Confinement in Nuclei and the Expanding Proton

TL;DR

This paper investigates how the internal structure of protons is altered when they are bound within nuclei, using Poincare-invariant models and lattice QCD to show that bound protons are larger than free ones, impacting nuclear physics measurements.

Contribution

The paper introduces simple, Poincare-invariant models respecting the Ward-Takahashi identity to demonstrate that medium modifications of proton structure are inevitable.

Findings

Bound protons are larger than free protons.

Medium modifications of proton structure are theoretically unavoidable.

Models confirm the necessity of considering nucleon size changes in nuclear physics.

Abstract

High-precision knowledge of electromagnetic form factors of nuclei is a subject of much current experimental and theoretical activity in nuclear and atomic physics. Such precision mandates that effects of the non-zero spatial extent of the constituent nucleons be handled in a manner that goes beyond the usual impulse approximation. A series of simple, Poincare-invariant, composite-proton models that respect the Ward-Takahashi identity and in which quarks are confined are used to study the validity of this approximation. The result of all of the models is a general theorem showing that medium modification of proton structure must occur. Combining this result with lattice QCD calculations leads to a conclusion that a bound proton must be larger than a free one.

Tables

Table 1. Table 1: Charge radii (in fm) for 3 H and 3 He. The first two rows are from experiment, the next two from theory.

Ref.	3H	3He
SACLAY Amroun:1994qj	1.76$\pm$ 0.09	1.96$\pm$0.03
Bates Beck:1987zz	1.68$\pm$ 0.0347	1.87$\pm$0.03
GFMC Pieper:2001ap	1.77$\pm$ 0.0131	1.97$\pm$0.0125
$\chi$EFT Piarulli:2012bn	1.756$\pm$0.006	1.962$\pm$ 0.004

Equations

$${\mathcal{F}}_A(Q^2)=F_A(Q^2)G_E(Q^2),$$

$$\Delta F_{1,2}=V\frac{\partial F_{1,2}}{\partial M^2}.$$

$$\Delta =xy{Q}^2+m_q^2(x+y)+z{m}_d^2-\frac{p^2+{p^{\prime }}^2}{2}z(1-z).$$

$$\Delta =(1-V\frac{\partial }{\partial M^2}){\Delta }_{\mathrm{on}}.$$

$$/p=M+\frac{p^2-M^2}{/p+M}\approx M+\frac{V}{2M}=(1+V\frac{\partial }{\partial M^2})M,$$

$$k\cdot k^{\prime }\to k^2+z^{2}p\cdot p^{\prime }+\frac{Q^2}{2}z(1-z).$$

$$q\cdot \Gamma (p+q,p)=S^{-1}(/p+/q)-S^{-1}(/p),$$

$$q\cdot {\Gamma }^{(q)}=\Sigma (/p)-\Sigma (/p^{\prime }),$$

$$S_C(p)=\sum _{\lambda =-1,1}\frac{1}{p^2+m_d^2+i\lambda ϵ}.$$

$$\delta r_E=V\frac{\partial r_E}{\partial m_{\pi }^2}\frac{\partial m_{\pi }^2}{\partial M^2}=-\frac{V}{M^2}(1.1\pm 0.1)\mathrm{fm},$$

Full text

Nucleons are composite particles made of quarks, gluons and quark-pairs bound by the confining forces of QCD. The composite nature means that nucleons bound in nuclei must be different than free ones Sargsian:2002wc . Many years of experiment and theory tells us the answer: the differences exist but are not very large. Early evidence of that was found in the EMC effect Aubert:1983xm ; Arnold:1983mw and also in kaon-nucleus scattering Mardor:1990cr . Recent reviews are in Refs. Hen:2016kwk ; Cloet:2019mql , and a recent update with many references is Wang:2018wfz . The present manuscript aims to present a new approach to medium modification of the nucleon wave function that is related both to experiment and to lattice QCD calculations. The key result of the presented arguments is that the proton gets bigger when it is bound in a nucleus.

