Dispersive corrections in elastic electron-nucleus scattering: an investigation in the intermediate energy regime and their impact on the nuclear matter

Abstract

Measurements of elastic electron scattering data within the past decade have highlighted two-photon exchange contributions as a necessary ingredient in theoretical calculations to precisely evaluate hydrogen elastic scattering cross sections. This correction can modify the cross section at the few percent level. In contrast, dispersive effects can cause significantly larger changes from the Born approximation. The purpose of this experiment is to extract the carbon-12 elastic cross section around the first diffraction minimum, where the Born term contributions to the cross section are small to maximize the sensitivity to dispersive effects. The analysis uses the LEDEX data from the high resolution Jefferson Lab Hall A spectrometers to extract the cross sections near the first diffraction minimum of $^{12}$C at beam energies of 362 MeV and 685 MeV. The results are in very good agreement with previous world data, although with less precision. The average deviation from a static nuclear charge distribution expected from linear and quadratic fits indicate a 30.6% contribution of dispersive effects to the cross section at 1 GeV. The magnitude of the dispersive effects near the first diffraction minimum of $^{12}$C has been confirmed to be large with a strong energy dependence and could account for a large fraction of the magnitude for the observed quenching of the longitudinal nuclear response. These effects could also be important for nuclei radii extracted from parity-violating asymmetries measured near a diffraction minimum.

Appendix A: Propagation of changes from the form factor to the charge radius

1.1 A.1: Formalism

We are interested in estimating how a change in the observed cross section, or the deduced form factor values, could impact the extracted radius $R_{ch}$.

The charge radius is a function of the M parameters of our model (15), in this case the M independent Bessel coefficients $a_\nu$, which in turn depend on the N experimentally extracted form factor values $y_i$. Therefore, through the coefficients $a_\nu$ the charge radius is a function of the experimental points and one can write a small change in $R_{ch}$ due to a given small change in the observations $(\delta y_1, \delta y_2, \ldots , \delta y_N)$ as:

$$\begin{aligned} \delta R_{ch} = \sum _i ^N \frac{\partial R_{ch}}{\partial y_i} \delta y_i = \sum _i ^N \left( \sum _\nu ^M \frac{\partial R_{ch}}{\partial a_\nu } \frac{\partial a_\nu }{\partial y_i} \right) \delta y_i. \end{aligned}$$

(30)

For M independent coefficients $a_\nu$, one has $M+1$ Bessel functions in our model due to the normalization constraint. The $a_{M+1}$ can be explicitly written by solving the constraint:

$$\begin{aligned} \left. \begin{array}{l} 4\pi \int \rho (r) r^2 dr = 1 , \\ \\ \sum _\nu ^{M+1} (-1)^{\nu +1}\frac{4\pi R_{cut}}{q_\nu ^2}a_\nu =1, \\ \\ a_{M+1} = (-1)^{M} \left( 1- \sum _\nu ^{M} (-1)^{\nu +1}\frac{4 \pi R_{cut}}{q_\nu ^2}a_\nu \right) \frac{(M+1)^2 \pi }{4 R_{cut}^3} \end{array} \right. . \end{aligned}$$

(31)

An alternative route would be to use Lagrange multipliers when making calculations for the data fit, which would allow to treat the $M+1$ coefficients independently. Following Eq. (18), and taking into account the normalization condition, the partial derivative of $R_\text {ch}$ with respect to a coefficient $a_\nu$ is given by:

$$\begin{aligned} \frac{\partial R_{ch}}{\partial a_\nu } = \frac{1}{2R_{ch}} 4\pi \frac{(-1)^\nu R^5 _{cut} (6-\nu ^2\pi ^2)}{\nu ^4\pi ^4} + \frac{\partial R_{ch}}{\partial a_{M+1}}\frac{\partial a_{M+1}}{\partial a_\nu }. \end{aligned}$$

(32)

The last term has to be included since $R_\text {ch}$ depends on the $M+1$ coefficients and $a_{M+1}$ depends linearly on the rest of the $a_\nu$, making the calculation straightforward from Eq. (31).

