Partial Wave Analysis of Nucleon-Nucleon Scattering below ...Enrique Ruiz Arriola Jos e Enrique Amaro Soriano University of Granada Atomic, Molecular and Nuclear Physics Department

Motivation Delta Shell Potential Fitting NN observables Calculations Chiral TPE Summary

Partial Wave Analysis of Nucleon-Nucleon Scatteringbelow pion production threshold

Rodrigo Navarro PérezEnrique Ruiz Arriola

José Enrique Amaro Soriano

University of GranadaAtomic, Molecular and Nuclear Physics Department

Raimondo Anni Nuclear Physics SchoolOtranto, May 2013

Navarro-Pérez R. (UGR) PWA of NN scattering below 350MeV Otranto, May 2013 1 / 19


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 NNinteraction

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 becomecomplicated

No statistical uncertainties estimates



Motivation

Effective coarse graining

Oscillator Shell ModelEuclidean Lattice EFTVlowk interaction

Characteristic distance ∼ 0.5− 1.0 fmNyquist 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



Delta Shell Potential

A sum of delta functions

V (r) =∑i

λi2µδ(r − ri)

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

Optimal and minimal sampling of the nuclear interactionPion production threshold ∆k ∼ 2 fm−1Optimal 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 fmShort range interactionNo delta shell (No repulsive core)

Intermediate region 0.5 ≤ r ≤ 3.0 fmUnknown interactionλi parameters fitted to scattering data

Outermost region r ≥ 3.0 fmLong 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)



Fitting NN observables

Database of NN scattering data obtainedtill 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 removepossible 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 parametersFixed and equidistant concentration radii ∆r = 0.6 fmEM 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 r3

[µ2pSij + 2(4µp − 1)L·S

]Navarro-Pérez R. (UGR) PWA of NN scattering below 350MeV Otranto, May 2013 10 / 19


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 uncertaintypropagated directly fromcovariance matrix

ǫ1

TLAB [MeV]

Phase

shift[deg]

350300250200150100500

6

5

4

3

2

1

0



Wolfenstein Parameters

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]



Wolfenstein Parameters

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 0.8846

rm(fm) 1.9645(9) 1.953(3) 1.9666 1.9675 1.9686 1.967 1.966QD(fm

2) 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 χ

2/ν NData Np χ2/ν

OPE 5766 31 1.10 6363 39 1.09 6438 46 1.06TPE(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 elseχTPE most of that something else



Summary

Sampling of the NN interaction by a delta shell potential

1/√mπM . ∆r . 1/mπ

3 well defined regions

Fit to NN scattering data

Good description of scattering observables (over 6400 data)

Statistical uncertainty propagation possibleδ-shell representation allows straightforward calculations

Separable in momentum spaceFinite nuclei Binding EnergyPhaseshiftsScattering amplitudesDeuteron properties and form factors

Comparing OPE and χTPEχTPE reduces number of parametersLess 3σ compatible data


MotivationDelta Shell PotentialFitting NN observablesCalculationsChiral TPESummary

Documents

Partial Wave Analysis of Nucleon-Nucleon Scattering below ...Enrique Ruiz Arriola Jos e Enrique Amaro Soriano University of Granada Atomic, Molecular and Nuclear Physics Department