Lower bound on the proton charge radius from electron scattering data

TL;DR

This paper establishes a model-independent lower bound on the proton charge radius using low-$Q^2$ electron-proton scattering data, reaffirming the discrepancy with muonic hydrogen measurements.

Contribution

It introduces a method to derive a rigorous lower bound on the proton radius directly from finite-$Q^2$ data, avoiding model-dependent extrapolations.

Findings

Lower bound on proton radius: R_E > 0.850(1) fm

Reaffirms discrepancy between electron scattering and muonic hydrogen results

Method bypasses assumptions in traditional form-factor extrapolations

Abstract

The proton charge-radius determinations from the electromagnetic form-factor measurements in electron-proton scattering require an extrapolation to zero momentum transfer ($Q^2=0$) which is prone to model-dependent assumptions. We show that the data at finite momentum transfer can be used to establish a rigorous lower bound on the proton charge radius. Using the available $ep$ data at low $Q^2$, we obtain $R_E > 0.850(1)$ fm as the lower bound on the proton radius. This reaffirms the discrepancy between the $ep$ and muonic-hydrogen values, while bypassing the model-dependent assumptions that go into the fitting and extrapolation of the $ep$ data.

Figures

Tables

Table 1. Table 1: The lower-bound value of the proton charge radius, R E subscript 𝑅 𝐸 R_{E} (in fm), from two experiments and three experimental data sets. The error corresponds to the 95% confidence interval (i.e., ± 2 ​ σ plus-or-minus 2 𝜎 \pm 2\sigma ), obtained from statistical errors alone. These results are represented by the bands in Figs.  2 , 3 and 4 with the corresponding color-coding.

Dataset		Raw Average	Normalization-free
	GeV2	$\overline{)0.857\pm 0.003}$	$\overline{)0.850\pm 0.001}$
Bernauer et al. [4]	subset “1:3”	$\overline{)0.864\pm 0.005}$	$\overline{)0.851\pm 0.003}$
Mihovilović et al. [30]	all data	$\overline{)0.842\pm 0.011}$	$\overline{)0.854\pm 0.014}$

Equations

$$G_E(Q^2)$$

$$G_M(Q^2)$$

$$G_E(Q^2)$$

$$\frac{G_M(Q^2)}{1+\kappa }$$

$${⟨r^k⟩}_E$$

$$G_E(Q^2)$$

$$G_E(Q^2)=1-\frac{Q^2}{\pi }\underset{4m_{\pi }^2}{\overset{\infty }{\int }}\mathrm{d}t\frac{\mathrm{Im}{G}_E(t)}{t(t+Q^2)},$$

$$R_E^2(Q^2)\equiv -\frac{6}{Q^2}\ln G_E(Q^2),$$

$$\lim_{Q^{2}\rightarrow 0}R_{E}^{2}(Q^{2})=-6\,\frac{G_{E}^{\prime}(Q^{2})}{G_{E}(Q^{2})}\bigg{|}_{Q^{2}=0}=R_{E}^{2}.$$

$$G_E(Q^2)\le 1,$$

$$R_E^2(Q^2)=\frac{1}{\pi }\underset{4m_{\pi }^2}{\overset{\infty }{\int }}\mathrm{d}t\frac{\mathrm{Im}{R}_E^2(t)}{t+Q^2},$$

$$R_E^2(Q^2)\le R_E^2,\text{mbox}for{Q}^2\ge 0.$$

$$R_{E}^{2}(Q^{2})=R_{E}^{2}-\left(\mbox{$\frac{1}{20}$}\left<r^{4}\right>_{E}-\mbox{$\frac{1}{12}$}R_{E}^{4}\right)Q^{2}+\mathcal{O}(Q^{4}),$$

$$\sqrt{\mbox{$\frac{3}{5}$}\left<r^{4}\right>_{E}}>R_{E}^{2}.$$

$$G_{E}(0)-G_{E}(Q^{2})=4\pi\int_{0}^{\infty}\!{\rm d}r\,r^{2}\big{[}1-j_{0}(Qr)\big{]}\,\rho_{E}(r),$$

