Transverse charge density and the radius of the proton
TTransverse Charge Density and the Radius of the Proton
Alexander V. Gramolin ∗ and Rebecca L. Russell Department of Physics, Boston University, Boston, Massachusetts 02215, USA The Charles Stark Draper Laboratory, Inc., Cambridge, Massachusetts 02139, USA
A puzzling discrepancy exists between the values of the proton charge radius obtained usingdifferent experimental techniques: elastic electron-proton scattering and spectroscopy of electronicand muonic hydrogen. The proton radius is defined through the slope of the electric form factor, G E ( Q ), at zero four-momentum transfer, which is inaccessible in scattering experiments. Wepropose a novel method for extracting the proton radius from scattering data that does not requiredetermining the slope of G E at Q = 0. This method relates the radius of the proton to its transversecharge density, which can be properly interpreted as a two-dimensional Fourier transform of the Diracform factor, F ( Q ), over the entire Q range. We apply our method to reanalyze the extensive dataobtained by the A1 Collaboration [J. C. Bernauer et al. , Phys. Rev. Lett. , 242001 (2010)] andextract a radius value, r E = 0 . stat (5) syst (4) model fm, that is consistent with the original result.We also provide new parametrizations for the Dirac and Pauli form factors and the transverse chargeand magnetization densities of the proton. Our reanalysis shows that the proton radius discrepancycannot be explained by issues with fitting and extrapolating the A1 data to Q = 0. Introduction.
Over a century after Rutherford’s dis-covery of the proton [1], some fundamental propertiesof this particle are still not well understood. In particu-lar, the proton charge radius, r E , remains experimentallypuzzling. Beginning with the pioneering research [2, 3], r E has long been measured in elastic electron-protonscattering experiments [4–10]. It has also been extractedfrom atomic transition frequencies in both electronic [11–15] and muonic [16, 17] hydrogen. The 2014 CODATArecommended value of r E , obtained from all non-muonicdata available at the time, is 0 . r E = 0 . r E has become known as the “proton radius puzzle” [18–20]. In this Letter, we propose a novel method for ex-tracting r E from scattering data and use it to reanalyzethe measurement reported in Refs. [5, 6].The electromagnetic structure of the proton is encodedin its Dirac and Pauli form factors, F ( Q ) and F ( Q ),which depend on the negative four-momentum transfersquared, Q = − q (see the textbooks [21–23]). Insteadof F and F , it is often more convenient to use the Sachselectric and magnetic form factors, defined as G E = F − Q M κF , G M = F + κF , (1)where M ≈ .
938 GeV is the mass of the proton and κ ≈ .
793 is its anomalous magnetic moment in unitsof the nuclear magneton. The Sachs form factors have asimple interpretation when considered in the Breit frame,where the exchanged virtual photon carries momentum q but no energy [21–23]. In this frame, Q = q and G E and G M can be interpreted as the three-dimensional Fourier transforms of the proton’s spatial charge andmagnetization densities, respectively. In particular, G E ( Q ) = 4 πQ ∞ Z rρ E ( r ) sin ( Qr ) dr, (2)where ρ E ( r ) is the spherically-symmetric charge densityof the proton. We can expand Eq. (2) as G E ( Q ) = 1 − h r E i Q + h r E i Q − . . . , (3)where h r nE i = 4 π ∞ Z r n +2 ρ E ( r ) dr (4)is the n -th moment of the charge distribution. As followsfrom Eq. (3), the second moment can also be determinedfrom the slope of G E at Q = 0: h r E i = − dG E ( Q ) dQ (cid:12)(cid:12)(cid:12)(cid:12) Q =0 . (5)This formula is accepted as the definition of r E in bothscattering and spectroscopic measurements [24].Unfortunately, the concept of the three-dimensionalspatial charge and magnetization densities is valid onlyin the non-relativistic limit Q (cid:28) M , when the Breitframe coincides with the proton rest frame [21, 24, 25].This is why the proton radius cannot be properly deter-mined through ρ E ( r ), as suggested by Eq. (4), but mustbe instead defined by Eq. (5). However, the definition (5)is particularly inconvenient for scattering experiments: itrequires measuring G E at the lowest achievable Q val-ues, extrapolating the data down to Q = 0, and inferringthe slope of G E at that point. Such a procedure is in-evitably model dependent, which greatly complicates the a r X i v : . [ nu c l - e x ] F e b extraction of the proton radius [26–31]. We propose toavoid these issues by using a relativistic analog of Eq. (4)formulated in terms of the transverse charge density. Transverse charge density.
