Towards an Estimation of Nuclear Forces and Nuclear Matrix ...theorie.ikp.physik.tu-darmstadt.de/ect13/Talks_files/ruiz-arriola.pdf · Matrix Elements Uncertainties: Chiral vs Non-Chiral

References Motivation Delta Shell Potential Fitting NN observables Calculations Chiral TPE Skyrme parameters Shell-Model Summary

Towards an Estimation of Nuclear Forces and NuclearMatrix Elements Uncertainties: Chiral vs Non-Chiral

Rodrigo Navarro PerezJose Enrique Amaro Soriano

Enrique Ruiz Arriola

University of GranadaAtomic, Molecular and Nuclear Physics Department

From Few-Nucleon Forces to Many-Nucleon StructureECT∗-Trento, 10 Jun 2013 to 14 Jun 2013

E. Ruiz Arriola (UGR) Errors in Nuclear Matrix Elements ECT* Trento, June 2013 1 / 47


1 References

2 Motivation

3 Delta Shell Potential

4 Fitting NN observables

5 Calculations

6 Chiral TPE

7 Skyrme parameters

8 Shell-Model

9 Summary



References

[1] Coarse graining Nuclear InteractionsERICE Summer School (Sep-2011)Prog. Part. Nucl. Phys. 67 (2012) 359 [arXiv:1111.4328 [nucl-th]].

[2] Phenomenological High Precision Neutron-Proton Delta-Shell PotentialPhys. Lett. B (2013) to appear, arXiv:1202.2689 [nucl-th].

[3] Error estimates on Nuclear Binding Energies from Nucleon-Nucleon uncertaintiesarXiv:1202.6624 [nucl-th].

[4] Nuclear Binding Energies and NN uncertaintiesQuark Nuclear Physics (May-2012)PoS QNP 2012 (2012) 145 [arXiv:1206.3508 [nucl-th]].

[5] Effective interactions in the delta-shells potentialInternational IUPAP Conference on Few-Body Problems in Physics, Aug-2012Few-Body Syst (2013), arXiv:1209.6269 [nucl-th].

[6] Nucleon-Nucleon Chiral Two Pion Exchange potential vs Coarse grained interactionsChiral Dynamics Aug-2012. arXiv:1301.6949 [nucl-th].

[7] Partial Wave Analysis of Nucleon-Nucleon Scattering below pion productionPhys. Rev. C (2013) to appear, arXiv:1304.0895 [nucl-th].



Bottomline

THE PROBLEM

GOAL: Estimate uncertainties from IGNORANCE of NN,3N,4N interactionReduce computational cost

Statistical Uncertainties: NN,3N,4N DataData abundance bias

Systematic Uncertainties: NN,3N,4N potentialMany forms of potentials possible

Confidence level of Imperfect theories vs Perfect experiments

OUR APPROACH

Start with NN

Fit data WITH ERRORS with a simple interaction

Compare different interactions (AV18,CDBonn,N3LO,Nijm,Spec)

Estimate uncertainties of Effective Interactions and Matrix elements



Error Analysis in Nuclear Structure

Theoretical Predictive Power Flow: From light to heavy nucleiExperiment much more precise than theoryHow to estimate theoretical errors based on INPUT data

INPUT = NN, 3N, · · · → OUTPUT = 4N, . . .

First Step: INPUT=NN scattering dataOUTPUT=NN scattering amplitudes

0 50 100 150 200 250 300 3500

20

40

60

80

100

120

140

160

180

Tlab [MeV]

θ [de

gree

s CM

]

NN-OnLine http://nn-online.org 7 June 2013

0 50 100 150 200 250 300 3500

20

40

60

80

100

120

140

160

180

Tlab [MeV]θ [

degr

ees

CM]

NN-OnLine http://nn-online.org 7 June 2013



Wolfenstein Parameters

PWA,NimjII,CDBonn,Spec,Reid93,AV18, χ2/dof ∼ 1

1801501209060300

0.08

0.06

0.04

0.02

0

1801501209060300

0.033

0.019

0.005

-0.009

-0.023

1801501209060300

0.1

0

-0.1

-0.2

-0.3

h[fm]

