the falsificacion of chiral nuclear forces · 2018. 11. 20. · e. ruiz arriola university of...

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The Falsificacion of Chiral Nuclear Forces

E. Ruiz Arriola

University of GranadaAtomic, Molecular and Nuclear Physics Department

XIIth Quark Confinement and the Hadron Spectrum, Thessaloniki (Greece)

September 1st 2016

with Rodrigo Navarro Pérez, José Enrique Amaro Soriano

E. Ruiz Arriola (UGR) The Falsification of Chiral Nuclear Forces Confinement 2016 1 / 60

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References

[1] Coarse graining Nuclear InteractionsProg. Part. Nucl. Phys. 67 (2012) 359

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

[3] Phenomenological High Precision Neutron-Proton Delta-Shell PotentialPhys.Lett. B724 (2013) 138-143.

[4] Nuclear Binding Energies and NN uncertaintiesPoS QNP 2012 (2012) 145

[5] Effective interactions in the delta-shells potentialFew Body Syst. 54 (2013) 1487.

[6] Nucleon-Nucleon Chiral Two Pion Exchange potential vs Coarse grained interactionsPoS CD12 (2013) 104.

[7] Partial Wave Analysis of Nucleon-Nucleon Scattering below pion productionPhys.Rev. C88 (2013) 024002, Phys.Rev. C88 (2013) 6, 069902.

[8] Coarse-grained potential analysis of neutron-proton and proton-proton scatteringbelow the pion production thresholdPhys.Rev. C88 (2013) 6, 064002, Phys.Rev. C91 (2015) 2, 029901.

[9] Coarse grained NN potential with Chiral Two Pion ExchangePhys.Rev. C89 (2014) 2, 024004.

[10] Error Analysis of Nuclear Matrix Elements Few Body Syst. 55 (2014) 977-981.

[11] Partial Wave Analysis of Chiral NN Interactions Few Body Syst. 55 (2014) 983-987.


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[12] Statistical error analysis for phenomenological nucleon-nucleon potentialsPhys.Rev. C89 (2014) 6, 064006.[13] Error analysis of nuclear forces and effective interactionsJ.P.G42(2015)3,034013.14] Bootstrapping the statistical uncertainties of NN scattering data Phys.Lett. B738(2014) 155-159.[15] Triton binding energy with realistic statistical uncertainties (with E. Garrido)Phys.Rev. C90 (2014) 4, 047001.[16] The Low energy structure of the Nucleon-Nucleon interaction: Statistical vsSystematic Uncertainties arXiv:1410.8097 [nucl-th] (Jour. Phys. G, in press).[17] Low energy chiral two pion exchange potential with statistical uncertaintiesPhys.Rev. C91 (2015) 5, 054002.[18] Minimally nonlocal nucleon-nucleon potentials with chiral two-pion exchangeincluding ∆ resonances (with M. Piarulli, L. Girlanda, R. Schiavilla).Phys.Rev. C91 (2015) 2, 024003.[19] The Falsification of Nuclear Forces EPJ Web Conf. 113 (2016) 04021[20] Statistical error propagation in ab initio no-core full configuration calculations oflight nuclei (with P. Maris, J.P. Vary) Phys.Rev. C92 (2015) no.6, 064003[21] Uncertainty quantification of effective nuclear interactionsInt.J.Mod.Phys. E25 (2016) no.05, 1641009[22] Binding in light nuclei: Statistical NN uncertainties vs Computational accuracy(with A. Nogga) arXiv:1604.00968[23] Precise Determination of Charge Dependent Pion-Nucleon-Nucleon CouplingConstants arXiv:1606.00592[24] Three pion nucleon coupling constantsMod.Phys.Lett.A,Vol.31,No.28(2016)1630027


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Error Analysis and Nuclear Structure

What is the predictive power of theoretical nuclear physics ?

INPUT from Experiment → CALCULATION → OUTPUT vs Experiment

Experiment much more precise than theory, but how much ?