In this section, we brieflyreview the definition of the transverse charge density andits relation to the proton radius [24, 25, 32–39]. We startwith a change of space-time coordinates from the usual( x , x , x , x ) to ( x + , x − , b ), where x ± = ( x ± x ) / √ b = ( x , x ) is the trans-verse position vector. By setting q + = 0, we specify theinfinite-momentum frame in which q µ has only transversecomponents: q µ = (0 , , q ⊥ ) and Q = q ⊥ . Then, theDirac form factor F ( Q ) can be related to a circularly-symmetric transverse charge density of the proton, ρ ( b ),by the following two-dimensional Fourier transform: F ( Q ) = 2 π ∞ Z bρ ( b ) J ( Qb ) db, (6)where b = | b | is the impact parameter and J denotesthe Bessel function of the first kind of order zero. Forexample, the m -pole form factor, F m -pole ( Q ) = (cid:18) Q Λ (cid:19) − m , (7)which is a generalization of the monopole ( m = 1) anddipole ( m = 2) form factors, corresponds to ρ m -pole ( b ) = Λ m +1 b m − m ( m − π K m − (Λ b ) , (8)where Λ is a scale parameter and K m − denotes the mod-ified Bessel function of the second kind of order m − J ( Qb ), we can rewrite Eq. (6) as F ( Q ) = 1 − h b i Q + h b i Q − . . . , (9)where h b n i = 2 π ∞ Z b n +1 ρ ( b ) db (10)is the n -th moment of ρ ( b ) and F (0) = h b i = 1. Themoment expansion (9) indicates that the mean-squaretransverse charge radius of the proton is h b i = − dF ( Q ) dQ (cid:12)(cid:12)(cid:12)(cid:12) Q =0 . (11)Equations (6) and (9)–(11) are the relativistic counter-parts of Eqs. (2) and (3)–(5), respectively.It is important to recognize that there is a simple con-nection between r E and h b i . Indeed, after differentiating Eq. (1) for G E with respect to Q , setting Q = 0, andsubstituting Eqs. (5) and (11), we obtain r E = r (cid:16) h b i + κM (cid:17) . (12)This equation defines the proton radius through the sec-ond moment (10) of the transverse charge density. Notethat we use Eq. (11) only to derive the relation (12) butnot to experimentally determine h b i . Series expansion for ρ ( b ) . To facilitate the extrac-tion of h b i from scattering data, we first expand thetransverse charge density in a series of functions such thatthe moments (10) are easy to calculate. Since F ( Q ) atsmall Q is close to the dipole form factor, we expect that ρ ( b ) can be approximated as ρ ( b ) times a polyno-mial in Λ b . Particularly suitable are the orthogonal poly-nomials P ( ν ) n defined by the orthonormality condition ∞ Z P ( ν ) m ( x ) P ( ν ) n ( x ) w ν ( x ) dx = δ mn , (13)where w ν ( x ) = 2Γ( ν + 1) x ν/ K ν (2 √ x ) (14)is the weight function, m and n are the degrees of thepolynomials, δ mn is the Kronecker delta, and Γ denotesthe gamma function. These polynomials have recentlybeen studied in Ref. [40] (note that we use a differentnormalization for the weight function). We choose ν = 1and x = Λ b / w ν ( x ) with ρ ( b ). The firstthree corresponding polynomials are P (1)0 ( x ) = 1 , P (1)1 ( x ) = x − √ , (15) P (1)2 ( x ) = x − x + 186 √ . (16)For more terms, see Supplemental Material [41].We can therefore approximate the transverse chargedensity as a truncated series ρ ( b ) ≈ ρ ( b ) N X n =0 α n P (1) n (cid:0) Λ b / (cid:1) , (17)where α n are the expansion coefficients. After substitut-ing Eq. (17) into Eq. (10), we find h b i = α , h b i = 8Λ (cid:0) α + √ α (cid:1) . (18)In general, h b n i is a linear combination of α , α , . . . , α n .Therefore, if Λ is fixed, there is a one-to-one correspon-dence between the expansion coefficients α n and the evenmoments of the transverse charge density (17). Parametrizations for F ( Q ) and F ( Q ) . After sub-stituting the series expansion (17) into Eq. (6), we obtainthe following parametrization for the Dirac form factor: F ( Q ) ≈ N X n =0 α n A n (cid:0) Q / Λ (cid:1) , (19)where A ( y ) = 1(1 + y ) , (20) A ( y ) = − y ( y + 4) √ y ) , (21) A ( y ) = y (cid:0) y + 22 y + 39 (cid:1) √
