coarse graining nuclear and hadronic interactions · pdf filefundamental vs effective forces...

Coarse graining nuclear and hadronic interactions

E. Ruiz Arriola

Universidad de GranadaAtomic, Molecular and Nuclear Physics Department

JLAB-Theory Seminar14 November 2016

Rodrigo Navarro Perez (Livermore)Jose Enrique Amaro Soriano, Ignacio Ruiz Simo (Granada),

Jacobo Ruiz de Elvira (Bern),Pedro Fernandez Soler (Valencia)

Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 1 / 73

FUNDAMENTAL VS EFFECTIVE FORCES


Fundamental Forces

Cavendish (1798) “Experiments to determine the Density of the Earth”Coulomb (1785) “Premier memoire sur l/’electricit et le magnetisme”


Strong Forces

Yukawa (1935)

V (r) = −f2 e−mr

r

Kemmer (1938) Isospin → π0

Bethe (1940) f2 = 0.077− 0.080 (Deuteron)

0.06

0.07

0.08

0.09

0.10

1950 1955 1960 1965 1970 1975 1980

YEAR

PION-NUCLEON COUPLING CONSTANT UNTIL 1980

0.07

0.08

1980 1985 1990 1995 2000

YEAR

PION-NUCLEON COUPLING CONSTANT AFTER 1980


Fundamental Forces beween protons

Gravitational

Electric

StrongWeak

10-4 0.01 1 100 10410-40

10-32

10-24

10-16

10-8

1

rHfmL

Vp

pHr

LHM

eV

L


Fundamental Forces-Elementary particles

The particle exchange picture “explains” all interactions for elementary particles

Quarks and gluons are elementary and interact strongly

Most of the masses of hadrons comes from the interaction

Hadrons are composite. When are they effectively elementary ?

Hadrons have a size. When are they effectively point-like ?

Hadrons can be internally excited. When are they effectively

Hadrons interact via van der Waals forces


Fundamental approach: QCD

Lattice form factor gπNN ∼ 10− 12

Lattice NN potential g2πNN/(4π) = 12.1± 2.7

QCD sum rules gπNN ∼ 13(1)


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.)Granada coarse grained analysis (2016) (isospin breaking !!)

g2p/(4π) = 13.72(7) 6= g2

n/(4π) = 14.91(39) 6= g2c/(4π) = 13.81(11)

0 50 100 150 200 250 300 3500

20

40

60

80

100

120

140

160

180

Tlab [MeV]

θ [de

grees

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

grees

CM]

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


Effective Field Theory Paradigm

Low energy physics (long wavelength λ re, a

Effective elementarity → ϕ(x) (effective field)

σ(x) = Zq(x)q(x) + Z2λ2qD2q + Z3λ

3[q(x)q(x)]2 + Z4λ

6[q(x)Dµq(x)]2 + . . .

Effective interactions

L =1

2

[(∂σ)2 −m2σ2

]+ C0λ

−2σ4 + C2λ−4σ2(∂σ)2 + . . .

Loops are suppressed pλ 1 → Power counting


EFT (polynomial power counting of potential)

Scattering problem: Lippmann-Schwinger equation

Tl(p′, p) = Vl(p

′, p) +2

π

∫ ∞0

dqVl(p′, q)

q2

k2 − q2Tl(q, p)

= V Λl (p′, p) +

2

π

∫ Λ

0dqV Λ

l (p′, q)q2

k2 − q2Tl(q, p)

In the partial wave amplitude we have a left-hand branch cut due to particle π exchange

|p|, |p′| < mπ/2 (1π) |p|, |p′| < mπ (2π) |p|, |p′| < 3mπ/2 (3π)

Low momentum expansion

Vl(p′, p) = p′lpl

[C0(mπ) + C2(mπ)(p2 + p′2) + . . .

]C0(m2

π) = c0 + c′0m2π + . . .

Count powers of Q = p, p′,mπ and use Λχ = 4πfπ ∼MN but keep Λ ∼ nmπ/2 forn− π exchange

Full chiral potential to O(Qn/Λnχ)

Vχ(p′, p)− V1π(p′p)− V2π(p′, p)− · · · = p′lpl[c0 + c2(p2 + p′2) + c′0m

2π + . . .

]Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 10 / 73

Fitting EFT

We do not know the EFT constants → fitting NDat data with NPar parameters

Oexpi = Oi(c1, c2, . . . )±∆Oexp

i ? i = 1, . . . , NDat

Standard χ2 minimization

χ2min = min

c1,c2,...χ2(c1, c2, . . . ) =

NDat∑i=1

[Oi(c1, c2, . . . )−Oexpi

∆Oexpi

]2

Probabilistic answer: p-value

p = 0.68 χ2min/ν = 1±

√2/ν ν = NPar −NDat

Confidence interval for parameters

ci = cFiti ±∆ci ∆χ2 = 1

What happens when there are incompatible data ?

TRUNCATED EFT (with ∆Oth) → Validate/falsify data vs validate/falsify EFT

Do we need NPar →∞ when ∆Oexp → 0 or NDat →∞ ?


Coarse graining approach

Particle exchange leads to small forces at long distances rmπ 1→ Perturbative definition of QM potential stemming from QFT

fBornQFT (θ, E) = −2µ

4π

∫d3xe−i

~k′·~xVQFT(~x, ~p)ei~k·~x ,

What is the elementarity radius re?

V (r) = VQFT(r) r > re

What is the maximum CM momentum pCM,max ?

∆r = ~/pCM,max → rn = n∆r lmax = pre

Centrifugal barrier

p2 >l(l + 1)

r2→ pr < l + 1/2

Use V (rn) as fitting parameters


Coarse graining approach

The complete fitting potential (delta-shells)

V (r) =

∑i

∆rV (ri)δ(r − ri)

︸︷︷︸Short(Coarse grained)

θ(re − r) + VQFT(r)︸︷︷︸Long(Particle exchange)

θ(r − re)

The number of fitting parameters

NPar =∑n,l

∼ 1

2(pmaxrc)

2 × spin, isospin

Compare to data with χ2. Check p-value

Confidence intervalV (rn)|Fit ±∆V (rn)

If NDat →∞ we expect NPar-FIXED but ∆V (rn)→ 0

THE NUMBER OF PARAMETERS IS ALWAYS THE SAME: WE CAN USE COARSEGRAINING TO FIT AND SELECT DATA


COARSE GRAINING ππ INTERACTIONS

J. Ruiz de Elvira and E.R.A.


Effective elementarity

Electromagnetic Pion form factors

〈π+(p′)|Jµ(0)|π+(p)〉 = (p′µ + pµ)Fem(q)

Vector meson dominance (VMD)

FV (q) =∑

V=ρ,ρ′,...

cVM2V

M2V + q2

≈m2ρ

m2ρ + q2

Electromagnetic Coulomb interaction ∆mπ |em = Vem(0) = 3MeV

Vem(r) =

∫d3q

(2π)3|Fem(q2)|2ei~q·~r =

1

r−e−mρr

[1

2mρ +

1

r

]∼ 1

rr > re ∼ 1−1.5fm

0 2 4 6 8 10 120.0

0.2

0.4

0.6

0.8

Q2 @GeV2D

Q2 F

ΠΠ

HQ2 L

@GeV

2 D

Pointlike pions

Extended pions

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

r@fmD

Vem

@rDHM

eVL


Effective elementarity

Gravitational Pion form factor

〈πb(p′) | Θµν(0) | πa(p)〉 =1

2δab[(gµνq2 − qµqν)Θ1(q2) + 4PµP νΘ2(q2)

]′µ)

Tensor meson dominance (TMD)

Θ1(q) =∑

T=f2,f′2,...

cTM2T

M2T + q2

≈m2f

m2f + q2

Gravitational Coulomb interaction

Vgrav(r) =

∫d3q

(2π)3|Θ(q2)|2ei~q·~r =

1

r−e−mf r

[1

2mf +

1

r

]∼ 1

rr > re ∼ 1−1.2fm

0 1 2 3 4 50.0

0.1

0.2

0.3

0.4

0.5

0.6

-t @GeV2D

H-tLA 2

0HtL

@GeV

2 D

Pointlike pions

Extended pions

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

r@fmD

Vem

@rDHM

eVL


ππ Phase shifts

Coarse graining

Vππ(r) =

∑i

V (ri)δ(r − ri)θ[re − r] + VQFT (r)θ[r − re]

Energy independent fits with N = 4, 5, 6 delta-shells

∆r = 0.3fm

0

40

80

120

160

300 400 500 600 700 800 900

â√s (MeV)

δ00(s)

δ004sh

δ005sh

δ006sh

0

40

80

120

160

400 600 800 1000 1200 1400

â√s (MeV)

δ11(s)

δ114sh

δ114sh

δ114sh

-40

-35

-30

-25

-20

-15

-10

-5

0

400 600 800 1000 1200 1400

â√s (MeV)

δ20(s)

δ204sh

δ205sh

δ206sh


ππ Phase shifts

Inelastic Coarse graining. Where does it reside the inelasticity ?

