SLIDE 1
Momentum-space treatment
- f the Coulomb force:
Screening and renormalization
- A. Deltuva
Momentum-space treatment of the Coulomb force: Screening and - - PowerPoint PPT Presentation
Momentum-space treatment of the Coulomb force: Screening and - - PowerPoint PPT Presentation
Momentum-space treatment of the Coulomb force: Screening and renormalization A. Deltuva Vilnius University In collaboration with A. C. Fonseca and P . U. Sauer Outline Momentum-space description of few-body scattering: screening and
SLIDE 2
SLIDE 3
Screened Coulomb
w
R(r) = w C(r)e−( r R)n
standard scattering theory
SLIDE 4
Screened Coulomb
w
R(r) = w C(r)e−( r R)n
standard scattering theory nature: Coulomb is screened at large distances large R: physical observables insensitive to screening, screened and full Coulomb physically indistinguishable
SLIDE 5
Screened Coulomb
w
R(r) = w C(r)e−( r R)n
standard scattering theory nature: Coulomb is screened at large distances large R: physical observables insensitive to screening, screened and full Coulomb physically indistinguishable in the R → ∞ limit physical results are recovered
SLIDE 6
Screened and full Coulomb physically indistinguishable
? p′|TR|p − − − →
R→∞
p′|T
C|p
SLIDE 7
Screened and full Coulomb physically indistinguishable
? e2iφRp′|TR|p − − − →
R→∞
p′|T
C|p
SLIDE 8
Screened and full Coulomb physically indistinguishable
initial physical state: wave packet ϕin(p)
- utgoing wave packet
ϕout(p′) =
- d3pp′|S|pϕin(p)
? ∼
- d2 ˆ
pe2iφRp′|TR|pϕin(p) − − − →
R→∞
- d2 ˆ
pp′|T
C|pϕin(p)
SLIDE 9
Screened and full Coulomb physically indistinguishable
initial physical state: wave packet ϕin(p)
- utgoing wave packet
ϕout(p′) =
- d3pp′|S|pϕin(p)
? ∼
- d2 ˆ
pe2iφRp′|TR|pϕin(p) − − − →
R→∞
- d2 ˆ
pp′|T
C|pϕin(p)
p′ = p : e2iφRp′|TR|p − − − →
R→∞ p′|T C|p
as distribution
SLIDE 10
Screened and full Coulomb physically indistinguishable
initial physical state: wave packet ϕin(p)
- utgoing wave packet
ϕout(p′) =
- d3pp′|S|pϕin(p)
? ∼
- d2 ˆ
pe2iφRp′|TR|pϕin(p) − − − →
R→∞
- d2 ˆ
pp′|T
C|pϕin(p)
p′ = p : e2iφRp′|TR|p − − − →
R→∞ p′|T C|p
as distribution φR− − − →
R→∞ [σL −ηLR]−
− − →
R→∞ αeM/p [ln(2pR)−C/n]
[J. R. Taylor, Nuovo Cimento B23, 313 (1974)]
SLIDE 11
Screened and full Coulomb wave functions
r < R : w
R(r) ≈w C(r)
⇓ eiφLRr|ψ(+)
LR (p) ≈r|ψ(+) LC (p)
SLIDE 12
Screened and full Coulomb wave functions
r < R : w
R(r) ≈w C(r)
⇓ eiφLRr|ψ(+)
LR (p) ≈r|ψ(+) LC (p)
eiφR|ψ(+)
R (p) −
− − →
R→∞ |ψ(+) C (p)
[ V. G. Gorshkov, Sov. Phys.-JETP 13, 1037 (1961)]
SLIDE 13
Screening and renormalization
Renormalization of the on-shell screened Coulomb transition matrix TR = wR +wRG0TR and wave function in the limit R → ∞ yields Coulomb amplitude and Coulomb wave function
TRz−1
R −
− − →
R→∞ T C as distribution
(1+G0TR)|pz−1/2
R
− − − →
R→∞ |ψ(+) C (p)
zR = e−2iφR
SLIDE 14
Two-particle scattering
transition matrix T (R) = v+wR +(v+wR)G0T (R)
SLIDE 15
Two-particle scattering
transition matrix T (R) = v+wR +(v+wR)G0T (R)
with long-range and Coulomb-distorted short-range parts T (R) = TR +(1+TRG0) ˜ T (R)(1+G0TR) ˜ T (R) = v+vGR ˜ T (R)