∆Mexp < 1KeV� ∆Mth =?

Theoretical Predictive Power Flow: From light to heavy nuclei

H(A) = T + V2N + V3N + V4N + · · · → E2, E3, E4, . . .

Chiral expansion allows to compute V2N , V3N , V4N . . . systematically so that one has thehierarchy

V2N � V3N � V4N � . . .

Forces depend on the renormalization scale allowing to adjust B2, B3, . . .


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Sources of uncertainties/bias

Input data from experimentChoose the form of the potentialHow 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 [de

gree

s CM

]



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The issue of predictive power in Chiral Approach

Chiral forces are UNIVERSAL at long distances

V χ(r) = V π(r) + V 2π(r) + V 3π(r) + . . . r � rc

Chiral forces are SINGULAR at short distances

V χ(r) =a1

f2πr3

+a2

f4πr5

+a3

f6πr7

+ . . . r � rc

They trade model independence for regulator dependence

What is the best theoretical accuracy we can get within “reasonable” cut-offs ?

What is a reasonable cut-off ?rc =?


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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

Estimate uncertainties of Effective Interactions and Matrix elements

Propagate errors to A=3,4, etc.


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Tjon-Line Correlation

α-triton calculations find a linear correlation between

BΑ= 4Bt -3BdBΑ= 4Bt -3Bd

++Exp.Exp.

CD-BonnCD-Bonn

AV18AV18

NijmINijmINijmIINijmII

Vlowk HAV18LVlowk HAV18L

SRG HN3LOLSRG HN3LOL

6 7 8 9 1020

22

24

26

28

30

BtHMeVL

BΑ

HMeV

L

SRG argument with on-shell nuclear forces Timoteo, Szpigel, ERA, Few Body 2013

Bα = 4Bt − 3Bd →∂Bα

∂Bt

∣∣∣Bd

= 4


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Tjon-Lines: numerical accuracy

(with A. Nogga)∆Estattriton = 15KeV ∆E

statα = 50KeV

-7.70 -7.68 -7.66 -7.64 -7.62 -7.60

-25.0

-24.9

-24.8

-24.7

-24.6

-24.5

EH3HL HMeVL

EH4HeLHM

eVL

-9.0 -8.8 -8.6 -8.4 -8.2 -8.0 -7.8 -7.6

-30.0

-29.5

-29.0

-28.5

-28.0

-27.5

-27.0

-26.5

-26.0

EH3HL HMeVL

EH4HeLHM

eVL

4-Body forces are masked by numerical noise in the 3 and 4 body calculation if

∆numt > 1KeV ∆numt > 20KeV


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SIMPLE ESTIMATE


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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)V̂ (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)


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Skyrme parameters as EFT counterterms

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

∫d3xr2V�1 (r)


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Scale dependence

Skyrme parameters fitting at different energy ranges

AV18DS Uncoupled S-Waves

DS Coupled S-Waves

kCM [MeV]

t 0[M

eVfm

3]

380360340320300280260240220200

0−200−400−600−800−1000−1200−1400


DS Coupled S-Waves

kCM [MeV]

x0

380360340320300280260240220200

1

0.5

0

−0.5

−1


DS Coupled S-Waves

kCM [MeV]

t 1[M

eVfm

5]

380360340320300280260240220200

3000

2500

2000

1500

1000

500

0


DS Coupled S-Waves

kCM [MeV]

x1

380360340320300280260240220200

0.1

0.05

0

−0.05

−0.1

AV18DS

kCM [MeV]

t 2[M

eVfm

5]

380360340320300280260240220200

2780

2775

2770

2765

2760

2755

2750

2745

AV18DS 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

AV18DS

kCM [MeV]

t V[M

eVfm

5]

380360340320300280260240220200

140

135

130

125

120

115

110

105

AV18DS

kCM [MeV]

t U[M

eVfm

5]