The specific focus here is on elastic electron-nucleus scattering. This reaction has the simplifying feature that the initial and final nuclei are in the same quantum state. Elastic electromagnetic form factors of nuclei can be compared with ab initio nuclear structure calculations. For example, Ruiz:2016gne measures isotope shifts in the radii of Ca isotopes to better than 1% accuracy. New muonic atom measurements Antognini:2015moa that determine the charge radii of light nuclei are now at about the 1% level. Furthermore, a current Jefferson Laboratory experiment 12 ; Gomez:2017cwj aims to measure the the difference between the charge radii of 3He and 3H to a precision of $\pm 0.02$ fm. The current high precision goals create a need to learn how to improve the treatment of the effects of the non-zero spatial extent of the constituent nucleons.

This is because the nuclear electromagnetic form factor ${\cal F}_{A}(Q^{2})$ is often approximated as:

[TABLE]

where a spin-0 nucleus absorbs a space-like photon of four momentum $q^{\mu}$ and $Q^{2}=-q^{2}$, $G_{E}(Q^{2})=F_{1}(Q^{2})-{Q^{2}\over 4M^{2}}F_{2}(Q^{2})$ is the proton Sachs electric form factor, where $F_{1,2}$ are Dirac and Pauli form factors, $F_{A}(Q^{2})$ is the probability amplitude for a point proton to absorb momentum without changing the nuclear state and $M$ is the proton mass. Effects of other charged particles are ignored here for simplicity. This product ansatz of Eq. (1) will be referred to as the factorization approximation.

The original derivation of Eq. (1) is ancient Amaldi1950 ; Villi1959 and is based on non-relativistic classical physics, A quantum mechanical derivation of the same formula can be obtained by assuming an impulse approximation in which only the free form factors, $F_{1,2}(Q^{2})$ appear. The factorization approximation cannot be 100% accurate because the struck protons are bound in nuclei. The factorization approximation has been widely used for a long time, with no examination appearing in the literature.

The approach taken here is to construct a diverse set of models of the free proton and then place that proton in the nucleus. Elastic electron-proton scattering is shown in Fig. 1a. In free space, $p^{2}={p^{\prime}}^{2}=M^{2}$. The initial and final protons are on their mass shell. Imposing parity, time-reversal invariance, Lorentz symmetries and current conservation the observable quantities are the Dirac (Pauli) form factor $F_{1,(2)}(Q^{2})$. Suppose instead the proton is bound in the nucleus (see Fig. 1b). Interactions with nuclei involve evaluating Feynman graphs containing an integral over the four-momentum $p$ of the initial nucleon that ranges over all possible values of $p^{2}$ from $-\infty$ to $\infty$. This means that the Einstein relation equality between square of the four-momentum and $M^{2}$ is not generally accurate for bound nucleons. In general, the nucleon form factors should depend on $\gamma\cdot p$ and $\gamma\cdot{p^{\prime}}$ and functions (such as $(\gamma\cdot p)^{2}=p^{2}$) thereof Naus:1987kv .

As a result medium modifications of nucleon structure are determined by the virtuality, $V\equiv p^{2}-M^{2}$ and/or ${p}^{\prime 2}-M^{2}$. For elastic scattering on nuclei Lorentz and time-reversal invariance insures that these two quantities are equal. The average value of the virtuality can be computed from the spectral function CiofidegliAtti:2007ork , but nuclear wave functions are not presented as a function of specific values of $V$. Therefore the technical procedure here is to use a first-order expansion in powers of $V$. At first glance, the fact that the binding energy is small compared to the nucleon mass might seem to make it reasonable to neglect the differences between $p^{2},\,{p^{\prime}}^{2}$ and $M^{2}$. However, a better estimate can be obtained from the Schroedinger equation. For example, within the nuclear Hartree-Fock approximation a single particle wave function obeys the Schroedinger equation, with a dominant central binding potential ${\cal U}\,(\ll M)$. Therefore $\vec{p}^{2}/(2M)+{\cal U}=-B$, where $B>0$ is the binding energy and $p^{0}=M-B$. Then $p^{2}-M^{2}=(M-B)^{2}-\vec{p}^{2}-M^{2}\approx 2M{\cal U}$. The dependence on the binding energy cancels to first-order in $B$. In the centers of typical nuclei ${\cal U}$ is about $-50\,$ MeV GFB72 , so that $(p^{2}-M^{2})/M^{2}\approx-0.1$, significantly different from zero, but small enough to be considered an expansion parameter.