26 (1 + y ) , . . . , (22) A N ( y ) = ∞ Z P (1) N ( x ) w ( x ) J (2 √ xy ) dx (23)are rational functions.Extending our formalism to the Pauli form factor, wecan represent it as F ( Q ) = 2 π ∞ Z bρ ( b ) J ( Qb ) db, (24)where ρ ( b ) is the transverse magnetization density [42].It is argued in Ref. [37] that ρ M = − b ( dρ /db ) is a betterdefinition for the proton magnetization density. (Thereis also a closely-related transverse charge density of a po-larized proton, see Refs. [25, 36].) However, in this Letterwe are not concerned with the physical interpretation of ρ ( b ) and use it only to parametrize F ( Q ).For reasons that will become clear shortly, we approx-imate ρ ( b ) as another truncated series, ρ ( b ) ≈ ρ ( b ) N X n =0 β n P (2) n (cid:0) Λ b / (cid:1) , (25)where β n are the expansion coefficients. After substitut-ing this into Eq. (24), we get F ( Q ) ≈ N X n =0 β n B n (cid:0) Q / Λ (cid:1) , (26)where B ( y ) = 1(1 + y ) , (27) B ( y ) = − √ y ( y + 5) √ y ) , (28) B ( y ) = y (cid:0) y + 64 y + 132 (cid:1) √
110 (1 + y ) , . . . , (29) B N ( y ) = ∞ Z P (2) N ( x ) w ( x ) J (2 √ xy ) dx. (30) We set α = β = 1 to ensure that F (0) = F (0) = 1.At high Q , our parametrizations have the asymptoticbehavior expected from the dimensional scaling laws [43]: F ∝ (Λ /Q ) and F ∝ (Λ /Q ) . This justifies our choiceof the series expansions for ρ ( b ) and ρ ( b ). Note that theterms A and B correspond to the dipole and “tripole”( m = 3) form factors (7), respectively. Extraction of the proton radius.
Based on the aboveresults, we propose the following method for extractingthe proton charge radius from elastic scattering data.First, the measured cross sections are fit with the Rosen-bluth formula [41] assuming the parametrizations (19)and (26) for the Dirac and Pauli form factors, where Λ, α , . . . , α N , and β , . . . , β N are 2 N + 1 free parameters.Then the mean-square transverse charge radius h b i iscalculated from Λ and α using Eq. (18). Finally, theproton radius is given by Eq. (12). Note that our methoddoes not require measuring the slope of G E at Q = 0and considers the data at all Q values. In addition, itallows one to extract the transverse densities ρ ( b ) and ρ ( b ) given by Eqs. (17) and (25).To illustrate our method, we apply it to the extensiveand precise elastic electron-proton scattering data ob-tained by the A1 Collaboration at the Mainz MicrotronMAMI [5, 6]. The collaboration measured 1422 crosssections at Q values spanning the range from 0.004 to1 GeV . Three magnetic spectrometers and six beam en-ergies (180, 315, 450, 585, 720, and 855 MeV) were used,resulting in 18 distinct data groups. To overcome theproblem of achieving the absolute normalization of themeasurement with sub-percent precision, they exploitedthe large redundancy of the data and introduced 31 freenormalization parameters. These parameters were de-termined using least-squares fits of different form factormodels to the measured cross sections. The A1 Collabo-ration obtained the following value for the proton chargeradius: r E = 0 . stat (4) syst (2) model (4) group fm , (31)where the numbers in parentheses represent the statis-tical, systematic, model, and group uncertainties. Thestatistical uncertainty accounts for all point-to-point er-rors of the cross sections, not only those due to countingstatistics. The group uncertainty was introduced becauseof an unexplained difference between the radii obtainedusing the spline and the polynomial groups of form factormodels.Following the original analysis [5, 6], we fit the data byminimizing the following objective function: χ = X i (cid:0) p i σ exp i − σ fit i (cid:1) ( p i ∆ σ i ) , (32)where σ exp i are the measured cross sections, ∆ σ i are theirpoint-to-point uncertainties, σ fit i are the model cross sec- TABLE I. Group-wise cross-validation results for different ex-pansion orders before ( λ = 0) and after ( λ >
0) regularizationwas applied. λ = 0 λ > N χ χ λ χ χ tions [41], and p i are known combinations of 31 free nor-malization parameters. The total number of fit parame-ters is 2 N +32, where N is the order of the form factor ex-pansions (19) and (26). When choosing the value of N , itis important to avoid both underfitting and overfitting—aproblem known as the bias-variance trade-off. The pop-ular reduced chi-square test is not appropriate for thispurpose because the number of degrees of freedom is ill-defined for a nonlinear fit [44]. Instead, we use cross-validation and regularization, which are standard tech-niques in statistical learning [45, 46].Careful cross-validation is critical to finding the rightbalance in the bias-variance trade-off. Typically, a modelis cross-validated by randomly dividing the data into k subsets, fitting the model to k − k times so that each of the subsets is used as a test set ex-actly once. However, this procedure assumes that errorson the data points are uncorrelated. We instead performcross-validation by holding out each of the 18 experimen-tal data groups in turn and testing on that group whiletraining on the others. Recall that the data groups corre-spond to different spectrometer and beam energy combi-nations. This 18-fold group cross-validation allows us tominimize overfitting to systematic artifacts, by ensuringthat the model generalizes well to unseen experimentalconditions.The cross-validation results for different orders N areshown in Table I, where χ and χ are the total chi-square values (32) obtained on the training and test sets,respectively. Note that each data point occurs only oncein test sets but 17 times in training sets. For this reason, χ has been divided by 17 to make it directly compa-rable to χ . While χ monotonically decreases as N increases and the model becomes more flexible, χ reaches a minimum at N = 5. This indicates underfittingfor N <
N > N ≥ TABLE II. Objective function values and extracted radii forthe regularized models trained on the full dataset.
N λ L χ h b i r E (cid:0) GeV − (cid:1) (fm)5 0.02 1584 1576 11.49 0.8896 0.07 1580 1573 11.42 0.8877 0.2 1579 1572 11.37 0.8858 0.4 1578 1571 11.32 0.883 function: L = χ + λ N X n =1 (cid:0) α n + β n (cid:1) , (33)where α n and β n are the expansion coefficients and λ is the regularization parameter. The second term inEq. (33) encourages the sum of the squares of the expan-sion coefficients to be small and thus reduces the flexi-bility of the model in a controlled way. We determinethe optimal regularization parameter for each order byscanning a range of λ values and choosing the one thatresults in the lowest χ . One can see from Table I thatregularization improves χ without significantly com-promising χ . As expected, the optimal λ value andthe improvement in χ increase with N .After the optimal values of λ are determined, we trainthe N ≥ N = 5 model as our main fit and the higherorders to estimate model misspecification uncertainty.Note that we achieve a similar χ value to that of Ref. [6]while using a more efficient parametrization of the formfactors (1576 for 11 parameters vs. 1565 for 16 parame-ters). Our cross section normalizations differ from thosedetermined in the original analysis by less than 0.3%.The form factors and the transverse densities given byour main fit are shown in Fig. 1 with 68% confidenceintervals. The point-to-point (statistical) uncertaintiesare determined by propagating the errors of the fit pa-rameters taking into account the full covariance matrix.To estimate the systematic uncertainties, we follow theoriginal analysis and refit our model using four modifi-cations of the cross section data. These modificationscorrespond to the upper and lower bounds of (1) the en-ergy cut in the elastic tail and (2) all other systematiceffects linear in the scattering angle [6]. We perform allfits with floating normalizations and use the largest de-viation from the primary fit as an uncertainty estimate.For further details on the data analysis, the reader isreferred to Supplemental Material [41] and our Pythoncode [47].Our final extraction of the proton charge radius fromthe full A1 data yields r E = 0 . stat (5) syst (4) model fm , (34) . . . . . . Q (cid:16) GeV (cid:17) . . . . . F , F F F (a) . . . . . b (fm) − − − ρ , ρ (cid:16) f m − (cid:17) ρ ρ (b) FIG. 1. Form factors and transverse densities extracted using our best model ( N = 5, λ = 0 . F (red) and F (blue) as functions of Q . The black dashed line is a tangent to F at Q = 0 corresponding to the mean-square transversecharge radius h b i = 11 .
49 GeV − . Note that our extraction of h b i is based on Eq. (10) rather than Eq. (11). (b) Transversedensities ρ (red) and ρ (blue) as functions of b . In both panels, lighter inner bands indicate the 68% statistical confidenceintervals of the corresponding quantities, while darker outer bands show the 68% statistical and systematic confidence intervalsadded in quadrature. where the model uncertainty is estimated based on thehigher-order values of r E listed in Table II. Our radius islarger by 0.01 fm than the original result (31), but bothvalues are consistent given their uncertainties. Therefore,we confirm that the A1 data imply a large proton radius,although the possibility of unrecognized systematic errorscan never be ruled out. Conclusion.
We have presented a novel method forextracting the proton charge radius from elastic scatter-ing data that does not require determining the slope of G E at Q = 0. The method is based on Eq. (12) relat-ing r E to the second moment of the transverse chargedensity ρ ( b ). This density is a proper two-dimensionalFourier transform of the Dirac form factor F ( Q ). As aconsequence, ρ ( b ) and r E can be determined by analyz-ing all available scattering data, not just those obtainedat low Q values. Another novelty is the use of F insteadof the usual G E to extract the proton radius. To facilitatethe analysis, we have proposed reasonable parametriza-tions not only for the form factors F ( Q ) and F ( Q ),but also for the transverse densities ρ ( b ) and ρ ( b ).We have applied our method to the extensive data ob-tained by the A1 Collaboration [5, 6]. To find the rightbalance between underfitting and overfitting, we haveused cross-validation and regularization—best practicesfrom the field of statistical learning often overlooked innuclear physics. Figure 1 shows the form factors and thetransverse densities that we have extracted. Our methodhas yielded the proton radius (34), which is consistentwith the A1 value (31) but larger by 0.01 fm. Therefore,our reanalysis has confirmed that the full A1 data lead to the proton charge radius that contradicts the muonichydrogen results [16, 17]. This means that the discrep-ancy cannot be explained by issues with data fitting andextrapolation. Further progress can be achieved by com-bining our approach with a careful reanalysis of all avail-able electron-proton scattering data. Finally, the methodcan be extended to better understand other properties ofthe proton such as its magnetic radius. ∗ [email protected][1] E. Rutherford, Collision of α particles with light atoms.IV. An anomalous effect in nitrogen, Philos. Mag. ,581 (1919).[2] R. Hofstadter and R. W. McAllister, Electron scatteringfrom the proton, Phys. Rev. , 217 (1955).[3] R. Hofstadter, Electron scattering and nuclear structure,Rev. Mod. Phys. , 214 (1956).[4] G. G. Simon, Ch. Schmitt, F. Borkowski, andV. H. Walther, Absolute electron-proton cross sectionsat low momentum transfer measured with a high pres-sure gas target system, Nucl. Phys. A , 381 (1980).[5] J. C. Bernauer et al. (A1 Collaboration), High-precisiondetermination of the electric and magnetic form factorsof the proton, Phys. Rev. Lett. , 242001 (2010).[6] J. C. Bernauer et al. (A1 Collaboration), Electric andmagnetic form factors of the proton, Phys. Rev. C ,015206 (2014).[7] X. Zhan et al. , High-precision measurement of the protonelastic form factor ratio µ p G E /G M at low Q , Phys. Lett.B , 59 (2011).[8] M. Mihoviloviˇc et al. , First measurement of proton’s charge form factor at very low Q with initial state radi-ation, Phys. Lett. B , 194 (2017).[9] M. Mihoviloviˇc et al. , The proton charge radius extractedfrom the Initial State Radiation experiment at MAMI,arXiv:1905.11182.[10] W. Xiong et al. , A small proton charge radius from anelectron-proton scattering experiment, Nature , 147(2019).[11] P. J. Mohr, D. B. Newell, and B. N. Taylor, CODATArecommended values of the fundamental physical con-stants: 2014, Rev. Mod. Phys. , 035009 (2016).[12] A. Beyer et al. , The Rydberg constant and proton sizefrom atomic hydrogen, Science , 79 (2017).[13] H. Fleurbaey, S. Galtier, S. Thomas, M. Bonnaud,L. Julien, F. Biraben, F. Nez, M. Abgrall, and J. Gu´ena,New measurement of the 1S-3S transition frequency ofhydrogen: Contribution to the proton charge radius puz-zle, Phys. Rev. Lett. , 183001 (2018).[14] N. Bezginov, T. Valdez, M. Horbatsch, A. Marsman,A. C. Vutha, and E. A. Hessels, A measurement of theatomic hydrogen Lamb shift and the proton charge ra-dius, Science , 1007 (2019).[15] A. Grinin, A. Matveev, D. C. Yost, L. Maisenbacher,V. Wirthl, R. Pohl, T. W. H¨ansch, and T. Udem, Two-photon frequency comb spectroscopy of atomic hydrogen,Science , 1061 (2020).[16] R. Pohl et al. , The size of the proton, Nature , 213(2010).[17] A. Antognini et al. , Proton structure from the measure-ment of 2S-2P transition frequencies of muonic hydrogen,Science , 417 (2013).[18] R. Pohl, R. Gilman, G. A. Miller, and K. Pachucki,Muonic hydrogen and the proton radius puzzle, Annu.Rev. Nucl. Part. Sci. , 175 (2013).[19] C. E. Carlson, The proton radius puzzle, Prog. Part.Nucl. Phys. , 59 (2015).[20] J.-P. Karr, D. Marchand, and E. Voutier, The protonsize, Nat. Rev. Phys. , 601 (2020).[21] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics (Pergamon Press, Oxford,1982).[22] F. Halzen and A. D. Martin,
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FUNCTIONS P ( ) n , P ( ) n , A n , AND B n Polynomials P ( ν ) n ( x ) are defined by the orthonormality condition given in Eq. (13) of the main text. The first sixterms of P ( ) n ( x ) are P ( ) ( x ) = , (S1) P ( ) ( x ) = x − √ , (S2) P ( ) ( x ) = x − x + √ , (S3) P ( ) ( x ) = x − x + x − √ , (S4) P ( ) ( x ) = x −
37 620 x +
997 200 x − x + √ , (S5) P ( ) ( x ) = x −
148 095 x + x −
169 866 000 x +
844 128 000 x −
572 702 4001 123 200 √ . (S6)The first six polynomials P ( ) n ( x ) are P ( ) ( x ) = , (S7) P ( ) ( x ) = x − √ , (S8) P ( ) ( x ) = x − x + √ , (S9) P ( ) ( x ) = x − x + x − √ , (S10) P ( ) ( x ) = x −
10 368 x +
327 420 x − x + √
11 938 638 , (S11) P ( ) ( x ) =
25 187 x − x +
433 086 780 x − x +
60 331 975 200 x −
57 256 264 800151 200 √
541 950 039 098 . (S12)The rational functions A n ( y ) are defined by Eq. (23) in the main text. The first six terms are A ( y ) = ( + y ) , (S13) A ( y ) = − y ( y + )√ ( + y ) , (S14) A ( y ) = y ( y + y + )√ ( + y ) , (S15) A ( y ) = − y ( y + y + y + )√ ( + y ) , (S16) A ( y ) = y ( y +
21 560 y +
104 885 y +
225 774 y +
182 520 ) √ ( + y ) , (S17) A ( y ) = − y (
13 257 y +
210 730 y + y + y + y + ) √ ( + y ) . (S18) a r X i v : . [ nu c l - e x ] F e b The rational functions B n ( y ) are defined by Eq. (30) in the main text. The first six terms are B ( y ) = ( + y ) , (S19) B ( y ) = − √ y ( y + )√ ( + y ) , (S20) B ( y ) = y ( y + y + )√ ( + y ) , (S21) B ( y ) = − y ( y + y + y + )√ ( + y ) , (S22) B ( y ) = √ y ( y +
10 324 y +
58 410 y +
141 476 y +
125 935 )√ ( + y ) , (S23) B ( y ) = − y (
378 679 y + y +
53 840 262 y +
189 977 062 y +
329 168 959 y +
225 929 067 )√