SLIDE 16
Two-particle scattering
transition matrix T (R) = v+wR +(v+wR)G0T (R)
with long-range and Coulomb-distorted short-range parts T (R) = TR +(1+TRG0) ˜ T (R)(1+G0TR) ˜ T (R) = v+vGR ˜ T (R) Renormalized amplitude: T (R)z−1
R −
− − →
R→∞ T = TC +ψ(−) C | ˜
T (C)|ψ(+)
C
SLIDE 17
Two-particle scattering
transition matrix T (R) = v+wR +(v+wR)G0T (R)
with long-range and Coulomb-distorted short-range parts T (R) = TR +(1+TRG0) ˜ T (R)(1+G0TR) ˜ T (R) = v+vGR ˜ T (R) Renormalized amplitude: T (R)z−1
R −
− − →
R→∞ T = TC +ψ(−) C | ˜
T (C)|ψ(+)
C
= TC + lim
R→∞z − 1
2
R [T (R) −TR]z − 1
2
R
short-range part: fast convergence with R
SLIDE 18
Test: convergence with R in pp scattering wR(r) wC(r) = e−( r
R)n
1 1 2
r/R
n = 1 n = 4 n → ∞
SLIDE 19
Test: convergence with R in pp scattering wR(r) wC(r) = e−( r
R)n
1 1 2
r/R
n = 1 n = 4 n → ∞ 50 52 10 20 30 40
η (deg) R (fm)
1S0 Ep = 3 MeV
↓ ↑
0.001%
exact n = 1 n = 4 n → ∞
- ptimal choice: 3 ≤ n ≤ 8
SLIDE 20
Limits of practical applicability
p → 0: κ = αM/p, σL = argΓ(1+L+iκ), and zR diverge, renormalization procedure ill-defined
SLIDE 21
Limits of practical applicability
p → 0: κ = αM/p, σL = argΓ(1+L+iκ), and zR diverge, renormalization procedure ill-defined ⇒ slow convergence with R at low relative energies
45.5 46.0
1 MeV
6.5 7.0
η (deg)
0.1 MeV
0.0 0.5 20 100 500
R (fm) 0.01 MeV
SLIDE 22
Three-particle scattering: short-range forces
Faddeev / Alt, Grassberger, and Sandhas equations Uβα = ¯ δβαG−1
0 +∑ σ
¯ δβσTσG0Uσα U0α = G−1
0 +∑ σ
TσG0Uσα Tσ = vσ +vσG0Tσ G0 = (E +i0−H0)−1 momentum-space partial-wave representation
SLIDE 23
AGS equations with 3BF
V3BF =
3
∑
α=1
uα Uβα = ¯ δβαG−1
0 +∑ γ
¯ δβγTγG0Uγα +uα +∑
γ
uγG0(1+TγG0)Uγα
SLIDE 24
Three-particle scattering: including screened Coulomb
Faddeev / Alt, Grassberger, and Sandhas equations U(R)
βα = ¯
δβαG−1
0 +∑ σ
¯ δβσT (R)
σ G0U(R) σα
U(R)
0α = G−1 0 +∑ σ
T (R)
σ G0U(R) σα
T (R)
σ
= vσ +wσR +(vσ +wσR)G0T (R)
σ
G0 = (E +i0−H0)−1 momentum-space partial-wave representation
SLIDE 25
Three-particle scattering: including screened Coulomb
Faddeev / Alt, Grassberger, and Sandhas equations U(R)
βα = ¯
δβαG−1
0 +∑ σ
¯ δβσT (R)
σ G0U(R) σα
U(R)
0α = G−1 0 +∑ σ
T (R)
σ G0U(R) σα
T (R)
σ
= vσ +wσR +(vσ +wσR)G0T (R)
σ
G0 = (E +i0−H0)−1 momentum-space partial-wave representation Additional difficulties: quasi-singular nature of screened Coulomb potential slow partial-wave convergence
SLIDE 26
Three-particle scattering: including screened Coulomb
Faddeev / Alt, Grassberger, and Sandhas equations U(R)
βα = ¯
δβαG−1
0 +∑ σ
¯ δβσT (R)
σ G0U(R) σα
U(R)
0α = G−1 0 +∑ σ
T (R)
σ G0U(R) σα
T (R)
σ
= vσ +wσR +(vσ +wσR)G0T (R)
σ
G0 = (E +i0−H0)−1 momentum-space partial-wave representation Additional difficulties: quasi-singular nature of screened Coulomb potential slow partial-wave convergence R → ∞ limit?
SLIDE 27
Three-particle scattering: R → ∞ limit
long-range part
- W c.m.
αR
T c.m.
αR = W c.m. αR +W c.m. αR G(R) α T c.m. αR
SLIDE 28
Three-particle scattering: R → ∞ limit
Split into long-range part
- W c.m.
