TALK BLED fin 1 Hikocrex.fmf.uni-lj.si/eep17/YYY_TALK_BLED_fin_1_Hiko.pdf · 2017-07-05 · CM CV...
Transcript of TALK BLED fin 1 Hikocrex.fmf.uni-lj.si/eep17/YYY_TALK_BLED_fin_1_Hiko.pdf · 2017-07-05 · CM CV...
Claudio Cio� degli Atti & Hiko Morita
Nu leon dynami s at short range: ground state energy and radii,
momentum distributions and spe tral fun tions of few-nu leon
systems and omplex nu lei
INTERNATIONAL WORKSHOP on (e, e′p) PROCESSES
July 2-6 - 2017 Bled, Slovenia
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
OUTLINE
1. The standard Many-body approa h to nu lei: ab-initio and varia-
tional al ulations.
2. Universality of SRCs in oordinate and momentum spa es: the
orrelation hole and the high momentum omponents of one- and
two-nu leon momentum disytributions.
3. Fa torization and the onvolution formula of the one-nu leon
spe tral fun tion for few-nu leon systems and omplex nu lei.
4. Summary and on lusions
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
1 THE STANDARD MANY BODY APPROACH TO NUCLEI:
ab-INITIO AND VARIATIONAL CALCULATIONS
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
E�e tive degrees of nu lei: nu leons and gauge bosons.
Redu tion of a �eld theoreti al des ription to an instantaneous
potential des ription (S hroedinger equation ) =⇒ two-body, three-
body,........,A-body potentials are generated.
HΨn =
−h2
2mN
∑
i
∇2i +
∑
i<j
v2(i, j) +∑
i<j<k
v3(i, j, k)
Ψn = EnΨn
Ψn ≡ Ψn(1 . . . A) i ≡ xi ≡ {σi, τi, ri}A∑
i=1
ri = 0
v2(i, j)- 18 omponents to explains free NN s attering data
v3(i, j, k)- ne essary to reprodu e binding energy of A = 3 nu lei
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
The NN intera tion has the following form (ARGONNE Family):
v2(i, j) =
18∑
n=1
v(n)(rij) O(n)ij ,
with
O(n)ij =
[
1 , σi · σj , Sij , (S ·L)ij , L2 , L2
σi · σj , (S ·L)2ij , ..]
⊗ [1 , τ i · τ j]
where σi and τi are Pauli matri es a ting in spin and isospin spa e, respe tively.
The most relevant omponents are the �rst six ones:
O(1)ij ≡ Oc
ij = 1 O(2)ij ≡ Oσ
ij = σi · σj
O(3)ij ≡ Oτ
ij = τ i · τ j O(4)ij ≡ Oσ τ
ij = (σi · σj) (τ i · τ j)
O(5)ij ≡ Ot
ij = Sij O(6)ij ≡ Ot τ
ij = Sij (τ i · τ j), .
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
CAN WE SOLVE THE MANY-BODY NUCLEAR PROBLEM?
• Exa t (ab-initio) solutions (Green Fun tion Monte Carlo) only for few-nu leon systems
• Light and heavy nu lei require a variational solution: �nd that Ψ0 whi h minimizes
〈 H 〉 =〈ψo| H |ψo〉
〈ψo |ψo〉≥ Eo .
The trial WF is a orrelated WF of the following form
ψo(x1, ...,xA) = F (x1, ...,xA)φo(x1, ...,xA) ,
where φo is a SM , mean-�eld WF des ribing the independent parti le motion, and F is a
symmetrized orrelation operator, whi h generates orrelations into the mean �eld WF. A-
dimensional integrals; enormous omputational e�orts; super omputers are required.
1. EXACT CALCULATION by MONTE-CARLO INTEGRATION (VMC): up to now only possible for A ≤ 12
(Argonne group Pieper, Wiringa and oworkers )
2. CLUSTER VARIATIONAL MONTE CARLO (CVMC): the Jastrow ( entral) ontribution is al ulated
exa tly by Monte Carlo integration and only few ontributions from non- entral intera tions are onsidered
by a linked luster expansion (Pandharipande (
16
O)) (Lonardoni et al. (
16O and
40Ca))
3. LINKED CLUSTER EXPANSION TECHNIQUE: (Various series expansions are developed for < H > and
any other operator < O > and all of them have the following basi prin iple: the 1st term is the MF
ntribution and the other terms ontain only linked ontributions (diagrams) Feenberg; Pandharipande,
Clark, Iwamoto-Yamada, Benhar, Fantoni, Ripka et and many others, in luding us)
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
FEW WORDS ABOUT CLUSTER EXPANSIONS
Consider only entral intera tions e.g, the Jastrow ase of a generi operator A :
< A >=< Ψ|A|Ψ >
< Ψ|Ψ >=< ΦMF |
∏
f(rij)A∏
f(rij)|ΦMF >
< ΦMF |∏
f(rij)2|ΦMF >
The numerator and the denominator ontains both linked (the good guys) and unlinked (the
bad guys) ontributions. The latter are bad be ause they make the expe tation value to diverge
with in reasing number of parti les a fa t whi h is known even from the theory of quantum
�uids (see Feenberg, Theory of quantum �uids, A ademi Press, 1969; J. -P. Blaizot, G.
Ripka, Quantum Theory of Finite Systems, MIT Press, Cambridge, MA, 1986). By
writing
f2ij = 1 + ηij
and expanding the denominator [1+x]−1 = 1− x+x2− ..., the unlinked terms in the numerator
exa tly an el out the ones arising from the denominator and a onvergent series expansion
ontaining only linked terms is obtained, e.g the η-expasion
〈A〉 = 〈A〉MFo + 〈A〉1 + 〈A〉2 + ... + 〈A〉n + ...
where the subs ripts denote the number of ηij in the given term and the �rst term of the
expansion represents the mean-�eld ontribution. To sum up, the main aim of any onvergent
luster expansion is to get rid of the expli it presen e of the denominator.
