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

1

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


2


1 THE STANDARD MANY BODY APPROACH TO NUCLEI:

ab-INITIO AND VARIATIONAL CALCULATIONS


3


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


4


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), .


5


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)


6


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.


7


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 〉


8


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


9


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


10


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


11


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


12


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!


13


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


14


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).


15


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


16


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]


17


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 !!


18


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


19


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.


20


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

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


21


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


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

npn

krel [fm-1]

npp+npn

4He


22


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.


23


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


24


UNIVERSALITY OF THE MOMENTUM DISTRIBUTIONS

THE DEUTERON AND HELIUM-4 AS REFERENCE NUCLEI FOR

p− n and p− p SRC NN PAIRS IN NUCLEI


25


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


26


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.


27


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)


28


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


29


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.


30


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


31


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)


32


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


33


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.


34


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)


35


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


36


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


37


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)


38


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


39


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


40


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.


41


5. SUMMARY & CONCLUSIONS


42


• 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.


43


ADDITIONAL SLIDES


44


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


45



46


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


47



48


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))


49


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.


50


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.


51


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


52


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


53


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. )


54


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.


55


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

.


56


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.


57


CdA, Kaptari, Morita, S opetta, Few-Body Systems 50(2011)243


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TALK BLED fin 1 Hikocrex.fmf.uni-lj.si/eep17/YYY_TALK_BLED_fin_1_Hiko.pdf · 2017-07-05 · CM CV...

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