ηI(b) ≡ ηl(s) l +1

2= bp ηI(b) = 0 b > 0.5fm→ V (r1, s), V (r2), . . .

Energy Dependent fits

0

40

80

120

160

200

240

280

320

400 600 800 1000 1200 1400

â√s (MeV)

δ00(s)

δ004shc

δ005shc

δ006shc

0

40

80

120

160

400 600 800 1000 1200 1400

â√s (MeV)

δ11(s)

δ114shc

δ114shc

δ114shc

-40

-35

-30

-25

-20

-15

-10

-5

0

400 600 800 1000 1200 1400

â√s (MeV)

δ20(s)

δ204shc

δ205shc

δ206shc

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

400 600 800 1000 1200 1400

â√s (MeV)

η00(s)

η004shc

η005shc

η006shc

0.88

0.9

0.92

0.94

0.96

0.98

1

400 600 800 1000 1200 1400

â√s (MeV)

η11(s)

η114shc

η114shc

η114shc

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

400 600 800 1000 1200 1400

â√s (MeV)

η20(s)

η204shc

η205shc

η206shc


Analyticity

Dispersion relations for the optical potential

ReU(r, s) = U(r) +1

π

∫ ∞4m2

π

ds′ImU(r, s)

s− s′ r = r1

-1

0

1

2

3

4

5

6

7

400 600 800 1000 1200 1400

â√s (MeV)

Im U00(s)

Im U004shc

Im U005shc

Im U006shc

0

2

4

6

8

10

12

14

400 600 800 1000 1200 1400

â√s (MeV)

Im U11(s)

Im U114shc

Im U116shc

Im U115shc

-0.5

0

0.5

1

1.5

2

2.5

400 600 800 1000 1200 1400

â√s (MeV)

Im U20(s)

Im U204shc

Im U205shc

Im U206shc

0

2

4

6

8

10

12

14

400 600 800 1000 1200 1400

â√s (MeV)

Re U00(s)

Re U004shc

Re U005shc

Re U006shc

0

5

10

15

20

25

400 600 800 1000 1200 1400

â√s (MeV)

Re U11(s)

Re U114shc

Re U116shc

Re U115shc

15

16

17

18

19

20

21

22

400 600 800 1000 1200 1400

â√s (MeV)

Re U20(s)

Re U204shc

Re U205shc

Re U206shc

Possibility to constrain fits (under consideration)


COARSE GRAINING NN BELOW PIONPRODUCTION


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 asymmetries

Partial 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 =

e2iδ

J,1J−1 cos 2εJ ie

i(δJ,1J−1

+δJ,1J+1

)sin 2εJ

iei(δJ,1J−1

+δJ,1J+1

)sin 2εJ e

2iδJ,1J+1 cos 2εJ

, (2)


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 localEnrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 22 / 73

Charge dependent One Pion Exchange

VOPE,pp(r) = f2ppVmπ0 ,OPE(r),

VOPE,np(r) = −fnnfppVmπ0 ,OPE(r) + (−)(T+1)2f2

c Vmπ± ,OPE(r),

where Vm,OPE is given by

Vm,OPE(r) =

(m

mπ±

)2 1

3m [Ym(r)σ1 · σ2 + Tm(r)S1,2] ,

S1,2 = 3σ1 · rσ2 · r − σ1 · σ2

Ym(r) =e−mr

mr

Tm(r) =e−mr

mr

[1 +

3

mr+

3

(mr)2

]


Small short range

OPE exchange

V1π(r) = −f2πNN

e−mηr

r

TPE exchange

η-exchange

Vη(r) = −f2ηNN

e−mηr

r

2Π

1Π

NN 1S0

3.0 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


Small but crucial long range

Coulomb interaction (pp) e/r

Magnetic moments ∼ µpµn/r3, µpµp/r3, µnµn/r3

Lowered 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


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|→ e2

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


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


Finite size effects

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 reproduced

Finite size effects set in at 2fm → exchange quark effects become explicit

High 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



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 = 50


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)


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 vs Fine 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


Delta Shell Potential

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)


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

]


STATISTICS


Self-consistent fits

We test the assumption

Oexpi = Oth

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

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

χ2(p) =N∑i=1

(Oexpi − Fi(p)

∆Oexpi

)2

→ minλi

χ2(pχ2(p0) (3)

Are residuals Gaussian ?

