Parity violating asymmetry,dipole polarizability, and theneutron skin thicknessin 48Ca and 208Pb
Xavier Roca-MazaUniversita degli Studi di Milano and INFNVia Celoria 16, I-20133, Milano (Italy)Calcium Radius Experiment workshop, 17-19 March 2013,
Newport News, USA.
Table of contents:
Motivation
Parity violating elastic electron scattering: Single angle measurement of Apv in 48Ca and 208Pb within the
Distorted Wave Born Approximation based on modernmean-field nucleon distributions
Uncertainty due to strange quark contributions on the weakneutral current nucleon form factors
Important effects on Apv for the case of the lighter 48Ca:spin-orbit and three-neutron forces
Isovector static dipole polarizability αD : Definition Hartree-Fock + Random Phase Approximation results for the
case of 48Ca and 208Pb
Conclusions
Motivation:
The importance of determining isovector properties in nuclei
In the past (and also in the present), neutron properties instable medium and heavy nuclei have been mainly measuredby using strongly interacting probes.
⇓Limited knowledge of isovector properties
At present, the use of rare ion beams has opened the possibility of
measuring properties of exotic nuclei parity violating elastic electron scattering (PVES), a
model independent technique, has allowed to estimate theneutron radius of a stable heavy nucleus like 208Pb
⇓Promising perspectives for the near future
Motivation:
It is possible to connect observables with general isovectorproperties of the nuclear effective interaction?
Example:Mean-Fieldpredictions show a clearcorrelation between∆rnp of a medium andheavy nucleus and thedensity slope of thesymmetry energy(L = 3ρ0∂ρS(ρ)|ρ0 =3ρ0p0).
R.J. Furnstahl, NPA, 706, 85 (2002)
Motivation:
More generally within MF, it has been found a semi-empiricallaw: asym(A) ≈ S(ρA) with ρA = ρ0 − ρ0/(1 + cA1/3) ⇒direct and clearconnection of any groundstate isospin sensitiveobservable with theparameters of the EoS.Following the sameexample: ∆r totalnp (A, I ) =
∆rbulknp (A, I ) +
∆r surfacenp (A, I )
MSk7
D1SSk-T6
SkM*
DD-ME2FSUGold Ska Sk-Rs
Sk-T4G2 NLC
NL3*
NL-ZNL1
0 50 100 150L (MeV)
0
0.1
0.2
0.3
∆rnp
of
208 P
b (
fm)
HFB-8
SGIIHFB-17
SLy4
SkSM*SkMP
NL-SH
D1N
TM1
NL3
NL-RA1
Total fit: r=0.992, slope=1.6 fm/GeVBulk fit: r=0.993, slope=1.4 fm/GeVSurface fit: r=0.602, slope=0.2 fm/GeV
∆rbulknp (A, I ) ≈ 2r03J
L(
1− ǫAKsym
2L
)
ǫAA1/3
(
I − IC)
M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda, Phys. Rev. Lett. 102, 122502 (2009); Phys. Rev. C 80
024316 (2009); Phys. Rev. C 81 054309 (2010) and Phys. Rev. C 82, 054314 (2010)
Motivation:Observables, processes and observations known to becorrelated with the isovector properties of the nucleareffective interaction
Binding energies
Neutron distributions (proton elastic scattering, antiprotonicatoms, parity violating asymmetry,...)
Giant Resonances: Giant Dipole, Gamow-Teller, IsobaricAnalog, Spin Dipole and Anti-analog of the Giant DipoleResonances (inelastic hadron-nucleus, nucleus-nucleus andγ-nucleus scattering).
Heavy Ion Collisions (EoS — transport models)
Neutron Star properties: mass-radius relation, transitiondensity crust-core, composition,... (observational data).
Low-energy dipole response (?)
Isovector GQR [see PRC 87, 034301 (2013)!]
Isoscalar Giant Resonances along isotopic chains (?)
...
Parity violating elastic electron scattering in48Ca and 208Pb
From previous talks, we have seen that,
Electrons interact by exchanging a γ or a Z0 boson.
While protons couple basically to γ, neutrons do it to Z0.
Ultra-relativistic electrons, depending on their helicity,interact with the nucleons V± = VCoulomb ± VWeak.
Ultra-relativistic electrons moving under the effect of V±
where Coulomb distortions are important ⇒ solution of theDirac equation via the Distorted Wave Born Approximation(DWBA).
Input for the calculation: ρn and ρp ... and nucleon formfactors for the e-m and the weak neutral current...
Refs: C. J. Horowitz, Phys. Rev. C 57 3430 (1998); C. J. Horowitz, S. J. Pollock, P. A. Souder, and R. Michaels,
Phys. Rev. C 63, 025501 (2001); M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda, Phys. Rev. C 82, 054314
(2010); X. Roca-Maza, M. Centelles, X. Vinas, and M. Warda, Phys. Rev. Lett. 106 252501 (2011) and (for the
electric proton and neutron form factors) J. Friedrich and Th. Walcher, Eur. Phys. J. A 17, 607623 (2003)
PREx and CREx measure: model-independently the parityviolating asymmetry,
Apv =dσ+
dΩ − dσ−
dΩdσ+
dΩ + dσ−
dΩ
at 1.06 GeV and for a single angle (∼ 5 deg.) in 208Pb and at 2.20GeV and for a single angle (∼ 4 deg.) in 48Ca
ρn of 208Pb and 48Ca are the quantities to be determined, aprecise determination of ∆rnp would constrain the densitydependence of the symmetry energy around saturation.