1801501209060300

0.015

-0.055

-0.125

-0.195

-0.265

0.104

0.072

0.04

0.008

-0.024

0.139

0.127

0.115

0.103

0.091

0.28

0.14

0

-0.14

-0.28

g[fm]

0.2

0.1

0

-0.1

-0.2

0.116

0.088

0.06

0.032

0.004

0.171

0.163

0.155

0.147

0.139

0.32

0.16

0

-0.16

-0.32

m[fm]

0.23

0.09

-0.05

-0.19

-0.33

0.225

0.175

0.125

0.075

0.025

0.099

0.077

0.055

0.033

0.011

0.064

0.032

0

-0.032

-0.064

c[fm]

0.025

0.015

0.005

-0.005

-0.015

Imaginary Part

TLAB = 200 MeV

0.53

0.39

0.25

0.11

-0.03

TLAB = 50 MeV

1.02

0.98

0.94

0.9

0.86

Real Part

TLAB = 200 MeV

0.9

0.7

0.5

0.3

0.1

TLAB = 50 MeV

a[fm]

0.9

0.8

0.7

0.6

0.5

θcm [deg]



Introduction

How much do we need to know light nuclei to predict heavy nuclei ?

Nucleon size a ∼ 1fm

Nuclear Force ∼ 1/mπ = 1.4fm

Nuclear matter (interparticle distance)

ρnm = 0.17fm−3 =1

(1.8fm)3

Fermi Momentum

kF = 270MeV λF = π/kF = 2.3fm� 1/√mπMN = 0.5fm

Can we ignore explicit core, finite nucleon size and explicit pions ?What is the confidence level for this scenario ?



Quark Cluster Dynamics (qcd)

Atomic analogue. Neutral atomsNon-overlapping atoms exchange TWO photons (Van der Waals force)Overlapping atoms are not locally neutral; ONE photon exchange is possible (Chemical bonding)

R/2

xi

yiηi

ξiR/2

c.m.

c.m.

ClusterB

ClusterA

Overlapping effects (quark exchange) constrain the applicability of Lagrangians



Quark Cluster Dynamics (qcd)

NN potential in the Born-Oppenheimer approximation Calle Cordon, RA, ’12

V 1π+2π+...NN,NN (r) = V 1π

NN,NN (r) + 2|V 1πNN,N∆(r)|2

MN −M∆+

1

2

|V 1πNN,∆∆(r)|2

MN −M∆+O(V 3) ,

Bulk of TWO-Pion Exchange Chiral forces reproducedFinite size effects set in at 2fm → exchange quark effects become explicitHigh quality potentials confirm these trends.

-10

-8

-6

-4

-2

0

1.6 1.8 2 2.2 2.4 2.6 2.8 3

VC

(r) [

MeV

]

r [fm]

BO-TPE no-FFBO-TPE axial-FF

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1.6 1.8 2 2.2 2.4 2.6 2.8 3

VS(r) [

MeV

]

r [fm]


-1

-0.8

-0.6

-0.4

-0.2

1.6 1.8 2 2.2 2.4 2.6 2.8 3

VT(r) [

MeV

]

r [fm]


-1

-0.8

-0.6

-0.4

-0.2

0

1.6 1.8 2 2.2 2.4 2.6 2.8 3

WC

(r) [

MeV

]

r [fm]


0

0.2

0.4

0.6

0.8

1

1.2

1.6 1.8 2 2.2 2.4 2.6 2.8 3

WS(r) [

MeV

]

r [fm]

OPE no-FFOPE axial-FF


-1

0

1

2

3

4

5

6

7

8

1.6 1.8 2 2.2 2.4 2.6 2.8 3

WT(r) [

MeV

]

r [fm]

OPE no-FFOPE axial-FF




Anatomy of the unknown NN interaction

At what distance look nucleons point-like ?

r > 2fm

When is OPE the ONLY contribution ?

