Connecting spatial moments and momentum densities
M. Hoballah, M.B. Barbaro, R. Kunne., M. Lassaut., D. Marchand, G. Quéméner, E. Voutier, J. van de Wiele
CConnecting spatial moments and momentum densities
M. Hoballah a , M.B. Barbaro b , R. Kunne. a , M. Lassaut. a , D. Marchand a , G. Qu´em´ener c , E. Voutier a , J. van de Wiele a a Universit´e Paris-Saclay, CNRS / IN2P3, IJCLab, 91405 Orsay, France b Dipartimento di Fisica, Universit´a di Torino and INFN Sezione di Torino, 10125 Torino, Italy c Normandie Univ, ENSICAEN, UNICAEN, CNRS / IN2P3, LPC Caen, 14000 Caen, France
Abstract
The precision of experimental data and analysis techniques is a key feature of any discovery attempt. A striking example is the pro-ton radius puzzle where the accuracy of the spectroscopy of muonic atoms challenges traditional electron scattering measurements.The present work proposes a novel method for the determination of spatial moments from densities expressed in the momentumspace. This method provides a direct access to even, odd, and more generally any real, negative and positive moment with orderlarger than −
3. As an illustration, the application of this method to the electric form factor of the proton is discussed in detail.
Keywords:PACS:
1. Introduction
The determination of the proton charge radius r E from theproton electric form factor measured experimentally throughthe elastic scattering of electrons o ff protons is the subject ofan intense scientific activity (see Ref. [1, 2] for recent reviews).According to the definition r E ≡ (cid:115) − G E ( k )d k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = , (1)the experimental method to determine r E in subatomic physicsconsists in the evaluation of the derivative of the electric formfactor of the proton G E ( k ) at zero-momentum transfer. Con-sequently, the method strongly relies on the zero-momentumextrapolation of the k -dependency of the electric form factormeasured in elastic lepton scattering o ff protons. In light of theproton radius puzzle [3] originating from the disagreement be-tween electron scattering [4] and muonic spectroscopy [5] mea-surements, this method has been scrutinized in every respect tosuggest that the extrapolation procedure of experimental datato zero-momentum transfer su ff ers from limited accuracy. Thederivative method is very sensitive to the functional used to per-form the extrapolation and to the upper limit of the k momen-tum domain considered for this purpose [6]. The significantdi ff erence between the proton charge radius obtained from elec-tron elastic scattering (0.879(8) fm [4]) and that obtained fromthe spectroscopy of muonic hydrogen (0.84184(67) fm [5]) im-plies such a small di ff erence in the electric form factor valuesat very low momentum transfers that it puts unbearable con-straints on the systematics of lepton scattering experiments [7].As a matter of fact, the precision of the highest quality electronscattering measurements (0.879(8) fm [4] and 0.831(14) fm [8])on that issue remains ∼
10 times worse than that of muonic atommeasurements [9, 10]. Improving the precision of the so-called derivative method to such a competitive level does not appearreachable with current knowledge and technologies [11].Within a non-relativistic description of the internal structureof the proton (see Ref. [12] for a recent discussion of relativistice ff ects), Eq. 1 can be recovered from the MacLaurin expansionof the electric form factor expressed as the Fourier transform ofthe proton charge density ρ E ( r ), G E ( k ) = (cid:90) IR d r e − i k · r ρ E ( r ) , (2)namely G E ( k ) = ∞ (cid:88) j = ( − j k j (2 j + (cid:104) r j (cid:105) (3)where k is the Euclidian norm of k . Here (cid:104) r j (cid:105) = ( − j (2 j + j ! d j G E ( k )d( k ) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = (4)relates the electric form factor to the even moments (cid:104) r j (cid:105) of thecharge density ρ E ( r ) (cid:104) r j (cid:105) ≡ ( r j , ρ E ) = (cid:90) IR d r r j ρ E ( r ) . (5)Consequently, the non-relativistic charge radius of the protonmay be expressed as r E = (cid:112) (cid:104) r (cid:105) . (6)The discrepancies between the latest scattering measurementsof the proton radius [4, 8, 13] clearly indicate the experimen-tal di ffi culty in measuring the first derivative of the form factor.Additionally, moments of the charge density beyond the secondorder are also of interest as they carry complementary infor-mation on the charge distribution inside the proton. However, Preprint submitted to Physics Letters B August 6, 2020 a r X i v : . [ nu c l - t h ] A ug eyond the limited precision of the experimental determinationof the j th derivative of the form factor, the derivative methodaccesses only even moments of the density.The purpose of the current work is to propose a new and in-trinsically more accurate method for the determination of thespatial moments of a density from momentum space experi-mental observables, assuming that only the Fourier transformof the probability density function is known. This method al-lows access to both odd and even, positive and negative, mo-ments of the distribution and it overcomes the limitations ofthe derivative technique. Its advantage lies in the more pre-cise determination of spatial moments through integral formsof the Fourier transform of the distribution. These are expectedto be less dependent on point-to-point systematics and hencemore precise. The validity of this approach is demonstrated onthe basis of generic densities, and its importance in the experi-mental determination of physics quantities is further discussed.The method for a generic probability distribution is described inSec. 2, presenting two di ff erent regularization schemes for theFourier transform yielding the spatial moments. The applicabil-ity of the method to a specific physical problem is discussed inSec. 3. The possible applications of the method to experimentaldata are outlined in Sec. 4, and conclusions are drawn in Sec. 5.