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
COMPARISON the (ηij) CLUSTER EXPANSION APPROACH with VMC and CVMC
1. INTERACTION
VMC/CVMC 2N (AV18) +3N (UIX)
OUR APPROACH 2N(AV8')
2. MEAN FIELD
VMC/CVMC Woods-Saxon
OUR APPROACH Woods-Saxon
3. VARIATIONAL WAVE FUNCTION
VMC/CVMC
|ΨV 〉 =
(
1 +∑
i<j<k
Uijk
)[
S∏
i<j
(
1 + U2−6ij
)
]
×
[
1 +∑
i<j
U7−8ij
][
∏
i<j
fc(rij)
]
|ΦMF 〉
OUR APPROACH
|ΨV 〉 =
[
S∏
i<j
(
1 + U2−6ij
)
][
∏
i<j
fc(rij)
]
|ΦMF 〉
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
Comparison of the linked- luster expansion [LCE℄ with the Variational Monte
Carlo (VMC) and Cluster Variational Monte Carlo [CVMC℄ approa h for
A = 16.
ALL THE THREE COMPARED APPROACH ARE GENUINE MANY-BODY APPROACHES IN THAT
THE WAVE FUNCTION RESULTS FROM THE MINIMIZATION OF THE EXPECTATION VALUE OF
THE HAMILTONIAN CONTAINING REALISTIC MODELS Of THE 2N AND 3N INTERACTION. NO
FREE PHENOMENOLOGICAL ADJUSTABLE PARAMETERS
Mean Field Approa h Potential (E/A) (E/A)exp < r2 >1/2
(< r2 >1/2
)exp
WS LCE AV8' -4.4 -7.98 2.64 2.69
WS CVMC AV18 -5.5 -7.98 2.54 2.69
WS CVCM AV18+UIX -5.15 -7.98 2.74 2.69
[LCE℄ Alvioli, CdA, Morita, Phys. Rev,C72 054310 (2005)
[CVMC℄ Lonardoni, Lovato, Pieper, Wiringa, arXiv.1705.04337v1 [nu l-th℄ 2017
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
COMPARISON OF THE OPERATOR TWO-BODY DENSITIES
0 1 2 3 4 5 6-1.5
-1.0
-0.5
0.0
0.516O - V8'
4 5 6
(2) (n
)(r) [fm
-3]
r [fm]
1 2 3
0 1 2 3 4 5 6
-3
-2
-1
0
140Ca - V8'
4 5 6
(2) (n
)(r) [fm
-3]
r [fm]
1 2 3
Alvioli, CdA Morita, Phys. Rev. C72 2005
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0
16O, AV18+UIX
ρp N
N(r
) (f
m-3
)
r (fm)
1
τ
σ
στ
t
tτ
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0
-4.5
-3.0
-1.5
0.0
1.5
0.0 1.0 2.0 3.0 4.0 5.0 6.0
40Ca, AV18+UIX
ρp N
N(r
) (f
m-3
)
r (fm)
1
τ
σ
στ
t
tτ-4.5
-3.0
-1.5
0.0
1.5
0.0 1.0 2.0 3.0 4.0 5.0 6.0
Lonardoni, Lovato, Pieper, Wiringa, arXiv:1705.04337v1 [nu l-th℄ 11 May 2017
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2 UNIVERSALITY OF SRCs in COORDINATE and
MOMENTUM SPACES: the CORRELATION HOLE and the
HIGH-MOMENTUM COMPONENTS of THE ONE- and
TWO-NUCLEON MOMENTUM DISTRIBUTIONS
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
2.1 The 2BDM ρ(2) in few-nu leon systems and omplex nu lei
Feldmeier, Horiu hi, Ne�, Suzuki, Phys. Rev. C84,054013(2011)
Alvioli, CdA, Morita, Phys. Rev., C72 0543 (2005); ArXiv: 0709:3989 (2007)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4
CN 0,
1 ρre
l0,
1 (r
) [fm
-3]
r [fm]
αt
h
α*
0 1 2 3 4 5 60.00.10.20.30.40.50.6
4He
12C
16O
40Ca(r) [fm
-3]
r [fm]
At r = |r1 − r2| < 1.0fm the 2BDM exhibits A-independen e and is similar to the
deuteron one
THE UNIVERSAL CORRELATION HOLE
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
ON THE EFFECTS OF 3N FORCES ON THE CORRELATION HOLE
[CVMC℄ Lonardoni, Lovato, Pieper, Wiringa,
arXiv.1705.04337v1 [nu l-th℄ 2017
0.00
0.10
0.20
0.30
0.40
0.50
0.0 1.0 2.0 3.0 4.0 5.0
16O
ρN
N(r
) (f
m-3
)
r (fm)
np, AV18
pp, AV18
np, AV18+UIX
pp, AV18+UIX
0.00
0.10
0.20
0.30
0.40
0.50
0.0 1.0 2.0 3.0 4.0 5.00.00
0.25
0.50
0. 5
1.00
1.25
0.0 1.0 2.0 3.0 4.0 5.0 6.0
40Ca
ρN
N)
(f3)
(f )
np, AV18
pp, AV18
np, AV18+UIX
pp, AV18+UIX
0.00
0.25
0.50
0. 5
1.00
1.25
0.0 1.0 2.0 3.0 4.0 5.0 6.0
3N FORCES DO NOT AFFECT THE CORRELATION HOLE!
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
2.2 THE SPIN-ISOSPIN STRUCTURE OF THE NUCLEAR
WAVE FUNCTION
Two nu leon system → Pauli Prin iple: L+S+T-odd
Shell Model (IPM):A ≤ 4: L = even,(10),(01)
A > 4: L = even,(10),(01); L = odd,(00),(11)
SRCs :
they reate states (00) and (11) (L-odd) also in A ≤ 4 nu lei and
de rease the per entage of (01) in favor of (11) state
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
The number of NN pairs in various spin-isospin (ST) states
(ST)
Nu leus (10) (01) (00) (11)
2
H 1 - - -
3
He IPM 1.50 1.50 - -
SRC (Present work) 1.488 1.360 0.013 0.139
SRC (Forest et al, 1996) 1.50 1.350 0.01 0.14
SRC (Feldmeier et al, 2011) 1.489 1.361 0.011 0.139
4
He IPM 3 3 - -
SRC (Present work) 2.99 2.57 0.01 0.43
SRC (Forestet al,1996) 3.02 2.5 0.01 0.47
SRC (Feldmeier et al, 2011) 2.992 2.572 0.08 0.428
16
O IPM 30 30 6 54
SRC (Present work) 29.8 27.5 6.075 56.7
SRC (Forest et al, 1996) 30.05 28.4 6.05 55.5
40
Ca IPM 165 165 45 405
SRC (Present work) 165.18 159.39 45.10 410.34
• NN intera tion doesn't pra ti ally a�e t the state (10) but appre iably redu es the state
(01) giving rise to a "visible" ontent of the (11) state; this is due to a three-body me hanism
originating from the tensor for e. J. L. Foster, et al, Phys. Rev. (1996) H. Feldemeier,
et al, Phys. Rev. (2011).
