Correlations with the weak charge formfactor
Discussion notes: skin collaboration Status 16. March 2013
1 Scope
Here come a few words about CREX and PREX and previous publications. We explain the two ways to correlation analysis, explicit trends with NMP and covariances.
2 Framework
We introduce the weak-charge formfactor in its simple form, only mentioning the spin- orbit contributions and the various models for the intrinsic nucleon formfactor; to minimize trouble, I propose that we stay with the old S&W formfactor for which our colleagues have already implemented the subroutine. We address the uncertainty in the s-quark strength. We explain covariance analysis.
0.22 0.225 0.23 0.235 0.24 0.245
220 230 240
FW(0.475)/FW(0)
incompressibility K [MeV]
with s-quark no s-quark
0.7 0.8 0.9 1 effective mass m*/m
with s-quark no s-quarkor 0.22
0.225 0.23 0.235 0.24 0.245
28 30 32 34
FW(0.475)/FW(0)
symmetry energy asym [MeV]
with s-quark no s-quarkor
0 0.2 0.4 0.6
TRK sum rule κ
trends of FW(0.475/fm) with NMP for 208Pb
with s-quark no s-quarkor
SV-min unconstr.
Figure 1: Results for the weak-charge formfactor FW(0.475/fm for208P b in a variety of Skyrme parametrizations. The rightmost panel shows the value obtained from SV-min, the unconstrained fit to finite nuclei only [3]. The other four panels show trends of FW
versus the four basic nuclear matter properties (NMP): incompressibility K, isoscalar effective mass m∗/m, symmetry energy asym, and TRK sum-rule enhancement κ (≡
isovector effective mass). Optimized parametrizations had been used which were fitted the the same ground state data as SV-min plus an additional constraint on the wanted nuclear matter property [3]. The black horizontal arrows indicate the “reasonable”
range of NMP for which the quality measure χ2 is less then than the optimum (for SV-min) plus 1. The error bars show the extrapolation errors from the χ2 fits. Two errors are shown: the extrapolation error without accounting for the s-quark (red) and the error which also includes the uncertainty about the s-quark strength (green).
3 Trends with varied nuclear matter properties
Figures 1 and 2 show trends versus all four NMP for case of SHF and the total error bars from the free fit SV 1-min. Goal is to show the particular sensitivity to asym
and the influence of the s-quark on the extrapolation errors. The error bars from SV-min together with the “reasonable” ranges of NMP demonstrate how the total
0.192 0.194 0.196 0.198 0.2 0.202 0.204 0.206 0.208
220 230 240
FW(0.778)/FW(0)
incompressibility K [MeV]
with s-quark no s-quark
0.7 0.8 0.9 1 effective mass m*/m
with s-quark no s-quark 0.192
0.194 0.196 0.198 0.2 0.202 0.204 0.206 0.208
28 30 32 34
FW(0.778)/FW(0)
symmetry energy asym [MeV]
with s-quark no s-quark
0 0.2 0.4 0.6
TRK sum rule κ
trends of FW(0.778/fm) with NMP for 48Ca
with s-quark no s-quark
SV-min unconstr.
Figure 2: Sames as figure 1 but for the case of FW(0.778/fm in 48Ca.
uncertainty is composed from some intrinsic uncertainty (mostly from shell effects) plus the uncertainty in the NMP.
Figure 3 compares the trends for various isovector observables and is the central figure on the analysis by dedicated variation of NMP. I think these five relations suffice.
It shows clearly that the neutron skin is closely related to FW in 208Pb the same way in all models.
13 14 15 16
100 150 200 250 300
polarizability(208 Pb) [fm2 /MeV]
r.m.s. skin [mfm]
SHF SV RMF DDME RMF NL RMF FSU 0.195
0.2 0.205 0.21
100 150 200 250 300
FW(0.778)/FW(0) for 48 Ca
r.m.s. skin [mfm]
SHF SV RMF DDME RMF NL RMF FSU
0.2 0.21 0.22 0.23 0.24 FW(0.475)/FW(0) for 208Pb
SHF SV RMF DDME RMF NL RMF FSU 100
150 200 250 300
rms skin [mfm] SHF SV
RMF DDME RMF NL RMF FSU SHF SV RMF DDME RMF NL RMF FSU
Figure 3: Trends of isovector observables, dipole polarizability αD in 208Pb, neutron r.m.s. skin in 208Pb, weak-charge formfactor FW(q = 0.475/fm)/FW(0) in 208Pb, and weak-charge formfactor FW(q = 0.778/fm)/FW(0) in 48Ca, for various model families as indicated. Trends are obtained by systematic variation of symmetry energy asym. Experimental values are indicated by black boxes and error bars, for skin and FW
from [7] and for αD from [8]. Lower left: polarizability versus skin. Lower right:
polarizability versus FW. Middle right: skin versus FW in 208Pb(this plot shows also extrapolation uncertainties for SHF). Upper right: FW in48Ca vs. 208Pb.