2. Spatial moments
Let f ( r ) be a fastly decreasing function in the 3-dimensionalspace. Without any loss of generality for the present discussion(see Appendix A), f ( r ) ≡ f ( r ) is assumed to be a pure radialfunction normalized to the constant ˜ f (cid:90) IR d r f ( r ) = π (cid:90) d r r f ( r ) = ˜ f . (7)Its Fourier transform˜ f ( k ) ≡ ˜ f ( k ) = (cid:90) IR d r e − i k · r f ( r ) (8)exists for any values of k . When ˜ f ( k ) is integrable over IR , theinverse Fourier transform exists and is defined by f ( r ) ≡ f ( r ) = π ) (cid:90) IR d k e i k · r ˜ f ( k ) . (9)The moments ( r λ , f ) of the operator r for the function f aredefined by [14] ( r λ , f ) = (cid:90) IR d r r λ f ( r ) . (10)Replacing f ( r ) with the inverse Fourier transform of ˜ f ( k )(Eq. (9)) and switching the integration order, Eq. 10 becomes( r λ , f ) = π ) (cid:90) IR d k ˜ f ( k ) (cid:90) IR d r e i k · r r λ . (11)The left-hand side of Eq. 11, the moment ( r λ , f ), is a finitequantity which represents a physics observable. However, theright-hand side of Eq. 11 contains the integral g λ ( k ) ≡ g λ ( k ) = (cid:90) IR d r e i k · r r λ , (12) that can be interpreted as the Fourier transform of the tempereddistribution r λ . This integral does not exist in a strict sense for λ ≥ − δ -distribution for λ =
0. Considering a real positive value t , the definition of g λ ( k )provides the property g λ ( t k ) = t λ + g λ ( k ) , (13)which is satisfied only by g λ ( k ) functions proportional to1 / k λ + [14, 15]. Eq. 11 can then be written as( r λ , f ) = N λ (cid:90) ∞ d k (cid:40) ˜ f ( k ) k λ + (cid:41) , (14)where N λ is the normalization coe ffi cient defined for λ (cid:44) , , ... as N λ = λ + √ π Γ ( λ + ) Γ ( − λ ) (15)in terms of the Γ function [16], with λ > −
3. The integral inEq. 14 is taken in the sense of distributions, i.e. the principalvalue of the integral defined from the regularization of the di-verging integrand at zero-momentum (cid:40) ˜ f ( k ) k λ + (cid:41) ≡ k λ + ˜ f ( k ) − n (cid:88) j = ˜ f j k j (16)with ˜ f j = j ! d j ˜ f ( k )d( k ) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = . (17)Here, n + f ( k ), where n = [ λ/
2] is the integer part of λ/ λ (cid:44) , , ... ). It is because ˜ f ( k ) originates from a pureradial function that this development is an even function of k .The right-hand side of Eq. 14 is a convergent quantity as awhole, i.e. divergences that may appear in the normalizationcoe ffi cient are compensated by the integral. The integral existsfor every λ in the domain n < λ/ < n + k → + andwhen k → ∞ . While the integrand diverges for even λ , evenmoments still accept a finite limit. Denoting for convenience λ = m - η with m integer, the moments ( r m − η , f ) write( r m − η , f ) = N m − η (cid:90) ∞ d k ˜ f ( k ) − (cid:80) nj = ˜ f j k j k m − η + (18)where n = [( m − /
2] with 0 < η < m ,and 0 ≤ η < m . Even (odd) moments areobtained taking the limit η → + (setting η = r m , f ) = lim η → + ( r m − η , f ) m even (19)( r m , f ) = ( r m − η , f ) | η = m odd . (20)The counterterms expansion of Eq. 18 is given in Tab. 1 for thefirst order moments.2 mm k m − η + k m − η + k m − η + nnn (cid:80) nj = ˜ f j k j (cid:80) nj = ˜ f j k j (cid:80) nj = ˜ f j k j -2 k − − η -2 --1 k − η -1 -0 k − η -1 -1 k − η f k − η f k − η f + ˜ f k k − η f + ˜ f k k − η f + ˜ f k + ˜ f k k − η f + ˜ f k + ˜ f k ... ... ... ... Table 1: Counterterms expansion of the moments of first orders.
The regularization procedure ensures the convergence of theintegrand in Eq. 18 over the integration domain. For values of m close to even integers, the logarithmic divergence of the in-tegral is balanced by the vanishing N λ to give a finite quantity.More precisely, considering ( r m − η , f ) for even m = p , the nor-malization coe ffi cient N p − η in the vicinity of η = + can bewritten as N p − η (cid:39) ( − p (2 p + η . (21)Introducing an intermediate momentum Q , the integral ofEq. 18 can be separated into a contribution dominated by thezero-momentum behaviour of the integrand and another de-pending on its infinite momentum behaviour. In the vicinityof zero-momentum, the integrand behaves as ˜ f p / k − η leading,after k -integration, to the contribution ˜ f p Q η /η . At large mo-mentum, the k -dependence of the integrand ensures a finite I Q value for the infinite momentum integral. Then, even momentscan be recast as( r p , f ) = lim η → + ( − p (2 p + η ˜ f p η Q η + I Q = ( − p (2 p + f p . (22)For instance, we have ( r , f ) = ˜ f , ( r , f ) = − f , ( r , f ) =
120 ˜ f ... as expected from the MacLaurin development of theFourier transform ˜ f ( k ).The regularization of the Fourier transform g λ ( k ) of the tem-pered distribution r λ is not unique. For instance, g λ ( k ) can alsobe given as a weak limit of the convergent integral g λ ( k ) = lim (cid:15) → + (cid:90) IR d r r λ e − (cid:15) r e i k · r = lim (cid:15) → + I λ ( k , (cid:15) ) (23)where the term e − (cid:15) r ensures the convergence of the integral I λ ( k , (cid:15) ). This is a standard technique used, for example, to reg-ularize the Fourier transform of the Coulomb potential [17, 18].The integration of Eq. 23 is analytical and yields for any λ > − λ (cid:44) − I λ ( k , (cid:15) ) = π Γ ( λ +
2) sin [( λ + k /(cid:15) )] k ( k + (cid:15) ) λ + (24) which accepts the limit (4 π/ k )Arctan ( k /(cid:15) ) at λ = −
2. The mo-ments defined in Eq. 11 can then be written as( r λ , f ) = π Γ ( λ + × (25)lim (cid:15) → + (cid:90) ∞ d k ˜ f ( k ) k sin [( λ + k /(cid:15) )]( k + (cid:15) ) λ/ + for any λ > − λ (cid:44) − λ ,the sine function in Eq. 25 can be developed in terms of a k /(cid:15) polynomial, such that Eq. 25 can be recast for λ = m as( r m , f ) = π ( m + × (26)lim (cid:15) → + (cid:15) m + (cid:90) ∞ d k ˜ f ( k ) k ( k + (cid:15) ) m + Φ m ( k /(cid:15) )with Φ m ( k /(cid:15) ) = m + (cid:88) j = sin (cid:18) j π (cid:19) ( m + j !( m + − j )! (cid:32) k (cid:15) (cid:33) j . (27)The formulations of Eq. 18 and Eq. 25 allow us to deter-mine the moments of a given operator directly in the momentumspace, for both integer and non-integer values of λ . For a given˜ f ( k ) functional form, the moments are numerically computedfrom these expressions and can also be obtained analyticallyfor specific cases.
3. Applicability and benefit of the integral method
The momentum integral determination of the moments out-lined in the previous section is a general approach that can beapplied to any relevant physics quantity. Without any restrictionon the applicability of the method, the specific case of the elec-tromagnetic form factors of the proton is considered hereafter.A typical function example is the radial density f D ( r ) = Λ π e − Λ r (28)leading to the well-known dipole parameterization˜ f D ( k ) = (cid:90) IR d r e − i k · r f D ( r ) = Λ ( k + Λ ) (29)where Λ represents the dipole mass parameter. The momentscan be determined directly in the configuration space, as( r λ , f D ) = (cid:90) IR d r r λ f D ( r ) = Γ ( λ + Λ λ . (30)Considering integer λ = m values, Eq. 26 can be written as( r m , f D ) = Γ ( m + π Λ m lim ˜ (cid:15) → + J m (˜ (cid:15) ) (31)with ˜ (cid:15) = (cid:15)/ Λ , and from Eq. 26 with the integral variable change z = k /(cid:15) J m (˜ (cid:15) ) = (cid:15) m m + (cid:88) j = sin (cid:18) j π (cid:19) ( m + j !( m + − j )! × (32) (cid:90) ∞ d z z j + (1 + ˜ (cid:15) z ) (1 + z ) m + = π m + + ˜ (cid:15) ) . - - - l ) l , f ) ( f m l (r - Dipole IM - Dipole IM - Kelly IM - Kelly IM - - - l D i po l e K e ll y Figure 1: λ -order moments of the proton electric form factor, determined fromthe integral method for the dipole ( Λ = − ) and the Kelly’s polynomialratio [19] parameterizations (top panel), and ratio between the two parameteri-zations (bottom panel). Evaluating the limit in Eq. 31, the momentum integral expres-sion of the moments becomes( r m , f D ) = Γ ( m + π Λ m π ( m + = Γ ( m + Λ m (33) i.e. identical to the result of Eq. 30 obtained from the con-figuration space integral. The same result is obtained for anyreal (integer and non-integer) λ value from the numerical eval-uation of the integrals in Eq. 18 and Eq. 25. The method hasbeen tested for di ff erent mathematical realizations of the radialfunction f ( r ) and several λ : the exponential form of Eq. 28,and a Yukawa-like form (see Appendix B) corresponding tothe parameterization of the proton electromagnetic form fac-tors in terms of a k -polynomial ratio, the Kelly’s parameteriza-tion [19]. In each case, the numerical evaluation of Eq. 18 andEq. 