Correlations with the weak charge formfactor

Correlations with the weak charge formfactor

Discussion notes: skin collaboration Status 1. March 2012

Contents

1 Scope and Framework 1

2 Trends with varied nuclear matter properties 2

3 Correlation analysis and variations of the fit data 5

4 Variations of the SHF fit and extrapolation errors 7

5 The sensitivity of F

as function of q 8

6 Outlook 9

1 Scope and Framework

The PREX measurements will deliver steadily improving information on the weak- charge formfactor F

(q) of nuclei. The hope is that F

provides direct access to the neutron formfactor, at least its basic form parameters as, e.g., the neutron r.m.s.

radius r

or the neutron diffraction radius R

(also denoted as box-equivalent radius). The aim of the present project is to explore quantitatively the physics content of the weak-charge formfactor F

. To this end, a brief summary on the computation of F

in connection with nuclear mean-field theory has been developed [1], based on previously published a detailed discussion of parity violating measurements [2].

There are various ways to evaluate the physics information content of a given ob- servable. We follow use here two strategies. The first one is a dedicated variation of key features of a mean-field parametrization, typically the basic nuclear matter prop- erties (NMP) incompressibility K, isoscalar effective mass m

/m, symmetry energy a

, and TRK sum-rule enhancement κ (≡ isovector effective mass). Such a variation has to be done with great care to produce a set of parametrizations with comparable quality but significantly different NMP. We employ here the sets of Skyrme-Hartree- Fock parametrizations from [3]. The second strategy is the correlation analysis in the context of a χ

fit, for a brief summary see the notes [4] and for a recent application [5].

This analysis requires a parametrization which is free from constraints on NMP. We

use the parametrization SV-min from [3] which was adjusted exclusively to measured

0.22 0.225 0.23 0.235 0.24 0.245

220 230 240

FW(0.475)/FW(0)

incompressibility K [MeV]

extr.err.+squark extrap.err 50

100 150 200

neutron skin in 208Pb [mfm]

rms skin diffr. skin

0.7 0.8 0.9 1 effective mass m^*/m extr.err.+squark

extrap.error rms skin diffr. skin

28 30 32 34

symmetry energy a_sym [MeV]

extr.err.+squark extrap.error rms skin diffr. skin

0 0.2 0.4 0.6

TRK sum rule κ extr.err.+squark

extrap.error rms skin diffr. skin

Figure 1: Trends of the weak-charge formfactor F

(q = 0.475fm

) (lower panels) and of the neutron skin (upper panels), both for

Pb, versus the four basic nuclear matter properties: incompressibility K, isoscalar effective mass m

/m, symmetry energy a

, and TRK sum-rule enhancement κ (≡ isovector effective mass). The neutron skin is shown for the r.m.s. radii, r

− r

, and for the diffraction radii, R

− R

. Optimized parametrizations with systematic variation of these nuclear matter properties from [3] had been used. The error bars show the extrapolation errors from the χ

fits. Two errors are shown for F

, the straightforward extrapolation error (red) and the error which also includes the uncertainty about the s-quark strength (green).

ground-state data from finite nuclei including binding energy, charge r.m.s. radius, charge diffraction radius, surface thickness of the charge distribution, selected pairing gaps, and some l ∗ s splittings. To explore the sensitivity to the adjustment strategy, we also explore fits without data on radii.

There is a subtle detail in the evaluation of F

: we encounter a non-negligible uncertainty in the strength ρ

of the contribution from the s quark [1]. We include this uncertainty into the standard χ

error analysis by adding ρ

as further “Skyrme parameter” in the fit and augmenting the total χ

as

χ

−→ χ

+ ρ

∆ρ

, ∆ρ

= 1.64 fm . (1)

This supplies the χ

scheme with the presently adopted uncertainty ∆ρ

[1]. The new parameter ρ

has influence on χ

and on the observable F

. This way the standard rules of error propagation automatically account for this additional uncertainty while all other observables remain untouched.

