Contents lists available atScienceDirect
Physics
Letters
B
www.elsevier.com/locate/physletb
Connecting
spatial
moments
and
momentum
densities
M. Hoballah
a,
∗
,
M.B. Barbaro
b,
R. Kunne
a,
M. Lassaut
a,
D. Marchand
a,
G. Quéméner
c,
E. Voutier
a,
J. van de Wiele
aaUniversitéParis-Saclay,CNRS/IN2P3,IJCLab,91405Orsay,France
bDipartimentodiFisica,UniversitádiTorinoandINFNSezionediTorino,10125Torino,Italy cNormandieUniv,ENSICAEN,UNICAEN,CNRS/IN2P3,LPCCaen,14000Caen,France
a
r
t
i
c
l
e
i
n
f
o
a
b
s
t
r
a
c
t
Articlehistory: Received20March2020
Receivedinrevisedform28July2020 Accepted28July2020
Availableonline3August2020 Editor:W.Haxton
Theprecisionofexperimentaldataandanalysistechniquesisakeyfeatureofanydiscoveryattempt.A strikingexampleistheprotonradiuspuzzlewhere theaccuracyofthespectroscopyofmuonicatoms challengestraditionalelectronscatteringmeasurements.Thepresentworkproposesanovelmethodfor thedetermination ofspatialmomentsfromdensitiesexpressedinthemomentumspace.Thismethod providesadirectaccesstoeven,odd,andmoregenerallyanyreal,negativeandpositivemomentwith orderlargerthan−3.Asanillustration,theapplicationofthismethodtotheelectricformfactorofthe protonisdiscussedindetail.
©2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
1. Introduction
ThedeterminationoftheprotonchargeradiusrE fromthe
pro-tonelectricformfactormeasuredexperimentallythroughthe elas-ticscatteringofelectrons offprotonsisthe subjectofan intense scientific activity(see Ref. [1,2] forrecent reviews).According to thedefinition rE
≡
−
6dGE(
k 2)
dk2 k2=0,
(1)the experimental method to determine rE in subatomic physics
consists in the evaluation of the derivative of the electric form factor of the proton GE
(
k2)
at zero-momentum transfer.Conse-quently,themethodstronglyreliesonthezero-momentum extrap-olationofthek2-dependencyoftheelectricformfactormeasured
inelasticleptonscatteringoffprotons.Theso-calledprotonradius puzzle [3], that originated fromthe disagreement between elec-tronscattering [4] andmuonicspectroscopy [5] measurements,has laid much critique on themethod suggestingthat the extrapola-tion procedure of experimental data to zero-momentum transfer suffersfromlimitedaccuracy. Thederivative methodis very sen-sitive tothe functional usedto perform theextrapolation andto theupperlimit ofthek2 momentumdomain consideredforthis
purpose [6].Thesignificantdifferencethatwas observedbetween
*
Correspondingauthor.E-mailaddress:hoballah@ipno.in2p3.fr(M. Hoballah).
the proton charge radius obtained from electron elastic scatter-ing (0.879(8) fm [4]) and that obtained from the spectroscopy of muonic hydrogen (0.84184(67) fm [5]) implies such a small difference inthe electricformfactor valuesatvery low momen-tum transfers that it puts unbearableconstraints on the system-atics of lepton scattering experiments [7]. As a matter of fact, the precision of the highestquality electron scattering measure-ments (0.879(8) fm [4] and0.831(14) fm [8])onthatissueremains
∼
10timesworsethanthatofmuonicatommeasurements [9,10]. Improvingtheprecisionoftheso-calledderivativemethodtosuch acompetitiveleveldoesnotappearreachablewithcurrent knowl-edge andtechnologies [11].While therecentPRadresult [8] and the recommended CODATA [12] and PDG [13] values of rE havereduced the tension with muonic atom measurements, improv-ingtheprecisionofscatteringexperimentsremainsahighpriority in light ofthe numerous discussions about thesensitivity ofthe derivativemethod(seeRef. [14] fornewdevelopments).
Within a non-relativistic description of the internal structure of theproton (see Ref. [15] for a recentdiscussion of relativistic effects),Eq. (1) canberecoveredfromtheMacLaurinexpansionof theelectricformfactorexpressedastheFouriertransformofthe protonchargedensity
ρ
E(r)
,GE
(
k2)
=
IR3 d3r e−ik·rρ
E(
r) ,
(2) namely https://doi.org/10.1016/j.physletb.2020.1356690370-2693/©2020TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.
GE
(
k2)
=
∞ j=0(
−
1)
j k 2 j(
2 j+
1)
!
r 2 j(3)
wherek istheEuclidiannormofk.Here
r2 j= (−
1)
j(
2 j+
1)
!
j!
djG E(
k2)
d(
k2)
j k2=0 (4) relatesthe electricformfactorto theeven moments r2 jofthe chargedensityρ
E(r)
r2 j≡ (
r2 j,
ρ
E)
=
IR3d3r r2 j
ρ
E(
r) .
(5)Consequently,thenon-relativisticchargeradiusoftheprotonmay beexpressedas
rE
=
r2.
(6)The discrepancies betweenthelatest scatteringmeasurements of the protonradius [4,8,16] clearly indicatethe experimental diffi-cultyinmeasuringthefirstderivativeoftheformfactor. Addition-ally,momentsofthe chargedensitybeyondthesecond orderare also ofinterest asthey carry complementary information onthe chargedistributioninsidetheproton.However,beyondthelimited precision oftheexperimental determination ofthe jth derivative
oftheformfactor,the derivativemethodaccesses onlyeven mo-mentsofthedensity.
