Effects of center-of-mass correction
and nucleon anomalous momenton nuclear charge radii
Yusuke Tanimura (Soongsil Univ./Tohoku Univ.)
Collaborator: Myung-Ki Cheoun (Soongsil Univ.)
Yusuke Tanimura and Myung-Ki Cheoun, arXiv:2312.15983 [nucl-th] (2023).

Nuclear charge radius• Measured accurately with electromagnetic (EM) probes• Information of nuclear structure and nuclear forceüShell effectsüPairing correlationüDeformationüSymmetry energy?

“Odd-even shape staggering” in Hg isotopes
Goodacre et al., PRL126, 032502 (’21).

• Theoretical interpretation of charge radiiüMany-body effectsüNuclear forceüInternal EM structure of nucleon“Anomalous behaviors” of rch in Ca isotopes
due to continuum effects?
Miller et al., Nat. Phys. 15, 432 (’19)

Charge radius with DFT• Nuclear DFT (mean-field model)• Universal “energy density functional”übinding energyünuclear size, EM moments, … (one-body observables)• Suitable for systematic study of charge radiiCM• Translational symmetry breaking• Nucleons are confined in a “mean-field potential”
fixed in space• Center of mass (CM) of the system is localizedaround the potential
à Fluctuation of CM motion gives spurious
contribution to radius (and any observable)Ormean fieldCharge radius with DFT• Nuclear DFT (mean-field model)• Universal “energy density functional”übinding energyünuclear size, EM moments, … (one-body observables)• Suitable for systematic study of charge radii
In this work, we study the charge radii withn Correction for center-of-mass motion of nucleus
n Electromagnetic properties of nucleon(anomalous magnetic moment and finite size)Mean-squared (MS) charge radiusMean-squared (MS) charge radius

ß point-proton densityMean-squared (MS) charge radius

ß point-proton density

ß Contribution of the (point) anomalous magnetic momentto charge density

m: nucleon massκp = 1.793, κn = -1.913

Mean-squared (MS) charge radius↑ Finite-size of nucleons
Independent of nuclear structureß point-proton density
ß Contribution of the (point) anomalous magnetic momentto charge density

m: nucleon massκp = 1.793, κn = -1.913

Center-of-mass correction of <r2>pPoint-protonsquared radius
w/o CM correction

Center-of-mass correction of <r2>pPoint-protonsquared radiuswith CM correction

1- and 2-body parts of the CM correction:α, β: canonical s.p. states,
vα, uα: occupation
amplitudes

Center-of-mass correction of <r2>pPoint-protonsquared radiuswith CM correction

1- and 2-body parts of the CM correction:

The CM correction has been
neglected, or taken into
account with• CM1 only
• harmonic-oscillator
approximation

CM correction

Anomalous
magnetic moment

Nucleon finite-size effect(independent of nuclear structure)

1. Decomposition of the correction terms <r2>κ, Δ(CM1), and Δ(CM2)
2. Comparison of RMS charge radii to experimental datafor isotope chains
4-8He, 10-22C, 12-28O, 36-56Ca, 50-80Ni, 78-112Zr, 100-134Sn, 180-220Pb
Model:• Relativistic Hartree-Bogoliubov (RHB) model• Meson-exchange interaction• DD-ME2 + Gogny D1S for pairing
Lalazissis et al., PRC71, 024312 (2005).

1. Decomposition of corrections for Pb isotopes

Experimental uncertainty
Anomalous magnetic moment
1-body CM
2-body CM• <r2>κ, Δ(CM1)p, and Δ(CM2)p are all of the same order and larger than theexperimental uncertainty
à None of them should be neglectedDecomposition• <r2>κ is sensitive to the shellstructure while Δ(CMi) arerather smooth• 2b-1b ratio• Δp(CM2)/Δp(CM1) ≈ 0 for lightnuclei• Δp(CM2)/Δp(CM1) à -1 for heavynuclei

Tanimura and Cheoun,arXiv:2312.15983 [nucl-th] (2023).

2. Absolute values of charge radii+(0.8)2: used for parameter fit+FF: with nucleon EM form factors+FF+CM: with CM correction

Sizable impacts of the corrections
from light to heavy nuclei

Tanimura and Cheoun,arXiv:2312.15983 [nucl-th] (2023).

Isotopic shifts

Tanimura and Cheoun,arXiv:2312.15983 [nucl-th] (2023).

Comparison withharmonic-oscillator model

Tanimura and Cheoun, arXiv:2312.15983 [nucl-th] (2023).

whereandComparison withharmonic-oscillator model

Tanimura and Cheoun, arXiv:2312.15983 [nucl-th] (2023).

HO model and Exp data

andComparison withharmonic-oscillator model
HO model is• not good for very light and/or near-dripline nuclei• nearly satisfactory for Ca and heavier nuclei

RHB

Comparison withharmonic-oscillator model
HO model is• not good for very light and/or near-dripline nuclei• nearly satisfactory for Ca and heavier nuclei

Connection with symmetry-restoration technique
HO approximation to our approach yields

cf. approximate momentum projection method
modifying the charge radius as$ℏ!
##!!" → !!" −&⟨( #⟩!"

Schmidt and Reinhard, NPA530, 283 (1991)
Reinhard and Nazarewicz, PRC103, 054310 (2021)

RHB

Summary• Charge radius with important corrections• CM correction (for symmetry-breaking in mean-field model)• EM structure of nucleon (directly modifies ρch)• Sizable impacts on charge radii• Matter radius and neutron skin?
• Refit of parameters in EDF?