Qualitatively,
Apv within the Plane Wave Born Approximation,
Apv =GFq
2
4πα√2
[
4 sin2 θW +Fn(q)− Fp(q)
Fp(q)
]
... which depends on Fn(q)− Fp(q). For q → 0, it isapproximately,
−q2
6
(
〈r2n 〉 − 〈r2p 〉)
= −q2
6
[
∆rnp(〈r2n 〉1/2 + 〈r2p 〉1/2)]
= −q2
6
(
2〈r2p 〉1/2∆rnp +∆r2np
)
variation of Apv at a fixed q dominated by the variation of∆rnp. Fp(q) well fixed by experiment
208Pb: directcorrelationsDWBA; no radiativecorrections or strangequark effects included
X. Roca-Maza, M. Centelles, X. Vinas, and
M. Warda, Phys. Rev. Lett. 106 252501
(2011)
v090M
Sk7H
FB-8
SkP
HFB
-17
SkM*
DD
-ME2
DD
-ME1
FSUG
oldD
D-PC
1Ska
PK1.s24 Sk-R
sN
L3.s25Sk-T4G
2
NL-SV2PK
1N
L3N
L3*
NL2
NL1
0 50 100 150 L (MeV)
0.1
0.15
0.2
0.25
0.3
∆rnp
(fm
)
Linear Fit, r = 0.979Mean Field
D1S
D1N
SG
II
Sk-T6
SkX S
Ly5
SLy4
MS
kAM
SL0
SIV
SkS
M*
SkM
P
SkI2S
V
G1
TM1
NL-S
HN
L-RA
1
PC
-F1
BC
P
RH
F-PK
O3
Sk-G
s
RH
F-PK
A1
PC
-PK
1
SkI5
v090HFB-8
D1S
D1N
SkXSLy5
MSkA
MSL0
DD-ME2
DD-ME1
Sk-Rs
RHF-PKA1
SVSkI2
Sk-T4
NL3.s25
G2SkI5
PK1
NL3*
NL2
0.1 0.15 0.2 0.25 0.3∆ r
np (fm)
6.8
7.0
7.2
7.4
107
Apv
Linear Fit, r = 0.995Mean Field From strong probes
MSk7
HFB-17SkP
SLy4
SkM*
SkSM*SIV
SkMP
Ska
Sk-Gs
PK1.s24
NL-SV2NL-SH
NL-RA1
TM1, NL3
NL1
SGIISk-T6
FSUGoldZenihiro
[6]
Klos [8]
Hoffmann [4
]
PC-F1
BCP
PC-PK1
RHF-PKO3
DD-PC1
G1
MF correlations allows todetermine ∆rnp and L
without direct assumptionson ρ, PREx-II andPV-RAPTOR expectedaccuracy → constrain on L
Different experiments onproton elastic scatteringand antirpotonic atomsagrees with the correlation
48Ca: direct correlations within MF includingradiative corrections and strange quark effects
DD
-ME
1
DD
-ME
2
DD
-ME
δ
FS
UG
old
G2
LNS
NL1
SG
IIS
III
SK
255
SK
I2
SLy
4
TM
1
0.15 0.2 0.25∆ r
np (fm)
2
2.1
2.2
2.3
2.4
Apv
(pp
m)
EDFs Gs
E(Q) = 0
r=0.98
EDFs Gs
E(Q) = −0.006±0.016 Q
2/(0.1 GeV
2)
48Ca @2.2 GeV, 4 deg.
SkM
*S
Ly5
SkI
3
PC
-F1
weightedaverage∆r
np(Exp.)
NL3
Apv decreases by around 0.005 ppm with an error of about 0.01 -0.02 ppm when G s
E (Q2) is included.
Used G sE (Q
2) from PRC 76, 025202 (2007) by Liu, McKeown, and Ramsey-Musolf Average ∆rnp from
hadronic probes: PRC12, 778 1978; PRL87, 08250113, 343 (2004); Phys. Rev. 174, 1380 (1968); Physics
Letters 57B 47 (1975); PRC 67, 054605 (2003) and PRC33 1624 (1986).
48Ca: estimation of spin-orbit effects
FS
UG
old
NL3
DD
-ME
1
DD
-ME
2
DD
-ME
δ
FS
UG
old
G2
LNS
NL1
SG
IIS
III
SK
255
SK
I2
SLy
4
TM
1
0.15 0.2 0.25∆ r
np (fm)
2
2.1
2.2
2.3
2.4
Apv
(pp
m)
Spin-Orbit included
48Ca @2.2 GeV, 4 deg.
SkM
*S
Ly5
SkI
3
PC
-F1
weightedaverage∆r
np(Exp.)