380360340320300280260240220200

148014701460145014401430142014101400

AV18DS

kCM [MeV]

t T[M

eVfm

5]

380360340320300280260240220200

−3600−3800−4000−4200−4400−4600


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Simple estimate

Fermi type shape density

ρ(x) =ρ0

1 + e(r−R)/a

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

∆t0 = 10MeVfm3 → 75MeVfm3

∆B/A = 0.6MeV → 2.5MeV


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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


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The voices of wisdom

“There is no way to deduce the uncertainty of calculated nuclear ground state energies,obtained in ab initio or any other many-body method, from the variance of a set of phaseshift equivalent nucleon-nucleon (NN) interactions.The paper does not make any sense to me and is not suitable for publication in anyjournal. ”Referee I

“ There is no doubt that even within a (hypothetical) possibility to solve the nuclearmany-body problem exactly, errors in knowing the many-body Hamiltonian must carryover to properties of finite nuclei. The idea is novel and very interesting.The paper raises a very important question of how the errors related to the estimation ofnuclear forces may propagate to the calculated biding energies of nuclei”.Referee II


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Our contribution

np+pp LARGEST database analysis so far = 6713

Scrupulous statistical error analysis

6 statistically equivalent chiral or non-chiral interactions

Propagation of errors to light nuclei

Upgrade of the simple calculation

∆B/A = 2− 2.4MeV


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The sophisticated calculation

Uncertainty analysis and order-by-order optimization of chiral nuclear interactionsB.D. Carlsson, A. Ekström, C. Forssén, D.Fahlin Strömberg, G.R. Jansen, O. Lilja, M.Lindby, B.A. Mattsson, K.A. WendtPhys.Rev. X6 (2016) no.1, 011019

Take into account 2N and 3N chiral forces order by order, different regularizations andestimates of higher orders

∆B16O16

= 4MeV

Much worse than the semiempirical mass formula

Most dependence on the cut-off Λ or the regularization

Is this the predictive power of theoretical nuclear physics ?

Conservative vs realistic uncertainty estimates


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ANATOMY OF NUCLEAR FORCES


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Fundamental approach: QCD

Lattice form factor gπNN ∼ 10− 12Lattice NN potential g2πNN/(4π) = 12.1± 2.7QCD sum rules gπNN ∼ 13(1)


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Long distances

Nucleons exchange JUST one pionp

p

p

p

p

p

p

p

p

p

n

n

n

n

n

n

ππ π

ππ

n n

n n

0

0 0

+ −

g −g

g g −g −g

2 g 2 g 2 g2 g

Low energies (about pion production) 8000 pp + np scattering data (polarizations etc.)

0 50 100 150 200 250 300 3500

20

40

60

80

100

120

140

160

180

Tlab [MeV]

θ [de

grees

CM]


0 50 100 150 200 250 300 3500

20

40

60

80

100

120

140

160

180

Tlab [MeV]

θ [de

grees

CM]



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Nucleon-Nucleon Scattering

Scattering amplitude

M = a+m(σ1 · n)(σ2 · n) + (g − h)(σ1 ·m)(σ2 ·m)+ (g + h)(σ1 · l)(σ2 · l) + c(σ1 + σ2) · n

l =kf + ki

|kf + ki|m =

kf − ki|kf − ki|

n =kf ∧ ki|kf ∧ ki|

5 complex amplitudes → 24 measurable cross-sections and polarization asymmetriesPartial Wave Expansion

Msm′s,ms(θ) =

1

2ik

∑J,l′,l

√4π(2l + 1)Y l

′m′s−ms (θ, 0)

× Cl′,s,Jms−m′s,m′s,ms

il−l′(SJ,sl,l′ − δl′,l)C

l,s,J0,ms,ms

, (1)

S-matrix

SJ =

e2iδJ,1J−1 cos 2�J iei(δJ,1J−1+δJ,1J+1) sin 2�Jiei(δJ,1J−1+δ

J,1J+1

)sin 2�J e

2iδJ,1J+1 cos 2�J

, (2)E. Ruiz Arriola (UGR) The Falsification of Chiral Nuclear Forces Confinement 2016 22 / 60

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Analytical Structure

s = 4(M2N + p2)→ ELAB = 2p2/MN

Partial Wave Scattering Amplitude analytical for |p| ≤ mπ/2

TJll′ (p) ≡ SJll′ (p)− δl,l′ = pl+l′∑n

Cn,l,l′p2n

Nucleons behave as elementary (AT WHAT SCALE ?)