25 provides with a very high accuracy the same results asthe configuration space integrals.Figure 1 shows the variation of the moments over a selected λ -range for two parameterizations of the electric form factor ofthe proton and both prescriptions of the integral method: theprincipal value regularization of Eq. 18 denoted IM , and theexponential regularization of Eq. 25 denoted IM . Particularly,the two di ff erent numerical evaluations are shown to deliver, asexpected, exactly the same results (top panel of Fig. 1). Becauseof a similar functional form, the polynomial ratio moments donot strongly di ff er from the dipole moments. Nevertheless, size-able di ff erences can be observed for negative λ ’s and high mo-ment orders (bottom panel of Fig. 1). Negative orders are rel-evant for the study of the high-momentum dependence of theform factor ( i.e. the central part of the corresponding density),and are of interest to probe its asymptotic behaviour, whereasthe high positive order moments probe the low-momentum be-haviour of the form factor (namely the density close to the nu-cleon’s surface).
4. Application to experimental data
The integral method described previously relies on integralsof Fourier transforms i.e. form factors for the present discus-sion. Unlike the derivative method, the integral method is lesssensitive to a very small variation of the form factor at lowmomentum, and a more stable behaviour with respect to thefunctional form can be expected. However, the evaluation ofmoments via this method requires an experimentally definedasymptotic limit which may be hardly obtained considering themomentum coverage of actual experimental data. The momen-tum dependence of the integrands of Eq. 18 and Eq. 25 providesthe solution to this issue. The denominator of the integrandsscales at large momentum like k λ + , meaning that the integralsare most likely to saturate at a momentum value well belowinfinity.Truncated moments, defined from Eq. 18 and Eq. 25 by re-placing the infinite integral boundary by a cut-o ff Q , allow us tounderstand the saturation behaviour of the moments. Consider-ing for sake of simplicity the case of integer λ = m values, theycan be written from Eq. 26( r m , f ) Q = π ( m + (cid:15) → + R m ( Q , (cid:15) ) (34)with R m ( Q , (cid:15) ) = (cid:15) m + (cid:90) Q d k ˜ f ( k ) k Φ m ( k /(cid:15) )( k + (cid:15) ) m + . (35)The integral is performed before taking the (cid:15) -limit, and obvi-ously lim Q →∞ ( r m , f ) Q = ( r m , f ) . (36)For the typical example of the dipole parameterization ofEq. 29, the integral for even and odd moments can be expressedas R p ( Q , (cid:15) ) = (cid:15) u p ( Q , (cid:15) ) + (cid:15) v p ( (cid:15) ) Arctan (cid:18) Q Λ (cid:19) + w p ( (cid:15) ) Arctan (cid:18) Q (cid:15) (cid:19) (37) R p + ( Q , (cid:15) ) = u p + ( Q , (cid:15) ) + v p + ( (cid:15) ) Arctan (cid:18) Q Λ (cid:19) + (cid:15) w p + ( (cid:15) ) Arctan (cid:18) Q (cid:15) (cid:19) . (38)The functions u i ’s, v i ’s, and w i ’s have finite limits when (cid:15) → + ,as well as when Q → ∞ for the u i ’s. Moreover, the v i ’s and w i ’s are independent of Q . The structure of Eq. 37 and Eq. 38exhibits three contributions with di ff erent Q -dependences: thefirst term (with u i ’s) corresponds to a ratio of Q -polynomialsand vanishes as 1 / Q at infinite cut-o ff ; the second term (with v i ’s) varies as Arctan( Q / Λ ) and is related to the k = ± i Λ com-plex pole of the ˜ f D ( k ) function; the last term (with w i ’s) satu-rates as Arctan( Q /(cid:15) ) and is associated to the k = ± i (cid:15) complexpole of the function that samples ˜ f D ( k ). The Q -convergenceof the two last terms is determined by the same asymptotic be-haviour lim x → + ∞ Arctan( x ) = π − x + x + O (cid:32) x (cid:33) . (39)4he (cid:15) factor in front of these contributions distinguishes thesaturation behaviour of even and odd moments. Particularly, inthe limit (cid:15) → + , the even truncated moments write( r p , f D ) Q = (2 p + w p (0 + ) = (2 p + Λ p (40)and are independent of Q , while the odd truncated moments( r p + , f D ) Q = π (2 p + × (41) (cid:20) u p + ( Q , + ) + v p + (0 + ) Arctan (cid:18) Q Λ (cid:19)(cid:21) are still depending on the cut-o ff . Indeed, Eq. 37 can be seenas a di ff erent realization of Eq. 22, similarly leading to the Q -independence of even moments. The u i ’s coe ffi cients behavelike 1 / Q functions at large cut-o ff , and consequently vanish forinfinite Q . For example, the first odd coe ffi cients write u ( Q , + ) = Λ + Q Q (cid:0) Λ + Q (cid:1) −−−−→ Q →∞ v (0 + ) = Λ (43) u ( Q , + ) = − Λ + Λ Q + Q Λ Q (cid:0) Λ + Q (cid:1) −−−−→ Q →∞ v (0 + ) = Λ . (45)Only the v i ’s remain in the infinite Q -limit, leading to the ex-pression of Eq. 33. Similar features are derived in Appendix Cfor the Kelly’s parameterization.The Q -convergence of truncated moments is shown in Fig. 2for selected moment orders, as determined for the two pre-scriptions of the integral method (IM and IM ) where the Q cut-o ff replaces the infinite boundary of the integrals. The Q -independence feature of even truncated moments is reproducedby each prescription (Fig. 2(a)). This is a general feature in-dependent of the specific form factor, as expressed by Eq. 22.In other words, the integral method for even moments recov-ers formally the same quantities as the derivative method. Inthe ideal world of perfect experiments, adjusting experimentaldata with the same function over a small or large k -domaina ff ects only the precision on the parameters of the function.In the context of the limited quality of real data, the integralmethod provides the mathematical support required to considerthe full k -unlimited domain of existing data, leading thereforeto a more accurate determination of the moments. The practi-cal constraint is to obtain an appropriate description of the dataover a large k -domain.Fig. 2(b) shows the Q -convergence of selected odd moment,comparing the integral method prescriptions. The di ff erent reg-ularizations of the g λ ( k ) integral lead to di ff erent saturation be-haviours. While the principal value regularization (IM ) asksfor large Q -values, the exponential regularization (IM ) rapidlysaturates about 6 fm − , i.e. in a momentum region well coveredby proton electromagnetic form factors data [20].Fig. 2(c) shows the Q -convergence of selected moments withnegative non-integer orders. For such orders, there are no coun-terterms for the principal value regularization (Tab. 1), and Q (1 / fm) ) l ( f m Q ) D , f l (r (a) Even Truncated Moments=2 l - IM =2 l - IM=4 l - IM =4 l - IM=6 l - IM =6 l - IM(a) Even Truncated Moments
Q (1 / fm) ) D , f l (r ⁄ Q ) D , f l (r (b) Odd Truncated Moments=1 l - IM =1 l - IM=3 l - IM =3 l - IM=5 l - IM =5 l - IM(b) Odd Truncated Moments
Q (1 / fm) ) D , f l (r ⁄ Q ) D , f l (r (c) Negative Truncated Moments=-0.5 l - IM =-0.5 l - IM=-1.5 l - IM =-1.5 l - IM=-2.5 l - IM =-2.5 l - IM(c) Negative Truncated Moments
Figure 2: Convergence of truncated moments of the proton electric form factorfor selected orders within the dipole parameterization: (a) positive even, (b)positive odd, and (c) negative non-integer. IM and IM denote the principalvalue and the exponential regularizations, respectively. the e ff ect of the exponential regularization term in Eq. 23 isstrongly suppressed since the integrand converges at infinity(for − < λ < − l ) G e V ( S a t. Q - -
10 110 Positive Moments Saturation at 98% (a) IM Dipole Kelly Positive Moments Saturation at 98% (a) IM l ) G e V ( S a t. Q Positive Moments Saturation at 99.5% (b) IM Dipole Kelly Positive Moments Saturation at 99.5% (b) IM l - - - - - - ) G e V ( S a t. Q - -
10 110 Negative Moments Saturation at 98% (c) IM Dipole Kelly Negative Moments Saturation at 98% (c) IM
Figure 3: Saturation momentum of the principal value (IM ) and exponential(IM ) regularizations of the integral method, for the dipole (solid line) andKelly [19] (circle and dashed line) parameterizations of the electric form factorof the proton: (a) 98% saturation of positive moments within the IM prescrip-tion, (b) 99.5% saturation of positive moments within the IM prescription,and (c) 98% saturation of negative moments. The latter is independent of theintegral method prescription. result: IM = IM for − < λ < Q S at . for each momentorder as the squared momentum transfer at which the truncatedmoment is some α -fraction of the true moment value obtainedin the limit Q → ∞ (Eq. 36), that is R λ Q Sat . = ( r λ , f ) Q Sat . ( r λ , f ) = α . (46)The variation of the saturation momentum as a function of themoment order is shown on Fig. 3 for both prescriptions ofthe integral method and two parameterizations of the electricform factor of the proton. The 98% saturation ( α = (Fig. 3(a)) is compared to the 99.5% saturation of IM (Fig. 3(b)), with respect to positive moments. The principalvalue regularization appears less performant than the exponen-tial regularization. The di ff erences between the integrands ofeach prescription is responsible for this behaviour. At a maxi-mum squared momentum transfer of 2 GeV , the IM prescrip-tion permits the determination of any positive moments, whilethe IM prescription is of very limited success, even when con-sidering a less demanding saturation and the full extension ofthe k -domain of existing data up to ∼