Thepurposeofthecurrentworkistoproposeanewand intrin-sicallymoreaccurate methodforthedeterminationofthespatial moments of a density from momentum space experimental ob-servables,assumingthat onlytheFouriertransformofthe proba-bilitydensityfunctionisknown.Thismethodallowsaccesstoboth oddandeven,positiveandnegative, momentsofthedistribution anditovercomesthelimitationsofthederivativetechnique.Its ad-vantageliesinthemoreprecisedeterminationofspatialmoments throughintegralformsoftheFouriertransformofthedistribution. These are expected to be less dependent on point-to-point sys-tematicsandhencemoreprecise.The validityofthisapproachis demonstratedonthebasisofgenericdensities,anditsimportance intheexperimental determinationofphysicsquantitiesisfurther discussed.Themethodforagenericprobability distributionis de-scribedinSec. 2,presentingtwo differentregularizationschemes fortheFouriertransformyieldingthespatialmoments.The appli-cabilityofthemethodtoa specificphysicalproblemisdiscussed inSec.3.Thepossibleapplicationsofthemethodtoexperimental dataareoutlinedinSec.4,andconclusionsaredrawninSec.5.
2. Spatialmoments
Let f
(
r)
be a fastly decreasing function in the 3-dimensional space. Without any loss of generality for the present discussion (seeAppendixA), f(
r)
≡
f(
r)
isassumedtobeapureradial func-tionnormalizedtotheconstant˜
f0 IR3d3r f
(
r)
=
4π
dr r2f
(
r)
= ˜
f0.
(7)ItsFouriertransform
˜
f
(
k)
≡ ˜
f(
k)
=
IR3d3r e−ik·rf
(
r)
(8)exists forany values ofk. When
˜
f(
k)
isintegrable over IR3,theinverseFouriertransformexistsandisdefinedby
f
(
r)
≡
f(
r)
=
1(
2π
)
3 IR3d3k eik·r
˜
f(
k) .
(9)The moments
(
rλ,
f)
oftheoperatorr for thefunction f arede-finedby [17]
(
rλ,
f)
=
IR3d3r rλ f
(
r) .
(10)Replacing f
(
r)
withtheinverseFouriertransformof˜
f(
k)
(Eq. (9)) andswitchingtheintegrationorder,Eq. (10) becomes(
rλ,
f)
=
1(
2π
)
3 IR3 d3k˜
f(
k)
IR3 d3r eik·rrλ.
(11)Theleft-handsideofEq. (11),themoment
(
rλ,
f)
,isafinitequan-tity which represents a physics observable. However, the right-handsideofEq. (11) containstheintegral
gλ
(
k)
≡
gλ(
k)
=
IR3d3r ei k·rrλ
,
(12)that can beinterpreted astheFouriertransformofthe tempered distribution rλ. This integral does not exist in a strict sense for
λ
≥ −
1 but can still be treated as a distribution; the finiteness of theleft-hand side ensures thephysical representativityofthis expression as well as the convergenceof the 6-fold integral. For instance,Eq. (12) correspondstotheDiracδ
-distributionforλ
=
0. Consideringarealpositivevaluet,thedefinitionofgλ(
k)
provides thepropertygλ
(
tk)
=
1
tλ+3gλ
(
k) ,
(13)which issatisfiedonlyby gλ
(
k)
functionsproportional to1/
kλ+3[17,18].Eq. (11) canthenbewrittenas
(
rλ,
f)
=
N
λ ∞ 0 dk˜
f(
k)
kλ+1,
(14)where
N
λisthenormalizationcoefficientdefinedforλ
=
0,
2,
4...
as
N
λ=
2λ+2√
π
(
λ+23)
(
−
λ2)
(15)in terms of the
function [19], with
λ
>
−
3. The integral in Eq. (14) is taken in the sense of distributions, i.e. the principal valueoftheintegraldefinedfromtheregularizationofthe diverg-ingintegrandatzero-momentum˜
f(
k)
kλ+1≡
1 kλ+1⎛
⎝ ˜
f(
k)
−
n j=0˜
f2 jk2 j⎞
⎠
(16) with˜
f2 j=
1 j!
dj˜
f(
k)
d(
k2)
j k=0.
(17)Here,n
+
1 isthenumberofcountertermsintheMacLaurin devel-opment of˜
f(
k)
,wheren= [λ/
2]
istheintegerpartofλ/
2 (withλ
=
0,
2,
4...
).Itisbecause˜
f(
k)
originatesfromapureradial func-tionthatthisdevelopmentisanevenfunctionofk.The right-hand side of Eq. (14) is a convergent quantity as a whole,i.e. divergencesthatmayappearinthenormalization coeffi-cientarecompensatedbytheintegral.Theintegralexistsforevery
Table 1
Countertermsexpansionofthemomentsoffirstorders.
m km−η+1 n n j=0˜f2 jk2 j −2 k−1−η −2 – −1 k−η −1 – 0 k1−η −1 – 1 k2−η 0 ˜f 0 2 k3−η 0 ˜f 0 3 k4−η 1 ˜f0+ ˜f2k2 4 k5−η 1 ˜f0+ ˜f2k2 5 k6−η 2 ˜f0+ ˜f2k2+ ˜f 4k4 6 k7−η 2 ˜f0+ ˜f2k2+ ˜f 4k4 . . . . . . . . . . . .