2 Trends with varied nuclear matter properties

In a first round, we perform the explicit correlation analysis, i.e. we track the con-

nection between the weak-charge formfactor F

and NMP by systematically varying

12.5 13 13.5 14 14.5 15 15.5 16 16.5 17

28 30 32 34 36 38

polarizability(208Pb) [fm2/MeV]

symmetry energy a_sym [MeV]

SHF SV RMF DDME RMF NL RMF FSU

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3

28 30 32 34 36 38

r.m.s. skin [fm]

symmetry energy a_sym [MeV]

SHF SV RMF DDME RMF NL RMF FSU

0.21 0.22 0.23 0.24

28 30 32 34 36 38

FW(0.475/fm)/FW(0)

symmetry energy a_sym [MeV]

SHF SV RMF DDME RMF NL RMF FSU

Figure 2: Trends with symmetry energy a

for three observables in

Pb: dipole polarizability α

(left), neutron r.m.s. skin (middle), and weak-charge formfactor F

(0.0475/fm)/F

(0). The SHF set is taken from [3]). The RMF sets are those used in [6].

the NMP. Figure 1 summarizes the trends for F

in

Pb. This is compared with the trends of the neutron skin in

Pb because this is the observable on which we want gain new information by measuring F

. The weak-charge formfactor is considered at transferred momentum q = 0.475fm

where PREX measurements are expected [7].

The neutron skin is the difference between neutron and proton radius whereby two choices are considered, the skin from r.m.s. radius and the skin from diffraction radius.

The figure shows also the extrapolation errors as deduced from the techniques of the χ

fits [3, 4].

The trends of skins and F

are much the same. All three quantities are rather insensitive to K, m

/m, and κ, but systematically varying with a

. The extrapolation errors are relatively larger for F

, where relative is meant with respect to the maximum variation given by the variation of a

. The case of F

shows, in fact, two errors. The smaller error values (red) stay for the straightforward extrapolation errors. The larger error bands (green) include the uncertainty from the s-quark strength in the evaluation of F

[1]. The growth of the error is visible but small. We have yet to see how the other analysis behaves in that respect.

Figure 2 shows trends with symmetry energy for three observables in

Pb and a variety of mean-field models as indicated. The SHF results show also the extrapolation errors. We can assume that these errors are of comparable size for the RMF. All models follow the same trends in all three observables. For the neutron skin (middle panel), the results agree with each other within the given uncertainties. However, even here we see a slightly different offset for the various mean-field families.

Larger differences between the mean-field models are seen for polarizability (left panel) and weak-charge formfactor (right panel). The differences exceed the uncer- tainty band and, again, we see systematically shifted offsets for the different mean-field models. The trends for F

are similar to those of the neutron skin with SHF being on one side and RMF-NL/RMF-FSU on the other while RMF-DDME stays in between.

The differences between RMF-DDME and the two other RMF models are most prob-

ably due to the different modeling of the density dependence. This may also be one

reason for the differences seen for the SV family.

100 150 200 250 300

0.2 0.21

0.22 0.23

0.24

rms skin [mfm]

F

(0.475)/F

(0) SV a

SV other RMF DDME RMF NL RMF FSU

Figure 3: Trends of the weak-charge formfactor F

(lower panels) versus neutron skin, both for

208

Pb. The results from all systematically varied SHF parametrizations (see figure 1 and [3]) have been united in one plot. Uncer- tainties are indicated for the SHF sets with varied a

(red). Experimental values from [7] are indi- cated by black boxes and error bars.

For the polarizability, the RMF curves have different slopes and cross each other.

Most puzzling is that the optimal parametrizations in each RMF model (the isolated points linked to the model by having the same color) have much different polarizability.

This case needs still some checking.

Figure 3 collects all data from figure 1 and plots them as r.m.s. neutron skin versus F

. This is a way to display the correlation between these two observables directly.

All data points line up nicely on a straight line. This demonstrates clearly that there is a strong correlation between F

and neutron skin in the studied class of SHF models.

One also sees that it is predominantly the variation of a

which delivers a large span of results while all other NMP variations change little on the neutron skin and F

. All three RMF models show the same trends and slopes, but have different offset. These differences seem to be of systematic nature as they are larger than the extrapolation uncertainties. As in the previous figure, we see RMF-DDME between the lines for SHF- SV and RMF-NL/RMF-FSU. The differences between SHF-SV and RMF-NL amount to about 0.045 fm for a given F

. This can be considered as the present limit of the determination of the neutron skin from given F

. However, this does not yet account for the uncertainties in the experimental F

. The present uncertainty, as indicated by the horizontal black dashed line, is too large to allow any reasonable conclusion on the neutron skin.

