Coarse graining nuclear and hadronic interactions · PDF fileFUNDAMENTAL VS EFFECTIVE FORCES...
Transcript of 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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 2 / 73
Fundamental Forces
Cavendish (1798) “Experiments to determine the Density of the Earth”Coulomb (1785) “Premier memoire sur l/’electricit et le magnetisme”
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 3 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 4 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 5 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 6 / 73
Fundamental approach: QCD
Lattice form factor gπNN ∼ 10− 12
Lattice NN potential g2πNN/(4π) = 12.1± 2.7
QCD sum rules gπNN ∼ 13(1)
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 7 / 73
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
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Tlab [MeV]
θ [de
grees
CM]
NN-OnLine http://nn-online.org 7 June 2013
0 50 100 150 200 250 300 3500
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120
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Tlab [MeV]
θ [de
grees
CM]
NN-OnLine http://nn-online.org 7 June 2013
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 8 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 9 / 73
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 →∞ ?
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 11 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 12 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 13 / 73
COARSE GRAINING ππ INTERACTIONS
J. Ruiz de Elvira and E.R.A.
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 14 / 73
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
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3.5
4.0
r@fmD
Vem
@rDHM
eVL
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 15 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 16 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 17 / 73
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
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400 600 800 1000 1200 1400
â√s (MeV)
δ00(s)
δ004shc
δ005shc
δ006shc
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400 600 800 1000 1200 1400
â√s (MeV)
δ11(s)
δ114shc
δ114shc
δ114shc
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-25
-20
-15
-10
-5
0
400 600 800 1000 1200 1400
â√s (MeV)
δ20(s)
δ204shc
δ205shc
δ206shc
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0.9
1
400 600 800 1000 1200 1400
â√s (MeV)
η00(s)
η004shc
η005shc
η006shc
0.88
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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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 18 / 73
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
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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)
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 19 / 73
COARSE GRAINING NN BELOW PIONPRODUCTION
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 20 / 73
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)
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 21 / 73
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
]
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 23 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 24 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 25 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 26 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 27 / 73
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]
BO-TPE no-FFBO-TPE axial-FF
-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]
BO-TPE no-FFBO-TPE axial-FF
-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]
BO-TPE no-FFBO-TPE axial-FF
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
BO-TPE no-FFBO-TPE 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
BO-TPE no-FFBO-TPE axial-FF
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 28 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 29 / 73
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)
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 30 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 31 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 32 / 73
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)
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 33 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 34 / 73
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
]
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 35 / 73
STATISTICS
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 36 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 37 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 38 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 39 / 73
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, ...
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 40 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 41 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 42 / 73
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]
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 43 / 73
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]
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 44 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 45 / 73
MORE STATISTICS
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 46 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 47 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 48 / 73
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 !!
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 49 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 50 / 73
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)
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 51 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 52 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 53 / 73
COUPLING CONSTANTS
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 54 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 55 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 56 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 57 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 58 / 73
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)
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 59 / 73
COARSE GRAINING SHORT RANGECORRELATIONS
I. Ruiz Simo, R. Navarro Perez, J. E. Amaro and E.R.A.
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 60 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 61 / 73
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)
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 62 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 63 / 73
REPULSIVE CORE VS STRUCTURAL CORE
P. Fernandez Soler and E.R.A.
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 64 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 65 / 73
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)
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 66 / 73
Coarse graining with square well potentials
Inverse scattering problem ambiguities for ≤ 1GeV
Attractive and Repulsive solutions. Which is the right one ?
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)
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 67 / 73
Coarse graining optical potential
Inverse scattering problem ambiguities for ≤ 3GeV
Attractive and Repulsive solutions. Which is the right one ?
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 68 / 73
UNCERTAINTIES IN A=2,3,4
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 69 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 70 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 71 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 72 / 73
CONCLUSIONS
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 73 / 73
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
Enrique Ruiz Arriola (UGR) Coarse graining nuclear and hadronic interactions 14 November 2016 74 / 73