λ
inthedomainn< λ/
2<
n+
1 [17,18], whichensuresthe con-vergenceof theintegrand both whenk→
0+ andwhen k→ ∞
. While theintegrand divergesfor evenλ
, even moments still ac-cept a finite limit. Denoting for convenienceλ
=
m−
η
with minteger,themoments
(
rm−η,
f)
write(
rm−η,
f)
=
Nm
−η ∞ 0 dk˜
f(
k)
−
n j=0
˜
f2 jk2 j km−η+1 (18)wheren
=
[(
m−
1)/
2] with0<
η
<
1 for even valuesof m, and 0≤
η
<
1 foroddvaluesofm. Even(odd)momentsareobtained takingthelimitη
→
0+(settingη
=
0).Respectively,(
rm,
f)
=
limη→0+
(
rm−η
,
f)
m even (19)(
rm,
f)
= (
rm−η,
f)
|
η=0 m odd.
(20)ThecountertermsexpansionofEq. (18) isgiveninTable1forthe firstordermoments.
Theregularizationprocedureensurestheconvergenceofthe in-tegrand inEq. (18) over theintegration domain.For valuesof m
closetoevenintegers,thelogarithmicdivergenceoftheintegralis balancedbythevanishing
N
λ togiveafinitequantity.Morepre-cisely, considering
(
rm−η,
f)
for even m=
2p, the normalizationcoefficient
N2p
−η inthevicinityofη
=
0+canbewrittenasN2p
−η(−
1)
p(
2p+
1)
!
η
.
(21)IntroducinganintermediatemomentumQ ,theintegralofEq. (18) can be separated into a contribution dominated by the zero-momentum behaviour of the integrand and another depending on its infinite momentum behaviour. In the vicinity of zero-momentum, the integrand behaves as
˜
f2p/k1−η leading, afterk-integration,tothecontribution
˜
f2pQ η/
η
.Atlargemomentum,thek-dependence ofthe integrand ensures a finite IQ value for the
infinitemomentumintegral.Then,evenmomentscanberecastas
(
r2p,
f)
=
lim η→0+(
−
1)
p(
2p+
1)
!
η
˜
f2pη
Q η+
I Q= (−
1)
p(
2p+
1)
! ˜
f2p.
(22)For instance, we have
(
r0,
f)
= ˜
f0,
(
r2,
f)
= −
6˜
f2,(
r4,
f)
=
120
˜
f4... as expected from the MacLaurin development of theFouriertransform
˜
f(
k)
.Theregularization of theFourier transform gλ
(
k)
of thetem-pereddistributionrλisnotunique.Forinstance,gλ
(
k)
canalsobegivenasaweaklimitoftheconvergentintegral
gλ
(
k)
=
lim →0+ IR3 d3r rλe−rei k·r=
lim →0+I
λ(
k,
)
(23)where the term e−r ensures the convergence of the integral
I
λ(
k,
)
. Thisisa standard technique used,forexample,toregu-larizetheFouriertransformoftheCoulombpotential [20,21].The integrationofEq. (23) isanalyticalandyieldsforany
λ
>
−
3 andλ
= −
2I
λ(
k,
)
=
4
π
(λ
+
2)
sin [(λ
+
2)
Arctan(
k/
)
]k
(
k2+
2
)
λ2+1(24) which accepts the limit
(
4π
/
k)
Arctan(
k/
)
atλ
= −
2. The mo-mentsdefinedinEq. (11) canthenbewrittenas(
rλ,
f)
=
2π
(λ
+
2)
×
(25) lim →0+ ∞ 0 dk˜
f(
k)
k sin [(λ
+
2)
Arctan(
k/
)
](
k2+
2
)
λ/2+1forany
λ
>
−
3 andλ
= −
2 value.Forintegervaluesofλ
,thesine functioninEq. (25) canbedevelopedintermsofak/
polynomial, suchthatEq. (25) canberecastfor
λ
=
m as(
rm,
f)
=
2π
(
m+
1)
! ×
(26) lim →0+m+2 ∞ 0 dk
˜
f(
k)
k(
k2+
2
)
m+2m
(
k/
)
withm
(
k/
)
=
m+2 j=0 sin j
π
2(
m+
2)
!
j!(m+
2−
j)
!
kj
.
(27)TheformulationsofEq. (18) andEq. (25) allowustodetermine themomentsofagivenoperatordirectlyinthemomentumspace, forbothintegerandnon-integervaluesof
λ
.Foragiven˜
f(
k)
func-tional form, the moments are numerically computed from these expressionsandcanalsobeobtainedanalyticallyforspecificcases. The generalization of Eq. (18) and Eq. (25) to a D-dimensionalcharge densityare further presented inAppendix Doffering the possibility, for example, to address the relativistic nature of the nucleonstructure [15].
3. Applicabilityandbenefitoftheintegralmethod
The momentum integral determination of the moments out-lined in the previous section is a general approach that can be applied to any relevant physics quantity. Without any restriction on the applicability ofthe method,the specific caseof the elec-tromagnetic formfactors ofthe protonis considered hereafter.A typicalfunctionexampleistheradialdensity
fD
(
r)
=
3
8
π
e−r (28)
leadingtothewell-knowndipoleparameterization
˜
fD(
k)
=
IR3 d3r e−ik·rfD(
r)
=
4
(
k2+
2)
2 (29)where
representsthedipolemassparameter.Themomentscan bedetermineddirectlyintheconfigurationspace,as
(
rλ,
fD)
=
IR3 d3r rλfD(
r)
=
(λ
+
3)
2 1λ
.
(30)Fig. 1.λ-ordermomentsoftheprotonelectricformfactor,determinedfromthe in-tegralmethodforthedipole(2=16.1 fm−2
)andtheKelly’spolynomialratio [22] parameterizations(toppanel),andratiobetweenthetwoparameterizations (bot-tompanel).