$$F_1(Q^2)=2\pi {\int }_0^{\infty }\mathrm{d}bb{J}_0(Qb){\rho }_{\perp }(b),$$

$$F_1(Q^2)\le 1.$$

$$F_1(Q^2)=4\pi {\int }_0^{\infty }\mathrm{d}r{r}^2{j}_0(Qr){\rho }_1(r),$$

$$J_0(x)$$

$$j_0(x)$$

$${\int }_0^{\infty }\mathrm{d}QQ{J}_l(Qb)J_l(Q{b}^{\prime })$$

$${\int }_0^{\infty }\mathrm{d}Q{Q}^2{j}_l(Qr)j_l(Q{r}^{\prime })$$

$${\rho }_{\perp }(b)$$

$$=$$

$$R_E^{2(\exp )}(Q^2)$$

$$R_{E}^{2\mathrm{(exp)}}(Q^{2})\nleq R_{E}^{2},\quad\mbox{for $Q^{2}<Q_{0}^{2}$},$$

$$R_E^2(Q_0^2)-\frac{6}{Q_0^2}\ln (1+ϵ)=R_E^2.$$

$$Q_{0}^{2}=\sqrt{\frac{-6\ln(1+\epsilon)}{\mbox{$\frac{1}{20}$}\left<r^{4}\right>_{E}-\mbox{$\frac{1}{12}$}R_{E}^{4}}}.$$

$$R_{ij}^2\equiv \frac{-6}{Q_j^2-Q_i^2}\ln \frac{G_E(Q_j^2)}{G_E(Q_i^2)}.$$

$$R_{ij}^2$$

$$R_{ij}^2<R_E^2(Q_i^2)<R_E^2.$$

Full text

1. 1 Introduction

2. 2 Basic ingredients of the radius extraction

3. 3 Positivity bounds

4. 4 Exploring the $ep$ scattering data

5. 4.1 Direct determination

6. 4.2 Overall normalization factor

7. 4.3 Towards normalization-free bound

8. 5 Conclusion

9. Acknowledgements

1 Introduction

The proton charge radius is traditionally accessed in elastic electron-proton ($ep$) scattering at small momentum transfers (low $Q$) [1, 2]. Recently, however, the accuracy of this method has been questioned in the context of the proton-radius puzzle, which is partially attributed to the discrepancy between the 2010 $ep$ scattering value of Bernauer et al. [3, 4] and the muonic-hydrogen ($\mu$H) extraction of the proton radius [5, 6], see Fig. 1. Meanwhile, as seen from the figure, the different extractions based on $ep$-scattering data have covered a whole range of values and hardly add-up into a coherent picture.

A “weak link” of the proton-radius extractions from $ep$ experiments is the extrapolation to zero momentum transfer. Namely, while the data taken in some finite-$Q^{2}$ range can directly be mapped into the proton (electric and magnetic) Sachs form factors $G_{E}(Q^{2})$ and $G_{M}(Q^{2})$, the radii extractions require the derivatives of those at $Q^{2}=0$, e.g.: $R_{E}=\sqrt{-6\,G_{E}^{\prime}(0)}$. As much as one believes that the slope at [math] is largely determined by the behavior at finite $Q^{2}$, it is not easy to quantify this relation with the necessary precision. The issues of fitting and extrapolation of the form-factor data have lately been under intense discussion, see, e.g., Refs. [14, 25, 26, 27]. Similar extrapolation problems should exist in the extractions based on lattice QCD, since the lowest momentum-transfer therein is severely limited by the finite volume.

Here, we show that the form-factor data at finite $Q^{2}$ provide a lower bound on the proton charge radius. A determination of this bound needs no extrapolation, therefore no major model assumptions, and should be based solely on experimental (or lattice) data. At the same time, given that some of the conventional extractions from $ep$ data show a considerably larger radius than the $\mu$H value, a strict lower bound, based purely on data, is potentially useful in understanding this discrepancy.

In what follows, we briefly recall the basic formulae in Sec. 2, introduce the quantity proposed to serve as the charge-radius bound in Sec. 3, obtain an empirical value for it based on proton electric form-factor data in Sec. 4 and conclude in Sec. 5.

2 Basic ingredients of the radius extraction

Let us recall that a spin-1/2 particle, such as the proton, has two electromagnetic form factors. These are either the Dirac and Pauli form factors: $F_{1}(Q^{2})$ and $F_{2}(Q^{2})$; or, the electric and magnetic Sachs form factors:

[TABLE]

with $M$ the particle mass. The Sachs form factors can be interpreted as the Fourier transforms of the charge and magnetization distributions, $\rho_{E}(\vec{r}\,)$ and $\rho_{M}(\vec{r}\,)$, in the Breit frame. Strictly speaking, this relation holds only for spherically symmetric densities, in which case one has, see e.g. Ref. [28]:

[TABLE]

where $j_{0}(x)=\frac{\sin x}{x}$ is the spherical Bessel function, and $\kappa$ is the anomalous magnetic moment of the proton. Note that these are Lorentz-invariant expressions, hence, the spherically symmetric charge and magnetization distributions are, just as the form factors, Lorentz-invariant quantities.