rc > 3fm

What is the minimal resolution where interaction is elastic ?

pmax ∼√MNmπ → ∆r = 1/pmax = 0.6fm

How many partial waves must be fitted ?

lmax = pmaxrcrc/∆r = 5

Minimal distance where centrifugal barrier dominates

l(l + 1)

r2min

≤ p2

How many parameters ?(1S0,3 S1) , (1P1,3 P0,3 P1,3 P2), (1D2,3 D1,3D2,3D3), (1F3,3 F2,3 F3,3 F4)

2× 5 + 4× 4 + 4× 3 + 4× 2 + 4× 1 = 50



Anatomy of the unknown NN interaction

0.6 1.2 1.8 2.4 30.0

FINITE NUCLEON SIZE

QUARK EXCHANGE

ONE PION EXCHANGE TPE

p=350 MeV

L=0

L=1

L=2

L=3

L=4

1S0,3S1

1G4,3G3,3G4,3G5

1P1,3P0,3P1,3P2

1D2,3D1,3D2,3D3

1F3,3F2,3F3,3F4

POINT−LIKE NUCLEON



Motivation

Study of the NN interaction for over 60 years

More than 7800 experimental scattering data from 1950 to 2013

Several partial wave analyses (PWA) and potentials since the 1950’s

Hamada Johnston, Yale, Paris, Bonn, Nijmegen, Argonne, ...χ2/d.o.f. ∼ 1 possible by 1993

[Stoks et al, Phys. Rev. C 48 (1993), 792]

Chiral potentials appear in the mid 1990’s



Motivation

ELAB [MeV]

0.1

350300250200150100500

ǫ4

0.13H4

0.23F4

0.23G4

0.11G4

0.3ǫ3

0.33G3

0.73D3

0.43F3

0.21F3

0.2ǫ2

0.43F2

1.03P2

0.63D2

0.71D2

0.5ǫ1

1.03D1

1.13S1

1.03P1

0.91P1

0.73P0

∆δ [deg.]0.71S0

No unique determination of the NN interaction

Different phenomenological potentials

Fitted to experimental scattering dataHigh accuracy χ2/d.o.f. ∼ 1Dispersion in PhaseshiftsOPE as the long range interaction∼ 40 parameters for the short andintermediate rangeRepulsive core for most of them

Short range correlations

Nuclear structure calculations become complicated

No statistical uncertainties estimates



Motivation

Effective coarse graining

Oscillator Shell ModelEuclidean Lattice EFTVlowk interaction

Characteristic distance ∼ 0.5− 1.0 fm

Nyquist Theorem

Optimal samplingFinite Bandwidth

∆r∆k ∼ 1

de Broglie wavelength of the mostenergetic particle

u350(r)u100(r)uk→0(r)

AV18

r [fm]

u(r)

V[fm

−1]

4

3

2

1

0

-1

54.543.532.521.510.50

4

3

2

1

0

-1

2fm



COARSE GRAINED INTERACTION



Delta Shell Potential

A sum of delta functions

V (r) =∑i

λi

2µδ(r − ri)

[Aviles, Phys.Rev. C6 (1972) 1467]

Optimal and minimal sampling of the nuclear interaction

Pion production threshold ∆k ∼ 2 fm−1

Optimal sampling, ∆r ∼ 0.5fm

Delta ShellAnalitic Potential

r [fm]

543210

0

−0.2

−0.4

−0.6

−0.8

−1

−1.2

−1.4

k = 1.1 fm−1, u(r)2

r [fm]

543210

2

1.5

1

0.5

0



Coarse Graining the AV18 potential

Delta ShellAV18

r [fm]

V[fm

−1]

43.532.521.510.50

4

3

2

1

0

-1

Delta ShellAV18

r [fm]

V[fm

−1]

43.532.521.510.50

4

3

2

1

0

-1

N = 600N = 54N = 42N = 30N = 18

ELAB [MeV]

δ[deg]

