Sensitivity of the elastic electron–proton cross section to the proton radius

Abstract

The precise determination of the proton radius from recent elastic scattering electron–proton data is discussed. The necessary precision on the elastic cross section to discriminate among the values coming from atomic spectroscopy is scrutinized in terms of the relevant quantity, i.e., the derivative of the form factor. It is shown that its uncertainty is two orders of magnitude larger that the error on the cross section, that is the measured observable. Different fits of the available data and of their discrete derivative, with analytical constraints are shown. The systematic error associated to the radius is evaluated taking into account the uncertainties from different sources, as the extrapolation to the static point, the choice of the class of fitting functions and the range of the data sample. This error is shown to be also orders of magnitude larger than commonly assumed.

Appendix A: The Schwarz’s inequality for mean values of even powers of the radius

For any pair functions f(x), $g(x)\in L^2(E)$, where $L^2(E)$ represents the set of functions which are quadratically Lebesgue integrable on $E\subset {\mathbb {R}}$, it holds the Schwarz’s inequality [31]

$$\begin{aligned} \left| \int _E f^*(x)g(x)dx\right| ^2 \le \int _E \left| f(x)\right| ^2dx \int _E \left| g(x)\right| ^2dx. \end{aligned}$$

(A.1)

Given a spherically symmetric density $\rho \left( \mathbf {r}\right) \equiv \rho (r)$, the mean value of $r^n$, i.e., the normalized n-th moment of the density around the origin, with $n\in {\mathbb {N}}$, is

$$\begin{aligned} \left\langle r^n\right\rangle= & {} \frac{\displaystyle \int r^n\rho (r) d^3\mathbf {r}}{\displaystyle \int \rho (r) d^3\mathbf {r}}= \frac{\displaystyle 4\pi \int _0^\infty r^{n+2}\rho (r) \, dr}{\displaystyle 4\pi \int _0^\infty r^{2} \rho (r) \, dr}. \end{aligned}$$

By defining the radial density $\eta (r)=4\pi r^2\rho (r)$, and assuming the unitary normalization

$$\begin{aligned} \int \rho (r) d^3\mathbf {r}= 4\pi \int _0^\infty r^{2} \rho (r) \, dr= \int _o^\infty \eta (r)dr= 1, \end{aligned}$$

the n-th moment around $r=0$ can be written as

$$\begin{aligned} \left\langle r^n\right\rangle =\int _0^\infty r^n\eta (r) \, dr. \end{aligned}$$

If the mean values of all integer powers of the radius, or, equivalently, all moments around the origin of the radial density $\eta (r)$ exist and are limited, then the function $f_n(r)=\sqrt{\eta (r)}\,r^n \in L^2(0,\infty )$, for all $n\in {\mathbb {N}}$, indeed

$$\begin{aligned} \int _0^\infty \left| f_n(r)\right| ^2dr= \int _0^\infty r^{2n}\eta (r)dr=\left\langle r^{2n}\right\rangle <\infty . \end{aligned}$$

Using the Schwarz’s inequality of Eq. (A.1), with $f(r)=f_n(r)=\sqrt{\eta (r)}\,r^n$ and $g(r)=f_0(r)=\sqrt{\eta (r)}$, we obtain the inequality $\left\langle r^n \right\rangle ^2\le \left\langle r^{2n}\right\rangle$, indeed

$$\begin{aligned} \left| \int _0^\infty \!\!\!\! f^*(r)g(r)dr\right| ^2\le & {} \int _0^\infty \!\!\!\! |f(r)|^2 dr \int _0^\infty \!\!\!\! |g(r)|^2 dr \nonumber \\ \left( \int _0^\infty \!\!\!\!\eta (r)\,r^n dr\right) ^2\le & {} \int _0^\infty \!\!\!\! \eta (r)\,r^{2n} dr \int _0^\infty \!\!\!\! \eta (r)dr \nonumber \\ \left\langle r^n \right\rangle ^2\le & {} \left\langle r^{2n}\right\rangle . \end{aligned}$$

(A.2)