(
rm,
fD)
=
2(
m+
2)
π
1m ˜lim→0+Jm
(
˜
)
(31)with
˜
=
/
,andfromEq. (26) withtheintegralvariablechangez
=
k/
Jm
(
˜
)
=
1˜
m m
+2 j=0 sin j
π
2(
m+
2)
!
j!(m+
2−
j)
!
×
(32) ∞ 0 dz z j+1(
1+ ˜
2z2
)
2(
1+
z2)
m+2=
π
4 m+
2(
1+ ˜
)
3.
EvaluatingthelimitinEq. (31),themomentumintegralexpression ofthemomentsbecomes
(
rm,
fD)
=
2(
m+
2)
π
m
π
(
m+
2)
4=
(
m+
3)
2 1m (33)
i.e. identical to the result of Eq. (30) obtained from the config-uration space integral. The same result is obtained for any real (integer and non-integer)
λ
value from the numerical evaluation of the integrals in Eq. (18) and Eq. (25). The method has been tested for different mathematical realizations of the radial func-tion f(
r)
andseveralλ
: the exponential form ofEq. (28), anda Yukawa-likeform(seeAppendix B)corresponding tothe parame-terizationoftheprotonelectromagneticformfactorsintermsofak2-polynomialratio,theKelly’sparameterization [22].Ineachcase, thenumericalevaluationofEq. (18) and Eq. (25) provideswitha veryhighaccuracythesameresultsastheconfigurationspace in-tegrals.
Fig.1 showsthe variation ofthe momentsover a selected
λ
-rangefortwoparameterizations oftheelectricformfactorofthe protonandbothprescriptionsoftheintegralmethod:theprincipal value regularization of Eq. (18) denoted IM1, and theexponen-tial regularization of Eq. (25) denoted IM2. Particularly, the two
differentnumericalevaluationsareshowntodeliver,asexpected, exactlythe same results(toppanel of Fig.1). Because ofa simi-larfunctionalform,thepolynomialratiomomentsdonotstrongly differfromthedipolemoments.Nevertheless,sizeabledifferences can be observed for negative
λ
’s and high moment orders (bot-tompanelofFig.1).Negativeordersare relevantforthestudy of thehigh-momentumdependenceoftheformfactor(i.e. thecentral partofthecorrespondingdensity),andareofinteresttoprobeits asymptotic behaviour, whereas the highpositive order moments probe the low-momentum behaviour of the formfactor (namely thedensityclosetothenucleon’ssurface).4. Applicationtoexperimentaldata
Theintegral methoddescribedpreviously reliesonintegralsof Fouriertransformsi.e. formfactorsforthepresentdiscussion. Un-likethederivativemethod,theintegralmethodislesssensitiveto a very smallvariation oftheform factoratlow momentum,and a more stablebehaviour with respectto the functionalform can beexpected.However,theevaluationofmomentsviathismethod requiresanexperimentallydefinedasymptoticlimitwhichmaybe hardlyobtainedconsideringthemomentumcoverageofactual ex-perimentaldata.The momentumdependenceoftheintegrandsof Eq. (18) andEq. (25) provides thesolution to thisissue.The de-nominator oftheintegrandsscalesatlarge momentumlikekλ+1,
meaningthattheintegralsaremostlikelytosaturateata momen-tumvaluewellbelowinfinity.
Truncated moments,definedfromEq. (18) andEq. (25) by re-placing theinfiniteintegralboundary bya cut-off Q , allowusto understandthesaturationbehaviourofthemoments.Considering forsakeofsimplicitythecaseofinteger
λ
=
m values,theycanbe writtenfromEq. (26)(
rm,
f)
Q=
2π
(
m+
1)
!
→lim0+Rm
(
Q,
)
(34) withRm
(
Q,
)
=
m+2 Q 0 dk
˜
f(
k)
km
(
k/
)
(
k2+
2
)
m+2.
(35)Theintegralisperformedbeforetakingthe
-limit,andobviously lim
Q→∞
(
r m,
f)
Q
= (r
m
,
f) .
(36)ForthetypicalexampleofthedipoleparameterizationofEq. (29), theintegralforevenandoddmomentscanbeexpressedas
R2p
(
Q,
)
=
u2p
(
Q,
)
+
v2p
(
)
Arctan Q+
w2p(
)
Arctan Q(37)
R2p
+1(
Q,
)
=
u2p+1(
Q,
)
+
v2p+1(
)
Arctan Q+
w2p+1
(
)
Arctan Q.
(38)The functionsui’s, vi’s,and wi’s havefinitelimitswhen
→
0+,aswellaswhen Q
→ ∞
forthe ui’s.Moreover,the vi’sandwi’sare independent of Q .The structure ofEq. (37) and Eq. (38) ex-hibitsthree contributionswithdifferent Q -dependences: thefirst term (with ui’s) corresponds to a ratio of Q -polynomials and
vanishes as 1
/
Q at infinite cut-off; the second term (with vi’s)varies as Arctan
(
Q/)
and is related to the k0= ±
icomplex
poleofthe
˜
fD(
k)
function;thelast term(with wi’s)saturatesasArctan
(
Q/
)
and is associated to the k0= ±
icomplex pole of
the function that samples
˜
fD(
k)
. The Q -convergence of the twolasttermsisdeterminedbythesameasymptoticbehaviour
lim x→+∞Arctan
(
x)
=
π
2−
1 x+
1 3x3+
O
1 x5.
(39)The
factorinfrontofthesecontributions distinguishesthe sat-uration behaviour ofeven and odd moments. Particularly, in the limit
→
0+,theeventruncatedmomentswrite(
r2p,
fD)
Q= (
2p+
1)
!
w2p(
0+)
=
(
2p+
2)
!