10 GeV . Noticeably, thesaturation momentum appears weakly dependent on the formfactor model (Fig. 3(a) and (b)).Negative moments are more di ffi cult to obtain very accuratelybut can still be determined with a few percents precision(Fig. 3(c)). The sensitivity to the form factor parameterizationis particularly remarkable. As noted previously in Sec. 3, neg-ative moments are sensitive to the high-momentum behaviourof the form factor which is only partly covered by actual data.Here, the di ff erence of interest between the parameterizations isthe sign change of G E ( k ) predicted at k = in Kelly’s.This results in a maximum ratio value at k such that R λ k > Q S at . < k ( Q S at . > k )when R λ k < − α ( R λ k > − α ). These two regimes are respon-sible for the discontinuity occuring about λ = -2.4 in Fig. 3(c).Note that the moment order corresponding to the discontinuityis not a constant but depends on the α saturation level. Negativemoments clearly magnify the impact of the change of the signof the form factor, and may be used to discriminate di ff erentform factor models.A closer look at the form factor parameterizations explainsfurther Fig. 3 behaviours. The k -dependences of the electricform factor of the proton within the Kelly and the dipole pa-rameterizations are compared in Fig. 4 for two di ff erent dipolemasses. Up to the momentum saturation of 2 GeV , the dif-ferences between the parameterizations are small ( ∼
10% atmost), which leads to the very similar saturation momentumbehaviour observed for moments of positive orders (Fig. 3).More precisely, the Kelly’s moments di ff er from the dipole ones(Fig. 1) but both kinds converge similarly towards the asymp-totic limit. Di ff erences only show up for the lowest order mo-ments (Fig. 3(b)) which succeed to catch changes in the k -dependences above ∼ . In the region between the satura-tion momentum and the zero-crossing momentum, the param-eterizations strongly di ff er in magnitudes and k -dependences(Fig. 4). This leads to the very di ff erent saturation momentumtrends observed in the moment region -2 . < λ ≤ ) GeV ( k - - -
10 1 10 D i po l e ) ( k E G ⁄ K e ll y ) ( k E G - - = 0.627 GeV -2 = 16.1 fm L = 0.710 GeV -2 = 18.2 fm L Figure 4: Kelly parameterization of the electric form factor of the proton nor-malized by the dipole parameterization for di ff erent dipole masses: the massused in the present work (solid line), and the historical parameterization mass(dashed line). The saturation momentum at 2 GeV (vertical dotted line) andthe zero-crossing momentum (vertical dash-dotted line) are also shown. When the moment order is large enough (-3 < λ < -2.4) to sam-ple the high- k region of the form factor where the parameter-izations have identical k -dependences (Fig. 4), the behavioursof the saturation momentum become similar (Fig. 3(c)).These features remain model-dependent in the sense that thehigh-momentum behaviour of the form factors is deduced frompredicted scaling laws [21] which, because of the limited exper-imental knowledge, are not confirmed by existing data. How-ever, the momentum range spanned by actual data, especiallyfor the proton, is large enough to su ffi ciently constrain anyphysical or phenomenological parameterization. Therefore amomentum saturation quasi-independent of the functional real-ization of the proton form factor can be determined for positivemoments. Major di ff erences attached to the high-momentumregion are specifically showing up for negative moments.
5. Conclusions
The present work proposes a new method to determine thespatial moments of densities expressed in the momentum space, i.e. form factors. The method provides a direct access to realmoments, both positive and negative, for any form factor func-tional. Particularly, it represents the only opportunity to accessspatial moments when the Fourier transform of a parameteri-zation cannot be performed. In addition, unlike the derivativemethod which is restricted to even moments, the so-called inte-gral method gives access to any moment order, especially oddmoments and more generally any real moment with λ > − . Negative moments require larger saturation momentabut remain quite accesssible with reduced accuray (a few per-cents) in the proton case.The integral method is not specific of the proton, and can alsobe applied to the neutron and nuclei electromagnetic form fac-tors. These applications will be presented elsewhere. Acknowledgements
This work was supported by the LabEx Physique des 2Infinis et des Origines (ANR-10-LABX-0038) in the frame-work (cid:28)
Investissements d’Avenir (cid:29) (ANR-11-IDEX-01), theFrench Ile-de-France region within the SESAME framework,the INFN under the Project Iniziativa Specifica MANYBODY,and the University of Turin under the Project BARM-RILO-19.This project has received funding from the European Unions’sHorizon 2020 research and innovation programme under grantagreement No 824093.