On the other hand, with $f(r)=\sqrt{\eta (r)}\,r^{n-1/2}$ and $g(r)=\sqrt{\eta (r)}\,r^{1/2}$, using Eq. (A.2) with $n=1$, i.e., $\left\langle r\right\rangle \le \left\langle r^2\right\rangle ^{1/2}$,

$$\begin{aligned} \left( \int _0^\infty \eta (r)\,r^n\, dr\right) ^2\le & {} \int _0^\infty \eta (r)\,r^{2n-1} dr \int _0^\infty \eta (r)\,r \,dr \\ \left\langle r^n \right\rangle ^2\le & {} \left\langle r^{2n-1}\right\rangle \left\langle r\right\rangle \le \left\langle r^{2n-1}\right\rangle \left\langle r^2\right\rangle ^{1/2} , \end{aligned}$$

so that

$$\begin{aligned} \left\langle r^{2n-1}\right\rangle \ge \left\langle r^n\right\rangle ^2 \left\langle r^2\right\rangle ^{-1/2}. \end{aligned}$$

(A.3)

Inequalities of Eqs. (A.2) and (A.3) can be generalized as it follows

$$\begin{aligned} \left\langle r^{2n-b}\right\rangle \ge \left\langle r^n\right\rangle ^2 \left\langle r^2\right\rangle ^{-b/2},\forall \,n\in {\mathbb {N}}, \end{aligned}$$

(A.4)

with $b\in \{0,1\}$. In order to apply iteratively the inequality of Eq. (A.4), we consider, for a generic integer $n\in {\mathbb {N}}$, the following expression in terms of powers of two

$$\begin{aligned} n= & {} 2^N-\sum _{j=0}^{N-2}b_j 2^{j} =2^N-\sum _{j=1}^{N-2}b_j 2^{j}-b_0 \nonumber \\= & {} 2\left( 2^{N-1}-\sum _{j=1}^{N-2}b_j 2^{j-1}\right) -b_0, \end{aligned}$$

(A.5)

where

$$\begin{aligned} N=\min _{j\in {\mathbb {N}}}\left\{ 2^j\ge n\right\} , \end{aligned}$$

(A.6)

and the $(N-1)$-tuple $\left( b_0,b_1,\ldots ,b_{N-2}\right) \in \left\{ 0,1\right\} ^{N-1}$. It is quite easy to show that such an expression always exists, indeed the sequence of 0’s and 1’s, represented by the $(N-1)$-tuple $\left( b_0,b_1,\ldots ,b_{N-2}\right)$, is the difference $2^N-n$ written in base-2 (binary notation), i.e.,

$$\begin{aligned} 2^N-n=\left( b_{N-2}b_{N-3}\ldots b_{1}b_{0}\right) _2, \end{aligned}$$

(A.7)

(the number “2” subscript on the parentheses indicates the base-2 notation). In particular it does contain only $N-1$ digits, because the difference between $2^N$ and a generic binary number with N digits is lower than $2^{N-1}$. In fact, if the difference of Eq. (A.7) would be a binary number with $N-1$ digits $\left( b_{N-1}b_{N-2}\ldots b_{1}b_{0}\right) _2$, where $b_{N-1}=1$, then $\left( b_{N-1}b_{N-2}\ldots b_{1}b_{0}\right) _2\ge 2^{N-1}$ and from Eq. (A.7)

$$\begin{aligned} n=2^N-\left( b_{N-1}b_{N-2}\ldots b_{1}b_{0}\right) _2 \le 2^N-2^{N-1}=2^{N-1}. \end{aligned}$$

But this is in contradiction with the definition of N given in Eq. (A.6), from which we would arrive at the absurd

$$\begin{aligned} N=\min _{j\in {\mathbb {N}}}\left\{ 2^j\ge n\right\} =N-1. \end{aligned}$$