350300250200150100500

706050403020100

-10-20-30

N = 5N = 4N = 3N = 2N = 1

ELAB [MeV]

δ[deg]

350300250200150100500

706050403020100

-10-20




Comparison with Vlowk

AV18Delta Shell

1S0

k [fm−1]

Vlo

wk

[fm

]

21.510.50

0.20

−0.2−0.4−0.6−0.8−1

−1.2−1.4−1.6−1.8−2

AV18Delta Shell

3S1

k [fm−1]

Vlo

wk

[fm

]

21.510.50

0

−0.5

−1

−1.5

−2

−2.5

Nuclear structure calculations

[Prog.Part.Nucl.Phys. 67 (2012) 359]

UCOMGFMCExp

4He

rm [fm]

B[M

eV]

21.81.61.41.21

−5

−10

−15

−20

−25

−30

−35

BHF CC

Exp

16O

rm [fm]

B[M

eV]

32.82.62.42.221.8

−20

−40

−60

−80

−100

−120

−140

Exp

40Ca

rm [fm]

B[M

eV]

43.83.63.43.232.82.6

−50

−100

−150

−200

−250

−300

−350

−400




3 well defined regions

Innermost region r ≤ 0.5 fm

Short range interactionNo delta shell (No repulsive core)

Intermediate region 0.5 ≤ r ≤ 3.0 fm

Unknown interactionλi parameters fitted to scattering data

Outermost region r ≥ 3.0 fm

Long range interactionDescribed by OPE and EM effects

Coulomb interaction VC1 and relativistic correction VC2 (pp)Vacuum polarization VV P (pp)Magnetic moment VMM (pp and np)



np AND pp PARTIAL WAVE ANALYSIS



Fitting NN observables

Database of NN scattering data obtained till 2013

http://nn-online.org/http://gwdac.phys.gwu.edu/NN provider for Android

Google Play Store

[J.E. Amaro, R. Navarro-Perez, and E. Ruiz-Arriola]

2868 pp data and 4991 np data

3σ criterion by Nijmegen to remove possible outliers



Fitting NN observables

Delta shell potential in every partial wave

V JSl,l′ (r) =1

2µαβ

N∑n=1

(λn)JSl,l′δ(r − rn) r ≤ rc = 3.0fm

Strength coefficients λn as fit parameters

Fixed and equidistant concentration radii ∆r = 0.6 fm

EM interaction is crucial for pp scattering amplitude

VC1(r) =α′

r,

VC2(r) ≈ − αα′

Mpr2,

VV P (r) =2αα′

3πr

∫ ∞1

dx e−2merx

[1 +

1

2x2

](x2 − 1)1/2

x2,

VMM (r) = − α

4M2p r

3

[µ2pSij + 2(4µp − 1)L·S

]



Scattering Observables

Comparing with Potentials and Experimental data

np data

Nijm PWAAV18

This workExperimental

P 325.0 MeV

1801501209060300

0.51

0.33

0.15

-0.03

-0.21

At 325.0 MeV

θc.m. [deg]

1801501209060300

0.7

0.5

0.3

0.1

-0.1Rt 325.0 MeV

1801501209060300

0.26

-0.02

-0.3

-0.58

-0.86

Dt 325.0 MeV

0.33

0.19

0.05

-0.09

-0.23

I0 324.1 MeV10.8

8.4

6

3.6

1.2

D 212.0 MeV1.48

0.96

0.44

-0.08

P 50.0 MeV

0.255

0.165

0.075

-0.015

-0.105

I0 50.0 MeV19

17

15

13

11

I0 25.8 MeV34.1

32.3

30.5

28.7

26.9



Scattering Observables

Comparing with Potentials and Experimental data

pp data

Nijm PWAAV18

This workExperimental

R 209.1 MeV

9060300

0.86

0.58

0.3

0.02

-0.26P 210.0 MeV

θc.m. [deg]