2 1
andareindependentofQ ,whiletheoddtruncatedmoments
(
r2p+1,
fD)
Q=
2π
(
2p+
2)
! ×
(41) u2p+1(
Q,
0+)
+
v2p+1(
0+)
Arctan Qare still depending on the cut-off. Indeed, Eq. (37) can be seen asa different realizationof Eq. (22), similarly leading to the Q
-independenceofeven moments. The ui’s coefficientsbehave like
1
/
Q functionsat large cut-off, andconsequently vanish for infi-nite Q .Forexample,thefirstoddcoefficientswriteu1
(
Q,
0+)
=
22
+
3Q2 2Q2
+
Q2−−−−→
Q→∞ 0 (42) v1(
0+)
=
3 2(43) u3
(
Q,
0+)
=
−
24
+
102Q2
+
15Q4 62Q3
2
+
Q2−−−−→
Q→∞ 0 (44) v3(
0+)
=
5 23
.
(45)Onlythevi’sremainintheinfiniteQ -limit,leadingtothe
expres-sionofEq. (33).SimilarfeaturesarederivedinAppendixCforthe Kelly’sparameterization.
TheQ -convergenceoftruncatedmomentsisshowninFig.2for selectedmomentorders, asdetermined forthetwo prescriptions of the integral method (IM1 and IM2) where the Q cut-off
re-placestheinfiniteboundaryoftheintegrals.The Q -independence
featureofeventruncatedmomentsisreproducedbyeach prescrip-tion(Fig.2(a)).Thisisageneralfeatureindependentofthespecific formfactor,asexpressedby Eq. (22).Inother words,theintegral methodforeven moments recovers formally thesame quantities asthederivativemethod.Intheidealworldofperfectexperiments, adjustingexperimental datawiththe samefunction over asmall orlarge k2-domain affects only the precision on the parameters
ofthefunction.In thecontextofthe limitedqualityofrealdata, the integral method provides the mathematical support required toconsiderthefull k2-unlimiteddomainofexisting data,leading thereforeto amore accurate determinationof themoments. The practicalconstraint isto obtainan appropriate description ofthe dataoveralargek2-domain.
Fig. 2(b) shows the Q -convergence of selected odd moment, comparing the integral method prescriptions. The different regu-larizations of the gλ
(
k)
integral lead to different saturationbe-haviours. While the principal value regularization (IM1) asks for
large Q -values,the exponentialregularization (IM2) rapidly
satu-rates about6 fm−1, i.e. ina momentum region well covered by protonelectromagneticformfactorsdata [23].
Fig.2(c)shows the Q -convergenceof selected moments with negative non-integer orders. For such orders, there are no coun-terterms for the principal value regularization (Table 1), andthe effectoftheexponentialregularizationterminEq. (23) isstrongly suppressedsincetheintegrandconvergesatinfinity(for
−
3< λ
<
−
1).Indeed,thereisnoneedofregularizationfornegativeorders andall prescriptions of the integral method should be identical. Thisisverified onFig. 2where thenumericalevaluationof each prescriptionis shown to provide the same result:IM1=
IM2 for−
3< λ
<
0.It is the essential benefit of the integral method to allow us todetermineoddandrealpositive andnegative spatial moments directlyfromexperimentaldatainthemomentumspace.
We define the saturation momentum QSat. for each moment
orderasthesquaredmomentum transferat whichthetruncated
Fig. 2. Convergenceoftruncatedmomentsofthe protonelectric formfactorfor selectedorderswithinthedipoleparameterization:(a)positiveeven,(b)positive odd,and(c)negativenon-integer.IM1andIM2denotetheprincipalvalueandthe exponentialregularizations,respectively.
momentissome
α
-fractionofthetruemomentvalueobtainedin thelimit Q→ ∞
(Eq. (36)),thatisRλQSat
.
=
(
rλ,
f)
QSat.Fig. 3. Saturationmomentumofthe principalvalue(IM1)andexponential(IM2) regularizationsoftheintegralmethod,forthedipole(solidline)andKelly [22] (cir-cleand dashedline)parameterizationsoftheelectric formfactoroftheproton: (a)98%saturationofpositivemomentswithintheIM1prescription,(b)99.5% sat-urationofpositivemomentswithintheIM2prescription,and(c)98%saturationof negativemoments.Thelatterisindependentoftheintegralmethodprescription.
The variation of the saturation momentum as a function of the momentorderisshownonFig.3forbothprescriptionsofthe inte-gralmethodandtwo parameterizationsoftheelectricformfactor
of theproton.The 98% saturation(
α
=
0.
98) of IM1 (Fig. 3(a)) iscomparedtothe99.5%saturationofIM2(Fig.3(b)),withrespectto
positive moments. Theprincipal value regularization appearsless performant than the exponential regularization. The differences betweentheintegrandsofeachprescriptionisresponsibleforthis behaviour.Atamaximumsquaredmomentumtransferof2 GeV2, theIM2prescriptionpermitsthedeterminationofanypositive
mo-ments,whiletheIM1 prescriptionisofverylimitedsuccess,even
whenconsideringalessdemandingsaturationandthefull exten-sionofthek2-domainofexistingdataupto
∼
10 GeV2.Noticeably,thesaturationmomentumappearsweaklydependentontheform factormodel(Fig.3(a)and(b)).
Negative moments are more difficult to obtain very accu-rately butcan still be determined with a few percents precision (Fig. 3(c)). The sensitivity to the form factor parameterization is particularlyremarkable.AsnotedpreviouslyinSec.3,negative mo-mentsaresensitivetothehigh-momentumbehaviouroftheform factorwhichisonlypartlycoveredbyactualdata.Here,the differ-enceofinterestbetweentheparameterizationsisthesignchange of GE
(
k2)
predicted atk02=
14.