8000 np+pp data

Π2Π3Π4Π5Π

Ρ,Ω,Σ

NNNNΠ

-0.4 -0.2 0.0 0.2 0.4-0.4

-0.2

0.0

0.2

0.4

Re ELAB HGeVL

ImE

LA

BHG

eVL

Nucleons are heavy → Local Potentials

Vnπ(r) ∼g2n

re−nmπr

Crucial long range electromagnetic effects are localE. Ruiz Arriola (UGR) The Falsification of Chiral Nuclear Forces Confinement 2016 23 / 60

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Charge dependent One Pion Exchange

VOPE,pp(r) = f2ppVmπ0 ,OPE(r),

VOPE,np(r) = −fnnfppVmπ0,OPE(r) + (−)(T+1)2f2c Vmπ± ,OPE(r),

where Vm,OPE is given by

Vm,OPE(r) =

(m

mπ±

)2 13m [Ym(r)σ1 · σ2 + Tm(r)S1,2] ,

S1,2 = 3σ1 · r̂σ2 · r̂ − σ1 · σ2

Ym(r) =e−mr

mr

Tm(r) =e−mr

mr

[1 +

3

mr+

3

(mr)2

]


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Small short range

OPE exchange

V1π(r) = −f2πNNe−mηr

r

TPE exchange

η-exchange

Vη(r) = −f2ηNNe−mηr

r

2Π

1Π

NN 1S03.0 3.5 4.0 4.5 5.0

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

rHfmL

VHrL

HMeV

L

3.0 3.5 4.0 4.5 5.0

-0.015

-0.010

-0.005

0.000

rHfmL

VΗ

HrLHM

eVL


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Small but crucial long range

Coulomb interaction (pp) e/r

Magnetic moments ∼ µpµn/r3, µpµp/r3, µnµn/r3Lowered χ2/ν ∼ 2→ χ2/ν ∼ 2→ 1Summing 1000-2000 partial waves

Vacuum polarization (Uehling potential,Lamb-shift)

Relativistic corrections 1/r2

Coulomb

Mag.Mom.

Rel.Coul.

Vac.Pol.

10 100 1000 104 10510-27

10-22

10-17

10-12

10-7

0.01

rHfmL

Vpp

,em

HrLHM

eVL


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Effective Elementary

When are two protons interacting as point-like particles ?

Electromagnetic Form factor

Fi(q) =

∫d3reiq·rρi(r)

Electrostatic interaction

V elpp(r) = e2

∫d3r1d

3r2ρp(r1)ρp(r2)

|~r1 − ~r2 − ~r|→ e

2

rr > re ∼ 2fm

Pointlike protons

Extended protons

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

rHfmL

Vpp

,em

HrLHM

eVL


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Quark Cluster Dynamics (qcd)

Atomic analogue. Neutral atoms

Non-overlapping atoms exchange TWO photons (Van der Waals force)

Overlapping atoms are not locally neutral; ONE photon exchange is possible (Chemicalbonding)

R/2

xi

yiηi

ξiR/2

c.m.

c.m.