Consider the sequence of inequalities that can be obtained using the inequality of Eq. (A.4) and the expression of n of Eq. (A.5)

$$\begin{aligned} \left\langle r^{2n}\right\rangle\ge & {} \left\langle r^n\right\rangle ^2 =\left\langle r^{2\left( 2^{N-1}-\sum _{j=1}^{N-2}b_j 2^{j-1}\right) -b_0}\right\rangle ^2 \\\ge & {} \left\langle r^{2^{N-1}-\sum _{j=1}^{N-1}b_j 2^{j-1}}\right\rangle ^{2^2} \left\langle r^2\right\rangle ^{-b_0} \\= & {} \left\langle r^{2\left( 2^{N-2}-\sum _{j=2}^{N-2}b_j 2^{j-2}\right) -b_1}\right\rangle ^{2^2} \left\langle r^2\right\rangle ^{-b_0} \\\ge & {} \left\langle r^{2^{N-2}-\sum _{j=2}^{N-2}b_j 2^{j-2}}\right\rangle ^{2^3} \left\langle r^2\right\rangle ^{-b_0-2b_1} \\= & {} \left\langle r^{2\left( 2^{N-3}-\sum _{j=3}^{N-2}b_j 2^{j-3}\right) -b_2 }\right\rangle ^{2^3} \left\langle r^2\right\rangle ^{-b_0-2b_1} \\\ge & {} \left\langle r^{2^{N-3}-\sum _{j=3}^{N-2}b_j 2^{j-3}}\right\rangle ^{2^4} \left\langle r^2\right\rangle ^{-b_0-2b_1-2^2b_2} \\= & {} \left\langle r^{2\left( 2^{N-4}-\sum _{j=4}^{N-2}b_j 2^{j-4}\right) -b_3 }\right\rangle ^{2^4} \left\langle r^2\right\rangle ^{-b_0-2b_1-2^2b_2} \\\ge & {} \cdots \\\ge & {} \left\langle r^{2^{N-k}-\sum _{j=k}^{N-2}b_j 2^{j-k}}\right\rangle ^{2^{k+1}} \left\langle r^2\right\rangle ^{-\sum _{j=0}^{k-1}b_j 2^j} \\\ge & {} \cdots \\\ge & {} \left\langle r^{2^{2}-\sum _{j=N-2}^{N-2}b_j 2^{j-N+2}}\right\rangle ^{2^{N-1}} \left\langle r^2\right\rangle ^{-\sum _{j=0}^{N-3}b_j 2^j} \\= & {} \left\langle r^{2^{2}-b_{N-2}}\right\rangle ^{2^{N-1}} \left\langle r^2\right\rangle ^{-\sum _{j=0}^{N-3}b_j 2^j} , k=N-2, \end{aligned}$$

where the last one is the $(N-2)$-th iteration. From Eq. (A.4) follows that the left-hand-side of the last inequality is greater than $\left\langle r^{2}\right\rangle ^{n}$, indeed

$$\begin{aligned} \begin{array}{rcl} \left\langle r^{2n}\right\rangle &{} \ge &{} \left\langle r^{2^{2}-b_{N-2}}\right\rangle ^{2^{N-1}} \left\langle r^2\right\rangle ^{-\sum _{j=0}^{N-3}b_j 2^j} \\ &{} \ge &{} \left( \left\langle r^{2}\right\rangle ^2 \left\langle r^{2}\right\rangle ^{-b_{N-2}/2}\right) ^{2^{N-1}} \left\langle r^2\right\rangle ^{-\sum _{j=0}^{N-3}b_j 2^j}\\ &{}=&{}\left\langle r^{2}\right\rangle ^{2^N-b_{N-2}2^{N-2}-\sum _{j=0}^{N-3}b_j 2^j} \\ &{}=&{}\left\langle r^{2}\right\rangle ^{2^N-\sum _{j=0}^{N-2}b_j 2^j}\\ \ &{}=&{}\left\langle r^{2}\right\rangle ^{n},\\ \end{array} \end{aligned}$$

where the last identity follows from Eq. (A.5), in summary we obtain the inequality

$$\begin{aligned} \left\langle r^{2n}\right\rangle \ge \left\langle r^{2}\right\rangle ^{n}. \end{aligned}$$

(A.8)