deg] P1/2 4 P1/2 4 P1/2 δ [deg] mix.par. [deg] 4 benchmark ε 1/2-5.5 ε 1/2-7.5 ε 1/ γ [fm -1 ] γ [fm -1 ] γ [fm -1 ] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

50 p-d J=1/2 + E lab =1 MeV E lab = 2 MeV E lab =3 MeV D1/2 4 D1/2 δ [deg] -1 4 D1/ δ [deg] S1/ S1/ S1/ mix.par. [deg] 1 ε 1/2+ benchmark 1 ε 1/2+ 1 ε 1/ γ [fm -1 ] γ [fm -1 ] γ [fm -1 ] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

51 2S+1 δj [deg] J=3/ E lab =1 MeV 2 D3/2 4 D3/ E lab = 2 MeV 2 D3/2 4 D3/ E lab =3 MeV 2 D3/2 4 D3/2 2S+1 δj [deg] S1/ S3/ S3/2 mix. ang. [deg] ξ 3/2+ ε 3/2+ η 3/2+ benchmark γ [fm -1 ] ξ 3/2+ ε 3/2+ η 3/ γ [fm -1 ] ξ 3/2+ ε 3/2+ η 3/ γ [fm -1 ] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

52 Scattering states from bound like states (A = 4) Ψ Jπ,T z 1 = X m,α,k < ρ, Ω m, α, K >= L (8) m (βρ) e βρ/2 X m,α,k A α,k m, α, K > m,1 h Y Lα [K α] (Ω) χ S α i A α,k m,1 < m, α, K H E 1 m, α, K >= 0 J π ζtα T z 0-1 J π =0 +, T=T z =1-2 E n [MeV] N2LO, m= β [fm -1 ] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

53 p 3 He scattering [ ] B 2nd = < Ψ n H E n F n > F n S [φ3 He χ 1 ] S F L (k n y) 2 [ ] A = < Ψ n H E n G n > G n S [φ3 He χ 1 ] S G γ L (k ny) 2 [R] 2nd = A 1 B 2nd J J -40 E lab =3.13 MeV -50 E lab =4.05 MeV -60 E lab =5.54 MeV δ [deg] S0 S0 S γ [fm -1 ] γ [fm -1 ] γ [fm -1 ] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

54 p d scattering from n d scattering Ψ LSJ (x, y) = Ψ C LSJ (x, y) + F LSJ(x, y) + L S J R SS LL G L S J(x, y) Ψ C LSJ (x, y) 0 as ρ F LSJ (x, y) = i F L(y i ) [ [φ d (x i ) 1 2 ] S Y L (ŷ i ) ] JJ z ξ TTz G LSJ (x, y) = i G L(y i ) [ [φ d (x i ) 1 2 ] S Y L (ŷ i ) ] JJ z ξ TTz n d scattering (r sc, n sc ) J B 2nd LS,L S = < Ψ rsc,nsc L S J H E F C LSJ > J A LS,L S =< Ψrsc,nsc L S J H E GC LSJ > [R] 2nd = A 1 B 2nd E N = 3.0 MeV, AV18, J = r sc = 50 fm, n sc = 5 p d Int.Rel. 4 D 1/2 = S 1/2 = η 1/2+ = A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

55 p d scattering from n d scattering Ψ LSJ (x, y) = Ψ C LSJ (x, y) + F LSJ(x, y) + L S J R SS LL G L S J(x, y) Ψ C LSJ (x, y) 0 as ρ F LSJ (x, y) = i F L(y i ) [ [φ d (x i ) 1 2 ] S Y L (ŷ i ) ] JJ z ξ TTz G LSJ (x, y) = i G L(y i ) [ [φ d (x i ) 1 2 ] S Y L (ŷ i ) ] JJ z ξ TTz n d scattering (r sc, n sc ) J B 2nd LS,L S = < Ψ rsc,nsc L S J H E F C LSJ > J A LS,L S =< Ψrsc,nsc L S J H E GC LSJ > [R] 2nd = A 1 B 2nd E N = 3.0 MeV, AV18, J = r sc = 50 fm, n sc = 5 p d Int.Rel. 4 D 1/2 = S 1/2 = η 1/2+ = A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