The detailed examination of the factorization approximation begins with a study of the Feynman graphs of Figs. 1 a,b. The aim is to compute the dependence on the off-mass-shell invariants that appear in the nucleus. The calculations are done so that the Ward-Takahashi identity, which guarantees current conservation, is respected. For the present models, including the diagram of Fig. 1c along with that of Fig. 1b is necessary for this to occur. Furthermore, the models must embody confinement. These two aspects are dealt with below in connection with arriving at the key result:

[TABLE]

At first glance, this equation seems very odd. How can a property of the proton depend upon its mass, when it is known to very high precision? The proton mass can be varied at will in different models. Moreover, results of fundamental lattice QCD calculations of nucleon properties depend implicitly on the proton mass via quark-mass dependence. The result Eq. (2) is obtained when $M^{2}$ is associated with the four-momentum squared that appears in propagators of the Bethe-Salpeter equations determining the wave functions of the various models used below.

The effects encapsulated in Eq. (2) can be recast in terms of specific two-body currents because the inverse of nucleon propagators contain a factor of $V$. The present formulation includes a variety of terms, is very compact and (as to be argued) arises naturally from considerations of relativistic dynamics.

The next step is to explain how Eq. (2) is derived. Five different models of the free proton are used:

• •

Quark-diquark, with spin 0 quarks and di-quarks with a scalar vertex function

• •

Quark-diquark with spin 1/2 quark, spin 0 di-quark with a scalar vertex function

• •

Quark-diquark with spin 1/2 quark, spin 1 di-quark with a vector vertex function (QED)

• •

Proton sometimes fluctuates into its neutron-$\pi^{+}$ component, with pseudovector coupling.

• •

Proton sometimes fluctuates into its $\Delta\pi$ components, with pseudovector coupling.

None of these models is realistic by itself, but each characterizes a significant aspect of proton structure.

Evaluating the Feynman graph of Fig. 1a for general off-shell kinematics renders it suitable for inclusion in Fig. 1b. The first-order approximation in $V$ allows the separate study of each term that contributes to medium modifications. The models employed here share common features, so that the generality of Eq. (2) can be displayed by discussing only the salient aspects of the models. For each model, the proton wave function involves a vertex function that converts a proton of momentum to a system of two constituents. One of the constituents, denoted by $q$ is charged and interacts with the photon, and the other, denoted by $d$ is a spectator. This notation is used for both quark-spectator models and pion-spectator models. In each model the three propagators provide a denominator of the form: D\equiv(k^{2}-m_{q}^{2})((k+q)^{2}-m_{q}^{2})((p-k)^{2}-m_{d}^{2}).\ These are combined with three Feynman parameters $x,y,z$ respectively, such that $x+y+z=1$, with a useful symmetry between $x$ and $y$. The factor $D$ can thus be re-written: $D\to(k^{2}-\Delta)^{3}$, as

[TABLE]

The on-mass-shell value of $\Delta$, denoted as $\Delta_{\rm on}$ is obtained by replacing $p^{2}$ and ${p^{\prime}}^{2}$ by $M^{2}$. By adding and subtracting the term $M^{2}z(1-z)$ one obtains the result that

[TABLE]

When used in a first-order expansion Eq. (4) gives one of the terms appearing in Eq. (2).

The terms in the numerator take many forms including: $\not{p},\,\not{p}^{\prime},p^{\mu},\,{p^{\prime}}^{\mu}=(p+q)^{\mu},\,2k^{\mu},\,k\cdot k^{\prime},\,\not{k}\not{k}^{\prime},\,(k^{\prime}\equiv k+q)$ where $\mu$ is the Lorentz-index of the photon-quark (or photon-pion) vertex. Let’s start with the term $\not{p}$, which is re-written as to first-order in $V$ as follows:

[TABLE]

and we see the pattern emerging. The same manipulations can be done for $\not{p}^{\prime}$. Another term that enters is $p^{\mu}$. Calculations are done in the Breit frame, with ${\mu}=0$ or in the Drell-Yan frame with ${\mu}=+$. Then the identity $2p^{\mu}=\gamma^{\mu}\not{p}+\not{p}^{\prime}\gamma^{\mu}+i\sigma^{{\mu}\nu}q_{\nu}$ is useful because the manipulations for $\not{p},\,\not{p}^{\prime}$ described above are applicable. The term involving $\sigma^{{\mu}\nu}$ contributes only to the on-mass-shell part of $F_{2}$.