9060300

0.305

0.215

0.125

0.035

-0.055

Cnn 143.0 MeV

9060300

1.04

0.72

0.4

0.08

-0.24

D 142.0 MeV0.86

0.58

0.3

0.02

-0.26P 142.0 MeV

0.245

0.135

0.025

-0.085

-0.195

R 141.0 MeV0.84

0.52

0.2

-0.12

-0.44

χ2/d.o.f. = 1.06 with N = 2747|pp + 3691|np

[arXiv:1304.0895]



Phase shifts

(u)

3D3

350250150500

5.4

4.2

3

1.8

0.6

(t)

3F2

TLAB [MeV]

35025015050

(s)

3F2

350250150500

1.53

1.19

0.85

0.51

0.17

(r)

3D1

-3

-9

-15

-21

-27

(q)ǫ2

(p)ǫ2

-0.35

-1.05

-1.75

-2.45

-3.15

(o)ǫ1

5.4

4.2

3

1.8

0.6(n)

3P2

(m)

3P2

18

14

10

6

2

(l)3S1

144

112

80

48

16

(k)

3P1

(j)

3P1

Phase

shift[deg]

-3.5

-10.5

-17.5

-24.5

-31.5

(i)

3D227

21

15

9

3

(h)

3P0

(g)

3P0

11

4

-3

-10

-17

(f)

1F3

-0.6

-1.8

-3

-4.2

-5.4(e)

1D2

(d)

1D2

10.8

8.4

6

3.6

1.2

(c)

1P1

np

-3.5

-10.5

-17.5

-24.5

-31.5

(b)

1S0

np

(a)

1S0

pp

63

45

27

9

-9

Phase shifts for every partial

Statistical uncertainty propagated directlyfrom covariance matrix

ǫ1

TLAB [MeV]

Phase

shift[deg]

350300250200150100500

6

5

4

3

2

1

0




A complete parametrization of the on-shell scattering amplitudes

Five independent complex quantities

Function of Energy and Angle

M(kf ,ki) = a+m(σ1,n)(σ2,n) + (g − h)(σ1,m)(σ2,m)

+(g + h)(σ1, l)(σ2, l) + c(σ1 + σ2,n)

Scattering observables can be calculated from M

[Bystricky, J. et al, Jour. de Phys. 39.1 (1978) 1]




TLAB = 200 MeV

(t)

1801501209060300

0.092

0.076

0.06

0.044

0.028

(s)

1801501209060300

-0.082

-0.126

-0.17

-0.214

-0.258

(r)

1801501209060300

0.08

0.06

0.04

0.02

0

(q)

h[fm]

1801501209060300

0.1

0

-0.1

-0.2

-0.3

(p)0.07

0.01

-0.05

-0.11

-0.17

(o)0.055

-0.035

-0.125

-0.215

-0.305(n)

0.104

0.072

0.04

0.008

-0.024

(m)

g[fm]

0.28

0.14

0

-0.14

-0.28

(l)0.006

-0.012

-0.03

-0.048

-0.066

(k)0.05

-0.05

-0.15

-0.25

-0.35(j)

0.116

0.088

0.06

0.032

0.004

(i)

m[fm]

0.32

0.16

0

-0.16

-0.32

(h)0.36

0.28

0.2

0.12

0.04(g)

-0.001

-0.003

-0.005

-0.007

-0.009

(f)0.225

0.175

0.125

0.075

0.025

(e)

c[fm]

0.064

0.032

0

-0.032

-0.064

(d)

Imaginary Part, pp

0.33

0.19

0.05

-0.09

-0.23

(c)

Real Part, pp

0.35

0.25

0.15

0.05

-0.05

(b)

Imaginary Part, np

0.53

0.39

0.25

0.11

-0.03

(a)

Real Part, np

a[fm]

0.9

0.7

0.5

0.3

0.1

θc.m. [deg]



Deuteron Properties

Delta Shell Empirical Nijm I Nijm II Reid93 AV18 CD-BonnEd(MeV) Input 2.224575(9) Input Input Input Input Input