Appendix A. Partial waves expansion of radial moments
This appendix demonstrates that only the spherical compo-nents of the form factor f ( r ) contribute to the radial momentsdefined in Eq. 10.Consider any real number λ and any function f ( r ) of thethree-dimensional variable r , and further assume that the in-tegral defined as I λ = (cid:90) IR f ( r ) r λ d r (A.1)is finite. Any function f ( r ) can be expanded in partial waves asfollows f ( r ) = ∞ (cid:88) (cid:96) = (cid:96) (cid:88) m = − (cid:96) β (cid:96) m ( r ) Y ∗ (cid:96) m (ˆ r ) (A.2)with β (cid:96) m ( r ) = (cid:90) f ( r ) Y (cid:96) m (ˆ r ) d ˆ r , (A.3)such that I λ = ∞ (cid:88) (cid:96) = (cid:96) (cid:88) m = − (cid:96) (cid:90) β (cid:96) m ( r ) r + λ Y ∗ (cid:96) m (ˆ r ) d ˆ r dr . (A.4)Using (cid:90) Y ∗ (cid:96) m (ˆ r ) d ˆ r = √ π δ (cid:96) δ m (A.5)we obtain I λ = ∞ (cid:88) (cid:96) = (cid:96) (cid:88) m = − (cid:96) [ I λ ] (cid:96) m = [ I λ ] (A.6)7here [ I λ ] = (cid:90) ∞ β ( r ) r + λ dr . (A.7)Therefore, I λ vanishes for any (cid:96) (cid:44) i.e. only the partial wave (cid:96) = (cid:96) =
0) term lead to a non-vanishing I λ . Moreover, theFourier transform of this spherical part will be induced only bythe j ( kr ) spherical Bessel function. Appendix B. Moments of a polynomial ratio form factor
This appendix discusses the determination in the configura-tion space of the moments of a function having Fourier trans-form in momentum space expressed as a polynomial ratio.These results serve the comparison with the moments obtainedin Sec. 3 from the momentum integral method.Considering the polynomial ratio function ˜ f K ( k ) expressed inmomentum space as˜ f K ( k ) ≡ ˜ f K ( k ) = + a k + b k + b k + b k , (B.1)its inverse Fourier transform writes f K ( r ) ≡ f K ( r ) = π r (cid:90) ∞ d k k ˜ f K ( k ) sin( kr ) . (B.2)˜ f K ( k ) is assumed to represent a regular physics quantity, forinstance the electromagnetic form factors of the nucleon [19],such that the denominator never vanishes for real k and thefunction accepts only complex poles. The product k ˜ f K ( k ) canthen be expanded in partial fractions as k ˜ f K ( k ) = (cid:88) i = A i k − k i + A i k − k i (B.3)where the k i ’s (with (cid:61) m[ k i ] >
0) are the poles of ˜ f ( k ), and A i = − i b (1 + a k i ) k i (cid:61) m[ k i ] (cid:30) (cid:89) j ( (cid:44) i ) = ( k i − k j )( k i − k j ) . (B.4)are the residues of the function k ˜ f K ( k ) at k = k i .The numericalvalues of the A i ’s and k i ’s corresponding to the parameteriza-tion of Ref. [19] for the electric and magnetic proton form fac-tors are listed in Tab. B.2. After integration, the radial functionwrites f K ( r ) = π r (cid:88) i = e −(cid:61) m[ k i ] r × (B.5) (cid:20) (cid:60) e[ A i ] cos (cid:0) (cid:60) e[ k i ] r (cid:1) − (cid:61) m[ A i ] sin (cid:0) (cid:60) e[ k i ] r (cid:1)(cid:21) . The absence of odd powers of k in the denominator of ˜ f K ( k )leads to the relationships (cid:88) i = (cid:60) e[ A i ] = (cid:88) i = (cid:60) e[ k i ] = G E p G E p G E p G M p G M p G M p /µ p /µ p /µ p iii k i k i k i (fm − − − ) A i A i A i (fm − − − ) k i k i k i (fm − − − ) A i A i A i (fm − − − ) (cid:60) e (cid:61) m (cid:60) e (cid:61) m (cid:60) e (cid:61) m (cid:60) e (cid:61) m Table B.2: Coe ffi cients of the partial fraction expansion for Kelly’s parameter-ization [19]. Note the unit change of the polynomial coe ffi cients as comparedto Kelly’s polynomial: a ≡ ( (cid:126) / M ) a , b ≡ ( (cid:126) / M ) b , b ≡ ( (cid:126) / M ) b , b ≡ ( (cid:126) / M ) b , where M is the proton mass. which ensure a finite value of f K ( r ) at r =