The radii are introduced through the density moments, which, for even $k$, can be given by the form-factor derivatives at [math]:

[TABLE]

and similarly for the magnetic radii with $\rho_{E}$ replaced by $\rho_{M}$, and $G_{E}$ replaced by $G_{M}/(1+\kappa)$, respectively. Therefore, the Taylor expansion of the form factor around $Q^{2}=0$ is written as:

[TABLE]

The subject of interest is the root-mean-square (rms) radius (or, simply the charge radius): $R_{E}=\sqrt{\left<r^{2}\right>_{E}}$. Ideally, it could be extracted by fitting the first few terms of the above Taylor expansion of the form factor to the experimental data at low $Q^{2}$. In practice, however, this does not work. The main reason is that the convergence radius of the Taylor expansion is limited by the onset of the pion-production branch cut for time-like photon momenta at $Q^{2}=-4m_{\pi}^{2}$ (the nearest singularity, as far as the strong interaction is concerned), and there are simply not many $ep$ data for $Q^{2}\ll 4m_{\pi}^{2}\approx 0.08$ GeV2.

A viable approach to fit to higher $Q^{2}$ is, instead of the Taylor expansion, to use a form which takes the singularities into account. This is done in the $z$-expansion [12] and dispersive fits [21, 23, 24]. These approaches have, however, other severe limitations. The $z$-expansion only deals with the first singularity and therefore extends the convergence radius to $9m_{\pi}^{2}$ only. The dispersive approach is based on an exact dispersion relation for the form factor:

[TABLE]

which, in principle, accounts for all singularities. Unfortunately, it requires the knowledge of the spectral function, $\operatorname{Im}G_{E}(t)$, which is not directly accessible in experiment, and needs to be modeled. Chiral perturbation theory can only provide a description of this function in the range of $t\ll 1$ GeV2. Despite the recent progress in the empirical description of the spectral function [29], the problem of model dependence of the radius extraction in the dispersive approach remains to be non-trivial.

3 Positivity bounds

Given the aforementioned issues in extracting the charge radius from form-factor data, we turn to establishing a bound on the radius, rather than the radius itself. The advantage is that the bound will follow from the finite-$Q^{2}$ data alone and needs no extrapolations or model assumptions.

To this end we consider the following quantity:

[TABLE]

which in the real-photon limit yields the radius squared:

[TABLE]

As will be argued in Sec. 3, the spacelike ($Q^{2}\geq 0$) proton form factor is bounded from above:

[TABLE]

and hence, the above log-function is positive, $R_{E}^{2}(Q^{2})\geq 0$. Furthermore, if $G_{E}$ falls with increasing $Q^{2}$ not faster than by a power law, then $R_{E}^{2}(Q^{2})$ falls as well. The analytic properties of $G_{E}$, in the absence of zeros, are inherited by its logarithm. The subtracted dispersion relation (5) for the form factor then leads to an unsubtracted one for $R_{E}^{2}$:

[TABLE]

where $\operatorname{Im}R^{2}_{E}(t)=(6/t)\varphi_{E}(t)$, and $\varphi_{E}(t)\geq 0$ is the phase defined through $G_{E}(t)=|G_{E}(t)|e^{i\varphi_{E}(t)}$. This dispersion relation shows that the function is monotonic in the spacelike region. The latter allows one to establish a lower bound on the radius:

[TABLE]

Substituting in here the Taylor expansion, Eq. (4), one has:

[TABLE]

and so, in order for the bound to hold at arbitrarily low $Q^{2}$, the fourth and second moments must satisfy the following inequality:111Based on Eq. (9), one can claim that $R^{2}_{E}(Q^{2})$ is completely monotonic, i.e.: $(-1)^{n}{\rm d}^{n}R_{E}^{2}(Q^{2})/{\rm d}(Q^{2})^{n}\geq 0$, from which the lower bounds on other radii can be derived. The lowest values of the radii are given in terms of the charge radius $R_{E}$, and can all be obtained from Taylor-expanding the following form of the form factor: G_{E}^{\mathrm{(min)}}(Q^{2})=\exp\!\big{(}\!-\frac{1}{6}R_{E}^{2}Q^{2}\big{)}.