The models involving struck pions contain a numerator term of the form $2(k^{\mu}+p^{\mu}z-q^{\mu})\to 2p^{\mu}\,z$ because of parity and the use of either of the two mentioned frames. The model with an intermediate $\Delta$ contains terms of the form $k\cdot k^{\prime}$ and $\not{k}\not{k}^{\prime}$. Upon applying the stated variable transformations, one finds

[TABLE]

The $k^{2}$ term is evaluated along with the denominators that are discussed above. The third term does not involve off-shell proton kinematics. The term $p\cdot{p^{\prime}}$ may be re-written as $q\cdot p+p^{2}={1\over 2}({p^{\prime}}^{2}+p^{2}-q^{2})$, and subtracting and adding $2M^{2}$ leads again to the result of Eq. (2). The manipulations needed to handle the term $\not{k}\not{k}^{\prime}$ are essentially the same, upon using Eq. (5).

The net result is that Eq. (2) emerges from each term. The general argument is that for each of the terms that enter one may add and subtract the on-shell expression. To first-order in $V$ all terms in the difference between the on-and off-mass-shell expressions can be expressed as a derivative.

The next step is to describe how the Ward-Takahashi (WT) identity Peskin:1995ev is maintained. If this is respected for all values of $p$ and ${p^{\prime}}$, electron-nucleus interactions will satisfy current conservation. This identity states that the amplitude ${\Gamma}^{\mu}(p+q,p)$ for a photon of momentum $q$ to be absorbed by a fermion of momentum $p$ is related to the fermion-propagator $S(\not{p})={1\over\not{p}-M_{0}-\Sigma(\not{p})},$ via

[TABLE]

where $M_{0}$ the bare mass and $\Sigma(p)$ the self-energy of the fermion. (A similar identity is obtained for photon absorption on a charged pion.) Satisfying the WT identity is absolutely necessary for high-precision nuclear calculations to be valid.

If one evaluates the term of Fig. 1 (a), in which the photon-quark interaction is denoted as ${\Gamma}^{(q)}$ one finds that

[TABLE]

in which the right-hand-side vanishes if $p^{2}=M^{2}$ and ${p^{\prime}}^{2}=M^{2}$. Thus the graph of Fig. 1a is a reasonable model for free protons. However, when the proton is bound in the nucleus, as in Fig. 1b, the momenta $p$ and ${p^{\prime}}$ are off the mass-shell and the WT identity is not respected. This problem is fixed by including the graph of Fig 1c. In that case one obtains $q\cdot\Gamma=(\not{p}+\not{q})-\not{p}-{\large(\Sigma((p+q)^{2})-\Sigma(p^{2}))\large)}=S^{-1}(p+q)-S^{-1}(p).$ The first two terms arise from Fig. 1c, and the next two from Fig. 1b.

The next step is to handle quark-confinement. Detailed evaluations of the Feynman graph of Fig. 1b fail dramatically to obey the factorization approximation, Eq. (1), if the quark propagator is taken to be that of a free quark. To see this, examine Eq. (3). For on shell kinematics with $p^{2}=M^{2},\,{p^{\prime}}^{2}=M^{2}$ the value of $\Delta$ is positive for all values of $x,y$ and $z$ provided the stability condition $M<m_{q}+m_{d}$ is obeyed. A similar stability condition holds for pion-baryon intermediate states. In evaluating the Feynman diagram of Fig. 1b, one integrates over all values of $p^{2}$, so that $\Delta$ can be negative. This means that the in medium proton form factor is complex-valued. The free form factor is real-valued, so the factorization approximation Eq. (1) must break down. Moreover, the singularity associated with lack of confinement makes plays havoc in numerical integration, and the existence of such singularities in models is unphysical because nuclei are stable. Finally, the appearance of zeros in the quantity $\Delta$ means that an expansion of nucleon properties in terms of the virtuality will not converge because the nuclear wave function will admit very large values.