αR
T c.m.
αR = W c.m. αR +W c.m. αR G(R) α T c.m. αR
and Coulomb-distorted short-range part U(R)
βα = δβαT c.m. αR +[1+T c.m. βR G(R) β ] ˜
U(R)
βα [1+G(R) α T c.m. αR ]
U(R)
0α = [1+TρRG0] ˜
U(R)
0α [1+G(R) α T c.m. αR ]
[ρ is neutral]
SLIDE 29
Three-particle scattering: R → ∞ limit
Split into long-range part
- W c.m.
αR
T c.m.
αR = W c.m. αR +W c.m. αR G(R) α T c.m. αR
and Coulomb-distorted short-range part U(R)
βα = δβαT c.m. αR +[1+T c.m. βR G(R) β ] ˜
U(R)
βα [1+G(R) α T c.m. αR ]
U(R)
0α = [1+TρRG0] ˜
U(R)
0α [1+G(R) α T c.m. αR ]
[ρ is neutral]
Renormalized amplitudes: Uβα = δβαT c.m.
αC + lim R→∞Z − 1
2
R f [U(R) βα −δβαT c.m. αR ]Z − 1
2
Ri
U0α = lim
R→∞z − 1
2
R U(R) 0α Z − 1
2
Ri
SLIDE 30
Three-particle scattering: R → ∞ limit
Split into long-range part
- W c.m.
αR
T c.m.
αR = W c.m. αR +W c.m. αR G(R) α T c.m. αR
and Coulomb-distorted short-range part U(R)
βα = δβαT c.m. αR +[1+T c.m. βR G(R) β ] ˜
U(R)
βα [1+G(R) α T c.m. αR ]
U(R)
0α = [1+TρRG0] ˜
U(R)
0α [1+G(R) α T c.m. αR ]
[ρ is neutral]
Renormalized amplitudes: Uβα = δβαT c.m.
αC + lim R→∞Z − 1
2
R f [U(R) βα −δβαT c.m. αR ]Z − 1
2
Ri
U0α = lim
R→∞z − 1
2
R U(R) 0α Z − 1
2
Ri
short-range part: fast convergence with R
SLIDE 31
r-space methods
Kohn VP + HH [Kievsky et al] differential Faddeev equations [Payne et al, Lazauskas et al, Suslov et al] integral Faddeev equations [Ishikawa]
SLIDE 32
Screening and renormalization: variations
separable potentials, quasiparticle equations, effective two-body potentials, Coulomb distorted ffs, ... [Alt et al] "rigorous Coulomb treatment" [Oryu et al] no/different renormalization [Witała et al] in progress: separable potentials, unscreened Coulomb representation [Mukhamedzhanov et al, TORUS]
SLIDE 33
Proton-deuteron scattering
Symmetrized Faddeev / AGS equations U(R) = PG−1
0 +PT (R)G0U(R)
U(R) = (1+P)G−1
0 +(1+P)T (R)G0U(R)
P = P12 P23 +P13 P23 Screening function with n = 4 Renormalized amplitudes: U = T c.m.
C
+ lim
R→∞Z−1 R [U(R) −T c.m. R
] U0 = lim
R→∞z − 1
2
R U(R) 0 Z − 1
2
R
SLIDE 34
pd elastic amplitude (spin-diagonal)
- 4
- 2
2
Re U(R)
non-renormalized
2 4 60 120 180
Im U(R) Θc.m. (deg)
Ep = 10 MeV
R = 5 fm R = 10 fm R = 15 fm R = 20 fm
- 4
- 2
2
Re U U = TC
c.m. + ZR
- 1(U(R)-TR
c.m.)