[TABLE]

We have checked that this non-trivial hierarchical condition on the radii, which follows from the lower bound Eq. (10), is verified in existing empirical parametrizations of the proton form factor, of which the dipole form, $G_{E}(Q^{2})=[1+Q^{2}/(0.71\,\mathrm{GeV}^{2})]^{-2}$, is the simplest one.

The fact that $R_{E}^{2}(Q^{2})$ is monotonically increasing towards $Q^{2}=0$ means that the best bound is obtained at lowest accessible $Q^{2}$. In practice, however, it depends on the size of the experimental errors, including the uncertainty in the overall normalization of the form factor. We discuss this in detail in Sec. 4, when obtaining the empirical value of the bound from experimental data. In the rest of this section we focus on the proof of Eq. (8).

The unitary bound on the proton form factor, given in Eq. (8), and subsequently the radius bound, given in Eq. (10), follow from positivity of the corresponding charge density distribution: $\rho_{E}(r)\geq 0$. Indeed, from Eq. (2a),

[TABLE]

with the property of the Bessel function $j_{0}(x)\leq 1$, and the positivity of $\rho_{E}(r)$, we can see see that the integrand on the right-hand side is positive definite, and Eq. (8) follows upon substituting $G_{E}(0)=1$ on the left-hand side.

There is a concern [31] that the proton charge density is not necessarily positive definite, and only the transverse charge density is ($\rho_{\perp}(b)\geq 0$). The latter relates to the Dirac form factor through the two-dimensional Fourier transform:

[TABLE]

where $J_{0}(x)$ is the cylindrical Bessel function. However, the positivity of the transverse charge density is sufficient to prove the unity bound of Eq. (8). To see this, one may apply the previous argument [cf. Eq. (13)] to Eq. (14) using $J_{0}(x)\leq 1$, and derive the bound on the Dirac form factor:

[TABLE]

Then, the unitary bound on $G_{E}$ follows from its definition in terms of the Dirac and Pauli form factors, see Eq. (1a), by taking into account the conditions $F_{1}(Q^{2})\leq 1$ and $F_{2}(Q^{2})\geq 0$. The latter is valid for the proton in at least the low-$Q$ region, as can be seen empirically from $F_{2}(0)=\kappa$, with $\kappa\simeq 1.79$ the anomalous magnetic moment of the proton.

While the unity bound on $G_{E}$ follows from the positivity of $\rho_{E}(r)$, the reverse is not necessarily true. Therefore, the proof based on the positivity of the transverse charge density $\rho_{\perp}(b)$ does not necessarily imply the positivity of $\rho_{E}(r)$. Introducing $\rho_{1}(r)$ as the three-dimensional Fourier-transform of the Dirac form factor, we have:

[TABLE]

and matching it to Eq. (14), we obtain its relation to the transverse density:222Here we recall the following relations between the spherical and cylindrical Bessel functions:

$\displaystyle J_{0}(x)$ $\displaystyle=$ $\displaystyle\frac{2}{\pi}\int_{x}^{\infty}{\rm d}x^{\prime}\,x^{\prime}\,\frac{j_{0}(x^{\prime})}{\sqrt{x^{\prime\,2}-x^{2}}},$

$\displaystyle j_{0}(x)$ $\displaystyle=$ $\displaystyle\frac{1}{x}\int_{0}^{x}{\rm d}x^{\prime}\,x^{\prime}\,\frac{J_{0}(x^{\prime})}{\sqrt{x^{2}-x^{\prime\,2}}},$

as well as their orthogonality:

$\displaystyle\int_{0}^{\infty}{\rm d}Q\,Q\,J_{l}(Qb)\,J_{l}(Qb^{\prime})$ $\displaystyle=$ $\displaystyle\frac{1}{b}\delta(b-b^{\prime}),$

$\displaystyle\int_{0}^{\infty}{\rm d}Q\,Q^{2}\,j_{l}(Qr)\,j_{l}(Qr^{\prime})$ $\displaystyle=$ $\displaystyle\frac{\pi}{2r^{2}}\delta(r-r^{\prime}).$

[TABLE]

The two are thus related by the Abel transform [32, p. 351 et seqq.]. It infers $\rho_{\perp}\geq 0$, for $\rho_{1}\geq 0$, while the reverse is not necessarily true.