ClusterB

ClusterA


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Finite size effects

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

V̄ 1π+2π+...NN,NN (r) = V1πNN,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 reproduced

Finite 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

) [M

eV]

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

) [M

eV]

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

) [M

eV]

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

) [M

eV]

r [fm]

OPE no-FFOPE axial-FF



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COARSE GRAINING


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The number of parameters (for ELAB ≤ 350 MeV)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 = pmaxrc = rc/∆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 = 50E. Ruiz Arriola (UGR) The Falsification of Chiral Nuclear Forces Confinement 2016 31 / 60

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0.6 1.2 1.8 2.4 30.0

FINITE NUCLEON SIZE

QUARK EXCHANGE

ONE PION EXCHANGE

p=350 MeV

L=0

L=1

L=2

L=3

L=4

1S0,3S1

1G4,3G3,3G4,3G5,E4

POINT−LIKE NUCLEON

1P1,3P0,3P1,3P2,E1

1D2,3D1,3D2,3D3,E2

1F3,3F2,3F3,3F4,E3

TPE

(ω, ρ, σ)


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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


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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


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Delta Shell Potential

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)


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STATISTICS


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Self-consistent fits

We test the assumption

Oexpi = Othi + ξi∆Oi i = 1, . . . , NData ξi ∈ N [0, 1]

Least squares minimization p = (p1, . . . ,)

χ2(p) =N∑i=1

(Oexpi − Fi(p)

∆Oexpi

)2→ min

λiχ2(pχ2(p0) (3)

Are residuals Gaussian ?

Ri =Oexpi −Othi

∆OiOthi = Fi(p0) i = 1, . . . , N (4)

If Ri ∈ N [0, 1] self-consistent fit.Normality test for a finite sample with N elements → Probability (Confidence level)p-value

χ2min = 1± σ√

2

νν = NDat −NPar p = 1−

∫ σσdte−t

2

√2π

Histograms, Moments, Kolmogorov-Smirnov, Tail Sentitive QQ-plots


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Normality tests

Does the sequencexexp1 ≤ x

exp2 ≤ · · · ≤ x

expN ∈ N [0, 1]

We compute the theoretical points

n

N + 1=

∫ xthn−∞

dte−t

2/2

√2π

The Q-Q plot is xthn vs xexpn

For large Nxthn − xexpn = O(1/

√N)

Complete database. N = 8125

420-2-4

0.5

0.4

0.3

0.2

0.1

0

Complete database

Theoretical Quantiles N(0, 1)

EmpiricalQuantiles

420-2-4

4

2

0

-2

-4 Kolmogorov-Smirnov

Tail Sensitive

All residuals

(a)

QTh

QEm

p−

QTh

43210-1-2-3-4

3

2

1

0

-1

-2

-3

-4

-5

-6


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Granada-2013 np+pp database

Selection criterium

Mutually incompatible data. Which experiment is correct? Is any of the two correct?

Maximization of experimental consensus

Exclude data sets inconsistent with the rest of the database

1 Fit to all data (χ2/ν > 1)2 Remove data sets with improbably high or low χ2 (3σ criterion)3 Refit parameters4 Re-apply 3σ criterion to all data5 Repeat until no more data is excluded or recovered

3σ consistent data. N = 6712

Complete database. N = 8125

420-2-4

0.5

0.4

0.3

0.2

0.1

0

3σ consistent data

Theoretical Quantiles N(0, 1)

EmpiricalQuantiles

420-2-4

4

2

0

-2

-4 Kolmogorov-Smirnov

Tail Sensitive

3σ residuals

(b)

QTh

QEm

p−

QTh

43210-1-2-3-4

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2


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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


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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 r3

[µ2pSij + 2(4µp − 1)L·S

]


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To believe or not to believe

0 2000 4000 6000 8000 100000.0

0.5

1.0

1.5

2.0

Ndat

Χ2Ndat

25Σ

2Σ

1ΣGr-All

TLAB < 350 MeV

SAID-2007

GrHSelectedL

NijmHSelectedL

AV18 CD-Bonn

χ2min/ν = 1±√

2/ν


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A=2 STATISTICAL UNCERTAINTIES


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To fit or not to fit

N- Experimental Data Oexpi ±∆Oi with i = 1, . . . , NTheory with P-parameters Othi = Oi(p1, . . . , pP )

Distance between theory and data

||Oth(p)−Oexp||2 =∑i

[Othi −O

expi

∆Oi

]2

Closest theory to data (Least Squares Method, Lagrange)

D2min = minp||Oth(p)−Oexp||2

Statistical interpretation: Maximum likelihood

Propose fits with natural parameters

Reject fits with bound states other than the deuteron

Check fit residuals a posteriori

χ2min/ν = 1±√

2/ν


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Fit Phases or fit data ?