0 0.2 0.4 0.6 0.8 1
FW(other) rrms,n(208Pb) skin(208Pb) skin(132Sn) skin(48Ca) polariz. 208Pb dρE/Aneut dρasym asym m*/m K kTRK
correlation with FW of 208Pb
0 0.2 0.4 0.6 0.8 1
correlation with FW of 48Ca
SV-minDDME-min1 PC-min1FSUgold
Figure 4: Correlations between the weak-charge formfactors and nuclear observables (FW, neutron radius rrms,n, skin=neutron minus proton radius, polarizability) as well as NMP (slope of neutron EoS ∂ρEneut/N ), symmetry energy asym and its slope ∂ρasym, effective mass m∗/m, incompressibility K, sum rule enhancement κTRK). Left: FW(q = 0.475fm−1) for 208Pb at the q value of PREX. Right: FW(q = 0.778fm−1) for 48Ca at the q value of CREX. Results are shown for four different models as indicated.
4 Correlation analysis and variations of the fit data
Figure 4 shows correlations between FW for48Ca (right panel) and208Pb (left panel) and a couple of typical observables. Results from the SHF parameterization SV-min and three RMF parameterizatrions are shown as indicated. There is a remarkable difference between the models, even within the RMF. Look, in particular, for the neutron radius and the neutron skin. For 208Pb, the correlations from PC-min1 are closer to those of FSU, both showing low correlation with the neutron skin while the correlation with the neutron radius is perfect. The correlations from DDME, on the other hand, are close to the SHF parameterization SV-min, booth leaving high correlation also for the neutron skins. For 48Ca, the results show even more differences. It is to be noted that
5.43 5.44 5.45 5.46 5.47 5.48 5.49 5.5 5.51 5.52 5.53
SV-min FSUgold PC-min1 DDME-min1
r.m.s. radius [fm]
5.56 5.58 5.6 5.62 5.64 5.66 5.68 5.7 5.72 5.74
SV-min FSUgold PC-min1 DDME-min1
diffr. radius [fm]
0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28
SV-min FSUgold PC-min1 DDME-min1
surfac thickn. [fm]
Figure 5: Form parameters of the charge distribution (r.m.s. radius, diffraction radius, surfacer thickness) for 208Pb and for the four parameterizations in this survey. The uncertainties from the fit are shown as errorbars.
DDME-min1 and PC-min1 were fitted with exactly the same data and strategy, both including binding energies, charge r.m.s. radii, charge diffraction radii, and charge surface thickness.
A quick glance at the uncertainties may help to understand this puzzling result.
Figure 5 shows them for the charge radii and surface thickness1. It is indeed so that DDME-min1 and SV-min have both low uncertainties in the charge r.m.s. radius.
This then makes the neutron skin practically following the neutron radius and pro- duces consequently similarly high correlations with FW. The FSUgold and PC-min1, on the other hand, show larger uncertainties in rrms which, in turn, decouples the neu- tron skin from the neutron radius. It remains to ask why DDME-min1 has smaller uncertainties than PC-min1. The answer is possibly very simple: PC-min1 has two more free parameters in the model (11 instead of 9).
5 The sensitivity of F
Was function of q
Figure 6 shows the weak-charge formfactor for 208Pb and its correlations with the neutron r.m.s. radius rn, both as function of transferred momentum q. The lower left panels shows the formfactor as such together with its uncertainties. Note that we had to amplify the uncertainties by a factor of 10 to make them conveniently visible in the plot. The upper panels shows the correlation of FW with rn. It is strong for low q. There are narrow regions of very low correlations. These emerge because the correlation coefficient is changing sign at this places. The breakdown seem to be related to maxima and minima in FW. It is clear that experiment should avoid these points. The effect is very similar for RMF and SHF (see right upper panel). The left upper panel does also show the effect of the uncertainty in the s-quark strength.