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
2.3 SRC IN MOMENTUM SPACE
2.3.1 ONE-BODY MOMENTUM DISTRIBUTIONS AND SRCs
ρ(r1, r′1) =
∫
Ψ∗0(r1, r2 . . . , rA) Ψ0(r
′1, r2 . . . , rA)
A∏
i=2
dri
n(k1) =
∫
e−ik1·(r1−r′1)ρ(r1, r
′1)dr1dr
′1
nA(k1) =∑
ST
n(ST )A (k1) =
=
∫
dr1 dr′1eik1·(r1−r
′1)∑
ST
∫
dr2ρN1N2ST (r1, r
′1; r2)
Alvioli, CdA, Kaptari, Mezzetti, Morita, Phys. Rev. C87 (2013) 709
lowest order linked luster expansion (four-nu leon luster) AV8' NN
intera tion
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
0 1 2 3 4 510-5
10-4
10-3
10-2
10-1
100
101
np(k
) [f
m3]
k [fm-1]
np(k) Argonne
np(k) Ours
4He
0 1 2 3 4 510-4
10-3
10-2
10-1
100
101
np(k)
[fm
3]
k [fm-1]
np(k) Argonne
np(k) Ours
16O
0 1 2 3 4 510-6
10-5
10-4
10-3
10-2
10-1
100
n(ST)
4(k
) [fm
3 ]
k [fm-1]
Full (ST)
(10) (00) (01) (11)
4HeAV8'
0 1 2 3 4 510-6
10-5
10-4
10-3
10-2
10-1
Full (ST)
(10) (00) (01) (11)
n(ST)
16(k
) [fm
3 ]
16OAV8'
k [fm-1]
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
Effe ts of 3N FORCES and Linked Cluster Expansion vs. CVMC
10-3
10-2
10-1
100
101
102
103
0.0 1.0 2.0 3.0 4.0 5.0
16
np
)/ (
f3)
(f1)
AV18
AV18+UIX
103
102
101
100
101
102
103
0.0 1.0 2.0 3.0 4.0 5.010
-3
10-2
10-1
100
101
102
103
0.0 1.0 2.0 3.0 4.0 5.0
40Ca
np
)/ (
f3)
(f1)
AV18
AV18+UIX
103
102
101
100
101
102
103
0.0 1.0 2.0 3.0 4.0 5.0
0 1 2 3 4 510-4
10-3
10-2
10-1
100
101
np(k)
[fm
3]
k [fm-1]
np(k) Argonne
np(k) Ours
16O
0 1 2 3 4 510-4
10-3
10-2
10-1
100
101
np(k)
[fm
3]
k [fm-1]
np(k) Argonne
np(k) Ours
40Ca
!! 3N For es do not affe t the high momentum ontent of the ground state wf and LCE OK !!
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
2.3.2 TWO-BODY MOMENTUM DISTRIBUTIONS
Alvioli, CdA, Morita Phys. Rev. C94,044309 (2016)
krel ≡ k =1
2(k1 − k2 ) Kc.m. ≡ K = k1 + k2
1. n(k1,k2) = n(krel,Kc.m.) = n(krel,Kc.m., θ) =
=1
(2π)6
∫
drdr′dRdR′ e−iKc.m.·(R−R′) e−ikrel·(r−r′)ρ(2)(r, r′;R,R′)
2. n(krel,Kc.m. = 0)
KCM = 0 =⇒ k2 = −k1,
ba k-to-ba k nu leons, like in the deuteron
3. nrel(k) =
∫
n(k,K) dK 4. nc.m.(K) =
∫
n(k,K) dk
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July 2-6, 2017 Bled, Slovenia
THE 3D PICTURE OF n(krel,KCM) = n(krel,KCM ,Θ)
! VERY IMPORTANT !
• If n(krel,KCM ,Θ) is θ independent, it means that n(krel,KCM )
= n(krel)n(KCM ) i.e. the relative and CM motions fa torize.
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July 2-6, 2017 Bled, Slovenia
0 1 2 3 4 510-10
10-8
10-6
10-4
10-2
100
102 Kc.m. = 0.0 Kc.m. = 0.5 Kc.m. = 1.0 Kc.m. = 3.0
npN A(k
rel,K
c.m
.,) [
fm6 ]
pn
3He
0 1 2 3 4 510-8
10-6
10-4
10-2
100
102
Kc.m.
= 0.0 K
c.m. = 0.5
Kc.m.
= 1.0 K
c.m. = 3.0
4He
pn0 1 2 3 4 5
10-6
10-4
10-2
100
102
pn
Kc.m.=0.0 Kc.m.=0.5 Kc.m.=1.0
12C
0 1 2 3 4 510-6
10-4
10-2
100
102
npp(Argonne)npn(Argonne)
npN(k
rel,K
c.m
.=0) [
fm6 ]
npp
npn
npn+npp
3He0 1 2 3 4 5
10-6
10-4
10-2
100
102
npp+npn
npp(Ref.)npn(Ref.)
npp
npn
4He
0 1 2 3 4 510-6
10-5
10-4
10-3
10-2
10-1
100
101
npp
npn
npp+npn
12C
0 1 2 3 4 510-6
10-4
10-2
100
102
krel [fm-1]
npN(k
rel) [fm
3 ]
npp
npn
npp+npn
3He
0 1 2 3 4 510-6
10-4
10-2
100
102
npp(Argonne)npn(Argonne)
npp
npn
krel [fm-1]
npp+npn
4He
0 1 2 3 4 510-6
10-4
10-2
100
102
npp+npn
npp
npn
npp(Argonne) npn(Argonne)
krel [fm-1]
12C
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
0 1 2 3 4 510-10
10-8
10-6
10-4
10-2
100
102 Kc.m. = 0.0 Kc.m. = 0.5 Kc.m. = 1.0 Kc.m. = 3.0
npN A
(kre
l,Kc.
m.,
) [fm
6 ]
pn
3He
0 1 2 3 4 510-8
10-6
10-4
10-2
100
102
Kc.m.
= 0.0 K
c.m. = 0.5
Kc.m.
= 1.0 K
c.m. = 3.0
4He
pn
0 1 2 3 4 510-6
10-4
10-2
100
102
npp(Argonne)npn(Argonne)
npN(k
rel,K
c.m
.=0) [
fm6 ]
npp
npn
npn+npp
3He0 1 2 3 4 5
10-6
10-4
10-2
100
102
npp+npn
npp(Ref.)npn(Ref.)
npp
npn
4He
0 1 2 3 4 510-6
10-4
10-2
100
102
krel [fm-1]
npN(k
rel) [fm
3 ]
npp
npn
npp+npn
3He
0 1 2 3 4 510-6
10-4
10-2
100
102
npp(Argonne)npn(Argonne)
npp
npn
krel [fm-1]
npp+npn
4He
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
What did we learn from these plots?
• npn(krel,Kc.m. = 0) >> npp(krel,Kc.m. = 0);
• npn(krel) > npp(krel);
•At Large krel ≥ 2 fm−1
and Small Kc.m. ≤ 1 fm−1we observe fa -
torization
npn(krel,Kc.m.) =⇒ npn(krel,Kc.m.) ≃ CpnA nD(krel)n
pnCM (Kc.m.)
•MOREOVER:Fa torization starts at values of krel ≡ k−rel(Kc.m.)
in reasing with in reasing values of Kcm!!!
• How do we get the values of CpnA ? The plots at various values
of Kc.m. and Θ tell it us.
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July 2-6, 2017 Bled, Slovenia
THE VALUE OF CpnA :
krel > k−rel ≃ 2 fm−1 ⇒npnA (krel,Kc.m.=0)
npnc.m.(Kc.m.=0)nD(krel)
⇒ Const ≡ CpnA
0 1 2 3 4 5100
101
102
103
Cpn3 (Ref. [11])
Cpn4 (Ref. [11])
npn A(k
rel,K
cm=0
)/nD(k
rel)n
cm(K
cm)
krel [fm-1]
Cpn3
Cpn4
Cpn12
Cpn16
Cpn40
2
H
3
He
4
He
12
C
16
O
40
Ca
Cpn2 = 1 Cpn
3 = 2 Cpn4 = 4 Cpn
12 = 20 Cpn16 = 24 Cpn
40 = 60
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
UNIVERSALITY OF THE MOMENTUM DISTRIBUTIONS
THE DEUTERON AND HELIUM-4 AS REFERENCE NUCLEI FOR
p− n and p− p SRC NN PAIRS IN NUCLEI
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July 2-6, 2017 Bled, Slovenia
npnA (krel, Kc.m. = 0)/[CpnA nc.m.(0)] ≃ nD(krel) when krel > 1.5 fm−1
0 1 2 3 4 510-6
10-4
10-2
100
102
npn(k
rel,K
cm=0
) /(C
A*n
pn cm(K
cm=0
)) [f
m3 ]
2H 12C 3He 16O 4He 40Ca
krel [fm-1]
(a)
npNA (krel, Kc.m. = 0)/[CpNA nc.m.(0)] ≃ n4(krel) when krel > 1.5(pn) fm−1 and 2.5(pp) fm−1
0 1 2 3 4 510-6
10-4
10-2
100
102
Cpn40=58
Cpn16=22
npn A(re
l)(k)/C
pn A [f
m3 ]
k [fm-1]
npn4(rel)(k)
npn12(rel)(k)/Cpn
12
npn16(rel)(k)/Cpn
16
npn40(rel)(k)/Cpn
40
pn Cpn
12=18
0 1 2 3 4 510-6
10-4
10-2
100
102
Cpp40=17
Cpp16=6.5
npp A(re
l)(k)/C
pp A [f
m3 ]
k [fm-1]
npp4(rel)(k)
npp12(rel)(k)/Cpp
12
npp16(rel)(k)/Cpp
16
npp40(rel)(k)/Cpp
40
pp Cpp
12=5
CdA, Mezzetti, Morita To Appear
Claudio Cio� degli Atti
26
July 2-6, 2017 Bled, Slovenia
4 FACTORIZATION AND THE CONVOLUTION MODEL OF
THE ONE-NUCLEON SPECTRAL FUNCTION OF
FEW-NUCLEON SYSTEMS AND COMPLEX NUCLEI
The Nu leon Spe tral fun tion is the basi nu lear stru ture element in e, A and
ν, A s attering pro esses.
Claudio Cio� degli Atti
27
July 2-6, 2017 Bled, Slovenia
THE FACTORIZATION REGIONS
Kc.m.= 03He
0.5
1.0
1.5
0 1 2 3 4 510-9
10-7
10-5
10-3
10-1
101
npn(k
rel,K
c.m
.,)
[fm6 ]
krel [fm-1]
3.0
The fa torization regions start at values of the relative momentum krel given by
krel > k−rel(Kc.m.) ≃ C1 + C2ΦA(Kc.m.) C1 ≃ 1fm−1 C2 = 0.5fm−1 ΦA = |Kc.m.|
CdA, Chiara Mezzetti, Hiko Morita, Phys. Rev. C95 044327 (2017) (arXiv:1701.08211v1[nu l-th℄ 27 Jan 2017)
Claudio Cio� degli Atti
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July 2-6, 2017 Bled, Slovenia
PNA (k1, E) =
1
2J + 1
∑
M,σ1
〈ΨJMA |a†k1σ1δ
(
E − (HA − EA))
ak1σ1|ΨJMA 〉
=1
2J + 1(2π)−3
∑
M,σ1
∑
∫
f
∣
∣
∣
∣
∫
dr1eik1·r1GMσ1
f (r1)
∣
∣
∣
∣
2
δ(
E − (EfA−1 − EA)
)
,
GMσ1f (r1) = 〈χ1/2
σ1,Ψf
A−1({x}A−1)|ΨJMA (r1, {x}A−1)〉,
4π
∫
PNA (k1, E) k2 d k1dE = 1
∫
PNA (k1, E) dE = nNA (k1)
.