Phase shifts are NOT experimental observables UNLESS we have a complete set of 10measurements (cross sections and polarization asymmetries) for a given energy.

Data is difficult since magnetic moments are 1/r3 difficult 1000-2000 partial waves inmomentum space approaches

Fitting phases is easier but

100

101

102

103

104

0 50 100 150 200 250 300 350

χ2/N

ELAB,Max (MeV)

LO pp

LO np

NLO pp

NLO np

N2LO pp

N2LO np

Local chiral effective field theory interactions and quantum Monte Carlo applications A. Gezerlis, I. Tews, E. Epelbaum,

M. Freunek, S. Gandolfi, K. Hebeler, A. Nogga and A. Schwenk, Phys. Rev. C 90 (2014) no.5, 054323


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Are residuals irrelevant ?

Even if you have a “good” fit , it can be inconsistent

Fit N2LO for TLAB125MeVOptimized Chiral Nucleon-Nucleon Interaction at Next-to-Next-to-Leading Order; A. Ekström, G. Baardsen, C. Forssén,

G. Hagen, M. Hjorth-Jensen, G.R. Jansen, R. Machleidt, W. Nazarewicz, T. Papenbrock, J. Sarich; Phys.Rev.Lett. 110

(2013) no.19, 192502

-2

-1

0

1

2

-4 -3 -2 -1 0 1 2 3 4

QEm

p−

QTh

QTh

Tail SensitiveKolmogorov-Smirnov

residualsscaled residuals

scaled standarized residuals

Gross Systematic errors !!


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Frequentist or Bayes ?

Frequentist: What is the probability that given a theory data are correct ?

Bayesian: What is the probability that given the data the theory is correct ? (priors,random parameters)

χ2augmented → χ2data + χ2parametersBoth approaches agree for NData � NParameters. We have NDat ∼ 7000 and NPar = 40


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Maximal energy vs shortest distance

The full potential is separated into two pieces

V (r) = Vshort(r)θ(rc − r) + V χlong(r)θ(r − rc)

Data are fitted up to a maximal TLAB

TLAB ≤ maxTLAB ↔ pCM ≤ Λ

Max TLAB rc c1 c3 c4 Highest χ2/ν

MeV fm GeV−1 GeV−1 GeV−1 counterterm

350 1.8 -0.4(11) -4.7(6) 4.3(2) F 1.08350 1.2 -9.8(2) 0.3(1) 2.84(5) F 1.26125 1.8 -0.3(29) -5.8(16) 4.2(7) D 1.03125 1.2 -0.92 -3.89 4.31 P 1.70125 1.2 -14.9(6) 2.7(2) 3.51(9) P 1.05

D-waves, forbidden by Weinberg counting, are indispensable !

Data and χ-N2LO do not support rc < 1.8fm !(Several χ-potentials take rc = 0.9− 1.1fm)


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Deconstructing chiral forces

The full potential is separated into two pieces

V (r) = Vshort(r)θ(rc − r) + V χlong(r)θ(r − rc)

Under what conditions are the short distance phases compatible with zero

|δshort| ≤ ∆δ rc = 1.8fm

3G3

1251007550250

0.04

-0.28

-0.6

-0.92

-1.24

ǫ3

1251007550250

4.05

3.15

2.25

1.35

0.45

3D3

TLAB [MeV]