56 p d scattering from n d scattering Ψ LSJ (x, y) = Ψ C LSJ (x, y) + F LSJ(x, y) + L S J R SS LL G L S J(x, y) Ψ C LSJ (x, y) 0 as ρ F LSJ (x, y) = i F L(y i ) [ [φ d (x i ) 1 2 ] S Y L (ŷ i ) ] JJ z ξ TTz G LSJ (x, y) = i G L(y i ) [ [φ d (x i ) 1 2 ] S Y L (ŷ i ) ] JJ z ξ TTz n d scattering (r sc, n sc ) J B 2nd LS,L S = < Ψ rsc,nsc L S J H E F C LSJ > J A LS,L S =< Ψrsc,nsc L S J H E GC LSJ > [R] 2nd = A 1 B 2nd E N = 3.0 MeV, AV18, J = r sc = 50 fm, n sc = 5 p d Int.Rel. 4 D 1/2 = S 1/2 = η 1/2+ = A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

57 Conclusions part I A variational description of continuum states is feasible The KVP principle can be put in terms of two (short-range) integrals They can be used to calculate the scattering matrix using bound-state like wave functions the only condition is that the wave function has to be a solution of (H E)Ψ in the interaction region Applications have been discussed in the case in which the Coulomb potential is present The determination of the breakup amplitude is in progress: already done for n d, the p d case is on the way A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

58 Universal aspects in few-nucleon systems The asymptotic constant (ANC) In the two-nucleon system there are two ANC s. Considering the L = 0 channel, it is defined as: e k d r Ψ D C 0 2kd [Y 0 χ 1 ] 1 ξ 0 r It can be calculated using the integral relations m < Ψ 2 D H E F >= C 0 with F = m < Ψ 2 D H E G >= 0 with G = It can be related to the effective range C k d r eff 1 4π 1 4π sinh(k d r) k d r e k d r k d r A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

59 Universal aspects in few-nucleon systems The asymptotic constant (ANC) In the two-nucleon system there are two ANC s. Considering the L = 0 channel, it is defined as: e k d r Ψ D C 0 2kd [Y 0 χ 1 ] 1 ξ 0 r It can be calculated using the integral relations m < Ψ 2 D H E F >= C 0 with F = m < Ψ 2 D H E G >= 0 with G = It can be related to the effective range C k d r eff 1 4π 1 4π sinh(k d r) k d r e k d r k d r A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

60 The expression C 2 0 = 1/(1 k dr eff ) This expression is obtained from the Effective Range Expansion (ERE): with k cotδ = 1 a r eff k r eff = 2 0 (φ 2 0 φ2 asymp)dr however as the deuteron is a shallow state r eff 2 0 (u 2 d u2 asymp )dr In fact the ERE expansion can be extended to negative (shallow) energies k d = 1 a r eff k 2 d A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

61 Numerical check From experiment we have: E d = MeV, a 1 = fm, r eff = fm, C 0 = using the E d and a 1 we predict: r eff = fm and C 0 = Due to the fact that the ratio r eff /a 1 is small (and even more in the singlet channel), the few-nucleon systems are close to the unitary limit: a 0, a 1 Consequence of the vicinity to the unitary limit: The two-nucleon system shows a Continous Scale Invariance (CSI): the low energy observables can be expanded in terms of the small quantity r λ eff /a λ The expansion converges very fast showing a universal behavior of the system. The observables are governed by the value of a. E d and C 0 are two examples A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

