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