4 Exploring the $ep$ scattering data

4.1 Direct determination

We now proceed to obtaining the lower bound on the proton charge radius from $ep$ scattering data. The first step is to convert the experimental data for $G_{E}(Q^{2})$ to $R_{E}^{2}(Q^{2})$, using the definition (6). The presently available data in the region well below the pion-pair production scale (here we chose $Q^{2}<0.02$ GeV2) are shown in Fig. 2. The light-blue points are from the dataset of Bernauer et al. [3, 4]. The light-red data points are from the recent initial-state radiation (ISR) experiment at MAMI [30]. In both cases we deal with the statistical error bars only. The two points at $Q^{2}=0$ indicate the muonic-hydrogen (green) and Bernauer’s $ep$-scattering (dark-blue) values of the proton charge radius.

In principle, every data point in Fig. 2, at finite $Q^{2}$, provides a lower bound on the proton charge radius. For a more accurate value, we can average over any subset of these data. In the figure, the horizontal blue band is the statistical average of Bernauer’s dataset, whereas the red band is the statistical average of the ISR dataset. The corresponding values for the lower bound are presented in the “Raw-average” column of Table 1.

This is how ideally the bound should be determined from the experimental data. However, the present experimental data have systematic uncertainties of which the most acute one is the unknown absolute normalization of the cross section. The Bernauer dataset, for example, is normalized in conjunction with the radius extraction. Thus, the data normalization and the extrapolation to $Q^{2}=0$ are done simultaneously in the same fit. Moreover, one can obtain an equally good representation of Bernauer’s data by using a lower value of the radius and different normalization factors [33, 34]. In what follows, we attempt to deal with this problem and construct a lower bound which is independent of the overall normalization.

4.2 Overall normalization factor

To see how the normalization uncertainty affects the bound, let us suppose the experimental form factor has a small normalization error $\epsilon$, such that $G_{E}^{\mathrm{(exp)}}=(1+\epsilon)\,G_{E}$, with $G_{E}$ having the usual interpretation. Then,

[TABLE]

If $\epsilon$ is positive, this is not a problem — the lower bound is preserved: $R_{E}^{2\mathrm{(exp)}}(Q^{2})\leq R_{E}^{2}$, for $\epsilon\geq 0$. In the case of $\epsilon<0$, in a certain low-$Q^{2}$ region, the bound is violated:

[TABLE]

where $Q_{0}$ is the root of the following equation:

[TABLE]

Assuming $Q_{0}$ is small, we can use the expanded form of $R_{E}^{2}(Q^{2})$ in Eq. (11), to find:

[TABLE]

For example, taking $\epsilon=-0.001$ and typical values of the radii [35], this equation gives $Q_{0}^{2}\approx 0.01$ GeV2. Therefore, one strategy for avoiding the possible normalization issue is to drop the data below a certain $Q^{2}$ value from the lower-bound evaluation. A more efficient strategy is to use values at different $Q^{2}$ to cancel the overall normalization, as illustrated in what follows.

4.3 Towards normalization-free bound

Having the form-factor data at a number of different points $Q^{2}_{i}$ (with $i=\overline{1,N}$), one may consider the following quantity:

[TABLE]

The obvious advantage of this form is that the overall-normalization uncertainty cancels out. At the same time, each element of the symmetric matrix $R_{ij}^{2}$ provides a lower bound: $R^{2}_{ij}<R_{E}^{2}$, for any $i,j$. This can be seen by rewriting it identically as:

[TABLE]

where $R^{2}_{E}(Q^{2})$ is the lower-bound function of Eq. (6). The second term is negative-definite, given that $R^{2}_{E}(Q^{2})$ is monotonically decreasing, and hence:

[TABLE]

Because of the first inequality, the bound obtained from $R_{ij}^{2}$ is lower than the one obtained from $R^{2}_{E}(Q^{2})$ and therefore is less optimal. Yet, it may be more precise when applied to real data, because of cancellation of systematic uncertainties which affect the absolute normalization of the experimental cross sections.

To illustrate the workings of this method, let us consider Fig. 3, where we plot the elements $R_{ij}$ for the two datasets, as a function of $\Delta Q^{2}=Q_{j}^{2}-Q_{i}^{2}$. The blue and red bands provide the two corresponding bounds obtained by fitting a horizontal line using the NonlinearModelFit routine of Mathematica [36]. The corresponding values are given in the “Normalization-free” column of Table 1.

Note that the error on $R_{ij}^{2}$ is decreasing with the increase of $\Delta Q^{2}$, and hence the obtained bounds are driven by the higher $\Delta Q^{2}$ interval. In fact, one can apply a cut on the lowest $\Delta Q^{2}$ values, without affecting the result.