62 Numerical check From experiment we have: E d = MeV, a 1 = fm, r eff = fm, C 0 = using the E d and a 1 we predict: r eff = fm and C 0 = Due to the fact that the ratio r eff /a 1 is small (and even more in the singlet channel), the few-nucleon systems are close to the unitary limit: a 0, a 1 Consequence of the vicinity to the unitary limit: The two-nucleon system shows a Continous Scale Invariance (CSI): the low energy observables can be expanded in terms of the small quantity r λ eff /a λ The expansion converges very fast showing a universal behavior of the system. The observables are governed by the value of a. E d and C 0 are two examples A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

63 Numerical check From experiment we have: E d = MeV, a 1 = fm, r eff = fm, C 0 = using the E d and a 1 we predict: r eff = fm and C 0 = Due to the fact that the ratio r eff /a 1 is small (and even more in the singlet channel), the few-nucleon systems are close to the unitary limit: a 0, a 1 Consequence of the vicinity to the unitary limit: The two-nucleon system shows a Continous Scale Invariance (CSI): the low energy observables can be expanded in terms of the small quantity r λ eff /a λ The expansion converges very fast showing a universal behavior of the system. The observables are governed by the value of a. E d and C 0 are two examples A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

64 Constructing (low energy) potential models Select a two-parameter potential, as for example a Gaussian: V(r) = V λ e (r/r λ) 2 Determine the coupling constant V 1 and the cuttof r 1 to describe E D and a 1 (for example V 1 = MeV and r 1 = 1.57 fm) Calculate r eff and C 0 : r eff = fm, C 0 = Very close to the values given by the most realistic potentials This very simple, spin-dependent potential, describes the low energy dynamics of the two-nucleon system with reasonable accuracy In the context of the pionless effective field theory this corresponds to a LO description, optimized due to a particular value of the cutoff. This particular value incorporates range corrections A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

65 Constructing (low energy) potential models Select a two-parameter potential, as for example a Gaussian: V(r) = V λ e (r/r λ) 2 Determine the coupling constant V 1 and the cuttof r 1 to describe E D and a 1 (for example V 1 = MeV and r 1 = 1.57 fm) Calculate r eff and C 0 : r eff = fm, C 0 = Very close to the values given by the most realistic potentials This very simple, spin-dependent potential, describes the low energy dynamics of the two-nucleon system with reasonable accuracy In the context of the pionless effective field theory this corresponds to a LO description, optimized due to a particular value of the cutoff. This particular value incorporates range corrections A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

66 The three-nucleon system Analyzing the results The two-body sector: Potential E 2 a 1 r (1) C eff 0 a 0 r (0) eff [MeV] [fm] [fm] [fm] [fm] Gaussian N3LO Exp (7) 1.753(8) 1.290(4) (2) 2.77(5) the three-body sector: Potential E 3 (2) a nd (4) a nd C 0 [MeV] [fm] [fm] TBG N3LO Exp (1) 6.35(2) Guidance from Efimov Physics: a three-body force is needed: V(ijk) = W 0 e r 2 ij /r 2 3 e r 2 ik /r 2 3 A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

67 Including a three-nucleon force The strenght W 0 and the range r 3 can be optimize to describe E 3 and the doublet scattering length (2) a nd : Potential E 3 (2) a nd (4) a nd C 0 E 4 [MeV] [fm] [fm] [MeV] TBG+3NF N3LO/N2LO Exp (1) 6.35(2) 28.3 n d scattering at low energies 0 Gaussian+3NF N3LO/N2LO -0.5 S k =k*cotanδ E cm [MeV] A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43

68 Conclusions part II Due to a particular situation the few-nucleon system is located close to the unitary limit The nucleons are most of the time outside the range of the NN interaction In this case concepts from Efimov physics can be applied The dynamics of the two-, three- and four-nucleon systems in the low energy regime is controlled by the scattering length values a λ with corrections in terms of the small parameter r (λ) eff /a λ It is possible to summarize this behavior with a two-parameter potential In this case a three-nucleon force is needed to stabilize the system against the Thomas collapse A very simple potential of this kind produce an acceptable description of these systems, even in the determination of the ANC s A. Kievsky (INFN-Pisa) Variatonal description of continuum states ECT*, October / 43