7 GeV2 in Kelly’s. This results ina maximum ratiovalue atk2 such that Rλ
k0
>
1, andprovides a saturation momentum QSat.<
k0 ( QSat.>
k0) when Rkλ0<
2−
α
(Rλ
k0
>
2−
α
).Thesetwo regimesareresponsibleforthe disconti-nuityoccurring aboutλ
= −
2.
4 inFig.3(c).Notethatthemoment ordercorrespondingtothediscontinuityisnotaconstantbut de-pendsontheα
saturationlevel.Negativemomentsclearlymagnify the impactofthechangeofthesignoftheformfactor, andmay beusedtodiscriminatedifferentformfactormodels.Acloserlookattheformfactorparameterizationsexplains fur-therFig.3behaviours.Thek2-dependencesoftheelectricform fac-toroftheprotonwithintheKellyandthedipoleparameterizations are comparedinFig.4fortwodifferentdipole masses.Up tothe momentumsaturationof2 GeV2,thedifferencesbetweenthe
pa-rameterizationsaresmall(
∼
10%atmost),whichleadstothevery similarsaturationmomentumbehaviourobservedformomentsof positive orders (Fig. 3). More precisely, the Kelly’s moments dif-ferfromthedipoleones(Fig.1)butbothkindsconvergesimilarly towards the asymptotic limit. Differences only show up for the lowest ordermoments (Fig.3(b))whichsucceedtocatch changes inthek2-dependencesabove∼
1 GeV2.Intheregionbetweenthesaturation momentum andthezero-crossing momentum,the pa-rameterizationsstronglydifferinmagnitudesandk2-dependences
(Fig. 4). This leads to the very different saturation momentum trends observed inthe momentregion
−
2.
4< λ
≤
0 in Fig. 3(c). Whenthemomentorderislargeenough(−
3< λ
<
−
2.
4)to sam-plethehigh-k2 regionoftheformfactorwherethe parameteriza-tionshaveidenticalk2-dependences(Fig.4),thebehavioursofthesaturationmomentumbecomesimilar(Fig.3(c)).
These features remainmodel-dependent inthe sense that the high-momentum behaviour of the form factors is deduced from predicted scaling laws [24] which, because ofthe limited exper-imental knowledge, arenot confirmed byexisting data.However, the momentum range spannedby actual data, especially for the proton, is large enough to sufficiently constrain any physical or phenomenologicalparameterization.Thereforeamomentum satu-rationquasi-independentofthefunctionalrealizationoftheproton form factor can be determined for positive moments. Major dif-ferences attached to the high-momentum region are specifically showingupfornegativemoments.
5. Conclusions
The present work proposes a new method to determine the spatial moments of densities expressedin themomentum space,
i.e. formfactors.The methodprovidesa directaccesstoreal mo-ments, bothpositive andnegative, foranyformfactorfunctional.
Fig. 4. Kellyparameterizationoftheelectricformfactoroftheprotonnormalized bythedipoleparameterizationfordifferentdipolemasses:themassusedinthe presentwork (solidline),andthehistoricalparameterizationmass(dashedline). Thesaturationmomentumat2 GeV2 (verticaldottedline)andthezero-crossing momentum(verticaldash-dottedline)arealsoshown.
Particularly, it represents the only opportunity to access spatial momentswhen the Fouriertransformof a parameterization can-notbeperformed.Inaddition,unlikethederivativemethodwhich isrestrictedtoevenmoments,theso-calledintegralmethodgives accessto anymoment order, especially odd moments and more generallyanyrealmomentwith
λ
>
−
3.Furthermore,itprovides theformal supporttotake intoaccount thefull rangeofexisting datafor thedetermination of evenmoments, allowing us to im-provetheiraccuracyascomparedtothederivativemethod.The integral method involves the regularization of integrals treatedasdistributions.Tworegularizationschemeswerestudied: the first one based on the principal value regularization, similar tothetechniqueusedtodetermineZemachmoments [25,26];the secondone involvinganexponentialregularization, similartothe techniqueusedtoregularizetheFouriertransformoftheCoulomb potential [20]. Thesetechniqueshavebeentestedwithrespectto thedipoleandKellyparameterizationsoftheelectromagneticform factorof the proton. The exponential regularization provides the most performant approach allowing us to determine accurately positive moments considering a squaredsaturation four momen-tumtransfer of2 GeV2. Negativemoments requirelarger satura-tionmomentabutremain quiteaccessible withreducedaccuracy (afewpercents)intheprotoncase.
Theintegralmethodisnotspecificoftheproton,andcanalso beappliedtotheneutronandnucleielectromagneticformfactors. Theseapplicationswillbepresentedelsewhere.
Declarationofcompetinginterest
Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.
