Four-nucleon scattering

Transcription

1 Four-nucleon scattering A. Deltuva Centro de Física Nuclear da Universidade de Lisboa

2 Hamiltonian H + i> j 4N scattering v i j 1 v Wave function: Schrödinger equation (HH + Kohn VP) [M. Viviani, A. Kievsky, L. E. Marcucci, S. Rosati, L. Girlanda] Wave function components: Faddeev-Yakubovsky equations [R. Lazauskas, J. Carbonell] Transition operators: Alt-Grassberger-Sandhas equations [AD, A. C. Fonseca]

3 HH + Kohn VP Ψ=Ψ A + Ψ C Ψ A = Ω SΩ + Ψ C = µ c µ Y µ [S]=S Ψ (H +V E) Ψ stationary

4 FY equations ( E H v s 12 v l.c i j K12,3 4 = i< j (v s 12+ v12 s.c [ )P 1 (1+ζP34 )K12,3+ 4 H12 34 ] ( ) E H v s 12 v l.c i j i< j ) H = (v s 12+ v12 s.c [ )P 2 (1+ζP34 )K12,3+ 4 H12 34 ]

5 AGS equations t = v+vg t G =(E+ iε H ) 1, ε + u j = P j G 1 + P j tg u j K 4 12,3 3+1 : P 1 = P 12 P 23 + P 13 P 23 H : P 2 = P 13 P 24 U 11 =(G tg ) 1 ζp 34 + ζp 34 u 1 G tg U 11 + u 2 G tg U 21 U 21 =(G tg ) 1 (1+ζP 34 )+(1+ζP 34 )u 1 G tg U 11 U 12 =(G tg ) 1 + ζp 34 u 1 G tg U 12 + u 2 G tg U 22 U 22 =(1+ζP 34 )u 1 G tg U 12 ζ= 1(+1) for fermions (bosons) basis states partially symmetrized

6 Wave function Ψ i =s i {[1+(1+P 1 )ζp 34 ](1+P 1 ) ψ 1,i +(1+P 1 )(1+P 2 ) ψ 2,i } with Faddeev-Yakubovsky components ψ 1,i K 4 12,3 ψ 2,i H ψ j,i =δ ji φ i +G tg u j G tg U ji φ i φ j =G tp j φ j Φ j =(1+P j ) φ j

7 Scattering amplitudes: E+ iε E+ i 2-cluster reactions: T f i = s f i φ f U f i φ i 3-cluster breakup/recombination: T 3i = s 3i φ 3 [(1+ζP 34 )u 1 G tg U 1i + u 2 G tg U 2i ] φ i 4-cluster breakup/recombination: T 4i = s 4i { φ 4 [1+(1+P 1 )ζp 34 ](1+P 1 )tg u 1 G tg U 1i φ i + φ 4 (1+P 1 )(1+P 2 )tg u 2 G tg U 2i φ i }

8 Solution of 4N AGS equations U 11 φ 1 = G 1 P 34P 1 φ 1 P 34 u 1 G tg U 11 φ 1 +u 2 G tg U 21 φ 1 1 lx l lx l l y 2 z y l z momentum-space partial-wave basis k x k y k z [l z ({l y [(l x S x ) j x s y ]S y }J y s z )S z ]JM,[(T x t y )T y t z ]T M T 1 k x k y k z [l z {(l x S x ) j x [l y (s y s z )S y ] j y }S z ]JM,[T x (t y t z )T y ]T M T 2 large system (up to 3) of coupled 3-variable integral equations with integrable singularities Coulomb interaction: screening and renormalization [PRC 75, 145, PRL 98, 16252]

9 Singularities of 4N AGS equations 3 H, 3 He, or d+d bound state poles G u j G P j φ j s j j φ j P j E+ iε E b j k2 z/2µ j treated by subtraction below 3-cluster threshold q p kz 2 F(k z ) dk z k 2 k2 z + i =P = q q p k 2 z dk z F(k z ) k 2 k2 z 1 2 iπk F(k ) kz 2 F(k z ) k 2 dk F(k ) z p k 2 k2 z 1 [ 2 k F(k ) iπ+ln (k + p)(q k ) ] (k p)(k + q)

10 n+ 3 H elastic scattering 5 1 MeV 2 MeV 3.5 MeV 6 MeV dσ/dω [mb/sr] A y A y -.1 θ c.m. [deg] θ c.m. [deg] θ c.m. [deg] 18 θ c.m. [deg] AGS/HH/FY [PRC 84, 541]

