Defining the Proton Radius: a Unified Treatment

TL;DR

This paper unifies hydrogen spectroscopy and parton distribution approaches to define the proton radius as a relativistically proper quantity, clarifying its measurement and theoretical interpretation.

Contribution

It introduces a relativistically correct, two-dimensional charge density and a new moment expansion (RME) to accurately define the proton radius.

Findings

Hydrogen spectroscopy and elastic lepton scattering measure the same proton radius.

The traditional derivation of the mean-square radius as a 3D density moment is incorrect.

A new relativistically invariant moment expansion (RME) is proposed.

Abstract

Background: There is significant current interest in knowing the value of the proton radius and also its proper definition. Purpose: Combine the disparate literatures of hydrogen spectroscopy and diverse modern parton distributions to show that the quantity $r_p^2\equiv -6 G_E'(0)$ is the relativistically proper definition that originates from the separate bodies of work. Methods: Use perturbation theory, light-front dynamics and elementary techniques to find relativistically correct definitions of the proton radius and charge density. Results: It is found that the very same proton radius is accessed by measurements of hydrogen spectroscopy and elastic lepton scattering. The derivation of the mean-square radius as a moment of a spherically symmetric three-dimensional density is shown to be incorrect. A relativistically-correct, two-dimensional charge density is related to the diverse modern literature of various parton distributions. Relativistically invariant moments thereof are derived in a new moment expansion, the RME.

Equations

$$-6G_E^{\prime }(0)\equiv r_p^2,$$

$$H_0=\frac{{\vec{p}}^2}{2m}+\frac{{\vec{p}}^2}{2M}-\frac{\alpha }{r}.$$

$$V_{\mathrm{Breit}}=\frac{\pi \alpha }{2}\left(\frac{1}{m^2}+\frac{1}{M^2}\right)\delta (\vec{r})-\frac{\alpha }{2mMr}\left({\vec{p}}^2+\frac{\vec{r}(\vec{r}\cdot \vec{p})\cdot \vec{p}}{r^2}\right)+\frac{\alpha }{r^3}\left(\frac{1}{4m^2}+\frac{1}{2mM}\right)[\vec{r}\times \vec{p}]\cdot \vec{\sigma }.$$

$$E_{nj}=m+M-\frac{m_r{\alpha }^2}{2n^2}-\frac{m_r{\alpha }^4}{2n^3}\left(\frac{1}{j+\frac{1}{2}}-\frac{3}{4n}+\frac{m_r}{4n(m+M)}\right)+\frac{{\alpha }^4{m}_r^3}{2n^3{M}^2}\left(\frac{1}{j+\frac{1}{2}}-\frac{1}{l+\frac{1}{2}}\right)(1-{\delta }_{l0}),$$

$${\Gamma }^{\mu }={\gamma }^{\mu }{F}_1(Q^2)+i\frac{{\sigma }^{\mu \nu }}{2M}\kappa F_2(Q^2),$$

$$G_E(Q^2)=F_1(Q^2)-\tau \kappa F_2,G_M(Q^2)=F_1(Q^2)+\kappa F_2(Q^2),$$

$$\bar{u}({\vec{p}}^{\prime },s^{\prime }){\Gamma }^{0}u(\vec{p},s)=(1-\frac{{\vec{q}}^2}{8M^2})G_E({\vec{q}}^2)$$

$$G_E({\vec{q}}^2)=1+{\vec{q}}^2{G}_E^{\prime }(0).$$

$$\bar{u}({\vec{p}}^{\prime },s^{\prime }){\Gamma }^{0}u(\vec{p},s)\to 1-\frac{{\vec{q}}^2}{8M^2}+{\vec{q}}^2{G}_E^{\prime }(0).$$

$$V_C({\vec{q}}^2)=-\frac{4\pi \alpha }{{\vec{q}}^2}(1+{\vec{q}}^2{G}_E^{\prime }(0))=-4\pi \alpha (\frac{1}{{\vec{q}}^2}+G_E^{\prime }(0))$$

$$V_C(r)=-\int \frac{d^{3}q}{(2\pi {)}^3}{e}^{-i\vec{q}\cdot \vec{r}}4\pi \alpha (\frac{1}{{\vec{q}}^2}+G_E^{\prime }(0))=-\frac{\alpha }{r}-4\pi \alpha G_E^{\prime }(0)\delta (\vec{r}).$$

$$\Delta V_C(\vec{r})=-4\pi \alpha G_E^{\prime }(0)\delta (\vec{r}).$$

$$\Delta E=⟨{\psi }_{nl}|\Delta V_C|{\psi }_{nl}⟩=-4\pi \alpha G_E^{\prime }(0)|{\psi }_{n0}(0){|}^2{\delta }_{l0}.$$

$$G_E^{\prime }(0)=-\frac{r_p^2}{6},$$

$$G_E(Q^2)=1-\frac{r_p^2}{6}{\vec{q}}^2$$

$$\Delta E=4\pi \alpha \frac{r_p^2}{6}|{\psi }_{n0}(0){|}^2{\delta }_{l0},$$

$$\frac{d\sigma }{d\Omega }={\left(\frac{d\sigma }{d\Omega }\right)}_M\times [G_E^2+\frac{\tau }{ϵ}{G}_M^2]\frac{1}{1+\tau },$$

$$\underset{Q_L^2\to 0}{\lim }F(Q_L^2)=1-\frac{Q_L^2{a}^2}{6}+\cdots ,$$

$$F(Q_L^2)={\int }_0^{\infty }\rho (\mathbf{r})e^{i{\vec{Q}}_L\cdot \mathbf{r}}{d}^{3}r,$$

$$F(Q_L^2)=1-\frac{Q_L^2}{6}⟨r^2⟩+\frac{Q_L^2}{120}⟨r^4⟩+\cdots$$

$$⟨r^n⟩=\int r^n\rho (\mathbf{r})d^{3}r.$$

$${\rho }_{\mathrm{NR}}(r)\equiv \int \frac{d^{3}Q}{(2\pi {)}^3}{e}^{-i\mathbf{Q}\cdot \mathbf{r}}{G}_E(Q^2).$$

$$⟨r^2{⟩}_{\mathrm{NR}}\equiv \int d^{3}r{r}^2{\rho }_{\mathrm{NR}}(r)=-6G_E^{\prime }(0)$$

$$⟨r^4{⟩}_{\mathrm{NR}}\equiv \int d^{3}r{r}^4{\rho }_{\mathrm{NR}}(r)=60G_E^{\prime \prime }(0).$$

$${\psi }_{B.S.}(P,p)=\frac{1}{p^2-m_q^2+iϵ}\frac{1}{(P-p{)}^2-M_S^2+iϵ}=\frac{1}{p^2-m_q^2+iϵ}\frac{1}{M^2-2P\cdot p+p^2-M_S^2+iϵ},$$

$$P\cdot p=\sqrt{M^2+{\mathbf{q}}^2/4}{p}^0+\mathbf{q}\cdot \mathbf{p}/2,P^{\prime }\cdot p=\sqrt{M^2+{\mathbf{q}}^2/4}{p}^0-\mathbf{q}\cdot \mathbf{p}/2$$

$$\rho \equiv \bar{u}(\vec{q}/2,s_z){\Gamma }^{0}u(-\vec{q}/2,s_z)=G_E(Q^2),$$

$$|\Psi ⟩=\int d^{3}Pg(\vec{P})|P,s⟩,$$

$$|g(\vec{P}){|}^2\to \delta (\vec{P}).$$

$$g(\vec{P})=\frac{R^{3/2}}{{\pi }^{3/4}}\exp [-{\vec{P}}^2{R}^2/2],$$

$$M^{(2)}\equiv \int d^{3}q\int d^{3}p{g}^{\ast }(\vec{p}+\vec{q}/2)g(\vec{p}-\vec{q}/2)\int d^{3}x{x}^2⟨P^{\prime },\lambda |j^0(x)|\vec{P},\lambda ⟩,$$

Full text

I Introduction

What is the value of the radius of the proton? This question has generated much interest since the publication of the results of the muon-hydrogen spectroscopy experiment in 2010 pohl:2010zza and its confirmation Antognini:1900ns . The proton radius was measured to be $r_{p}=0.84184(67)$ fm, which contrasted with the value obtained from electron-hydrogen spectroscopy $r_{p}=0.8768(69)$ fm. At that time the large value was consistent with that obtained (with much larger uncertainties Horbatsch:2015qda ) from electron scattering. This difference of about 4% has become known as the proton radius puzzle Miller:2011yw . The proton radius puzzle was reviewed in 2013 Pohl:2013yb and 2015 Carlson:2015jba . New experimental results for hydrogen have appeared since that time beyer:2017 ; Fleurbaey:2018fih without resolving the puzzle. More results are planned. The PRAD experiment Gasparian:2014rna seeks to make previous electron scattering determinations of $r_{p}$ more precise by making measurements at very small values of momentum transfer. A new measurement of the $2S_{1/2}-2P_{1/2}$ transition in hydrogen is expected to appear (E. A. Hessels, 2016 talk at ECT*). All possible explanations of the proton radius puzzle will be addressed in the Muon-Scattering Experiment (MUSE) by measuring $e^{\pm}-p$ and $\mu^{\pm}-p$ scattering Gilman:2013eiv .

One might wonder whether or not a 4% difference really matters. After all, 4% is pretty small and (at the present time) the value of $r_{p}$ cannot be calculated to that accuracy. Perhaps the most interesting issue is whether or not the fundamental electron-proton interaction is the same as the muon-proton interaction. To find that this is not the case is to discover a violation of the principle of lepton-universality, a cornerstone of the standard model.

But there is another basic question that must be addressed: what is the radius of the proton? How does one define the radius of a quantum-field theoretic system made of nearly massless quarks and gluons? This quantity can be measured in the hydrogen atom and also in electron-proton scattering. There is a separate, but clear, literature in the fields of atomic and nuclear physics. Both fields obtain the same answer that

[TABLE]

where $G_{E}(Q^{2})$ is the Sachs electric form factor. This form factor is a specifically defined probability amplitude that an interaction between a photon of four-momentum $q^{\mu}\,(Q^{2}=-q^{2})$ and a charged constituent of the proton can absorb such a momentum with the proton remaining in its ground state. The meaning of Eq. (1) is that the quantity $-6G_{E}^{\prime}(0)$ appears in both hydrogen spectroscopy and lepton-proton elastic scattering measurements. The quantity $r_{p}^{2}$ is merely an abbreviation. The expression Eq. (1) is uniquely used in spectroscopy and scattering experiments to determine the proton radius from the slope of $G_{E}$.

The aims of the present paper are:

• •

unite the hydrogen spectroscopy literature with that of lepton-proton scattering and show how Eq. (1) emerges from the separate bodies of work

• •

show that a three-dimensional charge density cannot be defined, so that $r_{p}^{2}$ is not a second moment of a density distribution.

• •

remind readers how it is that a two-dimensional charge density which is a matrix element of a density operator between identical initial and final states, can be defined and determined by the Dirac form factor, $F_{1}$

• •

place the two-dimensional charge density in the modern context of generalized parton distributions and Wigner functions

• •

derive a relativistically correct moment expansion of $F_{1}$.

The second item may be considered controversial by some. This is because the text-book interpretation BMot ; PB ; Frauenfelder:1979ii ; Cheng:1979ay ; Halzen:1984mc ; Wong:1998ex ; Thomas:2001kw ; Close:2007zzd of $G_{E}$ is that its Fourier transform is a three-dimensional charge density. This interpretation is deeply embedded in the thinking of nuclear and particle physicists and therefore continues to guide intuition, as it has since the days of the Nobel prize-winning work of Hofstadter Hofstadter:1956qs ; Hofstadter:1958 ; RHN . Nevertheless, the relativistic motion of the nearly massless fermionic constituents of the proton causes the text-book interpretation to be incorrect because relativistic invariance is ignored in defining the three-dimensional density.

The modern day literature regarding the measurable aspects of the proton, which is consistent with relativity, is much deeper than the understanding from 1956. The increasing availability of high energies and high luminosities at fixed target and collider experiments Boer:2011fh ; Dudek:2012vr allows for unprecedented access to the internal transverse spatial and momentum distributions of charge distributions inside nucleons and in nuclei. The standard framework Ji:2003ak is that of Wigner distributions Wigner:1932eb that allow simultaneous knowledge of both spatial and momentum aspects of the nucleon wave function. Knowledge of the Wigner distributions allows the construction of generalized parton distributions (GPDs) Mueller:1998fv ; Ji:1996nm ; Ji:1996ek ; Radyushkin:1996nd ; Radyushkin:1997ki ; Collins:1996fb ; Ji:1998pc ; Radyushkin:2000uy ; Goeke:2001tz ; Diehl:2003ny ; Ji:2004gf ; Belitsky:2005qn ; Burkardt:2002hr and transverse momentum distributions (TMDs) Ralston:1979ys ; Collins:1981uk ; Mulders:1995dh ; Boer:1997nt ; Belitsky:2002sm ; Miller:2008sq that are generalizations of the usual collinear parton distributions. The variables of the widely used relativistic formalism involve three dimensions– one is the longitudinal momentum of a parton and the other two involve either the transverse positions (GPD) or momenta (TMD). The longitudinal and transverse degrees of freedom are treated separately. This is necessary to maintain symmetries and sum rules provided by relativistic invariance. Electromagnetic form factors may be obtained by doing integrals over the longitudinal momentum coordinate of GPDs. These form factors must be described using the same variables as the other observables. Thus, only a two-dimensional charge density may be defined.

Let’s outline the remainder of this paper. The appearance of the proton radius, $r_{p}$, in hydrogen spectroscopy is discussed in Sect.II. It explains how the key points related to extracting the value of the proton radius were already clearly explained in Refs Eides:2000xc ; Eides:2007xc ; Eides:2014swa . It is nevertheless worthwhile to repeat, publicize this earlier discussion, try to re-emphasize the key points and strengthen the connection with treatments of lepton-proton scattering. Sect. III shows that the only existing derivation of a three-dimensional, spherically symmetric charge density is faulty. A properly defined relativistic three-dimensional charge density with modern formulations is discussed in Sect. IV. This quantity is intimately connected with modern formulations of the diverse set of possible parton distributions. The ensuing phenomenology is discussed in Sect. V in which a correctly defined moment expansion $\rm RME$ is derived. Some details are placed in Appendices.

II Hydrogen atom

This section is concerned with understanding the role of the proton radius in hydrogen spectroscopy. The starting point is to understand the leading relativistic corrections to the basic Dirac energy levels. The standard procedure is well-documented in Refs. Eides:2000xc ; Eides:2007xc , and their discussion is used here. In the center of mass system the non-relativistic Hamiltonian for a system of a proton and a lepton (of mass $m$) with a Coulomb interaction is given by

[TABLE]

In a non-relativistic loosely bound system an expansion in powers of $\alpha^{2}$ corresponds to an expansion in powers of $v^{2}/c^{2}$. To proceed one needs an effective Hamiltonian including terms of order $v^{2}/c^{2}$. Breit Breit:1929zz ; Breit:1930zza considered such a Hamiltonian, realizing that all corrections to the non-relativistic Hamiltonian of order $v^{2}/c^{2}$ may be obtained from the sum of the free relativistic Hamiltonians of each of the particles along with relativistic one-photon exchange between the fermions. An explicit expression for the resulting Breit potential was derived Barker:1955zz from the one-photon exchange amplitude using the Foldy-Wouthhuysen transformation. If hyperfine effects are ignored, the result to order $v^{2}/c^{2}$ is given by

[TABLE]

All contributions to the energy levels up to order $\alpha^{4}$ may be calculated from the total Hamiltonian $H_{0}+V_{\rm Breit}$. The corrections of order $\alpha^{4}$ are the first-order matrix elements of the Breit interaction between the Coulomb-Schroedinger eigenfunctions of $H_{0}$. The result is

[TABLE]

where the reduced mass $m_{r}=mM/(m+M)$ and $j$ in the total angular momentum quantum number of the lepton. Note the presence of the last term of Eq. (4) which removes the degeneracy of the Dirac spectrum between levels with the same $j$ and $l=j\pm{1\over 2}$.

The expressions Eq. (3) and Eq. (4) are obtained assuming the proton is a point-like proton. Electromagnetic form factors are introduced to include the effects of its non-zero spatial extent. The photon-proton vertex operator $\Gamma^{\mu}$ is given by

[TABLE]

where $Q^{2}>0$ is the negative of the square of the virtual space-like photon momentum, $M$ is the proton mass, $F_{1}$ is the Dirac form factor, $F_{2}$ is the Pauli form factor and $\kappa$ is the proton anomalous magnetic moment. It is useful to define the Sachs form factors:

[TABLE]

where $\tau\equiv{Q^{2}\over 4M^{2}}$. With this notation $F_{1,2}(0)=1$.

The photon-electron vertex function in a hydrogen-like atom is given Eides:2000xc ; Eides:2007xc as the matrix element: $\bar{u}(\vec{p}^{\prime},s^{\prime})\Gamma^{\mu}u(\vec{p},s)$ with spinors normalized as $u^{\dagger}u=1$. Calculation, see eg. Eides:2000xc ; Eides:2007xc ; Bertozzi:1972jff , to order $1/M^{2}$ (and ignoring a spin-orbit term) reveals that

[TABLE]

This key equation is derived in Appendix A. The left-hand side of Eq. (7) is the time component of a four-vector. The right-hand-side does not depend on $\vec{p}$; it is frame-independent, and may be used to identify the leading proton-radius effect.

One uses the lowest order Taylor expansion, keeping only the spin-independent term, to extract the proton radius from the measured energy levels. Thus one writes:

[TABLE]

The difference between $Q^{2}$ and $\vec{q}^{2}$, $-q_{0}^{2},$ is of order $\alpha^{2}m^{2}/M^{2}$ and is a higher order correction, and any correction arising from a $G_{E}^{\prime\prime}(0)$ term is completely negligible Stryker:2015ika . We then find (to order $\vec{q}^{2}$) that

[TABLE]

The term ${\vec{q}^{2}\over 8M^{2}}$ leads to the Darwin term in the lepton-proton interaction Barker:1955zz that provides the term proportional to $\delta_{l0}$ in Eq. (4), so it is already included. The net result is that the effect of the proton size is given simply by including the term $\vec{q}^{2}G_{E}^{\prime}(0)$ in the lepton-proton vertex function.

Keeping the non-zero size of the proton leads to the momentum space version of the Coulomb potential:

[TABLE]

The coordinate space potential $V_{C}(r)$ is given by the three-dimensional Fourier transform:

[TABLE]

Since $G_{E}$ falls with increasing $\vec{q}^{2}$ one finds a repulsive correction to the Coulomb potential, $\Delta V_{C}$, given by

[TABLE]

The typical value of $\vec{q}^{2}$ is of the order of the square of the inverse of the Bohr radius of the atom. The muonic hydrogen atom Bohr radius is about 200 times smaller than that for the electronic one. This huge difference does not influence the potential $\Delta V_{C}$ because of the cancellation of the factor $\vec{q}^{2}$ by its inverse that arises from the photon propagator. The net result is the delta function appearing in Eq. (12). The difference between Bohr radii would enter if one included the $\vec{q}^{4}$ term in the Taylor expansion of $G_{E}$, but such terms are smaller by the ratio of the proton size to the Bohr radius Stryker:2015ika .

The shift in the energy, $\Delta E$, is given by the matrix element:

[TABLE]

The net result is that the energy shift in the hydrogen atom is determined by the slope of $G_{E}$ at its origin. The effect, Eq. (13), is of order $\alpha^{4}$ because $|\psi_{n0}(0)|^{2}$ is of order $\alpha^{3}$, so that this term should be included with the others of order $\alpha^{4}$ displayed in Eq. (4). Effects of higher order in $\alpha$ are not considered here.

For historical reasons, to be discussed in the next Section, the slope is redefined as

[TABLE]

with

[TABLE]

for sufficiently small values of $\vec{q}^{2}$. This means that one may also write

[TABLE]

as is often done.

III Lepton-proton scattering

The electron-proton elastic scattering cross section, obtained under the assumption that the lepton-proton interaction is mediated by a single photon (and neglecting the electron mass) is most simply expressed in terms of $G_{E},G_{M}$ Rosenbluth:1950yq :

[TABLE]

where $\left({d\sigma\over d\Omega}\right)_{M}$ is the Mott cross-section in which the proton is treated as point-like ($F_{1,2}(Q^{2})=1)$, and $\epsilon$ is a kinematic factor. Including the non-zero value of the muon mass leads to a slightly more complicated expression Preedom:1987mx .

Elastic electron-proton scattering experiments were first performed at Stanford by Hofstadter and collaborators, and summarized in Ref. Hofstadter:1958 . This early work, which famously RHN discovered that the proton was not a point-particle, assumed that $F_{1}=F_{2}\equiv F$. Their analysis used the equations:

[TABLE]

in which $a$ was associated with the physical extent of the proton and $Q_{L}$ is the laboratory value of the magnitude of the three-momentum transfer. It is further asserted that

[TABLE]

where The authors are very careful about using this expression. They state that Eq. (19) “applies in the non-relativistic limit in which $\rho({\bf r})$ is the static density distribution”. This definition is frame-dependent and so violates the principle of relativity.

Expansion of the exponential appearing in Eq. (19) leads to the well-known moment expansion:

[TABLE]

The validity of this expansion depends upon the validity of the non-relativistic limit. But there is no reason to believe that any non-relativistic treatment is valid for treating the proton form factor because elastic electron-proton scattering proceeds mainly via the absorption of a virtual photon by a nearly massless up or down quark.

Sachs et. al. Ernst:1960zza ; Sachs:1962zzc introduced the so-called Breit frame in which $q^{0}=0$ so that here (and in all following equations) $Q^{2}$ is the Lorentz scalar quantity, $Q^{2}=-q^{2}$, and $q^{\mu}$ is the four-momentum of the single-photon mediator. Sachs Sachs:1962zzc argued that in this frame the charge density is given by the Fourier transform of $G_{E}$. The resulting non-relativistic (NR) density as $\rho_{\rm NR}(r)$ is defined by the equation:

[TABLE]

We shall show that the definition Eq. (22) (despite its wide use) has no connection with well-defined matrix elements of quantum field theory. Equation (22) can be used to obtain the results that

[TABLE]

Eq. (23) has often been used to analyze the charge distribution of the neutron $n$. The result is that the mean square charge radius, $\langle r^{2}_{n}\rangle_{\rm NR}$, is almost completely accounted for numerically by the anomalous magnetic term, $3\kappa_{n}/(2M^{2})$, arising from the $F_{2}$ contribution to $G_{E}$ Thomas:2001kw . This seems very strange and looks like a puzzle. The puzzle dissipears if one realizes that the slope of $G_{E}$ is not related to the expectation value of $r^{2}$ in a spherically symmetric charge density. The pion form factor is another example of oddity. The 3-dimensional Fourier transform of the monopole form factor (that approximates the data) is singular at the origin Miller:2009qu .

There is a more technical way of explaining why the non-relativistic approach fails. The difficulty comes because proton wave functions of differing momentum are different. These functions are related by a boost, which in general contains an exponential of an operator as complicated as the strong interaction Hamiltonian Gasiorowicz:1966xra ; Cahill:2013pez . To see this, consider a very simple example in which a scalar proton of mass $M$ is made of a scalar quark (of mass $m_{q}$ and a scalar di-quark (of mass $M_{S}$). In this case the Bethe-Salpeter wave function is given by

[TABLE]

in which various constants have not been displayed. This wave function is shown in Fig. 1.

Boosting the wave function in this simple model is achieved merely by changing the value of the four-momentum $P$. In lepton-proton elastic scattering the four momentum of the initial proton is $P$ and that of the final one is $P^{\prime}=P+q$. The initial-state wave function depends upon $P\cdot p$, which in the laboratory frame is given by $Mp^{0}$. The final-state wave function depends on $P^{\prime}\cdot p=(P+q)\cdot p=(M+q^{0})p^{0}-{\bf q}\cdot{\bf p}$. Thus electron scattering involves two different wave functions. A density requires the appearance of the square of a wave function, which does not appear here.

III.1 Breit Frame Falibility

The Breit frame (introduced in Ref. Ernst:1960zza ) is the one in which the three-momentum of the initial proton, $\vec{P}$, is -1/2 that of the incident virtual photon, $\vec{q}$, $\vec{P}=-\vec{q}/2$. The final proton momentum $\vec{P}^{\prime}=\vec{P}+\vec{q}=\vec{q}/2,$ and the initial and final protons have the same energy. However, the initial and final wave functions are different because the quantities $P\cdot p$ and $P^{\prime}\cdot p$ must appear in any relativistic wave function and

[TABLE]

These differ, so that again one is not dealing with the square of a given wave function.

Ref. Ernst:1960zza shows that the matrix element of the time component of $\Gamma^{\mu},\,\Gamma^{0}$ is proportional to $G_{E}$ if the helicity is changed (spin direction is not changed). This argument is also given in e.g. Halzen:1984mc . In that reference a quantity $\rho$ is defined as the matrix element of $\Gamma^{0}$:

[TABLE]

with the normalization again being $u^{\dagger}u=1$, and where the direction of $\vec{q}$ defines the $z$-axis. The equality follows from evaluating the spinor matrix element. However, no spatial dependence is implied by this definition of $\rho$.

To the best of my knowledge the derivation of a relationship between $G_{E}$ and a three-dimensional charge density is attempted only in Appendix II of Ref. Sachs:1962zzc . In that formalism the initial and final states are “brought to rest” through the use of narrow wave packets. This is an attempt to avoid the previously mentioned problem associated with the boosts.

Although Ref. Sachs:1962zzc made a strong attempt to derive the charge density as a three-dimensional Fourier transform of $G_{E}$, the derivation is simply wrong. To show this, I redo the calculation of Appendix II of that paper, avoiding an incorrect assumption used there.

Sachs writes the proton wave packet state as

[TABLE]

with $g(\vec{P})$ representing a narrow wave packet. The function $g(\vec{P})$ is sufficiently narrow so that

[TABLE]

It is worthwhile to use an explicit representation. Using a Gaussian

[TABLE]

where $R$ is assumed to be infinite, as a first choice is convenient. The meaning of the expression Eq. (30) is the standard one of a distribution in which the first step is to do all of the relevant integrals, keeping $R$ finite. Then, with the values of the integrals in hand take $R$ to infinity Born:1999ory ; AW2005 .

Sachs proceeded by computing moments of the putative charge distribution. Only the time-component, and a quadratic moment is relevant to display the so-called charge density. In this case:

[TABLE]

with the integration variables chosen as $\vec{p}$ and $\vec{q}$ with $\vec{P}^{\prime}=\vec{p}+\vec{q}/2,\vec{P}=\vec{p}-\vec{q}/2$, and $x^{2}\equiv\sum_{i}x_{i}^{2}$ with $x_{i}$ as the Cartesian component of the three vector $\vec{x}$.

The matrix element of the time-component of the current is given by

[TABLE]

which (after replacing $x^{2}$ by $-\nabla_{q}^{2}$, integration over $\vec{x}$, and integration by parts) leads to the expression:

[TABLE]

At this stage Sachs made the replacement $g^{*}(\vec{p}+\vec{q}/2)g(\vec{p}-\vec{q}/2)\rightarrow|g(p)|^{2}$. The justification is that “terms resulting from the shape of the wave packet are not of interest here and are therefore dropped.”. However, this is not a correct justification for the replacement because (as a $\delta$ function) the quantity $|g(P)|^{2}$ must vary rapidly. One needs to be careful about the derivatives. To see this, use the specific form of Eq. (30) in Eq. (33). Then

[TABLE]

The term ${R^{3}\over\pi^{3/2}}\exp{(-\vec{p}^{2}R^{2})}$ leads to a delta function setting $\vec{p}=0$, so that $q^{0}=0$ and the Breit-frame result, Eq. (27), for the matrix element of $\Gamma^{0}$ may be used. Then:

[TABLE]

This means that the quadratic moment is actually infinite! This moment expansion fails. The underlying reason is that $\nabla_{q}^{2}$ must involve the square of some distance and the infinite parameter $R^{2}$ must appear in addition to any length scales in the proton.

If one asserts that $G_{E}(\vec{q}^{2})=\int d^{3}xe^{-i\vec{q}\cdot\vec{x}}\rho_{\rm NR}|(|\vec{x})|$, then one finds

[TABLE]

which looks like the expressions of the usual literature. However, this term (which does appear in Eq. (37)) is overwhelmed by the infinite term that also appears. Note that the infinite result does not rely on using the specific Gaussian form of Eq. (30). It would occur with any specific representation of a delta function, as shown in Appendix B. Thus the derivation of Sachs is fatally flawed.111The evaluation of Eq. (34) proceeded by first obtaining $\delta(\vec{p})$ and then handling the dependence of $\vec{q}$. The same result, Eq. (37) is obtained if one first differentiates with respect to $\vec{q}$.

One may understand the failure of the Sachs procedure in simple terms. The wave function $|\Psi\rangle$ is meant to represent a proton of 0 three-momentum, so that (via the uncertainty principle) its position is totally undetermined. This is the origin of the infinite result. Another procedure would be to use Eq. (30) in the opposite limit that $R$ is very, very small. This would lead to a wave packet that is concentrated in a narrow region of space, taken as the origin. However, the use of such a wave packet in Eq. (31) would not allow the use of the Breit frame result because the integral over $\vec{p}$ would go over all of its values.

The net result of all of this is that the relation between $G_{E}^{\prime}(0)$ and $r_{p}^{2}$, Eq. (1), is merely a definition.

IV True Charge density

A proper determination of a charge density requires the measurement of a matrix element of a density operator taken between initial and final states that are the same. We show here that the proton form factor, $F_{1}$ is a specific integral of the three-dimensional charge density of partons in the infinite momentum frame, $\hat{\rho}_{\infty}(x^{-},{\bf b})$.

It is necessary to provide a brief introduction to light- front coordinates. Instead of the usual $x^{0}=ct,x^{3}=z$, the light front approach uses $x^{\pm}=(x^{-}\pm x^{3})/\sqrt{2}.$ By convention the term $x^{+}$ corresponds to the time and $x^{-}$ corresponds to the longitudinal distance coordinate.

In the infinite momentum frame, IMF, the time coordinate $ct=x^{0}/\sqrt{2}$ is expressed in a frame moving along the negative $z$ direction with a velocity nearly that of light using the Lorentz transformation as the variable $x^{+}=(x^{0}+x^{3})/\sqrt{2}$, with the usual $\gamma$ factor absorbed by a Jacobean of an integral over volume Susskind:1968zz . The $x^{+}$ variable is canonically conjugate to the minus-component of the momentum operator $p^{-}\equiv(p^{0}-p^{3})/\sqrt{2}$. The longitudinal spatial variable is $x^{-}=(x^{0}-x^{3})/\sqrt{2}$ and its conjugate momentum is $p^{+}=(p^{0}+p^{3})/\sqrt{2}$. It is this plus-component of momentum that is associated with the usual Bjkoren variable. The transverse coordinates $x,y$ are written as ${\bf b}$ with the conjugate momentum denoted ${\bf p}$. Boldface is used here to denote the two-dimensional transverse components of position and momentum vectors to distinguish these from the three-dimensional vectors (e.g. $\vec{q}$) of previous sections.

Light-front time-quantization, which sets $x^{+}$ and the plus-component of all spatial variables to zero, is used. This means that $x^{-}$ can be thought of as the longitudinal variable $-\sqrt{2}x^{3}.$ One extremely useful aspect of using these variables is that Lorentz transformations to frames moving with different transverse velocities do not depend on interactions. These transformations form the kinematic subgroup of the Poincaré group, so that boosts in the transverse direction are accomplished as in the non-relativistic theory; the dependence on the total transverse momentum of any system appears only as an overall phase factor.

This language may seem a bit abstract. All it means the wave function of a proton with a given $(p^{+},{\bf p})$ is related to the one of momentum ($p^{+},{\bf 0})$ by a factor that is independent of the relative momenta of the partons that make up the wave function. The necessary integrations to compute form factors (in a frame in which $Q^{2}={\bf q}^{2}$) only involve the relative variables that appear in light-front wave functions. Examples can be found in Brodsky:2000ii ; Miller:2009sg ; Miller:2010nz .

The density that is relevant here has been known for a long time Soper:1976jc and often been exploited Miller:2010nz ; Miller:2009qu ; Carlson:2007xd . In the IMF, the electromagnetic charge density $J^{0}$ operator becomes $J^{+}$ and

[TABLE]

where $q_{+}(x^{\mu})=\gamma^{0}\gamma^{+}/\sqrt{2}q(x^{\mu})$, the independent part of the quark-field operator $q(x^{\mu})$. The time variable, $x^{+}$ is set to zero. Note the appearance of the absolute square of quark field-operators, which is the signature of a true density operator. An analogous expression is widely used to describe color charge densities McLerran:1993ka ; Dumitru:2018vpr .

The purpose of this section is to show how matrix elements of $\hat{\rho}_{\infty}(x^{-},{\bf b})$ (which are true densities) emerge from modern quantum field theory treatments of nucleon structure. The vast literature concerning the diverse set of functions that are used to describe nucleon structure includes generalized parton distributions GPDs Mueller:1998fv ; Ji:1996nm ; Radyushkin:1997ki ; Collins:1996fb ; Ji:1998pc ; Radyushkin:2000uy ; Goeke:2001tz ; Diehl:2003ny ; Ji:2004gf ; Belitsky:2005qn ; Hagler:2004yt ; Boffi:2007yc transverse momentum distributions (TMDs) Collins:1981uw ; Ralston:1979ys ; Belyaev:1988xu ; Anselmino:1994gn ; Mulders:1995dh ; Chibisov:1995ss ; Pasquini:2009eb and, more recently generalized transverse momentum distributions (GTMDs) Lorce:2011kd ; Echevarria:2016mrc .

Generalized parton distributions are of high current interest because they can be related to the total angular momentum carried by quarks in the nucleon and can be determined using deeply virtual Compton scattering experiments Ji:1996nm . The opportunity of determining all of these is greatly enhanced by the possible creation of an electron-ion collider Accardi:2012qut .

These distributions are specific matrix elements of quark-field operators, between nucleon states, which in contrast to the usual quark distribution functions, do not necessarily have the same momenta. The specific case in which the longitudinal momentum transfer vanishes, and the initial and final states have the same helicity $\lambda^{\prime}=\lambda$ is relevant in the present context. Then, in the light-cone gauge, $A^{+}=0$, the matrix element defining the GPD, $H_{q}$ for a quark of flavor $q$ is

[TABLE]

where the normalization is $\langle{p^{\prime}}^{+},{\bf p}^{\prime},\lambda|{p}^{+},{\bf p},\lambda\rangle=2p^{+}(2\pi)^{3}\delta({p^{\prime}}^{+}-p^{+})\delta^{(2)}({{\bf p}}^{\prime}-{\bf p})$. The variable $\lambda$ denotes the helicity, and only the helicity non-flip term needed to compute $F_{1}$ appear here. The four-momentum transfer $q_{\alpha}=p^{\prime}_{\alpha}-p_{\alpha}$ is space-like, with the square of the space-like four-momentum transfer $q^{2}=-Q^{2}$ and use the Drell-Yan (DY) frame with $(q^{+}=0,Q^{2}={\bf q}^{2})$. No longitudinal momentum is transferred, so that initial and final states are related only by kinematic transformations. Moreover, the current operator links Fock-state components with the same number of constituents. The abbreviation $-t=-(p^{\prime}-p)^{2}=({\bf p}^{\prime}-{\bf p})^{2}=-q^{2}=Q^{2}$ is used. The presence of the operator $\gamma^{+}$ insures that independent field operators appear in the matrix element.

GPDs allow for a unified description of a number of hadronic properties Ji:1996ek . Notice that if $t=0$ they reduce to conventional PDFs $H_{q}(x,0)=q(x)$, and, of most relevance here, that the integration of $H_{q}$ over $x$ yields the nucleon electromagnetic form factor:

[TABLE]

with the defining equation

[TABLE]

The spatial structure of a nucleon can be examined if one uses the fact that transverse boosts are independent of interactions in the infinite momentum frame Kogut:1969xa ; Burkardt:2005td to define Soper:1976jc ; Burkardt:2002hr ; Diehl:2002he nucleonic states that are transversely localized. The state with transverse center of mass ${\bf R}$ set to 0 is formed by taking a linear superposition of states of transverse momentum. In particular,

[TABLE]

where $\left|p^{+},{\bf p},\lambda\right\rangle$ are light-cone helicity eigenstates Soper:1976jc and ${\cal N}$ is a normalization factor satisfying $\left|{\cal N}\right|^{2}\int\frac{d^{2}{\bf p}_{\perp}}{(2\pi)^{2}}=1$. References Burkardt:2000za ; Diehl:2000xz use wave packet treatments that avoid states normalized to $\delta$ functions, but this leads to the same results as using Eq. (43). Note however, the relevant range of integration in Eq. (43) must be restricted to $|{\bf p}|\ll p^{+}$ to maintain the interpretation of a nucleon moving with well-defined longitudinal momentum Burkardt:2000za . Thus the infinite momentum frame, with $p^{+}$ as the large momentum, is used. This is a frame in which the interpretation of a nucleon as a set of a large number of partons is valid.

Using Eq. (43) sets the transverse center of momentum of a state of total very large momentum $p^{+}$ to zero, so that a transverse distance ${\bf b}$ relative to ${\bf R}$ can be defined. To use this feature generalize the quark-field operator appearing in Eq. (40) by making a translation:

[TABLE]

The impact parameter dependent PDF is defined Burkardt:2000za as the matrix element of this operator in the state of Eq. (43):

[TABLE]

The use of Eq. (43) in Eq. (45) allows one to show that $q(x,{\bf b})$ is the two-dimensional Fourier transform of the GPD $H_{q}$:

[TABLE]

with $H_{q}$ appearing because the initial and final helicities are each $\lambda$. A complete determination of $H_{q}(x,t)$ (with $t\leq 0$) would determine $q(x,{\bf b})$.

One finds a probability interpretation Soper:1976jc by integrating $q(x,{\bf b})$ over all values of $x$. This sets the differences in longitudinal distances, appearing in Eq. (44), to 0. Then the use of translational invariance leads to the result

[TABLE]

This equation shows that the matrix element of a true density operator (square of a quark-field operator) taken between identical initial and final states is experimentally accessible.

Furthermore, multiplying Eq. (47) by the quark charge $e_{q}$ (in units of $e$), sums over quark flavors, uses Eq. (43) with $\hat{O}_{q}(x,{\bf b})=e^{-i\hat{{\bf p}}\cdot{\bf b}}\hat{O}_{q}(x,{\bf 0})e^{i\hat{{\bf p}}\cdot{\bf b}}$ along with Eq. (41), the resulting infinite-momentum-frame IMF parton charge density in transverse space is

[TABLE]

This relation shows that a properly-defined charge density, the transverse charge density, is obtained using the same formulations that is used to define generalized densities.

IV.1 Wigner distributions

There is now a broader perspective involving a diverse set of distributions that can be used to characterize nucleon structure Lorce:2011kd . This subsection is intended to place the transverse density in the context of Wigner distributions. Wigner distributions in QCD were first explored in Refs. Ji:2003ak ; Belitsky:2003nz . Neglecting relativistic effects, those authors used the standard three-dimensional Fourier transform in the Breit frame and introduced six-dimensional Wigner distributions (three position and three momentum coordinates). The modern perspective involves instead five-dimensional Wigner distributions (two position and three momentum coordinates) as seen from the infinite momentum frame (IMF). These three momentum variables of a quark are $k^{+},{\bf k}$, so there is no spherically-symmetric charge density. These light front variables were exploited Lorce:2011kd to arrive at a definition of Wigner distributions that is completely consistent with relativity.

The first step is to use Wigner operators for quarks of flavor $q$ at a fixed light-cone time $y^{+}=0$:

[TABLE]

with $y^{\mu}=[0,0,{\bf b}]$, $p^{+}$ is the average of the initial and final nucleon longitudinal momentum and $x=k^{+}/p^{+}$ is the average fraction of nucleon longitudinal momentum carried by the struck quark. The above equation is a specific Wigner operator that involves $\gamma^{+}$ that is relevant here. More generally one could use any twist-two Dirac operator $\Gamma=\gamma^{+},\gamma^{+}\gamma_{5},i\sigma^{j+}\gamma_{5}$ with $j=1,2$. A Wilson line, $\mathcal{W}$, ensures the color gauge invariance of the Wigner operator by connecting the points $(y-\tfrac{z}{2})$ and $(y+\tfrac{z}{2})$ see e.g. Meissner:2009ww .

Wigner distributions are defined as matrix elements of the Wigner operators sandwiched between nucleon states with polarization $\vec{S}$:

[TABLE]

Ref. Lorce:2011kd shows how four different three-dimensional densities can be defined. The task here is to connect the transverse density, $\rho(b)$, of Eq. (48) with the Wigner distribution defined above. This is done by integrating the quantity $\rho_{q}({\bf b},{\bf k},x,\lambda)$ over all values of $x$ and ${\bf k}$. This sets $z^{-}$ and $\bf z$ to 0, so that the Wilson line becomes unity, and the result is

[TABLE]

where the polarization vector $\vec{S}$ has been set to the light front helicity $\lambda$. Using translational invariance in the transverse direction and Eq. (50) shows that

[TABLE]

so that the charge density operator appears. Multiplying this expression by $e_{q}$, summing over quark flavors $q$ and using Eq. (42) and Eq. (48) shows that

[TABLE]

This means that the transverse charge density exhibits a specific aspect of quark Wigner distributions. Observe that proton electromagnetic form factors occupy a small, but important, corner of a vast field.

V True vs. Non-relativistic density

The quantity $\rho(b)$ of Eq. (48) is a true density. It is properly defined as the matrix element of a density operator between identical initial and final states. It depends only on the two-dimensional transverse variable ${\bf b}$ because boosts in the longitudinal momentum depend on interactions. Furthermore, it depends only on the magnitude $b$ because of its independence of $\lambda$.

The purpose of this Section is to understand the differences between $\rho(b)$ and $\rho_{\rm NR}(r)$ even though these are inherently different. The former is well-defined in quantum field theory, and the latter is defined as a three-dimensional Fourier transform of $G_{E}$. However, it is useful to compare $\rho(b)$ with a two-dimensional version of $\rho_{\rm NR}(r)$. A transverse non-relativistic density can be defined in analogy with the transverse densities that are used in relativistic heavy ion physics, and which generally appear in Glauber theory (eikonal approximation) calculations of scattering processes. To this end, one writes $\vec{r}=z\,\widehat{\bf k}+{\bf b}$ and then integrates $\rho_{\rm NR}(r)$ over all values of $z$. Then one obtains a non-relativistic transverse density

[TABLE]

The use of Eq. (22) in this equation followed by integration over angles yields the result

[TABLE]

where $J_{0}$ is a cylindrical Bessel function. The non-relativistic transverse density $\rho_{\rm NR,T}(b)$ may be compared with a more detailed version of the true transverse density of Eq. (48):

[TABLE]

The difference is simple: to obtain a relativistic transverse charge density one need only replace $G_{E}$ by $F_{1}$.

Next specific values of $F_{1}$ and $G_{E}$ are used to obtain an explicit comparison. The form factors $G_{E,M}$ have been measured Arrington:2006zm ; Punjabi:2015bba . Thus the quantity $F_{1}$ is readily available from the relation

[TABLE]

Then the functions $G_{E}$ and $F_{1}$, obtained from a recent parameterization Ye:2017gyb , are used to obtain the densities shown in Fig. 2. The non-relativistic transverse density is seen to be very different from the correct transverse density. In particular, the non-relativistic version has a larger spatial extent.

The spatial extent can be understood by computing the average value of $b^{2}$:

[TABLE]

The use of the relation Eq. (6) leads to the result:

[TABLE]

The difference between the true value and the non-relativistic one is very significant on the scale of distances relevant to the proton radius puzzle. But to be clear: hydrogen spectroscopy measures the slope of $G_{E}$ at its origin and relates that quantity to $r_{p}^{2}$.

V.1 Relativistic Moment Expansion- RME

A moment expansion analogous to Eq. (20) can be derived from the relation between $F_{1}(Q^{2})$ and the true density $\rho(b)$. Invert Eq. (48) to obtain

[TABLE]

Then expand the exponential in a power series in $i\bf Q\cdot{\bf b}$ and also expand $F_{1}(Q^{2})$ in powers of $Q^{2}$. Equating the two expansions gives the result

[TABLE]

where the notation $F_{1}^{(n)}(0)$ denotes taking the $n$’th derivative of $F_{1}$ with respect to $Q^{2}$ at $Q^{2}=0$ The first terms are given by

[TABLE]

Each of the moments $\langle b^{2n}\rangle$ is invariant under Lorentz transformations.

Hydrogen spectroscopy depends on the moments $\langle r^{2,4}\rangle_{\rm NR}$. The RME of Eq. (62) can be used to determine the true moments $\langle b^{2n}\rangle$ in terms of the non-relativistic ones. The use of Eq. (60) leads to the result:

[TABLE]

and

[TABLE]

where $\mu=1+\kappa,\,\langle r^{2}_{M}\rangle_{\rm NR}\equiv-6G_{M}^{\prime}(0)$.

The moments of $b$ are closely tied to GPDs, which are accessible experimentally and through lattice calculations.

VI Summary

This paper unites the hydrogen spectroscopy literature with that of lepton-proton scattering to show how Eq. (1) emerges from the separate literatures.

The appearance of the proton radius $r_{p}$ in hydrogen spectroscopy is discussed in Sect.II, which shows that the energy shift caused by the non-zero extent of the proton (Eq. (13)) is proportional to the slope of $G_{E}(Q^{2})$ at its origin. An explicit three-dimensional charge density does not appear. There is no need to define $r_{p}^{2}$ as a moment of such a density.

Sect. III begins with a brief historical review of how a non-relativistic, frame-dependent spherically-symmetric, three-dimensional charge density was postulated in the early work of Hofstadter and co-workers. There is only one attempted derivation of this density in the literature Sachs:1962zzc . This derivation is shown to be faulty because it used states of completely uncertain position, which leads to an infinite contribution, Eq. (37).

A properly defined relativistic three-dimensional charge density is discussed in Sect. IV. This quantity is intimately connected with modern formulations of the diverse set of possible parton distributions. It depends on longitudinal and transverse momentum or longitudinal and transverse position. The two-dimensional transverse density $(\rho(b))$, obtained as an integral over the longitudinal coordinate, is a two-dimensional Fourier transform of the Dirac form factor $F_{1}$, Eq. (48). The transverse density is shown to be related to specific integrals of a Wigner distribution, Eq. (53).

The phenomenology of $\rho(b)$ is discussed in Sect. V. The non-relativistic version is shown to be significantly different from the correct density, and a correctly defined moment expansion $\rm RME$ is derived, Eq. (62). These moments are related to the non-relativistic ones.

Appendix A-Derivation of Eq. (7)

The expression $\bar{u}(p^{\prime},s^{\prime})\Gamma^{0}u(p,s)$ (Eq. (5) with $\vec{p}^{\prime}=\vec{p}+\vec{q}$) is evaluated here. The spinor is given to order $1/M^{2}$ by

[TABLE]

in which the spin-dependent term is neglected. Then first evaluate

[TABLE]

Next evaluate the term proportional to $F_{2}$. Use $i\sigma^{0\nu}q_{n}=ii{1\over 2}[\gamma^{0},-\vec{\sigma}\cdot\vec{q}]=\gamma^{0}\vec{\gamma}\cdot\vec{q}$, so that

[TABLE]

in which a spin-dependent term is omitted. Combining the results Eq. (70) and Eq. (71) and recalling the definition Eq. (5) leads immediately to Eq. (7).

Appendix B-General Wave Packet

Sect. III showed that the derivation of the relation between $G_{E}$ and a three-dimensional charge density was incorrect because it ignores an infinite term . A specific representation, Eq. (30), of the delta function was used to demonstrate this flaw. Here we show that an infinite term appears for any representation of the delta function. The evaluation of Eq. (33) involves the combination $\nabla_{q}^{2}\int d^{3}p\,g(\vec{p}+\vec{q}/2)g^{*}(\vec{p}-\vec{q}/2)$. Other terms in which $\vec{\nabla}_{q}$ acts on the product of the $g_{R}$ factors times the matrix element of $\Gamma^{0}$ or only on $\Gamma^{0}$ are not infinite. But this term is actually infinite. To see this introduce the Fourier transform:

[TABLE]

The function $g(\vec{p})$ is very narrow in momentum space, so $\tilde{g}(\vec{x})$ must be very broad in coordinate space. Using Fourier transforms one finds

[TABLE]

Thus it is immediately plausible that this quantity is infinite for any representation of the delta function.

The infinite result can proved. Let the general form, dictated by dimensional analysis, of $g(\vec{p})$ be given by

[TABLE]

where $F$ is a dimensionless, real-valued function and we understand that the limit $R\to\infty$ is to be taken after doing the relevant integrals. Then

[TABLE]

and

[TABLE]

where $z$ is a dimensionless variable. The function $G$ is normalizable, and its $z^{2}$-weighted integral is finite, because it is a Fourier transform of a normalizable function. However, the entire expression is proportional to $R^{2}$ which is infinite.

The net result is that the wave packet treatment of Sachs is not feasible, no matter how a delta function is represented.

Acknowledgments

This work was stimulated by a suggestion from J. Bernauer. I thank E. Voutier for a useful discussion. This work was supported by the U. S. Department of Energy Office of Science, Office of Nuclear Physics under Award Number DE-FG02-97ER-41014.