The isovector dipole polarizability α

is another observable which is intimately connected to the symmetry energy [6] and thus also to neutron skin. Figure 4 shows a correlation plot of the connection between α

and neutron skin (left) and between α

and F

(right). As the polarizability has recently been measured with high precision

[8], we can now directly compare different measurements mediated by the prediction

of the SHF model with varied a

. As in the plots before, the different mean-field

models show the same trends and slopes but differ significantly in the offset whereby

the ordering of the lines is the same as in the previous two figures. This is a persistent

problem which we have to understand better. Besides that, all models show nicely

a strong correlation between polarizability α

and neutron skin as well as F

. The

13 13.5 14 14.5 15 15.5 16 16.5

100 150 200 250 300 polarizability(208 Pb) [fm2 /MeV]

r.m.s. skin [mfm]

SHF SV RMF DDME RMF NL RMF FSU

0.2 0.22

0.24

F_W(0.475)/F_W(0) SHF SV

RMF DDME RMF NL RMF FSU

Figure 4: Trends of isovector dipole polarizability α

in

Pb versus neutron skin (left) and F

(right) for the SHF family with varied a

. Experimental values are indicated by black boxes and error bars, for skin and F

from [7] and for α

from [8].

experimental mean values, however, do not match along any one of these correlation lines. While α

prefers a low symmetry energy a

= 28 MeV, the average of the PREX measurements pulls to very large a

= 38. However, we do presently not have a definite discrepancy when considering the associated error bars. Both measurements are compatible with the trend from all the mean fields when considered within their uncertainties. Let us speculate what future measurements could deliver. Assume that the discrepancy of the mean values persist with much shrunk error bars. For then, we have deep problem with mean-field theories which probably all predict the slope as shown in figure 4. However, we see a clear confirmation of mean field model if the r.m.s. radius steps down with decreasing error bars.

3 Correlation analysis and variations of the fit data

We now turn to correlation analysis as outlined in [4]. To this end, we employ parametrizations which include only information from finite nuclei in the fit. The typical representative for that is the force SV-min [3]. Figure 5 shows the correlation of F

(q = 0.475fm

) with a selection of those observables which are fairly well corre- lated. The standard SV-min (red lines) shows perfect correlation with both neutron radii, r

and R

, and the skin in

Pb. Even the connection to the neutron radius in

Sn is very tight. The next four observables are strongly related to time- even isovector response in bulk. All of them show still a considerably large correlation.

Remind that these four observables are strongly correlated with each other (see figure

3 in [4]). Thus having F

correlated with one of them will carry the other three with

it. That is what we see here. Finally, the skin in

Ca has less correlation which is

probably due to the fact that this observable is not as tightly determined by isovector

bulk properties and that shell effects may play a larger role here.

rrms(208 Pb) Rdiffr(208 Pb) skin(208 Pb) skin(132 Sn) polariz. 208 Pb dρasym dρE/Aneut asym skin(48 Ca) variation fit data

SV-min no R_diffr, σ no R_diffr, σ, rrms

SV-min, s-quark

0 0.2 0.4 0.6 0.8 1

rrms(208 Pb) Rdiffr(208 Pb) skin(208 Pb) skin(132 Sn) polariz. 208 Pb dρasym dρE/Aneut asym skin(48 Ca)

correlation with FW

impact of s-quark on F_W

Figure 5: The correlations between the weak-charge formfactor F

(q = 0.475fm

) and nine usually well correlated observables as indicated (“skin” means here skin from r.m.s. radii). The standard correlation from SV-min is shown in red. Correlations from SV-min including the uncertainty from the s-quark are shown in magenta (left).

Correlations from the fits without diffraction radius and surface thickness are shown in green, additionally without r.m.s radius in blue (right).