PNA (k1, E) = PN
0 (k1, E) + PN1 (k1, E) .
PN0 (k1, E) = (2π)−3(2J+1)−1
∑
M,σ,f≤F
∣
∣
∣
∣
∫
eik1·r1GMσf (r1) dr1
∣
∣
∣
∣
2
δ(E−Emin), Mean−Field
PN1 (k1, E) = (2π)−3(2J + 1)−1
∑
M,σ,f>F
∣
∣
∣
∣
∫
eik1·r1GMσf (r1) dr1
∣
∣
∣
∣
2
δ(E − EfA−1). S RC
Claudio Cio� degli Atti
29
July 2-6, 2017 Bled, Slovenia
Exa t relation between the 1- and 2-nu leon momentum distributions ( N1 6= N2)
nN1A (k1) =
1
A− 1
[∫
nN1N2A (k1,k2) dk2 + 2
∫
nN1N1A (k1,k2) dk2
]
be omes in the fa torization region
nN1A (k1) =
[∫
nN1N2rel (|k1 −
Kc.m.
2|)nN1N2
c.m. (Kc.m.) dKc.m.
+ 2
∫
nN1N1rel (|k1 −
Kc.m.
2|)nN1N1
c.m. (Kc.m.) dKc.m.
]
dKc.m. ≡ nn1ex(k1),
and the orrelation part of the nu leon spe tral fun tion is
PN11 (k1, E) =
∑
N2=p,n
CN1N2
∫
nN1N2rel (|k1 −
Kc.m.
2|)nN1N2
c.m. (Kc.m.)dKc.m.
× δ
(
E − Ethr −A− 2
2mN(A− 1)
[
k1 −(A− 1)Kc.m.
A− 2
]2)
where CN1=N2 = 2, CN1 6=N2 = 1. This is
the CONVOLUTION FORMULA OF THE SPECTRAL FUNCTION.
Claudio Cio� degli Atti
30
July 2-6, 2017 Bled, Slovenia
The onvolution formula dates ba k to 1996 (Frankfurt, Strikman
CdA,Simula)! It has been frequently used. However:
• it has been applied using phenomenologi al relative and .m. distributions
sin e the realisti ones were not known at that time;
• only 2N SRC have been onsidered;
• didn't take into a ount the onstraint on the values of Kc.m. imposed by
the variation of the fa torization region with in reasing values of Kc.m., whi h
has been onsidered only re ently CdA, Chiara Mezzetti, Hiko Morita, Phys.
Rev. C95, 044327 (2017)
It's time to re- al ulate the onvolution spe tral fun tions using realisti many-
body relative and .m. distributions taking into a ount the onstraint on Kc.m..
Using a realisti .m. momentum distribution one an also investigate the e�e ts
of both 2N and 3N SRCs. This has been re ently done in the ase of
3
He
CdA, Chiara Mezzetti, Hiko Morita, Phys. Rev. C95, 044327 (2017)
(arXiv :1701.08211v1[nu l-th℄ 27 Jan 2017) and TO APPEAR
Claudio Cio� degli Atti
31
July 2-6, 2017 Bled, Slovenia
CHECK OF THE CONVOLUTION FORMULA:
3
He (
3
He → n + (pp))
The ab initio neutron spe tral fun tion of
3
He (k1 ≡ k)
(a) RSC Intera tion (b) AV18 Intera tion
CdA, Pa e, Salme� Phys. Rev.C21 805(1980) CdA, Kaptari Phys. Rev. C71 024005 (2005)
(Kievsky, Rosati, Viviani Nu l. Phys. A551 241(1993))
FULL LINE: PWIA (exa t pp �nal state, neutron plane wave)
DASHED LINE: PWA (plane wave also for the pp �nal state)
Claudio Cio� degli Atti
32
July 2-6, 2017 Bled, Slovenia
0 100 200 300 40010-6
10-5
10-4
10-3
Pn 3 He(k
,E*)
[fm
4 ]
PWA PWIA
3He
k=2.5 fm-1
0 100 200 300 40010-6
10-5
10-4
10-3
k=3.0 fm-1
0 100 200 300 40010-6
10-5
10-4
10-3
Pn 3 He(k
,E*)
[fm
4 ]
k=3.5 fm-1
E* [MeV]0 100 200 300 400
10-6
10-5
10-4
10-3
E* [MeV]
k=4.0 fm-1
At high values of k PWA ≃ PWIA ( Ab-Initio No models )
Peaks lo ated at E∗ ≃ k2/(4mN) in agreement with non relativisti approa h
Claudio Cio� degli Atti
33
July 2-6, 2017 Bled, Slovenia
Let us onsider the neutron spe tral fun tion of
3
He: initial state:
3
He,�nal
state: two protons in the ontinuum with their ex itation energy being the
relative kineti energy
E∗ =1
4mN[k1 − 2KCM ]2
In
3
He ground state the neutron is orrelated with the two protons, thus the Neutron spe tral
fun tion in the SRC region will depend only upon the relative and .m. distributions of the pn
pairs and we obtain
P n(k1, E∗) = Cpn
3
∫
nD(|k1 −KCM
2|)nnpCM(KCM)dKCM ×
× δ
(
E∗ −1
4mN[k1 − 2KCM ]2
)
where the integration limits on Kc.m. must in lude only the fa torization regions where
krel =
∣
∣
∣
∣
k1 −Kc.m.
2
∣
∣
∣
∣
≥ k−rel(Kc.m.) = C1 + C2ΦA(Kc.m.).
Outside these regions the onvolution formula annot be applied.