η 0.02493(8) 0.0256(5) 0.0253 0.0252 0.0251 0.0250 0.0256

AS(fm1/2) 0.8829(4) 0.8781(44) 0.8841 0.8845 0.8853 0.8850 0.8846rm(fm) 1.9645(9) 1.953(3) 1.9666 1.9675 1.9686 1.967 1.966QD(fm2) 0.2679(9) 0.2859(3) 0.2719 0.2707 0.2703 0.270 0.270PD 5.62(5) 5.67(4) 5.664 5.635 5.699 5.76 4.85

〈r−1〉(fm−1) 0.4540(5) 0.4502 0.4515

(a)

q [MeV]

GC

10008006004002000

1

0.1

0.01

0.001

(c)

q [MeV]

GQ

10008006004002000

0.1

0.01

0.001

(b)

q [MeV]

MG

M/M

d

10008006004002000

1

0.1

0.01



Including Chiral Two Pion Exchange

Inclusion of χTPE interactions at long and intermediate ranges

pp PWA by the Nijmegen group

[Rentmeester et al, Phys. Rev. Lett. 82 (1999), 4992]

Improvement in the χ2 value compared to OPE onlyReduction of the number of parametersDetermination of chiral constants c1, c3, c4

Preliminary test using the δ-shell potential

OPE, TPE(l.o.) and TPE(s.o.)Different cut radius, rc = 3.0, 2.4, 1.8fm



Comparing OPE and χTPE

Fitting all NN data

rc [fm] 1.8 2.4 3.0Np χ2/ν Np χ2/ν Np χ2/ν

OPE 31 1.80 39 1.56 46 1.54TPE(l.o.) 31 1.72 38 1.56 46 1.52TPE(s.o.) 30+3 1.60 38+3 1.56 46+3 1.52

Fitting 3σ compatible NN data

NData Np χ2/ν NData Np χ2/ν NData Np χ2/νOPE 5766 31 1.10 6363 39 1.09 6438 46 1.06

TPE(l.o.) 5841 31 1.10 6432 38 1.10 6423 46 1.06TPE(s.o.) 6220 30+3 1.07 6439 38+3 1.10 6422 46+3 1.06

OPE only at 3.0fm describes the data

1.8 ≤ r ≤ 3.0fm OPE + something else

χTPE most of that something else



EFFECTIVE INTERACTIONS



Motivation

Effective Interaction [Skyrme, Moshinsky]

Useful simplifications in many body calculations [Brink, Vaughterin]

Power expansion in CM momenta

V (p′,p) =

∫d3xe−ix·(p

′−p)V (x)

= t0(1 + x0Pσ) +t1

2(1 + x1Pσ)(p′2 + p2)

+t2(1 + x2Pσ)p′ · p + 2itV S · (p′ ∧ p)

+tT

2

[σ1 · pσ2 · p + σ1 · p′σ2 · p′ −

1

3σ1σ2(p′2 + p2)

]+tU

2

[σ1 · pσ2 · p′ + σ1 · p′σ2 · p−

2

3σ1σ2p

′ · p]

+O(p4)



Skyrme parameters

Skyrme parameters in terms of paratial waves

Partial Wave potential in momentum space

V JSl′l′ (p′, p) =(4π)2

M

∫ ∞0

drr2jl′ (p′r)jl(pr)V

JSl′l (r)

Using the Bessel function expansion

jl(x) =xl

(2l + 1)!!

[1− x2

2(2l + 3)+ · · ·

]



Skyrme parameters

Comparing similar terms

(t0, x0t0) =1

2

∫d3x

[V3S1

(r)± V1S0(r)]

(t1, x1t1) = − 1

12

∫d3xr2

[V3S1

(r)± V1S0(r)]

(t2, x2t2) =1

54

∫d3xr2

[V3P0

(r) + 3V3P1(r) + 5V3P2

(r)± 9V1P1(r)]

tV =1

72

∫d3xr2

[2V3P0

(r) + 3V3P1(r)− 5V3P2

(r)]

tU =1

36

∫d3xr2

[−2V3P0

(r) + 3V3P1(r)− V3P2

(r)]

tT =1

5√

2

∫d3xr2Vε1 (r)