In order to explore the influence of modeling on the correlations, we have studied a few variants of the model. The right panel of figure 5 shows results from fits with the standard SHF functional but with reduced data sets. In a first step, we have skipped the diffraction radii R

and surface thickness σ from the fit data while still maintaining information on the charge r.m.s. radii. The results are shown in green. There is practically no loss in correlation as compared to SV-min (red). It is only the symmetry energy a

which is somewhat de-correlated, but not the slope thereof ∂

a

. The latter is always the quantity which is better correlated. In a next step, we also skip the charge r.m.s. radii from the fit data thus having no information at all on the nuclear size in the pool. This cause dramatic reduction of correlations. Nonetheless, the neutron radii as such remain perfectly correlated. The skin, of course, looses because there is no information any more one the proton radius. All isovector bulk properties are also becoming de-correlated. This may be plausible because isovector response lives from the difference between neutron and protons while we have discarded the proton information now. It is the more interesting that the dipole polarizability in

208

Pb maintains a significant correlation. This suggests that this observable has more relation to the neutron distribution than to the protons, a detail which deserves further inspection.

We have also checked the impact of the uncertainty on the strength of the s-quark.

The result is shown in the left panel of figure 5 (magenta). There is a sizable loss

of correlations on all observables. This seems surprising in view of the fact that the

extrapolation errors were not much influenced (see figure 1). The correlation analysis is

much more sensitive and more clearly reveals the limitations by remaining uncertainties

in determining the weak-charge formfactor.

40 60 80 100 120 140 160 180 200 220

SV-min no Rd,σ no rrms,Rd,σ

rrms skin 208 Pb [mfm]

5.48 5.5 5.52 5.54 5.56 5.58 5.6 5.62 5.64 5.66 5.68

SV-min no Rd,σ no rrms,Rd,σ

rrms,neutron(208 Pb) [fm]

0.22 0.23 0.24 0.25 0.26

FW(0.475/fm)/FW(0)

+ s-quark no s-quark

Figure 6: Three neutron sensitive observables, neutron r.m.s. skin (lower left), neutron r.m.s. radius (lower right), and weak-charge formfactor F

(q = 0.475fm

)/F

(0) (upper left), together with their extrapolation error for various options of the SHF fit. For F

and parametrization SV-min, the error including the uncertainty from the s-quark is also shown (green). The standard fit SV-min is the reference [3]. A similar fit was performed without the diffraction radius and surface thickness in the data set but with the r.m.s. radius; it is indicated by “no R

, σ”. In a further fit even the r.m.s. charge radius was ignored, indicated by “no r

, R

, σ”.

4 Variations of the SHF fit and extrapolation errors

It is interesting to check the effect of the variations in the fitting strategy on the extrapolation errors. This is shown in figure 6. As one could have expected, the three observables F

(upper left), r.m.s. skin (lower left), and neutron radius (lower right) behave very similar. The extrapolation errors grow a bit when skipping R

and σ.

They grow a sizable amount when skipping additionally r

. But they grow only by a

factor of two. This means that even the fit to energies only has a high predictive value

on these observables. Thus one can be optimistic that the correlation between F

and

neutron radius is robust. As great obstacle remains the uncertainty in the strength of

the s-quark contribution to F

.

5 The sensitivity of F _W as function of q

So far we have considered F

at one fixed momentum q. Results on the q-dependence of uncertainties and correlations are shown in figure 7. The weak-charge formfactor F

as such is rather robust. Differences between fits are not visible at plotting scale. It is quickly decreasing in size. That is why formfactors are usually plotted logarithmically.

The extrapolation errors on F

(second panels from below) show a strong q de- pendence. They are maximal at q ≈ 0.475 fm

, close to the fixed value used in the previous sections. They are still large at the first zero q

and they are minimal at the first maximum of F

. The further oscillations show always the minimal errors at the maxima of F

and maximal errors in between. The uncertainty in the s-quark strength (magenta, left panels) plays little role for the extrapolation errors. But the choice of fit data makes a difference. The error is doubled throughout when skipping information on the charge r.m.s. radius from the fit while the profile of the q-dependence remains the same. From the extrapolation errors we would conclude that most new information can be gathered at the maximum of the errors, i.e. at q ≈ 0.475 fm

in case of

Pb.