Ri =Oexpi −Oth

i

∆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−

∫ σ

σdte−t

2

√2π

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


Normality tests

Does the sequencexexp

1 ≤ xexp2 ≤ · · · ≤ xexp

N ∈ N [0, 1]

We compute the theoretical points

n

N + 1=

∫ xthn

−∞dte−t

2/2

√2π

The Q-Q plot is xthn vs xexp

n

For large Nxthn − xexp

n = 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


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


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

Charge dependence in OPE

Magnetic-Moments, Vacuum polarization, ...


Correlations

The strengths of the coarse grained potential are largely independent !!→ Good Fitting Parameters

1S0

1S0

3P0

1P1

3P1

3S1

ε1

3D1

1D2

3D2

3P2

ε2

3F2

1F3

3D3

1S01S0

3P01P1

3P13S1 ε1

3D11D2

3D23P2 ε2

3F21F3

3D3

OPE δ-shells

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 S 0

1 S 0

3 P 0

1 P 13 P 1

3 S 1

ε 1

3 D 11 D 2

3 D 2

3 P 2

ε 2

3 F 2

1 F 33 D 3

c i

1S0

1S0

3P0

1P1

3P1

3S1

ε1

3D1

1D2

3D2

3P2

ε2

3F2

1F3

3D3 c

i

TPE δ-shells

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


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


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 0.8846rm(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


MORE STATISTICS


To fit or not to fit

N- Experimental Data Oexpi ±∆Oi with i = 1, . . . , N

Theory 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


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


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. Ekstrom, G. Baardsen, C. Forssen,

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 !!


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 → χ2

data + χ2parameters

Both approaches agree for NData NParameters. We have NDat ∼ 7000 and NPar = 40


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/νMeV 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)


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


To count or not to count: The Falsification of Chiral Forces

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 Perez,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


COUPLING CONSTANTS


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

0 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 f2

0 f2c CD-waves χ2

pp χ2np NDat NPar χ2/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


Neutron-Neutron vs Proton-Proton (Polarized)

nn interaction is more intense than pp interaction

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


Arqueological Flashback

Can we witness isospin breaking in the couplings ?

dg

g

∣∣∣QCD

= O(α,mu −md

ΛQCD) = O(α,

Mn −Mp

ΛQCD) ∼ 0.01− 0.02

Statistically yes ! Granada NN N = 6713

dg

g

∣∣∣stat

= O(∆NDat

NDat) = O(

1√NDat

)→ N ∼ 7000− 10000

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


Chiral Two Pion Exchange from Granada-2013 np+ppdatabase

(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


fπN∆ from Granada-2013 np+pp database

π

N

N∆

N

π

N

N∆

∆

NN potential in the Born-Oppenheimer approximation

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 reproduced

Fit with re = 1.8fm to N = 6713pp+ np scattering data

fπN∆/fπNN = 2.178(14) χ2/ν = 1.12→ hA = 1.397(9)


COARSE GRAINING SHORT RANGECORRELATIONS

I. Ruiz Simo, R. Navarro Perez, J. E. Amaro and E.R.A.


Coarse graining vs fine graining

For S-wave and E = 0 we have the scattering problem

uk(r) = sin(kr) +2

π

N∑i=1

λi uk(ri)

∫ ∞0

dqsin(qr) sin(qri)

k2 − q2

→ A sin(kr + δ)

SAIDCoarse grained AV18

7 deltas6 deltas5 deltas4 deltas

TLAB [MeV]

δ N[degrees]

1S0 np

2000150010005000

80

60

40

20

0

-20

-40

-60

-80


Bethe-Goldstone equation

Independent pair approximation (Effective interaction G-matrix )

G = V +Q

E −H0G→ |Ψ〉 = |ϕ〉+

Q

E −H0|Ψ〉

For S-wave and E = 0 we have a N ×N matrix equation

uk(rn) = sin(krn) +2

π

N∑i=1

λi uk(ri)

∫ ∞kF

dqsin(qrn) sin(qri)

k2 − q2

In momentum space

Φk(p) =1

p

√4π

(2π)3

π

2δ(p− k) +

θ(p− kF )

k2 − p2

N∑i=1

λi uk(ri) sin(pri)


High momentum states

6 deltas are enough for p ≤ 1GeV!