11 p+ 3 He elastic scattering dσ/dω [mb/sr] MeV Famularo 1954 Fisher 26 I-N3LO AV18 low-k 4.5 MeV Mcdonald 1964 Fisher MeV Mcdonald Fisher 26 George 21 Fisher 26 Alley 1993 A y.2 18 A y.1 Daniels 21 Daniels 21 Alley 1993 Daniels 21 θ c.m. [deg] θ c.m. [deg] 18 θ c.m. [deg] AGS/HH/FY [PRC 84, 541]

12 [M. Viviani et al, arxiv: ] p+ 3 He A y -puzzle: Illinois-7 and N2LO 3NF 4 3 dσ/dω [mb/sr].5 A y A y.1 A yy A xx.1 A xz θ c.m. [deg] θ c.m. [deg]

13 -isobar excitation: effective 3N and 4N forces Fujita-Miyazawa higher order 3N force 4N force [PLB 66, 471]

14 n+ 3 He elastic scattering dσ/dω (mb/sr) AV18 N3LO CD Bonn CD Bonn + INOY4 E n = 1 MeV E n = 1 MeV E n = 3.7 MeV E n = 3.5 MeV A y [PRC 76, 211]

15 p+ 3 H elastic scattering dσ/dω (mb/sr) 4 2 N3LO INOY4 AV18 CD Bonn A y E p = 4.15 MeV p- 3 H max(a y ).4.2 p- 3 He CD Bonn E p (MeV)

16 Charge exchange reaction 3 H(p,n) 3 He dσ/dω (mb/sr) 8 4 AV18 N3LO CD Bonn CD Bonn + INOY4 E p = 2.48 MeV E p = 6 MeV.4 A y [PRC 76, 211]

17 d+d elastic scattering at E d = 3 MeV dσ/dω (mb/sr) N3LO INOY4 AV18 CD Bonn T [PLB 66, 471]

18 2 H(d,p) 3 H and 2 H(d,n) 3 He dσ/dω (mb/sr) 2 d+d p+ 3 H d+d n+ 3 He E d = 1.5 MeV dσ/dω (mb/sr) 4 2 AV18 N3LO CD Bonn CD Bonn + INOY4 E d = 3 MeV dσ/dω (mb/sr) 2 E d = 4 MeV [PRC 81, 542]

19 2 H(d,p) 3 H and 2 H(d,n) 3 He.2 E d = 1.5 MeV.2 d+d p+ 3 H d+d n+ 3 He it 11 T 21.. d+d p+ 3 H d+d n+ 3 He E d = 1.5 MeV it INOY4 CD Bonn + CD Bonn N3LO AV18 E d = 3 MeV E d = 4 MeV T AV18 N3LO CD Bonn CD Bonn + INOY4 E d = 3 MeV it 11 T 21.. E d = 4 MeV

20 Above breakup: complicated boundary conditions complex scaling method for solving FY equations: n+ 3 H elastic scattering with MT I-III potential [R. Lazauskas, PRC 86, 442] 1 dσ/dω (mb/sr) 1 Frenje Debertin θ c.m. (deg)

21 Above breakup: additional singularities in AGS equations deuteron bound state poles t v φ d φ d v E+ iε e d k 2 y/2µ y j k2 z/2µ j free resolvent G 1 E+ iε k 2 x/2µ x j k2 y/2µ y j k2 z/2µ j

22 Above breakup: additional singularities in AGS equations deuteron bound state poles t v φ d φ d v E+ iε e d k 2 y/2µ y j k2 z/2µ j free resolvent G 1 E+ iε k 2 x/2µ x j k2 y/2µ y j k2 z/2µ j treated by complex-energy method: 1. solve for U f i (E+ iε) with finite ε=ε 1,...,ε n 2. extrapolate to ε for physical amplitudes U f i (E+ i) [ L. Schlessinger, PR 167, 1411 (1968)] [ H. Kamada et al, Prog. Theor. Phys. 19, 869L (23)]

23 Integration with special weights accuracy & efficiency of the complex-energy method is greatly improved by a special integration b a f(x) x n + iy x n dx N j=1 f(x j )w j (n,x,y,a,b) where the quasi-singular factor is absorbed into special weights w j (n,x,y,a,b)= b a S j (x) x n + iy x n dx that may be calculated using spline functions {S j (x)} for standard Gaussian grid {x j } [PRC 86, 111]

24 Extrapolation ε : n+ 3 H at 22.1 MeV dσ/dω (mb/sr) [ε min,ε max ]/MeV: [1., 2.] [1.2, 2.] [1.4, 2.] [1.2, 1.8] ε = 1.4 MeV A y

25 Extrapolation ε : n+ 3 H at 22.1 MeV [ε min,ε max ] δ( 1 S ) η( 1 S ) δ( 3 P ) η( 3 P ) δ( 3 P 2 ) η( 3 P 2 ) [1., 2.] [1.2, 2.] [1.4, 2.] [1.2, 1.8] [PRC 86, 111]