Of course, this method only works if all the data points of a given dataset have the same normalization factor. In reality, the experiment of Bernauer et al. [4] has a complicated normalization procedure, involving 31 normalization factors, and one can manage to obtain significant shifts of the data points by a different fit of these factors [33, 34]. These shifts could then be considered as a systematic normalization uncertainty which is only partially attributed to an overall normalization.

Nonetheless, one can identify subsets where the difference in normalization is overall. In the experimental data of Bernauer et al. these are, for example, normalization sets (see Supplement in [4]):

3 (spectrometer A, $180$ MeV beam energy),

• 2.

1:3 (spectrometer B, $180$ MeV beam energy),

• 3.

6:9 (spectrometer B, $315$ MeV beam energy).

We have applied the $R_{ij}^{2}$ method to each of this subsets separately (for $Q^{2}\leq 0.02$ GeV2) and obtained the same results (within statistical errors). The most precise result is the one from the 1:3 subset, because it is the largest in this region. The results for this dataset are shown in Fig. 4 and the second row of Table 1. The latter value is indeed a normalization-free bound.

We obtained the same results with the two datasets of Higinbotham [33], generated from the data of Bernauer et al. [4], and corresponding to significantly different radii. A subset of 106 data points in the very low-$Q$ region ($Q^{2}\leq 0.012$ GeV2) differs in an overall factor between the two datasets. Applying our method to this subset leads to the lower-bound value of $0.851(3)$ fm, for both datasets, in exact agreement with our normalization-free result for the subset 1:3. We hence conclude that our method leads to a robust and accurate determination of the lower bound on $R_{E}$ from the form-factor data, even when they are prone to normalization uncertainties.

The lower bound resulting from the Bernauer dataset [$R_{E}>0.848$ fm at 95% confidence level (CL)] is very accurate, although we emphasize that the error is only statistical. It can be compared with the recent proton-radius extractions in Fig. 1. It is somewhat in conflict with the $\mu$H values [5, 6], and the Garching measurement of the $2S-4P$ transition frequency in H [9].

5 Conclusion

An extraction of the proton charge radius from $ep$ scattering requires an extrapolation to zero momentum transfer, which nowadays is entangled in the analysis of $ep$ data. We aim here to leave the extrapolation issues out of the interpretation of $ep$ data. We show that the $ep$ scattering may directly provide a lower bound on the proton charge radius, cf. Eq. (10) with Eq. (6). Thus, the lower bound is a directly observable quantity (to the extent that the form factor is), and is a more rigorous experimental outcome than the charge radius itself.

We have attempted a first determination of the lower bound on the proton charge radius from the available data in the region of $Q^{2}$ below 0.02 GeV2. Our results for the two presently available experiments are given in Table 1. The last column therein shows the lower-bound values with the overall-normalization uncertainty being canceled out.

The lower bound, $R_{E}>0.848$ fm (95% CL), resulting from our “normalization-free” analysis of the $ep$ data of Bernauer et al. [4], rules out the shaded area in Fig. 1. The figure also shows the results of recent proton-radius determinations. In particular, this $ep$ bound is in disagreement with the muonic-hydrogen values (green dots). We emphasize that the present determination of the lower bound does not involve any fitting of the $Q^{2}$-dependence with subsequent extrapolation to $Q^{2}=0$. On the other hand, the present analysis does not account for systematic errors in the experimental data, except for those that contribute to the overall normalization.

As the lower-bound function, defined in Eq. (6), is monotonically increasing with decreasing $Q^{2}$, the most stringent bound will be obtained from the lower $Q^{2}$ range, provided that the accuracy does not deteriorate with decreasing $Q^{2}$. Therefore, with the forthcoming results of the PRad experiment [37, 38], one hopes to obtain a much better determination of the lower bound. The PRad data will reach down to $2\times 10^{-4}\,\mathrm{GeV}^{2}$ and include a simultaneous measurement of the Møller scattering. The latter will allow to further reduce the systematic uncertainties.

Acknowledgements

We are grateful to Jan Bernauer, Michael Distler, Miha Mihovilović, and Thomas Walcher for sharing their data with us and helpful communications; to Douglas Higinbotham for checking some of our results and an interesting discussion; to Patricia Bickert, Ashot Gasparyan, Vadim Lensky, and Stefan Scherer for useful remarks on the manuscript. This work was supported by the Swiss National Science Foundation and the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center 1044 [The Low-Energy Frontier of the Standard Model].