The negative value of $\Delta$ can also be understood by examining the proton self-energy, $\Sigma({p^{\prime}}^{2})$ which involves the denominator $(k^{2}-m_{q}^{2})(({p^{\prime}}-k)^{2}-m_{d}^{2})\to k^{2}+{p^{\prime}}^{2}u(1-u)-m_{q}^{2}(1-u)-m_{d}^{2}u,$ where $u$ is another Feynman parameter with $0\leq u\leq 1$. This denominator has zeros for values of ${p^{\prime}}$ such that ${p^{\prime}}^{2}>(m_{q}+m_{d})^{2}$. This is also the condition required to knock a quark out of the proton. In Fig. 2, the final $q$ and $d$ can both be on the mass shell whenever ${p^{\prime}}^{2}>(m_{q}+m_{d})^{2}$. This feature arises from the lack of including effects of confinement.

Some means of implementing the main feature (no singularities) of confinement must be included in the present models. This can be done using an idea suggested by solutions to Dyson-Schwinger equations, see the review Roberts:1994dr . The procedure is to use quark (or di-quark) masses that occur in complex conjugate pairs. Complex conjugate singularities in quark propagators have been studied in connection with confinement in Refs. Atkinson:1978tk ; Cornwall:1980zw ; Munczek:1983dx ; Bhagwat:2002tx ; Bhagwat:2003vw , and Euclidean space lattice data can be modeled with propagators that have time-like complex conjugate singularities Alkofer:2003jj ; Alkofer:2003jk . Ref.  Tiburzi:2003ja used a model with complex conjugate poles to compute parton and generalized parton distributions.

The relevance here is that using a di-quark (spectator) propagator of the form

[TABLE]

in Euclidean space removes the unphysical singularities. The previous analysis of the effects of virtuality has been applied using Eq. (9) to the models discussed above with the result Eq. (2). Furthermore, detailed Euclidean space calculations using the models described above have shown that the results of using such a propagator can be obtained in Minkowski space by simply using a complex di-quark mass and obtaining the form factors by taking the real part of the computed amplitude. The net result is that using complex-valued quark masses removes the unphysical singularities initially present in the simple models used here. This is necessary to justify expansions in terms of virtuality.

The next step is to apply the key result, Eq. (2). It is worthwhile to study the proton charge radius defined Miller:2018ybm by ${r_{E}^{2}}\equiv-6G_{E}^{\prime}(0)$ in this first evaluation. Applying Eq. (2) then leads to a change in $r_{E}^{2}$ given by $\delta r_{E}^{2}={V}{}{\partial r_{E}^{2}\over\partial M^{2}}$ To evaluate this expression it is necessary to know how the proton radius depends on its mass. This derivative was explicitly computed to be negative for the five models discussed above. Given the negative values of the virtuality, the proton radius must expand when it is bound in the nuclear medium!

It is useful to take a more general approach by examining a broader range of models and lattice QCD calculations. For the models presented here an off-mass-shell proton is equivalent to an on-mass-shell proton of a mass less than $M$. This is because off-shell effects of the energy denominator $\Delta$ of Eq. (3) (in which $p^{2}$ replaces the $M^{2}$ that appears as an eigenvalue of the Bethe-Salpeter eqaution) and the off-shell effects of the numerator end up looking like Eq. (2). Based on obtaining Eq. (2) in all of the models, I assume that it is a generally valid, first-order treatment of the effects of virtuality and discuss the necessary derivative in a broader framework.

Let’s first examine the venerable MIT bag model Chodos:1974pn In its most simplistic version (with vanishing quark masses) the bag radius is inversely proportional to the mass of the nucleon. This translates to the result ${M^{2}\over r_{E}^{2}}{\partial r_{E}^{2}\over\partial M^{2}}=-1.$ The counterpoint to this model is the non-relativistic quark model as presented in e.g Ref. Isgur:1979be , in which harmonic oscillator confinement is used with the size parameter: $b^{2}\propto{1\over m_{q}}.$ This leads to ${M^{2}\over r_{E}^{2}}{\partial r_{E}^{2}\over\partial M^{2}}=-{1\over 2}$ because the $u,d$ quark mass $m_{q}\approx M/3$. There are other models in which pionic effects are very important. One example is found in the work of Beg:1973sc in which the dominant isovector contribution to the square of the nucleon radius is proportional to $\ln{M/m_{\pi}}$, where $m_{\pi}$ is the pion mass. This may be translated (using Eq. (4.2) of that paper) to ${M^{2}\over r_{E}^{2}}{\partial r_{E}^{2}\over\partial m_{\pi}^{2}}=-0.6{M^{2}\over m_{\pi}^{2}}$, potentially a very large effect. More detailed versions of this idea have been used to interpret the pion-mass dependence of the results of lattice QCD calculations, see e.g.  Hall:2013oga . If the proton mass increases with increasing pion mass (a reasonable expectation), the derivative ${\partial r_{E}^{2}\over\partial M^{2}}$ is again negative.