1251007550250

2.2

1.6

1

0.4

-0.23F3

1251007550250

0.6

0.2

-0.2

-0.6

-11F3

1251007550250

-0.1

-0.7

-1.3

-1.9

-2.5

3F21.06

0.78

0.5

0.22

-0.06ǫ2

-0.3

-0.9

-1.5

-2.1

-2.7

3P212.6

9.8

7

4.2

1.4

3D222.5

17.5

12.5

7.5

2.5

1D24.5

3.5

2.5

1.5

0.5

3D1

-1.6

-4.8

-8

-11.2

-14.4

ǫ112.6

9.8

7

4.2

1.43S1

146

118

90

62

343P1

-1.8

-5.4

-9

-12.6

-16.21P1

-1.8

-5.4

-9

-12.6

-16.2

3P0

10.4

7.2

4

0.8

-2.41S0

60

50

40

30

20

This is equivalent to set counterterms in given partial waves directly to cero

δshort = 0↔ Cshort = 0


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To count or not to count

We can fit CHIRAL forces to ANY energy and look if counterterms are compatible withzero within errors

We find that if ELAB ≤ 125MeV Weinberg counting is INCOMPATIBLE with data.

You have to promote D-wave counterterms.N2LO-Chiral TPE + N3LO-Counterterms → Residuals are normalPiarulli,Girlanda,Schiavilla,Navarro Pérez,Amaro,RA, PRC

We find that if ELAB ≤ 40MeV TPE is INVISIBLE

We find that peripheral waves predicted by 5th-order chiral perturbation theory ARE NOTconsistent with data within uncertainties

|δCh,N4LO − δPWA| > ∆δPWA,stat

Dominant contributions to the nucleon-nucleon interaction at sixth order of chiral perturbation theory D.R. Entem, N.

Kaiser, R. Machleidt, Y. Nosyk Phys.Rev. C92 (2015) no.6, 064001


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Statistical vs Systematics

Statistically equivalent interactions with χ2min/ν = 1±√

2/ν DO NOT overlapp

3I5

35025015050

-0.15

-0.45

-0.75

-1.05

-1.35

ǫ5

35025015050

3.15

2.45

1.75

1.05

0.35

3G5

TLAB (MeV)35025015050

0.2

0

-0.2

-0.4

-0.6

3H5

35025015050

-0.14

-0.42

-0.7

-0.98

-1.26

1H5

35025015050

-0.25

-0.75

-1.25

-1.75

-2.25

3H4

0.9

0.7

0.5

0.3

0.1

ǫ4

-0.2

-0.6

-1

-1.4

-1.8

3F4

3.6

2.8

2

1.2

0.4

3G4

9

7

5

3

1

1G4

2.25

1.75

1.25

0.75

0.25

3G3

-0.6

-1.8

-3

-4.2

-5.4

ǫ37.2

5.6

4

2.4

0.8

3D3

5.4

4.2

3

1.8

0.6

3F3

-0.4

-1.2

-2

-2.8

-3.6

1F3

-0.7

-2.1

-3.5

-4.9

-6.3

3F2

1.5

1.1

0.7

0.3

-0.1

ǫ2-0.35

-1.05

-1.75

-2.45

-3.15

3P2

18

14

10

6

2

3D227

21

15

9

3

1D2

δ(deg.)

10.8

8.4

6

3.6

1.2

3D1

-3

-9

-15

-21

-27ǫ1

5.4

4.2

3

1.8

0.6

3S1

144

112

80

48

16

3P1

-3.5

-10.5

-17.5

-24.5

-31.5

1P1

-3.5

-10.5

-17.5

-24.5

-31.5

3P0

11

4

-3

-10

-17

1S0

63

45

27

9

-9


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COUPLING CONSTANTS


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Arqueological Flashback

Chronological recreation of pion-nucleon coupling constants

Nijmegen determinations

f2 from NN data

f2p from pp data

Year

201020001990198019701960

0.084

0.082

0.08

0.078

0.076

0.074

0.072

0.07


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The pion-nucleon coupling constants f 2p , f20 and f