541 950 039 098 ( + y ) . (S24)The first ten terms of P ( ) n , P ( ) n , A n , and B n can be calculated using the following Mathematica [1] code: $ Assumptions = y > 0;w[ ν _] := 2*x^( ν /2)*BesselK[ ν , 2*Sqrt[x]]/Gamma[ ν +1]P1 = Together[Orthogonalize[x^Range[0, 9], Integrate[ { x, 0, Infinity } ] &]]P2 = Together[Orthogonalize[x^Range[0, 9], Integrate[ { x, 0, Infinity } ] &]]A = Together[Integrate[ { x, 0, Infinity } ] & /@ P1]B = Together[Integrate[ { x, 0, Infinity } ] & /@ P2] ELASTIC SCATTERING CROSS SECTION
We use the beam energy, E , and the negative four-momentum transfer squared, Q , as two independent kinematicvariables. The electron scattering angle, θ , can be determined from E and Q as θ = arccos [ − M Q E ( M E − Q ) ] . (S25)Also useful are the dimensionless kinematic variables τ and ε , defined as τ = Q M , (S26) ε = [ + ( + τ ) tan θ ] − . (S27)The differential cross section for unpolarized elastic electron-proton scattering is given by the Rosenbluth formula dσ d Ω = σ red ε ( + τ ) dσ Mott d Ω , (S28)where σ red = ε G E ( Q ) + τ G M ( Q ) = ε [ F ( Q ) − τ κF ( Q )] + τ [ F ( Q ) + κF ( Q )] (S29)is the so-called reduced cross section and dσ Mott / d Ω is the Mott cross section describing the scattering of electronson spinless point charged particles. The Sachs form factors G E and G M are often approximated as G E ( Q ) ≈ G dip ( Q ) , (S30) G M ( Q ) ≈ µ G dip ( Q ) , (S31)where G dip ( Q ) = ( + Q .
71 GeV ) − (S32)is the standard dipole form factor and µ = + κ is the magnetic moment of the proton. The corresponding reducedcross section is σ dip = ( ε + µ τ ) G ( Q ) . (S33) BEST-FIT RESULTS
Here we provide additional information on our best model ( N = λ = . TABLE S1. Expansion coefficients for our best fit. The scale parameter was found to be Λ = . ± .
03 GeV. n = n = n = n = n = n = α n . ± .
06 1 . ± .
14 6 . ± .
58 9 . ± . . ± . β n − . ± . − . ± . − . ± . − . ± . − . ± . . . . . . . Q (cid:16) GeV (cid:17) . . . . . G E / G d i p . . . . . . Q (cid:16) GeV (cid:17) . . . . . . G M / ( µ G d i p ) FIG. S1. Extracted electric (left panel) and magnetic (right panel) form factors as functions of Q . We determine G E and G M from F and F using Eq. (1) of the main text and scale them by the corresponding dipole form factors, Eqs. (S30) and (S31).The lighter inner bands around the black best-fit lines are the 68% statistical confidence intervals, while the darker outer bandsare the 68% statistical and systematic confidence intervals added in quadrature. We compare our extraction with the valuesof G E and G M obtained in Ref. [3] using the Rosenbluth separation technique (blue data points with error bars representingstatistical uncertainties). The Rosenbluth results are model-independent but based only on a subset of the cross section data.[1] Wolfram Research, Inc., Mathematica, Version 12.1, Champaign, IL (2021).[2] https://github.com/gramolin/radius/ [3] J. C. Bernauer et al. (A1 Collaboration), Electric and magnetic form factors of the proton, Phys. Rev. C , 015206 (2014). .
00 0 .
02 0 .
04 0 .
06 0 . Q (cid:16) GeV (cid:17) . . . . σ r e d / σ d i p
180 MeV .
00 0 .
05 0 .
10 0 .
15 0 . Q (cid:16) GeV (cid:17) . . . . . σ r e d / σ d i p
315 MeV . . . . Q (cid:16) GeV (cid:17) . . . . . . σ r e d / σ d i p
450 MeV . . . . . . Q (cid:16) GeV (cid:17) . . . . . . σ r e d / σ d i p
585 MeV . . . . Q (cid:16) GeV (cid:17) . . . . . . . . σ r e d / σ d i p
720 MeV . . . . . . Q (cid:16) GeV (cid:17) . . . . . σ r e d / σ d i p
855 MeV
FIG. S2. Experimental data [3] and our best fit as functions of Q for six different beam energies (180, 315, 450, 585, 720, and855 MeV). The data points are the measured reduced cross sections (S29) scaled by our best-fit normalizations and divided bythe corresponding dipole cross sections (S33). Different markers represent different spectrometers: ■ A, ● B, and ▲▲