Claudio Cio� degli Atti
34
July 2-6, 2017 Bled, Slovenia
THE PROTON-NEUTRON CENTER-of-MASS MOMEMTUM DISTRIBUTIONS IN
3
He
0 1 2 3 4 510-6
10-5
10-4
10-3
10-2
10-1
100
101
npn c.m
.(Kc.
m.) [
fm3 ]
Kc.m. [fm-1]
nAb-initio [5]c.m
nSoftc.m.
nHardc.m.
nAb-initio [3]c.m
3He
The .m. momentum distribution of the orrelated proton-neutron pair in
3
He al ulated
in Ref.[3℄ and [5℄ with ab-initio wave fun tions orresponding to the AV18 intera tion.
[3℄ARGONNE , [5℄ Alvioli, CdA, Morita Phys. Rev. C94, 044309 (2016). After: CdA,
Mezzetti, Morita Phys. Rev. C95, 044327 (2017)
Claudio Cio� degli Atti
35
July 2-6, 2017 Bled, Slovenia
NEUTRON and PROTON SPECTRAL FUNCTIONS of HELIUM-3
P nex(k1, E) =
∫
nnprel(|k1 −Kc.m.
2|)nnpc.m.(Kc.m.) dKc.m. × δ
(
E − Ethr −1
4mN[k1 − 2Kc.m.]
2
)
P pex(k1, E) = P n
ex(k1, E) + 2
∫
npprel(|k1 −Kc.m.pp
2|)nnpc.m.(Kc.m.) dKc.m. × δ
(
E − Ethr −1
4mN[k1 − 2Kc.m.]
2
)
0 100 200 300 400 50010-7
10-6
10-5
10-4
10-3
Pn(k
,E*) [f
m4]
E* [MeV]
PCONV
(k,E*)
krel
>1.0+0.5Kcm
PExact
(k,E*)
3Hek=3.5 fm-1
neutron
0 100 200 300 400 50010-7
10-6
10-5
10-4
10-3
proton
Pp (k
,E*) [f
m4]
E* [MeV]
PCONV
(k,E*)
krel
>1.0+0.5Kcm
PExact
(k,E*)
3Hek=3.5 fm-1
Claudio Cio� degli Atti
36
July 2-6, 2017 Bled, Slovenia
SPECTRAL FUNCTION of HELIUM-4 and COMPLEX NUCLEI
CdA, Mezzetti, Morita, to appear
0 100 200 300 400 50010-6
10-5
10-4
10-3
Pp (k
,E*) [f
m4]
E* [MeV]
PTotal
CONV(k,E*)
Ppn
CONV(k,E*)
Ppp
CONV(k,E*)
4He
k=3.5 fm-1
PTotal
CONV=Ppn
CONV+Ppp
CONV
0 100 200 300 400 50010-5
10-4
10-3
10-2
Pp (k
,E*) [f
m4]
E* [MeV]
PTotal
CONV(k,E*)
Ppn
CONV(k,E*)
Ppp
CONV(k,E*)
12Ck=3.5 fm-1
PTotal
CONV=Ppn
CONV+Ppp
CONV
0 100 200 300 400 50010-5
10-4
10-3
10-2
Pp (k
,E*) [f
m4]
E* [MeV]
PTotal
CONV(k,E*)
Ppn
CONV(k,E*)
Ppp
CONV(k,E*)
16Ok=3.5 fm-1
PTotal
CONV=Ppn
CONV+Ppp
CONV
0 100 200 300 400 50010-5
10-4
10-3
10-2
Pp (k
,E*) [f
m4]
E* [MeV]
PTotal
CONV(k,E*)
Ppn
CONV(k,E*)
Ppp
CONV(k,E*)
40Cak=3.5 fm-1
PTotal
CONV=Ppn
CONV+Ppp
CONV
Claudio Cio� degli Atti
37
July 2-6, 2017 Bled, Slovenia
COMPARISON WITH THE ORIGINAL EFFECTIVE CONVOLUTION SPECTRAL
FUNCTION
0 100 200 300 400 50010-5
10-4
10-3
10-2
Pp (k
,E*) [f
m4]
E* [MeV]
PCONV
(k,E*)
krel
>1.0+0.5Kcm
PCS
(k,E*)
12Ck=3.5 fm-1
0 100 200 300 400 50010-5
10-4
10-3
10-2
Pp (k
,E*) [f
m4]
E* [MeV]
PCONV
(k,E*)
krel
>1.0+0.5Kcm
PCS
(k,E*)
16Ok=3.5 fm-1
PCS(k,E) ⇒ CdA, S. SIMULA, Phys. Rev. C53 1689 (1996)
Claudio Cio� degli Atti
38
July 2-6, 2017 Bled, Slovenia
THE MOMENTUM SUM RULE nA(k) =∫
P(k,E)dE
0 1 2 3 4 510-5
10-4
10-3
10-2
10-1
100
101
),( EkdEP pCONV
np A(k
) [f
m3]
k [fm-1]
Exact Np
Total(k)
Exact np
0(k)
Exact np
1(k)
4He
krel
>1.0+0.5Kc.m.
0 1 2 3 4 510-5
10-4
10-3
10-2
10-1
100
101
),( EkdEP pCONV
np A(k
) [f
m3]
k [fm-1]
Exact Np
Total(k)
Exact np
0(k)
Exact np
1(k)
16O
krel
>1.0+0.5Kc.m.