2c

(a)

f2c × 102

f2 p×

102

7.727.77.687.667.647.627.67.587.56

7.64

7.63

7.62

7.61

7.6

7.59

7.58

7.57

7.56

7.55

7.54

7.53

(b)

f2c × 102

f2 0×

102

7.727.77.687.667.647.627.67.587.56

8.2

8.15

8.1

8.05

8

7.95

7.9

7.85

7.8

(c)

f2p × 102

f2 0×

102

7.647.637.627.617.67.597.587.577.567.557.547.53

8.2

8.15

8.1

8.05

8

7.95

7.9

7.85

7.8

Fits to the Granada-2013 database.f2 f20 f

2c CD-waves χ

2pp χ

2np NDat NPar χ

2/ν

0.075 idem idem 1S0 3051 3951 6713 46 1.0510.0761(3) idem idem 1S0 3051 3951 6713 46+1 1.051- - - 1S0, P 2999 3951.40 6713 46+3 1.0430.0759(4) 0.079(1) 0.0763(6) 1S0, P 3045 3870 6713 46+3+9 1.039


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The πNN vertices

N=1000

fΠ pp- fΠ nn

fΠ pn- fΠ nnfΠ pp- fΠ pn

0.270 0.275 0.280 0.285 0.290

0.270

0.275

0.280

0.285

0.290

p

p

p

n

n

n

π π πo o+

0.2755(7) 0.276(1) 0.287(3)


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Neutrons interact more strongly than protons

neutron-neutron

proton-proton

3.0 3.5 4.0 4.5 5.0

-0.4

-0.3

-0.2

-0.1

0.0

rHfmL

VΠ

HrLHM

eVL

proton-proton

neutron-neutron

3.0 3.5 4.0 4.5 5.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

rHfmL

VΠ

HrLHM

eVL

neutron-neutron

proton-proton

3.0 3.5 4.0 4.5 5.00.0

0.1

0.2

0.3

0.4

rHfmL

VΠ

HrLHM

eVL

neutron-neutron

proton-proton

3.0 3.5 4.0 4.5 5.0

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

rHfmL

VΠ

HrLHM

eVL


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Chiral Two Pion Exchange

Fit to from Granada-2013 np+pp database TLAB < 350MeV

(a)

c1 [GeV−1]

c3

[G

eV−

1]

3210-1-2-3-4

-2.5

-3

-3.5

-4

-4.5

-5

-5.5

-6

-6.5

(b)

c1 [GeV−1]

c4

[G

eV−

1]

3210-1-2-3-4

5.5

5

4.5

4

3.5

3

(c)

c3 [GeV−1]

c4

[G

eV−

1]

-2.5-3-3.5-4-4.5-5-5.5-6-6.5

5.5

5

4.5

4

3.5

3

Fit to from Granada-2013 np+pp database TLAB < 125MeV

-10

-8

-6

-4

-2

-10 -8 -6 -4 -2 0 2 4 6 8

c3[G

eV

−1]

c1 [GeV−1]

22.53

3.54

4.55

5.56

6.5

-10 -8 -6 -4 -2 0 2 4 6 8 10

c4[G

eV

−1]

c1 [GeV−1]

22.53

3.54

4.55

5.56

6.5

-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1

c4[G

eV

−1]

c3 [GeV−1]


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CONCLUSIONS


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Conclusions

Chiral nuclear forces have MANY advantages

Weinberg’s paper has over 1000 citations

Many calculations have been undertaken using 2,3,4 body forcesBUT:

Many forms of Chiral nuclear forces can be falsified

Only validated forces against data (and not phases) allow to make serious error analysis

One should be able to make REALISTIC error estimates not just conservative ones

Cut-off dependence is the largest source of error

We must not count our chickens before they are hatched !


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the falsificacion of chiral nuclear forces · 2018. 11. 20. · e. ruiz arriola university of...

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