0. The moments, de-termined from the configuration space integral of Eq. 10, canbe expressed as( r λ , f K ) = Γ ( λ + × (B.7) (cid:88) i = (cid:60) e[ A i ] cos( θ k i ) − (cid:61) m[ A i ] sin( θ k i ) | k i | λ + with λ > − θ k i = ( λ +
2) Arctan (cid:32) (cid:60) e[ k i ] (cid:61) m[ k i ] (cid:33) . (B.8) Appendix C. Truncated moments of a polynomial ratioform factor
Analytical expressions for truncated integer moments are de-rived hereafter for the polynomial ratio parameterization of theFourier transform ˜ f K ( k ) of Eq. B.1, within the exponential reg-ularization approach of Eq. 23.Following the discussion of Sec. 4, truncated integer mo-ments are defined for the cut-o ff Q by Eq. 34 and Eq. 35. Theintegral is performed before taking the (cid:15) -limit and takes thegeneric form R p ( Q , (cid:15) ) = (cid:15) u p ( Q , (cid:15) ) + (cid:15) (cid:88) i = i v p ( (cid:15) ) Arctan (cid:32) Q | k i | (cid:33) + w p ( (cid:15) ) Arctan (cid:18) Q (cid:15) (cid:19) (C.1) R p + ( Q , (cid:15) ) = u p + ( Q , (cid:15) ) + (cid:88) i = i v p + ( (cid:15) ) Arctan (cid:32) Q | k i | (cid:33) + (cid:15) w p + ( (cid:15) ) Arctan (cid:18) Q (cid:15) (cid:19) . (C.2)for even and odd truncated moments. Similarly to the dipole pa-rameterization, the u j ’s, i v j ’s, and w j ’s coe ffi cients accept finitelimits when (cid:15) →
0. The u j ’s are the only coe ffi cients dependingon the cut-o ff , and they vanish for infinite Q . The full expres-sion of these functions is too cumbersome to be reported here,but gets simplified when (cid:15) tends to zero.8he (cid:15) -dependence in Eq. C.1 and Eq. C.2 distinguishes the Q -saturation behaviour. In the (cid:15) → + limit, the even truncatedmoments become( r p , f K ) = (2 p + w p (0 + ) (C.3)independent of Q , while the odd truncated moments write( r p + , f K ) = π (2 p + × (C.4) u p + ( Q , + ) + (cid:88) i = i v p + (0 + ) Arctan (cid:32) Q | k i | (cid:33) still depending on the cut-o ff . For instance, the first even mo-ments can be expressed as( r , f K ) = r , f K ) =
3! ( b − a ) (C.6)( r , f K ) = (cid:16) b − a b − b (cid:17) (C.7)and the recurrence relation( r p , f K ) = (2 p + × (C.8) (cid:34) b ( r p − , f K )(2 p − − b ( r p − , f K )(2 p − + b ( r p − , f K )(2 p − (cid:35) , with p >
2, provides all the higher orders. The integrals corre-sponding to the first odd moments write R ( Q , + ) = Q − i A k Arctan (cid:32) Q | k | (cid:33) (C.9) − i A k Arctan (cid:32) Q | k | (cid:33) − i A k Arctan (cid:32) Q | k | (cid:33) R ( Q , + ) = b − a Q − Q (C.10) + i A k Arctan (cid:32) Q | k | (cid:33) + i A k Arctan (cid:32) Q | k | (cid:33) + i A k Arctan (cid:32) Q | k | (cid:33) R ( Q , + ) = b − a b − b Q − b − a Q + Q (C.11) − i A k Arctan (cid:32) Q | k | (cid:33) − i A k Arctan (cid:32) Q | k | (cid:33) − i A k Arctan (cid:32) Q | k | (cid:33) . The specific structure of ˜ f K ( k ) as a ratio of polynomials of evenpower of k with no poles on the real k -axis, leads either to pureimaginary poles or to relationship between A i ’s and k i ’s. Forinstance, in addition to the general properties of Eq. B.6 wehave for the proton electric form factor (Tab. B.2) | k | = | k | ⇒ | A | = | A | (C.12) k = − k ⇒ A = A (C.13) such that R p + ( Q , + ) are pure real quantities. In the limit Q →∞ , Eq. C.9-C.11 provide( r , f K ) = − i A k + A k + A k (C.14)( r , f K ) = i A k + A k + A k (C.15)( r , f K ) = − i A k + A k + A k , (C.16)and generally( r p + , f K ) = ( − p + i (2 p + (cid:88) i = A i k i p + . (C.17) References [1] C.E. Carlson, Prog. Part. Nucl. Phys. (2015) 59.[2] R.J. Hill, EPJ Web Conf. (2017) 01023.[3] J.C. Bernauer, R. Pohl, Sci. Am. (2014) 32.[4] (A1 Collaboration) J.C. Bernauer et al. Phys. Rev. Lett. (2010)242001.[5] (CREMA Collaboration) R. Pohl et al.
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