Acknowledgements
ThisworkwassupportedbytheLabExPhysiquedes2Infiniset desOrigines(ANR-10-LABX-0038) intheframework
Investisse-ments d’Avenir (ANR-11-IDEX-01), the French Ile-de-France re-gion within the SESAME framework, the INFN under the Project IniziativaSpecificaMANYBODY, andTheUniversityofTurinunder theProject BARM-RILO-19.Thisprojecthasreceivedfundingfrom theEuropeanUnions’sHorizon2020researchandinnovation pro-grammeundergrantagreementNo.824093.Appendix A. Partialwavesexpansionofradialmoments
Thisappendixdemonstratesthatonlythesphericalcomponents oftheform factor f
(
r)
contribute tothe radialmoments defined inEq. (10).Consideranyrealnumber
λ
andanyfunction f(
r)
ofthe three-dimensional variable r, and further assume that the integral de-finedasIλ
=
IR3f
(
r)
rλd3r (A.1)is finite. Any function f
(
r)
can be expandedin partial waves as follows f(
r)
=
∞ =0 m=−β
m(
r)
Y ∗m(
ˆ
r)
(A.2) withβ
m(
r)
=
f(
r)
Y m(
ˆ
r)
dˆr,
(A.3) suchthat Iλ=
∞ =0 m=−β
m(
r)
r2+λY ∗m(
rˆ
)
dˆrdr.
(A.4) Using Y ∗m(
ˆ
r)
dˆr=
√
4π
δ
0δ
m0 (A.5) weobtain Iλ=
∞ =0 m=− [Iλ] m=
[Iλ]00 (A.6) where [Iλ]00=
∞ 0β
00(
r)
r2+λdr.
(A.7)Therefore,Iλvanishesforany
=
0,i.e. onlythepartialwave=
0 contributestotheintegral.Consequently,anypureradialfunction or any function whose partial wave expansion have a spherical (=
0) term lead to a non-vanishing Iλ. Moreover, the Fouriertransformofthissphericalpartwillbeinducedonlybythe j0(kr
)
sphericalBesselfunction.
Appendix B. Momentsofapolynomialratioformfactor
Thisappendixdiscussesthedeterminationintheconfiguration space ofthe moments ofa function having Fouriertransform in momentum space expressed as a polynomial ratio. These results serve thecomparisonwiththe moments obtainedinSec. 3from themomentumintegralmethod.
Considering the polynomial ratio function
˜
fK(k)
expressed in momentumspaceas˜
fK(
k)
≡ ˜
fK(
k)
=
1
+
a1k21
+
b1k2+
b2k4+
b3k6,
(B.1)itsinverseFouriertransformwrites
fK
(
r)
≡
fK(
r)
=
1 2π
2 1 r ∞ 0 dk k˜
fK(
k)
sin(
kr) .
(B.2)Table B.2
CoefficientsofthepartialfractionexpansionforKelly’sparameterization [22].Note theunitchangeofthepolynomialcoefficientsascomparedtoKelly’spolynomial: a1≡ (¯h/2M)2a1,b1≡ (¯h/2M)2b1,b2≡ (¯h/2M)4b2,b3≡ (¯h/2M)6b3,whereM isthe protonmass. i GEp GMp/μp ki(fm−1) Ai(fm−2) ki(fm−1) Ai(fm−2) e m e m e m e m 1 0 3.02 5.12 0 0 3.18 6.38 0 2 4.41 6.43 −2.56 0.97 0 13.86 1.72 0 3 −4.41 6.43 −2.56 −0.97 0 7.62 −8.10 0
˜
fK(k
)
is assumedto representa regular physics quantity,for in-stancetheelectromagnetic formfactors ofthenucleon [22],such that the denominator never vanishes for realk and the function acceptsonly complexpoles. The product k˜
fK(k)
can then be ex-pandedinpartialfractionsask
˜
fK(
k)
=
3 i=1 Ai k−
ki+
Ai k−
ki (B.3)wheretheki’s(with
m[
ki]
>
0)arethepolesof˜
f(
k)
,and Ai= −
i 2b3(
1+
a1k2i)
ki m[
ki]
3 j( =i)=1
(
ki−
kj)(
ki−
kj) ,
(B.4)are the residues of thefunction k
˜
fK(k)
at k=
ki.The numericalvaluesofthe Ai’s andki’s corresponding tothe parameterization
ofRef. [22] for theelectricandmagneticproton formfactorsare listedinTableB.2.Afterintegration,theradialfunctionwrites
fK
(
r)
=
1 2π
1 r 3 i=1 e−m[ki]r×
(B.5) e[
Ai]
cos e[
ki]
r−
m[
Ai]
sin e[
ki]
r.
Theabsenceofoddpowersofk inthedenominatorof
˜
fK(k)
leads totherelationships 3 i=1 e[A
i] =
3 i=1 e[k
i] =
0 (B.6)which ensurea finite value of fK
(
r)
atr=
0.The moments,de-terminedfromtheconfigurationspaceintegral ofEq. (10),canbe expressedas
(
rλ,
fK)
=
2(λ
+
2)
×
(B.7) 3 i=1 e[
Ai]
cos(θ
ki)
−
m[
Ai]
sin(θ
ki)
|
ki|
λ+2 withλ
>
−
2 andθ
ki= (λ +
2)
Arctan e[
ki]
m[
ki]
.
(B.8)Appendix C. Truncatedmomentsofapolynomialratioform factor
Analytical expressions fortruncated integer moments are de-rived hereafter for the polynomial ratio parameterization of the Fouriertransform
˜
fK(
k)
of Eq. (B.1), within the exponentialreg-ularizationapproachofEq. (23).
Following thediscussion ofSec.4,truncatedinteger moments aredefinedforthecut-off Q byEq. (34) andEq. (35).Theintegral isperformedbeforetakingthe
-limitandtakesthegenericform
R2p
(
Q,
)
=
u2p
(
Q,
)
+
3 i=1 iv2p
(
)
Arctan Q|
ki|
+
w2p(
)
Arctan Q(C.1)
R2p
+1(
Q,
)
=
u2p+1(
Q,
)
+
3 i=1 iv2p+1(
)
Arctan Q|
ki|
+
w2p+1
(
)
Arctan Q,
(C.2)foreven andoddtruncatedmoments.Similarly tothe dipole pa-rameterization, the uj’s, ivj’s, and wj’s coefficientsaccept finite
limits when
→
0. The uj’s are the only coefficientsdependingonthecut-off,andthey vanishforinfinite Q .Thefull expression ofthesefunctionsistoocumbersometobereportedhere,butgets simplifiedwhen
tendstozero.