In each of model there is a connection between the proton mass and radius that occurs via a fundamental aspect of the model, such as bag radius, quark mass and the pion mass. The trend of all of these models is clear- ${\partial r_{E}^{2}\over\partial M^{2}}<0,$ which, along with the fact that $V<0$ means that the radius of a bound proton must be larger than that of a free one.

It is natural to turn to existing lattice QCD calculations of the proton radius because hadronic properties have long been computed as a function of quark masses (as expressed through the pion mass). The mass parameter that sets the mass scale is typically the square of the pion mass. In principle one can determine ${\partial r_{E}^{2}\over\partial m_{\pi}^{2}}$ and also $\partial M\over\partial m_{\pi}^{2}$ from lattice QCD calculations. The ratio of these two quantities gives the desired derivative.

Lattice QCD calculations of the proton charge radius have made significant recent progress Constantinou:2014tga ; Hasan:2017wwt ; Jang:2018djx ; Jang:2019jkn ; Alexandrou:2017ypw ; Alexandrou:2018sjm ; Bhattacharya:2013ehc ; Ishikawa:2018rew . Some difficulties involving lattice spacing, finite volume effects, disconnected graphs and the need to use extrapolations to extract charge radii. Current lattice results typically undershoot experiment by about 25 %. Many calculations focus on the dominant isovector radius because of the cancellation of disconnected graphs. Another persistent difficulty has been in making calculations at the small value of the physical pion mass. Ref. Alexandrou:2018sjm did calculations at $m_{\pi}=135$ MeV, and the recent calculation by the PACS collaboration (parallel array computer system) performed calculations at a pion mass of 146 MeV, using a large lattice size Ishikawa:2018rew . That work compiled recent results for $r_{E}^{2}$. Another calculation spanning pion masses from 135 to 320 MeV is that of Ref. Jang:2018djx . Ideally one could use the lattice calculations to determine ${\partial r_{E}^{2}\over\partial m_{\pi}^{2}}$ and ${\partial r_{E}^{2}\over\partial M^{2}}$. Using Fig 13 of Ref. Ishikawa:2018rew allows a determination that ${\partial r_{E}\over\partial m_{\pi}^{2}}\approx-3\,{\rm fm\,GeV^{2}}$ with a large uncertainty. This number is obtained by making a linear fit between the geometric mean of the values of the radii at the two lowest mass points  Ishikawa:2018rew ; Alexandrou:2018sjm and a precise value at $m_{\pi}=220$ MeV Bhattacharya:2013ehc . The more recent lattice data from Jang:2018djx ; Jang:2019jkn use an analytic parametrization of the $m_{\pi}^{2}$ dependence of their results. Using their formula and taking the isovector result to be dominant gives ${\partial r_{E}\over\partial m_{\pi}^{2}}\approx-2.6\,\pm 0.3\,{\rm fm\,GeV^{2}}$ The nucleon mass is well-described as a function of the pion mass as $M\approx M_{0}+1.14\,{\rm GeV}^{-1}m_{\pi}^{2}$ WalkerLoud:2008bp . See also  Alberg:2012wr for an interpretation of that formula. Using the results Jang:2018djx ; Jang:2019jkn one finds

[TABLE]

with the only source of uncertainty arising from ${\partial r_{E}\over\partial m_{\pi}^{2}}$. Taking $V/M^{2}=-0.1$ (from its value at the nuclear center) leads to an increase of the proton radius by about 0.11 fm or about 12%. The sign is well-determined as the product of two numbers that are each strongly constrained to be negative. The magnitude is less well determined, but is a reasonable estimate.