renormalized
2 4 60 120 180
Im U Θc.m. (deg)
R = 5 fm R = 10 fm R = 15 fm R = 20 fm
SLIDE 35
pd elastic amplitude (spin-nondiagonal)
- 0.2
0.0
Re U(R)
non-renormalized
- 0.2
0.0 60 120 180
Im U(R) Θc.m. (deg)
Ep = 10 MeV
R = 5 fm R = 10 fm R = 15 fm R = 20 fm
- 0.2
0.0
Re U U = ZR
- 1 U(R)
renormalized
- 0.2
0.0 60 120 180
Im U Θc.m. (deg)
R = 5 fm R = 10 fm R = 15 fm R = 20 fm
SLIDE 36
pd breakup amplitude
0.0 0.1
Re U0
(R)
non-renormalized
R = 10 fm R = 20 fm R = 30 fm R = 40 fm
- 0.1
0.0 0.1 5 10
Im U0
(R)
S (MeV)
Ep = 13 MeV
(50.5o,50.5o,120.0o)
0.0 0.1
Re U0
renormalized
U = zR
- 1/2 U0
(R) ZR
- 1/2
R = 10 fm R = 20 fm R = 30 fm R = 40 fm
- 0.1
0.0 0.1 5 10
Im U0 S (MeV)
SLIDE 37
Convergence with R: pd elastic scattering
100 200 400
dσ/dΩ (mb/sr)
no Coulomb
R = 10 fm R = 20 fm R = 30 fm
0.00 0.02 0.04 60 120 180
Ay (N) Θc.m. (deg) Ep = 3 MeV
0.00 0.02
T21 Ep = 3 MeV
- 0.05
0.00 0.05 60 120 180
T21 Θc.m. (deg) Ep = 10 MeV
SLIDE 38
45 90 135 180
θc.m.[deg]
1 10 100 100 100 1000 45 90 135 180
θc.m.[deg]
- 0.4
0.0 0.4 0.00 0.05 0.10 0.00 0.02 0.04 45 90 135 180
θc.m.[deg]
- 0.4
- 0.2
0.0 0.2 0.00 0.02 0.04 0.06 0.00 0.01 0.02
dσ/dΩ [mb/sr] 3 MeV dσ/dΩ [mb/sr] 10 MeV dσ/dΩ [mb/sr] 65 MeV
Ay Ay Ay
iT11 iT11 iT11
Comparison with configuration-space results
pd elastic scattering:
—— Kohn variational principle (Pisa) —— screening and renormalization
SLIDE 39
Convergence with R: pd breakup at Ep = 13 MeV
1.0 1.1 1.2 5 10
(50.5o,50.5o,120.0o)
no Coulomb
R = 10 fm R = 20 fm R = 30 fm
0.5 1.0 1.5 5 10
d5σ/dS dΩ1 dΩ2 (mb MeV-1sr-2)
(20.0o,35.0o,90.0o)
- 0.05
0.00 0.05 5 10 15
Ay (N)
S (MeV)
(39.0o,62.5o,180.0o)
SLIDE 40
Convergence with R: pd breakup at Ep = 13 MeV
1.0 1.1 1.2 5 10
(50.5o,50.5o,120.0o)
no Coulomb
R = 10 fm R = 20 fm R = 30 fm
0.5 1.0 1.5 5 10
d5σ/dS dΩ1 dΩ2 (mb MeV-1sr-2)
(20.0o,35.0o,90.0o)
- 0.05
0.00 0.05 5 10 15
Ay (N)
S (MeV)
(39.0o,62.5o,180.0o)
1 2 0.0 0.5 1.0
d5σ/dS dΩ1 dΩ2 (mb MeV-1sr-2)
Epp (MeV)
pp-FSI ↓
zR = exp(-2iαM/p [ln(2pR)-C/n])
(39.0o,39.0o,0.0o)
no Coulomb
R = 10 fm R = 20 fm R = 30 fm R = 40 fm R = 60 fm
SLIDE 41
Coulomb vs 3NF: 1H(d,pp)n at Ed = 130 MeV
0.0 0.2
(15o,15o,160o)
0.0 0.2
(20o,15o,160o)
0.0 0.2
(25o,15o,160o)
0.0 0.2 0.4 d5σ/dS dΩ1 dΩ2 (mb MeV-1sr-2)
(30o,15o,160o)
0.0 0.1
(20o,20o,160o)
0.0 0.1
(25o,20o,160o)
0.0 0.1 50 100 150 S (MeV)
(30o,20o,160o)
AV18(nd) AV18(pd) AV18+UIX(pd) 0.0 0.1 50 100 150 S (MeV)
(25o,25o,160o)
0.0 8.0 50 100 150 S (MeV)
(13o,13o,20o)
SLIDE 42
3He(γ, pn)p at Eγ = 55 MeV
40 80 120 160 200
d4σ/dΩ1 dΩ2 (µb/sr2)
Θp+Θn (deg)
Θp = 81.0o
↓
5 10 15 20 20 40 60
d5σ/dS dΩ1 dΩ2 (µb MeV-1sr-2)
S (MeV)
Θn = 86.0o
with Coulomb