Meanwhile, the change in the coefficient $a_\nu$ due to a change in $y_i$ is a little more challenging to compute. To do so, one must specify how exactly the coefficients where obtained from the experimental data. An usual way is by minimizing the sum of the squares of the residuals denoted by $\chi ^2$:

$$\begin{aligned} \chi ^2 \equiv \sum _i ^N \frac{[F(q_i,{\varvec{a}})-y_i]^2}{2 \varDelta y_i^2}, \end{aligned}$$

(33)

where $\varDelta y_i$ is the estimated error, or uncertainty, in the measurement $y_i$ and ${\varvec{a}}$ is the list of coefficients $a_\nu$. The optimal values of the parameters ${\varvec{a}}_{opt}$ is found by imposing the condition of a minimum:

$$\begin{aligned} \frac{\partial \chi ^2}{\partial a_\nu } \Big |_{a_{opt}} \equiv G_\nu ({\varvec{a}},{\varvec{y}}) \big |_{a_{opt}} =0. \end{aligned}$$

(34)

Now, the key point is that one has M different $G_\nu$ which are functions of the parameters ${\varvec{a}}$ and the observations $y_i$, and they all equal zero when evaluated at the optimal parameters ${\varvec{a}}_{opt}$. If the value of one observation $y_i$ changes by a small amount $\delta y_i$, the minimum of $\chi ^2$ will move in the parameter space by a small amount. One can calculate this displacement by noticing that all the parameter values $a_\nu$ would have to change accordingly in order to keep the values of each $G_\nu$ at zero. Quantitatively this implies: $\frac{\partial G_\nu }{\partial y_i} \delta y_i = - \sum _k ^M \frac{\partial G_\nu }{\partial a_k} \delta a_k$ for $\nu \in (1,\ldots , M)$, which can be put in a matrix equation:

$$\begin{aligned} \frac{\partial G_1}{\partial y_i}&\delta y_i = - \left( \frac{\partial G_1}{\partial a_1}\delta a_1+ \frac{\partial G_1}{\partial a_2} \delta a_2 \ \cdots \ + \frac{\partial G_1}{\partial a_M} \delta a_M \right) \\ \frac{\partial G_2}{\partial y_i}&\delta y_i = - \left( \frac{\partial G_2}{\partial a_1}\delta a_1+ \frac{\partial G_2}{\partial a_2} \delta a_2 \ \cdots \ + \frac{\partial G_2}{\partial a_M} \delta a_M \right) \\&\;\;\vdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \;\;\vdots \\ \frac{\partial G_M}{\partial y_i}&\delta y_i = - \left( \frac{\partial G_M}{\partial a_1}\delta a_1+ \frac{\partial G_M}{\partial a_2} \delta a_2 \ \cdots \ + \frac{\partial G_M}{\partial a_M} \delta a_M \right) \end{aligned}$$

resulting in:

$$\begin{aligned} \begin{aligned} \frac{\partial {\varvec{G}}}{\partial y_i} \delta y_i&= - \mathcal {H} {\varvec{\delta }}{\varvec{a}} \Rightarrow {\varvec{\delta }} {\varvec{a}} = - \left( \mathcal {H}^{-1} \right) \frac{\partial {\varvec{G}}}{\partial y_i} \delta y_i, \end{aligned} \end{aligned}$$

(35)

Since G was already first derivatives of $\chi ^2$ with respect to the parameters, the expression obtained is $\mathcal {H}_{[j,k]}\equiv \frac{\partial ^2 \chi ^2}{\partial a_j \partial a_k}$, the Hessian matrix which contains second derivatives of $\chi ^2$. From this equation one can finally extract how each parameter $a_\nu$ changes when an observation $y_i$ changes:

$$\begin{aligned} \frac{\partial a_\nu }{\partial y_i} = - \Bigg [ \Big (\mathcal {H}^{-1} \Big ) \frac{\partial {\varvec{G}}}{\partial y_i} \Bigg ]_{[\nu ]} = -\sum _k^M \mathcal {H}^{-1}_{[\nu ,k]} \frac{\partial G _k}{\partial y_i}. \end{aligned}$$

(36)

From the set of changes in the observations, $\delta y_i$, due to the dispersive corrections, one has all the ingredients needed to calculate the change in $R_{ch}$ from Eq. (30). In the following discussion, we apply this framework to the data set presented by Offermann et al. [4] under the convention that $\delta R_\text {ch} = R_\text {ch}^{stat} -R_\text {ch}^{disp}$, since we want to estimate the change in the radius once the corrections for the dispersive effects have been implemented.

1.2 A.2: Example: change in the nuclear radius of $^{12}\text {C}$

We use the work from [4] where the authors used 18 Bessel functions to fit cross section experimental data from $^{12}\text {C}$. To show our method, we use the values of their first 9 coefficients $a_\nu$$\nu \in \{1,9\}$ from their Table X second column (without dispersion corrections) to generate 9 values $y_\nu$ of the form factor according to the relation $a_\nu = F(q_\nu )q_\nu ^2/2\pi R_\text {cut}$ at those 9 special $q_\nu$ values with $R_\text {cut}= 8$ fm. For the error associated with each “observation” $y_\nu$, we use the adapted error $\varDelta y_\nu$ from their reported percentage error in $\varDelta a_\nu$. For the remaining 9 points $\nu \in \{10,18\}$, we center the observations $y_\nu$ at zero and add an error band associated with the form factor of the proton as the authors did following the recommendation in [43]. Since the normalization condition must be respected, only 17 from the 18 coefficients $a_\nu$ are independent. We identify therefore $N=18$ and $M=17$.