As we have learned in previous section, a much more sensitive test is provided by a correlation analysis. The aim is to determine the neutron radius. Thus we consider the correlation between the neutron r.m.s. radius (second panels from above) or the neutron diffraction radius (uppermost panels) and F

for a wide span of q. Let us first have a look at the standard case SV-min without contribution from the s quark (red). Both correlations are practically 1 up the first zero q

. Above that value, the correlations oscillate strongly. Although almost perfect correlations reappear, it is certainly safer to stay at zero q ≤ q

. It is interesting to note that the correlation drop to nearly zero just at the first maximum of F

where the extrapolation error is minimal. This coincidence is reasonable. The first maximum is already well determined by the fits and there remains not much to be correlated. Comparison of the trends for r

and R

shows a difference in detail. While the correlation for r

has its maximum at q = 0 and is slightly decreasing from then on, maximal correlation for R

is reached at q

, the point from which the diffraction radius is determined.

The fits with reduced information (right panels) show, of course, reduced correla- tions. But the reduction is surprisingly small which, in turn, means that the correlations and their trends are rather robust.

The correlations which include the uncertainty from the s-quark strength (magenta lines) show a significant reduction for small q while practically no harmful effect is seen at q

.

Thus the choice of the best q for measurements is open. Without the uncertainty

from the s quark, we would advocate a rather small q to have most predictive power

on the neutron r.m.s. radius. Including the s quark into the considerations points

clearly to q = q

≈ 0.65 fm

as best choice. The decision is unambiguous if we

aim to determine the diffraction radius where the optimum point with and without

considering the s quark is at q = q

≈ 0.65 fm

. This analysis has, of course, to be

checked also from smaller nuclei where q

becomes larger.

6 Outlook

The above first results indicates the direction for a continued collection of material for a study on the weak-charge formfactor. It is straightforward to collect data from more variants of models for the first type of analysis elaborating the trends with varied nuclear matter properties (K, m

/m, a

, κ

). Most of this part has already been done. The results reveal a systematic difference between the various mean-field models (SHF, three variants of RMF). The trends (and slopes) of the various correlations are the same in all models. But the offsets can differ significantly. This has, e.g., severe consequences for the deduction of a neutron skin from the weak-charge formfactor. The reasons for the discrepancies and have yet to be found out.

The second stage of the analysis if to perform the full correlation analysis for all mean-field models discussed so far. Presently, the analysis has only been performed for SHF. Similar data for RMF are urgently needed. For example, it is necessary to confirm the sensitivity analysis of section 5 for other models than SHF.

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 χ

-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

Pb through parity violation in electron scattering”, preprint 2012

[8] A. Tamii et al., Phys. Rev. Lett. 107, 062502 (2911).

-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]

∆F_W*10 0

0.1 0.2 0.3 0.4

∆FW

no s quark with s quark 0.6

0.7 0.8 0.9 1

correl: rrms,n vs FW 0.6 0.7 0.8 0.9 1

correl: Rdiffr,n vs FW

SV-min, ²⁰⁸Pb, with and without s-quark

-30 -25 -20 -15 -10 -5 0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 momentum q [fm^-1]

SV-min no R_driffr,σ R_driffr,σ,r_rms 0

0.1 0.2 0.3

0.4 SV-min

R_driffr,σ R_driffr,σ,r_rms 0.6

0.7 0.8 0.9 1 0.6 0.7 0.8 0.9 1

SV-min varied data, ²⁰⁸Pb, no s-quark

Figure 7: Extrapolation errors and sensitivity of F

for various transferred momenta

q. The lower panels show the formfactor F

as such. The first panels from below

show the extrapolation error on the prediction for F

. The two upper panels show

the correlation between the neutron r.m.s. radius (second from above) and neutron

diffraction radius (uppermost). Five different case are shown. The standard fit SV-

min is the reference [3] (red line). A similar fit was performed without the diffraction

radius and surface thickness in the data set but with the r.m.s. radius (green); it is

indicated by “no R

, σ”. In a further fit even the r.m.s. charge radius was ignored,

indicated by “no r

, R

, σ” (blue). Results for these four cases were produced

without accounting for the uncertainty in the s-quark strength. For SV-min, we also

considered an inclusion of this uncertainty (magenta lines). The faint vertical lines

indicate q

, the position of the first zero of F

.