SLIDE 43
3
He(
- γ,n)pp at Eγ = 12.8 MeV
(MeV)
n
E 2 2.5 3 3.5 4 4.5 5 b/sr MeV) µ (
n
dE
n
Ω d σ
3
d
- 10
10 20 30 40 50 60 70
- =75
n
θ
(MeV)
n
E 2 2.5 3 3.5 4 4.5 5 b/sr MeV) µ (
n
dE
n
Ω d σ
3
d
- 10
10 20 30 40 50 60 70
- =75
n
θ (MeV)
n
E 2 2.5 3 3.5 4 4.5 5 b/sr MeV) µ (
n
dE
n
Ω d σ
3
d
- 10
10 20 30 40 50 60 70
- =90
n
θ
(MeV)
n
E 2 2.5 3 3.5 4 4.5 5 b/sr MeV) µ (
n
dE
n
Ω d σ
3
d
- 10
10 20 30 40 50 60 70
- =90
n
θ
(MeV)
n
E 2 2.5 3 3.5 4 4.5 5 b/sr MeV) µ (
n
dE
n
Ω d σ
3
d
- 10
10 20 30 40 50 60
- =105
n
θ
(MeV)
n
E 2 2.5 3 3.5 4 4.5 5 b/sr MeV) µ (
n
dE
n
Ω d σ
3
d
- 10
10 20 30 40 50 60
- =105
n
θ
parallel antiparallel
- •
TUNL data [PRL 110, 202501]
- - -
CD Bonn + ∆ (with Coulomb) (Lisbon) — AV18 + UIX (no Coulomb) (Cracow)
SLIDE 44
4N scattering: symmetrized AGS equations
two-cluster 1+3 and 2+2 transition operators
U11 = −(G0 TG0)−1P34 −P34U1G0 TG0U11 +U2G0 TG0U21 U21 = (G0 TG0)−1(1−P34)+(1−P34)U1G0 TG0U11 U12 = (G0 TG0)−1 −P34U1G0 TG0U12 +U2G0 TG0U22 U22 = (1−P34)U1G0 TG0U12
Uj = PjG−1
0 +PjTG0Uj
P1 = P = P12 P23 +P13 P23 P2 = ˜ P = P13 P24 T = v+vG0T
scattering amplitude
T fi = Sfip fφf|Ufi|piφi
|φj = G0TPj|φj
SLIDE 45
Screening and renormalization in 4N scattering
v → v+wR T, Uj, Ufi, T fi → T (R), U(R)
j
, U(R)
fi , T (R) fi
isolate long-range interaction
- •
W c.m.
R
and Coulomb distortion between c.m. of two clusters
SLIDE 46
Screening and renormalization in 4N scattering
v → v+wR T, Uj, Ufi, T fi → T (R), U(R)
j
, U(R)
fi , T (R) fi
isolate long-range interaction
- •
W c.m.
R
and Coulomb distortion between c.m. of two clusters
Renormalization:
Tfi = lim
R→∞Z − 1
2
R f T (R) fi Z − 1
2
Ri
= δ fiT c.m.
Ci
+ lim
R→∞Z − 1
2
R f [T (R) fi
−δ fiT c.m.
Ri ]Z − 1
2
Ri
Coulomb-distorted short-range part: fast convergence with R
SLIDE 47
Convergence with R: p-3He scattering at Ep = 4 MeV
100 200 300 400
dσ/dΩ (mb/sr)
no Coulomb
R = 0 fm R = 6 fm R = 8 fm R = 10 fm R = 12 fm
0.0 0.1
Czx
0.0 0.2 0.4 60 120 180
Ay Θc.m. (deg)
0.0 0.1 60 120 180
Cyy Θc.m. (deg)
SLIDE 48
p-3He scattering
60 120 100 200 300 400 500 dσ/dΩ [mb/sr]
Famularo 1954 Fisher 2006 I-N3LO AV18 low-k
2.25 MeV 60 120 0.2 0.4 Ay0
Fisher 2006 George 2001
60 120 θc.m. [deg] 0.1 A0y
Daniels 2010
60 120
Mcdonald 1964 Fisher 2006
4.05 MeV 60 120
Fisher 2006
60 120 θc.m. [deg]
Daniels 2010
60 120 180
Mcdonald 1964
5.54 MeV 60 120 180
Alley 1993
60 120 180 θc.m. [deg]
Alley 1993 Daniels 2010
AGS/HH/FY (Lisbon/Pisa/Strasbourg, PRC 84, 054010)
SLIDE 49
d +d → N +[3N] transfer at Ed = 3 MeV
20 40
dσ/dΩ (mb/sr) d+d → p+3H
N3LO INOY04 AV18 CD Bonn
d+d → n+3He
0.0 0.2 50 100 150
iT11 Θc.m. (deg)
50 100 150
Θc.m. (deg)
SLIDE 50