Can this effect be measured? See Wang:2018wfz for a perspective. A 0.11 fm increase is actually a rather large effect, but the increase is much smaller at the edge of the nucleus. Tantalizing hints have been seen a several arenas. For example, in quasi-elastic electron nucleus scattering Strauch:2012nra and in the Coulomb sum rule (inclusive $(e,e^{\prime})$ scattering) Morgenstern:2001jt ; Paolone:2018jup . No definitive evidence for such a change has been seen so far, see e.g. Sick:1985ygc . In high-momentum transfer quasielastic scattering the final-state knocked out proton is essentially free, whereas for the present case of elastic scattering the initial and final nucleons are equally virtual. Thus the 12% reported here would be reduced by a factor of two, even before accounting for reductions caused when the reaction occurs near the nuclear surface. Moreover, the quasi-elastic measurements have complications due to the presence of final state interactions and related issues with current conservation.

The increase in radius seen here represents a violation of the factorization approximation Eq. (1). To understand its impact let’s examine the case of A=3. The current status of root-mean-square charge radii is summarized in Table I.

The nuclei 3H and 3He are related by isospin invariance, which is an approximate symmetry. Coulomb (and other smaller) effects cause 3He to be less bound and larger. The standard procedure, see e.g Pieper:2001mp for computing the charge radii of 3-body nuclei is to expand each of the terms of Eq. (1) to first-order in $Q^{2}$. This results in an expression: $R_{A}^{2}=R_{\rm pt}^{2}+r_{E}^{2}$ (if the neutron contribution is neglected). The proton radius $\sim 0.86$ fm is not small compared to the nuclear charge radii displayed in Table I. The average virtuality for 3He is reported in Ref. CiofidegliAtti:2007ork as $V/M^{2}=-0.073$. Given this number and Eq. (10) the resulting shift in the proton radius is about 0.08 fm. This corresponds to a 2% increase in the 3H charge radius comparable to the current experimental uncertainties, but within reach of present experimental goals. Furthermore the increase of 0.08 fm is much larger than that produced by the effects of meson exchange currents or variations in the cutoff of chiral perturbation theory found in Piarulli:2012bn . Note also that the difference in radii between 3He and 3H may be strongly affected by the effects of virtuality because a struck proton in 3He is influenced by one $pn$ interaction and one $pp$ interaction, while a struck proton in 3H is influenced by two $pn$ interactions. The $T=0$ $pn$ interactions are stronger than the $T=1$ $pp$ interactions, an important feature in a diverse set of reactions Hen:2016kwk ; Duer:2018sxh ; Schmookler:2019nvf . The net result is that the increase in the proton radius suggested here is relevant for understanding the properties of nuclei with $A=3$.

The calculations performed here are for nuclear medium effects of electromagnetic form factors. There are many general features of these calculations, so that one may speculate that the key result of Eq. (2) extends to other matrix elements of other one-body operators, $\cal O$ such that $\langle{\cal O}(p^{2})\rangle\approx\langle{\cal O}(M^{2})\rangle+(p^{2}-M^{2}){\partial\over\partial M^{2}}\langle{\cal O}(M^{2})\rangle$. The simplicity of this relation is very appealing. If it is valid, describing a wide variety of medium effects from the unified viewpoint of examining the dependence on virtuality may be possible.

I summarize. The field-theoretic calculations discussed here show that nucleons must be modified when bound in nuclei. Five different models yield the result Eq. (2). The necessary derivatives with respect to mass that appear in that equation may be computed using lattice QCD. Perhaps other proton properties can also be treated this way. The present work provides a new approach to understanding nuclear modifications of nucleon properties and strengthens the connection between lattice QCD calculations and nuclear physics.

Acknowledgments G.A. Miller would like to thank the Lab for Nuclear Science at MIT, the Southgate Fellowship of Adelaide University (Australia), the Bathsheba de Rothchild Fellowship of Hebrew University (Jerusalem), the Shaoul Fellowship of Tel Aviv University, the Physics Division of Argonne National Laboratory and the U. S. Department of Energy Office of Science, Office of Nuclear Physics under Award Number DE-FG02-97ER-41014 for support that enabled this work. I also thank P. C. Tandy, G.F. Bertsch, I. Cloët, M. Consantantinou, B. Holsetin, V. Koch, C. Monahan, C. Roberts, M.J. Savage, S. Sharpe, M. Strikman and S. Syritsyn for useful discussions. I thank H-W Lin, R. Gupta and the members of the PNDME collaboration for providing numerical data in advance of their arXiv posting.