Skyrme parameters

Straightforward for δ-shell potential

t ∝∑

λirni

Integrable for OPE starting at rc

t ∝ f2πNN

m2π

Γ(n,mπrc)

Where f2πNN/(4π) ∼ 0.08



Skyrme parameters

Skyrme parameters fitting at different energy ranges

AV18

DS Uncoupled S-WavesDS Coupled S-Waves

kCM [MeV]

t 0[M

eVfm

3]

380360340320300280260240220200

0−200−400−600−800−1000−1200−1400

AV18


kCM [MeV]

x0

380360340320300280260240220200

1

0.5

0

−0.5

−1

AV18


kCM [MeV]

t 1[M

eVfm

5]

380360340320300280260240220200

3000

2500

2000

1500

1000

500

0

AV18


kCM [MeV]

x1

380360340320300280260240220200

0.1

0.05

0

−0.05

−0.1



Skyrme parameters

Skyrme parameters fitting at different energy ranges

AV18

DS

kCM [MeV]

t 2[M

eVfm

5]

380360340320300280260240220200

2780

2775

2770

2765

2760

2755

2750

2745

AV18

DS Coupled S-Waves

kCM [MeV]

x2

380360340320300280260240220200

−0.82−0.825−0.83−0.835−0.84−0.845−0.85−0.855−0.86

AV18

DS

kCM [MeV]

t V[M

eVfm

5]

380360340320300280260240220200

140

135

130

125

120

115

110

105

AV18

DS

kCM [MeV]

t U[M

eVfm

5]

380360340320300280260240220200

148014701460145014401430142014101400

AV18

DS

kCM [MeV]

t T[M

eVfm

5]

380360340320300280260240220200

−3600

−3800

−4000

−4200

−4400

−4600



Skyrme Parameters

Fermi type shape density

ρ(x) =ρ0

1 + e(r−R)/a

R = r0A1/3, r0 = 1.1fm and a = 0.7fm

Error band for stable nuclei binding energy

0 50 100 150 200 250

0

2

4

6

8

10

A

-B(A)/AMeV

∆B

A=

3

8A∆t0

∫d3x ρ(x)2



Skyrme Parameters

Nuclear and Neutron matter

Error grows linearly with the density

∆Bn.m.A

=3

8∆t0ρ ∼ 3.75ρ

∆BnA

=1

4∆[t0(1 − x0)]ρn ∼ 3.5ρn

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0

20

40

60

80

100

120

�n fm

- 3

B/N

MeV

0.0 0.1 0.2 0.3 0.4 0.5 0.6

- 10

0

10

20

30

40

�n.m. fm

- 3

B/A

MeV



SHELL MODEL MATRIX ELEMENTS



Renormalization of Nuclear Matrix elements

Harmonic oscillator shell model

VHO(r) =r2

2Mb4→ εnl =

1

2Mb2(4n+ 2l − 1)

Distortion due to OPE and TPE → Energy shift ∆εnl

1S0

0.5 1.0 1.5 2.0 2.5 3.0-10

-9

-8

-7

-6

-5

rcHfmL

DE

HM

eV

L

∆εnl = 〈ϕnl|K(εnl + ∆εnl)|ϕnl〉

In order to see the differences we need to look into short distances.



Errors in Nuclear Matrix elements

(TLAB ≤ 350MeV,rc|OPE = 3fm, rc|TPE = 1.8fm )

OPEχTPE

TLAB [MeV]

δ1S0[degrees]

350300250200150100500

706050403020100

−10

OPEχTPE

TLAB [MeV]

δ3S1[degrees]

350300250200150100500

160140120100806040200

−20

OPEχTPE

TLAB [MeV]

δ1P1[degrees]

350300250200150100500

0

−5

−10

−15

−20

−25

−30

OPEχTPE

b [fm]

〈V1S0〉[M

eV]

2.42.221.81.61.41.210.80.6

6420

−2−4−6−8−10

OPEχTPE

b [fm]