The
-dependencein Eq. (C.1) and Eq. (C.2) distinguishes the
Q -saturation behaviour. Inthe
→
0+ limit, the eventruncated momentsbecome(
r2p,
fK)
= (
2p+
1)
!
w2p(
0+)
(C.3)independentofQ ,whiletheoddtruncatedmomentswrite
(
r2p+1,
fK)
=
2π
(
2p+
2)
! ×
(C.4) u2p+1(
Q,
0+)
+
3 i=1 iv2p+1(
0+)
Arctan Q|
ki|
stilldependingonthecut-off.Forinstance,thefirstevenmoments canbeexpressedas
(
r0,
fK)
=
1 (C.5)(
r2,
fK)
=
3! (
b1−
a1)
(C.6)(
r4,
fK)
=
5!
b21−
a1b1−
b2 (C.7) andtherecurrencerelation(
r2p,
fK)
= (
2p+
1)
! ×
(C.8) b1(
r2p−2,
fK)
(
2p−
1)
!
−
b2(
r2p−4,
fK)
(
2p−
3)
!
+
b3(
r2p−6,
fK)
(
2p−
5)
!
,
with p
>
2, provides all the higher orders. The integrals corre-spondingtothefirstoddmomentswriteR1
(
Q,
0+)
=
1 Q−
2i A1 k3 1 Arctan Q|
k1|
(C.9)−
2iA2 k3 2 Arctan Q|
k2|
−
2i A3 k3 3 Arctan Q|
k3|
R3
(
Q,
0+)
=
b1−
a1 Q−
1 3Q3 (C.10)+
2iA1 k51Arctan Q|k
1|
+
2i A2 k52Arctan Q|k
2|
+
2iA3 k53Arctan Q|k
3|
R5
(
Q,
0+)
=
b 2 1−
a1b1−
b2 Q−
b1−
a1 3Q3+
1 5Q5 (C.11)−
2iA1 k7 1 Arctan Q|
k1|
−
2i A2 k7 2 Arctan Q|
k2|
−
2iA3 k7 3 Arctan Q|
k3|
.
Thespecific structure of
˜
fK(
k)
as aratio ofpolynomials ofevenpowerofk with nopoleson therealk-axis,leads eitherto pure imaginarypoles or to relationship between Ai’sand ki’s. For
in-stance,inaddition tothe generalpropertiesofEq. (B.6) wehave fortheprotonelectricformfactor(TableB.2)
|
k2| = |
k3| ⇒ |
A2| = |
A3|
(C.12)k2
= −
k3⇒
A2=
A3 (C.13)suchthat
R2p
+1(Q,
0+)
arepurerealquantities.Inthelimit Q→
∞
,Eq. (C.9)-(C.11) provide(
r1,
fK)
= −
2i 2!
A1 k31+
A2 k32+
A3 k33 (C.14)(
r3,
fK)
=
2i 4!
A1 k51+
A2 k52+
A3 k53 (C.15)(
r5,
fK)
= −
2i 6!
A1 k7 1+
A2 k7 2+
A3 k7 3,
(C.16) andgenerally(
r2p+1,
fK)
= (−
1)
p+12i(
2p+
2)
!
3 i=1 Ai k2pi +3.
(C.17)Appendix D.D-dimensionalgeneralizationoftheintegral method
Thegeneralizationoftheprincipalvalueregularizationmethod IM1 to a D-dimensional spacemanifests mainlyin thechangeof
thenormalizationcoefficient
N
λ(Eq. (15)).InD dimensions,N
λ;Dwrites
N
λ;D=
2λ+1λ+2D
−
λ2D2 (D.1) andEq. (18) canbegeneralizedas
(
rλ,
fD)
=
N
λ;D ∞ 0 dk˜
fD(
k)
−
n j=0
˜
f2 j;Dk2 j kλ+1 (D.2)where
˜
fD is the D-dimensional Fourier transform of theD-dimensionalchargedensity fD
˜
fD(
k)
=
IRD dDr e−ik·rfD(
r)
(D.3) with˜
f2 j;D=
1 j!
djf˜
D(
k)
d(
k2)
j k=0.
(D.4)Theweak limitregularizationmethodIM2,asdefinedinEq. (23),
canbegeneralizedtoa D-dimensionalspaceas
(
rλ,
fD)
=
1(
2π
)
D IRD dDk˜
f(
k)
gλ;D(
k)
(D.5) where gλ;D(
k)
=
lim →0+I
λ;D(
k,
)
(D.6) withI
λ;D(
k,
)
=
IRD dDr rλe−rei k·r.
(D.7)ForD
≥
2,thisintegralcanbeexpressedasI
λ;D(
k,
)
=
Dπ
D/2(
k2+
2
)
(D+λ)/2(
D+ λ)
D2
+
1 (D.8)×
2F1 D+ λ
2,
−
λ
+
1 2;
D 2;
k2 k2+
2
where 2F1 isthe hypergeometricfunction [27]. Forinstance,this
expression can be used for the case D
=
2 of relevance for the relativisticapproachofthenucleonstructureproposedinRef. [15]. ForthecaseD=
3,Eq. (24) isrecovered.References
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