The onvolution formula does satisfy the momentum sum rule
Claudio Cio� degli Atti
39
July 2-6, 2017 Bled, Slovenia
ABOUT THE CONVERGENCE OF THE MOMENTUM SUM RULE
nE+(k) =∫ E+
0P(k,E∗)dE∗
0 1 2 3 4 510-5
10-4
10-3
10-2
10-1
100
101
E+=100(MeV)
E+=200(MeV)
E+=300(MeV)
E+=400(MeV)
np A(k
) [f
m3]
k [fm-1]
Exact Np
Total(k)
Exact np
0(k)
Exact np
1(k)
4He
E pCONV
pE EkPdEkn
0
** ),()(
0 1 2 3 4 510-6
10-5
10-4
10-3
10-2
10-1
100
101
E+=100(MeV)
E+=200(MeV)
E+=300(MeV)
E+=400(MeV)
np A(k
) [f
m3]
k [fm-1]
Exact np
Total(k)
Exact np
0(k)
Exact np
1(k)
16O
E pCONV
pE EkPdEkn
0
** ),()(
High momentum omponets ⇐⇒ High removal energies
Claudio Cio� degli Atti
40
July 2-6, 2017 Bled, Slovenia
THE ORIGIN OF FATORIZATION OF MOMENTUM DISTRIBUTIONS
The fa torized stru ture of two nu leon momentum distributions results from a general property
of the nu lear many-body wave fun tion, namely its fa torized form at short internu leon
distan es (Frankfurt-Strikman (1988), CdA-Simula (1995), Barnea & oworkers (2015)), namely
limrij→0
Ψ0({r}A) ≃
A{
χo(Rij)∑
n,fA−2
ao,n,fA−2
[
Φn(xij, rij)⊕ ΨfA−2({x}A−2, {r}A−2)
]}
,
Fa torized wave fun tions have been introdu ed in the past as physi ally sound approximations
of the unknown nu lear wave fun tion (see e.g. Levinger,(1951)), without however providing
any eviden e of the validity of su h an approximation due to the la k, at that time, of realisti
solutions of the nu lear many-body problem . These however be ame re ently available and
the validity of the fa torization approximation ould be quantitatively he ked . Indeed the
fa torization property of realisti many-body wave fun tions has been proved to hold in the
ase of ab initio wave fun tions of few-nu leon systems, �nite nu lei and nu lear matter.
Claudio Cio� degli Atti
41
July 2-6, 2017 Bled, Slovenia
5. SUMMARY & CONCLUSIONS
Claudio Cio� degli Atti
42
July 2-6, 2017 Bled, Slovenia
• A anoni al many-body variational approa h has been followed: 1. the 2N potential (Ar-
gonne family) and the general form of the many-body wave fun tion ΨA embodying entral,
spin,isospin, tensor et have been hosen; 2. the minimization of < ΨA|H|ΨA >< ΨA|ΨA >−1
has been performed and the parameters entering the wave fun tion were determined; 3.
the one, nA(k1) and two, nNNA (krel, Kc.m.,Θ), momentum distributions, free of any adjustable
parameter, have been al ulated.
• the one-nu leon distribution at k > kF shows high momentum ontents whi h annot be
re on iled with Hartree-Fo k or Brue kner-Hartree-Fo k type des riptions of nu lei;
• it is demonstrated that, starting from a ertain value of the relative momentum (depending
upon the value of the .m. momentum), the two-nu leon momentum distributions fa tor-
izes i.e. it obeys the relation nNNA (krel, Kc.m.,Θ) ≃ nN1N2
rel (krel)nN1N2c.m. (Kc.m.) and the region of
fa torizations and the expli it form of nN1N2
rel (krel) nN1N2c.m. (Kc.m.) have been obtained;
• the two-nu leon momentum distributions in the SRC region exhibits universality, i.e., apart
from a s aling fa tor, they are A independent; in ase of pn pairs their krel behavior is gov-
erned by the deuteron momentum distribution and their amplitude by the .m. momentum
distribution of the pair; as for the pp distributions their krel dependen e in a omplex nu-
leus is governed by the pp momentum distribution in
4He and their amplitude, as in the
ase of pn pairs, by the .m. distribution.
• using the above properties a model-independent parameter-free, fully mi ros opi spe tral
fun tion , the mi ros opi onvolution formula has been obtained.
Claudio Cio� degli Atti
43
July 2-6, 2017 Bled, Slovenia
ADDITIONAL SLIDES
Claudio Cio� degli Atti
44
July 2-6, 2017 Bled, Slovenia
0 1 2 3 4 5
10-4
10-3
10-2
10-1
100
101
102
103
104
q (fm-1)
ρS
T(q
) (f
m3)
4He(0+) - AV18+UX
ST=10ST=01ST=11ST=00
Claudio Cio� degli Atti
45
July 2-6, 2017 Bled, Slovenia
Claudio Cio� degli Atti
46
July 2-6, 2017 Bled, Slovenia
0 1 2 3 410-6
10-5
10-4
10-3
10-2
10-1
100
101
n 16(k
) [fm
3 ]
k [fm-1]
LDA 2NC V8' FHNC AV14
16O
Claudio Cio� degli Atti
47
July 2-6, 2017 Bled, Slovenia
Claudio Cio� degli Atti
48
July 2-6, 2017 Bled, Slovenia
COLLABORATORS
Hiko MORITA (Sapporo)
Massimiliano ALVIOLI (Perugia)
Chiara Benedetta MEZZETTI (Perugia)
HISTORICAL REVIEW PAPER ON SRCs
L. Frankfurt, M. Strikman, Physi s Report A 160, 235 (1988).
RECENT REVIEW PAPERS ON SRCs
L. Frankfurt, M. Sargsian and M. Strikman, Int. J. Mod. Phys. A 23, 2991 (2008).
J. Arrington, D. W. Higinbotham, G. Rosner and M. Sargsian, Prog. Part. Nu l. Phys. 67,
898 (2012).
O. Hen, D. W. Higinbotham, G. E. Miller, E. Piasetzky and L. B. Weinstein, Int. J. Mod.
Phys. E 22, 1330017 (2013).
M. Alvioli, C. Cio� degli Atti, L. P. Kaptari, C. B. Mezzetti and H. Morita, Int. J. Mod.
Phys. E 22, 1330021 (2013).
C. Cio� degli Atti, Phys. Rept. 590, 1 (2015).
O. Hen, G. A. Miller, E. Piasetzky, and L. B. Weinstein, arXiv: 1611.09748 ( Reviews of
Modern Physi s (2016))
Claudio Cio� degli Atti
49
July 2-6, 2017 Bled, Slovenia
When non- entral intera tions are onsidered the orrelation operator F is written as
F (x1,x2 ...xA) = SA∏
i<j
f (rij) f(rij) =N∑
n=1
f (n)(rij) f (n)(rij) = f (n)(rij) O(n)ij
〈A〉 =〈φo|F
† AF |φo〉
〈φo| F † F |φo〉= 〈φo|
∏
i<j
(1 + ηij) A|φo〉 · 〈φo|∏
i<j
(1 + ηij) |φo〉−1
is obtained. At 2nd order in η, one has, expli itly,
〈 A 〉0 = 〈φo| A |φo〉 , (1)
〈 A 〉1 = 〈φo|∑
i<j
ηij A |φo〉 − 〈A 〉o 〈φo|∑
i<j
ηij|φo〉 (2)
〈 A 〉2 = 〈φo|∑
(i<j)<(k<l)
ηij ηkl A |φo〉 − 〈φo|∑
i<j
ηij A|φo〉 〈φo|∑
i<j
ηij|φo〉 −
〈 A 〉o
〈φo|∑
(i<j)<(k<l)
ηij ηkl |φo〉 − 〈φo|∑
i<j
ηij|φo〉2
(3)
where the term of order n ontains ηij (fij) up to the n-th (2n-th) power.