Coarse grained AV187 deltas6 deltas5 deltas4 deltas

|Φk(p)|2

[fm

4]

1S0 np; k=40 MeV/c

1000900800700600500400300200

1e-05

1e-06

1e-07

1e-08

1e-09

1e-10

1e-11

1e-12

p [MeV/c]

|Φk(p)|2

[fm

4]

k=140 MeV/c

1000900800700600500400300200

0.0001

1e-05

1e-06

1e-07

1e-08

1e-09

1e-10

1e-11


REPULSIVE CORE VS STRUCTURAL CORE

P. Fernandez Soler and E.R.A.


The repulsive core: pros and cons

Flat differential pp cross section at ∼ 200MeV Jastrow (1950)

Stability of high density states

Lattice QCD provides a core

When V (r) > mπ pions should be produced


Coarse graining with square well potentials

Inverse scattering problem ambiguities for ≤ 350MeV

Attractive and Repulsive solutions. Which is the right one ?

0 1 2 3 4 5

020

040

060

080

0

r (fm)

V(r)

(M

eV)

0 1 2 3 4 5

020

060

010

00

r (fm)

V(r)

(M

eV)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

020

040

060

080

0

r (fm)

V(r)

MeV

0.0 0.5 1.0 1.5 2.0 2.5 3.0

−70

0−

500

−30

0−

100

0

r (fm)

V(r)

(M

eV)


Coarse graining with square well potentials

Inverse scattering problem ambiguities for ≤ 1GeV


0 200 400 600 800 1000

−60

−40

−20

020

4060

Tlab (MeV)

δ 0 (

grad

os)

δ0 pozosδ0 deltasδ0 datosδ0 CNS sol.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

010

0030

0050

00

r (fm)

V(r)

MeV

0.0 0.5 1.0 1.5 2.0 2.5 3.0

−14

00−

1000

−60

0−

200

0

r (fm)

V(r)

(M

eV)


Coarse graining optical potential

Inverse scattering problem ambiguities for ≤ 3GeV


Make V1 complex and energy dependent from data

0 500 1000 1500 2000 2500 3000

−50

050

100

Tlab (MeV)

δ 0 (

grad

os)

δ0 pozosδ0 deltasδ0 datosδ0 CNS sol.

300 400 500 600 700 800 900 1000

−50

050

100

Tlab (MeV)

δ 0 (

grad

os)

δ0 atractivoδ0 repulsivo

0 500 1000 1500 2000 2500 3000

−40

000

2000

4000

6000

Tlab (MeV)

V1

(MeV

)

Re RIm RReIm

The hard core is assumed to disappear with increasing energy and to be replaced byabsorption. (G. E. Brown, 1958)

There was never a hard core → smooth behaviour and adiabatic onset of inelasticity


UNCERTAINTIES IN A=2,3,4


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


Tjon-Lines: numerical accuracy

(with A. Nogga)∆Estat

triton = 15KeV ∆Estatα = 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 ∆num

t > 20KeV


Simple and sophisticated calculations

Error estimates on Nuclear Binding Energies from Nucleon-Nucleon uncertaintiesR.Navarro Perez,J. E. Amaro and r. Ruiz Arriola arXiv:1202.6624 [nucl-th].

∆B/A = 0.5MeV→ 2MeV

Uncertainty analysis and order-by-order optimization of chiral nuclear interactionsB.D. Carlsson, A. Ekstrom, C. Forssen, D.Fahlin Stromberg, 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

∆B16O

16= 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


CONCLUSIONS


Conclusions

Coarse graining is a simple method to analyze and select data using a statistical framework

Self-consistent databases allow to determine coupling constants

Validate power countings (Weinberg is not)

It could be a possible way to do Nuclear Physics


coarse graining nuclear and hadronic interactions · pdf filefundamental vs effective forces...

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