〈V3S1〉[M

eV]

2.42.221.81.61.41.210.80.6

20

15

10

5

0

−5

OPEχTPE

b [fm]

〈V1P1〉[M

eV]

21.91.81.71.61.51.41.3

987654321




(TLAB ≤ 350MeV,rc|OPE = 3fm, rc|TPE = 1.8fm )

OPEχTPE

TLAB [MeV]

δ3P0[degrees]

350300250200150100500

15

10

5

0

−5

−10

−15

OPEχTPE

TLAB [MeV]

δ3P1[degrees]

350300250200150100500

0

−5

−10

−15

−20

−25

−30

−35OPE

χTPE

TLAB [MeV]

δ3P2[degrees]

350300250200150100500

181614121086420

OPEχTPE

b [fm]

〈V3P0〉[M

eV]

21.91.81.71.61.51.41.3

−1.2−1.3−1.4−1.5−1.6−1.7−1.8−1.9−2

−2.1−2.2

OPEχTPE

b [fm]

〈V3P1〉[M

eV]

21.91.81.71.61.51.41.3

987654321

OPEχTPE

b [fm]

〈V3P2〉[M

eV]

21.91.81.71.61.51.41.3

0

−0.5

−1

−1.5

−2

−2.5

−3

−3.5




(TLAB ≤ 125MeV,rc|OPE = 1.8fm, rc|TPE = 1.8fm )

OPEχTPE

TLAB [MeV]

δ1S0[degrees]

120100806040200

6560555045403530252015

OPEχTPE

TLAB [MeV]

δ3S1[degrees]

120100806040200

160

140

120

100

80

60

40

20

OPEχTPE

TLAB [MeV]

δ1P1[degrees]

120100806040200

0−2−4−6−8−10−12−14−16−18

OPEχTPE

b [fm]

〈V1S0〉[M

eV]

2.42.221.81.61.41.210.80.6

40

30

20

10

0

−10

−20

−30

OPEχTPE

b [fm]

〈V3S1〉[M

eV]

2.42.221.81.61.41.210.80.6

20

15

10

5

0

−5

OPEχTPE

b [fm]

〈V1P1〉[M

eV]

21.91.81.71.61.51.41.3

987654321




(TLAB ≤ 125MeV,rc|OPE = 1.8fm, rc|TPE = 1.8fm )

OPEχTPE

TLAB [MeV]

δ3P0[degrees]

120100806040200

12

10

8

6

4

2

0

OPEχTPE

TLAB [MeV]

δ3P1[degrees]

120100806040200

0−2−4−6−8−10−12−14−16−18

OPEχTPE

TLAB [MeV]

δ3P2[degrees]

120100806040200

14

12

10

8

6

4

2

0

OPEχTPE

b [fm]

〈V3P0〉[M

eV]

21.91.81.71.61.51.41.3

−0.5−1

−1.5−2

−2.5−3

−3.5−4

−4.5

OPEχTPE

b [fm]

〈V3P1〉[M

eV]

21.91.81.71.61.51.41.3

14

12

10

8

6

4

2

0OPE

χTPE

b [fm]

〈V3P2〉[M

eV]

21.91.81.71.61.51.41.3

0−0.5−1

−1.5−2

−2.5−3

−3.5−4

−4.5



CONCLUSIONS



Summary

Sampling of the NN interaction by a delta shell potential

1/√mπM . ∆r . 1/mπ

Quantitative comparison of OPE and Chiral TPE → Reduccion of Parameters

Statistical uncertainty propagation possible

δ-shell representation allows straightforward calculations

Comparing OPE and χTPE matrix elements with errors

TAKE AWAY: Before cranking the machine accuracy make sure itdoes not exceed theoretical uncertainty


Towards an Estimation of Nuclear Forces and Nuclear Matrix ...theorie.ikp.physik.tu-darmstadt.de/ect13/Talks_files/ruiz-arriola.pdf · Matrix Elements Uncertainties: Chiral vs Non-Chiral

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