Claudio Cio� degli Atti
50
July 2-6, 2017 Bled, Slovenia
npnA (Kc.m.) =
∫
npn(krel,Kc.m.) d3 krel
0 1 2 310-4
10-3
10-2
10-1
100
101
3He
4He
npn
A(K
c.m
.)[fm
3]
Kc.m.
[fm-1]
12C
16O
40Ca
WARNING: This is the orre t many-body de�nition adopted e.g. by us and ARGONNE.
Claudio Cio� degli Atti
51
July 2-6, 2017 Bled, Slovenia
THE LOW MOMENTUM C.M. DISTRIBUTION of pn PAIRS in NUCLEI Frankfurt, Strikman, Cio�,
Simula wrote in 1992
nc.m.(Kc.m.) =(α
π
)1.5
e−α∗K2c.m. α =
3(A− 1)
4(A− 2)
1
2mN < T >SM
α(fm2) 4
He
12
C
16
O
40
Ca
1995 (Shell Model) 2.4 1.1 1.0 0.95
2017 (Many-Body)* 2.4 1.0 1.2 0.98
* CdA, Morita(to appear)
0.0 0.5 1.0 1.510-4
10-3
10-2
10-1
100
101
3He 4He
npn A
(Kc.
m.)[
fm3 ]
Kc.m.[fm-1]
12C 16O 40Ca
Claudio Cio� degli Atti
52
July 2-6, 2017 Bled, Slovenia
Moreover, if fa torization holds one should have:
krel > k−rel ≃ 2 fm−1 Rpnfact/exact
≡CpnA nD(krel)nc.m.(Kc.m.)
npnA (krel,Kcm,θ)
⇒ 1
2
4
6
8
10
0 1 2 3 4 5
2
4
6
8
10
1 2 3 4 5
4He
krel [fm-1]
n Apn(fa
c)(k
rel,K
cm)/n
Apn(k
rel,K
cm,
)
12C
16O
Kcm=0.0 Kcm=0.5 Kcm=1.0
40Ca
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2.4.2 The ratio nA(k)/nD(k) a ording to many-body al ulations
0 1 2 3 4 5012345678
np A(k)
/ n D
(k)
k [fm-1]
3He (AV18) 4He (AV8')
16O (AV8') 40Ca (AV8')
nA/nD 6= const
The in rease of the ratio with k originates from the spin-isospin dependen e of
the momentum distributions and from the CM motion of the pair in the
nu leus. )
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2.4.3 Non isos alar nu lei: the momentum distributions in
3
He
0 1 2 3 4 50.0
0.5
1.0
1.5
2.0
2.5
np 3(k)/n
D(k
)
AV18
n3(k)/nD(k)
Np(10)3 (k)/nD(k)
k [fm-1]
Np(01)3 (k)/nD(k)
Np(11)3 (k)/nD(k)
0 1 2 3 4 50.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
nn 3(k)/n
D(k
)
AV18
n3(k)/nD(k)
Nn(10)3 (k)/nD(k)
k [fm-1]
Nn(01)3 (k)/nD(k)
Nn(11)3 (k)/nD(k)
A proton is orrelated with one p-n and one p-p pair; a neutron
with two n-p pair → Tensor dominan e in neutron (proton)
distributions in
3He (3H) and in neutron-ri h nu lei.
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July 2-6, 2017 Bled, Slovenia
THE BACK-to-BACK and the Kc.m.-INTEGRATED MOMENTUM DISTRIBUTIONS
0 1 2 3 4 510-6
10-4
10-2
100
102
npn(k
rel,K
cm=0
) /(C
A*n
pn cm(K
cm=0
)) [f
m3 ]
2H 12C 3He 16O 4He 40Ca
krel [fm-1]
(a)
0 1 2 3 4 510-6
10-4
10-2
100
102
(b)
npn
A(k
rel)/
Cpn
A
[fm
3]
krel
[fm-1]
2H 12C
3He 16O
4He 40Ca
• npnA (krel,Kc.m. = 0)/[C
pnA nc.m.(0)] ∝ nD(krel) when krel > 1.5 −
2 fm−1
.
• npnA (krel) ∝ nD(krel) when krel > 3 fm−1
.
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July 2-6, 2017 Bled, Slovenia
THE ORIGIN OF FATORIZATION OF MOMENTUM DISTRIBUTIONS
The fa torized stru ture of two nu leon momentum distributions results from a general property
of the nu lear many-body wave fun tion,namely its fa torized form at short internu leon dis-
tan es (Frankfurt-Strikman (1988), CdA-Simula (1995), Barnea & oworkers (2015)), namely
limrij→0
Ψ0({r}A) ≃
A{
χo(Rij)∑
n,fA−2
ao,n,fA−2
[
Φn(xij, rij)⊕ ΨfA−2({x}A−2, {r}A−2)
]}
,
Fa torized wave fun tions have been introdu ed in the past as physi ally sound approximations
of the unknown nu lear wave fun tion (see e.g. Levinger,(1951)), without however providing
any eviden e of the validity of su h an approximation due to the la k, at that time, of realisti
solutions of the nu lear many-body problem . These however be ame re ently available and
the validity of the fa torization approximation ould be quantitatively he ked . Indeed the
fa torization property of realisti many-body wave fun tions has been proved to hold in the
ase of ab initio wave fun tions of �nite nu lei and nu lear matter.
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CdA, Kaptari, Morita, S opetta, Few-Body Systems 50(2011)243
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