26 n+ 3 H elastic scattering E n = 14.1 MeV dσ/dω (mb/sr) 1 1 INOY4 CD Bonn Frenje Debertin dσ/dω (mb/sr) dσ/dω (mb/sr) Debertin Seagrave E n = 18. MeV E n = 22.1 MeV Seagrave, 21 MeV Seagrave, 23 MeV A y.8.4. E n = 22.1 MeV INOY4 CD Bonn Seagrave INOY4, 14.1 MeV INOY4, 18. MeV 18

27 p+ 3 He elastic scattering E p (MeV) = dσ/dω (mb/sr) 1 1 INOY4 CD Bonn AV18 Clegg Hutson Murdoch [PRC 87, 542] 18

28 p+ 3 He elastic scattering: effects 1 E p = 3 MeV dσ/dω (mb/sr) 1 1 CD Bonn CD Bonn + CD Bonn + (2N) CD Bonn + (3N+4N) Murdoch

29 p+ 3 He elastic scattering A y.5 E p = 7. MeV Alley E p = 8.5 MeV INOY4 CD Bonn AV18 E p = 1. MeV. E p = 13.6 MeV E p = 19.4 MeV E p = 21.3 MeV.5 Jarmie Baker Birchall A y. E p = 25. MeV E p = 3. MeV E p = 35. MeV.5 A y. -.5

30 p+ 3 He elastic scattering A y.2. E p = 7. MeV Alley E p = 8.5 MeV INOY4 CD Bonn AV18 E p = 1. MeV E p = 13.6 MeV E p = 19.4 MeV E p = 21.3 MeV A y.2. Baker E p = 25. MeV E p = 3. MeV E p = 35. MeV A y.2. Muller McCamis -.2

31 p+ 3 He elastic scattering A yy.2.1 Alley INOY4 CD Bonn AV18..2 E p = 7. MeV E p = 1. MeV E p = 8.5 MeV E p = 19.4 MeV A yy. Baker

32 p+ 3 He elastic scattering 1. E p = 8.5 MeV E p = 1.8 MeV E p = 16.2 MeV.5 K x x. -.5 Weitkamp Hardekopf K z x. -.5 INOY4 CD Bonn AV18 1. K y y.8.6

33 n+ 3 H total and breakup cross sections 2.5 σ t (b) 2. AV18 N3LO CD Bonn CD Bonn + INOY4 σ t (b) 2 1 Phillips Battat INOY4 CD Bonn E n (MeV) E n (MeV) 6 σ b (mb) 4 2 INOY E n (MeV)

34 [PRC 87, 142] Recombination reaction 2 H+n+n n+ 3 H N A 2 K3 (cm 6 s -1 mol -2 ) E 3 (MeV) dρ t dt = K γ 2 ρ dρ n + K 3 ρ d ρ 2 n+...

35 Extension: 4-boson Efimov physics Energy 1/a a+t(n) IVS T(n,1) T(n,2) d+a+a d T(n,) d+d a+t(n 1) T(n 1,1) T(n 1,2)

36 Four-atom recombination at threshold Energy 1/a d+a+a 1 12 a+t(n) IVS T(n,2) d T(n,) T(n,1) 1 9 d+d K 4 m/- h a n n = 1 n = 2 n = 3 a n : b n = a+t(n 1) T(n 1,2) T(n 1,1) 1 n = 4 a n,k : B n,k = [PRA 85, 1278] a/a n a n,1 /a n =.4254 a n,2 /a n =.9125

37 Atom-trimer scattering length 2 Re A n /a n dd Im A n /a n dd Energy 1/a d+a+a A n /a n dd a+t(n) IVS T(n,1) T(n,2) d T(n,) -2 a+t(n 1) d+d T(n 1,1) T(n 1,2).1 1. a n dd /a a dd n : b n = 2b d [EPL 95, 432, PRA 85, 4275]

38 Summary: 4N scattering 4N scattering equations in coordinate (HH,FY) and momentum space (AGS) n+ 3 H and p+ 3 He scattering below breakup threshold (HH/FY/AGS) coupled p+ 3 H, n+ 3 He and d+d reactions below breakup threshold (AGS) n+ 3 H and p+ 3 He scattering above breakup threshold (AGS: complex-energy method + special integration)

39 Summary: 4N scattering 4N scattering equations in coordinate (HH,FY) and momentum space (AGS) n+ 3 H and p+ 3 He scattering below breakup threshold (HH/FY/AGS) coupled p+ 3 H, n+ 3 He and d+d reactions below breakup threshold (AGS) n+ 3 H and p+ 3 He scattering above breakup threshold (AGS: complex-energy method + special integration) universal & cold-atom physics