Figure 9 shows the matrix $\partial a_\nu / \partial y_i$ from Eq. (36) for the 18 observations $y_i$ and 17 + 1 coefficients $a_\nu$. Even though we are not treating $a_{18}$ as an independent variable since we solved the constraint explicitly, we can still calculate how much its value changes when any one of the observations $y_i$ changes. It can be seen that as $\nu$ increases, $a_\nu$ becomes more dependent on $y_\nu$ and less sensitive to other values of y. In principle, if the 18 coefficients were independent, each $a_\nu$ will only be sensitive to their corresponding $y_\nu$, but the normalization constraint introduces mixing.

In the third column of Table 6 are the numerical values of $\partial R_\text {ch}/ \partial y_i$ for the first 9 observations $y_i$. Each one of these numbers, when multiplied by a small change in their associated observation, will yield the corresponding small change in $R_\text {ch}$ as in Eq. (30). The fourth column shows the percentage change needed in observation $y_i$ to create a 1% change in the radius. Even though the values $\partial R_\text {ch}/ \partial y_i$ are roughly the same size for all the observations, this fourth column shows that $R_\text {ch}$ is more sensitive to percentage changes in the first observations.

Fig. 9

$\partial a_\nu / \partial y_i$ matrix for the data extracted from Offermann et al. [4]

Table 6 The first column shows the index number of the special momentum transfer $i\pi /R_\text {cut}$ and the second column its form factor value obtained from [4]. The third column shows the value of $\partial R_\text {ch}/ \partial y_i$ . The fourth column shows the percentage change needed in $y_i$ to generate an equivalent change of $1\%$ in the estimated charge radius

Fig. 10

$^{12}$C form factor expanded in the Bessel functions formalism using Offermann [4] coefficients without dispersive corrections. The circles in the q axis shows the special values of momentum transfer for the first 9 (red) from experimental data and the second 9 (black) from the extrapolation suggested in [43]. The dashed blue lines encloses the region of the data excluded from the analysis in [4]. The inset plot shows the three test forms for S(q) in addition to the empirical perturbation obtained directly from the data by third degree spline interpolation. The curves in the inset plot are the ones needed to obtain the corrected $F_\text {ch}^{stat}$ from the observed $F_\text {ch}^{disp}$ values

As previously stated in the main discussion, we assume in the calculation of $\delta y_i$ that we can separate the effects of the dispersive corrections on the form factor values as (Eq. (20)): $F_{disp}(q) =F(q)_{stat} [ 1 + \frac{1}{2}\delta (E_e)S(q) ]$ where $\delta (E_e)$ controls the overall strength of the perturbation and S(q) controls the impact this change would have on different q values. Table 5 in the main body shows the results for three different test perturbations S(q),in addition to an empirical one obtained from comparing columns 2 and 3 of Table X in [4], for the central values of the form factor. For the test perturbations, the central values of the form factor were modified assuming a constant high value of $\delta (E_e) = 30\%$, so that our analysis could serve as an upper bound.

The three test forms for S(q) consists of $\delta _4$, $\delta _5$ and Gaussian. The first two represent an up-shift of $15\%$ on the value of $F(q_\nu )$ for $\nu =4$ and $\nu =5$ alone respectively, while the Gaussian represents a Gaussian up-shift of amplitude $15\%$ at its peak, centered at the diffraction minimum $q=1.84$$\hbox {fm}^{-1}$ and with a standard deviation of 0.25 $\hbox {fm}^{-1}$. The functional forms of the three S(q) are shown in the inset of Fig. 10 as well as the empirical perturbation, while the outset plot shows the Bessel expanded form factor and the special values of the momentum transfer $q_\nu$.

In all three test cases for S(q) the change on the radius did not exceed $2\%$, which is still a substantial increase compared to Offermann result [4] of a $0.28\%$ increase. The empirical perturbation showed a change of $0.25\%$, consistent with the reported result [4]. This contrast suggests that our overall strength $\delta (E_e)=30\%$ was too large and could imply that for the data range in Offermann work [4] $\delta (E_e)S(q)\ll 30\%$, as can be inferred by the small size of the empirical perturbation.

This empirical perturbation was only calculated at the special values $q_\nu$ and interpolated using a third degree spline and therefore, is not discarded that it’s strength can reach a peak of  $30\%$ in the excluded region around the diffraction minimum $1.6<q<1.95$ fm $^{-1}$. Indeed, the authors excluded this data to perform their analysis and avoid as much as possible the dispersive effects.