NL3
In the two tested models, spin-orbit effects shifts to lower valuesthe Apv consistently by about 0.07 ppm. This predicts a reductionof ∆rnp of about 0.05 fm.Charge density distributions including spin orbit effects provided by
J. Piekarewicz (FSU).
48Ca: Estimation of three-neutron forceseffects in comparison with other corrections
Considering errors in Gs
E and no spin-orbit contributions //
No quark strange contributions and spin-orbit effects considered (filled in orange)No quark strange or spin-orbit contributions
Considering Gs
E and no spin-orbit contributions
FSUGold / FSUGold + Shell Model
a
b
cab
c
Self-consistent nucleon distributions in 48
Ca
Self-consistent 40
Ca core + Shell Model NN for 8 neutrons in excess
Self-consistent 40
Ca core + Shell Model NN+3N for 8 neutrons in excess
errors
spin-orbit mainly from ν1f7/2
0.05
ppm
±0.0
15 p
pm
0.07
ppm
rn ~ 0.01 fm
Shell Model calculations based on χEFT with NN to N3LO (fixedto scattering data) and 3N to N2LO (fixed to B tritium and R ofalpha particle) provided by J. Menendez (TU Darmstadt).Three-neutron forces used here shifts downwards the Apv by about0.05 ppm (very similar to spin-orbit effect)
Isovector static dipole polarizability
Definition: αD
The linear response or dynamic polarizability of a nuclearsystem excited from its g.s., |0〉, to an excited state, |ν〉, dueto the action of an external oscillating dipolar field of the form(Fe iwt + F †e−iwt):
FD =Z
A
N∑
i
rnY1M(rn)−N
A
Z∑
i
rpY1M(rp)
is proportional to the static dipole polarizability, αD , forsmall oscillations
αD =8π
9e2m−1 =
8π
9e2
∑
ν
|〈ν|FD |0〉|2E
where m−1 is the inverse energy weighted moment of thestrength function,
SD(E ) =∑
ν
|〈ν|FD |0〉|2δ(E − Eν)
Mean-Field + RPA results for 208Pb
J. Piekarewicz, B. K. Agrawal, G. Colo, W. Nazarewicz, N. Paar, P.-G. Reinhard, X. Roca-Maza and D. Vretenar,
Phys. Rev. C 85 041302 (2012) (R).
Mean-Field + RPA results for 48Ca
LNS
SGII
SIII
SK255
SkI3
SKM*
SLy4
SLy5
DD-M
E32
DD-M
E34
DD-M
E36
DD-M
E38
DD-M
E2
0.1 0.15 0.2 0.25 0.3∆r
np (fm)
2.2
2.3
2.4
2.5
2.6
2.7α D
(f
m3 )
48Ca r=0.281
Data on
relativistic models provided by N. Paar and D. Vretenar
Conclusions:
A precise and model-independent determination of ∆rnp in48Ca and 208Pb via PVES experiments would probe at thesame time the density dependence of the nuclear symmetryenergy and the relevance of three neutron-forces in 48Ca.Eventually, it can also provide indirect indications on theimpact of 3N in 208Pb.
We demonstrate a close linear correlation between Apv and∆rnp within the same framework in which the ∆rnp iscorrelated with L.
Other experiments fairly agree with the correlation betweenApv and ∆rnp.
Conclusions:
The estimated corrections to theApv ≈ A0
pv × [1− 0.005(strange)− 0.03(s− o)] where A0pv is
the result from DWBA calculations with a given neutron andproton density distributions convoluted with experimentalelectromagnetic form factors and weak neutral current formfactors including radiative corrections, indicate a reductionof about the 3%.
In addition, the inclusion of 3N-forces would change theneutron density producing a reduction in A0
pv of a few %.
Conclusions:
Families of modern energy density functionals show an almostlinear correlation between αD and ∆rnp while the correlationgets worst when models based on different grounds are alsotaken into account.
Apv and αD are complementary observables that may settight constraints on the density dependence of thesymmetry energy.
Collaborators:B. K. Agrawal1
G. Colo2,3
W. Nazarewicz4,5,6
N. Paar7
J. Piekarewicz8
P.-G. Reinhard9
D. Vretenar7
Michal Warda10
Mario Centelles11
Xavier Vinas11
1 Saha Institute of Nuclear Physics, Kolkata 700064, India2 Dipartimento di Fisica, Universita degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy3 INFN, Sezione di Milano, via Celoria 16, I-20133 Milano, Italy4 Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA5 Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA6 Institute of Theoretical Physics, University of Warsaw, ulitsa Hoa 69, PL-00-681 Warsaw, Poland7 Physics Department, Faculty of Science, University of Zagreb, Zagreb, Croatia8 Department of Physics, Florida State University, Tallahassee, Florida 32306, USA9 Institut fr Theoretische Physik II, Universitt Erlangen-Nrnberg, Staudtstrasse 7, D-91058 Erlangen, Germany10 Katedra Fizyki Teoretycznej, Uniwersytet Marii CurieSklodowskiej, ul. Radziszewskiego 10, PL-20-031 Lublin,Poland11 Departament dEstructura i Constituents de la Materia and Institut de Ciencies del Cosmos, Facultat de Fısica,Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain