NT@UW-09-18
Electromagnetic Form Factors and Charge Densities From Hadrons to Nuclei

The text-book interpretation of nucleon electromagnetic form factors is that their three-dimensional Fourier transforms are measurements of the charge and magnetization densities. This interpretation is deeply buried in our thinking and continues to guide intuition as it has since the days of the Nobel prize-winning work of HofstadterHofstadter:1956qs . Nevertheless, the relativistic motion of the constituents of the system causes the text-book interpretation to be incorrect.

The preceding statement leads to a number of questions, the first being: Is the statement correct? If correct, how relativistic does the motion of the constituents have to be? Why is it that the relativistic motion of the constituents and not that of the entire system that causes the non-relativistic approach to fail? It is probably true that the answers to these questions are displayed within the existing literature. However, obtaining general clear answers has been sufficiently difficult that posing even the first question of this paragraph would not lead to a unanimous answer by all professionals in the field.

This paper offers the strategy of using a simple model, a generalization of the $\phi^{3}$ model used by Weinberg Weinberg:1966jm to illustrate advantages of using the infinite momentum by choosing frame that was used by Gunion et al. Gunion:1973ex to explore form factors and hadronic interactions at high-momentum transfer. We take the interaction Lagrangian density to be of the form $g\Psi\phi\xi$ in which all of the fields are bosons.The $\Psi$ particle of mass $M$ represents the bound state of the two different constituents $\phi,\xi$ of masses $m_{1},m_{2}$ respectively. Thus the $\Psi$ represents the hadron or nucleus with the $\phi,\xi$ representing the quark or nucleonic constituents. Mass renormalization effects are ignored. We can choose either or both of the constituents to be charged and thus discuss charged and neutral $\Psi$ particles.

The motivation to pose questions regarding the meaning of electromagnetic form factors at this point in time arises from recent experimental work, especially the discovery that the ratio of the proton’s electric to magnetic Sachs form factors $G_{E}/G_{M}$ drops rapidly (please see the reviews reviews ) and from our recent finding Miller:2007uy , based on measurements and the use of the transverse density that the charge density at the neutron’s center is negative. The nucleon transverse density $\rho(b)$, the two-dimensional Fourier transform of $F_{1}$ is the infinite momentum frame charge densitynotation located at a transverse separation $b$ from the center of transverse momentumsoper1 ; mbimpact ; diehl2 ; Carlson:2008zc . This quantity has a direct relationship to matrix element of a density operator. The usual three-dimensional Fourier transforms of $G_{E}$ and $G_{M}$ do not because the initial- and final-state nucleons have different momentum, and therefore different wave functions. This is because the relativistic boost operator that transforms a nucleon at rest into a moving one changes the wave function in a manner that depends on the momentum of the nucleon. However, we expect that there are non-relativistic conditions for which the text-book interpretation is correct. We aim to explore those conditions by choosing appropriate values of the masses $m_{1},m_{2},M$.

Here is the outline we follow. The form factor for the situation in which the $\Psi$ and $\phi$ carry a single unit of charge, but the $\xi$ is neutral, is computed using an exact covariant calculation of the lowest-order triangle diagram in Sect. II. This is followed by a another derivation using the light-front technique of integrating over the minus-component of the virtual momentum in Sect. III that obtains the same form factor. This is also the result obtained originally by Gunion:1973ex by using time-ordered perturbation theory in the infinite-momentum-frame IMF. Thus three different approaches yield the same exact result for this model problem. Any approximation that does not yield the same form factor is simply not correct. The asymptotic limit of very high momentum transfer $Q^{2}$ is also studied. The next section (IV) explains the transverse density of the model, its central value, a general derivation of its relationship with the form factor using three dimensional spatial coordinates and the meaning of hadronic radii. Section V displays the spatial wave function in terms of three spatial coordinates. Section VI shows that the rest-frame charge distribution is generally not observable. Section VII is concerned with the question of whether neutral systems of two constituents obey the conventional lore that the heavier one is generally closer to the origin than the lighter one. The non-relativistic limit is defined and applied to a variety of examples in Section VIII. The exact formula for the form factor morphs into the familiar one of the three-dimensional Fourier transform for sufficiently large values of the constituent mass divided by the binding energy of the system. Examples that are motivated by the pion, deuterium, nucleon and heavy nuclei are provided. This work is summarized in Section IX.

The model Lagrangian density is given by $g\Psi\phi\;\xi$ where $\Psi,\phi$ and $\xi$ represent three scalar fields of masses $M,m_{1}$ and $m_{2}$ respectively and $g$ is a coupling constant. One can take two or three of these fields to carry charge to make up a system of definite charge (including the case when the hadron $\Psi$ is neutral). The effects of mass renormalization are not considered here because we aim to use a simple model to provide easily calculable examples and illustrate specific points. The condition $m_{1}+m_{2}>M$ is used to insure that the hadron $\Psi$ is stable.

We start with the situation in which the $\Psi,\phi$ each carry a single positive charge and $\xi$ is neutral. The form factor $F(q^{2})$ for a space-like incident photon of four-momentum $q^{\mu}\;(q^{2}<0,Q^{2}=-q^{2})$, incident on a target $\Psi$ of four-momentum $P^{\mu}$ interacting with the $\phi$ of mass $m_{1}$ is given, to lowest order in $g$, by the single Feynman diagram of Fig. 1. We take the model electromagnetic current $J^{\mu}$ (in units of the proton charge) as given by

and find

Proceed by combining the denominators using the Feynman procedure and shifting the origin of the convergent integral to find

where ${\cal M}^{2}=Q^{2}y(y+x-1)+M^{2}x(1-x)-m_{1}^{2}(1-x)-m_{2}^{2}.$ Use

to find

The integral over $y$ can be done in closed form with the result

The above expression shows that current conservation is satisfied and that the form factor can be obtained from any component of the current operator $J^{\mu}$. The final result for the form factor is

This closed form expression is the key result of this paper. It can be modified to describe a variety of different physical situations.

We derive the light front representation by starting with the form factor of Eq. (3) and integrating over $k^{-}$. This procedure is simplified by choosing $q^{+}=0$, so that $Q^{2}={\bf q}^{2}$ notation and evaluating the $+$ component of the electromagnetic current operator. In the present Section the convention is that $A^{\pm}=A^{0}\pm A^{3}$ for the four-vector $A^{\mu}$. Then Eq. (3) becomes

If we integrate over the upper half of the complex $k^{-}$ plane we find a non-zero contribution only for the case $0<k^{+}<P^{+}$. Carrying out the integral leads to

Next change variables by defining

so that

Further define the relative transverse momentum

so that the form factor can be re-expressed as

This is the expression obtained in Ref. Gunion:1973ex , by using time-order-perturbation theory in the infinite momentum frame. Integration over $k^{-}$ leads to the same result for this example.

It is useful to re-express the result Eq. (15) in terms of a wave function $\psi$ with

In that case

as found in Ref. Gunion:1973ex .

The integration over $\kappa$ is convergent so we carry out the integration over $\kappa$ by combining the propagators and shifting the origin. This gives

The integral over $z$ can be done with the result that

This is the same as our previous exactly computed result, Eq. (8). Thus evaluation in the infinite momentum frame or the equivalent (for this model) light front technique of integration over $k^{-}$ yields the exact result.

The expression Eq. (17) is noteworthy because the form factor is expressed as a three-dimensional integration that involves momentum-space wave functions evaluated at different initial and final momenta. If the $(1-x)$ factor multiplying ${\bf q}$ were replaced by a constant Eq. (17) would be similar to the usual expression for the form factor. We clarify this comparison by expressing the wave function of Eq. (16) and Eq. (17) in transverse position space, with ${\bf B}$ canonically conjugate to the transverse momentum variable $\kappa$:

with the phase space factor $\frac{1}{\sqrt{x(1-x)}}$ incorporated in the wave function. Then the form factor Eq. (17) can be re-expressed as

Further simplify by replacing the relative transverse position variable ${\bf B}$ by the transverse position variable of the charged parton ${\bf b}_{1}\equiv{\bf b}$. We have

where the middle equation sets the transverse center of $P^{+}$ momentum to zero. Use Eq. (28) in Eq. (25) to find

which can be re-written as

with the transverse density $\rho(b)$ given by

The transverse density $\rho(b)$ has been derived previously mbimpact ; diehl2 as the integral of the impact parameter generalized parton distributions GPD $\rho(x,b)$ over all values of $x$. The quantity $\rho(x,b)$ gives the probability that a quark of longitudinal momentum fraction $x$ resides at a transverse position mbimpact ; diehl2 . For the present model

The transverse density is also the integral of the three-dimensional infinite momentum frame density $\rho(x^{-},b)$ over all values of the longitudinal position coordinate miller:2009qu .

The transverse density is directly obtainable from experiment via the inverse Fourier transform of Eq. (30) provided the electromagnetic form factor is measured for sufficiently large values of $Q^{2}$. The momentum transfer is transverse in direction so that information about the longitudinal position or momentum is not available. There is no way to use only measured values of $F(Q^{2})$ to determine $\rho(x,b)$.

The previous section shows that the form factor is simply related to the three-dimensional coordinate-space density that depends on $(x^{-},{\bf b})$ in the infinite momentum frame. Given the simplicity of our model, we should be able to identify a wave function and density.

The basic idea is that the position variable of a particle $x^{-}$ is canonically conjugate to the plus-component of the momentum. The momentum of the charged constituent is $k^{+}=xP^{+}$, and its canonical longitudinal position variable is $x^{-}$ with $[x^{-},k^{+}]=i=[x^{-},x]P^{+}$ Pirner:2009zz . The canonical longitudinal position variable for the other particle can be taken as $-x^{-}$. So we can convert the wave function $\psi(x,{\bf B})$ of Eq. (24) to one expressed entirely in coordinate space. We find

which preserves the normalization condition that $F(Q^{2}=0)=1$. Note that ${\bf B}$ is a relative variable and that $x^{-}$ is the variable for the position of the charged constituent. This wave function displays no spherical symmetry–the longitudinal $x^{-}$ and transverse position dependence ${\bf B}$ are not related. Another point is that the wave function Eq. (45) explicitly depends on the momentum $P^{+}$. As $P^{+}$ approaches infinity, the value of $x^{-}$ must be very small to prevent $x^{-}P^{+}$ from being very large and causing the integral on Eq. (45) to vanish. This means that the system can be thought of as having a pancake or disc shape. For this reason, the position $b=0$ really does correspond to the center of the hadron. We shall show below, that in the non-relativistic limit, rotational symmetry emerges.

The wave function can be computed in closed form for the special case $M=0,m_{1}=m_{2}=m$. Using Eq. (24) in Eq. (45) with the stated parameters leads to the result

For this simple example, the $x^{-}$ and ${\bf B}$ dependence factorizes, showing the explicit violation of rotational symmetry. The formula Eq. (47) shows also how the spatial extent contracts with the increase in value $P^{+}$. We note that it is not useful to use the spatial wave function to compute the form factor because of the appearance of the factor $1-x$ in the exponential of Eq. (25).

The concept of a charge density that depends on three spatial variables, but not on the time, is inherently non-relativistic. This is because the use of only three variables involves replacing a four-dimensional quantity by one involving only three dimensions. One procedure, discussed above in Sect. II, is to evaluate the Feynman diagram of Fig. 1 by integrating over the $k^{-}$ component of the virtual momentum $k$. This leads to a formalism in which the form factor depends on a Fourier transform of the square of a wave function that depends both on position ${\bf B}$ and momentum $x$ variables.

One can try to recover the more familiar three-spatial dimension formalism by evaluating the Feynman diagram of Fig. 1 using the time-ordered-perturbation theory TOPT formalism in the rest frame. One proceeds by integrating over all times, with the exponential oscillating factors converted into energy denominators. In the TOPT formalism any given Feynman diagram is the sum of several TOPT diagrams. In the present case, the sum of the two TOPT diagrams of Fig. 2 leads to the Feynman diagram of Fig. 1. Only Fig. 2a corresponds to measuring a density. The term of Fig. 2b corresponds to the hadronic part of the incident photon wave function interacting with the target.

One can examine the contribution of the term of Fig. 2a, $F_{2a}(Q^{2})$. It is given Gunion:1973ex by

an expression that leads to the correct result in the infinite momentum frame ($P\rightarrow\infty$) Gunion:1973ex . This expression can be interpreted as involving an initial and a final state wave function if one interprets the energy denominators (multiplied by phase space factors) as wave function expressed in momentum space. The symbol $\rightarrow$ used above refers to the approximation of keeping only a single TOPT diagram. For simplicity we take the example, $m_{1}=m_{2}=m$ and also work in the target rest frame: ${\bf P}=\bf{0}$ to isolate the rest frame charge distribution. Then $E_{P}=\sqrt{P^{2}+M^{2}}=M,\;E_{{\bf P}+{\bf q}}=\sqrt{({\bf P}+{\bf q})^{2}+M^{2}}=\sqrt{{\bf q}^{2}+M^{2}},\;E_{1}=\sqrt{p^{2}+m^{2}},\;{E^{\prime}}_{1}=\sqrt{({\bf p}+{\bf q})^{2}+m^{2}}$. The four vector $(p^{\mu}_{1}+{p^{\prime}}^{\mu}_{1})=[E_{1}+{E^{\prime}}_{1},2{\bf p}+{\bf q}]$. There are three integrals appearing on the right-hand-side of Eq. (48):

Thus we arrive at the four-vector equality

Maintaining current conservation requires that the matrix element of $q_{\mu}J^{\mu}$ vanishes. Taking the scalar product of Eq. (52) with $q_{\mu}$ leads to the requirement:

where $q_{0}=\sqrt{{\bf q}^{2}+M^{2}}-M$. The right-hand-side of Eq. (53) is defined as $CC\;I_{1}(0)$ so that comparing $CC$ to unity provides a reasonable measure of the failure of this approximation to uphold current conservation. We express all momenta and mass in units of the target mass (=1), take as an examples $m=0.501,\;0.51$ and 0.6 and plot the numerical results in Fig. 3. Both $CC$ and ${\bf q}^{2}=Q^{2}$ are measured in units of the target mass $M$, taken as unity. Thus the natural scale of any quantity is unity. We see that current conservation is massively violated in the rest frame for systems in which $B/m=(2m-M)/m$ is not very small. In that case, the expression that potentially depends on the square of the wave function or density has no independent physical reality.

Previous work Miller:2007uy showed that the central transverse density of the neutron is negative. This contrasts with the long held view that there must be positive charge density at the center to neutralize the effects of a negatively charged pionic cloud that occupies the exterior. This result demands interpretation Miller:2008jc -Rinehimer:2009sz .

One relevant question is whether or not the intuition that a neutral system consisting of a heavy charge positively charged particle and a negatively charged lighter particle disobeys the standard intuition that the averaged squared charged radius is negative, when the charge density is evaluated in the infinite momentum frame. We examine this question in our model by taking the $\phi$ (of mass $m_{1}$) to be positively charged and the $\xi$ (of mass $m_{2}<m_{1}$ to be negatively charged.

The form factor of this model can be obtained by using Eq. (8) by including a second term obtained by interchanging $m_{1}$ and $m_{2}$ and putting a minus sign in front. That operation gives the result

The results of a numerical evaluation using $m_{1}=M$ and $m_{2}=0.14M$ are shown in Fig. 4. One observes the rise of $F(Q^{2})$ from zero, which is the effect expected from non-relativistic, rest-frame considerations. The effective squared radius, defined in Eq. (43) is indeed negative. This is the same as expected from the intuition that the negatively charged light particle resides on the outside edge of the system.

One obtains the analytic result for the charge radius by taking the limit of very low $Q^{2}$ in the expression for the form factor Eq. (54). The expression is simplified if one uses the (relevant for nucleon)assum case of $m_{1}=M$. Then one finds:

A very accurate approximation (better than 1% for $m^{2}_{2}\leq 0.14M^{2}$) is