The degradation of correlation is strongly q dependent. An optimally insensitive point
1PGR2all: I suggest to show only the uncertainties on rrmsand to skip the two other panels.
-30 -25 -20 -15 -10 -5 0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 FW(q)
momentum q [fm-1]
∆FW*10 0.6
0.7 0.8 0.9 1
correl: rrms,n vs FW
SV-min, 208Pb, with and without s-quark
no s with s
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 momentum q [fm-1]
FSU vs. SV-min, 208Pb, without s-quark
SV-min FSU
Figure 6: Extrapolation errors and sensitivity of FW for various transferred momenta q.
The left panel shows results from SHF for the parameterization SV-min. The left lower panel shows the formfactor FW as such. Error bars indicate the uncertainty multiplied by factor 10. The arrow indicates q = 0.475/fm, the momentum at which PREX is performed. The left upper panel shows the absolute value of correlation between the neutron r.m.s. radius rn and the FW(q). The first faint vertical line indicates q(0)1 , the position of the first zero of FW, the second faint line the position of formfactor maximum (from which the surface thickness is deduced). The right panel shows again correlations (absolute values) with the rn here comparing a result from RMF in FSU form with SHF.
appears for the SHF at q = 0.65/fm. The actual measuring point at q = 0.475/fm (see arrow) is plagued by the degradation which leaves a correlation of only 90%. This detail, however, should be counterchecked in other models.
The measuring point for 48Ca in CREX lies at q = 0.778/fm. One may one dis- quieted that this larger value comes into a not so well correlated region. To check this, we show in figure 7 the correlation with FW in 48Ca. The left panel compares results from SHF and RMF. The breakdown of correlations comes much later in this smaller nucleus and FW at the the measuring pont is safe. In fact, it resides even at a maximum of correlations. The results from SHF and RMF agree in that they find the CREX measuring point well suited. However, the curves as such differ in detail.
The regions of low correlations are much broad in SHF. Explanations for that have yet to be figured out. The right panel checks the influence of the uncertainty in the s-quark coupling. The difference is much mor dramatic for48Ca than it was for 208Pb.
Includiing the s-quark reduces the correlations to below 1/2. This casts serious doubty on the use of CREX measurements for determining the neutron radius.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.5 1 1.5 2
correl: rrms,n vs FW
momentum q [fm-1]
SV-min and FSU, 48Ca, without s-quark
qCREX SV-min
FSU
0 0.5 1 1.5 2
momentum q [fm-1]
SV-min, 48Ca, with + without s-quark
without with
Figure 7: Absolute value of correlation between the neutron r.m.s. radius rn in 48Ca and the FW(q) in48Ca. Left panel: Results from SHF (red line) and from RMF (green line), both computed without uncetaiinty from s-quark, are compared. The value of q for CREX is indicated by an arrow. Right panel: Resuls from SHF with and without s-quark are compared.
6 Outlook
...
References
[1] “Weak charge density and formfactor”, Discussion notes of the skin collaboration, status 5. January 2012.
[2] C. J. Horowitz, S. J. Pollock, P. A. Souder, and R. Michaels, Phys. Rev. C 63 (2001) 025501.
[3] P. Kl¨upfel, P.-G. Reinhard, T. B¨urvenich, and J.A. Maruhn, Phys. Rev. C 79 (2009) 034310.
[4] “Co-Variances in connection with χ2-fitting” P.–G. Reinhard and Witek Nazarewicz, Internal notes from the JUSTIPEN meeting: 19 March 2011
[5] W. Nazarewicz and P.–G. Reinhard, Phys. Rev. C, 81, 051303 (2010).
[6] J. Piekarewicz, B. K. Agrawal, G. Col`o, W. Nazarewicz, N. Paar, P.-G. Reinhard, X. Roca-Maza, and D. Vretenar, preprint 2012, submitted PRC
[7] C. Horowitz et al, “Weak charge form factor and radius of 208Pb through parity violation in electron scattering”, preprint 2012
[8] A. Tamii et al., Phys. Rev. Lett. 107, 062502 (2911).
